# Can I add 95% confidence ellipses around groups of data in a pca plot (biplot)?

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Heidi Hirsh on 1 Oct 2020
Commented: Adam Danz on 4 Oct 2020
I am plotting the results of principle component analysis using biplot. I am wondering if there is a way (in matlab) to add confidence ellipses around the groups of data. Maybe I have to use something instead of biplot?
In the example below I would want to draw an ellipse around the data for each site (green and blue points)
Here's how I am currently plotting the data:
[coeff,score,latent,tsquared,explained] = pca(X);
f1=figure(1)
h= biplot(coeff(:,1:2),'scores',score(:,1:2),'color','k','marker','.','markersize',17,'varlabels',...
{'DO','O2sat','pH','Temp','Sal','Depth','PAR','|Velocity|'});
%color by site
hID = get(h,'tag'); %identify handle
hPt = h(strcmp(hID,'obsmarker')); %isolate handles to scatter points
grp = findgroups(site);
grpID = 1:max(grp);
clrMap = winter(length(unique(grp)));
for i = 1:max(grp)
set(hPt(grp==i), 'Color', clrMap(i,:), 'DisplayName', sprintf('MSP%d', grpID(i)))
end
title('Full Deployment (12h)');
set(gca,'fontsize',18)
xlabel('Component 1 (55.26%)') % how can I add percent ('%s %', explained(1))')
ylabel('Component 2 (21.87%)')
[~, unqIdx] = unique(grp);
legend(hPt(unqIdx)) Heidi Hirsh on 1 Oct 2020
Here's a random example I found online for what I want to do: Adam Danz on 1 Oct 2020
Edited: Adam Danz on 4 Oct 2020
If you know the center of the clusters and the CI's along the x and y axes for each cluster, you can use this function to plot the ellipses
A comment below that answer points to an alternative solution as well.
Also check the file exchange.
The block of code below implements the ellipses outlined in the first link above with your data which is attached. I made additional changes from your version to clean some stuff up a bit.
Stars mark the center of each cluster using the mean (consider using the median instead since the data are not normally distributed). Major and minor axes of the ellipses represent the 95% CI using the percentile method which is a good method given that the data are not normally distributed. [coeff,score,latent,tsquared,explained] = pca(X);
f1=figure();
h= biplot(coeff(:,1:2),'scores',score(:,1:2),'color','k','marker','.','markersize',17,'varlabels',...
{'DO','O2sat','pH','Temp','Sal','Depth','PAR','|Velocity|'});
hold on
%color by site
hID = get(h,'tag'); %identify handle
hPt = h(strcmp(hID,'obsmarker')); %isolate handles to scatter points
[grp, grpID] = findgroups(site);
clrMap = winter(numel(grpID));
p = 95; % CI level
for i = 1:max(grpID)
set(hPt(grp==i), 'Color', clrMap(i,:), 'DisplayName', sprintf('MSP%d', grpID(i)))
% Compute centers (means)
allX = arrayfun(@(hh)hh.XData(1), hPt(grp==i));
allY = arrayfun(@(hh)hh.YData(1), hPt(grp==i));
centers(1) = mean(allX); % x mean
centers(2) = mean(allY); % y mean
% Plot centers, do they make sense?
plot(centers(1), centers(2), 'rp', 'MarkerFaceColor', clrMap(i,:), 'MarkerSize', 20, 'LineWidth', 1)
% Compute 95% CI using percentile method
CIx = prctile(allX, [(100-p)/2, p+(100-p)/2]); % x CI [left, right]
CIy = prctile(allY, [(100-p)/2, p+(100-p)/2]); % y CI [lower, upper]
CIrng(1) = CIx(2)-CIx(1); % CI range (x)
CIrng(2) = CIy(2)-CIy(1); % CI range (y)
% Draw ellipses
llc = [CIx(1), CIy(1)]; % (x,y) lower left corners
rectangle('Position',[llc,CIrng],'Curvature',[1,1], 'EdgeColor', clrMap(i,:));
end
title('Full Deployment (12h)');
set(gca,'fontsize',18)
xlabel('Component 1 (55.26%)') % how can I add percent ('%s %', explained(1))')
ylabel('Component 2 (21.87%)')
[~, unqIdx] = unique(grp);
legend(hPt(unqIdx))
Adam Danz on 4 Oct 2020