I'm able to use numpy.polynomial to fit terms to 1D polynomials like f(x) = 1 + x + x^2. How can I fit multidimensional polynomials, like f(x,y) = 1 + x + x^2 + y + yx + y x^2 + y^2 + y^2 x + y^2 x^2? It looks like numpy doesn't support multidimensional polynomials at all: is that the case? In my real application, I have 5 dimensions of input and I am interested in hermite polynomials. It looks like the polynomials in scipy.special are also only available for one dimension of inputs.
# One dimension of data can be fit
x = np.random.random(100)
y = np.sin(x)
params = np.polynomial.polynomial.polyfit(x, y, 6)
np.polynomial.polynomial.polyval([0, .2, .5, 1.5], params)
array([ -5.01799432e-08, 1.98669317e-01, 4.79425535e-01,
9.97606096e-01])
# When I try two dimensions, it fails.
x = np.random.random((100, 2))
y = np.sin(5 * x[:,0]) + .4 * np.sin(x[:,1])
params = np.polynomial.polynomial.polyvander2d(x, y, [6, 6])
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-13-5409f9a3e632> in <module>()
----> 1 params = np.polynomial.polynomial.polyvander2d(x, y, [6, 6])
/usr/local/lib/python2.7/site-packages/numpy/polynomial/polynomial.pyc in polyvander2d(x, y, deg)
1201 raise ValueError("degrees must be non-negative integers")
1202 degx, degy = ideg
-> 1203 x, y = np.array((x, y), copy=0) + 0.0
1204
1205 vx = polyvander(x, degx)
ValueError: could not broadcast input array from shape (100,2) into shape (100)
It doesn't look like polyfit supports fitting multivariate polynomials, but you can do it by hand, with linalg.lstsq. The steps are as follows:
Gather the degrees of monomials x**i * y**j you wish to use in the model. Think carefully about it: your current model already has 9 parameters, if you are going to push to 5 variables then with the current approach you'll end up with 3**5 = 243 parameters, a sure road to overfitting. Maybe limit to the monomials of __total_ degree at most 2 or three...
Plug the x-points into each monomial; this gives a 1D array. Stack all such arrays as columns of a matrix.
Solve a linear system with aforementioned matrix and with the right-hand side being the target values (I call them z because y is confusing when you also use x, y for two variables).
Here it is:
import numpy as np
x = np.random.random((100, 2))
z = np.sin(5 * x[:,0]) + .4 * np.sin(x[:,1])
degrees = [(i, j) for i in range(3) for j in range(3)] # list of monomials x**i * y**j to use
matrix = np.stack([np.prod(x**d, axis=1) for d in degrees], axis=-1) # stack monomials like columns
coeff = np.linalg.lstsq(matrix, z)[0] # lstsq returns some additional info we ignore
print("Coefficients", coeff) # in the same order as the monomials listed in "degrees"
fit = np.dot(matrix, coeff)
print("Fitted values", fit)
print("Original values", y)