通过 SVD 对卵巢癌数据进行 PCA答案

【问题标题】：PCA of Ovarian Cancer Data via SVD通过 SVD 对卵巢癌数据进行 PCA
【发布时间】：2021-01-12 18:45:35
【问题描述】：

我想用 PCA 分析 MATLAB 提供的卵巢癌数据。具体来说，我想可视化两个最大的主成分，并绘制两个对应的左奇异向量。据我了解，这些向量应该能够作为一个新的坐标系，与数据中的最大方差对齐。我最终要检查的是癌症患者是否与非癌症患者区分开来。

我的脚本中仍然存在错误的是左奇异向量。它们彼此之间不是 90 度角，如果我按各自的特征值缩放它们，它们的长度就会爆炸。我做错了什么？

%% PCA - Ovarian Cancer Data
close all;
clear all;

% obs is an NxM matrix, where ...
%   N = patients (216)
%   M = features - genes in this case (4000)
load ovariancancer.mat;

% Turn obs matrix, such that the rows represent the features
X = obs.';

[U, S, V] = svd(X, 'econ');

% Crop U, S and V, to visualize two largest principal components
U_crop = U(:, 1:2);
S_crop = S(1:2, 1:2);
V_crop = V(:, 1:2);

X_crop = U_crop * S_crop * V_crop.';

% Average over cancer patients
xC = mean(X_crop, 2); 

% Visualize two largest principal components as a data cloud
figure;
hold on;
for i = 1 : size(X, 2)
    if grp{i} == 'Cancer'
        plot(X_crop(1, i), X_crop(2, i), 'rx', 'LineWidth', 2);
    else
        plot(X_crop(1, i), X_crop(2, i), 'bo', 'LineWidth', 2);
    end
end
%scatter(X_crop(1, :), X_crop(2, :), 'k.', 'LineWidth', 2)
set(gca,'DataAspectRatio',[1 1 1])
xlabel('PC1')
ylabel('PC2')
grid on;

Xstd = U_crop; % * S_crop?
quiver([xC(1) xC(1)], [xC(2) xC(2)], Xstd(1, :), Xstd(2, :), 'green', 'LineWidth', 3);

【问题讨论】：

标签： matlab pca svd

【解决方案1】：

所以我的脚本中有多个错误。如果有人感兴趣，我将发布更正的代码（我现在正在绘制三台 PC）。 This 的帖子很有帮助。

% obs is an NxM matrix, where ...
%   N = patients (216)
%   M = features - genes in this case (4000)
load ovariancancer.mat;

% Let the data matrix X be of n×p size, where n is the number of samples and p is the number of variables
X = obs;

% Let us assume that it is centered, i.e. column means have been subtracted and are now equal to zero
Xavg = mean(X, 2);
%X = X - Xavg * ones(1, size(X, 2));

[U, S, V] = svd(X, 'econ');
PC = U * S;

% Visualize three largest principal components as a data cloud
% The j-th principal component is given by j-th column of XV. The coordinates of the i-th data point in the new PC space are given by the i-th row of XV
figure;
for i = 1 : size(PC, 2)
    if grp{i} == 'Cancer'
        plot3(PC(i, 1), PC(i, 2), PC(i, 3), 'rx', 'LineWidth', 2);
    else
        plot3(PC(i, 1), PC(i, 2), PC(i, 3), 'bo', 'LineWidth', 2);
    end
    hold on;
end
set(gca,'DataAspectRatio',[1 1 1])
xlabel('PC1')
ylabel('PC2')
zlabel('PC3')

【讨论】：