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Bk2_Ch16_三维等高线_07可视化三元高斯分布

《可视之美》

python

鸾尾花书

数据可视化

《可视之美》python鸾尾花书数据可视化

刀刀

发布于 2024-05-28

推荐镜像 :Basic Image:bohrium-notebook:2023-04-07

推荐机型 :c2_m4_cpu

可视化三元高斯分布

1. 自定义高斯分布概率密度函数

2. 生成数据

3. 绘制概率密度箱体外立面

4. 分层绘制等高线，沿x3

5. 将等高线展开，沿x3

6. 将等高线展开，沿x2

7. 将等高线展开，沿x1

©️ Copyright 2024 @ Authors
共享协议：本作品采用知识共享署名-非商业性使用-相同方式共享 4.0 国际许可协议进行许可。
作者信息：Notebook原地址为 https://github.com/Visualize-ML

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Chapter 16

可视化三元高斯分布

Book_2《可视之美》 | 鸢尾花书：从加减乘除到机器学习

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[1]

# 导入包

import matplotlib.pyplot as plt

import numpy as np

import os

# 如果文件夹不存在，创建文件夹

if not os.path.isdir("Figures"):

os.makedirs("Figures")

# 大家可以自己试着把下面可视化代码写成自定义函数

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1. 自定义高斯分布概率密度函数

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[2]

def Mahal_d_2_pdf(d,Sigma):

# 将马氏距离转化为概率密度

scale_1 = np.sqrt(np.linalg.det(Sigma))

scale_2 = (2*np.pi)**(3/2)

pdf = np.exp(-d**2/2)/scale_1/scale_2

return pdf

def Mahal_d(Mu, Sigma, x):

# 计算马哈距离

x_demeaned = x - Mu

# 中心化

inv_covmat = np.linalg.inv(Sigma)

# 矩阵逆

left = np.dot(x_demeaned, inv_covmat)

mahal = np.dot(left, x_demeaned.T)

return np.sqrt(mahal).diagonal()

# 也可以用这个函数：

# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.multivariate_normal.html

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2. 生成数据

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[3]

x1 = np.linspace(-2,2,31)

x2 = np.linspace(-2,2,31)

x3 = np.linspace(-2,2,31)

xxx1,xxx2,xxx3 = np.meshgrid(x1,x2,x3)

Mu = np.array([[0, 0, 0]])

Sigma = np.array([[1, 0.6, -0.4],

[0.6, 1.5, 1],

[-0.4, 1, 2]])

x_array = np.vstack([xxx1.ravel(),xxx2.ravel(), xxx3.ravel()]).T

# 首先计算马氏距离

d_array = Mahal_d(Mu, Sigma, x_array)

d_array = d_array.reshape(xxx1.shape)

# 将马氏距离转化成概率密度PDF

pdf_zz = Mahal_d_2_pdf(d_array, Sigma)

C:\Users\james\AppData\Local\Temp\ipykernel_7748\642966078.py:19: RuntimeWarning: invalid value encountered in sqrt

  return np.sqrt(mahal).diagonal()

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[5]

# 设定统一等高线分层

# levels = np.linspace(0,pdf_zz.max(),20)

levels_PDF = np.linspace(0,0.1,21)

levels_Mahal_D = np.linspace(0,10,21)

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3. 绘制概率密度箱体外立面

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[11]

# 定义等高线高度

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

ax = fig.add_subplot(111, projection='3d')

# 绘制三维等高线，填充

ax.contourf(xxx1[:, :, -1],

xxx2[:, :, -1],

pdf_zz[:, :, -1],

levels = levels_PDF,

zdir='z', offset=xxx3.max(),

cmap = 'turbo') # RdYlBu_r

ax.contour(xxx1[:, :, -1],

xxx2[:, :, -1],

pdf_zz[:, :, -1],

levels = levels_PDF,

zdir='z', offset=xxx3.max(),

linewidths = 0.25,

colors = '1')

ax.contourf(xxx1[0, :, :],

pdf_zz[0, :, :],

xxx3[0, :, :],

levels = levels_PDF,

zdir='y',

cmap = 'turbo',

offset=xxx2.min())

ax.contour(xxx1[0, :, :],

pdf_zz[0, :, :],

xxx3[0, :, :],

levels = levels_PDF,

zdir='y',

colors = '1',

linewidths = 0.25,

offset=xxx2.min())

CS = ax.contourf(pdf_zz[:, 0, :],

xxx2[:, 0, :],

xxx3[:, 0, :],

levels = levels_PDF,

cmap = 'turbo',

zdir='x',

offset=xxx1.min())

ax.contour(pdf_zz[:, 0, :],

xxx2[:, 0, :],

xxx3[:, 0, :],

levels = levels_PDF,

zdir='x',

colors = '1',

linewidths = 0.25,

offset=xxx1.min())

fig.colorbar(CS, ticks=np.linspace(0,0.1,6))

# Set limits of the plot from coord limits

xmin, xmax = xxx1.min(), xxx1.max()

ymin, ymax = xxx2.min(), xxx2.max()

zmin, zmax = xxx3.min(), xxx3.max()

ax.set(xlim=[xmin, xmax], ylim=[ymin, ymax], zlim=[zmin, zmax])

# 绘制框线

edges_kw = dict(color='0.6', linewidth=1, zorder=1e5)

# zorder 控制呈现 artist 的先后顺序

# zorder 越小，artist 置于越底层

# zorder 赋值很大的数，这样确保 zorder 置于最顶层

ax.plot([xmin, xmax], [ymin, ymin], [zmax, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymax], [zmax, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymin], [zmin, zmax], **edges_kw)

ax.set_xticks([-2,0,2])

ax.set_yticks([-2,0,2])

ax.set_zticks([-2,0,2])

ax.view_init(azim=-125, elev=30)

ax.set_proj_type('ortho')

ax.set_xlabel('$x_1$')

ax.set_ylabel('$x_2$')

ax.set_zlabel('$x_3$')

ax.set_box_aspect((1, 1, 1))

ax.grid(False)

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4. 分层绘制等高线，沿x3

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[130]

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

ax = fig.add_subplot(111, projection='3d')

for idx in np.arange(0,len(x3),5):

x3_idx = x3[idx]

ax.contourf(xxx1[:, :, idx],

xxx2[:, :, idx],

pdf_zz[:, :, idx],

levels = levels_PDF,

zdir='z',

offset=x3_idx,

cmap = 'turbo')

ax.plot([xmin, xmin], [ymin, ymax], [x3_idx, x3_idx], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [x3_idx, x3_idx], **edges_kw)

ax.set(xlim=[xmin, xmax], ylim=[ymin, ymax], zlim=[zmin, zmax])

ax.plot([xmin, xmin], [ymin, ymin], [zmin, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymax], [zmax, zmax], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [zmax, zmax], **edges_kw)

ax.set_xticks([-2,0,2])

ax.set_yticks([-2,0,2])

ax.set_zticks([-2,0,2])

ax.view_init(azim=-125, elev=30)

ax.set_proj_type('ortho')

ax.set_xlabel('$x_1$')

ax.set_ylabel('$x_2$')

ax.set_zlabel('$x_3$')

ax.set_box_aspect((1, 1, 1))

ax.grid(False)

fig.savefig('Figures/箱体分层x3_概率密度.svg', format='svg')

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5. 将等高线展开，沿x3

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[131]

fig = plt.figure(figsize=(6, 36))

for fig_idx,idx in enumerate(np.arange(0,len(x3),5)):

ax = fig.add_subplot(len(np.arange(0,len(x3),5)), 1, fig_idx + 1, projection='3d')

x3_idx = x3[idx]

ax.contourf(xxx1[:, :, idx],

xxx2[:, :, idx],

pdf_zz[:, :, idx],

levels = levels_PDF,

zdir='z',

offset=x3_idx,

cmap = 'turbo')

ax.contour(xxx1[:, :, idx],

xxx2[:, :, idx],

pdf_zz[:, :, idx],

levels = levels_PDF,

zdir='z',

offset=x3_idx,

linewidths = 0.25,

colors = '1')

ax.plot([xmin, xmin], [ymin, ymax], [x3_idx, x3_idx], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [x3_idx, x3_idx], **edges_kw)

ax.set(xlim=[xmin, xmax], ylim=[ymin, ymax], zlim=[zmin, zmax])

# 绘制框线

edges_kw = dict(color='0.5', linewidth=1, zorder=1e3)

ax.plot([xmin, xmin], [ymin, ymin], [zmin, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymax], [zmax, zmax], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [zmax, zmax], **edges_kw)

ax.view_init(azim=-125, elev=30)

ax.set_box_aspect(None)

ax.set_proj_type('ortho')

ax.set_xlabel('$x_1$')

ax.set_ylabel('$x_2$')

ax.set_zlabel('$x_3$')

ax.set_xticks([])

ax.set_yticks([])

ax.set_zticks([])

ax.set_box_aspect((1, 1, 1))

ax.grid(False)

fig.savefig('Figures/分层分图_x3_概率密度.svg', format='svg')

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6. 将等高线展开，沿x2

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[132]

fig = plt.figure(figsize=(6, 36))

for fig_idx,idx in enumerate(np.arange(0,len(x2),5)):

ax = fig.add_subplot(len(np.arange(0,len(x2),5)), 1, fig_idx + 1, projection='3d')

x2_idx = x2[idx]

ax.contourf(xxx1[idx, :, :],

pdf_zz[idx, :, :],

xxx3[idx, :, :],

levels = levels_PDF,

zdir='y',

offset=x2_idx,

cmap = 'turbo')

ax.contour(xxx1[idx, :, :],

pdf_zz[idx, :, :],

xxx3[idx, :, :],

levels = levels_PDF,

zdir='y',

offset=x2_idx,

linewidths = 0.25,

colors = '1')

ax.plot([xmin, xmin], [ymin, ymax], [x3_idx, x3_idx], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [x3_idx, x3_idx], **edges_kw)

ax.set(xlim=[xmin, xmax], ylim=[ymin, ymax], zlim=[zmin, zmax])

# Plot edges

edges_kw = dict(color='0.5', linewidth=1, zorder=1e3)

ax.plot([xmin, xmin], [ymin, ymin], [zmin, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymax], [zmax, zmax], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [zmax, zmax], **edges_kw)

# Set zoom and angle view

ax.view_init(azim=-125, elev=30)

ax.set_box_aspect(None)

ax.set_proj_type('ortho')

ax.set_xlabel('$x_1$')

ax.set_ylabel('$x_2$')

ax.set_zlabel('$x_3$')

ax.set_xticks([])

ax.set_yticks([])

ax.set_zticks([])

ax.set_box_aspect((1, 1, 1))

ax.grid(False)

fig.savefig('Figures/分层分图_x2_概率密度.svg', format='svg')

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7. 将等高线展开，沿x1

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[133]

fig = plt.figure(figsize=(6, 36))

for fig_idx,idx in enumerate(np.arange(0,len(x1),5)):

ax = fig.add_subplot(len(np.arange(0,len(x1),5)), 1, fig_idx + 1, projection='3d')

x1_idx = x1[idx]

ax.contourf(pdf_zz[:, idx, :],

xxx2[:, idx, :],

xxx3[:,idx, :],

levels = levels_PDF,

zdir='x',

offset=x1_idx,

cmap = 'turbo')

ax.contour(pdf_zz[:, idx, :],

xxx2[:, idx, :],

xxx3[:,idx, :],

levels = levels_PDF,

zdir='x',

offset=x1_idx,

linewidths = 0.25,

colors = '1')

ax.plot([xmin, xmin], [ymin, ymax], [x3_idx, x3_idx], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [x3_idx, x3_idx], **edges_kw)

ax.set(xlim=[xmin, xmax], ylim=[ymin, ymax], zlim=[zmin, zmax])

# 绘制框线

ax.plot([xmin, xmin], [ymin, ymin], [zmin, zmax], **edges_kw)

ax.plot([xmin, xmin], [ymin, ymax], [zmax, zmax], **edges_kw)

ax.plot([xmax, xmin], [ymin, ymin], [zmax, zmax], **edges_kw)

ax.view_init(azim=-125, elev=30)

ax.set_box_aspect(None)

ax.set_proj_type('ortho')

ax.set_xlabel('$x_1$')

ax.set_ylabel('$x_2$')

ax.set_zlabel('$x_3$')

ax.set_xticks([])

ax.set_yticks([])

ax.set_zticks([])

ax.set_box_aspect((1, 1, 1))

ax.grid(False)

fig.savefig('Figures/分层分图_x1_概率密度.svg', format='svg')

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[ ]

# Repo: https://github.com/Visualize-ML

# Book 2 Beauty of Visualization | From Basic Arithmetic to Machine Learning

# Published and copyrighted by Tsinghua University Press

# Beijing, China, 2023

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《可视之美》

python

鸾尾花书

数据可视化

《可视之美》python鸾尾花书数据可视化

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Book2《可视之美》 | 鸢尾花书：从加减乘除到机器学习

刀刀

更新于 2024-06-03

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