U 3d @sBdZddlZddlZddlZddlZddlmZmZddlZ ddl m Z ddl m Z mZmZddlmZdd lmZmZmZdd lmZmZdd lmZmZdd lmZdd lmZddl m!Z!m"Z"ddZ#ddZ$ddZ%ddddddde &e j'j(dd ddZ)GdddeZ*Gddde*Z+d$d d!Z,Gd"d#d#e*Z-dS)%zUGraphicalLasso: sparse inverse covariance estimation with an l1-penalized estimator. N)IntegralReal)linalg)empirical_covarianceEmpiricalCovariancelog_likelihood)ConvergenceWarning)_is_arraylike_not_scalarcheck_random_state check_scalar)delayedParallel)Interval StrOptions)_cd_fast)lars_path_gram)check_cvcross_val_scorecCsZ|jd}dt|||tdtj}||t|tt|7}|S)zEvaluation of the graphical-lasso objective function the objective function is made of a shifted scaled version of the normalized log-likelihood (i.e. its empirical mean over the samples) and a penalisation term to promote sparsity rr )shapernplogpiabssumdiag)Zmle precision_alphapcostr"C/tmp/pip-unpacked-wheel-zrfo1fqw/sklearn/covariance/_graph_lasso.py _objective$s "*r$cCsJt||}||jd8}||t|tt|7}|S)zExpression of the dual gap convergence criterion The specific definition is given in Duchi "Projected Subgradient Methods for Learning Sparse Gaussians". r)rrrrr)emp_covrrZgapr"r"r# _dual_gap1s*r&cCs4t|}d|jdd|jdd<tt|S)aFind the maximum alpha for which there are some non-zeros off-diagonal. Parameters ---------- emp_cov : ndarray of shape (n_features, n_features) The sample covariance matrix. Notes ----- This results from the bound for the all the Lasso that are solved in GraphicalLasso: each time, the row of cov corresponds to Xy. As the bound for alpha is given by `max(abs(Xy))`, the result follows. rNr)rcopyflatrmaxr)r%Ar"r"r# alpha_max=s r+cd-C6?dF) cov_initmodetolenet_tolmax_iterverbose return_costseps return_n_iterc CsD|j\} } |dkr|r|t|} dt|| }|| tdtj7}t|| | }| rl|| ||fdfS|| ||ffSn"| r|t|dfS|t|fS|dkr|}n|}|d9}|j dd| d}||j dd| d<t |} t | }t }|dkrt dd d }n t dd }ztj}tj|ddddfd d }t|D]R}t| D]}|dkr|d}||||k||<|dd|f||k|dd|f<n|ddddf|dd<||||kf}tjf||dkrH| ||k|f| ||fd|  }t||d|||||tdd \}} } } n(t|||j|| dd| ddd\} } }W5QRXd|||ft|||k|f|| ||f<| ||f || ||k|f<| ||f || |||kf<t||}|||||kf<||||k|f<qbt| s.tdt|| |}t|| |}|r^td|||f|rr|||ft||krqt|sT|dkrTtdqTtd||ft Wn:tk r}z|j!ddf|_!|W5d}~XYnX|r$| r|| ||dfS|| |fSn| r8|| |dfS|| fSdS)ac L1-penalized covariance estimator. Read more in the :ref:`User Guide `. .. versionchanged:: v0.20 graph_lasso has been renamed to graphical_lasso Parameters ---------- emp_cov : ndarray of shape (n_features, n_features) Empirical covariance from which to compute the covariance estimate. alpha : float The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf]. cov_init : array of shape (n_features, n_features), default=None The initial guess for the covariance. If None, then the empirical covariance is used. mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable. tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf]. enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf]. max_iter : int, default=100 The maximum number of iterations. verbose : bool, default=False If verbose is True, the objective function and dual gap are printed at each iteration. return_costs : bool, default=False If return_costs is True, the objective function and dual gap at each iteration are returned. eps : float, default=eps The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Default is `np.finfo(np.float64).eps`. return_n_iter : bool, default=False Whether or not to return the number of iterations. Returns ------- covariance : ndarray of shape (n_features, n_features) The estimated covariance matrix. precision : ndarray of shape (n_features, n_features) The estimated (sparse) precision matrix. costs : list of (objective, dual_gap) pairs The list of values of the objective function and the dual gap at each iteration. Returned only if return_costs is True. n_iter : int Number of iterations. Returned only if `return_n_iter` is set to True. See Also -------- GraphicalLasso : Sparse inverse covariance estimation with an l1-penalized estimator. GraphicalLassoCV : Sparse inverse covariance with cross-validated choice of the l1 penalty. Notes ----- The algorithm employed to solve this problem is the GLasso algorithm, from the Friedman 2008 Biostatistics paper. It is the same algorithm as in the R `glasso` package. One possible difference with the `glasso` R package is that the diagonal coefficients are not penalized. rrr Ngffffff?rr,raiseignore)Zoverinvalid)r:C)orderiFTlars)ZXyZGramZ n_samplesZ alpha_minZ copy_Gramr6methodZ return_pathg?z1The system is too ill-conditioned for this solverz<[graphical_lasso] Iteration % 3i, cost % 3.2e, dual gap %.3ezANon SPD result: the system is too ill-conditioned for this solverzDgraphical_lasso: did not converge after %i iteration: dual gap: %.3ez3. The system is too ill-conditioned for this solver)"rrinvrrrrrr'r(ZpinvhZarangelistdictinfrangeZerrstatecd_fastZenet_coordinate_descent_gramr rsizedotisfiniteFloatingPointErrorr&r$printappendrwarningswarnr args)r%rr/r0r1r2r3r4r5r6r7_Z n_featuresrr!Zd_gap covariance_ZdiagonalindicesZcostserrorsZsub_covarianceiidxZdirowZcoefser"r"r#graphical_lassoQsd        &          rVc s~eZdZUejeeddddgeeddddgeeddddgeddhgdgd Ze e d <e d dfdd Z Z S)BaseGraphicalLassorNrightclosedleftr,r=r4)r1r2r3r0r4_parameter_constraintsZstore_precisionr-r.Fcs0tj|d||_||_||_||_||_dS)Nassume_centered)super__init__r1r2r3r0r4)selfr1r2r3r0r4r^ __class__r"r#r`Is zBaseGraphicalLasso.__init__)r-r-r.r,FF)__name__ __module__ __qualname__rr\rrrrrA__annotations__popr` __classcell__r"r"rbr#rW>s   rWcsbeZdZUdZejdeeddddgiZee d<dd d d d d d d fdd Z dddZ Z S)GraphicalLassoa Sparse inverse covariance estimation with an l1-penalized estimator. Read more in the :ref:`User Guide `. .. versionchanged:: v0.20 GraphLasso has been renamed to GraphicalLasso Parameters ---------- alpha : float, default=0.01 The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf]. mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable. tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf]. enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf]. max_iter : int, default=100 The maximum number of iterations. verbose : bool, default=False If verbose is True, the objective function and dual gap are plotted at each iteration. assume_centered : bool, default=False If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation. Attributes ---------- location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean. covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. n_iter_ : int Number of iterations run. n_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 See Also -------- graphical_lasso : L1-penalized covariance estimator. GraphicalLassoCV : Sparse inverse covariance with cross-validated choice of the l1 penalty. Examples -------- >>> import numpy as np >>> from sklearn.covariance import GraphicalLasso >>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0], ... [0.0, 0.4, 0.0, 0.0], ... [0.2, 0.0, 0.3, 0.1], ... [0.0, 0.0, 0.1, 0.7]]) >>> np.random.seed(0) >>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0], ... cov=true_cov, ... size=200) >>> cov = GraphicalLasso().fit(X) >>> np.around(cov.covariance_, decimals=3) array([[0.816, 0.049, 0.218, 0.019], [0.049, 0.364, 0.017, 0.034], [0.218, 0.017, 0.322, 0.093], [0.019, 0.034, 0.093, 0.69 ]]) >>> np.around(cov.location_, decimals=3) array([0.073, 0.04 , 0.038, 0.143]) rrNrXrYr\{Gz?r,r-r.F)r0r1r2r3r4r^cs"tj||||||d||_dSN)r1r2r3r0r4r^)r_r`r)rarr0r1r2r3r4r^rbr"r#r`s zGraphicalLasso.__init__c Cs||j|ddd}|jr2t|jd|_n |d|_t||jd}t ||j |j |j |j |j|jdd\|_|_|_|S)aFit the GraphicalLasso model to X. Parameters ---------- X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate. y : Ignored Not used, present for API consistency by convention. Returns ------- self : object Returns the instance itself. r )ensure_min_featuresZensure_min_samplesrrr]Trr0r1r2r3r4r7)_validate_params_validate_datar^rzerosr location_meanrrVrr0r1r2r3r4rOrn_iter_)raXyr%r"r"r#fits"  zGraphicalLasso.fit)rk)N) rdrerf__doc__rWr\rrrArgr`rwrir"r"rbr#rjZs `rjc  Cs\td|d} t|} |dkr(| } n|} t} t} t}|dk rNt|}|D]}zFt| || ||||| d\} }| | | ||dk rt||}Wn4tk rtj }| tj | tj YnX|dk rt |stj }|||dkrt j dqR|dkrR|dk r2td||fqRtd|qR|dk rT| | |fS| | fS)al1-penalized covariance estimator along a path of decreasing alphas Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples, n_features) Data from which to compute the covariance estimate. alphas : array-like of shape (n_alphas,) The list of regularization parameters, decreasing order. cov_init : array of shape (n_features, n_features), default=None The initial guess for the covariance. X_test : array of shape (n_test_samples, n_features), default=None Optional test matrix to measure generalisation error. mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable. tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. The tolerance must be a positive number. enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. The tolerance must be a positive number. max_iter : int, default=100 The maximum number of iterations. This parameter should be a strictly positive integer. verbose : int or bool, default=False The higher the verbosity flag, the more information is printed during the fitting. Returns ------- covariances_ : list of shape (n_alphas,) of ndarray of shape (n_features, n_features) The estimated covariance matrices. precisions_ : list of shape (n_alphas,) of ndarray of shape (n_features, n_features) The estimated (sparse) precision matrices. scores_ : list of shape (n_alphas,), dtype=float The generalisation error (log-likelihood) on the test data. Returned only if test data is passed. rrN)rr/r0r1r2r3r4.z/[graphical_lasso_path] alpha: %.2e, score: %.2ez"[graphical_lasso_path] alpha: %.2e)r)rr'r@rVrJrrHrrBnanrGsysstderrwriterI)rualphasr/X_testr0r1r2r3r4 inner_verboser%rOZ covariances_Z precisions_Zscores_Z test_emp_covrr this_scorer"r"r#graphical_lasso_paths^C          rc seZdZUdZejeedddddgeeddddgdgedgdZee d <d d dd d d d dddd fdd Z dddZ Z S)GraphicalLassoCVaGSparse inverse covariance w/ cross-validated choice of the l1 penalty. See glossary entry for :term:`cross-validation estimator`. Read more in the :ref:`User Guide `. .. versionchanged:: v0.20 GraphLassoCV has been renamed to GraphicalLassoCV Parameters ---------- alphas : int or array-like of shape (n_alphas,), dtype=float, default=4 If an integer is given, it fixes the number of points on the grids of alpha to be used. If a list is given, it gives the grid to be used. See the notes in the class docstring for more details. Range is [1, inf) for an integer. Range is (0, inf] for an array-like of floats. n_refinements : int, default=4 The number of times the grid is refined. Not used if explicit values of alphas are passed. Range is [1, inf). cv : int, cross-validation generator or iterable, default=None Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 5-fold cross-validation, - integer, to specify the number of folds. - :term:`CV splitter`, - An iterable yielding (train, test) splits as arrays of indices. For integer/None inputs :class:`KFold` is used. Refer :ref:`User Guide ` for the various cross-validation strategies that can be used here. .. versionchanged:: 0.20 ``cv`` default value if None changed from 3-fold to 5-fold. tol : float, default=1e-4 The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf]. enet_tol : float, default=1e-4 The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode='cd'. Range is (0, inf]. max_iter : int, default=100 Maximum number of iterations. mode : {'cd', 'lars'}, default='cd' The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where number of features is greater than number of samples. Elsewhere prefer cd which is more numerically stable. n_jobs : int, default=None Number of jobs to run in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. .. versionchanged:: v0.20 `n_jobs` default changed from 1 to None verbose : bool, default=False If verbose is True, the objective function and duality gap are printed at each iteration. assume_centered : bool, default=False If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation. Attributes ---------- location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean. covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix. precision_ : ndarray of shape (n_features, n_features) Estimated precision matrix (inverse covariance). alpha_ : float Penalization parameter selected. cv_results_ : dict of ndarrays A dict with keys: alphas : ndarray of shape (n_alphas,) All penalization parameters explored. split(k)_test_score : ndarray of shape (n_alphas,) Log-likelihood score on left-out data across (k)th fold. .. versionadded:: 1.0 mean_test_score : ndarray of shape (n_alphas,) Mean of scores over the folds. .. versionadded:: 1.0 std_test_score : ndarray of shape (n_alphas,) Standard deviation of scores over the folds. .. versionadded:: 1.0 n_iter_ : int Number of iterations run for the optimal alpha. n_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 See Also -------- graphical_lasso : L1-penalized covariance estimator. GraphicalLasso : Sparse inverse covariance estimation with an l1-penalized estimator. Notes ----- The search for the optimal penalization parameter (`alpha`) is done on an iteratively refined grid: first the cross-validated scores on a grid are computed, then a new refined grid is centered around the maximum, and so on. One of the challenges which is faced here is that the solvers can fail to converge to a well-conditioned estimate. The corresponding values of `alpha` then come out as missing values, but the optimum may be close to these missing values. In `fit`, once the best parameter `alpha` is found through cross-validation, the model is fit again using the entire training set. Examples -------- >>> import numpy as np >>> from sklearn.covariance import GraphicalLassoCV >>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0], ... [0.0, 0.4, 0.0, 0.0], ... [0.2, 0.0, 0.3, 0.1], ... [0.0, 0.0, 0.1, 0.7]]) >>> np.random.seed(0) >>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0], ... cov=true_cov, ... size=200) >>> cov = GraphicalLassoCV().fit(X) >>> np.around(cov.covariance_, decimals=3) array([[0.816, 0.051, 0.22 , 0.017], [0.051, 0.364, 0.018, 0.036], [0.22 , 0.018, 0.322, 0.094], [0.017, 0.036, 0.094, 0.69 ]]) >>> np.around(cov.location_, decimals=3) array([0.073, 0.04 , 0.038, 0.143]) rNr[rYz array-likeZ cv_object)r~ n_refinementscvn_jobsr\r-r.r,F) r~rrr1r2r3r0rr4r^c s4tj||||| | d||_||_||_||_dSrl)r_r`r~rrr) rar~rrr1r2r3r0rr4r^rbr"r#r`$szGraphicalLassoCV.__init__c sjddjr0tjd_n d_tjd}t j |dd}t }j }t djdt|rj D]}t|dtdtjd d qj d}n:j}t|} d | } tt| t| |d d d t} t|D]} tBtdttjjdfdd||D} W5QRXt | \}}}t |}t |}|!t ||t"|t#$ddd}tj }d}t%|D]Z\}\}}}t|}|dt&tj'j(krtj)}t*|r|}||kr|}|}q|dkr$|dd} |dd} n||kr^|t+|dks^||d} ||dd} nP|t+|dkr||d} d ||d} n ||dd} ||dd} t|stt| t| |ddd jr|dkrt,d| d|t| fqt t |}t |d}t |d-d|-t.t/|jdt0|}dt0i_1t|jdD]$} |d d | fj1d| d<qtj|ddj1d<tj2|ddj1d<|}|_3t4||j5j6j7j8dd\_9_:_;S)aFit the GraphicalLasso covariance model to X. Parameters ---------- X : array-like of shape (n_samples, n_features) Data from which to compute the covariance estimate. y : Ignored Not used, present for API consistency by convention. Returns ------- self : object Returns the instance itself. r )rmrrr]F) classifierrrX)Zmin_valZmax_valZinclude_boundariesrkNr9)rr4c 3sF|]>\}}tt||jjjtdjdVqdS)皙?)r~rr0r1r2r3r4N)rrr0r1r2intr3).0Ztraintestrur~rrar"r# }s  z'GraphicalLassoCV.fit..T)keyreverserz8[GraphicalLassoCV] Done refinement % 2i out of %i: % 3is)rrr4r~splitZ _test_score)ZaxisZmean_test_scoreZstd_test_scorern)sR        n% w