>hlMSrSSKrSSKJr SSKJr SSKJr SSK J r Sr SSjr S r SS jrSS jrS rS rSrSrSrSrg)zL Created on Sun Nov 5 14:48:19 2017 Author: Josef Perktold License: BSD-3 N)stats)cov2corr) HolderTuple) array_likecF[RRU5S$)N)nplinalgslogdet)xs qC:\Users\julio\OneDrive\Documentos\Trabajo\Ideas Frescas\venv\Lib\site-packages\statsmodels/stats/multivariate.py_logdetrs 99  Q  ""c[R"U5nURupEURS5n[R"USSS9nXa- nXHR [R RXx55-n US- U-XE- - n X- n XTU- 4n [RRXSU S5n U(a[U U U U SS9nU$X4$)a[Hotellings test for multivariate mean in one sample Parameters ---------- data : array_like data with observations in rows and variables in columns mean_null : array_like mean of the multivariate data under the null hypothesis return_results : bool If true, then a results instance is returned. If False, then only the test statistic and pvalue are returned. Returns ------- results : instance of a results class with attributes statistic, pvalue, t2 and df (statistic, pvalue) : tuple If return_results is false, then only the test statistic and the pvalue are returned. rFrrowvarddofF statisticpvaluedft2distr) r asarrayshapemeancovdotr solverfsfr)data mean_nullreturn_resultsr nobsk_varsrrdiffrfactorrrrress r test_mvmeanr+s, 4A77LD 66!9D &&5q )C  D 34 4BQh& DM 2F I - B WWZZ a5"Q% 0FI!' # %    rc\[USSS9n[USSS9nURupEURupgXu:wa Sn[U5eURS5n URS5n [R "USSS 9n [R "USSS 9n XF-n US- U -US- U --U S- - nX- nXF-U - U-[R RX5-nU S- U-X- S- - nUU- nX]S- U- 4n[RRUUSUS5n[UUUUS S 9$) aHotellings test for multivariate mean in two independent samples The null hypothesis is that both samples have the same mean. The alternative hypothesis is that means differ. Parameters ---------- data1 : array_like first sample data with observations in rows and variables in columns data2 : array_like second sample data with observations in rows and variables in columns Returns ------- results : instance of a results class with attributes statistic, pvalue, t2 and df x1)ndimx2z4both samples need to have the same number of columnsrFrrrr) rr ValueErrorrr rr r rr!r"r)data1data2r-r0nobs1r'nobs2k_vars2msgmean1mean2cov1cov2nobs_t combined_covr(rr)rrrs r test_mvmean_2indepr>@sL$ E4a (B E4a (BHHMEXXNEDo GGAJE GGAJE 66"U +D 66"U +D ]FQY$&%!)t);; KL =D -6 !D (299??<+N NBzV#!(; B Irc [R"U5n[R"U5n[R"U5n[U5nUSLaUR U5nXaR UR 5R -R S5n US- n [RRUS- U 5n [R"X- 5U -n X- n X-nOUR U5nXaR UR 5R -R S5n US- U-X'- - U- nXrU- 4n [RRXJSU S5n[R"X-U-5n X- n X-nXU4$)aYConfidence interval for linear transformation of a multivariate mean Either pointwise or simultaneous confidence intervals are returned. Data is provided in the form of summary statistics, mean, cov, nobs. Parameters ---------- mean : ndarray cov : ndarray nobs : int lin_transf : array_like or None The linear transformation or contrast matrix for transforming the vector of means. If this is None, then the identity matrix is used which specifies the means themselves. alpha : float in (0, 1) confidence level for the confidence interval, commonly used is alpha=0.05. simult : bool If simult is False (default), then pointwise confidence interval is returned. Otherwise, a simultaneous confidence interval is returned. Warning: additional simultaneous confidence intervals might be added and the default for those might change. Notes ----- Pointwise confidence interval is based on Johnson and Wichern equation (5-21) page 224. Simultaneous confidence interval is based on Johnson and Wichern Result 5.3 page 225. This looks like Sheffe simultaneous confidence intervals. Bonferroni corrected simultaneous confidence interval might be added in future References ---------- Johnson, Richard A., and Dean W. Wichern. 2007. Applied Multivariate Statistical Analysis. 6th ed. Upper Saddle River, N.J: Pearson Prentice Hall. Frr.r) r r atleast_2dlenrTsumrtisfsqrtr!)rrr&r@rArBcr'values quad_formr t_critvalci_difflowuppr) f_critvals r rDrDsWZ ::d D **S/C j!A YF t',,Q/ AXGGKK 2. '').)I5t',,Q/ (f$ 6=Vm $GGKK!ube4 ''&,y89 V rc [R"U5US- -U- n[R"U5nURSnUnUS- nUSSU-S-SUS-- - SUS- -S- - - -n[U5[XfS- - U-5- nU[R"XfS- - [R R XC5-5U- - nXx-n XUS--S- n [RRX5n [U U U SSUS9$) u@One sample hypothesis test for covariance equal to null covariance The Null hypothesis is that cov = cov_null, against the alternative that it is not equal to cov_null Parameters ---------- cov : array_like Covariance matrix of the data, estimated with denominator ``(N - 1)``, i.e. `ddof=1`. nobs : int number of observations used in the estimation of the covariance cov_null : nd_array covariance under the null hypothesis Returns ------- res : instance of HolderTuple results with ``statistic, pvalue`` and other attributes like ``df`` References ---------- Bartlett, M. S. 1954. “A Note on the Multiplying Factors for Various Χ2 Approximations.” Journal of the Royal Statistical Society. Series B (Methodological) 16 (2): 296–98. Rencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher/Methods. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9781118391686. StataCorp, L. P. Stata Multivariate Statistics: Reference Manual. Stata Press Publication. rrg?r.chi2z equal value)rrrrnullcov_null) r rrrtracer r rrYr"r) rr&r[SS0knfactfact2rrrs r test_covrcsN 34!8$t+A H B ! A A "9DAQQ!a%[(Q!a%[1_= ==D BK'!1u+/2 2E RXXaq5kBIIOOB$:: ;a ??E I !eqB ZZ]]9 )F $#) (  rc[R"U5nURSnUS- SUS--U-S-SU-- - nX2[R"[R"U55-[ U5- U[R"U5-- -nX"S--S- S- n[ RRX45n[UUUSSS9$)uPOne sample hypothesis test that covariance matrix is spherical The Null and alternative hypotheses are .. math:: H0 &: \Sigma = \sigma I \\ H1 &: \Sigma \neq \sigma I where :math:`\sigma_i` is the common variance with unspecified value. Parameters ---------- cov : array_like Covariance matrix of the data, estimated with denominator ``(N - 1)``, i.e. `ddof=1`. nobs : int number of observations used in the estimation of the covariance Returns ------- res : instance of HolderTuple results with ``statistic, pvalue`` and other attributes like ``df`` References ---------- Bartlett, M. S. 1954. “A Note on the Multiplying Factors for Various Χ2 Approximations.” Journal of the Royal Statistical Society. Series B (Methodological) 16 (2): 296–98. Rencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher/Methods. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9781118391686. StataCorp, L. P. Stata Multivariate Statistics: Reference Manual. Stata Press Publication. rrr.rXrY sphericalrrrrrZ) r rrlogr\rrrYr"r)rr&r_rrrs r test_cov_sphericalrh@sT **S/C ! AqA1HqL1,Q77I RVVBHHSM**WS\9Aq MIII !eq1 B ZZ]]9 )F $#'  rc [R"U5nURSn[U5nUS- SU-S-S- - *[ U5-nX"S- -S- n[ R RXE5n[UUUSSS9$) aOne sample hypothesis test that covariance matrix is diagonal matrix. The Null and alternative hypotheses are .. math:: H0 &: \Sigma = diag(\sigma_i) \\ H1 &: \Sigma \neq diag(\sigma_i) where :math:`\sigma_i` are the variances with unspecified values. Parameters ---------- cov : array_like Covariance matrix of the data, estimated with denominator ``(N - 1)``, i.e. `ddof=1`. nobs : int number of observations used in the estimation of the covariance Returns ------- res : instance of HolderTuple results with ``statistic, pvalue`` and other attributes like ``df`` References ---------- Rencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher/Methods. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9781118391686. StataCorp, L. P. Stata Multivariate Statistics: Reference Manual. Stata Press Publication. rrr.rXrYdiagonalrf) r rrrrrrYr"r)rr&r_Rrrrs r test_cov_diagonalrmysF **S/C ! A A(a!eai1_,- :I !eqB ZZ]]9 )F $#&  rc6[U5n[R"U5nUSU:XaUSSnOUSU:a [S5e[R"[R "U5U5n/nUHnUR XSS2S4U45 M! XT4$)z$get diagonal blocks from matrix Nz)sum of block_len larger than shape of mat)rIr cumsumr1splitarangeappend)mat block_lenr_idx idx_blocksblocksiis r _get_blocksrzs CA ))I C 2w!|#2h R1DEE "))A,,J F cQW+r/*+  rc[R"U5n[X5SnURSnUVs/sHoURSPM nnU[ U5:wa Sn[ U5e[ SU55nUS-[ SU55- n US-[ SU55- n US- SU -SU --S U -- - n X[ U5- -n U S- n [RRX5n [U U U S S S 9$s snf) aOne sample hypothesis test that covariance is block diagonal. The Null and alternative hypotheses are .. math:: H0 &: \Sigma = diag(\Sigma_i) \\ H1 &: \Sigma \neq diag(\Sigma_i) where :math:`\Sigma_i` are covariance blocks with unspecified values. Parameters ---------- cov : array_like Covariance matrix of the data, estimated with denominator ``(N - 1)``, i.e. `ddof=1`. nobs : int number of observations used in the estimation of the covariance block_len : list list of length of each square block Returns ------- res : instance of HolderTuple results with ``statistic, pvalue`` and other attributes like ``df`` References ---------- Rencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher/Methods. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9781118391686. StataCorp, L. P. Stata Multivariate Statistics: Reference Manual. Stata Press Publication. rz8sample covariances and blocks do not have matching shapec38# UHn[U5v M g7f)Nr).0rOs r )test_cov_blockdiagonal..s7Jq Jr.c3*# UH oS-v M g7f)r.Nr~kis r rr-HbEHc3*# UH oS-v M g7f)rNrrs r rrrrrg@rYzblock-diagonalrf) r rrzrrKr1rrrYr"r)rr&ru cov_blocksr_rOk_blocksr7 logdet_blocksa2a3rrrs r test_cov_blockdiagonalrsJ **S/CS,Q/J ! A$./Jq JH/CMHo7J77M A-H-- -B A-H-- -BQVa"f_b99I --I aB ZZ]]9 )F $#,  0sDc[[[RU55n[ U5n[ U5nUSR Sn[ S[X555nXSU- -nX2- [U5-nU[ S[X555-n[ SU55SX2- - - nUSU-U-SU--S- SUS--US- -- -nUS- U-US--S- nSU- U-n [RRX5n [ S U55SX2- S-- - n XS- US--SUS- -- -n Un U S-[XS-- 5- n SU- X- - U - nSU- SU - -U - nXS-:aX-nOX-nX- U-SU-- nX4n[RR"U/UQ76n[UUUU U US UUUS S 9 $) a Multiple sample hypothesis test that covariance matrices are equal. This is commonly known as Box-M test. The Null and alternative hypotheses are .. math:: H0 &: \Sigma_i = \Sigma_j \text{ for all i and j} \\ H1 &: \Sigma_i \neq \Sigma_j \text{ for at least one i and j} where :math:`\Sigma_i` is the covariance of sample `i`. Parameters ---------- cov_list : list of array_like Covariance matrices of the sample, estimated with denominator ``(N - 1)``, i.e. `ddof=1`. nobs_list : list List of the number of observations used in the estimation of the covariance for each sample. Returns ------- res : instance of HolderTuple Results contains test statistic and pvalues for both chisquare and F distribution based tests, identified by the name ending "_chi2" and "_f". Attributes ``statistic, pvalue`` refer to the F-test version. Notes ----- approximations to distribution of test statistic is by Box References ---------- Rencher, Alvin C., and William F. Christensen. 2012. Methods of Multivariate Analysis: Rencher/Methods. Wiley Series in Probability and Statistics. 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