Polynomial fitting with the normal equations

Copyright (C) 2020 Andreas Kloeckner

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In [3]:
import numpy as np

import numpy.linalg as la

import matplotlib.pyplot as pt

In this demo, we will produce data from a simple parabola as a "model" and try to recover the "unknown" parameters $\alpha$, $\beta$, and $\gamma$ using least squares.

In [6]:
alpha = 3

beta = 2

gamma = 2



def f(x):

    return alpha*x**2 + beta*x + gamma



plot_grid = np.linspace(-3, 3, 100)



pt.plot(plot_grid, f(plot_grid))
Out[6]:
[<matplotlib.lines.Line2D at 0x7f6900e06748>]
In [7]:
npts = 5



np.random.seed(22)

points = np.linspace(-2, 2, npts) + np.random.randn(npts)

values = f(points) + 0.3*np.random.randn(npts)*f(points)



pt.plot(plot_grid, f(plot_grid))

pt.plot(points, values, "o")
Out[7]:
[<matplotlib.lines.Line2D at 0x7f6900de0ac8>]

Now build the Vandermonde matrix:

In [9]:
A = np.array([

    np.ones(npts),

    points,

    points**2

    ]).T

print(A)

[[ 1.         -2.09194992  4.37625447]
[ 1.         -2.46335065  6.06809644]
[ 1.          1.08179168  1.17027324]
[ 1.          0.76067483  0.5786262 ]
[ 1.          1.50887086  2.27669128]]

And solve for x using the normal equations:

In [10]:
#clear

x = la.solve(A.T@A, A.T@values)

x
Out[10]:
array([ 0.13178048,  1.84869187,  3.45132033])

Lastly, pick apart x into alpha_c, beta_c, and gamma_c:

In [11]:
#clear

gamma_c, beta_c, alpha_c = x
In [14]:
print(alpha, alpha_c)

print(beta, beta_c)

print(gamma, gamma_c)

3 3.45132033448
2 1.84869187248
2 0.131780484241
In [13]:
def f_c(x):

    return alpha_c*x**2 + beta_c*x + gamma_c



pt.plot(plot_grid, f(plot_grid), label="true")

pt.plot(points, values, "o", label="data")

pt.plot(plot_grid, f_c(plot_grid), label="found")

pt.legend()
Out[13]:
<matplotlib.legend.Legend at 0x7f6900cdfb70>

(Edit this cell for solution.)