PODusingPCA.md

POD of a scalar field using Matlabs's PCA function

clear; clc; close all;

Sample Data based on Fernando Zigunov's example

[Vfield_Snapshots, xdata, ydata, t] = GeneratePistonInDuctVelocityField;

Using PCA

The PCA method accepts data, X as an n-by-p matrix of n observations and p variables.

The snapshots above are 3-dimensional with the first dimension as time the first dimension.

Reshape the data so that there are n observations in time, and each pixel is a variable

By default, PCA centers the data

PCA returns the coefficients (space), scores (evolution), explained variance (energy)

X = reshape(Vfield_Snapshots,size(Vfield_Snapshots,1),[]);
[p,s,~,~,e]=pca(X);

PCA Relative explained variance (analogous to relative kinetic energy)

figure;
pareto(e/sum(e))
title('Explained Variance')

Principal component scores (analogous to POD time coefficients)

figure(Position=[969 102 811 357]);
hold on;
for moi = 1:3
    plot(t,s(:,moi),DisplayName="POM " + moi)
end
title('PCA scores')
hold off
legend('show')

Principal component coefficients (analogous to POD modes)

figure;
tl=tiledlayout('flow',TileSpacing="compact",Padding="compact");
for moi = 1:3
    nexttile;
    imagesc(xdata,ydata,reshape(p(:,moi), ...
        numel(ydata),numel(xdata)));
    colormap jet
    xlabel('x/L');
    ylabel('y/H');
    set(gca,'ydir','normal')
    title("POM " + moi);
end
title(tl,"PCA coefficients")