Perform Naive-Bayes classification(fitcnb) with non-zero off-diagonal covariance matrix

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I use a Bayesian classification model to generate class-conditional probability density functions (PDFs) from a Monte Carlo (MC) simulation (see Fig 1). The different classes have inter-variable correlations such that the covariance matrix has non-zeros on the off-diagonal elements. However, the Bayesian classification model seems to assume that the off-diagonal elements are zero, such that the PDFs for each class are not shaped according to the MC simulated data (see Fig 2); this makes the PDFs look like ellipsoids that are horizontally aligned.
So, how can I specify the covariance elements in the Bayesian classification model when I for instance want to use it to predict a new data set?
Fig 1:

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the cyclist
the cyclist am 18 Jan. 2018
Bearbeitet: the cyclist am 18 Jan. 2018
Disclaimer: I am not an expert on these methods.
Doesn't the "naive" in naive Bayes specifically mean that the model features are independent from each other (i.e. uncorrelated)? You might need a more sophisticated model.

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Ilya am 19 Jan. 2018
To estimate covariance per class, use fitcdiscr with discriminant type 'quadratic'.

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