Source code for statsmodels.multivariate.pca

"""Principal Component Analysis

Author: josef-pktd
Modified by Kevin Sheppard
"""
from __future__ import print_function, division

import numpy as np
import pandas as pd

from statsmodels.compat.python import range
from statsmodels.tools.sm_exceptions import (ValueWarning,
                                             EstimationWarning)

def _norm(x):
    return np.sqrt(np.sum(x * x))


[docs]class PCA(object): """ Principal Component Analysis Parameters ---------- data : array-like Variables in columns, observations in rows ncomp : int, optional Number of components to return. If None, returns the as many as the smaller of the number of rows or columns in data standardize: bool, optional Flag indicating to use standardized data with mean 0 and unit variance. standardized being True implies demean. Using standardized data is equivalent to computing principal components from the correlation matrix of data demean : bool, optional Flag indicating whether to demean data before computing principal components. demean is ignored if standardize is True. Demeaning data but not standardizing is equivalent to computing principal components from the covariance matrix of data normalize : bool , optional Indicates whether th normalize the factors to have unit inner product. If False, the loadings will have unit inner product. weights : array, optional Series weights to use after transforming data according to standardize or demean when computing the principal components. gls : bool, optional Flag indicating to implement a two-step GLS estimator where in the first step principal components are used to estimate residuals, and then the inverse residual variance is used as a set of weights to estimate the final principal components. Setting gls to True requires ncomp to be less then the min of the number of rows or columns method : str, optional Sets the linear algebra routine used to compute eigenvectors 'svd' uses a singular value decomposition (default). 'eig' uses an eigenvalue decomposition of a quadratic form 'nipals' uses the NIPALS algorithm and can be faster than SVD when ncomp is small and nvars is large. See notes about additional changes when using NIPALS tol : float, optional Tolerance to use when checking for convergence when using NIPALS max_iter : int, optional Maximum iterations when using NIPALS missing : string Method for missing data. Choices are 'drop-row' - drop rows with missing values 'drop-col' - drop columns with missing values 'drop-min' - drop either rows or columns, choosing by data retention 'fill-em' - use EM algorithm to fill missing value. ncomp should be set to the number of factors required tol_em : float Tolerance to use when checking for convergence of the EM algorithm max_em_iter : int Maximum iterations for the EM algorithm Attributes ---------- factors : array or DataFrame nobs by ncomp array of of principal components (scores) scores : array or DataFrame nobs by ncomp array of of principal components - identical to factors loadings : array or DataFrame ncomp by nvar array of principal component loadings for constructing the factors coeff : array or DataFrame nvar by ncomp array of principal component loadings for constructing the projections projection : array or DataFrame nobs by var array containing the projection of the data onto the ncomp estimated factors rsquare : array or Series ncomp array where the element in the ith position is the R-square of including the fist i principal components. Note: values are calculated on the transformed data, not the original data ic : array or DataFrame ncomp by 3 array containing the Bai and Ng (2003) Information criteria. Each column is a different criteria, and each row represents the number of included factors. eigenvals : array or Series nvar array of eigenvalues eigenvecs : array or DataFrame nvar by nvar array of eigenvectors weights : array nvar array of weights used to compute the principal components, normalized to unit length transformed_data : array Standardized, demeaned and weighted data used to compute principal components and related quantities cols : array Array of indices indicating columns used in the PCA rows : array Array of indices indicating rows used in the PCA Examples -------- Basic PCA using the correlation matrix of the data >>> import numpy as np >>> from statsmodels.multivariate.pca import PCA >>> x = np.random.randn(100)[:, None] >>> x = x + np.random.randn(100, 100) >>> pc = PCA(x) Note that the principal components are computed using a SVD and so the correlation matrix is never constructed, unless method='eig'. PCA using the covariance matrix of the data >>> pc = PCA(x, standardize=False) Limiting the number of factors returned to 1 computed using NIPALS >>> pc = PCA(x, ncomp=1, method='nipals') >>> pc.factors.shape (100, 1) Notes ----- The default options perform principal component analysis on the demeanded, unit variance version of data. Setting standardize to False will instead onle demean, and setting both standardized and demean to False will not alter the data. Once the data have been transformed, the following relationships hold when the number of components (ncomp) is the same as tne minimum of the number of observation or the number of variables. .. math: X' X = V \\Lambda V' .. math: F = X V .. math: X = F V' where X is the `data`, F is the array of principal components (`factors` or `scores`), and V is the array of eigenvectors (`loadings`) and V' is the array of factor coefficients (`coeff`). When weights are provided, the principal components are computed from the modified data .. math: \\Omega^{-\\frac{1}{2}} X where :math:`\\Omega` is a diagonal matrix composed of the weights. For example, when using the GLS version of PCA, the elements of :math:`\\Omega` will be the inverse of the variances of the residuals from .. math: X - F V' where the number of factors is less than the rank of X .. [*] J. Bai and S. Ng, "Determining the number of factors in approximate factor models," Econometrica, vol. 70, number 1, pp. 191-221, 2002 """ def __init__(self, data, ncomp=None, standardize=True, demean=True, normalize=True, gls=False, weights=None, method='svd', missing=None, tol=5e-8, max_iter=1000, tol_em=5e-8, max_em_iter=100, ): self._index = None self._columns = [] if isinstance(data, pd.DataFrame): self._index = data.index self._columns = data.columns self.data = np.asarray(data) # Store inputs self._gls = gls self._normalize = normalize self._tol = tol if not 0 < self._tol < 1: raise ValueError('tol must be strictly between 0 and 1') self._max_iter = max_iter self._max_em_iter = max_em_iter self._tol_em = tol_em # Prepare data self._standardize = standardize self._demean = demean self._nobs, self._nvar = self.data.shape if weights is None: weights = np.ones(self._nvar) else: weights = np.array(weights).flatten() if weights.shape[0] != self._nvar: raise ValueError('weights should have nvar elements') weights = weights / np.sqrt((weights ** 2.0).mean()) self.weights = weights # Check ncomp against maximum min_dim = min(self._nobs, self._nvar) self._ncomp = min_dim if ncomp is None else ncomp if self._ncomp > min_dim: import warnings warn = 'The requested number of components is more than can be ' \ 'computed from data. The maximum number of components is ' \ 'the minimum of the number of observations or variables' warnings.warn(warn, ValueWarning) self._ncomp = min_dim self._method = method # Workaround to avoid instance methods in __dict__ if self._method not in ('eig', 'svd', 'nipals'): raise ValueError('method {0} is not known.'.format(method)) self.rows = np.arange(self._nobs) self.cols = np.arange(self._nvar) # Handle missing self._missing = missing self._adjusted_data = self.data if missing is not None: self._adjust_missing() # Update size self._nobs, self._nvar = self._adjusted_data.shape if self._ncomp == np.min(self.data.shape): self._ncomp = np.min(self._adjusted_data.shape) elif self._ncomp > np.min(self._adjusted_data.shape): raise ValueError('When adjusting for missing values, user ' 'provided ncomp must be no larger than the ' 'smallest dimension of the ' 'missing-value-adjusted data size.') # Attributes and internal values self._tss = 0.0 self._ess = None self.transformed_data = None self._mu = None self._sigma = None self._ess_indiv = None self._tss_indiv = None self.scores = self.factors = None self.loadings = None self.coeff = None self.eigenvals = None self.eigenvecs = None self.projection = None self.rsquare = None self.ic = None # Prepare data self.transformed_data = self._prepare_data() # Perform the PCA self._pca() if gls: self._compute_gls_weights() self.transformed_data = self._prepare_data() self._pca() # Final calculations self._compute_rsquare_and_ic() if self._index is not None: self._to_pandas() def _adjust_missing(self): """ Implements alternatives for handling missing values """ def keep_col(x): index = np.logical_not(np.any(np.isnan(x), 0)) return x[:, index], index def keep_row(x): index = np.logical_not(np.any(np.isnan(x), 1)) return x[index, :], index if self._missing == 'drop-col': self._adjusted_data, index = keep_col(self.data) self.cols = np.where(index)[0] self.weights = self.weights[index] elif self._missing == 'drop-row': self._adjusted_data, index = keep_row(self.data) self.rows = np.where(index)[0] elif self._missing == 'drop-min': drop_col, drop_col_index = keep_col(self.data) drop_col_size = drop_col.size drop_row, drop_row_index = keep_row(self.data) drop_row_size = drop_row.size if drop_row_size > drop_col_size: self._adjusted_data = drop_row self.rows = np.where(drop_row_index)[0] else: self._adjusted_data = drop_col self.weights = self.weights[drop_col_index] self.cols = np.where(drop_col_index)[0] elif self._missing == 'fill-em': self._adjusted_data = self._fill_missing_em() else: raise ValueError('missing method is not known.') if self._index is not None: self._columns = self._columns[self.cols] self._index = self._index[self.rows] # Check adjusted data size if self._adjusted_data.size == 0: raise ValueError('Removal of missing values has eliminated all data.') def _compute_gls_weights(self): """ Computes GLS weights based on percentage of data fit """ errors = self.transformed_data - np.asarray(self.projection) if self._ncomp == self._nvar: raise ValueError('gls can only be used when ncomp < nvar ' 'so that residuals have non-zero variance') var = (errors ** 2.0).mean(0) weights = 1.0 / var weights = weights / np.sqrt((weights ** 2.0).mean()) nvar = self._nvar eff_series_perc = (1.0 / sum((weights / weights.sum()) ** 2.0)) / nvar if eff_series_perc < 0.1: eff_series = int(np.round(eff_series_perc * nvar)) import warnings warn = 'Many series are being down weighted by GLS. Of the ' \ '{original} series, the GLS estimates are based on only ' \ '{effective} (effective) ' \ 'series.'.format(original=nvar, effective=eff_series) warnings.warn(warn, EstimationWarning) self.weights = weights def _pca(self): """ Main PCA routine """ self._compute_eig() self._compute_pca_from_eig() self.projection = self.project() def __repr__(self): string = self.__str__() string = string[:-1] string += ', id: ' + hex(id(self)) + ')' return string def __str__(self): string = 'Principal Component Analysis(' string += 'nobs: ' + str(self._nobs) + ', ' string += 'nvar: ' + str(self._nvar) + ', ' if self._standardize: kind = 'Standardize (Correlation)' elif self._demean: kind = 'Demean (Covariance)' else: kind = 'None' string += 'transformation: ' + kind + ', ' if self._gls: string += 'GLS, ' string += 'normalization: ' + str(self._normalize) + ', ' string += 'number of components: ' + str(self._ncomp) + ', ' string += 'method: ' + 'Eigenvalue' if self._method == 'eig' else 'SVD' string += ')' return string def _prepare_data(self): """ Standardize or demean data. """ adj_data = self._adjusted_data if np.all(np.isnan(adj_data)): return np.empty(adj_data.shape[1]).fill(np.nan) self._mu = np.nanmean(adj_data, axis=0) self._sigma = np.sqrt(np.nanmean((adj_data - self._mu) ** 2.0, axis=0)) if self._standardize: data = (adj_data - self._mu) / self._sigma elif self._demean: data = (adj_data - self._mu) else: data = adj_data return data / np.sqrt(self.weights) def _compute_eig(self): """ Wrapper for actual eigenvalue method This is a workaround to avoid instance methods in __dict__ """ if self._method == 'eig': return self._compute_using_eig() elif self._method == 'svd': return self._compute_using_svd() else: # self._method == 'nipals' return self._compute_using_nipals() def _compute_using_svd(self): """SVD method to compute eigenvalues and eigenvecs""" x = self.transformed_data u, s, v = np.linalg.svd(x) self.eigenvals = s ** 2.0 self.eigenvecs = v.T def _compute_using_eig(self): """ Eigenvalue decomposition method to compute eigenvalues and eigenvectors """ x = self.transformed_data self.eigenvals, self.eigenvecs = np.linalg.eigh(x.T.dot(x)) def _compute_using_nipals(self): """ NIPALS implementation to compute small number of eigenvalues and eigenvectors """ x = self.transformed_data if self._ncomp > 1: x = x + 0.0 # Copy tol, max_iter, ncomp = self._tol, self._max_iter, self._ncomp vals = np.zeros(self._ncomp) vecs = np.zeros((self._nvar, self._ncomp)) for i in range(ncomp): max_var_ind = np.argmax(x.var(0)) factor = x[:, [max_var_ind]] _iter = 0 diff = 1.0 while diff > tol and _iter < max_iter: vec = x.T.dot(factor) / (factor.T.dot(factor)) vec = vec / np.sqrt(vec.T.dot(vec)) factor_last = factor factor = x.dot(vec) / (vec.T.dot(vec)) diff = _norm(factor - factor_last) / _norm(factor) _iter += 1 vals[i] = (factor ** 2).sum() vecs[:, [i]] = vec if ncomp > 1: x -= factor.dot(vec.T) self.eigenvals = vals self.eigenvecs = vecs def _fill_missing_em(self): """ EM algorithm to fill missing values """ non_missing = np.logical_not(np.isnan(self.data)) # If nothing missing, return without altering the data if np.all(non_missing): return self.data # 1. Standardized data as needed data = self.transformed_data = np.asarray(self._prepare_data()) ncomp = self._ncomp # 2. Check for all nans col_non_missing = np.sum(non_missing, 1) row_non_missing = np.sum(non_missing, 0) if np.any(col_non_missing < ncomp) or np.any(row_non_missing < ncomp): raise ValueError('Implementation requires that all columns and ' 'all rows have at least ncomp non-missing values') # 3. Get mask mask = np.isnan(data) # 4. Compute mean mu = np.nanmean(data, 0) # 5. Replace missing with mean projection = np.ones((self._nobs, 1)) * mu projection_masked = projection[mask] data[mask] = projection_masked # 6. Compute eigenvalues and fit diff = 1.0 _iter = 0 while diff > self._tol_em and _iter < self._max_em_iter: last_projection_masked = projection_masked # Set transformed data to compute eigenvalues self.transformed_data = data # Call correct eig function here self._compute_eig() # Call function to compute factors and projection self._compute_pca_from_eig() projection = np.asarray(self.project(transform=False, unweight=False)) projection_masked = projection[mask] data[mask] = projection_masked delta = last_projection_masked - projection_masked diff = _norm(delta) / _norm(projection_masked) _iter += 1 # Must copy to avoid overwriting original data since replacing values data = self._adjusted_data + 0.0 projection = np.asarray(self.project()) data[mask] = projection[mask] return data def _compute_pca_from_eig(self): """ Compute relevant statistics after eigenvalues have been computed """ # Ensure sorted largest to smallest vals, vecs = self.eigenvals, self.eigenvecs indices = np.argsort(vals) indices = indices[::-1] vals = vals[indices] vecs = vecs[:, indices] if (vals <= 0).any(): # Discard and warn num_good = vals.shape[0] - (vals <= 0).sum() if num_good < self._ncomp: import warnings warn = 'Only {num:d} eigenvalues are positive. The is the ' \ 'maximum number of components that can be extracted.' warnings.warn(warn.format(num=num_good), EstimationWarning) self._ncomp = num_good vals[num_good:] = np.finfo(np.float64).tiny # Use ncomp for the remaining calculations vals = vals[:self._ncomp] vecs = vecs[:, :self._ncomp] self.eigenvals, self.eigenvecs = vals, vecs # Select correct number of components to return self.scores = self.factors = self.transformed_data.dot(vecs) self.loadings = vecs self.coeff = vecs.T if self._normalize: self.coeff = (self.coeff.T * np.sqrt(vals)).T self.factors /= np.sqrt(vals) self.scores = self.factors def _compute_rsquare_and_ic(self): """ Final statistics to compute """ # TSS and related calculations # TODO: This needs careful testing, with and without weights, gls, standardized and demean weights = self.weights ss_data = self.transformed_data * np.sqrt(weights) self._tss_indiv = np.sum(ss_data ** 2, 0) self._tss = np.sum(self._tss_indiv) self._ess = np.zeros(self._ncomp + 1) self._ess_indiv = np.zeros((self._ncomp + 1, self._nvar)) for i in range(self._ncomp + 1): # Projection in the same space as transformed_data projection = self.project(ncomp=i, transform=False, unweight=False) indiv_rss = (projection ** 2).sum(axis=0) rss = indiv_rss.sum() self._ess[i] = self._tss - rss self._ess_indiv[i, :] = self._tss_indiv - indiv_rss self.rsquare = 1.0 - self._ess / self._tss # Information Criteria ess = self._ess invalid = ess <= 0 # Prevent log issues of 0 if invalid.any(): last_obs = (np.where(invalid)[0]).min() ess = ess[:last_obs] log_ess = np.log(ess) r = np.arange(ess.shape[0]) nobs, nvar = self._nobs, self._nvar sum_to_prod = (nobs + nvar) / (nobs * nvar) min_dim = min(nobs, nvar) penalties = np.array([sum_to_prod * np.log(1.0 / sum_to_prod), sum_to_prod * np.log(min_dim), np.log(min_dim) / min_dim]) penalties = penalties[:, None] ic = log_ess + r * penalties self.ic = ic.T
[docs] def project(self, ncomp=None, transform=True, unweight=True): """ Project series onto a specific number of factors Parameters ---------- ncomp : int, optional Number of components to use. If omitted, all components initially computed are used. Returns ------- projection : array nobs by nvar array of the projection onto ncomp factors transform : bool Flag indicating whether to return the projection in the original space of the data (True, default) or in the space of the standardized/demeaned data unweight : bool Flag indicating whether to undo the effects of the estimation weights Notes ----- """ # Projection needs to be scaled/shifted based on inputs ncomp = self._ncomp if ncomp is None else ncomp if ncomp > self._ncomp: raise ValueError('ncomp must be smaller than the number of ' 'components computed.') factors = np.asarray(self.factors) coeff = np.asarray(self.coeff) projection = factors[:, :ncomp].dot(coeff[:ncomp, :]) if transform or unweight: projection *= np.sqrt(self.weights) if transform: # Remove the weights, which do not depend on transformation if self._standardize: projection *= self._sigma if self._standardize or self._demean: projection += self._mu if self._index is not None: projection = pd.DataFrame(projection, columns=self._columns, index=self._index) return projection
def _to_pandas(self): """ Returns pandas DataFrames for all values """ index = self._index # Principal Components num_zeros = np.ceil(np.log10(self._ncomp)) comp_str = 'comp_{0:0' + str(int(num_zeros)) + 'd}' cols = [comp_str.format(i) for i in range(self._ncomp)] df = pd.DataFrame(self.factors, columns=cols, index=index) self.scores = self.factors = df # Projections df = pd.DataFrame(self.projection, columns=self._columns, index=index) self.projection = df # Weights df = pd.DataFrame(self.coeff, index=cols, columns=self._columns) self.coeff = df # Loadings df = pd.DataFrame(self.loadings, index=self._columns, columns=cols) self.loadings = df # eigenvals self.eigenvals = pd.Series(self.eigenvals) self.eigenvals.name = 'eigenvals' # eigenvecs vec_str = comp_str.replace('comp', 'eigenvec') cols = [vec_str.format(i) for i in range(self.eigenvecs.shape[1])] self.eigenvecs = pd.DataFrame(self.eigenvecs, columns=cols) # R2 self.rsquare = pd.Series(self.rsquare) self.rsquare.index.name = 'ncomp' self.rsquare.name = 'rsquare' # IC self.ic = pd.DataFrame(self.ic, columns=['IC_p1', 'IC_p2', 'IC_p3']) self.ic.index.name = 'ncomp'
[docs] def plot_scree(self, ncomp=None, log_scale=True, cumulative=False, ax=None): """ Plot of the ordered eigenvalues Parameters ---------- ncomp : int, optional Number of components ot include in the plot. If None, will included the same as the number of components computed log_scale : boot, optional Flag indicating whether ot use a log scale for the y-axis cumulative : bool, optional Flag indicating whether to plot the eigenvalues or cumulative eigenvalues ax : Matplotlib axes instance, optional An axes on which to draw the graph. If omitted, new a figure is created Returns ------- fig : figure Handle to the figure """ import statsmodels.graphics.utils as gutils fig, ax = gutils.create_mpl_ax(ax) ncomp = self._ncomp if ncomp is None else ncomp vals = np.asarray(self.eigenvals) vals = vals[:self._ncomp] if cumulative: vals = np.cumsum(vals) if log_scale: ax.set_yscale('log') ax.plot(np.arange(ncomp), vals[: ncomp], 'bo') ax.autoscale(tight=True) xlim = np.array(ax.get_xlim()) sp = xlim[1] - xlim[0] xlim += 0.02 * np.array([-sp, sp]) ax.set_xlim(xlim) ylim = np.array(ax.get_ylim()) scale = 0.02 if log_scale: sp = np.log(ylim[1] / ylim[0]) ylim = np.exp(np.array([np.log(ylim[0]) - scale * sp, np.log(ylim[1]) + scale * sp])) else: sp = ylim[1] - ylim[0] ylim += scale * np.array([-sp, sp]) ax.set_ylim(ylim) ax.set_title('Scree Plot') ax.set_ylabel('Eigenvalue') ax.set_xlabel('Component Number') fig.tight_layout() return fig
[docs] def plot_rsquare(self, ncomp=None, ax=None): """ Box plots of the individual series R-square against the number of PCs Parameters ---------- ncomp : int, optional Number of components ot include in the plot. If None, will plot the minimum of 10 or the number of computed components ax : Matplotlib axes instance, optional An axes on which to draw the graph. If omitted, new a figure is created Returns ------- fig : figure Handle to the figure """ import statsmodels.graphics.utils as gutils fig, ax = gutils.create_mpl_ax(ax) ncomp = 10 if ncomp is None else ncomp ncomp = min(ncomp, self._ncomp) # R2s in rows, series in columns r2s = 1.0 - self._ess_indiv / self._tss_indiv r2s = r2s[1:] r2s = r2s[:ncomp] ax.boxplot(r2s.T) ax.set_title('Individual Input $R^2$') ax.set_ylabel('$R^2$') ax.set_xlabel('Number of Included Principal Components') return fig
[docs]def pca(data, ncomp=None, standardize=True, demean=True, normalize=True, gls=False, weights=None, method='svd'): """ Principal Component Analysis Parameters ---------- data : array Variables in columns, observations in rows. ncomp : int, optional Number of components to return. If None, returns the as many as the smaller to the number of rows or columns of data. standardize: bool, optional Flag indicating to use standardized data with mean 0 and unit variance. standardized being True implies demean. demean : bool, optional Flag indicating whether to demean data before computing principal components. demean is ignored if standardize is True. normalize : bool , optional Indicates whether th normalize the factors to have unit inner product. If False, the loadings will have unit inner product. weights : array, optional Series weights to use after transforming data according to standardize or demean when computing the principal components. gls : bool, optional Flag indicating to implement a two-step GLS estimator where in the first step principal components are used to estimate residuals, and then the inverse residual variance is used as a set of weights to estimate the final principal components method : str, optional Determines the linear algebra routine uses. 'eig', the default, uses an eigenvalue decomposition. 'svd' uses a singular value decomposition. Returns ------- factors : array or DataFrame nobs by ncomp array of of principal components (also known as scores) loadings : array or DataFrame ncomp by nvar array of principal component loadings for constructing the factors projection : array or DataFrame nobs by var array containing the projection of the data onto the ncomp estimated factors rsquare : array or Series ncomp array where the element in the ith position is the R-square of including the fist i principal components. The values are calculated on the transformed data, not the original data. ic : array or DataFrame ncomp by 3 array containing the Bai and Ng (2003) Information criteria. Each column is a different criteria, and each row represents the number of included factors. eigenvals : array or Series nvar array of eigenvalues eigenvecs : array or DataFrame nvar by nvar array of eigenvectors Notes ----- This is a simple function wrapper around the PCA class. See PCA for more information and additional methods. """ pc = PCA(data, ncomp=ncomp, standardize=standardize, demean=demean, normalize=normalize, gls=gls, weights=weights, method=method) return (pc.factors, pc.loadings, pc.projection, pc.rsquare, pc.ic, pc.eigenvals, pc.eigenvecs)