# 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

If you look at the derivation of CUSUM from first principles, the idea is to construct a likelihood ratio test statistic and wait till it exceeds a threshold. In case of CUSUM, the assumption is that samples are uncorrelated and so the test statistic simplifies to a sum. Since you are dealing with a different statistical change model, you can try writing a likelihood ratio statistic and see if you can simplify it to anything useful. There are also other algorithms out there for testing "shifts in variance" or "shifts in spectral content" for ARMA processes which you may want to read about.

See also: Ch 6 and 7 Detection of abrupt changes: theory and application M Basseville, IV Nikiforov - 1993.

• Hi: if you knew the underlying correlation structure, you could use the standard GLS transformation in order to make the observations "independent". ( see generalized least squares for how that's done ). but, it's rarely known and trying to estimate the structure is often an exercise in futility so I would go with one of the other suggestions. – mark leeds May 9 '18 at 22:41

You can also use a whitening filter. The book mentioned in the other answer by Basseville and Nikorov is good and free, although a little old.

CUSUM can tolerate some correlation.