M/PhUddlZddlZddlmZmZmZmZddlZ ddl m Z ddl m Z ddlmZddlmZddlmZdd lmZmZd d iZd Zd ZGddeZGddZdS)N)eighinvnorm matrix_rank)minimize)cache_readonly)Model)summary2) _import_mpl)rotate_factorspromaxgtolgHz>cd}||t|||tjd|dkrtd|z||tjddSdSdS)N&Either endog or corr must be provided.zHBoth endog and corr are provided, corr will be used for factor analysis.rz&n_factor must be larger than 0! %d < 0z&nobs is ignored when endog is provided) ValueErrorwarningswarn)endogn_factorcorrnobsmsgs _/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/multivariate/factor.py _check_args_1rs 2C T-oo } ? @ @ @1}}A"$%% % E- >?????--c<||krtd||fztjtjtj|dz dkrtd|jd|jdkrtd|jzdS)Nz@n_factor cannot be greater than the number of variables! %d > %dr 绽|=z!corr must be a correlation matrixrz=Correlation matrix corr must be a square (rows %d != cols %d))rnpmaxabsdiagshape)rrrrk_endogs r _check_args_2r%$s'2"G,-.. . vbfRWT]]Q&''((500<=== z!} 1 %%026*=>> >&%rceZdZdZ dfd ZedZejd Z ddZddZ dZ dZ dZ dZ dZddZdZxZS)FactoraD Factor analysis Parameters ---------- endog : array_like Variables in columns, observations in rows. May be `None` if `corr` is not `None`. n_factor : int The number of factors to extract corr : array_like Directly specify the correlation matrix instead of estimating it from `endog`. If provided, `endog` is not used for the factor analysis, it may be used in post-estimation. method : str The method to extract factors, currently must be either 'pa' for principal axis factor analysis or 'ml' for maximum likelihood estimation. smc : True or False Whether or not to apply squared multiple correlations (method='pa') endog_names : str Names of endogenous variables. If specified, it will be used instead of the column names in endog nobs : int The number of observations, not used if endog is present. Needs to be provided for inference if endog is None. missing : 'none', 'drop', or 'raise' Missing value handling for endog, default is row-wise deletion 'drop' If 'none', no nan checking is done. If 'drop', any observations with nans are dropped. If 'raise', an error is raised. Notes ----- **Experimental** Supported rotations: 'varimax', 'quartimax', 'biquartimax', 'equamax', 'oblimin', 'parsimax', 'parsimony', 'biquartimin', 'promax' If method='ml', the factors are rotated to satisfy condition IC3 of Bai and Li (2012). This means that the scores have covariance I, so the model for the covariance matrix is L * L' + diag(U), where L are the loadings and U are the uniquenesses. In addition, L' * diag(U)^{-1} L must be diagonal. References ---------- .. [*] Hofacker, C. (2004). Exploratory Factor Analysis, Mathematical Marketing. http://www.openaccesstexts.org/pdf/Quant_Chapter_11_efa.pdf .. [*] J Bai, K Li (2012). Statistical analysis of factor models of high dimension. Annals of Statistics. https://arxiv.org/pdf/1205.6617.pdf Nr paTdropc t|||||ct|d||j}|jd} |jd}t j|dx}|_nH|5t j|x}|_|jjd} d|_nd} t| t||||| ||_ d|_ d|_ ||_||_||_||_||_| |_|.t%|dr|j}t%|dr|j}||_dS)N)exogmissingr r)rowvarrindexcolumns)rsuper__init__rr#rcorrcoefrasarrayrr%rloadings communalitymethodsmcrr$hasattrr.r/ endog_names) selfrrrr6r7r9rr,r$r __class__s rr1zFactor.__init__isX eXtT222   GG  Uw  ? ? ?JEk!nG;q>D!{5;;; ;D499  !z$// /D49ioa(GDJJ:CS// !eXtT7;;;         tW%% )"j tY'' +"l &rc|j|jS|j |jjSd|jjddz }|dkrdz |dz}|dkfdt |jjdDS)zNames of endogenous variablesNrr c@g|]}dtzdz|zS)zvar%0d)str).0ir?s r z&Factor.endog_names..s><<<!3q66)C/14<<BFGSc|j}|dkr|||S|dkr|||||Sd|jz}t |)a* Estimate factor model parameters. Parameters ---------- maxiter : int Maximum number of iterations for iterative estimation algorithms tol : float Stopping criteria (error tolerance) for iterative estimation algorithms start : array_like Starting values, currently only used for ML estimation opt_method : str Optimization method for ML estimation opt : dict-like Keyword arguments passed to optimizer, only used for ML estimation em_iter : int The number of EM iterations before starting gradient optimization, only used for ML estimation. Returns ------- FactorResults Results class instance. r()maxitertolmlz'Unknown factor extraction approach '%s')r6lower_fit_pa_fit_mlr) r:rQrRstart opt_methodoptem_iterr6rs rfitz Factor.fitso6""$$ T>><<S<99 9 t^^<<w C@@ @;dkICS// !rc |j}t||_|j|jkrt d|j|jfz|dkrt d|z|dks|dkrt d|z|jr(ddtjt|z z }n!tj t|}d}t|D]c}tt|D]}|||||f<t|d \}} tj|} tj|} | ddd } || }|dk} | dd| f} tj|}tj| |jg} tjtj|d| }| ddd| f} | |}tj|d d }t+| |z |krne||_||_d|z |_||_t5|S) a Extract factors using the iterative principal axis method Parameters ---------- maxiter : int Maximum number of iterations for communality estimation tol : float If `norm(communality - last_communality) < tolerance`, estimation stops Returns ------- results : FactorResults instance z?n_factor must be smaller or equal to the rank of endog! %d > %drz(n_max_iter must be larger than 0! %d < 0g{Gz?zEtolerance must be larger than 0 and smaller than 0.01! Got %f insteadr NU)UPLOaxis)rcopyrn_comprrr7rr"ronesrJrGrarrayargsortsumminsqrtdotpowerr eigenvalsr5 uniquenessr4 FactorResults)r:rQrRRcrmrBjLVc_lastindn_posrHsLAs rrUzFactor._fit_pas" INN  "!nn =4; & &2"mT[9:;; ; a<<G%'(( ( !88sTzz58;=>> > 8 AA''AAAA w  A3q66]]  A$!Q$$$$DAqXa[[F*Q--Cddd)C#AUKKMME!!!S& A It}-..A2A2((B!!!RaR%Ab AA"""**AFQJ#%%&#a% T"""rc|d|jdztj||jdd|jfjfS)Nrr`r_)r$rreshapeT)r:pars r_unpackzFactor._unpacksEAdlN#Q& 3t|}}-DL/ABBDF Frcftjtj||jjfSN)r concatenaterjr|flat)r:loaduniqs r_packz Factor._pack s#~rwt}}dfk:;;;rct|tjur||\}}n|d|d}}||dddfz }tj|j|}|jdd|jddzxxdz cc<tj |\}}tj tj ||z}tj d|z } tj|j|j |dddfz } tj || } tj|| } | tj| z} || z d|jzz S)a Evaluate the log-likelihood function. Parameters ---------- par : ndarray or tuple of 2 ndarray's The model parameters, either a packed representation of the model parameters or a 2-tuple containing a `k_endog x n_factor` matrix of factor loadings and a `k_endog` vector of uniquenesses. Returns ------- float The value of the log-likelihood evaluated at par. rr Nr`)typerndarrayr~rkr|rr#linalgslogdetrhlogrsolvetracer$) r:r}rrloadulul_ldvwbs rloglikezFactor.loglike#s\$ 99 " "c**JD$$QQ$DtAAAtG}$fTVU## !!39Q<>!"""a'""" !!#&&2 F26$<< 2 % F1t8   F4649tAAAtG}4 5 5 IOOC # # F5!   RXa[[Qx1T\>**rct|tjur||\}}n|d|d}}||dddfz }tj|j|}|jdd|jddzxxdz cc<tj ||j}tj||}|j |z |dddfz }tj||} tj|j|} tj|| } dtj |zd|z ||jz d|dzz z z} d|tj||z z} tj||}tj||j}| dtj |ztj |dtj |zz tj |zzz} | d| zz} | dtj|| zz } | d| zz } | dtj|| zz} tj| | jjf d|jzz S)a Evaluate the score function (first derivative of loglike). Parameters ---------- par : ndarray or tuple of 2 ndarray's The model parameters, either a packed representation of the model parameters or a 2-tuple containing a `k_endog x n_factor` matrix of factor loadings and a `k_endog` vector of uniquenesses. Returns ------- ndarray The score function evaluated at par. rr Nr`)rrrr~rkr|rr#rrrrjrhr"rr$)r:r}rrrrqr?ludcurrluzdudlhfs rscorez Factor.scoreOsF$ 99 " "c**JD$$QQ$DtAAAtG}$ F465 ! ! ~~A~!# IOOAtv & &fUAi$$qqq$w- / F2t  fSUD!!fRoorwt}}_$!df*)9)9!)<)BBrc|/|\}}||}nt|dkr^t|d|djdkrd}t ||d|d}nt dfd}fd} |t }t ||| || } | jstj d | j } | \}}| d krtj d  ||}|_d|z _|_| _t%S) z7estimate Factor model using Maximum Likelihood Nr`r rz,Starting values have incompatible dimensionszInvalid starting valuesc0| Sr)rr}r:s rnloglikez Factor._fit_ml..nloglikesLL%%% %rc0| Sr)rrs rnscorezFactor._fit_ml..nscoresJJsOO# #r)jacr6optionszFitting did not convergerz!Some uniquenesses are nearly zero) _fit_ml_emrrJr#r _opt_defaultsrsuccessrrxr~ri_rotaternr5r4 mle_retvalsro) r:rWrZrXrYrrrrrrr}s ` rrVzFactor._fit_mls =11JD$JJtT**EE ZZ1__58}}aq 111D oo%JJuQxq22EE677 7 & & & & & $ $ $ $ $ ;C Xu&  " " "y 6 M4 5 5 5c\\#&& d 88::   M= > > >||D$''t8 T"""rc|tjd}d||j|jfz}dtj|jz}t|D]}||dddfz }tj|j |}|j dd|j ddzxxdz cc<tj ||j }tj|j |} tj||} |tj|| z } | |dddfz} tj| |} tj| |j} tj|j |j| z }tj|| }|tj|j | z}|j dd|j ddzxxdz cc<tj|j| }tj ||j j }tj|j||j zdz }||fS)z1estimate Factor model using EM algorithm Nic g?)sizeg?rr )rrandom RandomStatestandard_normalr$rrerGrkr|rr#rrrr"rh)r:iter random_staterrkrrrqrrqger?ars rrzFactor._fit_ml_ems  90066L\11 dm7T1UUURWT\***t < >QWQZ\> " " "a ' " " " 57++Auw%%Aq$ArvdA&A aaag Aq! Aq$)$$Auw **Q.Aq! A "" "A F>>QWQZ\> " " "a ' " " "ty!$$A9??1ac**,D749%%(8(8(;(;;DDTzrc<tj|d\}}}||z}|jd}n|j}tj|j||dddfz |z }tj|\}}tj||}|S)z rotate loadings for MLE rNr )rrsvdrrkr|eig)r:rrsrrcmrs rrzFactor._rotatesY]]4++ a   9 DD9D VDFD44=0 1 1D 8y}}R  1vdA r)Nr Nr(TNNr))rLrMNrNNrO)rLrMr)__name__ __module__ __qualname____doc__r1propertyr9setterr[rUr~rrrrVrr __classcell__)r;s@rr'r'3s=44jBF@F$'$'$'$'$'$'L<<X< %%%@F""""""""HE#E#E#E#VFFF <<<*+*+*+X3C3C3Cl+#+#+#Z$$$$Lrr'ceZdZdZdZdZdZdZddZdd Z d Z ddZ ddZ ddZ edZeddZedZdS)roa Factor results class For result summary, scree/loading plots and factor rotations Parameters ---------- factor : Factor Fitted Factor class Attributes ---------- uniqueness : ndarray The uniqueness (variance of uncorrelated errors unique to each variable) communality : ndarray 1 - uniqueness loadings : ndarray Each column is the loading vector for one factor loadings_no_rot : ndarray Unrotated loadings, not available under maximum likelihood analysis. eigenvals : ndarray The eigenvalues for a factor analysis obtained using principal components; not available under ML estimation. n_comp : int Number of components (factors) nbs : int Number of observations fa_method : str The method used to obtain the decomposition, either 'pa' for 'principal axes' or 'ml' for maximum likelihood. df : int Degrees of freedom of the factor model. Notes ----- Under ML estimation, the default rotation (used for `loadings`) is condition IC3 of Bai and Li (2012). Under this rotation, the factor scores are iid and standardized. If `G` is the canonical loadings and `U` is the vector of uniquenesses, then the covariance matrix implied by the factor analysis is `GG' + diag(U)`. Status: experimental, Some refactoring will be necessary when new features are added. c||_|j|_|j|_t |dr |j|_|j|_|j|_d|_|j |_ |jj d|_ |j |_ ||_t |dr |j|_|jj \}}||z dz||zz dz|_|j|_t#j|j |_dS)Nrmr rr`)modelr9r4loadings_no_rotr8rmr5rnrotation_methodr6 fa_methodr#rdr_factorrdfreyerotation_matrix)r:factorprs rr1zFactorResults.__init__s !-% 6; ' ' .#-DN!- +#o+A. K  6= ) ) 2%1D #)1EA:Q'A- !vdk22rcN|Sr)summary__str__)r:s rrzFactorResults.__str__3s||~~%%'''rc>||_|dvrtd|z|dvrt|j|\|_}nV|dkrt|jd\|_}n2|dkrt |j\|_}ntd||_dS) a Apply rotation, inplace modification of this Results instance Parameters ---------- method : str Rotation to be applied. Allowed methods are varimax, quartimax, biquartimax, equamax, oblimin, parsimax, parsimony, biquartimin, promax. Returns ------- None : nothing returned, modifications are inplace Notes ----- Warning: 'varimax', 'quartimax' and 'oblimin' are verified against R or Stata. Some rotation methods such as promax do not produce the same results as the R or Stata default functions. See Also -------- factor_rotation : subpackage that implements rotation methods ) varimax quartimax biquartimaxequamaxobliminparsimax parsimony biquartiminrzUnknown rotation method %s)rrrrrrrr quartiminrzrotation method not recognizedN)rrr rr4rr)r:r6r|s rrotatezFactorResults.rotate6s4 & 3 3 39VDEE E > > >-d.BFKK DM11 y -d.B.9 ; ; DM11 x  %d&:;; DM11=>> > rcH|j}|j|}|S)a8correlation of factors implied by rotation If the rotation is oblique, then the factors are correlated. currently not cached Returns ------- corr_f : ndarray correlation matrix of rotated factors, assuming initial factors are orthogonal )rr|rk)r:r|corr_fs r _corr_factorszFactorResults._corr_factorscs"   rbartlettc X|j}|jj}d|jz }|dkrftj|j||dddfz |j|z j}n|drq|j j }| }||jtj|j}n|dkrX|j j }| }|tj |j}n|dkr|j j }| }tjtj||j||dddfz z}||j|z j}ntd|S)a8 Compute factor scoring coefficient matrix The coefficient matrix is not cached. Parameters ---------- method : 'bartlett' or 'regression' Method to use for factor scoring. 'regression' can be abbreviated to `reg` Returns ------- coeff_matrix : ndarray matrix s to compute factors f from a standardized endog ys. ``f = ys dot s`` Notes ----- The `regression` method follows the Stata definition. Method bartlett and regression are verified against Stats. Two unofficial methods, 'ols' and 'gls', produce similar factor scores but are not verified. See Also -------- statsmodels.multivariate.factor.FactorResults.factor_scoring r rNregolsglsz3method not available, use "bartlett or "regression")r4rr|r5rrrrk startswithrrrpinvr)r:r6rsr|unis_matrrs rfactor_score_paramsz!FactorResults.factor_score_paramsts: M  "$"" Z  IMM!#''!S4[/":":;;??c JJLEE   u % % 0:?D''))FJJqswwry}}T':':;;<<>EE u__:?D''))FJJry~~a00113EE u__:?D''))FIMM")--"7"7!#''!S4[/:R:R"RSSEIIacCi((*EE/00 0 rNTc|dur|tj|}n|jjd|jjd}|jjdd}| |jj}n$tj|}nt d||z |z }||}||}|S)a factor scoring: compute factors for endog If endog was not provided when creating the factor class, then a standarized endog needs to be provided here. Parameters ---------- method : 'bartlett' or 'regression' Method to use for factor scoring. 'regression' can be abbreviated to `reg` transform : bool If transform is true and endog is provided, then it will be standardized using mean and scale of original data, which has to be available in this case. If transform is False, then a provided endog will be used unchanged. The original endog in the Factor class will always be standardized if endog is None, independently of `transform`. Returns ------- factor_score : ndarray estimated factors using scoring matrix s and standarized endog ys ``f = ys dot s`` Notes ----- Status: transform option is experimental and might change. See Also -------- statsmodels.multivariate.factor.FactorResults.factor_score_params FNrr )ddofrbzPIf transform is True, then `endog` needs to be available in the Factor instance.)r6) rr3rrmeanstdrrrk)r:rr6 transformmrrfactorss rfactor_scoringzFactorResults.factor_scoringsF   %"3Ju%%EEz+J$))!,,J$((aa(88= J,EEJu--EE "KLLLQY!OE(((77))E""rc*tj}|dtj|jdt |jjdD|j}t|drOtj|j g|jdg}| ddi| |tj|j g|jdg}| ddi| ddi| || ddi| dd i| || ddi|j|tj|jd t |jjdD|j}| dd |jzi| ||S) SummaryzFactor analysis resultscg|]}d|zSz factor %drArBs rrCz)FactorResults.summary..s3DDD!A&DDDrr r/r.rm Eigenvalues CommunalityzPre-rotated loadingsNcg|]}d|zSrrrs rrCz)FactorResults.summary..3AAA%*AAArz%s rotated loadings)r r add_titlepd DataFramerrGr#r9r8rmadd_dictadd_dfr5rr4)r:summrrmr5r4s rrzFactorResults.summarys6!! 0111,  DD#D$8$>q$ABBDDD"     4 % % #  $*:2$HHHI MM2}- . . . KK " " "lD$4#5+/+;B4III  r2h r=)*** K    r2h r12333 O$$$ r2h   +| AA"' (;A(>"?"?AAA& H MM248LMN O O O KK ! ! ! rdisplay333333?yellowchtj|jdt|jjdD|j}|dvrd}t ||dkr|S|dur|} t| } tj | j  d} | | d<tj |j tj | | f| d <| dd gdd gd | dd gd }|d krd} dkrKfd} |j| } n*#t$$r|j| } YnwxYw|dur%fd}| |j} | |d } |!| |j} | d|z| |S| S|dkrV|}|||}|t0}dkrd|| k<|SdS)a| get loadings matrix as DataFrame or pandas Styler Parameters ---------- style : 'display' (default), 'raw' or 'strings' Style to use for display * 'raw' returns just a DataFrame of the loadings matrix, no options are applied * 'display' add sorting and styling as defined by other keywords * 'strings' returns a DataFrame with string elements with optional sorting and suppressing small loading coefficients. sort_ : bool If True, then the rows of the DataFrame is sorted by contribution of each factor. applies if style is either 'display' or 'strings' threshold : float If the threshold is larger than zero, then loading coefficients are either colored white (if style is 'display') or replace by empty string (if style is 'strings'). highlight_max : bool This add a background color to the largest coefficient in each row. color_max : html color default is 'yellow'. color for background of row maximum decimals : None or int If None, then pandas default precision applies. Otherwise values are rounded to the specified decimals. If style is 'display', then the underlying dataframe is not changed. If style is 'strings', then values are rounded before conversion to strings. Returns ------- loadings : DataFrame or pandas Styler instance The return is a pandas Styler instance, if style is 'display' and at least one of highlight_max, threshold or decimals is applied. Otherwise, the returned loadings is a DataFrame. Examples -------- >>> mod = Factor(df, 3, smc=True) >>> res = mod.fit() >>> res.get_loadings_frame(style='display', decimals=3, threshold=0.2) To get a sorted DataFrame, all styling options need to be turned off: >>> df_sorted = res.get_loadings_frame(style='display', ... highlight_max=False, decimals=None, threshold=0) Options except for highlighting are available for plain test or Latex usage: >>> lds = res_u.get_loadings_frame(style='strings', decimals=3, ... threshold=0.3) >>> print(lds.to_latex()) cg|]}d|zSrrrs rrCz4FactorResults.get_loadings_frame..Hrrr r)rawrstringsz2style has to be one of 'raw', 'display', 'strings'r ThighlargestF)by ascendinginplacerarNrcFtj|krdnd}d|zS)z Takes a scalar and returns a string with the css property `'color: white'` for small values, black otherwise. takes threshold from outer scope whiteblackz color: %s)rr!)valcolor thresholds rcolor_white_smallz;FactorResults.get_loadings_frame..color_white_smallas+(*vc{{Y'>'>GGGE&..rcxtj|}||k}fd|DS)zS highlight the maximum in a Series yellow. c"g|] }|rdznd S)zbackground-color: rr)rAr color_maxs rrCzKFactorResults.get_loadings_frame..highlight_max..ws)YYYQqH0);;bYYYr)rr!r )ris_maxrs r highlight_maxz7FactorResults.get_loadings_frame..highlight_maxqs>q A!%%''\FYYYYRXYYYYrz{:.%sf}r r)rrr4rGr#r9rrcrJrr!valuesargmaxarange sort_valuesr)stylemapAttributeErrorapplymapapplyformatroundastyper@)r:r#sort_rrrdecimals loadings_dfr loadings_df2n_frstyrrs ` ` rget_loadings_framez FactorResults.get_loadings_frame svl AA"' (;A(>"?"?AAA&   5 5 5FCS// ! E>>  D==&++--Ll##C6,-..55a88D#'L &(f[-? #PT@T-U&V&VL #  $ $(;e}^b $ c c c&++VY,?a+HHK I  C1}}/////H%+//0ABBCC%HHH%+445FGGCCCH$$ZZZZZ;%+Cii Ai66#;%+C 9x/000{"" I  B#XXh''3B1}}46;??$$y01I  s?E$FFcXtddlm}||j|j|S)a Plot of the ordered eigenvalues and variance explained for the loadings Parameters ---------- ncomp : int, optional Number of loadings to include in the plot. If None, will included the same as the number of maximum possible loadings Returns ------- Figure Handle to the figure. r ) plot_scree)r plotsr3rmrd)r:ncompr3s rr3zFactorResults.plot_screes7  %%%%%%z$.$+u===rFctddlm}|jd}|r|jn|j}|rd}n d|jz}|j|jz dz}|||||j|S) aZ Plot factor loadings in 2-d plots Parameters ---------- loading_pairs : None or a list of tuples Specify plots. Each tuple (i, j) represent one figure, i and j is the loading number for x-axis and y-axis, respectively. If `None`, all combinations of the loadings will be plotted. plot_prerotated : True or False If True, the loadings before rotation applied will be plotted. If False, rotated loadings will be plotted. Returns ------- figs : a list of figure handles r ) plot_loadingsNTzPrerotated Factor Patternz%s Rotated Factor Patternd) loading_pairstitle row_namespercent_variance) r r4r7rrr4rmrdr9)r:r9plot_prerotatedr7r4r: var_explaineds rr7zFactorResults.plot_loadingss$  ((((((   '"O+:M4''   I/EE/43GHE4s: }X]#(D4D.;=== =rctj|j|jj}|jdd|jddzxx|jz cc<|S)z7 Returns the fitted covariance matrix. Nrr )rrkr4r|rr#rn)r:rqs r fitted_covzFactorResults.fitted_covsP F4=$-/ 2 2 ~~A~$/1rrc|jdkrd}t||jd}t||jdzd|zz}t j||jz S)a The standard errors of the uniquenesses. Parameters ---------- kurt : float Excess kurtosis Notes ----- If excess kurtosis is known, provide as `kurt`. Standard errors are only available if the model was fit using maximum likelihood. If `endog` is not provided, `nobs` must be provided to obtain standard errors. These are asymptotic standard errors. See Bai and Li (2012) for conditions under which the standard errors are valid. The standard errors are only applicable to the original, unrotated maximum likelihood solution. rS2Standard errors only available under ML estimationN+nobs is required to obtain standard errors.r`)rrTrrrnrrj)r:kurtrrs r uniq_stderrzFactorResults.uniq_stderrsr0 >   ! !T ) )FCS// ! 9 ?CS// ! OQ !d( +wq49}%%%rc>|jdkrd}t||jd}t|t j|jt j|jj d}t j ||jz S)a The standard errors of the loadings. Standard errors are only available if the model was fit using maximum likelihood. If `endog` is not provided, `nobs` must be provided to obtain standard errors. These are asymptotic standard errors. See Bai and Li (2012) for conditions under which the standard errors are valid. The standard errors are only applicable to the original, unrotated maximum likelihood solution. rSrBNrCr ) rrTrrrouterrnrer4r#rj)r:rrs r load_stderrzFactorResults.load_stderrs >   ! !T ) )FCS// ! 9 ?CS// ! HT_bgdm.A!.D&E&E F Fwq49}%%%r)r)NrT)rTrTr Nr)NF)r)rrrrr1rrrrrrr1r3r7rr@rErHrrrroros<..^3332(((+!+!+!Z"8888t7777r"""HIL9A$(EEEEN>>>>& = = = =D^ & & &^ &D&&^&&&rro)rnumpyr numpy.linalgrrrrpandasrscipy.optimizerstatsmodels.tools.decoratorsrstatsmodels.base.modelr statsmodels.iolibr statsmodels.graphics.utilsr factor_rotationr rrrr%r'rorrrrRsW555555555555######777777((((((&&&&&&22222233333333 @@@" > > >tttttUtttn c&c&c&c&c&c&c&c&c&c&r