Gaussian Distributions

The Gaussian distribution is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). These determine the centre and spread of the distribution, respectively. The probability density function (PDF) of the Gaussian distribution describes the likelihood of observing a value x given the distribution's parameters.

Standard Normal Probability Density Function (PDF)

In one dimension, the standard normal probability density function $f(x)$ describes the probability density of observing a value $x$ from a standard normal distribution. It is defined as:

$f(x\;|\;\mu,\;\sigma)=\frac{1}{\sigma\sqrt{2\pi}}exp[-\frac{x-\mu^{2}}{2\sigma^{2}}]$

where:

• x is the value at which to evaluate the probability density function.
• $\mu$ is the mean of the distribution.
• $\sigma$ is the standard deviation of the distribution.

Function arguments

This script requires the following function arguments:

• make_gauss: Plot multiple Gaussian curves by defining a Gaussian function factory.
• N (amplitudo): Controls the overall height of the Gaussian.
• sig (standard deviation, $\sigma$): Determines the spread of the Gaussian.
• mu (mean, $\mu$): Sets the center of the Gaussian.
• return lambda: creates a function of x that computes the Gaussian formula.

This Python script demonstrates the Gaussian distribution function, also known as the normal distribution.

import numpy as np
import matplotlib.pyplot as plt

def make_gauss(N, sig, mu):
    return lambda x: N/(sig * np.sqrt(2*np.pi)) * np.exp(-(x-mu)**2/(2 * sig**2))

def main():
    fig, ax = plt.subplots()
    x = np.arange (-5, 5, 0.01)

    variances = [0.2, 1, 5, 0.5]
    sigmas = np.sqrt(variances)
    mus = [0, 0, 0, -2]
    colors = ['b', 'r', 'y', 'g']

    for var, sig, mu, color in zip(variances, sigmas, mus, colors):
        gauss = make_gauss(1, sig, mu)(x)
        ax.plot(x, gauss, color, linewidth=2, label=f'$\\sigma^{2}={var}, \\mu={mu}$') 

    ax.set_xlim(-5, 5)
    ax.set_ylim(0, 1)
    ax.set_ylabel(r'$\phi_{\mu,\sigma^{2}}(x)$', fontsize=14)
    ax.set_xlabel(r'$x$', fontsize=14)
    ax.legend()
    ax.grid(True, linestyle='--', alpha=0.6)
    ax.set_title("Gaussian Functions with Different Variable")
    plt.savefig('Gaussian Functions with Different Variable.svg', bbox_inches='tight')
    plt.show()

main()

Two-dimensional Gaussian Function

In two dimensions, the circular Gaussian is the distribution function for uncorrelated variates $x$ and $y$ having a bivariate normal distribution and equal standard deviation $\sigma=\sigma_{x}=\sigma_{y}$.

$(x,\;y)=exp(-x^{2}-y^{2})$

$f(x,\;y)=A\;exp(-(\frac{(x-x_{0})^{2}}{2\sigma_{X}^{2}}\;+\;\frac{(y-y_{0})^{2}}{2\sigma_{Y}^{2}}))$

where:

• A is Amplitude
• $x_{0}$ and $y_{0}$ are the centre
• $\sigma_{x}$ and $\sigma_{y}$ are the X and Y spreads of the blob.

Function arguments

This script requires the following function arguments:

• X, Y: are the meshgrid coordinates.
• Z: is the function value at X, Y.
• fig.add_subplot(111): is to set up a 3D subplot with 1 row, 1 column, and the first subplot.
• projection='3d': is matplotlib to make a 3D axis.

This Python script demonstrates the Gaussian distribution function, also known as the normal distribution.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm

# Define grid
x = y = np.linspace(0, 1, 40)
X, Y = np.meshgrid(x, y)

# Gaussian parameters
x0 = y0 = 0.5
sig_X, sig_Y = 0.2, 0.4

# Gaussian function
f = np.exp(-(X-x0)**2/sig_X**2 - (Y-y0)**2/sig_Y**2)

# Create 3D plot
fig = plt.figure(figsize=(7, 5))
ax = fig.add_subplot(111, projection='3d')

# Plot surface 
surf = ax.plot_surface(X, Y, f, cmap=cm.twilight, edgecolor = 'none')

# Add label
ax.set_xlabel('X position [a.u.]')
ax.set_ylabel('Y position [a.u.]')
ax.set_title('2D Gaussian Surface', fontsize=14)
ax.grid(True, alpha=0.6)

# Add color bar
cbar= fig.colorbar(surf, shrink=0.5, aspect=10)
cbar.set_label('Amplitudo [a.u.]')
plt.savefig('2D Gaussian surface.svg', bbox_inches='tight')
plt.show()