# Solved – Time varying Gaussian distribution

Does there exist time varying Gaussian model?

To be specific, a 2D Gaussian model whose mean and covariance matrix is varying with another parameter called time. The \$mu\$(mean) and \$sigma\$(covariance matrix) is changing with according to time.

Then, we say the distribution is \$p(x_1, x_2, t)\$ with \$mu(t)\$ and \$sigma(t)\$. Given \$(x_1, x_2, t)\$, we can get its probability at \$(x_1, x_2)\$ with specific \$t\$.

For original Gaussian distribution, if I have i items: \$(x_1^1, x_2^1), (x_1^2, x_2^2) dots (x_1^i, x_2^i)\$, and I want to calculate the probability of a specific item \$P(x_1^i, x_2^i)\$, I can first calculate the mean and covariance matrix of all the items, and then calculate the probability by applying Gaussian distribution.

Now, the problem is that I have the data: \$(x_1^1, x_2^1, t^1), (x_1^2, x_2^2, t^2) dots (x_1^i, x_2^i, t^i)\$, and want the mean and covariance matrix can be different values according to \$t\$, say \$mu(t)\$ and \$sigma(t)\$. Which means at different time \$t\$, the Gaussian distribution has different \$mu\$ and \$sigma\$. And at certain \$t\$, the distribution is still a normal Guassian distribution.

\$(x_1, x_2)\$ are two Double value, and \$t\$ is radian. I hope to cope with \$t\$ in directional statistics way.

If there exists such kind of model, how to inference its parameters, the relationship between its mean and covariance matrix with time?

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