Efficient method/technique to update covariance matrix

Question

A covariance matrix of multivariate random variable can be constructed given a time-series random variables.

Eg. If you observe a student's performance in different objects (Math, English, Physics, etc) for a period of time; then you can construct the covariance matrix for those objects for that specific student.

However, the random variable can be statistical unstable. Hence, the covariance matrix must be updated each time a new value of randome variable is observed.

I'm looking for an efficient method/technique to update that covariance matrix.

Although just the name for the method is enough, if you have a link to a tutorial, lecture note or a paper on the topic, it will be much more helpful too.

Thanks,

PS: Please correct me if I use some terminologies incorrectly. (I'm not a mathematician or statistician).

Answer 1

I suggest you a very simple online update formulas:

At each time $t$ you provide me a point of a time series $x$ of size $N$, then

(1) $\hat\mu=\hat\mu+\frac{1}{1+t}(x-\hat\mu)$

(2) $\hat\Sigma=\frac{t}{1+t}\hat\Sigma+\frac{t}{(1+t)^2}(x-\hat\mu)^\prime (x-\hat\mu)$ See for instance this paper.