不,一般来说没有简单的方法。任意多项式的闭式解不可用 https://en.wikipedia.org/wiki/Closed-form_expression#Example:_roots_of_polynomials对于七阶多项式。
可以进行相反方向的拟合,但仅限于原始多项式的单调变化区域。如果原始多项式在您感兴趣的域上具有最小值或最大值,那么即使 y 是 x 的函数,x 也不能是 y 的函数,因为它们之间不存在 1 对 1 的关系。
如果您(i)可以重新执行拟合过程,并且(ii)可以一次在拟合的单个单调区域上分段工作,那么您可以执行以下操作:
-
import numpy as np
# generate a random coefficient vector a
degree = 1
a = 2 * np.random.random(degree+1) - 1
# an assumed true polynomial y(x)
def y_of_x(x, coeff_vector):
"""
Evaluate a polynomial with coeff_vector and degree len(coeff_vector)-1 using Horner's method.
Coefficients are ordered by increasing degree, from the constant term at coeff_vector[0],
to the linear term at coeff_vector[1], to the n-th degree term at coeff_vector[n]
"""
coeff_rev = coeff_vector[::-1]
b = 0
for a in coeff_rev:
b = b * x + a
return b
# generate some data
my_x = np.arange(-1, 1, 0.01)
my_y = y_of_x(my_x, a)
# verify that polyfit in the "traditional" direction gives the correct result
# [::-1] b/c polyfit returns coeffs in backwards order rel. to y_of_x()
p_test = np.polyfit(my_x, my_y, deg=degree)[::-1]
print p_test, a
# fit the data using polyfit but with y as the independent var, x as the dependent var
p = np.polyfit(my_y, my_x, deg=degree)[::-1]
# define x as a function of y
def x_of_y(yy, a):
return y_of_x(yy, a)
# compare results
import matplotlib.pyplot as plt
%matplotlib inline
plt.plot(my_x, my_y, '-b', x_of_y(my_y, p), my_y, '-r')
注意:此代码不检查单调性,而只是假设它。
通过玩弄价值degree
,您应该看到该代码仅适用于所有随机值a
when degree=1
。它有时对于其他度数也可以,但当有很多最小值/最大值时就不行了。它永远不会完美地完成degree > 1
因为用平方根函数逼近抛物线并不总是有效,等等。