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?

Take a look at Wiener process, Brownian motion and Ito calculus. For instance, a particular form of the process with time-varying mean and variance is Heston model used a lot in finance: $dS_t=mu S_tdt+sqrt{nu_t}S_tdW_t^S$

$dnu_t=kappa (theta-nu_t)dt+xisqrtnu_tdW_t^nu$.

Here, in the first equation you see how both mean and the variance of $dS_t$ changes with time.

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