statsmodels.multivariate.pca.PCA¶

class
statsmodels.multivariate.pca.
PCA
(data, ncomp=None, standardize=True, demean=True, normalize=True, gls=False, weights=None, method='svd', missing=None, tol=5e08, max_iter=1000, tol_em=5e08, max_em_iter=100)[source]¶ Principal Component Analysis
 Parameters
 dataarraylike
Variables in columns, observations in rows
 ncompint, 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
 demeanbool, 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
 normalizebool , optional
Indicates whether th normalize the factors to have unit inner product. If False, the loadings will have unit inner product.
 weightsarray, optional
Series weights to use after transforming data according to standardize or demean when computing the principal components.
 glsbool, optional
Flag indicating to implement a twostep 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
 methodstr, 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
 tolfloat, optional
Tolerance to use when checking for convergence when using NIPALS
 max_iterint, optional
Maximum iterations when using NIPALS
 missingstring
Method for missing data. Choices are ‘droprow’  drop rows with missing values ‘dropcol’  drop columns with missing values ‘dropmin’  drop either rows or columns, choosing by data retention ‘fillem’  use EM algorithm to fill missing value. ncomp should be set to the number of factors required
 tol_emfloat
Tolerance to use when checking for convergence of the EM algorithm
 max_em_iterint
Maximum iterations for the EM algorithm
Notes
The default options perform principal component analysis on the demeaned, unit variance version of data. Setting standardize to False will instead only 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.
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
where \(\Omega\) is a diagonal matrix composed of the weights. For example, when using the GLS version of PCA, the elements of \(\Omega\) will be the inverse of the variances of the residuals from
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. 191221, 2002
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)
 Attributes
 factorsarray or DataFrame
nobs by ncomp array of of principal components (scores)
 scoresarray or DataFrame
nobs by ncomp array of of principal components  identical to factors
 loadingsarray or DataFrame
ncomp by nvar array of principal component loadings for constructing the factors
 coeffarray or DataFrame
nvar by ncomp array of principal component loadings for constructing the projections
 projectionarray or DataFrame
nobs by var array containing the projection of the data onto the ncomp estimated factors
 rsquarearray or Series
ncomp array where the element in the ith position is the Rsquare of including the fist i principal components. Note: values are calculated on the transformed data, not the original data
 icarray 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.
 eigenvalsarray or Series
nvar array of eigenvalues
 eigenvecsarray or DataFrame
nvar by nvar array of eigenvectors
 weightsarray
nvar array of weights used to compute the principal components, normalized to unit length
 transformed_dataarray
Standardized, demeaned and weighted data used to compute principal components and related quantities
 colsarray
Array of indices indicating columns used in the PCA
 rowsarray
Array of indices indicating rows used in the PCA
Methods
plot_rsquare
([ncomp, ax])Box plots of the individual series Rsquare against the number of PCs
plot_scree
([ncomp, log_scale, cumulative, ax])Plot of the ordered eigenvalues
project
([ncomp, transform, unweight])Project series onto a specific number of factors