In this post, we will approach the problem of convolution from a matricial point of view.

What is convolution?

What we mean by convolution, is in the composing of two different functions to have a third one. The particularity of the convolution is that we have one function moving in the domain space of another.

Convolution is a useful resource in many data science aspects; from cross-correlation computing to image filtering. There’s also a great relationship with the “stencil pattern” in a parallel programming problem. This pattern consists of picking up a surrounding domain, to the point that we study our problem in order to apply a reduction and move it into the problem domain.

We define a convolution like this:

In a discrete problem, our convolution will turn into this next formula:

The previous formula is just a generalization of a well-known technical indicator: mobile averages. So regarding the convolution, we could use it to make a smoother curve or, as in this case, to calculate an exponential average over a prices serie.

But what about our matricial approach?

Well, the idea is performing the convolution through a Toeplitz matrix in order to achieve a more efficient calculation. We can achieve our convolution (y) through the inner product between a Toepliz matrix built onto the h function (convolution function in this specific domain), and the serie from which we desire the convolution (see the next formula):

Let’s check out the python code to achieve it:

import numpy as np
from scipy.linalg import toeplitz

# Convolution matrix functions:
convolution_matrix = lambda h,dim: np.tril(toeplitz(np.r_[h[::-1],[0]*(dim-f.shape[0])]))
convolution = lambda f,g: np.dot(convolution_matrix(f,g.shape[0]),g)

# Convolution functions:
hN = lambda n: (lambda x: np.exp(x)/np.exp(x).sum())(np.r_[0:1+(1./n):1./n])

# Convolution over ~3 months:
np.random.seed(19940907)
prices = np.cumsum(np.random.randn(10000,1))
convolution(hN(63),prices)

The results of applying the convolution with 63 evaluation points in the convolution matrix are:

The red axis is the exponential function used to perform the convolution over the blue price serie, and we get the green curve as a result of the convolution.