How to fit parametric equations to data points in Python

问题

I am looking for a way to fit parametric equations to a set of data points, using Python.

As a simple example, given is the following set of data points:

import numpy as np
x_data = np.array([1, 2, 3, 4, 5])
y_data = np.array([2, 0, 3, 7, 13])

Using t as the parameter, I want to fit the following parametric equation to the data points,

t = np.arange(0, 5, 0.1)
x = a1*t + b1
y = a2*t**2 + b2*t + c2

that is, have Python find the values for the coefficients a1, b1, a2, b2, c2 that fits (x,y) best to the data points (x_data, y_data).

Note that the y(t) and x(t) functions above only serve as examples of parametric equations. The actual functions I want to fit my data to are much more complex, and in those functions, it is not trivial to express y as a function of x.

Help will be appreciated - thank you!

回答1:

You can use polyfit, but please take care that the length of t must match the length of data points

import numpy as np

tt = np.linspace(0, 5, len(x_data))

x_params = np.polyfit(tt, x_data, 1)
y_params = np.polyfit(tt, y_data, 2)

Change the third parameter to the degree that you think fits your data.

To get the function you can use

y = np.poly1d(y_params)

t = np.arange(0, 5, 0.1)

plot(t, y(t))
plot(tt, y_data, 'o')

回答2:

Since the relation between x and y is a quadratic one, you can use np.polyfit to get the coefficients. According to your equations, your x and y relation is:

y = a2*((x-b1)/a1)**2 + b2*((x-b1)/a1) + c2

Using polyfit, we get

y_data = np.array([2, 0, 3, 7, 13])
x_data = np.array([1, 2, 3, 4, 5])
np.polyfit(x_data,y_data,2)
[p2,p1,p0] = list(array([ 1.21428571, -4.38571429,  4.8       ]))

The values of a1, b1, a2, b2, c2 can be obtained by solving the following eqns

p2 = a2/a1**2

p1 = -2b1*a2/a1**2 + b2/a1

p0 = a2*b1**2/a1**2 -b2*b1/a1 + c2

来源：https://stackoverflow.com/questions/38221050/how-to-fit-parametric-equations-to-data-points-in-python