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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Filter Example
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.
Step1:  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 $F$.
Step2:  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.
Step3:  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.
Step4:  Fit the model using the data generated by $F$.
Step5:  To check for internal consistency, we can compare the predicted
output with the original for a few values
Step6:  With a slight linear manipulation of the coefficients, they agree perfectly with the shape of the filter, $F$.
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Python Code:
if __import__('pyfftw'):
    import pyfftw
%matplotlib inline
%load_ext autoreload
%autoreload 2
import numpy as np
import matplotlib.pyplot as plt
Explanation: Filter Example
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.
End of explanation
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')
Explanation: 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 $F$.
End of explanation
import scipy.ndimage
n_space = 101
n_sample = 50
np.random.seed(201)
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])
Explanation: 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.
End of explanation
from pymks import MKSLocalizationModel
from pymks import PrimitiveBasis
p_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
model = MKSLocalizationModel(basis=p_basis)
Explanation: 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.
End of explanation
model.fit(X, y)
Explanation: Fit the model using the data generated by $F$.
End of explanation
y_pred = model.predict(X)
print y[0, :4]
print y_pred[0, :4]
Explanation: To check for internal consistency, we can compare the predicted
output with the original for a few values
End of explanation
plt.plot(x, F(x), label=r'$F$', color='#1a9850')
plt.plot(x, -model.coef_[:,0] + model.coef_[:, 1], 
         'k--', label=r'$\alpha$')
l = plt.legend()
Explanation: With a slight linear manipulation of the coefficients, they agree perfectly with the shape of the filter, $F$.
End of explanation
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