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58 lines
1.7 KiB
Python
58 lines
1.7 KiB
Python
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"""
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Test the transformation of a PDF.
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Author:
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Ilias Bilionis
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Date:
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8/4/2015
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"""
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import sys
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import os
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# Make sure we load the GP that is here
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sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), '..')))
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import GPy
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import matplotlib.pyplot as plt
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import numpy as np
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import scipy.stats as st
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import scipy.integrate as integrate
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if __name__ == '__main__':
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p_theta = st.lognorm(.9)
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# Plot the PDF of theta
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fig, ax = plt.subplots()
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theta = np.linspace(0.0001, 8., 100)
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ax.plot(theta, p_theta.pdf(theta), linewidth=2, label='True PDF')
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ax.set_xlabel(r'$\theta$', fontsize=16)
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ax.set_ylabel(r'$p(\theta)$', fontsize=16)
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# Now let's look at the transformation phi = log(exp(theta - 1))
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t = GPy.constraints.Logexp()
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t.plot()
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# Plot the transformed probability density
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phi = np.linspace(-8, 8, 100)
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fig, ax = plt.subplots()
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ax.plot(phi, p_theta.pdf(t.f(phi)) * t.jacobianfactor(t.f(phi)), linewidth=2,
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label='Transformed PDF')
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# Now find the normalization constant for the naive transformation of the
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# PDF
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p_phi_prop = lambda(phi): p_theta.pdf(t.f(phi))
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c = integrate.quad(p_phi_prop, -np.inf, np.inf)[0]
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p_phi = lambda(phi): p_phi_prop(phi) / c
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ax.plot(phi, p_phi(phi), '--', linewidth=2, label='Naively transformed PDF')
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# Now let's draw some samples of theta and transform them so that we see
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# which one is right
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theta_s = p_theta.rvs(100000)
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phi_s = t.finv(theta_s)
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ax.hist(phi_s, normed=True, bins=100, alpha=0.25, label='Empirical')
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ax.set_xlim(-3, 10)
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ax.set_xlabel(r'transformed $\theta$', fontsize=16)
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ax.set_ylabel('PDF', fontsize=16)
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plt.legend(loc='best')
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plt.show(block=True)
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