Distributions in 2D

Exploring bivariate probability distributions including 2D normal and uniform distributions, with practical examples using PyTorch and real-world height-weight data

Keywords

bivariate distributions, 2D normal, multivariate normal, uniform distribution, PyTorch, probability density function

import matplotlib.pyplot as plt
import numpy as np
print(np.__version__)
import torch 
import torch.nn as nn

import pandas as pd
# Retina mode
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

1.26.4

dist = torch.distributions.Normal(0, 1)
x = dist.sample((1000,))
plt.hist(x.numpy(), bins=50, density=True)

x_range = torch.linspace(-3, 3, 1000)
y = dist.log_prob(x_range).exp()
plt.plot(x_range.numpy(), y.numpy())

dist.sample([10])

tensor([-1.9083,  0.3758,  0.0051,  0.5140,  0.9852, -0.5989,  0.5222, -0.7744,
         0.9462, -1.7868])

dist_2d_normal = torch.distributions.MultivariateNormal(torch.tensor([0.0, 0.0]), torch.eye(2))
#dist_2d_normal = torch.distributions.MultivariateNormal(torch.tensor([0.0, 0.0]), torch.tensor([[1.0, 0.5], [0.5, 1.0]]))

dist_2d_normal.sample([10])

tensor([[ 0.0438, -0.0310],
        [ 0.0487, -0.3790],
        [-0.7872,  0.9880],
        [ 1.0010, -0.9025],
        [ 0.5449,  0.1047],
        [ 1.6466,  0.0925],
        [ 0.9357,  0.2228],
        [-1.2721,  2.5194],
        [-0.3306, -0.1152],
        [ 1.2249, -1.7330]])

plt.scatter(*dist_2d_normal.sample([1000]).numpy().T, alpha=0.2, color='k')

# Plot 2D normal distribution surface plot of PDF
from mpl_toolkits.mplot3d import Axes3D

x = torch.linspace(-3, 3, 100)
y = torch.linspace(-3, 3, 100)

X, Y = torch.meshgrid(x, y)
xy = torch.stack([X, Y], 2)
z = dist_2d_normal.log_prob(xy).exp()

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X.numpy(), Y.numpy(), z.numpy())

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('PDF')

fig.tight_layout()

df = pd.read_html("http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Data_Dinov_020108_HeightsWeights.html")

store_df = df[0]
store_df.columns = store_df.iloc[0]
store_df = store_df.iloc[1:]
store_df = store_df.astype(float)
store_df = store_df.drop(columns=["Index"])
store_df = store_df.dropna()

store_df.head()

	Height(Inches)	Weight(Pounds)
1	65.78331	112.9925
2	71.51521	136.4873
3	69.39874	153.0269
4	68.21660	142.3354
5	67.78781	144.2971

### Fiting a bi-variate normal distribution to the data
data = torch.tensor(store_df.values)
mean = data.mean(0)
cov = torch.cov(data.T)
dist = torch.distributions.MultivariateNormal(mean, cov)

dist.loc

tensor([ 67.9931, 127.0794], dtype=torch.float64)

dist.covariance_matrix

tensor([[  3.6164,  11.1510],
        [ 11.1510, 135.9765]], dtype=torch.float64)

# Plot the data

plt.scatter(data[:, 0], data[:, 1], alpha=0.1, color='k', facecolors='k')
plt.xlabel("Height")
plt.ylabel("Weight")

Text(0, 0.5, 'Weight')

# plot the PDF
x = torch.linspace(50, 80, 100)
y = torch.linspace(80, 280, 100)
X, Y = torch.meshgrid(x, y)
xy = torch.stack([X, Y], 2)
z = dist.log_prob(xy).exp()

import plotly.graph_objects as go

# Create surface plot with custom hover labels
fig = go.Figure(data=[go.Surface(
    x=X, y=Y, z=z, colorscale="viridis",
    hovertemplate="Height: %{x:0.2f}<br>Weight: %{y:0.2f}<br>PDF: %{z:0.5f}<extra></extra>"
)])

# Maximize figure size and reduce whitespace
fig.update_layout(
    autosize=True,
    width=1200,  # Set wider figure
    height=700,  # Set taller figure
    margin=dict(l=0, r=0, t=40, b=0),  # Remove extra whitespace
    title="2D Gaussian PDF",
    scene=dict(
        xaxis_title="Height",
        yaxis_title="Weight",
        zaxis_title="PDF"
    )
)

# Show plot
fig.show()

# uniform distribution
dist_uniform = torch.distributions.Uniform(0, 1)
x = dist_uniform.sample((1000,))
plt.hist(x.numpy(), bins=50, density=True)

x_range = torch.linspace(0, 1, 1000)
y = dist_uniform.log_prob(x_range).exp()
plt.plot(x_range.numpy(), y.numpy())

dist_uniform_2d = torch.distributions.Uniform(torch.tensor([0.0, 0.0]), torch.tensor([1.0, 1.0]))
dist_uniform_2d.sample([10])

tensor([[0.5493, 0.3478],
        [0.7661, 0.2568],
        [0.7199, 0.2975],
        [0.9114, 0.2916],
        [0.0045, 0.4948],
        [0.0156, 0.7434],
        [0.6856, 0.1037],
        [0.4446, 0.1913],
        [0.1995, 0.5009],
        [0.0716, 0.6085]])

plt.scatter(*dist_uniform_2d.sample([1000]).numpy().T, alpha=0.5)

plt.scatter(*dist_uniform_2d.sample([10000]).numpy().T, alpha=0.1)

# surface plot of PDF

x = torch.linspace(0.0, 1.0, 100)
y = torch.linspace(0.0, 1.0, 100)

X, Y = torch.meshgrid(x, y)
xy = torch.stack([X, Y], 2)

## Important:
## f(x, y) = f(x) * f(y) for independent random variables
## log(f(x, y)) = log(f(x)) + log(f(y))
z1 = dist_uniform_2d.log_prob(xy).sum(-1).exp()
z2 = dist_uniform.log_prob(X).exp() * dist_uniform.log_prob(Y).exp()
assert torch.allclose(z1, z2) 


fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X.numpy(), Y.numpy(), z1.numpy())
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('PDF')

Text(0.5, 0, 'PDF')