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I'd like to find a least-squares solution for the a coefficients in
z = (a0 + a1*x + a2*y + a3*x**2 + a4*x**2*y + a5*x**2*y**2 + a6*y**2 + a7*x*y**2 + a8*x*y) given arrays x, y, and z of length 20. Basically I'm looking for the equivalent of numpy.polyfit but for a 2D polynomial.
This question is similar, but the solution is provided via MATLAB.
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Method: Scipy.polyfit( ) or numpy.polyfit( ) This is a pretty general least squares polynomial fit function which accepts the data set and a polynomial function of any degree (specified by the user), and returns an array of coefficients that minimizes the squared error.
polyfit() method takes a few parameters and returns a vector of coefficients p that minimizes the squared error in the order deg, deg-1, … 0. It least squares the polynomial fit. It fits a polynomial p(X) of degree deg to points (X, Y).
In python, Numpy polyfit() is a method that fits the data within a polynomial function. That is, it least squares the function polynomial fit. For example, a polynomial p(X) of deg degree fits the coordinate points (X, Y).
Here is an example showing how you can use numpy.linalg.lstsq for this task:
import numpy as np x = np.linspace(0, 1, 20) y = np.linspace(0, 1, 20) X, Y = np.meshgrid(x, y, copy=False) Z = X**2 + Y**2 + np.random.rand(*X.shape)*0.01 X = X.flatten() Y = Y.flatten() A = np.array([X*0+1, X, Y, X**2, X**2*Y, X**2*Y**2, Y**2, X*Y**2, X*Y]).T B = Z.flatten() coeff, r, rank, s = np.linalg.lstsq(A, B) the adjusting coefficients coeff are:
array([ 0.00423365, 0.00224748, 0.00193344, 0.9982576 , -0.00594063, 0.00834339, 0.99803901, -0.00536561, 0.00286598]) Note that coeff[3] and coeff[6] respectively correspond to X**2 and Y**2, and they are close to 1. because the example data was created with Z = X**2 + Y**2 + small_random_component.
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