Approximation
To begin with, let us import the libraries that we need for the moment — NumPy and
matplotlib.pyplot:
In [ 1 ]: import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
Throughout this discussion, the main example function we will use is the following, which
is comprised of a trigonometric term and a linear term:
In [ 2 ]: def f(x):
return np.sin(x) + 0.5 * x
The main focus is the approximation of this function over a given interval by regression
and interpolation. First, let us generate a plot of the function to get a better view of what
exactly the approximation shall achieve. The interval of interest shall be [–2,2].
Figure 9-1 displays the function over the fixed interval defined via the linspace function.
np.linspace(start, stop, num) returns num points beginning with start and ending
with stop, with the subintervals between two consecutive points being evenly spaced:
In [ 3 ]: x = np.linspace(- 2 * np.pi, 2 * np.pi, 50 )
In [ 4 ]: plt.plot(x, f(x), ‘b’)
plt.grid(True)
plt.xlabel(‘x’)
plt.ylabel(‘f(x)’)
Figure 9-1. Example function plot
Regression
Regression is a rather efficient tool when it comes to function approximation. It is not only
suited to approximate one-dimensional functions but also works well in higher
dimensions. The numerical techniques needed to come up with regression results are
easily implemented and quickly executed. Basically, the task of regression, given a set of
so-called basis functions bd, d ∈ {1,...,D}, is to find optimal parameters
according to Equation 9-1, where yi ≡ f(xi) for i ∈ {1,⋯, I} observation points. The xi are
considered independent observations and the yi dependent observations (in a functional or
statistical sense).
Equation 9-1. Minimization problem of regression
