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

给定两个变量(X，Y)，如何在香草MATLAB中(无工具箱)生成和绘制经验联合PDF和CDF?

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)?

推荐答案

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X，Y的向量.
• bin在x，y轴上的边缘:由向量x_axis，y_axis定义.边缘显然必须增加，但不必均匀间隔.

• 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.

生成的PDF和CDF定义在由x和y边缘确定的矩形的中心.

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

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

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'

Matlab R2015b包括一个 histogram2 函数工作.它会自动进行规范化以获取PDF(具有适当的输入标志)，甚至是CDF.

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.

使用与上述相同的示例，

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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