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/srv/reproducible-results/rbuild-debian/r-b-build.OsaIbmml/b2/statsmodels_0.14.4+dfsg-1_amd64.changes ¶

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python-statsmodels-doc_0.14.4+dfsg-1_all.deb ¶

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file list ¶

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control.tar.xz ¶

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control.tar ¶

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./control ¶

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./md5sums ¶

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./md5sums ¶

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data.tar ¶

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1	Package:·python-statsmodels-doc	1	Package:·python-statsmodels-doc
2	Source:·statsmodels	2	Source:·statsmodels
3	Version:·0.14.4+dfsg-1	3	Version:·0.14.4+dfsg-1
4	Architecture:·all	4	Architecture:·all
5	Maintainer:·Debian·Science·Maintainers·<debian-science-maintainers@lists.alioth.debian.org>	5	Maintainer:·Debian·Science·Maintainers·<debian-science-maintainers@lists.alioth.debian.org>
6	Installed-Size:·154897	6	Installed-Size:·155872
7	Depends:·libjs-sphinxdoc·(>=·8.1),·libjs-requirejs,·libjs-mathjax	7	Depends:·libjs-sphinxdoc·(>=·8.1),·libjs-requirejs,·libjs-mathjax
8	Suggests:·python3-statsmodels,·python3-doc,·python-numpy-doc,·python-patsy-doc,·python-pandas-doc,·python-scipy-doc	8	Suggests:·python3-statsmodels,·python3-doc,·python-numpy-doc,·python-patsy-doc,·python-pandas-doc,·python-scipy-doc
9	Breaks:·python-scikits-statsmodels-doc,·python-scikits.statsmodels-doc,·python-statsmodels·(<<·0.9.0-3~)	9	Breaks:·python-scikits-statsmodels-doc,·python-scikits.statsmodels-doc,·python-statsmodels·(<<·0.9.0-3~)
10	Replaces:·python-scikits-statsmodels-doc,·python-scikits.statsmodels-doc,·python-statsmodels·(<<·0.9.0-3~)	10	Replaces:·python-scikits-statsmodels-doc,·python-scikits.statsmodels-doc,·python-statsmodels·(<<·0.9.0-3~)
11	Section:·doc	11	Section:·doc
12	Priority:·optional	12	Priority:·optional
13	Homepage:·https://www.statsmodels.org	13	Homepage:·https://www.statsmodels.org


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78	················{	78	················{


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9	\x8\x8\x8\x8\x8\x8·L\x8Le\x8ea\x8as\x8st\x8t·s\x8sq\x8qu\x8ua\x8ar\x8re\x8es\x8s·f\x8fi\x8it\x8tt\x8ti\x8in\x8ng\x8g·o\x8of\x8f·m\x8mo\x8od\x8de\x8el\x8ls\x8s·t\x8to\x8o·d\x8da\x8at\x8ta\x8a_\x8?\x8¶·\x8\x8\x8\x8\x8\x8	9	\x8\x8\x8\x8\x8\x8·L\x8Le\x8ea\x8as\x8st\x8t·s\x8sq\x8qu\x8ua\x8ar\x8re\x8es\x8s·f\x8fi\x8it\x8tt\x8ti\x8in\x8ng\x8g·o\x8of\x8f·m\x8mo\x8od\x8de\x8el\x8ls\x8s·t\x8to\x8o·d\x8da\x8at\x8ta\x8a_\x8?\x8¶·\x8\x8\x8\x8\x8\x8
10	This·is·a·quick·introduction·to·statsmodels·for·physical·scientists·(e.g.	10	This·is·a·quick·introduction·to·statsmodels·for·physical·scientists·(e.g.
11	physicists,·astronomers)·or·engineers.	11	physicists,·astronomers)·or·engineers.
12	Why·is·this·needed?	12	Why·is·this·needed?
13	Because·most·of·statsmodels·was·written·by·statisticians·and·they·use·a	13	Because·most·of·statsmodels·was·written·by·statisticians·and·they·use·a
14	different·terminology·and·sometimes·methods,·making·it·hard·to·know·which	14	different·terminology·and·sometimes·methods,·making·it·hard·to·know·which
15	classes·and·functions·are·relevant·and·what·their·inputs·and·outputs·mean.	15	classes·and·functions·are·relevant·and·what·their·inputs·and·outputs·mean.
16	[·]:	16	[1]:
17	import·numpy·as·np	17	import·numpy·as·np
18	import·pandas·as·pd	18	import·pandas·as·pd
19	import·statsmodels.api·as·sm	19	import·statsmodels.api·as·sm
20	\x8\x8\x8\x8\x8·L\x8Li\x8in\x8ne\x8ea\x8ar\x8r·m\x8mo\x8od\x8de\x8el\x8ls\x8s_\x8?\x8¶·\x8\x8\x8\x8\x8	20	\x8\x8\x8\x8\x8·L\x8Li\x8in\x8ne\x8ea\x8ar\x8r·m\x8mo\x8od\x8de\x8el\x8ls\x8s_\x8?\x8¶·\x8\x8\x8\x8\x8
21	Assume·you·have·data·points·with·measurements·y·at·positions·x·as·well·as	21	Assume·you·have·data·points·with·measurements·y·at·positions·x·as·well·as
22	measurement·errors·y_err.	22	measurement·errors·y_err.
23	How·can·you·use·statsmodels·to·fit·a·straight·line·model·to·this·data?	23	How·can·you·use·statsmodels·to·fit·a·straight·line·model·to·this·data?
24	For·an·extensive·discussion·see·_\x8H_\x8o_\x8g_\x8g_\x8·_\x8e_\x8t_\x8·_\x8a_\x8l_\x8._\x8·_\x8(_\x82_\x80_\x81_\x80_\x8)_\x8,_\x8·_\x8“_\x8D_\x8a_\x8t_\x8a_\x8·_\x8a_\x8n_\x8a_\x8l_\x8y_\x8s_\x8i_\x8s_\x8·_\x8r_\x8e_\x8c_\x8i_\x8p_\x8e_\x8s_\x8:	24	For·an·extensive·discussion·see·_\x8H_\x8o_\x8g_\x8g_\x8·_\x8e_\x8t_\x8·_\x8a_\x8l_\x8._\x8·_\x8(_\x82_\x80_\x81_\x80_\x8)_\x8,_\x8·_\x8“_\x8D_\x8a_\x8t_\x8a_\x8·_\x8a_\x8n_\x8a_\x8l_\x8y_\x8s_\x8i_\x8s_\x8·_\x8r_\x8e_\x8c_\x8i_\x8p_\x8e_\x8s_\x8:
25	_\x8F_\x8i_\x8t_\x8t_\x8i_\x8n_\x8g_\x8·_\x8a_\x8·_\x8m_\x8o_\x8d_\x8e_\x8l_\x8·_\x8t_\x8o_\x8·_\x8d_\x8a_\x8t_\x8a_\x8”·…·we’ll·use·the·example·data·given·by·them·in·Table·1.	25	_\x8F_\x8i_\x8t_\x8t_\x8i_\x8n_\x8g_\x8·_\x8a_\x8·_\x8m_\x8o_\x8d_\x8e_\x8l_\x8·_\x8t_\x8o_\x8·_\x8d_\x8a_\x8t_\x8a_\x8”·…·we’ll·use·the·example·data·given·by·them·in·Table·1.
26	So·the·model·is·f(x)·=·a·*·x·+·b·and·on·Figure·1·they·print·the·result·we·want	26	So·the·model·is·f(x)·=·a·*·x·+·b·and·on·Figure·1·they·print·the·result·we·want
27	to·reproduce·…·the·best-fit·parameter·and·the·parameter·errors·for·a·“standard	27	to·reproduce·…·the·best-fit·parameter·and·the·parameter·errors·for·a·“standard
28	weighted·least-squares·fit”·for·this·data·are:··a·=·2.24·+-·0.11··b·=·34·+-	28	weighted·least-squares·fit”·for·this·data·are:··a·=·2.24·+-·0.11··b·=·34·+-
29	18	29	18
30	[·]:	30	[2]:
31	data·=·"""	31	data·=·"""
32	··x···y·y_err	32	··x···y·y_err
33	201·592····61	33	201·592····61
34	244·401····25	34	244·401····25
35	·47·583····38	35	·47·583····38
36	287·402····15	36	287·402····15
37	203·495····21	37	203·495····21
Offset 56, 34 lines modified		Offset 56, 71 lines modified
56	except·ImportError:	56	except·ImportError:
57	····from·io·import·StringIO	57	····from·io·import·StringIO
58	data·=·pd.read_csv(StringIO(data),·delim_whitespace=True).astype(float)	58	data·=·pd.read_csv(StringIO(data),·delim_whitespace=True).astype(float)

59	#·Note:·for·the·results·we·compare·with·the·paper·here,·they·drop·the·first	59	#·Note:·for·the·results·we·compare·with·the·paper·here,·they·drop·the·first
60	four·points	60	four·points
61	data.head()	61	data.head()
		62	/tmp/ipykernel_880521/2751177292.py:28:·FutureWarning:·The·'delim_whitespace'
		63	keyword·in·pd.read_csv·is·deprecated·and·will·be·removed·in·a·future·version.
		64	Use·``sep='\s+'``·instead
		65	··data·=·pd.read_csv(StringIO(data),·delim_whitespace=True).astype(float)
		66	[2]:
		67	·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·
		68	\|_\x8·_\x8\|_\x8·_\x8·_\x8·_\x8·_\x8x\x8x_\x8\|_\x8·_\x8·_\x8·_\x8·_\x8y\x8y_\x8\|_\x8y\x8y_\x8_\x8__\x8e\x8e_\x8r\x8r_\x8r\x8r\|
		69	\|_\x80\x80_\x8\|_\x82_\x80_\x81_\x8._\x80_\x8\|_\x85_\x89_\x82_\x8._\x80_\x8\|_\x86_\x81_\x8._\x80_\x8·\|
		70	\|_\x81\x81_\x8\|_\x82_\x84_\x84_\x8._\x80_\x8\|_\x84_\x80_\x81_\x8._\x80_\x8\|_\x82_\x85_\x8._\x80_\x8·\|
		71	\|_\x82\x82_\x8\|_\x84_\x87_\x8._\x80_\x8·_\x8\|_\x85_\x88_\x83_\x8._\x80_\x8\|_\x83_\x88_\x8._\x80_\x8·\|
		72	\|_\x83\x83_\x8\|_\x82_\x88_\x87_\x8._\x80_\x8\|_\x84_\x80_\x82_\x8._\x80_\x8\|_\x81_\x85_\x8._\x80_\x8·\|
		73	\|_\x84\x84_\x8\|_\x82_\x80_\x83_\x8._\x80_\x8\|_\x84_\x89_\x85_\x8._\x80_\x8\|_\x82_\x81_\x8._\x80_\x8·\|
62	To·fit·a·straight·line·use·the·weighted·least·squares·class·_\x8W_\x8L_\x8S·…·the	74	To·fit·a·straight·line·use·the·weighted·least·squares·class·_\x8W_\x8L_\x8S·…·the
63	parameters·are·called:··exog·=·sm.add_constant(x)··endog·=·y·*·weights·=·1·/	75	parameters·are·called:··exog·=·sm.add_constant(x)··endog·=·y·*·weights·=·1·/
64	sqrt(y_err)	76	sqrt(y_err)
65	Note·that·exog·must·be·a·2-dimensional·array·with·x·as·a·column·and·an·extra	77	Note·that·exog·must·be·a·2-dimensional·array·with·x·as·a·column·and·an·extra
66	column·of·ones.·Adding·this·column·of·ones·means·you·want·to·fit·the·model·y·=	78	column·of·ones.·Adding·this·column·of·ones·means·you·want·to·fit·the·model·y·=
67	a··x·+·b,·leaving·it·off·means·you·want·to·fit·the·model·y·=·a··x.	79	a··x·+·b,·leaving·it·off·means·you·want·to·fit·the·model·y·=·a··x.
68	And·you·have·to·use·the·option·cov_type='fixed·scale'·to·tell·statsmodels·that	80	And·you·have·to·use·the·option·cov_type='fixed·scale'·to·tell·statsmodels·that
69	you·really·have·measurement·errors·with·an·absolute·scale.·If·you·do·not,	81	you·really·have·measurement·errors·with·an·absolute·scale.·If·you·do·not,
70	statsmodels·will·treat·the·weights·as·relative·weights·between·the·data·points	82	statsmodels·will·treat·the·weights·as·relative·weights·between·the·data·points
71	and·internally·re-scale·them·so·that·the·best-fit·model·will·have·chi**2·/·ndf	83	and·internally·re-scale·them·so·that·the·best-fit·model·will·have·chi**2·/·ndf
72	=·1.	84	=·1.
73	[·]:	85	[3]:
74	exog·=·sm.add_constant(data["x"])	86	exog·=·sm.add_constant(data["x"])
75	endog·=·data["y"]	87	endog·=·data["y"]
76	weights·=·1.0·/·(data["y_err"]·**·2)	88	weights·=·1.0·/·(data["y_err"]·**·2)
77	wls·=·sm.WLS(endog,·exog,·weights)	89	wls·=·sm.WLS(endog,·exog,·weights)
78	results·=·wls.fit(cov_type="fixed·scale")	90	results·=·wls.fit(cov_type="fixed·scale")
79	print(results.summary())	91	print(results.summary())
		92	····························WLS·Regression·Results
		93	==============================================================================
		94	Dep.·Variable:······················y···R-squared:·······················0.400
		95	Model:····························WLS···Adj.·R-squared:··················0.367
		96	Method:·················Least·Squares···F-statistic:·····················193.5
		97	Date:················Sat,·05·Oct·2024···Prob·(F-statistic):···········4.52e-11
		98	Time:························10:43:25···Log-Likelihood:················-119.06
		99	No.·Observations:··················20···AIC:·····························242.1
		100	Df·Residuals:······················18···BIC:·····························244.1
		101	Df·Model:···························1
		102	Covariance·Type:··········fixed·scale
		103	==============================================================================
		104	·················coef····std·err··········z······P>\|z\|······[0.025······0.975]
		105	------------------------------------------------------------------------------
		106	const········213.2735·····14.394·····14.817······0.000·····185.062·····241.485
		107	x··············1.0767······0.077·····13.910······0.000·······0.925·······1.228
		108	==============================================================================
		109	Omnibus:························0.943···Durbin-Watson:···················2.901
		110	Prob(Omnibus):··················0.624···Jarque-Bera·(JB):················0.181
		111	Skew:··························-0.205···Prob(JB):························0.914
		112	Kurtosis:·······················3.220···Cond.·No.·························575.
		113	==============================================================================

		114	Notes:
		115	[1]·Standard·Errors·are·based·on·fixed·scale
80	\x8\x8\x8\x8·C\x8Ch\x8he\x8ec\x8ck\x8k·a\x8ag\x8ga\x8ai\x8in\x8ns\x8st\x8t·s\x8sc\x8ci\x8ip\x8py\x8y.\x8.o\x8op\x8pt\x8ti\x8im\x8mi\x8iz\x8ze\x8e.\x8.c\x8cu\x8ur\x8rv\x8ve\x8e_\x8_f\x8fi\x8it\x8t_\x8?\x8¶·\x8\x8\x8\x8	116	\x8\x8\x8\x8·C\x8Ch\x8he\x8ec\x8ck\x8k·a\x8ag\x8ga\x8ai\x8in\x8ns\x8st\x8t·s\x8sc\x8ci\x8ip\x8py\x8y.\x8.o\x8op\x8pt\x8ti\x8im\x8mi\x8iz\x8ze\x8e.\x8.c\x8cu\x8ur\x8rv\x8ve\x8e_\x8_f\x8fi\x8it\x8t_\x8?\x8¶·\x8\x8\x8\x8
81	[·]:	117	[4]:
82	#·You·can·use·`scipy.optimize.curve_fit`·to·get·the·best-fit·parameters·and	118	#·You·can·use·`scipy.optimize.curve_fit`·to·get·the·best-fit·parameters·and
83	parameter·errors.	119	parameter·errors.
84	from·scipy.optimize·import·curve_fit	120	from·scipy.optimize·import·curve_fit


85	def·f(x,·a,·b):	121	def·f(x,·a,·b):
86	····return·a·*·x·+·b	122	····return·a·*·x·+·b
Offset 93, 16 lines modified		Offset 130, 18 lines modified
93	ydata·=·data["y"]	130	ydata·=·data["y"]
94	p0·=·[0,·0]··#·initial·parameter·estimate	131	p0·=·[0,·0]··#·initial·parameter·estimate
95	sigma·=·data["y_err"]	132	sigma·=·data["y_err"]
96	popt,·pcov·=·curve_fit(f,·xdata,·ydata,·p0,·sigma,·absolute_sigma=True)	133	popt,·pcov·=·curve_fit(f,·xdata,·ydata,·p0,·sigma,·absolute_sigma=True)
97	perr·=·np.sqrt(np.diag(pcov))	134	perr·=·np.sqrt(np.diag(pcov))
98	print("a·=·{0:10.3f}·+-·{1:10.3f}".format(popt[0],·perr[0]))	135	print("a·=·{0:10.3f}·+-·{1:10.3f}".format(popt[0],·perr[0]))
99	print("b·=·{0:10.3f}·+-·{1:10.3f}".format(popt[1],·perr[1]))	136	print("b·=·{0:10.3f}·+-·{1:10.3f}".format(popt[1],·perr[1]))
		137	a·=······1.077·+-······0.077
		138	b·=····213.273·+-·····14.394
100	\x8\x8\x8\x8·C\x8Ch\x8he\x8ec\x8ck\x8k·a\x8ag\x8ga\x8ai\x8in\x8ns\x8st\x8t·s\x8se\x8el\x8lf\x8f-\x8-w\x8wr\x8ri\x8it\x8tt\x8te\x8en\x8n·c\x8co\x8os\x8st\x8t·f\x8fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8n_\x8?\x8¶·\x8\x8\x8\x8	139	\x8\x8\x8\x8·C\x8Ch\x8he\x8ec\x8ck\x8k·a\x8ag\x8ga\x8ai\x8in\x8ns\x8st\x8t·s\x8se\x8el\x8lf\x8f-\x8-w\x8wr\x8ri\x8it\x8tt\x8te\x8en\x8n·c\x8co\x8os\x8st\x8t·f\x8fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8n_\x8?\x8¶·\x8\x8\x8\x8
101	[·]:	140	[5]:
102	#·You·can·also·use·`scipy.optimize.minimize`·and·write·your·own·cost·function.	141	#·You·can·also·use·`scipy.optimize.minimize`·and·write·your·own·cost·function.
103	#·This·does·not·give·you·the·parameter·errors·though·...·you'd·have	142	#·This·does·not·give·you·the·parameter·errors·though·...·you'd·have
104	#·to·estimate·the·HESSE·matrix·separately·...	143	#·to·estimate·the·HESSE·matrix·separately·...
105	from·scipy.optimize·import·minimize	144	from·scipy.optimize·import·minimize


106	def·chi2(pars):	145	def·chi2(pars):
Offset 112, 16 lines modified		Offset 151, 18 lines modified
112	····return·np.sum(chi·**·2)	151	····return·np.sum(chi·**·2)


113	result·=·minimize(fun=chi2,·x0=[0,·0])	152	result·=·minimize(fun=chi2,·x0=[0,·0])
114	popt·=·result.x	153	popt·=·result.x
115	print("a·=·{0:10.3f}".format(popt[0]))	154	print("a·=·{0:10.3f}".format(popt[0]))
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8	············]	8	············]
9	········},	9	········},
10	········{	10	········{
11	············"cell_type":·"code",	11	············"cell_type":·"code",
12	············"execution_count":·1,	12	············"execution_count":·1,
13	············"metadata":·{	13	············"metadata":·{
14	················"execution":·{	14	················"execution":·{
15	····················"iopub.execute_input":·"2025-05-23T22:55:03.~~7112~~98Z",	15	····················"iopub.execute_input":·"2026-06-26T06:53:08.673397Z",
16	····················"iopub.status.busy":·"2025-05-23T22:55:03.710~~849~~Z",	16	····················"iopub.status.busy":·"2026-06-26T06:53:08.671791Z",
17	····················"iopub.status.idle":·"2025-05-23T22:55:04.~~8169~~59Z",	17	····················"iopub.status.idle":·"2026-06-26T06:53:09.960059Z",
18	····················"shell.execute_reply":·"2025-05-23T22:55:04.~~815~~892Z"	18	····················"shell.execute_reply":·"2026-06-26T06:53:09.958835Z"
19	················}	19	················}
20	············},	20	············},
21	············"outputs":·[],	21	············"outputs":·[],
22	············"source":·[	22	············"source":·[
23	················"import·matplotlib.pyplot·as·plt\n",	23	················"import·matplotlib.pyplot·as·plt\n",
24	················"import·numpy·as·np\n",	24	················"import·numpy·as·np\n",
25	················"import·pandas·as·pd\n",	25	················"import·pandas·as·pd\n",
Offset 42, 18 lines modified		Offset 42, 18 lines modified
42	············]	42	············]
43	········},	43	········},
44	········{	44	········{
45	············"cell_type":·"code",	45	············"cell_type":·"code",
46	············"execution_count":·2,	46	············"execution_count":·2,
47	············"metadata":·{	47	············"metadata":·{
48	················"execution":·{	48	················"execution":·{
49	····················"iopub.execute_input":·"2025-05-23T22:55:04.~~822051~~Z",	49	····················"iopub.execute_input":·"2026-06-26T06:53:09.965292Z",
50	····················"iopub.status.busy":·"2025-05-23T22:55:04.~~821154~~Z",	50	····················"iopub.status.busy":·"2026-06-26T06:53:09.964239Z",
51	····················"iopub.status.idle":·"2025-05-23T22:55:05.082381Z",	51	····················"iopub.status.idle":·"2026-06-26T06:53:10.268919Z",
52	····················"shell.execute_reply":·"2025-05-23T22:55:05.~~081334~~Z"	52	····················"shell.execute_reply":·"2026-06-26T06:53:10.265742Z"
53	················}	53	················}
54	············},	54	············},
55	············"outputs":·[	55	············"outputs":·[
56	················{	56	················{
57	····················"data":·{	57	····················"data":·{
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167	············},	167	············},
168	············"outputs":·[	168	············"outputs":·[
169	················{	169	················{
170	····················"name":·"stdout",	170	····················"name":·"stdout",
171	····················"output_type":·"stream",	171	····················"output_type":·"stream",
172	····················"text":·[	172	····················"text":·[
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203	········},	203	········},
204	········{	204	········{
205	············"cell_type":·"code",	205	············"cell_type":·"code",
206	············"execution_count":·7,	206	············"execution_count":·7,
207	············"metadata":·{	207	············"metadata":·{
208	················"execution":·{	208	················"execution":·{
209	····················"iopub.execute_input":·"2025-05-23T22:55:01.490~~612~~Z",	209	····················"iopub.execute_input":·"2026-06-26T06:53:04.840398Z",
210	····················"iopub.status.busy":·"2025-05-23T22:55:01.~~490173~~Z",	210	····················"iopub.status.busy":·"2026-06-26T06:53:04.839982Z",
211	····················"iopub.status.idle":·"2025-05-23T22:55:01.53~~4701~~Z",	211	····················"iopub.status.idle":·"2026-06-26T06:53:04.932569Z",
212	····················"shell.execute_reply":·"2025-05-23T22:55:01.53~~371~~8Z"	212	····················"shell.execute_reply":·"2026-06-26T06:53:04.931408Z"
213	················}	213	················}
214	············},	214	············},
215	············"outputs":·[	215	············"outputs":·[
216	················{	216	················{
217	····················"name":·"stdout",	217	····················"name":·"stdout",
218	····················"output_type":·"stream",	218	····················"output_type":·"stream",
219	····················"text":·[	219	····················"text":·[
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10	············]	10	············]
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12	········{	12	········{
13	············"cell_type":·"code",	13	············"cell_type":·"code",
14	············"execution_count":·1,	14	············"execution_count":·1,
15	············"metadata":·{	15	············"metadata":·{
16	················"execution":·{	16	················"execution":·{
17	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~460691~~Z",	17	····················"iopub.execute_input":·"2026-06-26T06:53:00.323030Z",
18	····················"iopub.status.busy":·"2025-05-23T22:54:58.~~460442~~Z",	18	····················"iopub.status.busy":·"2026-06-26T06:53:00.322606Z",
19	····················"iopub.status.idle":·"2025-05-23T22:54:59.642~~233~~Z",	19	····················"iopub.status.idle":·"2026-06-26T06:53:01.632166Z",
20	····················"shell.execute_reply":·"2025-05-23T22:54:59.641339Z"	20	····················"shell.execute_reply":·"2026-06-26T06:53:01.631382Z"
21	················}	21	················}
22	············},	22	············},
23	············"outputs":·[],	23	············"outputs":·[],
24	············"source":·[	24	············"source":·[
25	················"import·numpy·as·np\n",	25	················"import·numpy·as·np\n",
26	················"from·scipy.stats.distributions·import·norm\n",	26	················"from·scipy.stats.distributions·import·norm\n",
27	················"from·statsmodels.base.distributed_estimation·import·DistributedModel\n",	27	················"from·statsmodels.base.distributed_estimation·import·DistributedModel\n",
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61	············]	61	············]
62	········},	62	········},
63	········{	63	········{
64	············"cell_type":·"code",	64	············"cell_type":·"code",
65	············"execution_count":·2,	65	············"execution_count":·2,
66	············"metadata":·{	66	············"metadata":·{
67	················"execution":·{	67	················"execution":·{
68	····················"iopub.execute_input":·"2025-05-23T22:54:59.64~~677~~7Z",	68	····················"iopub.execute_input":·"2026-06-26T06:53:01.640627Z",
69	····················"iopub.status.busy":·"2025-05-23T22:54:59.64~~6314~~Z",	69	····················"iopub.status.busy":·"2026-06-26T06:53:01.640078Z",
70	····················"iopub.status.idle":·"2025-05-23T22:54:59.6~~53027~~Z",	70	····················"iopub.status.idle":·"2026-06-26T06:53:01.647853Z",
71	····················"shell.execute_reply":·"2025-05-23T22:54:59.6~~52060~~Z"	71	····················"shell.execute_reply":·"2026-06-26T06:53:01.646905Z"
72	················}	72	················}
73	············},	73	············},
74	············"outputs":·[],	74	············"outputs":·[],
75	············"source":·[	75	············"source":·[
76	················"X·=·np.random.normal(size=(1000,·25))\n",	76	················"X·=·np.random.normal(size=(1000,·25))\n",
77	················"beta·=·np.random.normal(size=25)\n",	77	················"beta·=·np.random.normal(size=25)\n",
78	················"beta·*=·np.random.randint(0,·2,·size=25)\n",	78	················"beta·*=·np.random.randint(0,·2,·size=25)\n",
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88	············]	88	············]
89	········},	89	········},
90	········{	90	········{
91	············"cell_type":·"code",	91	············"cell_type":·"code",
92	············"execution_count":·3,	92	············"execution_count":·3,
93	············"metadata":·{	93	············"metadata":·{
94	················"execution":·{	94	················"execution":·{
95	····················"iopub.execute_input":·"2025-05-23T22:54:59.65~~6725~~Z",	95	····················"iopub.execute_input":·"2026-06-26T06:53:01.653393Z",
96	····················"iopub.status.busy":·"2025-05-23T22:54:59.65~~6320~~Z",	96	····················"iopub.status.busy":·"2026-06-26T06:53:01.652484Z",
97	····················"iopub.status.idle":·"2025-05-23T22:55:00.222912Z",	97	····················"iopub.status.idle":·"2026-06-26T06:53:02.109727Z",
98	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~221~~830Z"	98	····················"shell.execute_reply":·"2026-06-26T06:53:02.108825Z"
99	················}	99	················}
100	············},	100	············},
101	············"outputs":·[],	101	············"outputs":·[],
102	············"source":·[	102	············"source":·[
103	················"debiased_OLS_mod·=·DistributedModel(m)\n",	103	················"debiased_OLS_mod·=·DistributedModel(m)\n",
104	················"debiased_OLS_fit·=·debiased_OLS_mod.fit(\n",	104	················"debiased_OLS_fit·=·debiased_OLS_mod.fit(\n",
105	················"····zip(_endog_gen(y,·m),·_exog_gen(X,·m)),·fit_kwds={\"alpha\":·0.2}\n",	105	················"····zip(_endog_gen(y,·m),·_exog_gen(X,·m)),·fit_kwds={\"alpha\":·0.2}\n",
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114	············]	114	············]
115	········},	115	········},
116	········{	116	········{
117	············"cell_type":·"code",	117	············"cell_type":·"code",
118	············"execution_count":·4,	118	············"execution_count":·4,
119	············"metadata":·{	119	············"metadata":·{
120	················"execution":·{	120	················"execution":·{
121	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~227~~692Z",	121	····················"iopub.execute_input":·"2026-06-26T06:53:02.113668Z",
122	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~22722~~8Z",	122	····················"iopub.status.busy":·"2026-06-26T06:53:02.113048Z",
123	····················"iopub.status.idle":·"2025-05-23T22:55:01.37~~1355~~Z",	123	····················"iopub.status.idle":·"2026-06-26T06:53:04.179033Z",
124	····················"shell.execute_reply":·"2025-05-23T22:55:01.37~~0319~~Z"	124	····················"shell.execute_reply":·"2026-06-26T06:53:04.177746Z"
125	················}	125	················}
126	············},	126	············},
127	············"outputs":·[],	127	············"outputs":·[],
128	············"source":·[	128	············"source":·[
129	················"from·statsmodels.genmod.generalized_linear_model·import·GLM\n",	129	················"from·statsmodels.genmod.generalized_linear_model·import·GLM\n",
130	················"from·statsmodels.genmod.families·import·Gaussian\n",	130	················"from·statsmodels.genmod.families·import·Gaussian\n",
131	················"\n",	131	················"\n",
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145	············]	145	············]
146	········},	146	········},
147	········{	147	········{
148	············"cell_type":·"code",	148	············"cell_type":·"code",
149	············"execution_count":·5,	149	············"execution_count":·5,
150	············"metadata":·{	150	············"metadata":·{
151	················"execution":·{	151	················"execution":·{
152	····················"iopub.execute_input":·"2025-05-23T22:55:01.~~379221~~Z",	152	····················"iopub.execute_input":·"2026-06-26T06:53:04.183616Z",
153	····················"iopub.status.busy":·"2025-05-23T22:55:01.~~378780~~Z",	153	····················"iopub.status.busy":·"2026-06-26T06:53:04.182477Z",
154	····················"iopub.status.idle":·"2025-05-23T22:55:01.~~735195~~Z",	154	····················"iopub.status.idle":·"2026-06-26T06:53:04.429662Z",
155	····················"shell.execute_reply":·"2025-05-23T22:55:01.~~734155~~Z"	155	····················"shell.execute_reply":·"2026-06-26T06:53:04.428774Z"
156	················}	156	················}
157	············},	157	············},
158	············"outputs":·[],	158	············"outputs":·[],
159	············"source":·[	159	············"source":·[
160	················"from·statsmodels.base.distributed_estimation·import·_est_regularized_naive,·_join_naive\n",	160	················"from·statsmodels.base.distributed_estimation·import·_est_regularized_naive,·_join_naive\n",
161	················"\n",	161	················"\n",
162	················"\n",	162	················"\n",
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176	············]	176	············]
177	········},	177	········},
178	········{	178	········{
179	············"cell_type":·"code",	179	············"cell_type":·"code",
180	············"execution_count":·6,	180	············"execution_count":·6,
181	············"metadata":·{	181	············"metadata":·{
182	················"execution":·{	182	················"execution":·{
183	····················"iopub.execute_input":·"2025-05-23T22:55:01.73~~9931~~Z",	183	····················"iopub.execute_input":·"2026-06-26T06:53:04.434202Z",
184	····················"iopub.status.busy":·"2025-05-23T22:55:01.739455Z",	184	····················"iopub.status.busy":·"2026-06-26T06:53:04.433757Z",
185	····················"iopub.status.idle":·"2025-05-23T22:55:01.759356Z",	185	····················"iopub.status.idle":·"2026-06-26T06:53:04.453374Z",
186	····················"shell.execute_reply":·"2025-05-23T22:55:01.758293Z"	186	····················"shell.execute_reply":·"2026-06-26T06:53:04.452337Z"
187	················}	187	················}
188	············},	188	············},
189	············"outputs":·[],	189	············"outputs":·[],
190	············"source":·[	190	············"source":·[
191	················"from·statsmodels.base.distributed_estimation·import·(\n",	191	················"from·statsmodels.base.distributed_estimation·import·(\n",
192	················"····_est_unregularized_naive,\n",	192	················"····_est_unregularized_naive,\n",
193	················"····DistributedResults,\n",	193	················"····DistributedResults,\n",


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21	············]	21	············]
22	········},	22	········},
23	········{	23	········{
24	············"cell_type":·"code",	24	············"cell_type":·"code",
25	············"execution_count":·1,	25	············"execution_count":·1,
26	············"metadata":·{	26	············"metadata":·{
27	················"execution":·{	27	················"execution":·{
28	····················"iopub.execute_input":·"2025-05-23T22:55:04.~~0897~~06Z",	28	····················"iopub.execute_input":·"2026-06-26T06:53:00.242804Z",
29	····················"iopub.status.busy":·"2025-05-23T22:55:04.~~089184~~Z",	29	····················"iopub.status.busy":·"2026-06-26T06:53:00.242395Z",
30	····················"iopub.status.idle":·"2025-05-23T22:55:05.~~990616~~Z",	30	····················"iopub.status.idle":·"2026-06-26T06:53:03.634399Z",
31	····················"shell.execute_reply":·"2025-05-23T22:55:05.~~989554~~Z"	31	····················"shell.execute_reply":·"2026-06-26T06:53:03.631462Z"
32	················}	32	················}
33	············},	33	············},
34	············"outputs":·[],	34	············"outputs":·[],
35	············"source":·[	35	············"source":·[
36	················"import·numpy·as·np\n",	36	················"import·numpy·as·np\n",
37	················"import·matplotlib.pyplot·as·plt\n",	37	················"import·matplotlib.pyplot·as·plt\n",
38	················"import·pandas·as·pd\n",	38	················"import·pandas·as·pd\n",
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43	············]	43	············]
44	········},	44	········},
45	········{	45	········{
46	············"cell_type":·"code",	46	············"cell_type":·"code",
47	············"execution_count":·2,	47	············"execution_count":·2,
48	············"metadata":·{	48	············"metadata":·{
49	················"execution":·{	49	················"execution":·{
50	····················"iopub.execute_input":·"2025-05-23T22:55:05.~~995130~~Z",	50	····················"iopub.execute_input":·"2026-06-26T06:53:03.642569Z",
51	····················"iopub.status.busy":·"2025-05-23T22:55:05.~~994573~~Z",	51	····················"iopub.status.busy":·"2026-06-26T06:53:03.641810Z",
52	····················"iopub.status.idle":·"2025-05-23T22:55:05.9~~993~~13Z",	52	····················"iopub.status.idle":·"2026-06-26T06:53:03.652138Z",
53	····················"shell.execute_reply":·"2025-05-23T22:55:05.998407Z"	53	····················"shell.execute_reply":·"2026-06-26T06:53:03.650860Z"
54	················}	54	················}
55	············},	55	············},
56	············"outputs":·[],	56	············"outputs":·[],
57	············"source":·[	57	············"source":·[
58	················"plt.rcParams[\"figure.figsize\"]·=·(12,·8)"	58	················"plt.rcParams[\"figure.figsize\"]·=·(12,·8)"
59	············]	59	············]
60	········},	60	········},
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82	············]	82	············]
83	········},	83	········},
84	········{	84	········{
85	············"cell_type":·"code",	85	············"cell_type":·"code",
86	············"execution_count":·3,	86	············"execution_count":·3,
87	············"metadata":·{	87	············"metadata":·{
88	················"execution":·{	88	················"execution":·{
89	····················"iopub.execute_input":·"2025-05-23T22:55:06.005830Z",	89	····················"iopub.execute_input":·"2026-06-26T06:53:03.656397Z",
90	····················"iopub.status.busy":·"2025-05-23T22:55:06.005445Z",	90	····················"iopub.status.busy":·"2026-06-26T06:53:03.655849Z",
91	····················"iopub.status.idle":·"2025-05-23T22:55:06.309534Z",	91	····················"iopub.status.idle":·"2026-06-26T06:53:04.075099Z",
92	····················"shell.execute_reply":·"2025-05-23T22:55:06.~~308436~~Z"	92	····················"shell.execute_reply":·"2026-06-26T06:53:04.073759Z"
93	················}	93	················}
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189	</div>	189	</div>
190	</div>	190	</div>
191	<div·class="nboutput·nblast·docutils·container">	191	<div·class="nboutput·nblast·docutils·container">
192	<div·class="prompt·empty·docutils·container">	192	<div·class="prompt·empty·docutils·container">
193	</div>	193	</div>
194	<div·class="output_area·stderr·docutils·container">	194	<div·class="output_area·stderr·docutils·container">
195	<div·class="highlight"><pre>	195	<div·class="highlight"><pre>
196	/tmp/ipykernel_3~~674049~~/3508103541.py:24:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.	196	/tmp/ipykernel_881382/3508103541.py:24:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.
197	··index·=·pd.date_range(start="1996",·end="2008",·freq="Y")	197	··index·=·pd.date_range(start="1996",·end="2008",·freq="Y")
198	/tmp/ipykernel_3~~674049~~/3508103541.py:44:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.	198	/tmp/ipykernel_881382/3508103541.py:44:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.
199	··index·=·pd.date_range(start="1990",·end="2005",·freq="Y")	199	··index·=·pd.date_range(start="1990",·end="2005",·freq="Y")
200	/tmp/ipykernel_3~~674049~~/3508103541.py:80:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.	200	/tmp/ipykernel_881382/3508103541.py:80:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.
201	··index·=·pd.date_range(start="1970",·end="2001",·freq="Y")	201	··index·=·pd.date_range(start="1970",·end="2001",·freq="Y")
202	/tmp/ipykernel_3~~674049~~/3508103541.py:84:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.	202	/tmp/ipykernel_881382/3508103541.py:84:·FutureWarning:·'Y'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'YE'·instead.
203	··index·=·pd.date_range(start="2001",·end="2008",·freq="Y")	203	··index·=·pd.date_range(start="2001",·end="2008",·freq="Y")
204	</pre></div></div>	204	</pre></div></div>
205	</div>	205	</div>
206	</section>	206	</section>
207	<section·id="Simple-Exponential-Smoothing">	207	<section·id="Simple-Exponential-Smoothing">
208	<h2>Simple·Exponential·Smoothing<a·class="headerlink"·href="#Simple-Exponential-Smoothing"·title="Link·to·this·heading">¶</a></h2>	208	<h2>Simple·Exponential·Smoothing<a·class="headerlink"·href="#Simple-Exponential-Smoothing"·title="Link·to·this·heading">¶</a></h2>
209	<p>Lets·use·Simple·Exponential·Smoothing·to·forecast·the·below·oil·data.</p>	209	<p>Lets·use·Simple·Exponential·Smoothing·to·forecast·the·below·oil·data.</p>


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126	····59.7668,	126	····59.7668,
127	····35.1919,	127	····35.1919,
128	····44.3197,	128	····44.3197,
129	····47.9137,	129	····47.9137,
130	]	130	]
131	index·=·pd.date_range(start="2005",·end="2010-Q4",·freq="QS-OCT")	131	index·=·pd.date_range(start="2005",·end="2010-Q4",·freq="QS-OCT")
132	aust·=·pd.Series(data,·index)	132	aust·=·pd.Series(data,·index)
133	/tmp/ipykernel_3~~674049~~/3508103541.py:24:·FutureWarning:·'Y'·is·deprecated·and	133	/tmp/ipykernel_881382/3508103541.py:24:·FutureWarning:·'Y'·is·deprecated·and
134	will·be·removed·in·a·future·version,·please·use·'YE'·instead.	134	will·be·removed·in·a·future·version,·please·use·'YE'·instead.
135	··index·=·pd.date_range(start="1996",·end="2008",·freq="Y")	135	··index·=·pd.date_range(start="1996",·end="2008",·freq="Y")
136	/tmp/ipykernel_3~~674049~~/3508103541.py:44:·FutureWarning:·'Y'·is·deprecated·and	136	/tmp/ipykernel_881382/3508103541.py:44:·FutureWarning:·'Y'·is·deprecated·and
137	will·be·removed·in·a·future·version,·please·use·'YE'·instead.	137	will·be·removed·in·a·future·version,·please·use·'YE'·instead.
138	··index·=·pd.date_range(start="1990",·end="2005",·freq="Y")	138	··index·=·pd.date_range(start="1990",·end="2005",·freq="Y")
139	/tmp/ipykernel_3~~674049~~/3508103541.py:80:·FutureWarning:·'Y'·is·deprecated·and	139	/tmp/ipykernel_881382/3508103541.py:80:·FutureWarning:·'Y'·is·deprecated·and
140	will·be·removed·in·a·future·version,·please·use·'YE'·instead.	140	will·be·removed·in·a·future·version,·please·use·'YE'·instead.
141	··index·=·pd.date_range(start="1970",·end="2001",·freq="Y")	141	··index·=·pd.date_range(start="1970",·end="2001",·freq="Y")
142	/tmp/ipykernel_3~~674049~~/3508103541.py:84:·FutureWarning:·'Y'·is·deprecated·and	142	/tmp/ipykernel_881382/3508103541.py:84:·FutureWarning:·'Y'·is·deprecated·and
143	will·be·removed·in·a·future·version,·please·use·'YE'·instead.	143	will·be·removed·in·a·future·version,·please·use·'YE'·instead.
144	··index·=·pd.date_range(start="2001",·end="2008",·freq="Y")	144	··index·=·pd.date_range(start="2001",·end="2008",·freq="Y")
145	\x8\x8\x8\x8\x8·S\x8Si\x8im\x8mp\x8pl\x8le\x8e·E\x8Ex\x8xp\x8po\x8on\x8ne\x8en\x8nt\x8ti\x8ia\x8al\x8l·S\x8Sm\x8mo\x8oo\x8ot\x8th\x8hi\x8in\x8ng\x8g_\x8?\x8¶·\x8\x8\x8\x8\x8	145	\x8\x8\x8\x8\x8·S\x8Si\x8im\x8mp\x8pl\x8le\x8e·E\x8Ex\x8xp\x8po\x8on\x8ne\x8en\x8nt\x8ti\x8ia\x8al\x8l·S\x8Sm\x8mo\x8oo\x8ot\x8th\x8hi\x8in\x8ng\x8g_\x8?\x8¶·\x8\x8\x8\x8\x8
146	Lets·use·Simple·Exponential·Smoothing·to·forecast·the·below·oil·data.	146	Lets·use·Simple·Exponential·Smoothing·to·forecast·the·below·oil·data.
147	[2]:	147	[2]:
148	ax·=·oildata.plot()	148	ax·=·oildata.plot()
149	ax.set_xlabel("Year")	149	ax.set_xlabel("Year")


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652	-rw-r--r--···0·root·········(0)·root·········(0)····68890·2024-10-05·10:43:25.000000·./usr/share/doc/python-statsmodels-doc/html/_images/examples_notebooks_generated_tsa_dates_12_0.png	665	-rw-r--r--···0·root·········(0)·root·········(0)····68890·2024-10-05·10:43:25.000000·./usr/share/doc/python-statsmodels-doc/html/_images/examples_notebooks_generated_tsa_dates_12_0.png


Offset 79, 18 lines modified		Offset 79, 18 lines modified
79	The·estimated·covariance·parameters·should·be·similar·to·groups_var,	79	The·estimated·covariance·parameters·should·be·similar·to·groups_var,
80	level1_var,·etc.·as·defined·above.	80	level1_var,·etc.·as·defined·above.
81	[13]:	81	[13]:
82	r.cov_struct.summary()	82	r.cov_struct.summary()
83	[13]:	83	[13]:
84	·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·	84	·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·
85	\|_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8\|_\x8V\x8V_\x8a\x8a_\x8r\x8r_\x8i\x8i_\x8a\x8a_\x8n\x8n_\x8c\x8c_\x8e\x8e\|	85	\|_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8·_\x8\|_\x8V\x8V_\x8a\x8a_\x8r\x8r_\x8i\x8i_\x8a\x8a_\x8n\x8n_\x8c\x8c_\x8e\x8e\|
86	\|_\x8g\x8g_\x8r\x8r_\x8o\x8o_\x8u\x8u_\x8p\x8p_\x8s\x8s_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x81_\x8._\x80_.3_.0_.3_.4_.4\|	86	\|_\x8g\x8g_\x8r\x8r_\x8o\x8o_\x8u\x8u_\x8p\x8p_\x8s\x8s_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x81_\x8._\x81_.0_.7_.0_.6_.5\|
87	\|_\x8l\x8l_\x8e\x8e_\x8v\x8v_\x8e\x8e_\x8l\x8l_\x81\x81_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x81_\x8._\x89_.7_.1_.0_.1_.6\|	87	\|_\x8l\x8l_\x8e\x8e_\x8v\x8v_\x8e\x8e_\x8l\x8l_\x81\x81_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x81_\x8._\x88_.8_.7_.3_.7_.8\|
88	\|_\x8l\x8l_\x8e\x8e_\x8v\x8v_\x8e\x8e_\x8l\x8l_\x82\x82_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x82_\x8._\x89_\x87_.6_.7_.9_.8\|	88	\|_\x8l\x8l_\x8e\x8e_\x8v\x8v_\x8e\x8e_\x8l\x8l_\x82\x82_\x8_\x8__\x8i\x8i_\x8x\x8x_\x8\|_\x82_\x8._\x89_\x88_.0_.7_.2_.7\|
89	\|_\x8R\x8R_\x8e\x8e_\x8s\x8s_\x8i\x8i_\x8d\x8d_\x8u\x8u_\x8a\x8a_\x8l\x8l_\x8·_\x8\|_\x84_\x8._\x80_\x81_\x80_.7_.2_.9\|	89	\|_\x8R\x8R_\x8e\x8e_\x8s\x8s_\x8i\x8i_\x8d\x8d_\x8u\x8u_\x8a\x8a_\x8l\x8l_\x8·_\x8\|_\x84_\x8._\x80_\x81_\x83_.1_.1_.7\|
90	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]	90	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]
91	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8	91	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8
92	····*·_\x8I_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s	92	····*·_\x8I_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s
93	····*·_\x8G_\x8e_\x8t_\x8t_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8r_\x8t_\x8e_\x8d	93	····*·_\x8G_\x8e_\x8t_\x8t_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8r_\x8t_\x8e_\x8d
94	····*·_\x8U_\x8s_\x8e_\x8r_\x8·_\x8G_\x8u_\x8i_\x8d_\x8e	94	····*·_\x8U_\x8s_\x8e_\x8r_\x8·_\x8G_\x8u_\x8i_\x8d_\x8e
95	····*·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s	95	····*·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s
96	····*·_\x8A_\x8P_\x8I_\x8·_\x8R_\x8e_\x8f_\x8e_\x8r_\x8e_\x8n_\x8c_\x8e	96	····*·_\x8A_\x8P_\x8I_\x8·_\x8R_\x8e_\x8f_\x8e_\x8r_\x8e_\x8n_\x8c_\x8e


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543	</div>	543	</div>
544	</div>	544	</div>
545	<div·class="nboutput·docutils·container">	545	<div·class="nboutput·docutils·container">
546	<div·class="prompt·empty·docutils·container">	546	<div·class="prompt·empty·docutils·container">
547	</div>	547	</div>
548	<div·class="output_area·stderr·docutils·container">	548	<div·class="output_area·stderr·docutils·container">
549	<div·class="highlight"><pre>	549	<div·class="highlight"><pre>
550	/tmp/ipykernel_367~~2673~~/427128218.py:3:·UserWarning:·set_ticklabels()·should·only·be·used·with·a·fixed·number·of·ticks,·i.e.·after·set_ticks()·or·using·a·FixedLocator.	550	/tmp/ipykernel_877145/427128218.py:3:·UserWarning:·set_ticklabels()·should·only·be·used·with·a·fixed·number·of·ticks,·i.e.·after·set_ticks()·or·using·a·FixedLocator.
551	··ax.set_xticklabels(dta.columns.values[::10])	551	··ax.set_xticklabels(dta.columns.values[::10])
552	</pre></div></div>	552	</pre></div></div>
553	</div>	553	</div>
554	<div·class="nboutput·nblast·docutils·container">	554	<div·class="nboutput·nblast·docutils·container">
555	<div·class="prompt·empty·docutils·container">	555	<div·class="prompt·empty·docutils·container">
556	</div>	556	</div>
557	<div·class="output_area·docutils·container">	557	<div·class="output_area·docutils·container">


Offset 230, 27 lines modified		Offset 230, 27 lines modified
230	······<th></th>	230	······<th></th>
231	······<th>Variance</th>	231	······<th>Variance</th>
232	····</tr>	232	····</tr>
233	··</thead>	233	··</thead>
234	··<tbody>	234	··<tbody>
235	····<tr>	235	····<tr>
236	······<th>groups_ix</th>	236	······<th>groups_ix</th>
237	······<td>1.030344</td>	237	······<td>1.107065</td>
238	····</tr>	238	····</tr>
239	····<tr>	239	····<tr>
240	······<th>level1_ix</th>	240	······<th>level1_ix</th>
241	······<td>1.~~971016~~</td>	241	······<td>1.887378</td>
242	····</tr>	242	····</tr>
243	····<tr>	243	····<tr>
244	······<th>level2_ix</th>	244	······<th>level2_ix</th>
245	······<td>2.976798</td>	245	······<td>2.980727</td>
246	····</tr>	246	····</tr>
247	····<tr>	247	····<tr>
248	······<th>Residual</th>	248	······<th>Residual</th>
249	······<td>4.01~~0729~~</td>	249	······<td>4.013117</td>
250	····</tr>	250	····</tr>
251	··</tbody>	251	··</tbody>
252	</table>	252	</table>
253	</div></div>	253	</div></div>
254	</div>	254	</div>
255	</section>	255	</section>


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12	············]	12	············]
13	········},	13	········},
14	········{	14	········{
15	············"cell_type":·"code",	15	············"cell_type":·"code",
16	············"execution_count":·1,	16	············"execution_count":·1,
17	············"metadata":·{	17	············"metadata":·{
18	················"execution":·{	18	················"execution":·{
19	····················"iopub.execute_input":·"2025-05-23T22:55:05.~~263396~~Z",	19	····················"iopub.execute_input":·"2026-06-26T06:53:00.590801Z",
20	····················"iopub.status.busy":·"2025-05-23T22:55:05.~~262949~~Z",	20	····················"iopub.status.busy":·"2026-06-26T06:53:00.590162Z",
21	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~464980~~Z",	21	····················"iopub.status.idle":·"2026-06-26T06:53:02.233145Z",
22	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~463950~~Z"	22	····················"shell.execute_reply":·"2026-06-26T06:53:02.232113Z"
23	················}	23	················}
24	············},	24	············},
25	············"outputs":·[],	25	············"outputs":·[],
26	············"source":·[	26	············"source":·[
27	················"import·numpy·as·np\n",	27	················"import·numpy·as·np\n",
28	················"import·pandas·as·pd\n",	28	················"import·pandas·as·pd\n",
29	················"import·statsmodels.api·as·sm"	29	················"import·statsmodels.api·as·sm"
Offset 37, 18 lines modified		Offset 37, 18 lines modified
37	············]	37	············]
38	········},	38	········},
39	········{	39	········{
40	············"cell_type":·"code",	40	············"cell_type":·"code",
41	············"execution_count":·2,	41	············"execution_count":·2,
42	············"metadata":·{	42	············"metadata":·{
43	················"execution":·{	43	················"execution":·{
44	····················"iopub.execute_input":·"2025-05-23T22:55:08.~~471630~~Z",	44	····················"iopub.execute_input":·"2026-06-26T06:53:02.238427Z",
45	····················"iopub.status.busy":·"2025-05-23T22:55:08.~~4710~~18Z",	45	····················"iopub.status.busy":·"2026-06-26T06:53:02.237816Z",
46	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~476930~~Z",	46	····················"iopub.status.idle":·"2026-06-26T06:53:02.243574Z",
47	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~47584~~0Z"	47	····················"shell.execute_reply":·"2026-06-26T06:53:02.241960Z"
48	················}	48	················}
49	············},	49	············},
50	············"outputs":·[],	50	············"outputs":·[],
51	············"source":·[	51	············"source":·[
52	················"p·=·5"	52	················"p·=·5"
53	············]	53	············]
54	········},	54	········},
Offset 60, 18 lines modified		Offset 60, 18 lines modified
60	············]	60	············]
61	········},	61	········},
62	········{	62	········{
63	············"cell_type":·"code",	63	············"cell_type":·"code",
64	············"execution_count":·3,	64	············"execution_count":·3,
65	············"metadata":·{	65	············"metadata":·{
66	················"execution":·{	66	················"execution":·{
67	····················"iopub.execute_input":·"2025-05-23T22:55:08.484558Z",	67	····················"iopub.execute_input":·"2026-06-26T06:53:02.248051Z",
68	····················"iopub.status.busy":·"2025-05-23T22:55:08.~~484090~~Z",	68	····················"iopub.status.busy":·"2026-06-26T06:53:02.247638Z",
69	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~490271~~Z",	69	····················"iopub.status.idle":·"2026-06-26T06:53:02.254737Z",
70	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~489289~~Z"	70	····················"shell.execute_reply":·"2026-06-26T06:53:02.253715Z"
71	················}	71	················}
72	············},	72	············},
73	············"outputs":·[],	73	············"outputs":·[],
74	············"source":·[	74	············"source":·[
75	················"groups_var·=·1\n",	75	················"groups_var·=·1\n",
76	················"level1_var·=·2\n",	76	················"level1_var·=·2\n",
77	················"level2_var·=·3\n",	77	················"level2_var·=·3\n",
Offset 86, 18 lines modified		Offset 86, 18 lines modified
86	············]	86	············]
87	········},	87	········},
88	········{	88	········{
89	············"cell_type":·"code",	89	············"cell_type":·"code",
90	············"execution_count":·4,	90	············"execution_count":·4,
91	············"metadata":·{	91	············"metadata":·{
92	················"execution":·{	92	················"execution":·{
93	····················"iopub.execute_input":·"2025-05-23T22:55:08.~~494426~~Z",	93	····················"iopub.execute_input":·"2026-06-26T06:53:02.258333Z",
94	····················"iopub.status.busy":·"2025-05-23T22:55:08.~~493~~979Z",	94	····················"iopub.status.busy":·"2026-06-26T06:53:02.257940Z",
95	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~50239~~9Z",	95	····················"iopub.status.idle":·"2026-06-26T06:53:02.263409Z",
96	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~501220~~Z"	96	····················"shell.execute_reply":·"2026-06-26T06:53:02.262596Z"
97	················}	97	················}
98	············},	98	············},
99	············"outputs":·[],	99	············"outputs":·[],
100	············"source":·[	100	············"source":·[
101	················"n_groups·=·100"	101	················"n_groups·=·100"
102	············]	102	············]
103	········},	103	········},
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109	············]	109	············]
110	········},	110	········},
111	········{	111	········{
112	············"cell_type":·"code",	112	············"cell_type":·"code",
113	············"execution_count":·5,	113	············"execution_count":·5,
114	············"metadata":·{	114	············"metadata":·{
115	················"execution":·{	115	················"execution":·{
116	····················"iopub.execute_input":·"2025-05-23T22:55:08.~~507853~~Z",	116	····················"iopub.execute_input":·"2026-06-26T06:53:02.266706Z",
117	····················"iopub.status.busy":·"2025-05-23T22:55:08.506803Z",	117	····················"iopub.status.busy":·"2026-06-26T06:53:02.266301Z",
118	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~512420~~Z",	118	····················"iopub.status.idle":·"2026-06-26T06:53:02.270831Z",
119	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~511408~~Z"	119	····················"shell.execute_reply":·"2026-06-26T06:53:02.269619Z"
120	················}	120	················}
121	············},	121	············},
122	············"outputs":·[],	122	············"outputs":·[],
123	············"source":·[	123	············"source":·[
124	················"group_size·=·20\n",	124	················"group_size·=·20\n",
125	················"level1_size·=·10\n",	125	················"level1_size·=·10\n",
126	················"level2_size·=·5"	126	················"level2_size·=·5"
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134	············]	134	············]
135	········},	135	········},
136	········{	136	········{
137	············"cell_type":·"code",	137	············"cell_type":·"code",
138	············"execution_count":·6,	138	············"execution_count":·6,
139	············"metadata":·{	139	············"metadata":·{
140	················"execution":·{	140	················"execution":·{
141	····················"iopub.execute_input":·"2025-05-23T22:55:08.~~51764~~3Z",	141	····················"iopub.execute_input":·"2026-06-26T06:53:02.274273Z",
142	····················"iopub.status.busy":·"2025-05-23T22:55:08.~~517194~~Z",	142	····················"iopub.status.busy":·"2026-06-26T06:53:02.273876Z",
143	····················"iopub.status.idle":·"2025-05-23T22:55:08.52~~321~~6Z",	143	····················"iopub.status.idle":·"2026-06-26T06:53:02.278561Z",
144	····················"shell.execute_reply":·"2025-05-23T22:55:08.521712Z"	144	····················"shell.execute_reply":·"2026-06-26T06:53:02.277603Z"
145	················}	145	················}
146	············},	146	············},
147	············"outputs":·[],	147	············"outputs":·[],
148	············"source":·[	148	············"source":·[
149	················"n·=·n_groups··group_size··level1_size·*·level2_size"	149	················"n·=·n_groups··group_size··level1_size·*·level2_size"
150	············]	150	············]
151	········},	151	········},
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157	············]	157	············]
158	········},	158	········},
159	········{	159	········{
160	············"cell_type":·"code",	160	············"cell_type":·"code",
161	············"execution_count":·7,	161	············"execution_count":·7,
162	············"metadata":·{	162	············"metadata":·{
163	················"execution":·{	163	················"execution":·{
164	····················"iopub.execute_input":·"2025-05-23T22:55:08.~~527332~~Z",	164	····················"iopub.execute_input":·"2026-06-26T06:53:02.284473Z",
165	····················"iopub.status.busy":·"2025-05-23T22:55:08.~~526667~~Z",	165	····················"iopub.status.busy":·"2026-06-26T06:53:02.284045Z",
166	····················"iopub.status.idle":·"2025-05-23T22:55:08.~~5610~~57Z",	166	····················"iopub.status.idle":·"2026-06-26T06:53:02.315858Z",
167	····················"shell.execute_reply":·"2025-05-23T22:55:08.~~559966~~Z"	167	····················"shell.execute_reply":·"2026-06-26T06:53:02.314874Z"
168	················}	168	················}
169	············},	169	············},
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14	············]	14	············]
15	········},	15	········},
16	········{	16	········{
17	············"cell_type":·"code",	17	············"cell_type":·"code",
18	············"execution_count":·1,	18	············"execution_count":·1,
19	············"metadata":·{	19	············"metadata":·{
20	················"execution":·{	20	················"execution":·{
21	····················"iopub.execute_input":·"2025-05-23T22:55:02.2~~61437~~Z",	21	····················"iopub.execute_input":·"2026-06-26T06:53:00.218958Z",
22	····················"iopub.status.busy":·"2025-05-23T22:55:02.2~~60972~~Z",	22	····················"iopub.status.busy":·"2026-06-26T06:53:00.218448Z",
23	····················"iopub.status.idle":·"2025-05-23T22:55:04.430~~372~~Z",	23	····················"iopub.status.idle":·"2026-06-26T06:53:02.410953Z",
24	····················"shell.execute_reply":·"2025-05-23T22:55:04.429~~447~~Z"	24	····················"shell.execute_reply":·"2026-06-26T06:53:02.409750Z"
25	················}	25	················}
26	············},	26	············},
27	············"outputs":·[],	27	············"outputs":·[],
28	············"source":·[	28	············"source":·[
29	················"import·pandas·as·pd\n",	29	················"import·pandas·as·pd\n",
30	················"import·numpy·as·np\n",	30	················"import·numpy·as·np\n",
31	················"from·scipy.stats.distributions·import·norm,·poisson\n",	31	················"from·scipy.stats.distributions·import·norm,·poisson\n",
Offset 42, 18 lines modified		Offset 42, 18 lines modified
42	············]	42	············]
43	········},	43	········},
44	········{	44	········{
45	············"cell_type":·"code",	45	············"cell_type":·"code",
46	············"execution_count":·2,	46	············"execution_count":·2,
47	············"metadata":·{	47	············"metadata":·{
48	················"execution":·{	48	················"execution":·{
49	····················"iopub.execute_input":·"2025-05-23T22:55:04.4~~34443~~Z",	49	····················"iopub.execute_input":·"2026-06-26T06:53:02.416760Z",
50	····················"iopub.status.busy":·"2025-05-23T22:55:04.4~~33991~~Z",	50	····················"iopub.status.busy":·"2026-06-26T06:53:02.416124Z",
51	····················"iopub.status.idle":·"2025-05-23T22:55:04.43~~865~~8Z",	51	····················"iopub.status.idle":·"2026-06-26T06:53:02.430408Z",
52	····················"shell.execute_reply":·"2025-05-23T22:55:04.4~~37843~~Z"	52	····················"shell.execute_reply":·"2026-06-26T06:53:02.424350Z"
53	················},	53	················},
54	················"lines_to_next_cell":·1	54	················"lines_to_next_cell":·1
55	············},	55	············},
56	············"outputs":·[],	56	············"outputs":·[],
57	············"source":·[	57	············"source":·[
58	················"def·negbinom(u,·mu,·scale):\n",	58	················"def·negbinom(u,·mu,·scale):\n",
59	················"····p·=·(scale·-·1)·/·scale\n",	59	················"····p·=·(scale·-·1)·/·scale\n",
Offset 70, 18 lines modified		Offset 70, 18 lines modified
70	············]	70	············]
71	········},	71	········},
72	········{	72	········{
73	············"cell_type":·"code",	73	············"cell_type":·"code",
74	············"execution_count":·3,	74	············"execution_count":·3,
75	············"metadata":·{	75	············"metadata":·{
76	················"execution":·{	76	················"execution":·{
77	····················"iopub.execute_input":·"2025-05-23T22:55:04.4~~41837~~Z",	77	····················"iopub.execute_input":·"2026-06-26T06:53:02.436919Z",
78	····················"iopub.status.busy":·"2025-05-23T22:55:04.4~~41456~~Z",	78	····················"iopub.status.busy":·"2026-06-26T06:53:02.436171Z",
79	····················"iopub.status.idle":·"2025-05-23T22:55:04.44~~9088~~Z",	79	····················"iopub.status.idle":·"2026-06-26T06:53:02.448174Z",
80	····················"shell.execute_reply":·"2025-05-23T22:55:04.447~~959~~Z"	80	····················"shell.execute_reply":·"2026-06-26T06:53:02.447043Z"
81	················}	81	················}
82	············},	82	············},
83	············"outputs":·[],	83	············"outputs":·[],
84	············"source":·[	84	············"source":·[
85	················"#·Sample·size\n",	85	················"#·Sample·size\n",
86	················"n·=·1000\n",	86	················"n·=·1000\n",
87	················"\n",	87	················"\n",
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106	············]	106	············]
107	········},	107	········},
108	········{	108	········{
109	············"cell_type":·"code",	109	············"cell_type":·"code",
110	············"execution_count":·4,	110	············"execution_count":·4,
111	············"metadata":·{	111	············"metadata":·{
112	················"execution":·{	112	················"execution":·{
113	····················"iopub.execute_input":·"2025-05-23T22:55:04.453315Z",	113	····················"iopub.execute_input":·"2026-06-26T06:53:02.457916Z",
114	····················"iopub.status.busy":·"2025-05-23T22:55:04.45~~2849~~Z",	114	····················"iopub.status.busy":·"2026-06-26T06:53:02.457481Z",
115	····················"iopub.status.idle":·"2025-05-23T22:55:04.460353Z",	115	····················"iopub.status.idle":·"2026-06-26T06:53:02.464357Z",
116	····················"shell.execute_reply":·"2025-05-23T22:55:04.4~~59191~~Z"	116	····················"shell.execute_reply":·"2026-06-26T06:53:02.463529Z"
117	················}	117	················}
118	············},	118	············},
119	············"outputs":·[],	119	············"outputs":·[],
120	············"source":·[	120	············"source":·[
121	················"#·Build·a·design·matrix·for·the·alternative·(more·complex)·model\n",	121	················"#·Build·a·design·matrix·for·the·alternative·(more·complex)·model\n",
122	················"x·=·np.random.normal(size=(n,·p))\n",	122	················"x·=·np.random.normal(size=(n,·p))\n",
123	················"x[:,·0]·=·1"	123	················"x[:,·0]·=·1"
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131	············]	131	············]
132	········},	132	········},
133	········{	133	········{
134	············"cell_type":·"code",	134	············"cell_type":·"code",
135	············"execution_count":·5,	135	············"execution_count":·5,
136	············"metadata":·{	136	············"metadata":·{
137	················"execution":·{	137	················"execution":·{
138	····················"iopub.execute_input":·"2025-05-23T22:55:04.464906Z",	138	····················"iopub.execute_input":·"2026-06-26T06:53:02.468507Z",
139	····················"iopub.status.busy":·"2025-05-23T22:55:04.46~~4469~~Z",	139	····················"iopub.status.busy":·"2026-06-26T06:53:02.468100Z",
140	····················"iopub.status.idle":·"2025-05-23T22:55:04.47~~3315~~Z",	140	····················"iopub.status.idle":·"2026-06-26T06:53:02.474287Z",
141	····················"shell.execute_reply":·"2025-05-23T22:55:04.469293Z"	141	····················"shell.execute_reply":·"2026-06-26T06:53:02.473233Z"
142	················}	142	················}
143	············},	143	············},
144	············"outputs":·[],	144	············"outputs":·[],
145	············"source":·[	145	············"source":·[
146	················"x0·=·x[:,·0:3]"	146	················"x0·=·x[:,·0:3]"
147	············]	147	············]
148	········},	148	········},
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154	············]	154	············]
155	········},	155	········},
156	········{	156	········{
157	············"cell_type":·"code",	157	············"cell_type":·"code",
158	············"execution_count":·6,	158	············"execution_count":·6,
159	············"metadata":·{	159	············"metadata":·{
160	················"execution":·{	160	················"execution":·{
161	····················"iopub.execute_input":·"2025-05-23T22:55:04.4~~80821~~Z",	161	····················"iopub.execute_input":·"2026-06-26T06:53:02.477945Z",
162	····················"iopub.status.busy":·"2025-05-23T22:55:04.4~~80363~~Z",	162	····················"iopub.status.busy":·"2026-06-26T06:53:02.477518Z",
163	····················"iopub.status.idle":·"2025-05-23T22:55:04.484994Z",	163	····················"iopub.status.idle":·"2026-06-26T06:53:02.483198Z",
164	····················"shell.execute_reply":·"2025-05-23T22:55:04.48~~394~~4Z"	164	····················"shell.execute_reply":·"2026-06-26T06:53:02.482154Z"
165	················}	165	················}
166	············},	166	············},
167	············"outputs":·[],	167	············"outputs":·[],
168	············"source":·[	168	············"source":·[
169	················"#·Scale·parameter·for·negative·binomial·distribution\n",	169	················"#·Scale·parameter·for·negative·binomial·distribution\n",
170	················"scale·=·10"	170	················"scale·=·10"
171	············]	171	············]
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178	············]	178	············]
179	········},	179	········},
180	········{	180	········{
181	············"cell_type":·"code",	181	············"cell_type":·"code",
182	············"execution_count":·7,	182	············"execution_count":·7,
183	············"metadata":·{	183	············"metadata":·{
184	················"execution":·{	184	················"execution":·{
185	····················"iopub.execute_input":·"2025-05-23T22:55:04.48~~9044~~Z",	185	····················"iopub.execute_input":·"2026-06-26T06:53:02.487663Z",
186	····················"iopub.status.busy":·"2025-05-23T22:55:04.48~~8602~~Z",	186	····················"iopub.status.busy":·"2026-06-26T06:53:02.487011Z",
187	····················"iopub.status.idle":·"2025-05-23T22:55:04.494~~576~~Z",	187	····················"iopub.status.idle":·"2026-06-26T06:53:02.494305Z",
188	····················"shell.execute_reply":·"2025-05-23T22:55:04.493566Z"	188	····················"shell.execute_reply":·"2026-06-26T06:53:02.491578Z"
189	················}	189	················}
190	············},	190	············},
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27	············]	27	············]
28	········},	28	········},
29	········{	29	········{
30	············"cell_type":·"code",	30	············"cell_type":·"code",
31	············"execution_count":·1,	31	············"execution_count":·1,
32	············"metadata":·{	32	············"metadata":·{
33	················"execution":·{	33	················"execution":·{
34	····················"iopub.execute_input":·"2025-05-23T22:54:58.5~~2904~~3Z",	34	····················"iopub.execute_input":·"2026-06-26T06:53:08.533803Z",
35	····················"iopub.status.busy":·"2025-05-23T22:54:58.5~~28599~~Z",	35	····················"iopub.status.busy":·"2026-06-26T06:53:08.533361Z",
36	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~230053~~Z",	36	····················"iopub.status.idle":·"2026-06-26T06:53:10.349389Z",
37	····················"shell.execute_reply":·"2025-05-23T22:55:00.228~~998~~Z"	37	····················"shell.execute_reply":·"2026-06-26T06:53:10.348325Z"
38	················},	38	················},
39	················"jupyter":·{	39	················"jupyter":·{
40	····················"outputs_hidden":·false	40	····················"outputs_hidden":·false
41	················}	41	················}
42	············},	42	············},
43	············"outputs":·[],	43	············"outputs":·[],
44	············"source":·[	44	············"source":·[
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56	············]	56	············]
57	········},	57	········},
58	········{	58	········{
59	············"cell_type":·"code",	59	············"cell_type":·"code",
60	············"execution_count":·2,	60	············"execution_count":·2,
61	············"metadata":·{	61	············"metadata":·{
62	················"execution":·{	62	················"execution":·{
63	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~237872~~Z",	63	····················"iopub.execute_input":·"2026-06-26T06:53:10.352998Z",
64	····················"iopub.status.busy":·"2025-05-23T22:55:00.237233Z",	64	····················"iopub.status.busy":·"2026-06-26T06:53:10.352470Z",
65	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~258689~~Z",	65	····················"iopub.status.idle":·"2026-06-26T06:53:10.371743Z",
66	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~257~~759Z"	66	····················"shell.execute_reply":·"2026-06-26T06:53:10.369475Z"
67	················},	67	················},
68	················"jupyter":·{	68	················"jupyter":·{
69	····················"outputs_hidden":·false	69	····················"outputs_hidden":·false
70	················}	70	················}
71	············},	71	············},
72	············"outputs":·[	72	············"outputs":·[
73	················{	73	················{
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113	············]	113	············]
114	········},	114	········},
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39	················"import·numpy·as·np\n",	39	················"import·numpy·as·np\n",
40	················"import·statsmodels.api·as·sm\n",	40	················"import·statsmodels.api·as·sm\n",
41	················"from·scipy·import·stats\n",	41	················"from·scipy·import·stats\n",
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143	················"data·=·sm.datasets.star98.load()\n",	143	················"data·=·sm.datasets.star98.load()\n",
144	················"data.exog·=·sm.add_constant(data.exog,·prepend=False)"	144	················"data.exog·=·sm.add_constant(data.exog,·prepend=False)"
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249	········},	249	········},
250	········{	250	········{
251	············"cell_type":·"code",	251	············"cell_type":·"code",
252	············"execution_count":·7,	252	············"execution_count":·7,
253	············"metadata":·{	253	············"metadata":·{
254	················"execution":·{	254	················"execution":·{
255	····················"iopub.execute_input":·"2025-05-23T22:55:05.~~542~~173Z",	255	····················"iopub.execute_input":·"2026-06-26T06:53:13.739178Z",
256	····················"iopub.status.busy":·"2025-05-23T22:55:05.~~541808~~Z",	256	····················"iopub.status.busy":·"2026-06-26T06:53:13.738723Z",
257	····················"iopub.status.idle":·"2025-05-23T22:55:05.57~~6765~~Z",	257	····················"iopub.status.idle":·"2026-06-26T06:53:13.771618Z",
258	····················"shell.execute_reply":·"2025-05-23T22:55:05.57~~581~~7Z"	258	····················"shell.execute_reply":·"2026-06-26T06:53:13.770357Z"
259	················}	259	················}
260	············},	260	············},
261	············"outputs":·[	261	············"outputs":·[
262	················{	262	················{
263	····················"name":·"stdout",	263	····················"name":·"stdout",
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Offset 230, 17 lines modified		Offset 230, 17 lines modified
230	</pre></div></div>	230	</pre></div></div>
231	</div>	231	</div>
232	<div·class="nboutput·nblast·docutils·container">	232	<div·class="nboutput·nblast·docutils·container">
233	<div·class="prompt·empty·docutils·container">	233	<div·class="prompt·empty·docutils·container">
234	</div>	234	</div>
235	<div·class="output_area·stderr·docutils·container">	235	<div·class="output_area·stderr·docutils·container">
236	<div·class="highlight"><pre>	236	<div·class="highlight"><pre>
237	/tmp/ipykernel_3~~672640~~/1000445862.py:1:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`	237	/tmp/ipykernel_883332/1000445862.py:1:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`
238	··print(mod1.params[1])	238	··print(mod1.params[1])
239	/tmp/ipykernel_3~~672640~~/1000445862.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`	239	/tmp/ipykernel_883332/1000445862.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`
240	··print(mod2.params[1]·*·2)	240	··print(mod2.params[1]·*·2)
241	</pre></div></div>	241	</pre></div></div>
242	</div>	242	</div>
243	</section>	243	</section>


244	············<div·class="clearer"></div>	244	············<div·class="clearer"></div>


Offset 176, 20 lines modified		Offset 176, 20 lines modified
176	As·expected,·the·coefficient·for·double_it(LOWINC)·in·the·second·model·is·half	176	As·expected,·the·coefficient·for·double_it(LOWINC)·in·the·second·model·is·half
177	the·size·of·the·LOWINC·coefficient·from·the·first·model:	177	the·size·of·the·LOWINC·coefficient·from·the·first·model:
178	[4]:	178	[4]:
179	print(mod1.params[1])	179	print(mod1.params[1])
180	print(mod2.params[1]·*·2)	180	print(mod2.params[1]·*·2)
181	-0.020395987154755987	181	-0.020395987154755987
182	-0.02039598715475644	182	-0.02039598715475644
183	/tmp/ipykernel_3~~672640~~/1000445862.py:1:·FutureWarning:·Series.__getitem__	183	/tmp/ipykernel_883332/1000445862.py:1:·FutureWarning:·Series.__getitem__
184	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys	184	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys
185	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To	185	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To
186	access·a·value·by·position,·use·`ser.iloc[pos]`	186	access·a·value·by·position,·use·`ser.iloc[pos]`
187	··print(mod1.params[1])	187	··print(mod1.params[1])
188	/tmp/ipykernel_3~~672640~~/1000445862.py:2:·FutureWarning:·Series.__getitem__	188	/tmp/ipykernel_883332/1000445862.py:2:·FutureWarning:·Series.__getitem__
189	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys	189	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys
190	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To	190	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To
191	access·a·value·by·position,·use·`ser.iloc[pos]`	191	access·a·value·by·position,·use·`ser.iloc[pos]`
192	··print(mod2.params[1]·*·2)	192	··print(mod2.params[1]·*·2)
193	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]	193	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]
194	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8	194	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8
195	····*·_\x8I_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s	195	····*·_\x8I_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8i_\x8n_\x8g_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s


Offset 17, 18 lines modified		Offset 17, 18 lines modified
17	············]	17	············]
18	········},	18	········},
19	········{	19	········{
20	············"cell_type":·"code",	20	············"cell_type":·"code",
21	············"execution_count":·1,	21	············"execution_count":·1,
22	············"metadata":·{	22	············"metadata":·{
23	················"execution":·{	23	················"execution":·{
24	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~5107~~47Z",	24	····················"iopub.execute_input":·"2026-06-26T06:53:10.458947Z",
25	····················"iopub.status.busy":·"2025-05-23T22:54:58.510290Z",	25	····················"iopub.status.busy":·"2026-06-26T06:53:10.458528Z",
26	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~125389~~Z",	26	····················"iopub.status.idle":·"2026-06-26T06:53:13.072691Z",
27	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~124283~~Z"	27	····················"shell.execute_reply":·"2026-06-26T06:53:13.071551Z"
28	················},	28	················},
29	················"jupyter":·{	29	················"jupyter":·{
30	····················"outputs_hidden":·false	30	····················"outputs_hidden":·false
31	················}	31	················}
32	············},	32	············},
33	············"outputs":·[],	33	············"outputs":·[],
34	············"source":·[	34	············"source":·[
Offset 69, 18 lines modified		Offset 69, 18 lines modified
69	············]	69	············]
70	········},	70	········},
71	········{	71	········{
72	············"cell_type":·"code",	72	············"cell_type":·"code",
73	············"execution_count":·2,	73	············"execution_count":·2,
74	············"metadata":·{	74	············"metadata":·{
75	················"execution":·{	75	················"execution":·{
76	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~133339~~Z",	76	····················"iopub.execute_input":·"2026-06-26T06:53:13.077126Z",
77	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~132909~~Z",	77	····················"iopub.status.busy":·"2026-06-26T06:53:13.076698Z",
78	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~227~~226Z",	78	····················"iopub.status.idle":·"2026-06-26T06:53:13.135284Z",
79	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~226046~~Z"	79	····················"shell.execute_reply":·"2026-06-26T06:53:13.134452Z"
80	················},	80	················},
81	················"jupyter":·{	81	················"jupyter":·{
82	····················"outputs_hidden":·false	82	····················"outputs_hidden":·false
83	················}	83	················}
84	············},	84	············},
85	············"outputs":·[	85	············"outputs":·[
86	················{	86	················{
Offset 139, 18 lines modified		Offset 139, 18 lines modified
139	············]	139	············]
140	········},	140	········},
141	········{	141	········{
142	············"cell_type":·"code",	142	············"cell_type":·"code",
143	············"execution_count":·3,	143	············"execution_count":·3,
144	············"metadata":·{	144	············"metadata":·{
145	················"execution":·{	145	················"execution":·{
146	····················"iopub.execute_input":·"2025-05-23T22:55:00.309619Z",	146	····················"iopub.execute_input":·"2026-06-26T06:53:13.139118Z",
147	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~309145~~Z",	147	····················"iopub.status.busy":·"2026-06-26T06:53:13.138306Z",
148	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~381024~~Z",	148	····················"iopub.status.idle":·"2026-06-26T06:53:13.199372Z",
149	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~379971~~Z"	149	····················"shell.execute_reply":·"2026-06-26T06:53:13.198605Z"
150	················},	150	················},
151	················"jupyter":·{	151	················"jupyter":·{
152	····················"outputs_hidden":·false	152	····················"outputs_hidden":·false
153	················}	153	················}
154	············},	154	············},
155	············"outputs":·[	155	············"outputs":·[
156	················{	156	················{
Offset 215, 18 lines modified		Offset 215, 18 lines modified
215	············]	215	············]
216	········},	216	········},
217	········{	217	········{
218	············"cell_type":·"code",	218	············"cell_type":·"code",
219	············"execution_count":·4,	219	············"execution_count":·4,
220	············"metadata":·{	220	············"metadata":·{
221	················"execution":·{	221	················"execution":·{
222	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~38950~~5Z",	222	····················"iopub.execute_input":·"2026-06-26T06:53:13.204015Z",
223	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~389~~132Z",	223	····················"iopub.status.busy":·"2026-06-26T06:53:13.203160Z",
224	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~3961~~11Z",	224	····················"iopub.status.idle":·"2026-06-26T06:53:13.214410Z",
225	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~3952~~07Z"	225	····················"shell.execute_reply":·"2026-06-26T06:53:13.213707Z"
226	················},	226	················},
227	················"jupyter":·{	227	················"jupyter":·{
228	····················"outputs_hidden":·false	228	····················"outputs_hidden":·false
229	················}	229	················}
230	············},	230	············},
231	············"outputs":·[	231	············"outputs":·[
232	················{	232	················{
Offset 237, 17 lines modified		Offset 237, 17 lines modified
237	························"-0.02039598715475644\n"	237	························"-0.02039598715475644\n"
238	····················]	238	····················]
239	················},	239	················},
240	················{	240	················{
241	····················"name":·"stderr",	241	····················"name":·"stderr",
242	····················"output_type":·"stream",	242	····················"output_type":·"stream",
243	····················"text":·[	243	····················"text":·[
244	························"/tmp/ipykernel_3~~672640~~/1000445862.py:1:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`\n",	244	························"/tmp/ipykernel_883332/1000445862.py:1:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`\n",
245	························"··print(mod1.params[1])\n",	245	························"··print(mod1.params[1])\n",
246	························"/tmp/ipykernel_3~~672640~~/1000445862.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`\n",	246	························"/tmp/ipykernel_883332/1000445862.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`\n",
247	························"··print(mod2.params[1]·*·2)\n"	247	························"··print(mod2.params[1]·*·2)\n"
248	····················]	248	····················]
249	················}	249	················}
250	············],	250	············],
251	············"source":·[	251	············"source":·[
252	················"print(mod1.params[1])\n",	252	················"print(mod1.params[1])\n",
253	················"print(mod2.params[1]·*·2)"	253	················"print(mod2.params[1]·*·2)"


Offset 18, 18 lines modified		Offset 18, 18 lines modified
18	············]	18	············]
19	········},	19	········},
20	········{	20	········{
21	············"cell_type":·"code",	21	············"cell_type":·"code",
22	············"execution_count":·1,	22	············"execution_count":·1,
23	············"metadata":·{	23	············"metadata":·{
24	················"execution":·{	24	················"execution":·{
25	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~506668~~Z",	25	····················"iopub.execute_input":·"2026-06-26T06:53:07.755925Z",
26	····················"iopub.status.busy":·"2025-05-23T22:54:58.~~506288~~Z",	26	····················"iopub.status.busy":·"2026-06-26T06:53:07.754519Z",
27	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~196739~~Z",	27	····················"iopub.status.idle":·"2026-06-26T06:53:09.474924Z",
28	····················"shell.execute_reply":·"2025-05-23T22:55:00.195614Z"	28	····················"shell.execute_reply":·"2026-06-26T06:53:09.469588Z"
29	················}	29	················}
30	············},	30	············},
31	············"outputs":·[],	31	············"outputs":·[],
32	············"source":·[	32	············"source":·[
33	················"import·pandas·as·pd\n",	33	················"import·pandas·as·pd\n",
34	················"import·numpy·as·np\n",	34	················"import·numpy·as·np\n",
35	················"import·statsmodels.api·as·sm\n",	35	················"import·statsmodels.api·as·sm\n",
Offset 44, 18 lines modified		Offset 44, 18 lines modified
44	············]	44	············]
45	········},	45	········},
46	········{	46	········{
47	············"cell_type":·"code",	47	············"cell_type":·"code",
48	············"execution_count":·2,	48	············"execution_count":·2,
49	············"metadata":·{	49	············"metadata":·{
50	················"execution":·{	50	················"execution":·{
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54	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~221177~~Z"	54	····················"shell.execute_reply":·"2026-06-26T06:53:09.486810Z"
55	················}	55	················}
56	············},	56	············},
57	············"outputs":·[],	57	············"outputs":·[],
58	············"source":·[	58	············"source":·[
59	················"np.random.seed(3424)"	59	················"np.random.seed(3424)"
60	············]	60	············]
61	········},	61	········},
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67	············]	67	············]
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69	········{	69	········{
70	············"cell_type":·"code",	70	············"cell_type":·"code",
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77	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~234824~~Z"	77	····················"shell.execute_reply":·"2026-06-26T06:53:09.499409Z"
78	················}	78	················}
79	············},	79	············},
80	············"outputs":·[],	80	············"outputs":·[],
81	············"source":·[	81	············"source":·[
82	················"n·=·1000"	82	················"n·=·1000"
83	············]	83	············]
84	········},	84	········},
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92	········{	92	········{
93	············"cell_type":·"code",	93	············"cell_type":·"code",
94	············"execution_count":·4,	94	············"execution_count":·4,
95	············"metadata":·{	95	············"metadata":·{
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101	················},	101	················},
102	················"lines_to_next_cell":·1	102	················"lines_to_next_cell":·1
103	············},	103	············},
104	············"outputs":·[],	104	············"outputs":·[],
105	············"source":·[	105	············"source":·[
106	················"exp·=·np.random.normal(size=n)"	106	················"exp·=·np.random.normal(size=n)"
107	············]	107	············]
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114	············]	114	············]
115	········},	115	········},
116	········{	116	········{
117	············"cell_type":·"code",	117	············"cell_type":·"code",
118	············"execution_count":·5,	118	············"execution_count":·5,
119	············"metadata":·{	119	············"metadata":·{
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124	····················"shell.execute_reply":·"2025-05-23T22:55:00.254~~186~~Z"	124	····················"shell.execute_reply":·"2026-06-26T06:53:09.542790Z"
125	················},	125	················},
126	················"lines_to_next_cell":·1	126	················"lines_to_next_cell":·1
127	············},	127	············},
128	············"outputs":·[],	128	············"outputs":·[],
129	············"source":·[	129	············"source":·[
130	················"def·gen_mediator():\n",	130	················"def·gen_mediator():\n",
131	················"····mn·=·np.exp(exp)\n",	131	················"····mn·=·np.exp(exp)\n",
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144	············]	144	············]
145	········},	145	········},
146	········{	146	········{
147	············"cell_type":·"code",	147	············"cell_type":·"code",
148	············"execution_count":·6,	148	············"execution_count":·6,
149	············"metadata":·{	149	············"metadata":·{
150	················"execution":·{	150	················"execution":·{
151	····················"iopub.execute_input":·"2025-05-23T22:55:00.269~~404~~Z",	151	····················"iopub.execute_input":·"2026-06-26T06:53:09.549661Z",
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154	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~2864~~47Z"	154	····················"shell.execute_reply":·"2026-06-26T06:53:09.562743Z"
155	················},	155	················},
156	················"lines_to_next_cell":·1	156	················"lines_to_next_cell":·1
157	············},	157	············},
158	············"outputs":·[],	158	············"outputs":·[],
159	············"source":·[	159	············"source":·[
160	················"def·gen_outcome(otype,·mtime0):\n",	160	················"def·gen_outcome(otype,·mtime0):\n",
161	················"····if·otype·==·\"full\":\n",	161	················"····if·otype·==·\"full\":\n",
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180	············]	180	············]
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183	············"cell_type":·"code",	183	············"cell_type":·"code",
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191	················},	191	················},
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24	············]	24	············]
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29	············"metadata":·{	29	············"metadata":·{
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31	····················"iopub.execute_input":·"2025-05-23T22:55:05.39~~9843~~Z",	31	····················"iopub.execute_input":·"2026-06-26T06:53:08.791952Z",
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34	····················"shell.execute_reply":·"2025-05-23T22:55:06.24~~8574~~Z"	34	····················"shell.execute_reply":·"2026-06-26T06:53:09.745059Z"
35	················}	35	················}
36	············},	36	············},
37	············"outputs":·[],	37	············"outputs":·[],
38	············"source":·[	38	············"source":·[
39	················"%matplotlib·inline"	39	················"%matplotlib·inline"
40	············]	40	············]
41	········},	41	········},
42	········{	42	········{
43	············"cell_type":·"code",	43	············"cell_type":·"code",
44	············"execution_count":·2,	44	············"execution_count":·2,
45	············"metadata":·{	45	············"metadata":·{
46	················"execution":·{	46	················"execution":·{
47	····················"iopub.execute_input":·"2025-05-23T22:55:06.25~~3751~~Z",	47	····················"iopub.execute_input":·"2026-06-26T06:53:09.755296Z",
48	····················"iopub.status.busy":·"2025-05-23T22:55:06.253243Z",	48	····················"iopub.status.busy":·"2026-06-26T06:53:09.754747Z",
49	····················"iopub.status.idle":·"2025-05-23T22:55:09.0~~53233~~Z",	49	····················"iopub.status.idle":·"2026-06-26T06:53:11.007985Z",
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51	················}	51	················}
52	············},	52	············},
53	············"outputs":·[],	53	············"outputs":·[],
54	············"source":·[	54	············"source":·[
55	················"import·numpy·as·np\n",	55	················"import·numpy·as·np\n",
56	················"import·pandas·as·pd\n",	56	················"import·pandas·as·pd\n",
57	················"from·scipy·import·stats,·optimize\n",	57	················"from·scipy·import·stats,·optimize\n",
Offset 80, 18 lines modified		Offset 80, 18 lines modified
80	············]	80	············]
81	········},	81	········},
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85	············"metadata":·{	85	············"metadata":·{
86	················"execution":·{	86	················"execution":·{
87	····················"iopub.execute_input":·"2025-05-23T22:55:09.061931Z",	87	····················"iopub.execute_input":·"2026-06-26T06:53:11.015388Z",
88	····················"iopub.status.busy":·"2025-05-23T22:55:09.0~~61257~~Z",	88	····················"iopub.status.busy":·"2026-06-26T06:53:11.014621Z",
89	····················"iopub.status.idle":·"2025-05-23T22:55:09.0~~76429~~Z",	89	····················"iopub.status.idle":·"2026-06-26T06:53:11.038082Z",
90	····················"shell.execute_reply":·"2025-05-23T22:55:09.0~~75354~~Z"	90	····················"shell.execute_reply":·"2026-06-26T06:53:11.034640Z"
91	················}	91	················}
92	············},	92	············},
93	············"outputs":·[	93	············"outputs":·[
94	················{	94	················{
95	····················"data":·{	95	····················"data":·{
96	························"text/plain":·[	96	························"text/plain":·[
97	····························"['Carroll',·'Grant',·'Peck',·'Donat',·'Stewart',·'Young']"	97	····························"['Carroll',·'Grant',·'Peck',·'Donat',·'Stewart',·'Young']"
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118	············]	118	············]
119	········},	119	········},
120	········{	120	········{
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125	····················"iopub.execute_input":·"2025-05-23T22:55:09.0~~81358~~Z",	125	····················"iopub.execute_input":·"2026-06-26T06:53:11.045509Z",
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129	················}	129	················}
130	············},	130	············},
131	············"outputs":·[	131	············"outputs":·[
132	················{	132	················{
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135	····························"['Carroll',·'Grant',·'Peck',·'Donat',·'Stewart',·'Young']"	135	····························"['Carroll',·'Grant',·'Peck',·'Donat',·'Stewart',·'Young']"
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149	············]	149	············]
150	········},	150	········},
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157	····················"iopub.status.busy":·"2025-05-23T22:55:09.0~~98183~~Z",	157	····················"iopub.status.busy":·"2026-06-26T06:53:11.063044Z",
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160	················}	160	················}
161	············},	161	············},
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164	····················"data":·{	164	····················"data":·{
165	························"text/plain":·[	165	························"text/plain":·[
166	····························"array([120,·130,··80,·400,··95,·170])"	166	····························"array([120,·130,··80,·400,··95,·170])"
Offset 183, 18 lines modified		Offset 183, 18 lines modified
183	············]	183	············]
184	········},	184	········},
185	········{	185	········{
186	············"cell_type":·"code",	186	············"cell_type":·"code",
187	············"execution_count":·6,	187	············"execution_count":·6,
188	············"metadata":·{	188	············"metadata":·{
189	················"execution":·{	189	················"execution":·{
190	····················"iopub.execute_input":·"2025-05-23T22:55:09.~~111824~~Z",	190	····················"iopub.execute_input":·"2026-06-26T06:53:11.079040Z",
191	····················"iopub.status.busy":·"2025-05-23T22:55:09.~~111410~~Z",	191	····················"iopub.status.busy":·"2026-06-26T06:53:11.078667Z",
192	····················"iopub.status.idle":·"2025-05-23T22:55:09.116579Z",	192	····················"iopub.status.idle":·"2026-06-26T06:53:11.086277Z",
193	····················"shell.execute_reply":·"2025-05-23T22:55:09.~~115622~~Z"	193	····················"shell.execute_reply":·"2026-06-26T06:53:11.084435Z"
194	················}	194	················}
195	············},	195	············},
196	············"outputs":·[],	196	············"outputs":·[],
197	············"source":·[	197	············"source":·[
198	················"eff,·var_eff·=·effectsize_smd(mean2,·sd2,·nobs2,·mean1,·sd1,·nobs1)"	198	················"eff,·var_eff·=·effectsize_smd(mean2,·sd2,·nobs2,·mean1,·sd1,·nobs1)"
199	············]	199	············]
200	········},	200	········},
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209	············]	209	············]
210	········},	210	········},
211	········{	211	········{
212	············"cell_type":·"code",	212	············"cell_type":·"code",
213	············"execution_count":·7,	213	············"execution_count":·7,
214	············"metadata":·{	214	············"metadata":·{
215	················"execution":·{	215	················"execution":·{
216	····················"iopub.execute_input":·"2025-05-23T22:55:09.~~121267~~Z",	216	····················"iopub.execute_input":·"2026-06-26T06:53:11.097692Z",
217	····················"iopub.status.busy":·"2025-05-23T22:55:09.120824Z",	217	····················"iopub.status.busy":·"2026-06-26T06:53:11.096108Z",
218	····················"iopub.status.idle":·"2025-05-23T22:55:09.138322Z",	218	····················"iopub.status.idle":·"2026-06-26T06:53:11.124393Z",
219	····················"shell.execute_reply":·"2025-05-23T22:55:09.1~~3703~~0Z"	219	····················"shell.execute_reply":·"2026-06-26T06:53:11.122600Z"
220	················}	220	················}
221	············},	221	············},
222	············"outputs":·[	222	············"outputs":·[
223	················{	223	················{
224	····················"name":·"stdout",	224	····················"name":·"stdout",
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Offset 126, 16 lines modified		Offset 126, 16 lines modified
126	lines·=·ax.plot(pca_model.factors.iloc[:,·:3],·lw=4,·alpha=0.6)	126	lines·=·ax.plot(pca_model.factors.iloc[:,·:3],·lw=4,·alpha=0.6)
127	ax.set_xticklabels(dta.columns.values[::10])	127	ax.set_xticklabels(dta.columns.values[::10])
128	ax.set_xlim(0,·51)	128	ax.set_xlim(0,·51)
129	ax.set_xlabel("Year",·size=17)	129	ax.set_xlabel("Year",·size=17)
130	fig.subplots_adjust(0.1,·0.1,·0.85,·0.9)	130	fig.subplots_adjust(0.1,·0.1,·0.85,·0.9)
131	legend·=·fig.legend(lines,·["PC·1",·"PC·2",·"PC·3"],·loc="center·right")	131	legend·=·fig.legend(lines,·["PC·1",·"PC·2",·"PC·3"],·loc="center·right")
132	legend.draw_frame(False)	132	legend.draw_frame(False)
133	/tmp/ipykernel_367~~2673~~/427128218.py:3:·UserWarning:·set_ticklabels()·should	133	/tmp/ipykernel_877145/427128218.py:3:·UserWarning:·set_ticklabels()·should·only
134	~~only·~~be·used·with·a·fixed·number·of·ticks,·i.e.·after·set_ticks()·or·using·a	134	be·used·with·a·fixed·number·of·ticks,·i.e.·after·set_ticks()·or·using·a
135	FixedLocator.	135	FixedLocator.
136	··ax.set_xticklabels(dta.columns.values[::10])	136	··ax.set_xticklabels(dta.columns.values[::10])
137	[../../../_images/examples_notebooks_generated_pca_fertility_factors_15_1.png]	137	[../../../_images/examples_notebooks_generated_pca_fertility_factors_15_1.png]
138	To·better·understand·what·is·going·on,·we·will·plot·the·fertility·trajectories	138	To·better·understand·what·is·going·on,·we·will·plot·the·fertility·trajectories
139	for·sets·of·countries·with·similar·PC·scores.·The·following·convenience	139	for·sets·of·countries·with·similar·PC·scores.·The·following·convenience
140	function·produces·such·a·plot.	140	function·produces·such·a·plot.
141	[8]:	141	[8]:


Offset 481, 17 lines modified		Offset 481, 17 lines modified
481	</div>	481	</div>
482	<div·class="output_area·stderr·docutils·container">	482	<div·class="output_area·stderr·docutils·container">
483	<div·class="highlight"><pre>	483	<div·class="highlight"><pre>
484	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'	484	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'
485	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'	485	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'
486	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'	486	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'
487	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'	487	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'
488	/tmp/ipykernel_3~~67261~~8/1205084488.py:18:·SyntaxWarning:·invalid·escape·sequence·'\$'	488	/tmp/ipykernel_881958/1205084488.py:18:·SyntaxWarning:·invalid·escape·sequence·'\$'
489	··plot_opts["bean_legend_text"]·=·"Income·<·\$30k"	489	··plot_opts["bean_legend_text"]·=·"Income·<·\$30k"
490	/tmp/ipykernel_3~~67261~~8/1205084488.py:30:·SyntaxWarning:·invalid·escape·sequence·'\$'	490	/tmp/ipykernel_881958/1205084488.py:30:·SyntaxWarning:·invalid·escape·sequence·'\$'
491	··plot_opts["bean_legend_text"]·=·"Income·>·\$50k"	491	··plot_opts["bean_legend_text"]·=·"Income·>·\$50k"
492	</pre></div></div>	492	</pre></div></div>
493	</div>	493	</div>
494	<div·class="nboutput·docutils·container">	494	<div·class="nboutput·docutils·container">
495	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[15]:	495	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[15]:
496	</pre></div>	496	</pre></div>
497	</div>	497	</div>


Offset 238, 18 lines modified		Offset 238, 18 lines modified

238	#·Show·all·plots.	238	#·Show·all·plots.
239	#·plt.show()	239	#·plt.show()
240	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'	240	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'
241	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'	241	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'
242	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'	242	<>:18:·SyntaxWarning:·invalid·escape·sequence·'\$'
243	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'	243	<>:30:·SyntaxWarning:·invalid·escape·sequence·'\$'
244	/tmp/ipykernel_3~~67261~~8/1205084488.py:18:·SyntaxWarning:·invalid·escape·sequence	244	/tmp/ipykernel_881958/1205084488.py:18:·SyntaxWarning:·invalid·escape·sequence
245	'\$'	245	'\$'
246	··plot_opts["bean_legend_text"]·=·"Income·<·\$30k"	246	··plot_opts["bean_legend_text"]·=·"Income·<·\$30k"
247	/tmp/ipykernel_3~~67261~~8/1205084488.py:30:·SyntaxWarning:·invalid·escape·sequence	247	/tmp/ipykernel_881958/1205084488.py:30:·SyntaxWarning:·invalid·escape·sequence
248	'\$'	248	'\$'
249	··plot_opts["bean_legend_text"]·=·"Income·>·\$50k"	249	··plot_opts["bean_legend_text"]·=·"Income·>·\$50k"
250	[15]:	250	[15]:
251	Text(0.5,·1.0,·"US·national·election·'96·-·Age·&·Party·Identification")	251	Text(0.5,·1.0,·"US·national·election·'96·-·Age·&·Party·Identification")
252	[../../../_images/examples_notebooks_generated_plots_boxplots_22_2.png]	252	[../../../_images/examples_notebooks_generated_plots_boxplots_22_2.png]
253	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]	253	_\x8[_\x8L_\x8o_\x8g_\x8o_\x8·_\x8o_\x8f_\x8·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g_\x8]
254	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8	254	\x8\x8\x8\x8·_\x8T\x8T_\x8a\x8a_\x8b\x8b_\x8l\x8l_\x8e\x8e_\x8·_\x8o\x8o_\x8f\x8f_\x8·_\x8C\x8C_\x8o\x8o_\x8n\x8n_\x8t\x8t_\x8e\x8e_\x8n\x8n_\x8t\x8t_\x8s\x8s·\x8\x8\x8\x8


Offset 59, 23 lines modified		Offset 59, 23 lines modified
59	······<div·class="documentwrapper">	59	······<div·class="documentwrapper">
60	········<div·class="bodywrapper">	60	········<div·class="bodywrapper">
61	··········<div·class="body"·role="main">	61	··········<div·class="body"·role="main">
62	············	62	············
63	··<section·id="M-Estimators-for-Robust-Linear-Modeling">	63	··<section·id="M-Estimators-for-Robust-Linear-Modeling">
64	<h1>M-Estimators·for·Robust·Linear·Modeling<a·class="headerlink"·href="#M-Estimators-for-Robust-Linear-Modeling"·title="Link·to·this·heading">¶</a></h1>	64	<h1>M-Estimators·for·Robust·Linear·Modeling<a·class="headerlink"·href="#M-Estimators-for-Robust-Linear-Modeling"·title="Link·to·this·heading">¶</a></h1>
65	<div·class="nbinput·nblast·docutils·container">	65	<div·class="nbinput·nblast·docutils·container">
66	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	66	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[1]:
67	</pre></div>	67	</pre></div>
68	</div>	68	</div>
69	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%</span><span·class="k">matplotlib</span>·inline	69	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="o">%</span><span·class="k">matplotlib</span>·inline
70	</pre></div>	70	</pre></div>
71	</div>	71	</div>
72	</div>	72	</div>
73	<div·class="nbinput·nblast·docutils·container">	73	<div·class="nbinput·nblast·docutils·container">
74	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	74	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[2]:
75	</pre></div>	75	</pre></div>
76	</div>	76	</div>
77	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.compat</span>·<span·class="kn">import</span>·<span·class="n">lmap</span>	77	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="kn">from</span>·<span·class="nn">statsmodels.compat</span>·<span·class="kn">import</span>·<span·class="n">lmap</span>
78	<span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>	78	<span·class="kn">import</span>·<span·class="nn">numpy</span>·<span·class="k">as</span>·<span·class="nn">np</span>
79	<span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>	79	<span·class="kn">from</span>·<span·class="nn">scipy</span>·<span·class="kn">import</span>·<span·class="n">stats</span>
80	<span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>	80	<span·class="kn">import</span>·<span·class="nn">matplotlib.pyplot</span>·<span·class="k">as</span>·<span·class="nn">plt</span>

Offset 94, 23 lines modified		Offset 94, 23 lines modified
94	<li><p><span·class="math·notranslate·nohighlight">\(s\)</span>·is·an·estimate·of·scale.</p></li>	94	<li><p><span·class="math·notranslate·nohighlight">\(s\)</span>·is·an·estimate·of·scale.</p></li>
95	<li><p>The·robust·estimates·<span·class="math·notranslate·nohighlight">\(\hat{\beta}\)</span>·are·computed·by·the·iteratively·re-weighted·least·squares·algorithm</p></li>	95	<li><p>The·robust·estimates·<span·class="math·notranslate·nohighlight">\(\hat{\beta}\)</span>·are·computed·by·the·iteratively·re-weighted·least·squares·algorithm</p></li>
96	</ul>	96	</ul>
97	<ul·class="simple">	97	<ul·class="simple">
98	<li><p>We·have·several·choices·available·for·the·weighting·functions·to·be·used</p></li>	98	<li><p>We·have·several·choices·available·for·the·weighting·functions·to·be·used</p></li>
99	</ul>	99	</ul>
100	<div·class="nbinput·nblast·docutils·container">	100	<div·class="nbinput·nblast·docutils·container">
101	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	101	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[3]:
102	</pre></div>	102	</pre></div>
103	</div>	103	</div>
104	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">norms</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">robust</span><span·class="o">.</span><span·class="n">norms</span>	104	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">norms</span>·<span·class="o">=</span>·<span·class="n">sm</span><span·class="o">.</span><span·class="n">robust</span><span·class="o">.</span><span·class="n">norms</span>
105	</pre></div>	105	</pre></div>
106	</div>	106	</div>
107	</div>	107	</div>
108	<div·class="nbinput·nblast·docutils·container">	108	<div·class="nbinput·nblast·docutils·container">
109	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	109	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[4]:
110	</pre></div>	110	</pre></div>
111	</div>	111	</div>
112	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="k">def</span>·<span·class="nf">plot_weights</span><span·class="p">(</span><span·class="n">support</span><span·class="p">,</span>·<span·class="n">weights_func</span><span·class="p">,</span>·<span·class="n">xlabels</span><span·class="p">,</span>·<span·class="n">xticks</span><span·class="p">):</span>	112	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="k">def</span>·<span·class="nf">plot_weights</span><span·class="p">(</span><span·class="n">support</span><span·class="p">,</span>·<span·class="n">weights_func</span><span·class="p">,</span>·<span·class="n">xlabels</span><span·class="p">,</span>·<span·class="n">xticks</span><span·class="p">):</span>
113	····<span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">plt</span><span·class="o">.</span><span·class="n">figure</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>	113	····<span·class="n">fig</span>·<span·class="o">=</span>·<span·class="n">plt</span><span·class="o">.</span><span·class="n">figure</span><span·class="p">(</span><span·class="n">figsize</span><span·class="o">=</span><span·class="p">(</span><span·class="mi">12</span><span·class="p">,</span>·<span·class="mi">8</span><span·class="p">))</span>
114	····<span·class="n">ax</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">111</span><span·class="p">)</span>	114	····<span·class="n">ax</span>·<span·class="o">=</span>·<span·class="n">fig</span><span·class="o">.</span><span·class="n">add_subplot</span><span·class="p">(</span><span·class="mi">111</span><span·class="p">)</span>
115	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">plot</span><span·class="p">(</span><span·class="n">support</span><span·class="p">,</span>·<span·class="n">weights_func</span><span·class="p">(</span><span·class="n">support</span><span·class="p">))</span>	115	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">plot</span><span·class="p">(</span><span·class="n">support</span><span·class="p">,</span>·<span·class="n">weights_func</span><span·class="p">(</span><span·class="n">support</span><span·class="p">))</span>
116	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">set_xticks</span><span·class="p">(</span><span·class="n">xticks</span><span·class="p">)</span>	116	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">set_xticks</span><span·class="p">(</span><span·class="n">xticks</span><span·class="p">)</span>
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118	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">set_ylim</span><span·class="p">(</span><span·class="o">-</span><span·class="mf">0.1</span><span·class="p">,</span>·<span·class="mf">1.1</span><span·class="p">)</span>	118	····<span·class="n">ax</span><span·class="o">.</span><span·class="n">set_ylim</span><span·class="p">(</span><span·class="o">-</span><span·class="mf">0.1</span><span·class="p">,</span>·<span·class="mf">1.1</span><span·class="p">)</span>
119	····<span·class="k">return</span>·<span·class="n">ax</span>	119	····<span·class="k">return</span>·<span·class="n">ax</span>
120	</pre></div>	120	</pre></div>
121	</div>	121	</div>
122	</div>	122	</div>
123	<section·id="Andrew's-Wave">	123	<section·id="Andrew's-Wave">
124	<h2>Andrew’s·Wave<a·class="headerlink"·href="#Andrew's-Wave"·title="Link·to·this·heading">¶</a></h2>	124	<h2>Andrew’s·Wave<a·class="headerlink"·href="#Andrew's-Wave"·title="Link·to·this·heading">¶</a></h2>
125	<div·class="nbinput·~~nblast·~~docutils·container">	125	<div·class="nbinput·docutils·container">
126	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	126	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[5]:
127	</pre></div>	127	</pre></div>
128	</div>	128	</div>
129	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">help</span><span·class="p">(</span><span·class="n">norms</span><span·class="o">.</span><span·class="n">AndrewWave</span><span·class="o">.</span><span·class="n">weights</span><span·class="p">)</span>	129	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">help</span><span·class="p">(</span><span·class="n">norms</span><span·class="o">.</span><span·class="n">AndrewWave</span><span·class="o">.</span><span·class="n">weights</span><span·class="p">)</span>
130	</pre></div>	130	</pre></div>
131	</div>	131	</div>
132	</div>	132	</div>
133	<div·class="nbinput·nblast·docutils·container">	133	<div·class="nboutput·nblast·docutils·container">
		134	<div·class="prompt·empty·docutils·container">
		135	</div>
		136	<div·class="output_area·docutils·container">
		137	<div·class="highlight"><pre>
		138	Help·on·function·weights·in·module·statsmodels.robust.norms:

		139	weights(self,·z)
		140	····Andrew's·wave·weighting·function·for·the·IRLS·algorithm

		141	····The·psi·function·scaled·by·z

		142	····Parameters
		143	····----------
		144	····z·:·array_like
		145	········1d·array

		146	····Returns
		147	····-------
		148	····weights·:·ndarray
		149	········weights(z)·=·sin(z/a)·/·(z/a)·····for·\\|z\\|·<=·a*pi

		150	········weights(z)·=·0····················for·\\|z\\|·>·a*pi

		151	</pre></div></div>
		152	</div>
		153	<div·class="nbinput·docutils·container">
134	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[·]:	154	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[6]:
135	</pre></div>	155	</pre></div>
136	</div>	156	</div>
137	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">a</span>·<span·class="o">=</span>·<span·class="mf">1.339</span>	157	<div·class="input_area·highlight-ipython3·notranslate"><div·class="highlight"><pre><span></span><span·class="n">a</span>·<span·class="o">=</span>·<span·class="mf">1.339</span>
138	<span·class="n">support</span>·<span·class="o">=</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">linspace</span><span·class="p">(</span><span·class="o">-</span><span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="mi">100</span><span·class="p">)</span>	158	<span·class="n">support</span>·<span·class="o">=</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">linspace</span><span·class="p">(</span><span·class="o">-</span><span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="mi">100</span><span·class="p">)</span>
139	<span·class="n">andrew</span>·<span·class="o">=</span>·<span·class="n">norms</span><span·class="o">.</span><span·class="n">AndrewWave</span><span·class="p">(</span><span·class="n">a</span><span·class="o">=</span><span·class="n">a</span><span·class="p">)</span>	159	<span·class="n">andrew</span>·<span·class="o">=</span>·<span·class="n">norms</span><span·class="o">.</span><span·class="n">AndrewWave</span><span·class="p">(</span><span·class="n">a</span><span·class="o">=</span><span·class="n">a</span><span·class="p">)</span>
140	<span·class="n">plot_weights</span><span·class="p">(</span>	160	<span·class="n">plot_weights</span><span·class="p">(</span>
141	····<span·class="n">support</span><span·class="p">,</span>·<span·class="n">andrew</span><span·class="o">.</span><span·class="n">weights</span><span·class="p">,</span>·<span·class="p">[</span><span·class="s2">"$-\pia$"</span><span·class="p">,</span>·<span·class="s2">"0"</span><span·class="p">,</span>·<span·class="s2">"$\pia$"</span><span·class="p">],</span>·<span·class="p">[</span><span·class="o">-</span><span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="mi">0</span><span·class="p">,</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">]</span>	161	····<span·class="n">support</span><span·class="p">,</span>·<span·class="n">andrew</span><span·class="o">.</span><span·class="n">weights</span><span·class="p">,</span>·<span·class="p">[</span><span·class="s2">"$-\pia$"</span><span·class="p">,</span>·<span·class="s2">"0"</span><span·class="p">,</span>·<span·class="s2">"$\pia$"</span><span·class="p">],</span>·<span·class="p">[</span><span·class="o">-</span><span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">,</span>·<span·class="mi">0</span><span·class="p">,</span>·<span·class="n">np</span><span·class="o">.</span><span·class="n">pi</span>·<span·class="o"></span>·<span·class="n">a</span><span·class="p">]</span>
142	<span·class="p">)</span>	162	<span·class="p">)</span>
143	</pre></div>	163	</pre></div>
144	</div>	164	</div>
145	</div>	165	</div>
		166	<div·class="nboutput·docutils·container">
		167	<div·class="prompt·empty·docutils·container">
		168	</div>
		169	<div·class="output_area·stderr·docutils·container">
		170	<div·class="highlight"><pre>
		171	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		172	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		173	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		174	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		175	/tmp/ipykernel_877395/3509132601.py:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		176	··support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi··a,·0,·np.pi··a]
		177	/tmp/ipykernel_877395/3509132601.py:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		178	··support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi··a,·0,·np.pi··a]
		179	</pre></div></div>
		180	</div>
		181	<div·class="nboutput·docutils·container">
		182	<div·class="prompt·highlight-none·notranslate"><div·class="highlight"><pre><span></span>[6]:
		183	</pre></div>
		184	</div>
		185	<div·class="output_area·docutils·container">
		186	<div·class="highlight"><pre>
		187	<Axes:·>
		188	</pre></div></div>
		189	</div>
		190	<div·class="nboutput·nblast·docutils·container">
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3	····*·_\x8m_\x8o_\x8d_\x8u_\x8l_\x8e_\x8s·\|	3	····*·_\x8m_\x8o_\x8d_\x8u_\x8l_\x8e_\x8s·\|
4	····*·_\x8n_\x8e_\x8x_\x8t·\|	4	····*·_\x8n_\x8e_\x8x_\x8t·\|
5	····*·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·\|	5	····*·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·\|
6	····*·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g·»	6	····*·_\x8s_\x8t_\x8a_\x8t_\x8s_\x8m_\x8o_\x8d_\x8e_\x8l_\x8s_\x8·_\x80_\x8._\x81_\x84_\x8._\x84_\x8+_\x8d_\x8f_\x8s_\x8g·»
7	····*·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s·»	7	····*·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s·»
8	····*·M-Estimators·for·Robust·Linear·Modeling	8	····*·M-Estimators·for·Robust·Linear·Modeling
9	\x8\x8\x8\x8\x8\x8·M\x8M-\x8-E\x8Es\x8st\x8ti\x8im\x8ma\x8at\x8to\x8or\x8rs\x8s·f\x8fo\x8or\x8r·R\x8Ro\x8ob\x8bu\x8us\x8st\x8t·L\x8Li\x8in\x8ne\x8ea\x8ar\x8r·M\x8Mo\x8od\x8de\x8el\x8li\x8in\x8ng\x8g_\x8?\x8¶·\x8\x8\x8\x8\x8\x8	9	\x8\x8\x8\x8\x8\x8·M\x8M-\x8-E\x8Es\x8st\x8ti\x8im\x8ma\x8at\x8to\x8or\x8rs\x8s·f\x8fo\x8or\x8r·R\x8Ro\x8ob\x8bu\x8us\x8st\x8t·L\x8Li\x8in\x8ne\x8ea\x8ar\x8r·M\x8Mo\x8od\x8de\x8el\x8li\x8in\x8ng\x8g_\x8?\x8¶·\x8\x8\x8\x8\x8\x8
10	[·]:	10	[1]:
11	%matplotlib·inline	11	%matplotlib·inline
12	[·]:	12	[2]:
13	from·statsmodels.compat·import·lmap	13	from·statsmodels.compat·import·lmap
14	import·numpy·as·np	14	import·numpy·as·np
15	from·scipy·import·stats	15	from·scipy·import·stats
16	import·matplotlib.pyplot·as·plt	16	import·matplotlib.pyplot·as·plt

17	import·statsmodels.api·as·sm	17	import·statsmodels.api·as·sm
18	····*·An·M-estimator·minimizes·the·function	18	····*·An·M-estimator·minimizes·the·function
19	\[Q(e_i,·\rho)·=·\sum_i~\rho·\left·(\frac{e_i}{s}\right·)\]	19	\[Q(e_i,·\rho)·=·\sum_i~\rho·\left·(\frac{e_i}{s}\right·)\]
20	where·\(\rho\)·is·a·symmetric·function·of·the·residuals	20	where·\(\rho\)·is·a·symmetric·function·of·the·residuals
21	····*·The·effect·of·\(\rho\)·is·to·reduce·the·influence·of·outliers	21	····*·The·effect·of·\(\rho\)·is·to·reduce·the·influence·of·outliers
22	····*·\(s\)·is·an·estimate·of·scale.	22	····*·\(s\)·is·an·estimate·of·scale.
23	····*·The·robust·estimates·\(\hat{\beta}\)·are·computed·by·the·iteratively·re-	23	····*·The·robust·estimates·\(\hat{\beta}\)·are·computed·by·the·iteratively·re-
24	······weighted·least·squares·algorithm	24	······weighted·least·squares·algorithm
25	····*·We·have·several·choices·available·for·the·weighting·functions·to·be·used	25	····*·We·have·several·choices·available·for·the·weighting·functions·to·be·used
26	[·]:	26	[3]:
27	norms·=·sm.robust.norms	27	norms·=·sm.robust.norms
28	[·]:	28	[4]:
29	def·plot_weights(support,·weights_func,·xlabels,·xticks):	29	def·plot_weights(support,·weights_func,·xlabels,·xticks):
30	····fig·=·plt.figure(figsize=(12,·8))	30	····fig·=·plt.figure(figsize=(12,·8))
31	····ax·=·fig.add_subplot(111)	31	····ax·=·fig.add_subplot(111)
32	····ax.plot(support,·weights_func(support))	32	····ax.plot(support,·weights_func(support))
33	····ax.set_xticks(xticks)	33	····ax.set_xticks(xticks)
34	····ax.set_xticklabels(xlabels,·fontsize=16)	34	····ax.set_xticklabels(xlabels,·fontsize=16)
35	····ax.set_ylim(-0.1,·1.1)	35	····ax.set_ylim(-0.1,·1.1)
36	····return·ax	36	····return·ax
37	\x8\x8\x8\x8\x8·A\x8An\x8nd\x8dr\x8re\x8ew\x8w?\x8’s\x8s·W\x8Wa\x8av\x8ve\x8e_\x8?\x8¶·\x8\x8\x8\x8\x8	37	\x8\x8\x8\x8\x8·A\x8An\x8nd\x8dr\x8re\x8ew\x8w?\x8’s\x8s·W\x8Wa\x8av\x8ve\x8e_\x8?\x8¶·\x8\x8\x8\x8\x8
38	[·]:	38	[5]:
39	help(norms.AndrewWave.weights)	39	help(norms.AndrewWave.weights)
		40	Help·on·function·weights·in·module·statsmodels.robust.norms:

		41	weights(self,·z)
		42	····Andrew's·wave·weighting·function·for·the·IRLS·algorithm

		43	····The·psi·function·scaled·by·z

		44	····Parameters
		45	····----------
		46	····z·:·array_like
		47	········1d·array

		48	····Returns
		49	····-------
		50	····weights·:·ndarray
		51	········weights(z)·=·sin(z/a)·/·(z/a)·····for·\\|z\\|·<=·a*pi

		52	········weights(z)·=·0····················for·\\|z\\|·>·a*pi
40	[·]:	53	[6]:
41	a·=·1.339	54	a·=·1.339
42	support·=·np.linspace(-np.pi··a,·np.pi··a,·100)	55	support·=·np.linspace(-np.pi··a,·np.pi··a,·100)
43	andrew·=·norms.AndrewWave(a=a)	56	andrew·=·norms.AndrewWave(a=a)
44	plot_weights(	57	plot_weights(
45	····support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi·*·a,·0,	58	····support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi·*·a,·0,
46	np.pi·*·a]	59	np.pi·*·a]
47	)	60	)
		61	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		62	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		63	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		64	<>:5:·SyntaxWarning:·invalid·escape·sequence·'\p'
		65	/tmp/ipykernel_877395/3509132601.py:5:·SyntaxWarning:·invalid·escape·sequence
		66	'\p'
		67	··support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi·*·a,·0,·np.pi
		68	*·a]
		69	/tmp/ipykernel_877395/3509132601.py:5:·SyntaxWarning:·invalid·escape·sequence
		70	'\p'
		71	··support,·andrew.weights,·["$-\pia$",·"0",·"$\pia$"],·[-np.pi·*·a,·0,·np.pi
		72	*·a]
		73	[6]:
		74	<Axes:·>
		75	[../../../_images/examples_notebooks_generated_robust_models_1_9_2.png]
48	\x8\x8\x8\x8\x8·H\x8Ha\x8am\x8mp\x8pe\x8el\x8l?\x8’s\x8s·1\x817\x87A\x8A_\x8?\x8¶·\x8\x8\x8\x8\x8	76	\x8\x8\x8\x8\x8·H\x8Ha\x8am\x8mp\x8pe\x8el\x8l?\x8’s\x8s·1\x817\x87A\x8A_\x8?\x8¶·\x8\x8\x8\x8\x8
49	[·]:	77	[7]:
50	help(norms.Hampel.weights)	78	help(norms.Hampel.weights)
		79	Help·on·function·weights·in·module·statsmodels.robust.norms:

		80	weights(self,·z)
		81	····Hampel·weighting·function·for·the·IRLS·algorithm

		82	····The·psi·function·scaled·by·z

		83	····Parameters
		84	····----------
		85	····z·:·array_like
		86	········1d·array

		87	····Returns
		88	····-------
		89	····weights·:·ndarray
		90	········weights(z)·=·1································for·\\|z\\|·<=·a

		91	········weights(z)·=·a/\\|z\\|··························for·a·<·\\|z\\|·<=·b

		92	········weights(z)·=·a(c·-·\\|z\\|)/(\\|z\\|(c-b))······for·b·<·\\|z\\|·<=·c

		93	········weights(z)·=·0································for·\\|z\\|·>·c
51	[·]:	94	[8]:
52	c·=·8	95	c·=·8
53	support·=·np.linspace(-3··c,·3··c,·1000)	96	support·=·np.linspace(-3··c,·3··c,·1000)
54	hampel·=·norms.Hampel(a=2.0,·b=4.0,·c=c)	97	hampel·=·norms.Hampel(a=2.0,·b=4.0,·c=c)
55	plot_weights(support,·hampel.weights,·["3c",·"0",·"3c"],·[-3··c,·0,·3··c])	98	plot_weights(support,·hampel.weights,·["3c",·"0",·"3c"],·[-3··c,·0,·3··c])
		99	[8]:
		100	<Axes:·>
		101	[../../../_images/examples_notebooks_generated_robust_models_1_12_1.png]
56	\x8\x8\x8\x8\x8·H\x8Hu\x8ub\x8be\x8er\x8r?\x8’s\x8s·t\x8t_\x8?\x8¶·\x8\x8\x8\x8\x8	102	\x8\x8\x8\x8\x8·H\x8Hu\x8ub\x8be\x8er\x8r?\x8’s\x8s·t\x8t_\x8?\x8¶·\x8\x8\x8\x8\x8
57	[·]:	103	[9]:
58	help(norms.HuberT.weights)	104	help(norms.HuberT.weights)
59	[·]:	105	Help·on·function·weights·in·module·statsmodels.robust.norms:

		106	weights(self,·z)
		107	····Huber's·t·weighting·function·for·the·IRLS·algorithm

		108	····The·psi·function·scaled·by·z

		109	····Parameters
		110	····----------
		111	····z·:·array_like
		112	········1d·array

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16	········},	16	········},
17	········{	17	········{
18	············"cell_type":·"code",	18	············"cell_type":·"code",
19	············"execution_count":·1,	19	············"execution_count":·1,
20	············"metadata":·{	20	············"metadata":·{
21	················"collapsed":·false,	21	················"collapsed":·false,
22	················"execution":·{	22	················"execution":·{
23	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~402226~~Z",	23	····················"iopub.execute_input":·"2026-06-26T06:53:07.091037Z",
24	····················"iopub.status.busy":·"2025-05-23T22:54:58.~~401976~~Z",	24	····················"iopub.status.busy":·"2026-06-26T06:53:07.090589Z",
25	····················"iopub.status.idle":·"2025-05-23T22:54:59.~~400149~~Z",	25	····················"iopub.status.idle":·"2026-06-26T06:53:08.027950Z",
26	····················"shell.execute_reply":·"2025-05-23T22:54:59.~~399215~~Z"	26	····················"shell.execute_reply":·"2026-06-26T06:53:08.026940Z"
27	················},	27	················},
28	················"jupyter":·{	28	················"jupyter":·{
29	····················"outputs_hidden":·false	29	····················"outputs_hidden":·false
30	················}	30	················}
31	············},	31	············},
32	············"outputs":·[],	32	············"outputs":·[],
33	············"source":·[	33	············"source":·[
Offset 36, 18 lines modified		Offset 36, 18 lines modified
36	········},	36	········},
37	········{	37	········{
38	············"cell_type":·"code",	38	············"cell_type":·"code",
39	············"execution_count":·2,	39	············"execution_count":·2,
40	············"metadata":·{	40	············"metadata":·{
41	················"collapsed":·false,	41	················"collapsed":·false,
42	················"execution":·{	42	················"execution":·{
43	····················"iopub.execute_input":·"2025-05-23T22:54:59.403~~549~~Z",	43	····················"iopub.execute_input":·"2026-06-26T06:53:08.034024Z",
44	····················"iopub.status.busy":·"2025-05-23T22:54:59.403145Z",	44	····················"iopub.status.busy":·"2026-06-26T06:53:08.031243Z",
45	····················"iopub.status.idle":·"2025-05-23T22:55:00.970~~382~~Z",	45	····················"iopub.status.idle":·"2026-06-26T06:53:09.760651Z",
46	····················"shell.execute_reply":·"2025-05-23T22:55:00.969~~274~~Z"	46	····················"shell.execute_reply":·"2026-06-26T06:53:09.759518Z"
47	················},	47	················},
48	················"jupyter":·{	48	················"jupyter":·{
49	····················"outputs_hidden":·false	49	····················"outputs_hidden":·false
50	················}	50	················}
51	············},	51	············},
52	············"outputs":·[],	52	············"outputs":·[],
53	············"source":·[	53	············"source":·[
Offset 61, 18 lines modified		Offset 61, 18 lines modified
61	········},	61	········},
62	········{	62	········{
63	············"cell_type":·"code",	63	············"cell_type":·"code",
64	············"execution_count":·3,	64	············"execution_count":·3,
65	············"metadata":·{	65	············"metadata":·{
66	················"collapsed":·false,	66	················"collapsed":·false,
67	················"execution":·{	67	················"execution":·{
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69	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~981340~~Z",	69	····················"iopub.status.busy":·"2026-06-26T06:53:09.768079Z",
70	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~986736~~Z",	70	····················"iopub.status.idle":·"2026-06-26T06:53:09.773089Z",
71	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~985723~~Z"	71	····················"shell.execute_reply":·"2026-06-26T06:53:09.772117Z"
72	················},	72	················},
73	················"jupyter":·{	73	················"jupyter":·{
74	····················"outputs_hidden":·false	74	····················"outputs_hidden":·false
75	················}	75	················}
76	············},	76	············},
77	············"outputs":·[],	77	············"outputs":·[],
78	············"source":·[	78	············"source":·[
Offset 88, 18 lines modified		Offset 88, 18 lines modified
88	········},	88	········},
89	········{	89	········{
90	············"cell_type":·"code",	90	············"cell_type":·"code",
91	············"execution_count":·4,	91	············"execution_count":·4,
92	············"metadata":·{	92	············"metadata":·{
93	················"collapsed":·false,	93	················"collapsed":·false,
94	················"execution":·{	94	················"execution":·{
95	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~993152~~Z",	95	····················"iopub.execute_input":·"2026-06-26T06:53:09.777913Z",
96	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~992645~~Z",	96	····················"iopub.status.busy":·"2026-06-26T06:53:09.777406Z",
97	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~998675~~Z",	97	····················"iopub.status.idle":·"2026-06-26T06:53:09.783401Z",
98	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~997~~620Z"	98	····················"shell.execute_reply":·"2026-06-26T06:53:09.782462Z"
99	················},	99	················},
100	················"jupyter":·{	100	················"jupyter":·{
101	····················"outputs_hidden":·false	101	····················"outputs_hidden":·false
102	················}	102	················}
103	············},	103	············},
104	············"outputs":·[	104	············"outputs":·[
105	················{	105	················{
Offset 125, 18 lines modified		Offset 125, 18 lines modified
125	········},	125	········},
126	········{	126	········{
127	············"cell_type":·"code",	127	············"cell_type":·"code",
128	············"execution_count":·5,	128	············"execution_count":·5,
129	············"metadata":·{	129	············"metadata":·{
130	················"collapsed":·false,	130	················"collapsed":·false,
131	················"execution":·{	131	················"execution":·{
132	····················"iopub.execute_input":·"2025-05-23T22:55:01.~~077721~~Z",	132	····················"iopub.execute_input":·"2026-06-26T06:53:09.792939Z",
133	····················"iopub.status.busy":·"2025-05-23T22:55:01.077215Z",	133	····················"iopub.status.busy":·"2026-06-26T06:53:09.792017Z",
134	····················"iopub.status.idle":·"2025-05-23T22:55:01.~~09801~~4Z",	134	····················"iopub.status.idle":·"2026-06-26T06:53:09.819934Z",
135	····················"shell.execute_reply":·"2025-05-23T22:55:01.~~091394~~Z"	135	····················"shell.execute_reply":·"2026-06-26T06:53:09.818759Z"
136	················},	136	················},
137	················"jupyter":·{	137	················"jupyter":·{
138	····················"outputs_hidden":·false	138	····················"outputs_hidden":·false
139	················}	139	················}
140	············},	140	············},
141	············"outputs":·[],	141	············"outputs":·[],
142	············"source":·[	142	············"source":·[
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145	········},	145	········},
146	········{	146	········{
147	············"cell_type":·"code",	147	············"cell_type":·"code",
148	············"execution_count":·6,	148	············"execution_count":·6,
149	············"metadata":·{	149	············"metadata":·{
150	················"collapsed":·false,	150	················"collapsed":·false,
151	················"execution":·{	151	················"execution":·{
152	····················"iopub.execute_input":·"2025-05-23T22:55:01.~~10333~~5Z",	152	····················"iopub.execute_input":·"2026-06-26T06:53:09.824175Z",
153	····················"iopub.status.busy":·"2025-05-23T22:55:01.~~10289~~4Z",	153	····················"iopub.status.busy":·"2026-06-26T06:53:09.823364Z",
154	····················"iopub.status.idle":·"2025-05-23T22:55:01.~~116750~~Z",	154	····················"iopub.status.idle":·"2026-06-26T06:53:09.848023Z",
155	····················"shell.execute_reply":·"2025-05-23T22:55:01.~~115683~~Z"	155	····················"shell.execute_reply":·"2026-06-26T06:53:09.846970Z"
156	················},	156	················},
157	················"jupyter":·{	157	················"jupyter":·{
158	····················"outputs_hidden":·false	158	····················"outputs_hidden":·false
159	················}	159	················}
160	············},	160	············},
161	············"outputs":·[],	161	············"outputs":·[],
162	············"source":·[	162	············"source":·[
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166	········},	166	········},
167	········{	167	········{
168	············"cell_type":·"code",	168	············"cell_type":·"code",
169	············"execution_count":·7,	169	············"execution_count":·7,
170	············"metadata":·{	170	············"metadata":·{
171	················"collapsed":·false,	171	················"collapsed":·false,
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174	····················"iopub.status.busy":·"2025-05-23T22:55:01.120877Z",	174	····················"iopub.status.busy":·"2026-06-26T06:53:09.860217Z",
175	····················"iopub.status.idle":·"2025-05-23T22:55:01.55~~4087~~Z",	175	····················"iopub.status.idle":·"2026-06-26T06:53:10.350341Z",
176	····················"shell.execute_reply":·"2025-05-23T22:55:01.~~5528~~79Z"	176	····················"shell.execute_reply":·"2026-06-26T06:53:10.349275Z"
177	················},	177	················},
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Offset 76, 15 lines modified		Offset 76, 15 lines modified
76	</div>	76	</div>
77	</div>	77	</div>
78	<div·class="nboutput·nblast·docutils·container">	78	<div·class="nboutput·nblast·docutils·container">
79	<div·class="prompt·empty·docutils·container">	79	<div·class="prompt·empty·docutils·container">
80	</div>	80	</div>
81	<div·class="output_area·stderr·docutils·container">	81	<div·class="output_area·stderr·docutils·container">
82	<div·class="highlight"><pre>	82	<div·class="highlight"><pre>
83	/tmp/ipykernel_3672723/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'QE'·instead.	83	/tmp/ipykernel_877326/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'QE'·instead.
84	··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')	84	··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')
85	</pre></div></div>	85	</pre></div></div>
86	</div>	86	</div>
87	<section·id="Introduction">	87	<section·id="Introduction">
88	<h2>Introduction<a·class="headerlink"·href="#Introduction"·title="Link·to·this·heading">¶</a></h2>	88	<h2>Introduction<a·class="headerlink"·href="#Introduction"·title="Link·to·this·heading">¶</a></h2>
89	<p>(much·of·this·is·based·on·Harvey·(1989);·see·especially·section·3.4)</p>	89	<p>(much·of·this·is·based·on·Harvey·(1989);·see·especially·section·3.4)</p>
90	<p>State·space·models·can·generically·be·written·as·follows·(here·we·focus·on·time-invariant·state·space·models,·but·similar·results·apply·also·to·time-varying·models):</p>	90	<p>State·space·models·can·generically·be·written·as·follows·(here·we·focus·on·time-invariant·state·space·models,·but·similar·results·apply·also·to·time-varying·models):</p>
Offset 331, 19 lines modified		Offset 331, 19 lines modified
331	</pre></div></div>	331	</pre></div></div>
332	</div>	332	</div>
333	<div·class="nboutput·nblast·docutils·container">	333	<div·class="nboutput·nblast·docutils·container">
334	<div·class="prompt·empty·docutils·container">	334	<div·class="prompt·empty·docutils·container">
335	</div>	335	</div>
336	<div·class="output_area·stderr·docutils·container">	336	<div·class="output_area·stderr·docutils·container">
337	<div·class="highlight"><pre>	337	<div·class="highlight"><pre>
338	/tmp/ipykernel_3672723/976557282.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`	338	/tmp/ipykernel_877326/976557282.py:2:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`
339	··print('var.level·····=·%.5f'·%·res.params[0])	339	··print('var.level·····=·%.5f'·%·res.params[0])
340	/tmp/ipykernel_3672723/976557282.py:3:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`	340	/tmp/ipykernel_877326/976557282.py:3:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`
341	··print('var.irregular·=·%.5f'·%·res.params[1])	341	··print('var.irregular·=·%.5f'·%·res.params[1])
342	/tmp/ipykernel_3672723/976557282.py:7:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`	342	/tmp/ipykernel_877326/976557282.py:7:·FutureWarning:·Series.__getitem__·treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys·will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To·access·a·value·by·position,·use·`ser.iloc[pos]`
343	··print('h··scale·····=·%.5f'·%·(res_conc.params[0]··res_conc.scale))	343	··print('h··scale·····=·%.5f'·%·(res_conc.params[0]··res_conc.scale))
344	</pre></div></div>	344	</pre></div></div>
345	</div>	345	</div>
346	</section>	346	</section>
347	</section>	347	</section>
348	<section·id="Example:-SARIMAX">	348	<section·id="Example:-SARIMAX">
349	<h2>Example:·SARIMAX<a·class="headerlink"·href="#Example:-SARIMAX"·title="Link·to·this·heading">¶</a></h2>	349	<h2>Example:·SARIMAX<a·class="headerlink"·href="#Example:-SARIMAX"·title="Link·to·this·heading">¶</a></h2>


Offset 12, 15 lines modified		Offset 12, 15 lines modified
12	[1]:	12	[1]:
13	import·numpy·as·np	13	import·numpy·as·np
14	import·pandas·as·pd	14	import·pandas·as·pd
15	import·statsmodels.api·as·sm	15	import·statsmodels.api·as·sm

16	dta·=·sm.datasets.macrodata.load_pandas().data	16	dta·=·sm.datasets.macrodata.load_pandas().data
17	dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')	17	dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')
18	/tmp/ipykernel_3672723/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and	18	/tmp/ipykernel_877326/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and
19	will·be·removed·in·a·future·version,·please·use·'QE'·instead.	19	will·be·removed·in·a·future·version,·please·use·'QE'·instead.
20	··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')	20	··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')
21	\x8\x8\x8\x8\x8·I\x8In\x8nt\x8tr\x8ro\x8od\x8du\x8uc\x8ct\x8ti\x8io\x8on\x8n_\x8?\x8¶·\x8\x8\x8\x8\x8	21	\x8\x8\x8\x8\x8·I\x8In\x8nt\x8tr\x8ro\x8od\x8du\x8uc\x8ct\x8ti\x8io\x8on\x8n_\x8?\x8¶·\x8\x8\x8\x8\x8
22	(much·of·this·is·based·on·Harvey·(1989);·see·especially·section·3.4)	22	(much·of·this·is·based·on·Harvey·(1989);·see·especially·section·3.4)
23	State·space·models·can·generically·be·written·as·follows·(here·we·focus·on	23	State·space·models·can·generically·be·written·as·follows·(here·we·focus·on
24	time-invariant·state·space·models,·but·similar·results·apply·also·to·time-	24	time-invariant·state·space·models,·but·similar·results·apply·also·to·time-
25	varying·models):	25	varying·models):
Offset 246, 25 lines modified		Offset 246, 25 lines modified
246	Original·model	246	Original·model
247	var.level·····=·0.74469	247	var.level·····=·0.74469
248	var.irregular·=·3.37330	248	var.irregular·=·3.37330

249	Concentrated·model	249	Concentrated·model
250	scale·········=·0.74472	250	scale·········=·0.74472
251	h·*·scale·····=·3.37338	251	h·*·scale·····=·3.37338
252	/tmp/ipykernel_3672723/976557282.py:2:·FutureWarning:·Series.__getitem__	252	/tmp/ipykernel_877326/976557282.py:2:·FutureWarning:·Series.__getitem__
253	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys	253	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys
254	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To	254	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To
255	access·a·value·by·position,·use·`ser.iloc[pos]`	255	access·a·value·by·position,·use·`ser.iloc[pos]`
256	··print('var.level·····=·%.5f'·%·res.params[0])	256	··print('var.level·····=·%.5f'·%·res.params[0])
257	/tmp/ipykernel_3672723/976557282.py:3:·FutureWarning:·Series.__getitem__	257	/tmp/ipykernel_877326/976557282.py:3:·FutureWarning:·Series.__getitem__
258	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys	258	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys
259	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To	259	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To
260	access·a·value·by·position,·use·`ser.iloc[pos]`	260	access·a·value·by·position,·use·`ser.iloc[pos]`
261	··print('var.irregular·=·%.5f'·%·res.params[1])	261	··print('var.irregular·=·%.5f'·%·res.params[1])
262	/tmp/ipykernel_3672723/976557282.py:7:·FutureWarning:·Series.__getitem__	262	/tmp/ipykernel_877326/976557282.py:7:·FutureWarning:·Series.__getitem__
263	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys	263	treating·keys·as·positions·is·deprecated.·In·a·future·version,·integer·keys
264	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To	264	will·always·be·treated·as·labels·(consistent·with·DataFrame·behavior).·To
265	access·a·value·by·position,·use·`ser.iloc[pos]`	265	access·a·value·by·position,·use·`ser.iloc[pos]`
266	··print('h··scale·····=·%.5f'·%·(res_conc.params[0]··res_conc.scale))	266	··print('h··scale·····=·%.5f'·%·(res_conc.params[0]··res_conc.scale))
267	\x8\x8\x8\x8\x8·E\x8Ex\x8xa\x8am\x8mp\x8pl\x8le\x8e:\x8:·S\x8SA\x8AR\x8RI\x8IM\x8MA\x8AX\x8X_\x8?\x8¶·\x8\x8\x8\x8\x8	267	\x8\x8\x8\x8\x8·E\x8Ex\x8xa\x8am\x8mp\x8pl\x8le\x8e:\x8:·S\x8SA\x8AR\x8RI\x8IM\x8MA\x8AX\x8X_\x8?\x8¶·\x8\x8\x8\x8\x8
268	By·default·in·SARIMAX·models,·the·variance·term·is·chosen·by·numerically	268	By·default·in·SARIMAX·models,·the·variance·term·is·chosen·by·numerically
269	maximizing·the·likelihood·function,·but·an·option·has·been·added·to·allow	269	maximizing·the·likelihood·function,·but·an·option·has·been·added·to·allow


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8	············]	8	············]
9	········},	9	········},
10	········{	10	········{
11	············"cell_type":·"code",	11	············"cell_type":·"code",
12	············"execution_count":·1,	12	············"execution_count":·1,
13	············"metadata":·{	13	············"metadata":·{
14	················"execution":·{	14	················"execution":·{
15	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~539~~704Z",	15	····················"iopub.execute_input":·"2026-06-26T06:53:00.178777Z",
16	····················"iopub.status.busy":·"2025-05-23T22:54:58.~~539246~~Z",	16	····················"iopub.status.busy":·"2026-06-26T06:53:00.178368Z",
17	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~115502~~Z",	17	····················"iopub.status.idle":·"2026-06-26T06:53:01.836484Z",
18	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~114475~~Z"	18	····················"shell.execute_reply":·"2026-06-26T06:53:01.835366Z"
19	················}	19	················}
20	············},	20	············},
21	············"outputs":·[	21	············"outputs":·[
22	················{	22	················{
23	····················"name":·"stderr",	23	····················"name":·"stderr",
24	····················"output_type":·"stream",	24	····················"output_type":·"stream",
25	····················"text":·[	25	····················"text":·[
26	························"/tmp/ipykernel_3672723/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'QE'·instead.\n",	26	························"/tmp/ipykernel_877326/2378637424.py:6:·FutureWarning:·'Q'·is·deprecated·and·will·be·removed·in·a·future·version,·please·use·'QE'·instead.\n",
27	························"··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')\n"	27	························"··dta.index·=·pd.date_range(start='1959Q1',·end='2009Q4',·freq='Q')\n"
28	····················]	28	····················]
29	················}	29	················}
30	············],	30	············],
31	············"source":·[	31	············"source":·[
32	················"import·numpy·as·np\n",	32	················"import·numpy·as·np\n",
33	················"import·pandas·as·pd\n",	33	················"import·pandas·as·pd\n",
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89	············]	89	············]
90	········},	90	········},
91	········{	91	········{
92	············"cell_type":·"code",	92	············"cell_type":·"code",
93	············"execution_count":·2,	93	············"execution_count":·2,
94	············"metadata":·{	94	············"metadata":·{
95	················"execution":·{	95	················"execution":·{
96	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~242413~~Z",	96	····················"iopub.execute_input":·"2026-06-26T06:53:01.920609Z",
97	····················"iopub.status.busy":·"2025-05-23T22:55:00.241897Z",	97	····················"iopub.status.busy":·"2026-06-26T06:53:01.920152Z",
98	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~271~~702Z",	98	····················"iopub.status.idle":·"2026-06-26T06:53:01.928702Z",
99	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~2605~~59Z"	99	····················"shell.execute_reply":·"2026-06-26T06:53:01.927756Z"
100	················}	100	················}
101	············},	101	············},
102	············"outputs":·[],	102	············"outputs":·[],
103	············"source":·[	103	············"source":·[
104	················"class·LocalLevel(sm.tsa.statespace.MLEModel):\n",	104	················"class·LocalLevel(sm.tsa.statespace.MLEModel):\n",
105	················"····_start_params·=·[1.,·1.]\n",	105	················"····_start_params·=·[1.,·1.]\n",
106	················"····_param_names·=·['var.level',·'var.irregular']\n",	106	················"····_param_names·=·['var.level',·'var.irregular']\n",
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135	············]	135	············]
136	········},	136	········},
137	········{	137	········{
138	············"cell_type":·"code",	138	············"cell_type":·"code",
139	············"execution_count":·3,	139	············"execution_count":·3,
140	············"metadata":·{	140	············"metadata":·{
141	················"execution":·{	141	················"execution":·{
142	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~276638~~Z",	142	····················"iopub.execute_input":·"2026-06-26T06:53:01.934304Z",
143	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~276181~~Z",	143	····················"iopub.status.busy":·"2026-06-26T06:53:01.933867Z",
144	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~346212~~Z",	144	····················"iopub.status.idle":·"2026-06-26T06:53:01.991441Z",
145	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~34524~~7Z"	145	····················"shell.execute_reply":·"2026-06-26T06:53:01.990537Z"
146	················}	146	················}
147	············},	147	············},
148	············"outputs":·[	148	············"outputs":·[
149	················{	149	················{
150	····················"name":·"stdout",	150	····················"name":·"stdout",
151	····················"output_type":·"stream",	151	····················"output_type":·"stream",
152	····················"text":·[	152	····················"text":·[
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191	········},	191	········},
192	········{	192	········{
193	············"cell_type":·"code",	193	············"cell_type":·"code",
194	············"execution_count":·4,	194	············"execution_count":·4,
195	············"metadata":·{	195	············"metadata":·{
196	················"execution":·{	196	················"execution":·{
197	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~353805~~Z",	197	····················"iopub.execute_input":·"2026-06-26T06:53:02.002289Z",
198	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~353362~~Z",	198	····················"iopub.status.busy":·"2026-06-26T06:53:02.001837Z",
199	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~362920~~Z",	199	····················"iopub.status.idle":·"2026-06-26T06:53:02.007243Z",
200	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~361736~~Z"	200	····················"shell.execute_reply":·"2026-06-26T06:53:02.006468Z"
201	················}	201	················}
202	············},	202	············},
203	············"outputs":·[	203	············"outputs":·[
204	················{	204	················{
205	····················"name":·"stdout",	205	····················"name":·"stdout",
206	····················"output_type":·"stream",	206	····················"output_type":·"stream",
207	····················"text":·[	207	····················"text":·[
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245	········},	245	········},
246	········{	246	········{
247	············"cell_type":·"code",	247	············"cell_type":·"code",
248	············"execution_count":·5,	248	············"execution_count":·5,
249	············"metadata":·{	249	············"metadata":·{
250	················"execution":·{	250	················"execution":·{
251	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~366889~~Z",	251	····················"iopub.execute_input":·"2026-06-26T06:53:02.012156Z",
252	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~366~~514Z",	252	····················"iopub.status.busy":·"2026-06-26T06:53:02.011511Z",
253	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~374467~~Z",	253	····················"iopub.status.idle":·"2026-06-26T06:53:02.019315Z",
254	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~3736~~25Z"	254	····················"shell.execute_reply":·"2026-06-26T06:53:02.018424Z"
255	················}	255	················}
256	············},	256	············},
257	············"outputs":·[],	257	············"outputs":·[],
258	············"source":·[	258	············"source":·[
259	················"class·LocalLevelConcentrated(sm.tsa.statespace.MLEModel):\n",	259	················"class·LocalLevelConcentrated(sm.tsa.statespace.MLEModel):\n",
260	················"····_start_params·=·[1.]\n",	260	················"····_start_params·=·[1.]\n",
261	················"····_param_names·=·['ratio.irregular']\n",	261	················"····_param_names·=·['ratio.irregular']\n",
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289	············]	289	············]
290	········},	290	········},
291	········{	291	········{
292	············"cell_type":·"code",	292	············"cell_type":·"code",
293	············"execution_count":·6,	293	············"execution_count":·6,
294	············"metadata":·{	294	············"metadata":·{
295	················"execution":·{	295	················"execution":·{
296	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~378~~158Z",	296	····················"iopub.execute_input":·"2026-06-26T06:53:02.024119Z",
297	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~377802~~Z",	297	····················"iopub.status.busy":·"2026-06-26T06:53:02.023448Z",
298	····················"iopub.status.idle":·"2025-05-23T22:55:00.~~423104~~Z",	298	····················"iopub.status.idle":·"2026-06-26T06:53:02.074692Z",
299	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~422083~~Z"	299	····················"shell.execute_reply":·"2026-06-26T06:53:02.073704Z"
300	················}	300	················}
301	············},	301	············},
302	············"outputs":·[	302	············"outputs":·[
303	················{	303	················{
304	····················"name":·"stdout",	304	····················"name":·"stdout",
305	····················"output_type":·"stream",	305	····················"output_type":·"stream",
306	····················"text":·[	306	····················"text":·[
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351	········},	351	········},
352	········{	352	········{
353	············"cell_type":·"code",	353	············"cell_type":·"code",
354	············"execution_count":·7,	354	············"execution_count":·7,
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50	········},	50	········},
51	········{	51	········{
52	············"cell_type":·"code",	52	············"cell_type":·"code",
53	············"execution_count":·1,	53	············"execution_count":·1,
54	············"id":·"ba21553a-e571-42ac-b166-b625a50509fe",	54	············"id":·"ba21553a-e571-42ac-b166-b625a50509fe",
55	············"metadata":·{	55	············"metadata":·{
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57	····················"iopub.execute_input":·"2025-05-23T22:54:58.~~508512~~Z",	57	····················"iopub.execute_input":·"2026-06-26T06:53:05.794668Z",
58	····················"iopub.status.busy":·"2025-05-23T22:54:58.~~50820~~8Z",	58	····················"iopub.status.busy":·"2026-06-26T06:53:05.793538Z",
59	····················"iopub.status.idle":·"2025-05-23T22:55:00.331734Z",	59	····················"iopub.status.idle":·"2026-06-26T06:53:06.621284Z",
60	····················"shell.execute_reply":·"2025-05-23T22:55:00.330475Z"	60	····················"shell.execute_reply":·"2026-06-26T06:53:06.620427Z"
61	················}	61	················}
62	············},	62	············},
63	············"outputs":·[],	63	············"outputs":·[],
64	············"source":·[	64	············"source":·[
65	················"%matplotlib·inline"	65	················"%matplotlib·inline"
66	············]	66	············]
67	········},	67	········},
68	········{	68	········{
69	············"cell_type":·"code",	69	············"cell_type":·"code",
70	············"execution_count":·2,	70	············"execution_count":·2,
71	············"id":·"fe284c44-b750-4e6e-94d0-b238f364cd7b",	71	············"id":·"fe284c44-b750-4e6e-94d0-b238f364cd7b",
72	············"metadata":·{	72	············"metadata":·{
73	················"execution":·{	73	················"execution":·{
74	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~337781~~Z",	74	····················"iopub.execute_input":·"2026-06-26T06:53:06.626533Z",
75	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~337210~~Z",	75	····················"iopub.status.busy":·"2026-06-26T06:53:06.625978Z",
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77	····················"shell.execute_reply":·"2025-05-23T22:55:00.~~870572~~Z"	77	····················"shell.execute_reply":·"2026-06-26T06:53:07.161659Z"
78	················}	78	················}
79	············},	79	············},
80	············"outputs":·[],	80	············"outputs":·[],
81	············"source":·[	81	············"source":·[
82	················"import·numpy·as·np\n",	82	················"import·numpy·as·np\n",
83	················"import·pandas·as·pd\n",	83	················"import·pandas·as·pd\n",
84	················"\n",	84	················"\n",
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95	········},	95	········},
96	········{	96	········{
97	············"cell_type":·"code",	97	············"cell_type":·"code",
98	············"execution_count":·3,	98	············"execution_count":·3,
99	············"id":·"f02b87c6-ee8f-4bb1-bf08-d252b9277733",	99	············"id":·"f02b87c6-ee8f-4bb1-bf08-d252b9277733",
100	············"metadata":·{	100	············"metadata":·{
101	················"execution":·{	101	················"execution":·{
102	····················"iopub.execute_input":·"2025-05-23T22:55:00.~~881587~~Z",	102	····················"iopub.execute_input":·"2026-06-26T06:53:07.167254Z",
103	····················"iopub.status.busy":·"2025-05-23T22:55:00.~~88100~~2Z",	103	····················"iopub.status.busy":·"2026-06-26T06:53:07.166652Z",
104	····················"iopub.status.idle":·"2025-05-23T22:55:01.8~~350~~83Z",	104	····················"iopub.status.idle":·"2026-06-26T06:53:08.089738Z",
105	····················"shell.execute_reply":·"2025-05-23T22:55:01.~~834189~~Z"	105	····················"shell.execute_reply":·"2026-06-26T06:53:08.088732Z"
106	················}	106	················}
107	············},	107	············},
108	············"outputs":·[],	108	············"outputs":·[],
109	············"source":·[	109	············"source":·[
110	················"from·statsmodels.tsa.api·import·SARIMAX,·AutoReg\n",	110	················"from·statsmodels.tsa.api·import·SARIMAX,·AutoReg\n",
111	················"from·statsmodels.tsa.arima.model·import·ARIMA"	111	················"from·statsmodels.tsa.arima.model·import·ARIMA"
112	············]	112	············]
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121	········},	121	········},
122	········{	122	········{
123	············"cell_type":·"code",	123	············"cell_type":·"code",
124	············"execution_count":·4,	124	············"execution_count":·4,
125	············"id":·"7200d248-bd47-4c95-9f1c-6daaf1e09bb1",	125	············"id":·"7200d248-bd47-4c95-9f1c-6daaf1e09bb1",
126	············"metadata":·{	126	············"metadata":·{
127	················"execution":·{	127	················"execution":·{
128	····················"iopub.execute_input":·"2025-05-23T22:55:01.~~845811~~Z",	128	····················"iopub.execute_input":·"2026-06-26T06:53:08.094033Z",
129	····················"iopub.status.busy":·"2025-05-23T22:55:01.~~845205~~Z",	129	····················"iopub.status.busy":·"2026-06-26T06:53:08.092862Z",
130	····················"iopub.status.idle":·"2025-05-23T22:55:02.368386Z",	130	····················"iopub.status.idle":·"2026-06-26T06:53:08.616809Z",
131	····················"shell.execute_reply":·"2025-05-23T22:55:02.~~367322~~Z"	131	····················"shell.execute_reply":·"2026-06-26T06:53:08.615561Z"
132	················}	132	················}
133	············},	133	············},
134	············"outputs":·[],	134	············"outputs":·[],
135	············"source":·[	135	············"source":·[
136	················"ar0_res·=·SARIMAX(y,·order=(0,·0,·0),·trend=\"c\").fit()\n",	136	················"ar0_res·=·SARIMAX(y,·order=(0,·0,·0),·trend=\"c\").fit()\n",
137	················"sarimax_res·=·SARIMAX(y,·order=(1,·0,·0),·trend=\"c\").fit()\n",	137	················"sarimax_res·=·SARIMAX(y,·order=(1,·0,·0),·trend=\"c\").fit()\n",
138	················"arima_res·=·ARIMA(y,·order=(1,·0,·0),·trend=\"c\").fit()\n",	138	················"arima_res·=·ARIMA(y,·order=(1,·0,·0),·trend=\"c\").fit()\n",
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149	········},	149	········},
150	········{	150	········{
151	············"cell_type":·"code",	151	············"cell_type":·"code",
152	············"execution_count":·5,	152	············"execution_count":·5,
153	············"id":·"8ff07d0e-6754-4664-93e4-0f9299096868",	153	············"id":·"8ff07d0e-6754-4664-93e4-0f9299096868",
154	············"metadata":·{	154	············"metadata":·{
155	················"execution":·{	155	················"execution":·{
156	····················"iopub.execute_input":·"2025-05-23T22:55:02.~~375985~~Z",	156	····················"iopub.execute_input":·"2026-06-26T06:53:08.624297Z",
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158	····················"iopub.status.idle":·"2025-05-23T22:55:02.~~398005~~Z",	158	····················"iopub.status.idle":·"2026-06-26T06:53:08.669526Z",
159	····················"shell.execute_reply":·"2025-05-23T22:55:02.~~397159~~Z"	159	····················"shell.execute_reply":·"2026-06-26T06:53:08.666345Z"
160	················}	160	················}
161	············},	161	············},
162	············"outputs":·[	162	············"outputs":·[
163	················{	163	················{
164	····················"data":·{	164	····················"data":·{
165	························"text/html":·[	165	························"text/html":·[
166	····························"<div>\n",	166	····························"<div>\n",
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268	········},	268	········},
269	········{	269	········{
270	············"cell_type":·"code",	270	············"cell_type":·"code",
271	············"execution_count":·6,	271	············"execution_count":·6,
272	············"id":·"c18adf81-1ad9-4d11-a23b-e6d139c1fa3e",	272	············"id":·"c18adf81-1ad9-4d11-a23b-e6d139c1fa3e",
273	············"metadata":·{	273	············"metadata":·{
274	················"execution":·{	274	················"execution":·{
275	····················"iopub.execute_input":·"2025-05-23T22:55:02.~~404~~969Z",	275	····················"iopub.execute_input":·"2026-06-26T06:53:08.677939Z",
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277	····················"iopub.status.idle":·"2025-05-23T22:55:02.~~649967~~Z",	277	····················"iopub.status.idle":·"2026-06-26T06:53:08.994034Z",
278	····················"shell.execute_reply":·"2025-05-23T22:55:02.~~648763~~Z"	278	····················"shell.execute_reply":·"2026-06-26T06:53:08.993151Z"
279	················}	279	················}
280	············},	280	············},
281	············"outputs":·[	281	············"outputs":·[
282	················{	282	················{
283	····················"name":·"stdout",	283	····················"name":·"stdout",
284	····················"output_type":·"stream",	284	····················"output_type":·"stream",
285	····················"text":·[	285	····················"text":·[
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335	········},	335	········},
336	········{	336	········{
337	············"cell_type":·"code",	337	············"cell_type":·"code",
338	············"execution_count":·7,	338	············"execution_count":·7,
339	············"id":·"8978b4c9-05cb-4674-9c67-53eccd8302a0",	339	············"id":·"8978b4c9-05cb-4674-9c67-53eccd8302a0",
340	············"metadata":·{	340	············"metadata":·{
341	················"execution":·{	341	················"execution":·{
342	····················"iopub.execute_input":·"2025-05-23T22:55:02.~~656~~850Z",	342	····················"iopub.execute_input":·"2026-06-26T06:53:09.002185Z",
343	····················"iopub.status.busy":·"2025-05-23T22:55:02.~~656383~~Z",	343	····················"iopub.status.busy":·"2026-06-26T06:53:09.001429Z",
344	····················"iopub.status.idle":·"2025-05-23T22:55:02.~~662130~~Z",	344	····················"iopub.status.idle":·"2026-06-26T06:53:09.010262Z",
345	····················"shell.execute_reply":·"2025-05-23T22:55:02.~~660892~~Z"	345	····················"shell.execute_reply":·"2026-06-26T06:53:09.009078Z"
346	················}	346	················}
347	············},	347	············},
348	············"outputs":·[],	348	············"outputs":·[],
349	············"source":·[	349	············"source":·[
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