Chi distribution
When studying a multi-dimensional random variable, if these are guassian the norm if the vector follows a $\chi$ distribution (see http://en.m.wikipedia.org/wiki/Chi_distribution).
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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
This is important when one considers for instance a 2D vector which is normally distributed:
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N = 1e3
x = np.random.randn(2, N)
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_ = plt.scatter(x[0], x[1])
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d = np.sqrt(x[0]**2+x[1]**2)
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hist, bins, _ = plt.hist(d, 20)
This should be well fitted with the $\chi$-dstribution with n=2
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p = bins * np.exp(- bins**2 / 2)
plt.bar(bins[:-1], hist/hist.sum(), width=bins[1]-bins[0])
plt.plot(bins, p/p.sum(), 'r')
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Hurray! Other visualizations for this distribution:
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_ = plt.semilogy(bins, p/p.sum())
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_ = plt.semilogx(bins, p/p.sum())
Comparison to other distirbutions
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_ = plt.loglog(bins, bins, 'r--', label='lin')
_ = plt.loglog(bins, np.exp(- bins**2 / 2), 'g--', label='gaussian')
_ = plt.loglog(bins, np.exp(- np.log(bins)**2 / 2), 'r--', label='log-gaussian')
_ = plt.loglog(bins, (bins+.1)**-2, 'b--', label='power-law')
_ = plt.loglog(bins, p, label=r'$\chi$')
_ = plt.legend(loc='lower center' )
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plt.legend?
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