#!/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