pca versus princomp diferences in signs of coeffs (loadings) for cities dataset

So with the latest install of MatLab (2019B), I no longer have access to the function princomp from the old statistics toolbox. I ran into a problem switching to the recommended function pca and tried to diagnose it using the cities dataset (example dataset from princomp documentation). The principal component loadings are different between princomp and pca. The magnitude of the loadings are the same but the signs are sometimes different: pc1 same, pc2 - pc4 opposite signs, pc5-8 same and pc9 opposite sign. The pareto variance explained by each component plots are the same. Has anyone else noticed this? My code follows:

--------------------

close all

clear

clc

% PCA analysis of MatLab "cities" data

load cities

% display categories

categories

%%

% display names

names(1:3,:)

names(end-2:end,:)

%%

% display ratings

ratings(1:3,:)

%%

figure(1)

boxplot(ratings,0,'+',0)

set(gca,'Yticklabel',categories)

%%

stdr = std(ratings); % standard deviation of each rating category

sr = ratings./stdr(ones(329,1),:); % "standardize" each rating

figure(3)

boxplot(sr,0,'+',0)

set(gca,'Yticklabel',categories)

figure(4)

set(gcf, 'paperposition', [0.5 0.5 10.5 7.5])

set(gcf, 'paperorientation', 'landscape')

% use pca

subplot(2,1,1)

[pcs, newdata variances, t2, explained] = pca(sr,'Algorithm','svd','centered','on'); % principal components analysis

%

p3 = pcs(:,1:3); %subset the first three PCs

I = p3'*p3 % shows that the PCs are orthogonal*p3

bar(pcs) % plot the coefficients/loadings for the first 3 PCs;

set(gca,'Xticklabel',categories)

legend('PC1','PC2','PC3','PC4','PC5','PC6','PC7','PC8','PC9')

title('pca')

%use princomp

subplot(2,1,2)

clear pcs neqwdata variances t2

[pcs, newdata, variances, t2] = princomp(sr); % principal components analysis

% pcs = principal component coefficients, aka "loadings"

%

p3 = pcs(:,1:3); %subset the first three PCs

I = p3'*p3 % shows that the PCs are orthogonal*p3

bar(pcs) % plot the coefficients/loadings for the first 3 PCs;

set(gca,'Xticklabel',categories)

legend('PC1','PC2','PC3','PC4','PC5','PC6','PC7','PC8','PC9')

title('princomp')

%%

% [pcs, newdata, variances, t2] = pca(sr); % principal components analysis

% newdata = principal component coefficients, aka "loadings"

%

figure(5)

percent_explained = 100.*variances/sum(variances);

pareto(percent_explained) % make a “scree plot”

xlabel('Principal Component')

ylabel('Percent Explained, %')

[Spencer Chen]

I have used both, but I haven't compared them. You should find the corresponding factors also negated, in which case, the output is alright.

Blessings,

Spencer

Answer 1

[John D'Errico]

The vectors generated are not unique to within a sign change. They can arbitrarily change sign, and nothing matters. (Not just one element can change, but the entire vector can have a sign flip.) This has been true since time began, well, at least since MATLAB began. Not a question of anyone "noticing". This is a known characteristic of these algorithms.

Answer 2

[Brian Heikes]

Spencer and John, Thank-you.