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我很困惑试图在一组已读入 numpy 数组的空间网格上运行 PCA。作为数组，它们看起来像这样，其中 mdata[0] 表示单个网格中的行和列的集合，每个值代表一个网格单元：

mdata = np.array([ [[3, 4, 6, 4],
                    [4, 9, 2, 7],
                    [8, 7, 2, 2]],

                   [[1, 3, 7, 9],
                    [3, 1, 8, 8],
                    [2, 6, 7, 5]],

                   [[1, 8, 5, 2],
                    [6, 4, 7, 3],
                    [3, 1, 4, 6]],

                   [[5, 8, 2, 4],
                    [3, 7, 7, 7],
                    [1, 4, 8, 2]],

                   [[4, 5, 6, 4],
                    [1, 7, 5, 7],
                    [2, 1, 3, 3]] ])

然后我将数组展平，以便可以在整个集合中正确计算协方差，因此输入数组如下所示：

[[3 4 6 4 4 9 2 7 8 7 2 2]
 [1 3 7 9 3 1 8 8 2 6 7 5]
 [1 8 5 2 6 4 7 3 3 1 4 6]
 [5 8 2 4 3 7 7 7 1 4 8 2]
 [4 5 6 4 1 7 5 7 2 1 3 3]]

我需要生成一组代表主成分的输出数组（最终转换回网格）。输出数组是通过将特征向量与输入网格集（即输入数组）上的每个数组元素相乘并将它们加在一起而产生的。每个输出网格将代表一个主成分，按解释方差的降序排列。

所以最终的方程看起来像这样：

First principal component (uses first eigenvector array row??):
  (input_grid_1 * evec_row_1_val_1) + (input_grid_2 * evec_row_1_val_2) ... (input_grid_n * evec_row_1_val_n)

Second principal component (uses second eigenvector array row??):
  (input_grid_1 * evec_row_2_val_1) + (input_grid_2 * evec_row_2_val_2) ... (input_grid_n * evec_row_2_val_n)

etc...

我的问题是......通过 numpy.linalg.eig 生成特征向量后，我不确定它们的顺序。evec[0] 中的值（见下面的输出）是否代表我应该用来计算第一个主成分输出数组的特征向量？或者我应该从底行开始（即，下面的 evec[11]）？或者特征向量是否在 numpy.linalg.eig 的输出中按列排列？

感谢您的帮助，对于本文的篇幅，我深表歉意。

这是我正在使用的一些示例代码，以及结果输出：

print "flattened mdata:\n", mdata2, "\n"
mdata2 -= np.mean(mdata2, axis=0)
print "mean centered data:\n", mdata2, "\n"
covar = np.cov(mdata2, rowvar=0)
print "covariance matrix:\n", covar, "\n"
eval, evec = la.eig(covar)
print "eigenvectors:\n", evec, "\n"

flattened mdata:
[[3 4 6 4 4 9 2 7 8 7 2 2]
 [1 3 7 9 3 1 8 8 2 6 7 5]
 [1 8 5 2 6 4 7 3 3 1 4 6]
 [5 8 2 4 3 7 7 7 1 4 8 2]
 [4 5 6 4 1 7 5 7 2 1 3 3]]

mean centered data:
[[ 0 -1  0  0  0  3 -3  0  4  3 -2 -1]
 [-1 -2  1  4  0 -4  2  1 -1  2  2  1]
 [-1  2  0 -2  2 -1  1 -3  0 -2  0  2]
 [ 2  2 -3  0  0  1  1  0 -2  0  3 -1]
 [ 1  0  0  0 -2  1  0  0 -1 -2 -1  0]]

covariance matrix:
[[ 1.7   0.95 -1.65 -0.6  -1.    2.   -0.3   0.6  -1.   -0.55  0.65 -1.3 ]
 [ 0.95  3.2  -1.9  -3.1   1.    1.25  0.7  -1.9  -1.5  -2.8   0.9   0.2 ]
 [-1.65 -1.9   2.3   1.2   0.   -1.75 -0.15  0.05  1.25  0.6  -1.55  1.1 ]
 [-0.6  -3.1   1.2   4.8  -1.   -3.5   1.4   2.7  -1.    2.9   1.8  -0.1 ]
 [-1.    1.    0.   -1.    2.   -1.    0.5  -1.5   0.5   0.    0.5   1.  ]
 [ 2.    1.25 -1.75 -3.5  -1.    7.   -4.25 -0.25  3.25  0.25 -3.   -2.5 ]
 [-0.3   0.7  -0.15  1.4   0.5  -4.25  3.7  -0.15 -4.   -1.8   3.15  1.45]
 [ 0.6  -1.9   0.05  2.7  -1.5  -0.25 -0.15  2.3  -0.25  2.1   0.7  -1.15]
 [-1.   -1.5   1.25 -1.    0.5   3.25 -4.   -0.25  5.5   3.   -3.75 -0.75]
 [-0.55 -2.8   0.6   2.9   0.    0.25 -1.8   2.1   3.    5.2  -0.1  -1.3 ]
 [ 0.65  0.9  -1.55  1.8   0.5  -3.    3.15  0.7  -3.75 -0.1   4.3   0.15]
 [-1.3   0.2   1.1  -0.1   1.   -2.5   1.45 -1.15 -0.75 -1.3   0.15  1.7 ]]

eigenvectors:
[[-0.05226275 +0.00000000e+00j  0.14243804 +0.00000000e+00j
   0.40669907 +0.00000000e+00j -0.08874805 +0.00000000e+00j
   0.24983235 +0.00000000e+00j -0.07788035 +0.00000000e+00j
  -0.20417938 +8.64129842e-02j -0.20417938 -8.64129842e-02j
   0.17142025 -1.53009928e-02j  0.17142025 +1.53009928e-02j
  -0.07359380 -5.42353843e-02j -0.07359380 +5.42353843e-02j]
 [ 0.01313939 +0.00000000e+00j  0.46207747 +0.00000000e+00j
   0.04053819 +0.00000000e+00j  0.24412764 +0.00000000e+00j
  -0.29669270 +0.00000000e+00j -0.11848560 +0.00000000e+00j
  -0.00129189 -1.08527235e-01j -0.00129189 +1.08527235e-01j
   0.24951441 -4.06744086e-02j  0.24951441 +4.06744086e-02j
  -0.08613229 -4.16143549e-02j -0.08613229 +4.16143549e-02j]
 [ 0.01212470 +0.00000000e+00j -0.23533069 +0.00000000e+00j
  -0.38627976 +0.00000000e+00j -0.29050062 +0.00000000e+00j
  -0.04031410 +0.00000000e+00j  0.11914628 +0.00000000e+00j
  -0.03397920 -4.04899870e-01j -0.03397920 +4.04899870e-01j
  -0.02317506 +1.11224412e-01j -0.02317506 -1.11224412e-01j
   0.16309959 +1.28363333e-02j  0.16309959 -1.28363333e-02j]
 [ 0.25764595 +0.00000000e+00j -0.49039161 +0.00000000e+00j
   0.17582472 +0.00000000e+00j -0.08694414 +0.00000000e+00j
  -0.49772683 +0.00000000e+00j  0.04915654 +0.00000000e+00j
  -0.22312499 +1.60190725e-02j -0.22312499 -1.60190725e-02j
  -0.13318403 -2.09741709e-01j -0.13318403 +2.09741709e-01j
  -0.06711105 +1.03673483e-01j -0.06711105 -1.03673483e-01j]
 [ 0.04158039 +0.00000000e+00j  0.08806516 +0.00000000e+00j
  -0.28689676 +0.00000000e+00j  0.55985795 +0.00000000e+00j
   0.16139009 +0.00000000e+00j -0.31020587 +0.00000000e+00j
   0.06748141 -1.77634044e-02j  0.06748141 +1.77634044e-02j
  -0.06824769 -3.42630421e-01j -0.06824769 +3.42630421e-01j
  -0.01037536 -1.65831604e-01j -0.01037536 +1.65831604e-01j]
 [-0.55737468 +0.00000000e+00j  0.20542936 +0.00000000e+00j
   0.32117797 +0.00000000e+00j -0.04132469 +0.00000000e+00j
  -0.38616878 +0.00000000e+00j -0.07888938 +0.00000000e+00j
  -0.18663356 -2.26138514e-01j -0.18663356 +2.26138514e-01j
  -0.48555065 +0.00000000e+00j -0.48555065 +0.00000000e+00j
  -0.05777089 +8.39094379e-02j -0.05777089 -8.39094379e-02j]
 [ 0.44645722 +0.00000000e+00j  0.09345400 +0.00000000e+00j
  -0.01958382 +0.00000000e+00j  0.00350717 +0.00000000e+00j
  -0.49307361 +0.00000000e+00j -0.19705809 +0.00000000e+00j
  -0.26393665 +1.65676777e-01j -0.26393665 -1.65676777e-01j
  -0.07097582 +1.77803457e-01j -0.07097582 -1.77803457e-01j
   0.31461793 -8.41424299e-02j  0.31461793 +8.41424299e-02j]
 [ 0.03562756 +0.00000000e+00j -0.29183500 +0.00000000e+00j
   0.34969990 +0.00000000e+00j -0.16770853 +0.00000000e+00j
   0.18499349 +0.00000000e+00j -0.27031753 +0.00000000e+00j
   0.04149287 -1.82755302e-01j  0.04149287 +1.82755302e-01j
   0.27944344 -1.37299309e-01j  0.27944344 +1.37299309e-01j
  -0.09637630 -4.39851477e-01j -0.09637630 +4.39851477e-01j]
 [-0.46659969 +0.00000000e+00j -0.23849866 +0.00000000e+00j
  -0.26877609 +0.00000000e+00j  0.24093339 +0.00000000e+00j
  -0.35050288 +0.00000000e+00j  0.44403480 +0.00000000e+00j
  -0.43610595 +0.00000000e+00j -0.43610595 +0.00000000e+00j
   0.37131095 +1.81269636e-01j  0.37131095 -1.81269636e-01j
   0.29146786 -2.22035640e-01j  0.29146786 +2.22035640e-01j]
 [-0.13939660 +0.00000000e+00j -0.51461522 +0.00000000e+00j
   0.15383834 +0.00000000e+00j  0.51033734 +0.00000000e+00j
   0.13867222 +0.00000000e+00j -0.36401539 +0.00000000e+00j
   0.19655716 +8.68943320e-02j  0.19655716 -8.68943320e-02j
  -0.16900231 +9.74521096e-02j -0.16900231 -9.74521096e-02j
  -0.20194713 +1.85123869e-01j -0.20194713 -1.85123869e-01j]
 [ 0.39116291 +0.00000000e+00j  0.04835035 +0.00000000e+00j
   0.32293099 +0.00000000e+00j  0.42167036 +0.00000000e+00j
  -0.04351453 +0.00000000e+00j  0.61425451 +0.00000000e+00j
  -0.20683758 -3.83249003e-01j -0.20683758 +3.83249003e-01j
  -0.02544963 +2.31011753e-01j -0.02544963 -2.31011753e-01j
   0.13276711 -1.26161298e-04j  0.13276711 +1.26161298e-04j]
 [ 0.16560064 +0.00000000e+00j  0.04924581 +0.00000000e+00j
  -0.38012669 +0.00000000e+00j -0.03145605 +0.00000000e+00j
   0.06088502 +0.00000000e+00j -0.20480777 +0.00000000e+00j
  -0.23192947 -1.44999255e-01j -0.23192947 +1.44999255e-01j
  -0.27861106 -9.88363770e-05j -0.27861106 +9.88363770e-05j
  -0.60558363 +0.00000000e+00j -0.60558363 +0.00000000e+00j]]
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1 回答 1

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特征向量按列排列。例如，如果你得到一个特征向量矩阵 [[0, 1], [-1, 0]]

你的特征向量是 V1 = [0, -1]; V2 = [1, 0]。

只需在斐波那契数字上尝试一下：

>>> fib = numpy.array([[1, 1],[1, 0]])
>>> numpy.linalg.eig(fib)
(array([ 1.61803399, -0.61803399]), array([[ 0.85065081, -0.52573111],
       [ 0.52573111,  0.85065081]]))

第一个数组是特征值，下一个包含列中的特征向量。比较： http: //mathproofs.blogspot.com/2005/04/nth-term-of-fibonacci-sequence.html

祝你好运！

于 2012-09-12T21:02:15.710 回答