ó ‡ˆ\c@sdZddlmZmZddlZddlZddlmZddl m Z ddl m Z ddl mZdd lmZd d lmZdd lmZdd lmZddlmZmZmZddlmZddlmZd„Zd„Zdefd„ƒYZdS(s Principal Component Analysis iÿÿÿÿ(tlogtsqrtN(tlinalg(tgammaln(tissparse(tsvdsi(tsixi(t_BasePCA(tcheck_random_state(t check_array(t fast_logdettrandomized_svdtsvd_flip(t stable_cumsum(tcheck_is_fittedc Cs|t|ƒkr!tdƒ‚n| tdƒ}xEt|ƒD]7}|t||dƒttjƒ||d7}q?Wtjtj|| ƒƒ}| |d}||krÀd}d}n;tj||ƒ||}tj|ƒ |||d}||||dd} tdtjƒ| |dd} d} |jƒ} || ||+xvt|ƒD]h}x_t|dt|ƒƒD]D} | t|||| d| | d| |ƒt|ƒ7} q„WqdW|||| | d|t|ƒd}|S(sZCompute the likelihood of a rank ``rank`` dataset The dataset is assumed to be embedded in gaussian noise of shape(n, dimf) having spectrum ``spectrum``. Parameters ---------- spectrum : array of shape (n) Data spectrum. rank : int Tested rank value. n_samples : int Number of samples. n_features : int Number of features. Returns ------- ll : float, The log-likelihood Notes ----- This implements the method of `Thomas P. Minka: Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604` s5The tested rank cannot exceed the rank of the datasetg@iigð?g( tlent ValueErrorRtrangeRtnptpitsumtcopy(tspectrumtrankt n_samplest n_featurestputitpltpvtvtmtpptpat spectrum_tjtll((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyt_assess_dimension_ s0!   #   1,cCsUt|ƒ}tj|ƒ}x-t|ƒD]}t||||ƒ||`. Parameters ---------- n_components : int, float, None or string Number of components to keep. if n_components is not set all components are kept:: n_components == min(n_samples, n_features) If ``n_components == 'mle'`` and ``svd_solver == 'full'``, Minka's MLE is used to guess the dimension. Use of ``n_components == 'mle'`` will interpret ``svd_solver == 'auto'`` as ``svd_solver == 'full'``. If ``0 < n_components < 1`` and ``svd_solver == 'full'``, select the number of components such that the amount of variance that needs to be explained is greater than the percentage specified by n_components. If ``svd_solver == 'arpack'``, the number of components must be strictly less than the minimum of n_features and n_samples. Hence, the None case results in:: n_components == min(n_samples, n_features) - 1 copy : bool (default True) If False, data passed to fit are overwritten and running fit(X).transform(X) will not yield the expected results, use fit_transform(X) instead. whiten : bool, optional (default False) When True (False by default) the `components_` vectors are multiplied by the square root of n_samples and then divided by the singular values to ensure uncorrelated outputs with unit component-wise variances. Whitening will remove some information from the transformed signal (the relative variance scales of the components) but can sometime improve the predictive accuracy of the downstream estimators by making their data respect some hard-wired assumptions. svd_solver : string {'auto', 'full', 'arpack', 'randomized'} auto : the solver is selected by a default policy based on `X.shape` and `n_components`: if the input data is larger than 500x500 and the number of components to extract is lower than 80% of the smallest dimension of the data, then the more efficient 'randomized' method is enabled. Otherwise the exact full SVD is computed and optionally truncated afterwards. full : run exact full SVD calling the standard LAPACK solver via `scipy.linalg.svd` and select the components by postprocessing arpack : run SVD truncated to n_components calling ARPACK solver via `scipy.sparse.linalg.svds`. It requires strictly 0 < n_components < min(X.shape) randomized : run randomized SVD by the method of Halko et al. .. versionadded:: 0.18.0 tol : float >= 0, optional (default .0) Tolerance for singular values computed by svd_solver == 'arpack'. .. versionadded:: 0.18.0 iterated_power : int >= 0, or 'auto', (default 'auto') Number of iterations for the power method computed by svd_solver == 'randomized'. .. versionadded:: 0.18.0 random_state : int, RandomState instance or None, optional (default None) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. Used when ``svd_solver`` == 'arpack' or 'randomized'. .. versionadded:: 0.18.0 Attributes ---------- components_ : array, shape (n_components, n_features) Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by ``explained_variance_``. explained_variance_ : array, shape (n_components,) The amount of variance explained by each of the selected components. Equal to n_components largest eigenvalues of the covariance matrix of X. .. versionadded:: 0.18 explained_variance_ratio_ : array, shape (n_components,) Percentage of variance explained by each of the selected components. If ``n_components`` is not set then all components are stored and the sum of the ratios is equal to 1.0. singular_values_ : array, shape (n_components,) The singular values corresponding to each of the selected components. The singular values are equal to the 2-norms of the ``n_components`` variables in the lower-dimensional space. mean_ : array, shape (n_features,) Per-feature empirical mean, estimated from the training set. Equal to `X.mean(axis=0)`. n_components_ : int The estimated number of components. When n_components is set to 'mle' or a number between 0 and 1 (with svd_solver == 'full') this number is estimated from input data. Otherwise it equals the parameter n_components, or the lesser value of n_features and n_samples if n_components is None. noise_variance_ : float The estimated noise covariance following the Probabilistic PCA model from Tipping and Bishop 1999. See "Pattern Recognition and Machine Learning" by C. Bishop, 12.2.1 p. 574 or http://www.miketipping.com/papers/met-mppca.pdf. It is required to compute the estimated data covariance and score samples. Equal to the average of (min(n_features, n_samples) - n_components) smallest eigenvalues of the covariance matrix of X. References ---------- For n_components == 'mle', this class uses the method of `Minka, T. P. "Automatic choice of dimensionality for PCA". In NIPS, pp. 598-604` Implements the probabilistic PCA model from: `Tipping, M. E., and Bishop, C. M. (1999). "Probabilistic principal component analysis". Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 611-622. via the score and score_samples methods. See http://www.miketipping.com/papers/met-mppca.pdf For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`. For svd_solver == 'randomized', see: `Halko, N., Martinsson, P. G., and Tropp, J. A. (2011). "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions". SIAM review, 53(2), 217-288.` and also `Martinsson, P. G., Rokhlin, V., and Tygert, M. (2011). "A randomized algorithm for the decomposition of matrices". Applied and Computational Harmonic Analysis, 30(1), 47-68.` Examples -------- >>> import numpy as np >>> from sklearn.decomposition import PCA >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]]) >>> pca = PCA(n_components=2) >>> pca.fit(X) PCA(copy=True, iterated_power='auto', n_components=2, random_state=None, svd_solver='auto', tol=0.0, whiten=False) >>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS [0.9924... 0.0075...] >>> print(pca.singular_values_) # doctest: +ELLIPSIS [6.30061... 0.54980...] >>> pca = PCA(n_components=2, svd_solver='full') >>> pca.fit(X) # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE PCA(copy=True, iterated_power='auto', n_components=2, random_state=None, svd_solver='full', tol=0.0, whiten=False) >>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS [0.9924... 0.00755...] >>> print(pca.singular_values_) # doctest: +ELLIPSIS [6.30061... 0.54980...] >>> pca = PCA(n_components=1, svd_solver='arpack') >>> pca.fit(X) PCA(copy=True, iterated_power='auto', n_components=1, random_state=None, svd_solver='arpack', tol=0.0, whiten=False) >>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS [0.99244...] >>> print(pca.singular_values_) # doctest: +ELLIPSIS [6.30061...] See also -------- KernelPCA SparsePCA TruncatedSVD IncrementalPCA tautogcCsC||_||_||_||_||_||_||_dS(N(t n_componentsRtwhitent svd_solverttoltiterated_powert random_state(tselfR,RR-R.R/R0R1((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyt__init__9s      cCs|j|ƒ|S(sjFit the model with X. Parameters ---------- X : array-like, shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features. y : Ignored Returns ------- self : object Returns the instance itself. (t_fit(R2tXty((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pytfitDs cCss|j|ƒ\}}}|dd…d|j…f}|jr^|t|jddƒ9}n|||j 9}|S(s”Fit the model with X and apply the dimensionality reduction on X. Parameters ---------- X : array-like, shape (n_samples, n_features) Training data, where n_samples is the number of samples and n_features is the number of features. y : Ignored Returns ------- X_new : array-like, shape (n_samples, n_components) Nii(R4t n_components_R-Rtshape(R2R5R6tUtStV((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyt fit_transformWs  cCs|t|ƒrtdƒ‚nt|dtjtjgdtd|jƒ}|jdkrŽ|j dkrxt |j ƒ}q—t |j ƒd}n |j}|j |_ |j dkrt|j ƒdksÓ|d krßd |_ q|dkr|d t |j ƒkrd |_ qd |_ n|j d kr;|j||ƒS|j dkr`|j|||j ƒStd j|j ƒƒ‚dS(s?Dispatch to the right submethod depending on the chosen solver.sOPCA does not support sparse input. See TruncatedSVD for a possible alternative.tdtypet ensure_2dRtarpackiR+iôtmletfullgš™™™™™é?t randomizedsUnrecognized svd_solver='{0}'N(R@RC(Rt TypeErrorR Rtfloat64tfloat32tTrueRR,tNoneR.tminR9t_fit_svd_solvertmaxt _fit_fullt_fit_truncatedRtformat(R2R5R,((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyR4ss, !   ! %   cCsN|j\}}|dkr9||krÉtdƒ‚qÉnd|koYt||ƒkns€td|t||ƒfƒ‚nI|dkrÉt|tjtjfƒsÉtd|t|ƒfƒ‚qÉntj |ddƒ|_ ||j 8}t j |dt ƒ\}}}t||ƒ\}}|}|d |d} | jƒ} | | } |jƒ} |dkr|t| ||ƒ}nAd|ko“d knr½t| ƒ} tj| |ƒd}n|t||ƒkrè| |j ƒ|_n d |_|||_|_|| |_||_| | |_| | |_| | |_|||fS( s(Fit the model by computing full SVD on XRAs?n_components='mle' is only supported if n_samples >= n_featuresisZn_components=%r must be between 0 and min(n_samples, n_features)=%r with svd_solver='full'isSn_components=%r must be of type int when greater than or equal to 1, was of type=%rtaxist full_matricesigð?g(R9RRIt isinstancetnumberstIntegralRtintegerttypetmeantmean_RtsvdtFalseR RRR)R t searchsortedtnoise_variance_t n_samples_t n_features_t components_R8texplained_variance_texplained_variance_ratio_tsingular_values_(R2R5R,RRR:R;R<R^R_t total_varR`Rat ratio_cumsum((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyRLsF  %             c Cs|j\}}t|tjƒr:td||fƒ‚nÊd|koZt||ƒkns„td|t||ƒ|fƒ‚n€t|tjtj fƒs¾td|t |ƒfƒ‚nF|dkr|t||ƒkrtd|t||ƒ|fƒ‚nt |j ƒ}tj |ddƒ|_||j8}|dkrà|jd dd t|jƒƒ}t|d |d |jd |ƒ\}} } | ddd …} t|dd…ddd …f| ddd …ƒ\}} n?|dkrt|d|d|jdtd|ƒ\}} } n|||_|_| |_||_| d|d|_tj|ddddƒ} |j| jƒ|_| jƒ|_|jt||ƒkrì| jƒ|jjƒ|_ |j t||ƒ|:_ n d|_ || | fS(sXFit the model by computing truncated SVD (by ARPACK or randomized) on X s7n_components=%r cannot be a string with svd_solver='%s'isXn_components=%r must be between 1 and min(n_samples, n_features)=%r with svd_solver='%s'sSn_components=%r must be of type int when greater than or equal to 1, was of type=%rR@s]n_components=%r must be strictly less than min(n_samples, n_features)=%r with svd_solver='%s'ROiiÿÿÿÿtsizetkR/tv0NRCR,tn_itert flip_signR1itddofg(!R9RQRt string_typesRRIRRRSRRTRURR1RVRWtuniformRR/R R R0RGR\R]R^R8R_tvarRR`RRaR[( R2R5R,R.RRR1RfR:R;R<Rb((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyRMÜsT%   !*>      cCs’t|dƒt|ƒ}||j}|jd}|jƒ}d|tj||ƒjddƒ}|d|tdtj ƒt |ƒ8}|S(sÁReturn the log-likelihood of each sample. See. "Pattern Recognition and Machine Learning" by C. Bishop, 12.2.1 p. 574 or http://www.miketipping.com/papers/met-mppca.pdf Parameters ---------- X : array, shape(n_samples, n_features) The data. Returns ------- ll : array, shape (n_samples,) Log-likelihood of each sample under the current model RWigà¿ROgà?g@( RR RWR9t get_precisionRtdotRRRR (R2R5tXrRt precisiontlog_like((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyt score_samples#s     &cCstj|j|ƒƒS(sÒReturn the average log-likelihood of all samples. See. "Pattern Recognition and Machine Learning" by C. Bishop, 12.2.1 p. 574 or http://www.miketipping.com/papers/met-mppca.pdf Parameters ---------- X : array, shape(n_samples, n_features) The data. y : Ignored Returns ------- ll : float Average log-likelihood of the samples under the current model (RRVRr(R2R5R6((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pytscore?sN(t__name__t __module__t__doc__RHRGRYR3R7R=R4RLRMRrRs(((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyR*jsÍ     * ? G ( RvtmathRRRRtnumpyRtscipyRt scipy.specialRt scipy.sparseRtscipy.sparse.linalgRt externalsRtbaseRtutilsRR t utils.extmathR R R R tutils.validationRR%R)R*(((s8lib/python2.7/site-packages/sklearn/decomposition/pca.pyts"    >