Bivariate normal value standardization
显示 更早的评论
I want to standardize a bivariate normal CDF. I tried with inverse square root of covariance matrix and with Cholesky decomposition. The results are always different across all 3. I don't know why.
sigma=[1,0.5;0.5,1];
X = [1,1];
z=mvncdf(X,[0,0],sigma);
%%method 1
X1=X*sqrtm(inv(sigma));
z1=mvncdf(X1,[0,0],[1,0;0,1]);
%method 2
L = chol(sigma, 'lower');
X11=X*inv(L);
z11=mvncdf(X11,[0,0],[1,0;0,1]);
%results
disp([z,z1,z11])
1 个评论
Umar
2024-7-28
Hi CJ,
To ensure consistency in standardization, you can try using a standardized input vector by transforming X using the mean and standard deviation of the bivariate normal distribution.
采纳的回答
Hi CJ,
In short, the area of integration for the X1 case is no longer a rectangle as is assumed by mvncdf
Define the original distribution of an MVN vector X
Sigma = [1,0.5;0.5,1];
mu = [0 0];
Find the probability that -inf < X1 < 1 & -inf < X2 < 1
X = [1,1];
p1 = mvncdf(X,mu,Sigma)
This probablity can also be computed by integrating under the pdf of X.
Find the pdf of X
x = -3:.01:3;
[X1,X2] = meshgrid(x);
pdf1 = reshape(mvnpdf([X1(:),X2(:)],mu,Sigma),size(X1));
Plot it and add the limits at X1 = 1 and X2 = 1;
figure
pcolor(X1,X2,pdf1),shading interp
xline(1,'w');yline(1,'w');
We can approximate the probability by numerical integration under the pdf over the lower left square of the plot. Of course we are not capturing the tails of the density.
mask = (X1 <= 1) & (X2 <= 1);
trapz(x,trapz(x,pdf1.*mask,2),1)
Same (close enough) result as above.
Let Z be standard MVN, we have X = A*Z, where
A = sqrtm(Sigma)
Plot the pdf of Z
z = -3:.01:3;
[Z1,Z2] = meshgrid(z);
pdf2 = reshape(mvnpdf([Z1(:),Z2(:)],mu,eye(2)),size(Z1));
figure
pcolor(Z1,Z2,pdf2),shading interp,colorbar
Now, to properly compute the probability we need to find the region in the Z-plane that maps through
X = A*Z
to the lower left square above in the X-plane.
% X = A*Z
mask = reshape(all((A*[Z1(:),Z2(:)].').' <= [1 1],2),size(Z1));
hold on
Overlay the mask on the Z-plane to visualize the region of integration (which extends down and left to infinity)
plotmask = double(mask);
plotmask(plotmask == 0) = nan;
scatter3(Z1(1:10:end,1:10:end),Z2(1:10:end,1:10:end),plotmask(1:10:end,1:10:end),'w.'),view(2)
Compute the probability in z-space
trapz(z,trapz(z,pdf2.*mask,2),1)
3 个评论
CJ
2024-7-28
Paul
2024-7-28
You're very welcome.
As far as I know, mvncdf can only be used over rectangular regions, possibly extending to -inf in two directions.
I'm not sure what the issue is. If you have a non-standard normal vector, like X above, and want to find the probability over a rectangular region, why not just use mvncdf? Why transform to a standard normal vector?
CJ
2024-7-28
更多回答（0 个）
类别
在 帮助中心 和 File Exchange 中查找有关 Random Number Generation 的更多信息
Translated by 
