# Alternative tests for determining whiteness of a residual time series that does not result from an ARIMA process?

I’m doing some work that involves time series analysis of climate data, and I’m trying to figure out the best way to test for whiteness in a residual time series.

In the course of this work I’m subjecting a 119 year time series to a low pass Lanczos filter (Duchon 1979), then I subtract the resulting low frequency series from the original series to form a high pass residual series.

I want to test whether the high pass residuals is consistent with white noise. After some digging around I found the the paper by Ljung and Box (1978), and am trying to figure out how Ljung-Box or Box-Pierce statistics might be used to test for whether these residual series are white.

The Ljung-Box test involves calculating lagged autocorrelations of the residual series, then squaring and summing those values to form a chi-squared statistic. The number of degrees of freedom the chi-2 statistic should be h-(p+q) where h = maximum lag of the autocorrelations and p + q are the total number of parameters in an ARIMA model.

The hitch is that the high passed residual series is not an error series from an ARIMA model. In my case h = 10, but the only values I can think of that are analogous to p+q are the total number of weights in the Lanczos filter.

The main problem here is that Lanczos filters involve a lot of weights. The general idea behind these filters is a Fourier convolution Y(f) = R(f)*X(f) where X is the original series, Y the filtered version, and R the frequency response function of the filter. How it works by using lots of coefficients calculated to produce a selective, i.e. sharp, R frequency response function. The filter I’m using has 35 weights, so h – 35 in this case = -25. A negative number of degrees of freedom doesn’t make sense, so for the time being I using Box-Pierce stats, which assumes h = 10 degrees of freedom in my case.

Any ideas on this? Are there alternative tests for determining whiteness of a residual time series that does not result from an ARIMA process?

Duchon, C.1979: Lanczos Filtering in One and Two Dimensions, J. Appl. Clim. and Met., 18, 1016-1022.

Ljung, G. M. and Box, G. E. P.: 1978. On a measure of lack of fit in time series models. Biometrika, 65, 297-303.

• See also "Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey" which shows that Ljung-Box test is inapplicable on residuals from ARIMA models, while Breusch-Godfrey test could be used instead. – Richard Hardy Nov 23 '16 at 17:10
• It could help if you explained how the Duchon filter works so as to make it easier for potential contributors to this thread. On a first look, Duchon's filter looks like double-sided exponentially-weighted moving average, but I guess it is a bit more complicated than that. – Richard Hardy Nov 23 '16 at 17:17
• Thanks for the reply - I'll check out the post you suggested. The general idea behind a Lanczos filter is a Fourier convolution Y(f) = R(f)*X(f) where X is the original series, Y the filtered version, and R the frequency response function of the filter. How it works is similar to what you describe, but with with coefficients calculated to produce a selective, i.e. sharp, R frequency response function. I can send you paper and or code if you like. How is that usually done on CV? – Steven Mauget Nov 28 '16 at 16:04
• We try to keep it public so that everyone can contribute and benefit. You could post a brief summary of the method by editing your post, for example. The summary is intended to save time and distile knowledge (in contrast to having to read a paper) and thus foster more/quicker answers. – Richard Hardy Nov 28 '16 at 18:16