Plot normal distribution in 3D

Question

I am trying to plot the comun distribution of two normal distributed variables.

The code below plots one normal distributed variable. What would the code be for plotting two normal distributed variables?

import matplotlib.pyplot as plt
import numpy as np
import matplotlib.mlab as mlab
import math

mu = 0
variance = 1
sigma = math.sqrt(variance)
x = np.linspace(-3, 3, 100)
plt.plot(x,mlab.normpdf(x, mu, sigma))

plt.show()

Solution 1

It sounds like what you're looking for is a Multivariate Normal Distribution. This is implemented in scipy as scipy.stats.multivariate_normal. It's important to remember that you are passing a covariance matrix to the function. So to keep things simple keep the off diagonal elements as zero:

[X variance ,     0    ]
[     0     ,Y Variance]

Here is an example using this function and generating a 3D plot of the resulting distribution. I add the colormap to make seeing the curves easier but feel free to remove it.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal
from mpl_toolkits.mplot3d import Axes3D

#Parameters to set
mu_x = 0
variance_x = 3

mu_y = 0
variance_y = 15

#Create grid and multivariate normal
x = np.linspace(-10,10,500)
y = np.linspace(-10,10,500)
X, Y = np.meshgrid(x,y)
pos = np.empty(X.shape + (2,))
pos[:, :, 0] = X; pos[:, :, 1] = Y
rv = multivariate_normal([mu_x, mu_y], [[variance_x, 0], [0, variance_y]])

#Make a 3D plot
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, rv.pdf(pos),cmap='viridis',linewidth=0)
ax.set_xlabel('X axis')
ax.set_ylabel('Y axis')
ax.set_zlabel('Z axis')
plt.show()

Giving you this plot:

Edit the method used below was deprecated in Matplotlib v2.2 and removed in v3.1

A simpler version is available through matplotlib.mlab.bivariate_normal It takes the following arguments so you don't need to worry about matrices matplotlib.mlab.bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0, mux=0.0, muy=0.0, sigmaxy=0.0) Here X, and Y are again the result of a meshgrid so using this to recreate the above plot:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.mlab import bivariate_normal
from mpl_toolkits.mplot3d import Axes3D

#Parameters to set
mu_x = 0
sigma_x = np.sqrt(3)

mu_y = 0
sigma_y = np.sqrt(15)

#Create grid and multivariate normal
x = np.linspace(-10,10,500)
y = np.linspace(-10,10,500)
X, Y = np.meshgrid(x,y)
Z = bivariate_normal(X,Y,sigma_x,sigma_y,mu_x,mu_y)

#Make a 3D plot
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z,cmap='viridis',linewidth=0)
ax.set_xlabel('X axis')
ax.set_ylabel('Y axis')
ax.set_zlabel('Z axis')
plt.show()

Giving:

Solution 2

The following adaption to @Ianhi's code above returns a contour plot version of the 3D plot above.

import matplotlib.pyplot as plt
from matplotlib import style
style.use('fivethirtyeight')
import numpy as np
from scipy.stats import multivariate_normal




#Parameters to set
mu_x = 0
variance_x = 3

mu_y = 0
variance_y = 15

x = np.linspace(-10,10,500)
y = np.linspace(-10,10,500)
X,Y = np.meshgrid(x,y)

pos = np.array([X.flatten(),Y.flatten()]).T



rv = multivariate_normal([mu_x, mu_y], [[variance_x, 0], [0, variance_y]])


fig = plt.figure(figsize=(10,10))
ax0 = fig.add_subplot(111)
ax0.contour(X, Y, rv.pdf(pos).reshape(500,500))

plt.show()

Solution 3

While the other answers are great, I wanted to achieve similar results while also illustrating the distribution with a scatter plot of the sample.

More details can be found here: Python 3d plot of multivariate gaussian distribution

The results looks like:

And is generated using the following code:

from mpl_toolkits.mplot3d import Axes3D
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.stats import multivariate_normal


# Sample parameters
mu = np.array([0, 0])
sigma = np.array([[0.7, 0.2], [0.2, 0.3]])
rv = multivariate_normal(mu, sigma)
sample = rv.rvs(500)

# Bounds parameters
x_abs = 2.5
y_abs = 2.5
x_grid, y_grid = np.mgrid[-x_abs:x_abs:.02, -y_abs:y_abs:.02]

pos = np.empty(x_grid.shape + (2,))
pos[:, :, 0] = x_grid
pos[:, :, 1] = y_grid

levels = np.linspace(0, 1, 40)

fig = plt.figure()
ax = fig.gca(projection='3d')

# Removes the grey panes in 3d plots
ax.xaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.yaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.zaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))

# The heatmap
ax.contourf(x_grid, y_grid, 0.1 * rv.pdf(pos),
            zdir='z', levels=0.1 * levels, alpha=0.9)

# The wireframe
ax.plot_wireframe(x_grid, y_grid, rv.pdf(
    pos), rstride=10, cstride=10, color='k')

# The scatter. Note that the altitude is defined based on the pdf of the
# random variable
ax.scatter(sample[:, 0], sample[:, 1], 1.05 * rv.pdf(sample), c='k')

ax.legend()
ax.set_title("Gaussian sample and pdf")
ax.set_xlim3d(-x_abs, x_abs)
ax.set_ylim3d(-y_abs, y_abs)
ax.set_zlim3d(0, 1)

plt.show()

Comments

[Vlad]

It should be from matplotlib.mlab import bivariate_normal.

[mgalardini]

It seems like bivariate_normal will eventually be removed though: MatplotlibDeprecationWarning: The bivariate_normal function was deprecated in Matplotlib 2.2 and will be removed in 3.1.

[vicemagui]

Let's say that the Gaussian function is my likelihood, but I want to multiply it by its associated prior (Pa). How can I include Pa in this code?