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我试图实现一个标准的多项式拟合程序，遇到了一个我无法理解的问题。这是我有一个样本 x 和 y 数据的代码，我使用正规方程进行拟合，并且还使用了 numpy 中的 polyfit 函数。

def mytest(N):
    print "\nPolynomial order (N): {}".format(N)
    mVals = [5, 10, 15, 20]
    a_orig = [198.764, 13.5, 0.523]
    for M in mVals:
        x = arange(-M, M+1)
        y = matrix(a_orig[0]+ a_orig[1]*x + a_orig[2]*x**2).T

        # Code implementing the solution from the normal equations
        nArray = arange(N+1)    
        A = matrix([[n**i for i in nArray] for n in x])
        B = (A.T*A).I
        a_myfit = B*(A.T*y)

        # numpy's polyfit    
        a_polyfit = polyfit(x, y, N)

        print "M: {}".format(M)
        print ["{0:0.3f}".format(i) for i in a_orig] 
        print ["{0:0.3f}".format(i) for i in array(a_myfit)[:,0]]
        print ["{0:0.3f}".format(i) for i in list(array(a_polyfit)[:,0])[::-1]]

mytest(N=5)
mytest(N=6)

这是输出

Polynomial order (N): 5
M: 5
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '0.000', '0.000', '-0.000']
['198.764', '13.500', '0.523', '-0.000', '0.000', '0.000']
M: 10
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '-0.000', '0.000', '-0.000']
['198.764', '13.500', '0.523', '0.000', '0.000', '0.000']
M: 15
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '0.000', '0.000', '-0.000']
['198.764', '13.500', '0.523', '-0.000', '-0.000', '0.000']
M: 20
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '0.000', '-0.000', '-0.000']
['198.764', '13.500', '0.523', '0.000', '0.000', '0.000']

Polynomial order (N): 6
M: 5
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '0.000', '0.000', '-0.000', '-0.000']
['198.764', '13.500', '0.523', '-0.000', '0.000', '0.000', '-0.000']
M: 10
['198.764', '13.500', '0.523']
['198.764', '13.500', '0.523', '-0.000', '-0.000', '-0.000', '-0.000']
['198.764', '13.500', '0.523', '0.000', '0.000', '-0.000', '-0.000']
M: 15
['198.764', '13.500', '0.523']
['294.451', '13.500', '-0.061', '0.000', '-0.001', '-0.000', '-0.000']
['198.764', '13.500', '0.523', '0.000', '0.000', '0.000', '-0.000']
M: 20
['198.764', '13.500', '0.523']
['369.135', '13.500', '-0.046', '0.000', '-0.000', '-0.000', '-0.000']
['198.764', '13.500', '0.523', '0.000', '0.000', '-0.000', '-0.000']

对于 N > 5 和 M > 13，多项式拟合的值给出了错误的值。

我哪里错了？polyfit 实现有什么不同？

4

1 回答 1

4

A是 dtype int32。

可表示的最大值为：

In [10]: np.iinfo(np.dtype('int32')).max
Out[10]: 2147483647

当指数非常大时，n**i可以变得大于 2147483647。此时你得到一个不正确的结果。

NumPy 不检查算术溢出（在对数组执行操作时），因为这会影响性能。您可以选择避免算术溢出的 dtype。

要解决此问题，请声明一个可以表示更大数字的不同 dtype：

A = np.matrix([[n**i for i in nArray] for n in x], dtype='float64')

请注意，问题仍然存在，它只会出现在更大的值N.

于 2013-07-12T11:37:42.330 回答