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?