# Applying the CUSUM algorithm to a correlated random process

As far as I know, the CUSUM algorithm is meant for detecting change points on discrete-time uncorrelated random processes.

For instance, to apply the CUSUM algorithm to a discrete Gaussian process, we must know for sure that each sample is statistically independent from the others. I have seen this assumption in the CUSUM algorithm demonstration.

However, I have not found the CUSUM alternative for a discrete-time correlated random process. Let's say, for a realization of a band-pass filtered Gaussian process $X[n]$ with a changing variance at a certain sample $n_c$.

What to do in such a scenario? Can I still apply CUSUM?

• Why not try it? – Ben May 7 '18 at 18:41
• I am a bit confused about the second paragraph: "for a discrete Gaussian process, it is assumed that each sample is statistically independent from the others" No! What you mean is white, I think, not Gaussian. Gaussian typically characterizes the distribution of amplitudes of each sample (individually), and statistical independence says something about the temporal properties of the random process (as a whole). There's white processes that are not Gaussian, and there's Gaussian processes that are not white. – Marcus Müller May 7 '18 at 19:11
• by the way, the question seems to be far more interesting than I first thought - the fact that the variance changes at some point breaks all forms of stationarity, and hence, we can't use Wiener-Khinchin to find the PSD of the process; "windowing" / limiting the period where we consider the ACF might be the solution hereto, since it doesn't seem that easy to analytically give the let's call it "short-time PSD" of windows into which that switchover falls. – Marcus Müller May 7 '18 at 20:37
• @MarcusMüller, you misunderstood my message. Why I meant was that to apply the CUSUM algorithm to a Gaussian process, it is assumed that each sample is statistically independent from the others. I did not mean that every Gaussian process is white. Let me edit my post to make it more clear. – Luis M Gato May 8 '18 at 14:09