List of distributions#

import jax; jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
from jaxkuramoto import distribution

import matplotlib.pyplot as plt

xs = jnp.arange(-5, 5, 0.01)

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No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

Normal distribution#

  • Probability density function

\[ p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]

  • parameters

    • \(\mu\): mean (loc)

    • \(\sigma\): standard deviation (scale)

loc = 0.0
scale = 1.0

dist = distribution.Normal(loc=loc, scale=scale)
plt.plot(xs, dist.pdf(xs), label="pdf")

[<matplotlib.lines.Line2D at 0x7fbb3ff62f40>]
_images/distributions_3_1.png

Cauchy distribution#

  • Probability density function

\[ p(x)=\frac{\gamma}{\pi}\frac{1}{(x-\mu)^2+\gamma^2} \]

  • parameters

    • \(\mu\): location (loc)

    • \(\gamma\): scale (gamma)

loc = 0.0
gamma = 1.0

dist = distribution.Cauchy(loc=loc, gamma=gamma)
plt.plot(xs, dist.pdf(xs), label="pdf")

[<matplotlib.lines.Line2D at 0x7fbb3c3ebfd0>]
_images/distributions_5_1.png

Uniform distribution#

  • Probability density function for \(a\leq x\leq b\)

\[ p(x)=\frac{1}{b-a} \]

  • parameters

    • \(a\): lower bound (low)

    • \(b\): upper bound (high)

low = -1.0
high = 1.0

dist = distribution.Uniform(low=low, high=high)
plt.plot(xs, dist.pdf(xs), label="pdf")

[<matplotlib.lines.Line2D at 0x7fbb3c3a32b0>]
_images/distributions_7_1.png

Generalized Normal distribution#

  • Probability density function

\[ p(x)=\frac{n\gamma}{\Gamma(1/(2n))}\exp(-\gamma^{2n}(x-\mu)^{2n}) \]

  • parameters

    • \(\mu\): location (loc)

    • \(\gamma\): scale (gamma)

    • \(n\): shape (n)

  • Note:

    • \(n=1\) is the normal distribution

    • \(n\to\infty\) is uniform distribution (range of \(x\) is \(|x-\mu|<\gamma\))

    \[ \frac{n\gamma}{\Gamma(1/(2n))}\exp(-\gamma^{2n}(x-\mu)^{2n})\to\frac{1}{2\gamma} \]
loc = 0.0
gamma = 1.0

for n in [1,2,3,10]:
    dist = distribution.GeneralNormal(loc=loc, gamma=gamma, n=n)
    plt.plot(xs, dist.pdf(xs), label=f"n={n}")
dist_uniform = distribution.Uniform(low=-gamma, high=gamma)
plt.plot(xs, dist_uniform.pdf(xs), label="Uniform", linestyle="--")
plt.legend()

<matplotlib.legend.Legend at 0x7fbb3c2b2f10>
_images/distributions_9_1.png

Generalized Cauchy distribution#

  • Probability density function

\[ p(x)=\frac{n\sin(\pi/(2n))}{\pi}\frac{\gamma^{2n-1}}{(x-\mu)^{2n}+\gamma^{2n}} \]

  • parameters

    • \(\mu\): location (loc)

    • \(\gamma\): scale (gamma)

    • \(n\): shape (n)

  • Note:

    • \(n=1\) is Cauchy distribution

    • \(n\to\infty\) is uniform distribution (range of \(x\) is \(|x-\mu|<\gamma\))

    \[ \frac{n\sin(\pi/(2n))}{\pi}\frac{\gamma^{2n-1}}{x^{2n}+\gamma^{2n}}\to\frac{1}{2\gamma} \]
loc = 0.0
gamma = 1.0

for n in [1,2,3,10]:
    dist = distribution.GeneralCauchy(loc=loc, gamma=gamma, n=n)
    plt.plot(xs, dist.pdf(xs), label=f"n={n}")
dist_uniform = distribution.Uniform(low=-gamma, high=gamma)
plt.plot(xs, dist_uniform.pdf(xs), label="Uniform", linestyle="--")
plt.legend()

<matplotlib.legend.Legend at 0x7fbb3c2812e0>
_images/distributions_11_1.png

Muliplied Cauchy distribution#

  • Probability density function

\[ p(x)=\frac{\gamma_{1}\gamma_{2}[(\gamma_{1}+\gamma_{2})^{2}+4\Omega^{2}]}{\pi(\gamma_{1}+\gamma_{2})}\frac{1}{[(x-\Omega)^{2}+\gamma_{1}^{2}][(x+\Omega)^{2}+\gamma_{2}^{2}]} \]

  • parameters

    • \(\Omega\): location (Omega)

    • \(\gamma_{1}\): scale (gamma1)

    • \(\gamma_{2}\): scale (gamma2)

# Omega, gamma1, gamma2
params = [
    (0.6, 1.0, 1.0), (1.5, 0.6, 1.0), (3.0, 1.0, 1.0),
    (3.0, 0.8, 1.0), (1.8, 0.9, 1.0)
]
for Omega, gamma1, gamma2 in params:
    dist = distribution.CauchyMultiply(Omega=Omega, gamma1=gamma1, gamma2=gamma2)
    plt.plot(xs, dist.pdf(xs), label=rf"$\Omega={Omega}, \gamma_{1}={gamma1}$")
plt.legend()

<matplotlib.legend.Legend at 0x7fbb3c1fb160>
_images/distributions_13_1.png

Finite-differentiable distribution#

  • Probability density function for \(|x-\mu|\leq\gamma\)

\[ p(x)=\frac{1}{\gamma B(n+2, 1/2)}\left[1 - \left(\frac{x-\mu}{\gamma}\right)^{2}\right]^{n+1} \]

  • parameters

    • \(\mu\): mean (loc)

    • \(\gamma\): represent the scale of the distribution (scale)

  • Note:

    • This distribution has the smoothness of \(C^{n}\) but not \(C^{n+1}\) for finite \(n\).

    • \(B(x, y)\) is the beta function.

    • By scaling \(\gamma=\sqrt{n+1}\gamma\), this distribution goes to the noraml distribution with the standard deviation \(\gamma/\sqrt{2}\) in the limit of \(n\to\infty\).

    \[ \frac{1}{\sqrt{n+1}\gamma B(n+2, 1/2)}\left[1 - \frac{1}{n+1}\left(\frac{x-\mu}{\gamma}\right)^{2}\right]^{n+1}\to\frac{1}{\gamma\sqrt{\pi}}\exp\left(-\frac{(x-\mu)^{2}}{\gamma^{2}}\right) \]
loc = 0.0
gamma = 1.0
for n in [1,2,3,10]:
    scale = gamma * jnp.sqrt(n + 1)
    dist = distribution.FiniteDifferential(loc=loc, scale=scale, n=n)
    plt.plot(xs, dist.pdf(xs), label=f"n={n}")
dist_normal = distribution.Normal(loc=loc, scale=gamma / jnp.sqrt(2))
plt.plot(xs, dist_normal.pdf(xs), label="Normal", linestyle="--")
plt.legend()

<matplotlib.legend.Legend at 0x7fbb1eff3610>
_images/distributions_15_1.png