# Computing CDF of multivariate Gaussian from eigendecomposition

I have a multivariate Gaussian for a set of data, and I'd like to compute the confidence interval for that data sample.

In hopes of finding an elegant solution, I did an eigendecomposition and transformed the data into the basis defined by the eigenvectors where the eigenvalues are the variance of that dimension. What's nice about this basis is there is 0 covariance, and each axis is "independent". I compute the CDF of each Gaussian independently and from that determine the confidence interval for each dimension, but now I run into the problem of how to combine them.

In some sense, if these de-correlated dimensions are independent, I ought to multiply the probabilities, but it doesn't make sense to me to multiply the confidence intervals. I thought about averaging them, but that also doesn't make sense because if one dimension has a confidence interval of $\sim1$, then averaging doesn't take into account the exponential nature of this math. The last thing I considered was an L2 norm, but that doesn't work at all.

Anyway, I'd appreciate some help figuring this out. Specifically, how can I determine the multivariate Gaussian CDF from the CDFs of the individual de-correlated Gaussian dimensions? Or confidence interval instead of CDF – basically the same information I am interested in.

I posted this in the Math forum before I discovered this one. I hope that's kosher:

• what do you mean by confidence interval for a joint distribution? Mar 3, 2014 at 1:17
• Reposts from other sites in the SE network are explicitly discouraged in the help. Instead, what it suggests is that when you decide a post is better placed on a different site, you flag it for moderator attention and ask for it to be moved. Mar 3, 2014 at 4:57
• Same concept as with a 1D Gaussian distribution. I'd like a measure of likelihood that this ND distribution lies within the multidimensional Gaussian.
– Chet
Mar 3, 2014 at 6:36
• Glen_b - not sure how I do that in my other post...
– Chet
Mar 3, 2014 at 6:37
• Detailed information about confidence sets for the mean of multivariate Gaussians appears at stats.stackexchange.com/questions/9898 and mathematica.stackexchange.com/questions/21396. More related information is available from a search on confidence ellipse. General information about the distribution of quadratic forms of Normally distributed variables can be found with an appropriate search.
– whuber
Mar 3, 2014 at 15:26