Incremental PCA (Python recipe) by Micha Kalfon

ActiveState Code (http://code.activestate.com/recipes/577213/)

#!/usr/bin/env python

"""
Incremental PCA calculation module.

Based on P.Hall, D. Marshall and R. Martin "Incremental Eigenalysis for 
Classification" which appeared in British Machine Vision Conference, volume 1,
pages 286-295, September 1998.

Principal components are updated sequentially as new observations are 
introduced. Each new observation (x) is projected on the eigenspace spanned by
the current principal components (U) and the residual vector (r = x - U(U.T*x))
is used as a new principal component (U' = [U r]). The new principal components
are then rotated by a rotation matrix (R) whose columns are the eigenvectors
of the transformed covariance matrix (D=U'.T*C*U) to yield p + 1 principal 
components. From those, only the first p are selected.

"""

__author__ = "Micha Kalfon"

import numpy as np

_ZERO_THRESHOLD = 1e-9      # Everything below this is zero


class IPCA(object):
    """Incremental PCA calculation object.
    
    General Parameters:
        m - Number of variables per observation
        n - Number of observations
        p - Dimension to which the data should be reduced
    """

    def __init__(self, m, p):
        """Creates an incremental PCA object for m-dimensional observations
        in order to reduce them to a p-dimensional subspace.

        @param m: Number of variables per observation.
        @param p: Number of principle components.

        @return: An IPCA object.
        """
        self._m = float(m)
        self._n = 0.0
        self._p = float(p)
        self._mean = np.matrix(np.zeros((m , 1), dtype=np.float64))
        self._covariance = np.matrix(np.zeros((m, m), dtype=np.float64))
        self._eigenvectors = np.matrix(np.zeros((m, p), dtype=np.float64))
        self._eigenvalues = np.matrix(np.zeros((1, p), dtype=np.float64))

    def update(self, x):
        """Updates with a new observation vector x.

        @param x: Next observation as a column vector (m x 1).
        """
        m = self._m
        n = self._n
        p = self._p
        mean = self._mean
        C = self._covariance
        U = self._eigenvectors
        E = self._eigenvalues

        if type(x) is not np.matrix or x.shape != (m, 1):
            raise TypeError('Input is not a matrix (%d, 1)' % int(m))
      
        # Update covariance matrix and mean vector and centralize input around
        # new mean
        oldmean = mean
        mean = (n*mean + x) / (n + 1.0)
        C = (n*C + x*x.T + n*oldmean*oldmean.T - (n+1)*mean*mean.T) / (n + 1.0)
        x -= mean
        
        # Project new input on current p-dimensional subspace and calculate
        # the normalized residual vector
        g = U.T*x
        r = x - (U*g)
        r = (r / np.linalg.norm(r)) if not _is_zero(r) else np.zeros_like(r)
     
        # Extend the transformation matrix with the residual vector and find
        # the rotation matrix by solving the eigenproblem DR=RE
        U = np.concatenate((U, r), 1)
        D = U.T*C*U
        (E, R) = np.linalg.eigh(D)

        # Sort eigenvalues and eigenvectors from largest to smallest to get the
        # rotation matrix R
        sorter = list(reversed(E.argsort(0)))
        E = E[sorter]
        R = R[:,sorter]

        # Apply the rotation matrix
        U = U*R       

        # Select only p largest eigenvectors and values and update state
        self._n += 1.0
        self._mean = mean
        self._covariance = C
        self._eigenvectors = U[:, 0:p]
        self._eigenvalues = E[0:p]
    
    @property
    def components(self):
        """Returns a matrix with the current principal components as columns.
        """
        return self._eigenvectors

    @property
    def variances(self):
        """Returns a list with the appropriate variance along each principal
        component.
        """
        return self._eigenvalues

def _is_zero(x):
    """Return a boolean indicating whether the given vector is a zero vector up
    to a threshold.
    """
    return np.fabs(x).min() < _ZERO_THRESHOLD


if __name__ == '__main__':
    import sys

    def pca_svd(X):
        X = X - X.mean(0).repeat(X.shape[0], 0)
        [_, _, V] = np.linalg.svd(X)
        return V

    N = 1000
    obs = np.matrix([np.random.normal(size=10) for _ in xrange(N)])
       
    V = pca_svd(obs)
    print V[0:2]

    pca = IPCA(obs.shape[1], 2)
    for i in xrange(obs.shape[0]):
        x = obs[i,:].transpose()
        pca.update(x)
    U = pca.components
    print U