Filter ExampleΒΆ

This example demonstrates the connection between MKS and signal processing for a 1D filter. It shows that the filter is in fact the same as the influence coefficients and, thus, applying the predict method provided by the MKSLocalizationnModel is in essence just applying a filter.

%matplotlib inline
%load_ext autoreload
%autoreload 2

import numpy as np
import matplotlib.pyplot as plt
The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

Here we construct a filter, \(F\), such that

\[F\left(x\right) = e^{-|x|} \cos{\left(2\pi x\right)}\]

We want to show that if \(F\) is used to generate sample calibration data for the MKS, then the calculated influence coefficients are in fact just \(F\).

x0 = -10.
x1 = 10.
x = np.linspace(x0, x1, 1000)
def F(x):
    return np.exp(-abs(x)) * np.cos(2 * np.pi * x)
p = plt.plot(x, F(x), color='#1a9850')

Next we generate the sample data (X, y) using scipy.ndimage.convolve. This performs the convolution

\[p\left[ s \right] = \sum_r F\left[r\right] X\left[r - s\right]\]

for each sample.

import scipy.ndimage

n_space = 101
n_sample = 50
x = np.linspace(x0, x1, n_space)
X = np.random.random((n_sample, n_space))
y = np.array([scipy.ndimage.convolve(xx, F(x), mode='wrap') for xx in X])

For this problem, a basis is unnecessary as no discretization is required in order to reproduce the convolution with the MKS localization. Using the ContinuousIndicatorBasis with n_states=2 is the equivalent of a non-discretized convolution in space.

from pymks import MKSLocalizationModel
from pymks import PrimitiveBasis

prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
model = MKSLocalizationModel(basis=prim_basis)

Fit the model using the data generated by \(F\)., y)

To check for internal consistency, we can compare the predicted output with the original for a few values

y_pred = model.predict(X)
print y[0, :4]
print y_pred[0, :4]
[-0.41059557  0.20004566  0.61200171  0.5878077 ]
[-0.41059556  0.20004566  0.6120017   0.58780769]

With a slight linear manipulation of the coefficients, they agree perfectly with the shape of the filter, \(F\).

plt.plot(x, F(x), label=r'$F$', color='#1a9850')
plt.plot(x, -model.coeff[:,0] + model.coeff[:, 1],
         'k--', label=r'$\alpha$')
l = plt.legend()

Some manipulation of the coefficients is required to reproduce the filter. Remember the convolution for the MKS is

\[p \left[s\right] = \sum_{l=0}^{L-1} \sum_{r=0}^{S - 1} \alpha[l, r] m[l, s - r]\]

However, when the primitive basis is selected, the MKSLocalizationModel solves a modified form of this. There are always redundant coefficients since

\[\sum\limits_{l=0}^{L-1} m[l, s] = 1\]

Thus, the regression in Fourier space must be done with categorical variables, and the regression takes the following form.

\[\begin{split} \begin{split} p [s] & = \sum_{l=0}^{L - 1} \sum_{r=0}^{S - 1} \alpha[l, r] m[l, s -r] \\ P [k] & = \sum_{l=0}^{L - 1} \beta[l, k] M[l, k] \\ &= \beta[0, k] M[0, k] + \beta[1, k] M[1, k] \end{split}\end{split}\]


\[\begin{split}\beta[0, k] = \begin{cases} \langle F(x) \rangle ,& \text{if } k = 0\\ 0, & \text{otherwise} \end{cases}\end{split}\]

This removes the redundancies from the regression, and we can reproduce the filter.