生成并绘制经验联合 pdf 和 CDF [英] Generate and plot the empirical joint pdf and CDF

问题描述

给定一对两个变量 (X,Y)，如何在 vanilla MATLAB(无工具箱)中生成和绘制经验联合 PDF 和 CDF?

解决方案

原始答案(Matlab R2015a 或更低版本)

数据为:

• 随机变量X、Y:定义为样本向量X、Y.
• x、y 轴上的 bin 边缘:由向量 x_axis、y_axis 定义.边缘显然必须增加，但不必均匀间隔.

生成的 PDF 和 CDF 定义在由 x 和 y 边确定的矩形的中心.

要以 3D 形式绘制结果，请使用 surf(...) 而不是 imagesc(...).

清除所有%//数据(示例):X = randn(1,1e5);%//随机变量.Y = randn(1,1e5);x_axis = -3:.2:3;%//为 x 轴定义 bin 的边缘.列向量y_axis = -3:.2:3;%//y 轴相同%//计算 2D-bins 的角点:[x_mesh_upper,y_mesh_upper] = meshgrid(x_axis(2:end),y_axis(2:end));[x_mesh_lower,y_mesh_lower] = meshgrid(x_axis(1:end-1),y_axis(1:end-1));%//1D-bins的计算中心:x_centers = (x_axis(2:end)+x_axis(1:end-1))/2;y_centers = (y_axis(2:end)+y_axis(1:end-1))/2;%//计算pdf:pdf = mean( bsxfun(@le, X(:), x_mesh_upper(:).') ...&bsxfun(@gt, X(:), x_mesh_lower(:).') ...&bsxfun(@le, Y(:), y_mesh_upper(:).') ...&bsxfun(@gt, Y(:), y_mesh_lower(:).'));pdf = reshape(pdf,length(x_axis)-1,length(y_axis)-1);%//pdf 值在%//x_centers, y_centers 定义的网格点pdf = pdf ./(y_mesh_upper-y_mesh_lower) ./(x_mesh_upper-x_mesh_lower);%//将 pdf 归一化为单位积分%//计算 cdf:cdf = mean( bsxfun(@le, X(:), x_mesh_upper(:).') ...&bsxfun(@le, Y(:), y_mesh_upper(:).'));cdf = reshape(cdf,length(x_axis)-1,length(y_axis)-1);%//绘制pdf数字imagesc(x_centers,y_centers,pdf)xy轴轴相等颜色条标题'pdf'%//绘制 cdf数字图像c(x_centers，y_centers，cdf)xy轴轴相等颜色条标题'cdf'

编辑答案(Matlab R2015b 或更高版本)

Matlab R2015b 包括一个

Given a pair of two variables (X,Y), how can you generate and plot the empirical joint PDF and CDF in vanilla MATLAB (no toolboxes)?

解决方案

Original answer (Matlab R2015a or lower)

The data are:

• The random variables X, Y: defined as vectors of samples X, Y.
• The bin edges at the x, y axes: defined by vectors x_axis, y_axis. The edges must obviously be increasing, but need not be uniformly spaced.

The resulting PDF and CDF are defined at the centers of the rectangles determined by the x and y edges.

To plot the results in 3D, use surf(...) instead of imagesc(...).

clear all

%// Data (example):
X = randn(1,1e5); %// random variables.
Y = randn(1,1e5);

x_axis = -3:.2:3; %// Define edges of bins for x axis. Column vector
y_axis = -3:.2:3; %// Same for y axis

%// Compute corners of 2D-bins:
[x_mesh_upper,y_mesh_upper] = meshgrid(x_axis(2:end),y_axis(2:end));
[x_mesh_lower,y_mesh_lower] = meshgrid(x_axis(1:end-1),y_axis(1:end-1));

%// Compute centers of 1D-bins:
x_centers = (x_axis(2:end)+x_axis(1:end-1))/2;
y_centers = (y_axis(2:end)+y_axis(1:end-1))/2;

%// Compute pdf:
pdf = mean( bsxfun(@le, X(:), x_mesh_upper(:).') ...
    & bsxfun(@gt, X(:), x_mesh_lower(:).') ...
    & bsxfun(@le, Y(:), y_mesh_upper(:).') ...
    & bsxfun(@gt, Y(:), y_mesh_lower(:).') );
pdf = reshape(pdf,length(x_axis)-1,length(y_axis)-1); %// pdf values at the
%// grid points defined by x_centers, y_centers
pdf = pdf ./ (y_mesh_upper-y_mesh_lower) ./ (x_mesh_upper-x_mesh_lower);
%// normalize pdf to unit integral

%// Compute cdf:
cdf = mean( bsxfun(@le, X(:), x_mesh_upper(:).') ...
    & bsxfun(@le, Y(:), y_mesh_upper(:).') );
cdf = reshape(cdf,length(x_axis)-1,length(y_axis)-1);

%// Plot pdf
figure
imagesc(x_centers,y_centers,pdf)
axis xy
axis equal
colorbar
title 'pdf'

%// Plot cdf
figure
imagesc(x_centers,y_centers,cdf)
axis xy
axis equal
colorbar
title 'cdf'

Edited answer (Matlab R2015b or higher)

Matlab R2015b includes an histogram2 function that does all the work. It automatically does the normalization to obtain the PDF (given the appropriate input flag), or even the CDF.

Using the same example as above,

clear all

%// Data (example):
X = randn(1,1e5); % random variables.
Y = randn(1,1e5);

x_axis = -3:.2:3; % Define edges of bins for x axis. Column vector
y_axis = -3:.2:3; % Same for y axis

%// Compute and plot pdf
figure
histogram2(X, Y, x_axis, y_axis, 'Normalization', 'pdf')

%// Compute and plot cdf
figure
histogram2(X, Y, x_axis, y_axis, 'Normalization', 'cdf')

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