My aim is to fit functions to covariance matrices. Furthermore I would like to have these functions positive definite.
For example the figure below shows a fitted covariance matrix modeled using a Gaussian function centered at point [4,4] with a given SD. This is excellent because this matrix is positive definite, thus I can factor it with Cholesky decomposition for finding the best fit to the observed data (not shown).
I also have observations which exhibits circular similarity. Below is an example of observed 8x8 covariance matrix. Here the circularity of similarity between the 8 different dataset can be appreciated by high covariance along the off-diagonals, as well as high covariance between first vs. 8th datasets (right upper corner).
My question is what function can I use to model this kind of similarity structure, which also ensures positive definiteness?