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I'm new to Kalman filters so this might be a stupid question.

I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a random walk.

However, I also want to be able to estimate the standard deviation of my observations, similar to how I'm using it to estimate the mean of a time series. I know I have to provide the filter with a constant observation covariance matrix, but I don't know if there's a way I can get an estimate of the observation's variance (so I can compute the standard deviation).

Is there a way I can extract this information from my Kalman filter? Is there a better way to estimate the standard deviation of a time series using a Kalman filter?

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