File:Chernoff-bound.svg
Summary
| Description |
English: Chernoff bound (upper and lower) for the CDF of a chi-square random variable with ten degrees of freedom |
| Date | |
| Source | Own work |
| Author | Nicoguaro |
| SVG development | |
| Source code | Python codefrom __future__ import division
import numpy as np
from scipy.special import gammaincc
import matplotlib.pyplot as plt
plt.style.use(u"seaborn-white")
def chifunc(x, n):
return gammaincc(n/2, x*n/2)
red = "#e41a1c"
blue = "#377eb8"
green = "#4daf4a"
M = 10
z = np.linspace(0.01, 10, 1000)
a = (1 - 2/(9*M))**3
nx = z <= a
nx2 = z >= a
chernoff = (z * np.exp(1 - z))**(M/2)
plt.figure(figsize=(5, 3))
plt.loglog(z[nx], 1 - chifunc(z[nx], M), color=red)
plt.loglog(z[nx2], chifunc(z[nx2], M), color=blue, linestyle="dashed")
plt.loglog(z, chernoff, color=green, linestyle="dotted")
plt.xlim(0.01, 10)
plt.ylim(1e-15, 10)
plt.yticks(np.logspace(-15, 0, 4))
plt.legend(["CDF", "1 - CDF", "Chernoff bound"], loc="best")
plt.xlabel(r"$z$", fontsize=15)
plt.savefig("Chernoff-bound.svg", bbox_inches="tight")
|
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