Determining 1, 2 and 3 sigma radii for a bivariate Gaussian histogram

Hi all,

I've a 2d scatter plot which I've converted into a 2d histogram using scatter_kde (credit: Nils). I'd like to know whether it's possible to determine the 1, 2 and 3 sigma radii for the 2d histogram? It would also be useful if I could quantify the "roundness" of those circles.

Thanks and compliments of the season.

Tim

Answer 1

[Star Strider]

If the point positions are in matrix format (or can be interpolated to matrix format using griddata or a similar function), the contour function would likely be appropriate. Set the contours to be whatever values are desired.

Example —

xv = linspace(-30, 30);
yv = linspace(-40, 40);
[Xm,Ym] = ndgrid(xv, yv);
zfcn = @(x,y) exp(-(x/15).^2) + exp(-(y/20).^2);
Zm = zfcn(Xm,Ym) + randn(size(Xm))*0.25;

Xm(Zm < 1) = NaN;
Ym(Zm < 1) = NaN;
Zm(Zm < 1) = NaN;

Zmm = mean(Zm(:),'omitnan');
sgma1 = std(Zm,1,1,'omitnan');
sgma1m = mean(sgma1);
sgma2 = std(Zm,1,2,'omitnan');
sgma2m = mean(sgma2);

NoNaN = ~isnan(Zm);
Xmn = Xm(NoNaN);
Ymn = Ym(NoNaN);
Zmn = Zm(NoNaN);

xvi = linspace(-30, 30, 25);
yvi = linspace(-40, 40, 25);
[Xi,Yi] = ndgrid(xvi, yvi);
Zi = griddata(Xmn(:), Ymn(:), Zmn(:), Xi, Yi);

figure
scatter3(Xm(:), Ym(:), Zm(:), [], Zm(:), 'Marker','.')
xlabel('X')
ylabel('Y')
hold on
surf(Xi, Yi, Zi, 'FaceAlpha',0.75, 'EdgeColor','k')
contour3(Xi, Yi, Zi, Zmm+[0.15 0.30 0.45], '-r', 'LineWidth',5)
hold off
grid on

figure
scatter3(Xm(:), Ym(:), Zm(:), [], Zm(:), 'Marker','.')
xlabel('X')
ylabel('Y')
view(0,90)
hold on
contour3(Xi, Yi, Zi, Zmm+[0.15 0.30 0.45], '-r', 'LineWidth',2.5)
hold off
grid on

This is definitely not a perfect illustration of the data, however it illustrates the approach.

.

[Tim Fulcher]

Thanks Star Strider. I've not finished implementation yet but so far...so good.

Regards and compliments of the season.

[Star Strider]

As always, my pleasure!

Thank you! And to you, too!

.