[PYTHON] Versuchen Sie, mit matplotlib eine Normalverteilung zu zeichnen

Überblick

Ich möchte die Normalverteilung eher wie ein Bild erfassen, machen Sie sich also eine Notiz, während Sie mit matplotlib zeichnen.

Bibliothek verwendet

Python 3.8.3
matplotlib 3.3.2
numpy 1.19.0
seaborn 0.10.1
tensorflow 2.2.0
tensorflow-probability 0.10.1

Ich möchte auch die Tensorflow-Wahrscheinlichkeit ausprobieren, also werde ich sie dieses Mal verwenden.

Umgebungseinstellung

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import tensorflow as tf
import tensorflow_probability as tfp
from matplotlib.gridspec import GridSpec
from mpl_toolkits.axes_grid1 import make_axes_locatable
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d.art3d import Poly3DCollection

tfd, tfb = tfp.distributions, tfp.bijectors

rc = {
    'font.family': ['sans-serif'],
    'font.sans-serif': ['Open Sans', 'Arial Unicode MS'],
    'font.size': 12,
    'figure.figsize': (8, 6),
    'grid.linewidth': 0.5,
    'legend.fontsize': 10,
    'legend.frameon': True,
    'legend.framealpha': 0.6,
    'legend.handletextpad': 0.2,
    'lines.linewidth': 1,
    'axes.facecolor': '#fafafa',
    'axes.labelsize': 11,
    'axes.titlesize': 14,
    'axes.linewidth': 0.5,
    'xtick.labelsize': 11,
    'xtick.major.width': 0.5,
    'ytick.labelsize': 11,
    'ytick.major.width': 0.5,
    'figure.titlesize': 14,
}
sns.set('notebook', 'whitegrid', rc=rc)

Eindimensionale Normalverteilung

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

fig, ax = plt.subplots(figsize=(8, 5))

x = np.linspace(-5, 5, 500)
for loc, std in [[-2,1.0],[2,1.2],[0,0.8],[-1,0.5]]:
    normal = tfd.Normal(loc, std)
    label = '$\mathcal{{N}}({:.1f},{:.1f})$'.format(loc, std)
    ax.plot(x, normal.prob(x), label=label)

ax.set(xlabel='$x$',ylabel='$p(x)$',xlim=(-5,5))
ax.legend()

plt.tight_layout()
plt.show()

Multivariate Normalverteilung

p(\boldsymbol{x}; \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \frac{1}{\sqrt{(2\pi)^n|\boldsymbol{\Sigma}|}} \exp\left( -\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right),

nd1 = tfd.MultivariateNormalTriL(loc=[0., 0.], scale_tril=tf.linalg.cholesky([[1.,0.],[0., 1.]]))
nd2 = tfd.MultivariateNormalTriL(loc=[0., 1.], scale_tril=tf.linalg.cholesky([[1.,-0.5],[-0.5, 1.5]]))

X1, X2 = np.meshgrid(np.linspace(-3,3,100), np.linspace(-3,3,100))
X = np.dstack((X1, X2))
data = [["Independent", nd1.prob(X)], ["Correlated", nd2.prob(X)]]

fig = plt.figure(figsize=(14, 6))

for i, (title, y) in enumerate(data):
    ax = fig.add_subplot(1, 2, i + 1, projection='3d')
    ax.plot_surface(X1, X2, y, rstride=2, cstride=2, linewidth=0.1, antialiased=True, cmap='cividis')
    ax.contourf(X1, X2, y, 15, zdir='z', offset=-0.15, cmap='cividis')
    ax.view_init(30, -21)
    ax.tick_params(color='0.8')
    ax.set(title=title,
           zlim=(-0.15, 0.2),
           xlabel='$x_1$',ylabel="$x_2$",zlabel="$p(x_1,x_2)$",
           fc='none')
    
fig.suptitle("Bivariate Normal Distributions")

plt.tight_layout()
plt.show()

Probenahme

nd = tfd.MultivariateNormalTriL(loc=[0., 0.], scale_tril=tf.linalg.cholesky([[1.0,-0.2],[-0.8,1.]]))

X1, X2 = np.meshgrid(np.linspace(-3,3,100), np.linspace(-3,3,100))
Z = nd.prob(np.dstack((X1, X2)))
Y = nd.sample(500)

fig, ax = plt.subplots(figsize=(8, 6))

ax.plot(Y[:,0], Y[:,1], 'ro',alpha=.7, markeredgecolor='k', markeredgewidth=0.5)

con = ax.contourf(X1, X2, Z, 100, cmap='YlGnBu')
ax.set(xlabel='$x_1$',ylabel='$x_2$', axisbelow=False,
       xlim=(-3,3),ylim=(-3,3),
       title='Samples From Bivariate Normal')

cbar = fig.colorbar(con)
cbar.ax.set_ylabel('$p(x_1, x_2)$')

plt.tight_layout()
plt.show()

Periphere Verteilung


\begin{bmatrix}
\mathbf{x} \\
\mathbf{y} 
\end{bmatrix}
\sim
\mathcal{N}\left(
\begin{bmatrix}
\mu_{\mathbf{x}} \\
\mu_{\mathbf{y}}
\end{bmatrix},
\begin{bmatrix}
A & C \\
C^\top & B
\end{bmatrix}
\right)
= \mathcal{N}(\mu, \Sigma)
, \qquad 
\Sigma^{-1} = \Lambda = 
\begin{bmatrix}
\tilde{A} & \tilde{C} \\
\tilde{C}^\top & \tilde{B}
\end{bmatrix} \\

p(\mathbf{x}) = \mathcal{N}(\mu_{\mathbf{x}}, A) \\

p(\mathbf{y}) = \mathcal{N}(\mu_{\mathbf{y}}, B)

Daten vorbereiten

mean = np.array([0., 0.])
cov = np.array([[1.0,0.7],[0.7,1.]])
nd = tfd.MultivariateNormalTriL(loc=mean, scale_tril=tf.linalg.cholesky(cov))
nx, ny = tfd.Normal(mean[0], cov[1, 1]), tfd.Normal(mean[1], cov[0, 0])

x, y = np.linspace(-3,3,100), np.linspace(-3,3,100)
X1, X2 = np.meshgrid(x, y)
Z = nd.prob(np.dstack((X1, X2)))

Zeichnung

fig = plt.figure(figsize=(8, 8))

gs = GridSpec(2, 2, width_ratios=[8,2], height_ratios=[8,2])

ax = plt.subplot(gs[0])
con = ax.contourf(X1, X2, Z, 100, cmap='YlGnBu')
ax.set(axisbelow=False)
ax.autoscale(tight=True)

ax = plt.subplot(gs[1])
ax.plot(ny.prob(y), y, 'b--', label='$p(y)$')
ax.set(ylabel='$y$')
ax.yaxis.set_label_position('right')
ax.yaxis.set_ticks_position('right')
ax.tick_params(color="0.8")
ax.autoscale(tight=True)
ax.legend()

ax = plt.subplot(gs[2])
ax.plot(x, nx.prob(x), 'r--', label='$p(x_1)$')
ax.set(xlabel='$x$')
ax.autoscale(tight=True)
ax.legend()

ax = plt.subplot(gs[3])
ax.set_visible(False)
cax = make_axes_locatable(ax).append_axes('left', size='20%', pad=0)
cbar = fig.colorbar(con, cax=cax)
cbar.ax.set_ylabel('$p(x, xy)$')

fig.suptitle('Marginal Distributions')

plt.tight_layout(rect=[0, 0, 1, 0.97])
plt.show()

In drei Dimensionen

x1, y1, z1 = np.repeat(-3, 100), y, ny.prob(y).numpy()
x2, y2, z2 = x, np.repeat(3, 100), nx.prob(x).numpy()

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(projection='3d')

ax.plot_surface(X1, X2, Z, rstride=2, cstride=2, linewidth=0.1, antialiased=True, cmap='YlGnBu')
ax.plot(x1, y1, z1, 'b--',zorder=-1, label='$p(y)$')
ax.plot(x2, y2, z2, 'r--',zorder=-1, label='$p(x)$')
ax.legend()

ax.tick_params(color='0.8')
ax.set(xlabel='$x$',ylabel="$y$",zlabel="$p(x,y)$",fc='none')
    
fig.suptitle("Marginal Distributions")

plt.tight_layout()
plt.show()

Bedingte Verteilung

p(\mathbf{x} \mid \mathbf{y}) = \mathcal{N}(\mu_{x \mid y}, \Sigma_{x \mid y}) \\

p(\mathbf{y} \mid \mathbf{x}) = \mathcal{N}(\mu_{y \mid x}, \Sigma_{y \mid x}) \\

\begin{split}
\mu_{x \mid y} & = \mu_x + CB^{-1}(\mathbf{y}-\mu_y), \quad 
\Sigma_{x \mid y} & = A - CB^{-1}C^\top = \tilde{A}^{-1}
\end{split} \\

\begin{split}
\mu_{y \mid x} & = \mu_y + C^\top A^{-1}(\mathbf{x}-\mu_x), \quad 
\Sigma_{y \mid x} & = B - C^\top A^{-1}C = \tilde{B}^{-1}
\end{split}

Daten vorbereiten

A, B, C = cov[0, 0], cov[1, 1], cov[0, 1]

x_cond, y_cond = -1, 1

mean_xy = mean[0] + (C * (1/B) * (y_cond - mean[1]))
cov_xy = A - C * (1/B) * C

mean_yx = mean[1] + (C * (1/A) * (x_cond - mean[0]))
cov_yx = B - (C * (1/A) * C)

nyx, nxy = tfd.Normal(mean_yx, cov_yx), tfd.Normal(mean_xy, cov_xy)

zeichnen

fig = plt.figure(figsize=(8, 8))

gs = GridSpec(2, 2, width_ratios=[8,2], height_ratios=[8,2])

ax = plt.subplot(gs[0])
con = ax.contourf(X1, X2, Z, 100, cmap='YlGnBu')
ax.plot([x_cond,x_cond], [-3,3], 'b--', lw=1.5, label='$p(y \mid x={:.1f})$'.format(x_cond))
ax.plot([-3,3], [y_cond,y_cond], 'r--', lw=1.5, label='$p(x \mid y={:.1f})$'.format(y_cond))
ax.set(axisbelow=False)
ax.legend(loc="lower right")
ax.autoscale(tight=True)

ax = plt.subplot(gs[1])
ax.plot(nyx.prob(y), y, 'b--')
ax.set(ylabel='$y$')
ax.yaxis.set_label_position('right')
ax.yaxis.set_ticks_position('right')
ax.autoscale(tight=True)

ax = plt.subplot(gs[2])
ax.plot(x, nxy.prob(x), 'r--')
ax.set(xlabel='$x$')
ax.autoscale(tight=True)

ax = plt.subplot(gs[3])
ax.set_visible(False)
cax = make_axes_locatable(ax).append_axes('left', size='20%',pad=0)
cbar = fig.colorbar(con, cax=cax)
cbar.ax.set_ylabel('$p(x, xy)$')

fig.suptitle('Conditional Distributions')

plt.tight_layout(rect=[0, 0, 1, 0.97])
plt.show()

Dreidimensionale Menschen

x1, y1, z1 = x, np.repeat(y_cond,100), nyx.prob(y).numpy()
x2, y2, z2 = np.repeat(x_cond,100), y, nxy.prob(x).numpy()

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(projection='3d')

ax.plot(x1, y1, z1, 'r--', lw=2, label='$p(y \mid y={:.1f})$'.format(y_cond),zorder=2)
ax.plot(x2, y2, z2, 'b--', lw=2, label='$p(x \mid x={:.1f})$'.format(x_cond),zorder=2)
ax.add_collection3d(Poly3DCollection([list(zip(x1, y1, z1))],color='r',alpha=0.4))
ax.add_collection3d(Poly3DCollection([list(zip(x2, y2, z2))],color='b',alpha=0.4))
ax.contourf(X1, X2, Z, 15, zdir='z', offset=0, cmap='YlGnBu')

ax.tick_params(color='0.8')
ax.set(xlabel='$x$',ylabel="$y$",zlabel="$p(x,y)$",fc='none')
ax.legend()
    
fig.suptitle("Conditional Distributions")

plt.tight_layout()
plt.show()

Zusammenfassung

Ich habe auch den gesamten Code in Colab eingefügt, siehe hier, wenn Sie interessiert sind

https://colab.research.google.com/github/stwind/notebooks/blob/master/normal-dist.ipynb