巨蟒主成分分析

我想用主成分分析降维。是否 numpy 或 scypy 已经有它，或者我必须卷我自己使用 numpy.linalg.eigh？

I don't just want to use singular value decomposition (SVD) because my input data are quite high-dimensional (~460 dimensions), so I think SVD will be slower than computing the eigenvectors of the covariance matrix.

我希望能够找到一个预制的、调试过的实现，它已经为何时使用哪个方法做出了正确的决定，并且可能还做了其他我不知道的优化。

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最佳答案

你可以看看 MDP。

我还没有机会亲自测试它，但是我已经为 PCA 功能准确地标记了它。

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这里 是另一个使用 numpy、 scypy 和 C 扩展的 python PCA 模块实现。该模块采用奇异值分解(SVD)算法和非线性迭代偏最小二乘(NIPALS)算法进行主成分分析，并在 C 语言中实现。

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Mlab 有一个 PCA 的实施。

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我刚读完这本书。本书中的所有代码示例都是由 Python 编写的(几乎都是用 Numpy 编写的)。10.2主成分分析的代码片段可能值得一读。它使用 numpy.linalg.eig。
顺便说一下，我认为 SVD 可以很好地处理460 * 460维。我用 numpy/scypy.linalg.SVD 在一台非常老的 PC 上计算了一个6500 * 6500 SVD: Pentium III 733mHz。老实说，这个脚本需要大量的内存(大约1。XG)和大量的时间(大约30分钟)来得到 SVD 结果。 But I think 460*460 on a modern PC will not be a big problem unless u need do SVD a huge number of times.

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SVD 在460维的情况下应该可以正常工作。在我的 Atom 上网本上大约需要7秒钟。Eig ()方法需要花费 更多时间(因为它应该使用更多的浮点操作) ，并且几乎总是不那么精确。

如果你有少于460个例子，那么你要做的就是对角化散布矩阵(x-datamean) ^ T (x-mean) ，假设你的数据点是列，然后向左乘以(x-datamean)。在维数多于数据的情况下，might会更快。

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几个月后，这是一个小班级的 PCA，还有一张照片:

#!/usr/bin/env python
""" a small class for Principal Component Analysis
Usage:
p = PCA( A, fraction=0.90 )
In:
A: an array of e.g. 1000 observations x 20 variables, 1000 rows x 20 columns
fraction: use principal components that account for e.g.
90 % of the total variance


Out:
p.U, p.d, p.Vt: from numpy.linalg.svd, A = U . d . Vt
p.dinv: 1/d or 0, see NR
p.eigen: the eigenvalues of A*A, in decreasing order (p.d**2).
eigen[j] / eigen.sum() is variable j's fraction of the total variance;
look at the first few eigen[] to see how many PCs get to 90 %, 95 % ...
p.npc: number of principal components,
e.g. 2 if the top 2 eigenvalues are >= `fraction` of the total.
It's ok to change this; methods use the current value.


Methods:
The methods of class PCA transform vectors or arrays of e.g.
20 variables, 2 principal components and 1000 observations,
using partial matrices U' d' Vt', parts of the full U d Vt:
A ~ U' . d' . Vt' where e.g.
U' is 1000 x 2
d' is diag([ d0, d1 ]), the 2 largest singular values
Vt' is 2 x 20.  Dropping the primes,


d . Vt      2 principal vars = p.vars_pc( 20 vars )
U           1000 obs = p.pc_obs( 2 principal vars )
U . d . Vt  1000 obs, p.obs( 20 vars ) = pc_obs( vars_pc( vars ))
fast approximate A . vars, using the `npc` principal components


Ut              2 pcs = p.obs_pc( 1000 obs )
V . dinv        20 vars = p.pc_vars( 2 principal vars )
V . dinv . Ut   20 vars, p.vars( 1000 obs ) = pc_vars( obs_pc( obs )),
fast approximate Ainverse . obs: vars that give ~ those obs.




Notes:
PCA does not center or scale A; you usually want to first
A -= A.mean(A, axis=0)
A /= A.std(A, axis=0)
with the little class Center or the like, below.


See also:
http://en.wikipedia.org/wiki/Principal_component_analysis
http://en.wikipedia.org/wiki/Singular_value_decomposition
Press et al., Numerical Recipes (2 or 3 ed), SVD
PCA micro-tutorial
iris-pca .py .png


"""


from __future__ import division
import numpy as np
dot = np.dot
# import bz.numpyutil as nu
# dot = nu.pdot


__version__ = "2010-04-14 apr"
__author_email__ = "denis-bz-py at t-online dot de"


#...............................................................................
class PCA:
def __init__( self, A, fraction=0.90 ):
assert 0 <= fraction <= 1
# A = U . diag(d) . Vt, O( m n^2 ), lapack_lite --
self.U, self.d, self.Vt = np.linalg.svd( A, full_matrices=False )
assert np.all( self.d[:-1] >= self.d[1:] )  # sorted
self.eigen = self.d**2
self.sumvariance = np.cumsum(self.eigen)
self.sumvariance /= self.sumvariance[-1]
self.npc = np.searchsorted( self.sumvariance, fraction ) + 1
self.dinv = np.array([ 1/d if d > self.d[0] * 1e-6  else 0
for d in self.d ])


def pc( self ):
""" e.g. 1000 x 2 U[:, :npc] * d[:npc], to plot etc. """
n = self.npc
return self.U[:, :n] * self.d[:n]


# These 1-line methods may not be worth the bother;
# then use U d Vt directly --


def vars_pc( self, x ):
n = self.npc
return self.d[:n] * dot( self.Vt[:n], x.T ).T  # 20 vars -> 2 principal


def pc_vars( self, p ):
n = self.npc
return dot( self.Vt[:n].T, (self.dinv[:n] * p).T ) .T  # 2 PC -> 20 vars


def pc_obs( self, p ):
n = self.npc
return dot( self.U[:, :n], p.T )  # 2 principal -> 1000 obs


def obs_pc( self, obs ):
n = self.npc
return dot( self.U[:, :n].T, obs ) .T  # 1000 obs -> 2 principal


def obs( self, x ):
return self.pc_obs( self.vars_pc(x) )  # 20 vars -> 2 principal -> 1000 obs


def vars( self, obs ):
return self.pc_vars( self.obs_pc(obs) )  # 1000 obs -> 2 principal -> 20 vars




class Center:
""" A -= A.mean() /= A.std(), inplace -- use A.copy() if need be
uncenter(x) == original A . x
"""
# mttiw
def __init__( self, A, axis=0, scale=True, verbose=1 ):
self.mean = A.mean(axis=axis)
if verbose:
print "Center -= A.mean:", self.mean
A -= self.mean
if scale:
std = A.std(axis=axis)
self.std = np.where( std, std, 1. )
if verbose:
print "Center /= A.std:", self.std
A /= self.std
else:
self.std = np.ones( A.shape[-1] )
self.A = A


def uncenter( self, x ):
return np.dot( self.A, x * self.std ) + np.dot( x, self.mean )




#...............................................................................
if __name__ == "__main__":
import sys


csv = "iris4.csv"  # wikipedia Iris_flower_data_set
# 5.1,3.5,1.4,0.2  # ,Iris-setosa ...
N = 1000
K = 20
fraction = .90
seed = 1
exec "\n".join( sys.argv[1:] )  # N= ...
np.random.seed(seed)
np.set_printoptions( 1, threshold=100, suppress=True )  # .1f
try:
A = np.genfromtxt( csv, delimiter="," )
N, K = A.shape
except IOError:
A = np.random.normal( size=(N, K) )  # gen correlated ?


print "csv: %s  N: %d  K: %d  fraction: %.2g" % (csv, N, K, fraction)
Center(A)
print "A:", A


print "PCA ..." ,
p = PCA( A, fraction=fraction )
print "npc:", p.npc
print "% variance:", p.sumvariance * 100


print "Vt[0], weights that give PC 0:", p.Vt[0]
print "A . Vt[0]:", dot( A, p.Vt[0] )
print "pc:", p.pc()


print "\nobs <-> pc <-> x: with fraction=1, diffs should be ~ 0"
x = np.ones(K)
# x = np.ones(( 3, K ))
print "x:", x
pc = p.vars_pc(x)  # d' Vt' x
print "vars_pc(x):", pc
print "back to ~ x:", p.pc_vars(pc)


Ax = dot( A, x.T )
pcx = p.obs(x)  # U' d' Vt' x
print "Ax:", Ax
print "A'x:", pcx
print "max |Ax - A'x|: %.2g" % np.linalg.norm( Ax - pcx, np.inf )


b = Ax  # ~ back to original x, Ainv A x
back = p.vars(b)
print "~ back again:", back
print "max |back - x|: %.2g" % np.linalg.norm( back - x, np.inf )


# end pca.py

enter image description here

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你不需要全奇异值分解(SVD)来计算所有的特征矢量，对于大矩阵来说可能是禁止的。 Scypy 及其稀疏模块提供了一般的线性代数函数，它们同时适用于稀疏矩阵和稠密矩阵，其中包括 eig * 族函数:

Http://docs.scipy.org/doc/scipy/reference/sparse.linalg.html#matrix-factorizations

Scikit-learn 提供了一个目前只支持稠密矩阵的 Python PCA 实现。

时间:

In [1]: A = np.random.randn(1000, 1000)


In [2]: %timeit scipy.sparse.linalg.eigsh(A)
1 loops, best of 3: 802 ms per loop


In [3]: %timeit np.linalg.svd(A)
1 loops, best of 3: 5.91 s per loop
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使用 numpy.linalg.svd的 PCA 非常简单，下面是一个简单的演示:

import numpy as np
import matplotlib.pyplot as plt
from scipy.misc import lena


# the underlying signal is a sinusoidally modulated image
img = lena()
t = np.arange(100)
time = np.sin(0.1*t)
real = time[:,np.newaxis,np.newaxis] * img[np.newaxis,...]


# we add some noise
noisy = real + np.random.randn(*real.shape)*255


# (observations, features) matrix
M = noisy.reshape(noisy.shape[0],-1)


# singular value decomposition factorises your data matrix such that:
#
#   M = U*S*V.T     (where '*' is matrix multiplication)
#
# * U and V are the singular matrices, containing orthogonal vectors of
#   unit length in their rows and columns respectively.
#
# * S is a diagonal matrix containing the singular values of M - these
#   values squared divided by the number of observations will give the
#   variance explained by each PC.
#
# * if M is considered to be an (observations, features) matrix, the PCs
#   themselves would correspond to the rows of S^(1/2)*V.T. if M is
#   (features, observations) then the PCs would be the columns of
#   U*S^(1/2).
#
# * since U and V both contain orthonormal vectors, U*V.T is equivalent
#   to a whitened version of M.


U, s, Vt = np.linalg.svd(M, full_matrices=False)
V = Vt.T


# PCs are already sorted by descending order
# of the singular values (i.e. by the
# proportion of total variance they explain)


# if we use all of the PCs we can reconstruct the noisy signal perfectly
S = np.diag(s)
Mhat = np.dot(U, np.dot(S, V.T))
print "Using all PCs, MSE = %.6G" %(np.mean((M - Mhat)**2))


# if we use only the first 20 PCs the reconstruction is less accurate
Mhat2 = np.dot(U[:, :20], np.dot(S[:20, :20], V[:,:20].T))
print "Using first 20 PCs, MSE = %.6G" %(np.mean((M - Mhat2)**2))


fig, [ax1, ax2, ax3] = plt.subplots(1, 3)
ax1.imshow(img)
ax1.set_title('true image')
ax2.imshow(noisy.mean(0))
ax2.set_title('mean of noisy images')
ax3.imshow((s[0]**(1./2) * V[:,0]).reshape(img.shape))
ax3.set_title('first spatial PC')
plt.show()
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你可以很容易地“滚”你自己使用 scipy.linalg(假设一个预中心的数据集 data) :

covmat = data.dot(data.T)
evs, evmat = scipy.linalg.eig(covmat)

然后 evs是你的特征值，evmat是你的投影矩阵。

如果你想保持 d尺寸，使用第一个 d特征值和第一个 d特征向量。

考虑到 scipy.linalg具有分解和矩阵乘法的麻烦，您还需要什么？

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你可以使用 sklearn:

import sklearn.decomposition as deco
import numpy as np


x = (x - np.mean(x, 0)) / np.std(x, 0) # You need to normalize your data first
pca = deco.PCA(n_components) # n_components is the components number after reduction
x_r = pca.fit(x).transform(x)
print ('explained variance (first %d components): %.2f'%(n_components, sum(pca.explained_variance_ratio_)))
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If you're working with 3D vectors, you can apply SVD concisely using the toolbelt VG. It's a light layer on top of numpy.

import numpy as np
import vg


vg.principal_components(data)

There's also a convenient alias if you only want the first principal component:

vg.major_axis(data)

我在上一次创业时创建了这个库，它的动机是这样的: 在 NumPy 中冗长或不透明的简单想法。


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