# Testing for changes in auto-covariance

I am working with uniformly-spaced time series data where I am interested in knowing whether there are changes in temporal auto-covariance. The mean can be assumed constant. Visually, there are no abrupt/obvious changes in variance or temporal correlation.

Intuitively, it seems to make sense to divide up the data into contiguous blocks, compute for each block the sample auto-covariance at some number of lags, and compare the sample auto-covariances for each block. I suppose that one could apply procedures to see if the per-block auto-covariance estimates come from the same distribution (which, I guess, would imply stationarity).

• Can anyone point me to a some papers where this sort of thing has been done?

• Also, are there other methods out there which could apply?

• I have seen unit root tests. While an AR process may be appropriate, I'm not sure if the non-stationarity can be described by a unit root--perhaps instead slowly time-varying AR coefficients?

I have also seen some wavelet approaches, which seem to focus on detecting abrupt changes, which is not the case here.

## 1 Answer

I'd recommend looking into the relation of correlation and covariance; the one is just the "bias-corrected" version of the other.

Then, use the Wiener-Khinchin¹ theorem: If, and only if, the signal is weak-sense stationary, the Fourier transform of the autocorrelation of that signal is the same as Expectation value of the magnitude-squared Fourier transform of the signal itself.

So, I think, you're on the right track: Just use (potentially even overlapping) Short-term Fourier Transforms to get an estimate for the Fourier transform of the autocorrelation; if you know the mean is fixed, that should be hold the same amount of information as auto-covariance. Compare those over time.

In fact, with what is often known as a spectrogram in plotting libraries, you'd get something that visually represents that pretty well.

¹ I say this every time: There's more spellings to Tschinshin's name than there's letters in his originally cyrillic name.

• Thanks for your input. Actually, I found something I currently implementing based on the paper by Lund et al: TESTING EQUALITY OF STATIONARY AUTOCOVARIANCES. This is a time-domain approach which, at least for the cases considered in the paper, performed quite well wrt to frequency domain techniques. – rhz Sep 12 '16 at 5:02