""" Principal Component Analysis. """ # Author: Alexandre Gramfort # Olivier Grisel # Mathieu Blondel # Denis A. Engemann # Michael Eickenberg # Giorgio Patrini # # License: BSD 3 clause from math import log, sqrt from numbers import Integral, Real import numpy as np from scipy import linalg from scipy.special import gammaln from scipy.sparse import issparse from scipy.sparse.linalg import svds from ._base import _BasePCA from ..utils import check_random_state from ..utils._arpack import _init_arpack_v0 from ..utils.deprecation import deprecated from ..utils.extmath import fast_logdet, randomized_svd, svd_flip from ..utils.extmath import stable_cumsum from ..utils.validation import check_is_fitted from ..utils._param_validation import Interval, StrOptions def _assess_dimension(spectrum, rank, n_samples): """Compute the log-likelihood of a rank ``rank`` dataset. The dataset is assumed to be embedded in gaussian noise of shape(n, dimf) having spectrum ``spectrum``. This implements the method of T. P. Minka. Parameters ---------- spectrum : ndarray of shape (n_features,) Data spectrum. rank : int Tested rank value. It should be strictly lower than n_features, otherwise the method isn't specified (division by zero in equation (31) from the paper). n_samples : int Number of samples. Returns ------- ll : float The log-likelihood. References ---------- This implements the method of `Thomas P. Minka: Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604 `_ """ n_features = spectrum.shape[0] if not 1 <= rank < n_features: raise ValueError("the tested rank should be in [1, n_features - 1]") eps = 1e-15 if spectrum[rank - 1] < eps: # When the tested rank is associated with a small eigenvalue, there's # no point in computing the log-likelihood: it's going to be very # small and won't be the max anyway. Also, it can lead to numerical # issues below when computing pa, in particular in log((spectrum[i] - # spectrum[j]) because this will take the log of something very small. return -np.inf pu = -rank * log(2.0) for i in range(1, rank + 1): pu += ( gammaln((n_features - i + 1) / 2.0) - log(np.pi) * (n_features - i + 1) / 2.0 ) pl = np.sum(np.log(spectrum[:rank])) pl = -pl * n_samples / 2.0 v = max(eps, np.sum(spectrum[rank:]) / (n_features - rank)) pv = -np.log(v) * n_samples * (n_features - rank) / 2.0 m = n_features * rank - rank * (rank + 1.0) / 2.0 pp = log(2.0 * np.pi) * (m + rank) / 2.0 pa = 0.0 spectrum_ = spectrum.copy() spectrum_[rank:n_features] = v for i in range(rank): for j in range(i + 1, len(spectrum)): pa += log( (spectrum[i] - spectrum[j]) * (1.0 / spectrum_[j] - 1.0 / spectrum_[i]) ) + log(n_samples) ll = pu + pl + pv + pp - pa / 2.0 - rank * log(n_samples) / 2.0 return ll def _infer_dimension(spectrum, n_samples): """Infers the dimension of a dataset with a given spectrum. The returned value will be in [1, n_features - 1]. """ ll = np.empty_like(spectrum) ll[0] = -np.inf # we don't want to return n_components = 0 for rank in range(1, spectrum.shape[0]): ll[rank] = _assess_dimension(spectrum, rank, n_samples) return ll.argmax() class PCA(_BasePCA): """Principal component analysis (PCA). Linear dimensionality reduction using Singular Value Decomposition of the data to project it to a lower dimensional space. The input data is centered but not scaled for each feature before applying the SVD. It uses the LAPACK implementation of the full SVD or a randomized truncated SVD by the method of Halko et al. 2009, depending on the shape of the input data and the number of components to extract. It can also use the scipy.sparse.linalg ARPACK implementation of the truncated SVD. Notice that this class does not support sparse input. See :class:`TruncatedSVD` for an alternative with sparse data. Read more in the :ref:`User Guide `. Parameters ---------- n_components : int, float or 'mle', default=None 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, 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 : {'auto', 'full', 'arpack', 'randomized'}, default='auto' If 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. If full : run exact full SVD calling the standard LAPACK solver via `scipy.linalg.svd` and select the components by postprocessing If arpack : run SVD truncated to n_components calling ARPACK solver via `scipy.sparse.linalg.svds`. It requires strictly 0 < n_components < min(X.shape) If randomized : run randomized SVD by the method of Halko et al. .. versionadded:: 0.18.0 tol : float, default=0.0 Tolerance for singular values computed by svd_solver == 'arpack'. Must be of range [0.0, infinity). .. versionadded:: 0.18.0 iterated_power : int or 'auto', default='auto' Number of iterations for the power method computed by svd_solver == 'randomized'. Must be of range [0, infinity). .. versionadded:: 0.18.0 n_oversamples : int, default=10 This parameter is only relevant when `svd_solver="randomized"`. It corresponds to the additional number of random vectors to sample the range of `X` so as to ensure proper conditioning. See :func:`~sklearn.utils.extmath.randomized_svd` for more details. .. versionadded:: 1.1 power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto' Power iteration normalizer for randomized SVD solver. Not used by ARPACK. See :func:`~sklearn.utils.extmath.randomized_svd` for more details. .. versionadded:: 1.1 random_state : int, RandomState instance or None, default=None Used when the 'arpack' or 'randomized' solvers are used. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. .. versionadded:: 0.18.0 Attributes ---------- components_ : ndarray of shape (n_components, n_features) Principal axes in feature space, representing the directions of maximum variance in the data. Equivalently, the right singular vectors of the centered input data, parallel to its eigenvectors. The components are sorted by decreasing ``explained_variance_``. explained_variance_ : ndarray of shape (n_components,) The amount of variance explained by each of the selected components. The variance estimation uses `n_samples - 1` degrees of freedom. Equal to n_components largest eigenvalues of the covariance matrix of X. .. versionadded:: 0.18 explained_variance_ratio_ : ndarray of 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_ : ndarray of 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. .. versionadded:: 0.19 mean_ : ndarray of 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. n_features_ : int Number of features in the training data. n_samples_ : int Number of samples in the training data. 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. 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 -------- KernelPCA : Kernel Principal Component Analysis. SparsePCA : Sparse Principal Component Analysis. TruncatedSVD : Dimensionality reduction using truncated SVD. IncrementalPCA : Incremental Principal Component Analysis. References ---------- For n_components == 'mle', this class uses the method from: `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. For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`. For svd_solver == 'randomized', see: :doi:`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. <10.1137/090771806>` and also :doi:`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. <10.1016/j.acha.2010.02.003>` 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(n_components=2) >>> print(pca.explained_variance_ratio_) [0.9924... 0.0075...] >>> print(pca.singular_values_) [6.30061... 0.54980...] >>> pca = PCA(n_components=2, svd_solver='full') >>> pca.fit(X) PCA(n_components=2, svd_solver='full') >>> print(pca.explained_variance_ratio_) [0.9924... 0.00755...] >>> print(pca.singular_values_) [6.30061... 0.54980...] >>> pca = PCA(n_components=1, svd_solver='arpack') >>> pca.fit(X) PCA(n_components=1, svd_solver='arpack') >>> print(pca.explained_variance_ratio_) [0.99244...] >>> print(pca.singular_values_) [6.30061...] """ _parameter_constraints: dict = { "n_components": [ Interval(Integral, 0, None, closed="left"), Interval(Real, 0, 1, closed="neither"), StrOptions({"mle"}), None, ], "copy": ["boolean"], "whiten": ["boolean"], "svd_solver": [StrOptions({"auto", "full", "arpack", "randomized"})], "tol": [Interval(Real, 0, None, closed="left")], "iterated_power": [ StrOptions({"auto"}), Interval(Integral, 0, None, closed="left"), ], "n_oversamples": [Interval(Integral, 1, None, closed="left")], "power_iteration_normalizer": [StrOptions({"auto", "QR", "LU", "none"})], "random_state": ["random_state"], } def __init__( self, n_components=None, *, copy=True, whiten=False, svd_solver="auto", tol=0.0, iterated_power="auto", n_oversamples=10, power_iteration_normalizer="auto", random_state=None, ): self.n_components = n_components self.copy = copy self.whiten = whiten self.svd_solver = svd_solver self.tol = tol self.iterated_power = iterated_power self.n_oversamples = n_oversamples self.power_iteration_normalizer = power_iteration_normalizer self.random_state = random_state # TODO(1.4): remove in 1.4 # mypy error: Decorated property not supported @deprecated( # type: ignore "Attribute `n_features_` was deprecated in version 1.2 and will be " "removed in 1.4. Use `n_features_in_` instead." ) @property def n_features_(self): return self.n_features_in_ def fit(self, X, y=None): """Fit the model with X. Parameters ---------- X : array-like of 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 Ignored. Returns ------- self : object Returns the instance itself. """ self._validate_params() self._fit(X) return self def fit_transform(self, X, y=None): """Fit the model with X and apply the dimensionality reduction on X. Parameters ---------- X : array-like of 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 Ignored. Returns ------- X_new : ndarray of shape (n_samples, n_components) Transformed values. Notes ----- This method returns a Fortran-ordered array. To convert it to a C-ordered array, use 'np.ascontiguousarray'. """ self._validate_params() U, S, Vt = self._fit(X) U = U[:, : self.n_components_] if self.whiten: # X_new = X * V / S * sqrt(n_samples) = U * sqrt(n_samples) U *= sqrt(X.shape[0] - 1) else: # X_new = X * V = U * S * Vt * V = U * S U *= S[: self.n_components_] return U def _fit(self, X): """Dispatch to the right submethod depending on the chosen solver.""" # Raise an error for sparse input. # This is more informative than the generic one raised by check_array. if issparse(X): raise TypeError( "PCA does not support sparse input. See " "TruncatedSVD for a possible alternative." ) X = self._validate_data( X, dtype=[np.float64, np.float32], ensure_2d=True, copy=self.copy ) # Handle n_components==None if self.n_components is None: if self.svd_solver != "arpack": n_components = min(X.shape) else: n_components = min(X.shape) - 1 else: n_components = self.n_components # Handle svd_solver self._fit_svd_solver = self.svd_solver if self._fit_svd_solver == "auto": # Small problem or n_components == 'mle', just call full PCA if max(X.shape) <= 500 or n_components == "mle": self._fit_svd_solver = "full" elif 1 <= n_components < 0.8 * min(X.shape): self._fit_svd_solver = "randomized" # This is also the case of n_components in (0,1) else: self._fit_svd_solver = "full" # Call different fits for either full or truncated SVD if self._fit_svd_solver == "full": return self._fit_full(X, n_components) elif self._fit_svd_solver in ["arpack", "randomized"]: return self._fit_truncated(X, n_components, self._fit_svd_solver) def _fit_full(self, X, n_components): """Fit the model by computing full SVD on X.""" n_samples, n_features = X.shape if n_components == "mle": if n_samples < n_features: raise ValueError( "n_components='mle' is only supported if n_samples >= n_features" ) elif not 0 <= n_components <= min(n_samples, n_features): raise ValueError( "n_components=%r must be between 0 and " "min(n_samples, n_features)=%r with " "svd_solver='full'" % (n_components, min(n_samples, n_features)) ) # Center data self.mean_ = np.mean(X, axis=0) X -= self.mean_ U, S, Vt = linalg.svd(X, full_matrices=False) # flip eigenvectors' sign to enforce deterministic output U, Vt = svd_flip(U, Vt) components_ = Vt # Get variance explained by singular values explained_variance_ = (S**2) / (n_samples - 1) total_var = explained_variance_.sum() explained_variance_ratio_ = explained_variance_ / total_var singular_values_ = S.copy() # Store the singular values. # Postprocess the number of components required if n_components == "mle": n_components = _infer_dimension(explained_variance_, n_samples) elif 0 < n_components < 1.0: # number of components for which the cumulated explained # variance percentage is superior to the desired threshold # side='right' ensures that number of features selected # their variance is always greater than n_components float # passed. More discussion in issue: #15669 ratio_cumsum = stable_cumsum(explained_variance_ratio_) n_components = np.searchsorted(ratio_cumsum, n_components, side="right") + 1 # Compute noise covariance using Probabilistic PCA model # The sigma2 maximum likelihood (cf. eq. 12.46) if n_components < min(n_features, n_samples): self.noise_variance_ = explained_variance_[n_components:].mean() else: self.noise_variance_ = 0.0 self.n_samples_ = n_samples self.components_ = components_[:n_components] self.n_components_ = n_components self.explained_variance_ = explained_variance_[:n_components] self.explained_variance_ratio_ = explained_variance_ratio_[:n_components] self.singular_values_ = singular_values_[:n_components] return U, S, Vt def _fit_truncated(self, X, n_components, svd_solver): """Fit the model by computing truncated SVD (by ARPACK or randomized) on X. """ n_samples, n_features = X.shape if isinstance(n_components, str): raise ValueError( "n_components=%r cannot be a string with svd_solver='%s'" % (n_components, svd_solver) ) elif not 1 <= n_components <= min(n_samples, n_features): raise ValueError( "n_components=%r must be between 1 and " "min(n_samples, n_features)=%r with " "svd_solver='%s'" % (n_components, min(n_samples, n_features), svd_solver) ) elif svd_solver == "arpack" and n_components == min(n_samples, n_features): raise ValueError( "n_components=%r must be strictly less than " "min(n_samples, n_features)=%r with " "svd_solver='%s'" % (n_components, min(n_samples, n_features), svd_solver) ) random_state = check_random_state(self.random_state) # Center data self.mean_ = np.mean(X, axis=0) X -= self.mean_ if svd_solver == "arpack": v0 = _init_arpack_v0(min(X.shape), random_state) U, S, Vt = svds(X, k=n_components, tol=self.tol, v0=v0) # svds doesn't abide by scipy.linalg.svd/randomized_svd # conventions, so reverse its outputs. S = S[::-1] # flip eigenvectors' sign to enforce deterministic output U, Vt = svd_flip(U[:, ::-1], Vt[::-1]) elif svd_solver == "randomized": # sign flipping is done inside U, S, Vt = randomized_svd( X, n_components=n_components, n_oversamples=self.n_oversamples, n_iter=self.iterated_power, power_iteration_normalizer=self.power_iteration_normalizer, flip_sign=True, random_state=random_state, ) self.n_samples_ = n_samples self.components_ = Vt self.n_components_ = n_components # Get variance explained by singular values self.explained_variance_ = (S**2) / (n_samples - 1) # Workaround in-place variance calculation since at the time numpy # did not have a way to calculate variance in-place. N = X.shape[0] - 1 np.square(X, out=X) np.sum(X, axis=0, out=X[0]) total_var = (X[0] / N).sum() self.explained_variance_ratio_ = self.explained_variance_ / total_var self.singular_values_ = S.copy() # Store the singular values. if self.n_components_ < min(n_features, n_samples): self.noise_variance_ = total_var - self.explained_variance_.sum() self.noise_variance_ /= min(n_features, n_samples) - n_components else: self.noise_variance_ = 0.0 return U, S, Vt def score_samples(self, X): """Return 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-like of shape (n_samples, n_features) The data. Returns ------- ll : ndarray of shape (n_samples,) Log-likelihood of each sample under the current model. """ check_is_fitted(self) X = self._validate_data(X, dtype=[np.float64, np.float32], reset=False) Xr = X - self.mean_ n_features = X.shape[1] precision = self.get_precision() log_like = -0.5 * (Xr * (np.dot(Xr, precision))).sum(axis=1) log_like -= 0.5 * (n_features * log(2.0 * np.pi) - fast_logdet(precision)) return log_like def score(self, X, y=None): """Return 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-like of shape (n_samples, n_features) The data. y : Ignored Ignored. Returns ------- ll : float Average log-likelihood of the samples under the current model. """ return np.mean(self.score_samples(X)) def _more_tags(self): return {"preserves_dtype": [np.float64, np.float32]}