Question

如何计算非方阵的Cholesky分解，以便用numpy计算马哈拉诺比斯距离？

def get_fitting_function(G):
    print(G.shape) #(14L, 11L) --> 14 samples of dimension 11
    g_mu = G.mean(axis=0) 
    #Cholesky decomposition uses half of the operations as LU
    #and is numerically more stable.
    L = np.linalg.cholesky(G)

    def fitting_function(g):
        x = g - g_mu
        z = np.linalg.solve(L, x)
        #Mahalanobis Distance 
        MD = z.T*z
        return math.sqrt(MD)
    return fitting_function  


C:\Users\Matthias\CV\src\fitting_function.py in get_fitting_function(G)
     22     #Cholesky decomposition uses half of the operations as LU
     23     #and is numerically more stable.
---> 24     L = np.linalg.cholesky(G)
     25 
     26     def fitting_function(g):

C:\Users\Matthias\AppData\Local\Enthought\Canopy\User\lib\site-packages\numpy\linalg\linalg.pyc in cholesky(a)
    598     a, wrap = _makearray(a)
    599     _assertRankAtLeast2(a)
--> 600     _assertNdSquareness(a)
    601     t, result_t = _commonType(a)
    602     signature = 'D->D' if isComplexType(t) else 'd->d'

C:\Users\Matthias\AppData\Local\Enthought\Canopy\User\lib\site-packages\numpy\linalg\linalg.pyc in _assertNdSquareness(*arrays)
    210     for a in arrays:
    211         if max(a.shape[-2:]) != min(a.shape[-2:]):
--> 212             raise LinAlgError('Last 2 dimensions of the array must be square')
    213 
    214 def _assertFinite(*arrays):

LinAlgError: Last 2 dimensions of the array must be square 


    LinAlgError: Last 2 dimensions of the array must be square

基于Matlab实现：Mahalanobis distance inverting the covariance matrix

修改：chol(a) = linalg.cholesky(a).T 矩阵的cholesky分解（matlab中的chol(a)返回一个上三角矩阵，但linalg.cholesky(a)返回一个下三角矩阵）（来源：http://wiki.scipy.org/NumPy_for_Matlab_Users）

EDIT2：

G -= G.mean(axis=0)[None, :]
C = (np.dot(G, G.T) / float(G.shape[0]))
#Cholesky decomposition uses half of the operations as LU
#and is numerically more stable.
L = np.linalg.cholesky(C).T

所以如果D = x ^ t.S ^ -1.x = x ^ t。（L.L ^ t）^ - 1.x = x ^ t.L.L ^ t.x = z ^ t.z

Answer 1

我不相信你可以。 Cholesky分解不仅需要一个方阵，而且需要一个Hermitian矩阵和一个唯一性的正定矩阵。它基本上是LU分解，条件是L = U'。事实上，该算法经常被用作数字检查给定矩阵是正定的方法。见Wikipedia。

也就是说，根据定义，协方差矩阵是对称的正半定，所以你应该能够对它进行cholesky。

编辑：在您计算时，您的矩阵C=np.dot(G, G.T)应该是对称的，但也许是错误的。您可以尝试强制对称C = ( C + C.T) /2.0，然后再次尝试chol(C)。

如何计算非方矩阵的Cholesky分解，以便用'numpy'计算马哈拉诺比斯距离？

1 个答案: