# Why is prewhitening important?

I am writing code, geophysical time series processing. First step is to prewhiten values in time domain. Why is this step important?

For example, I have found this on sas.com

If, as is usually the case, an input series is autocorrelated, the direct cross-correlation function between the input and response series gives a misleading indication of the relation between the input and response series.

I do not understand, in my case all values are E field measurements over time. What means that input series is autocorrelated?

How will it influence Fourier transform on the next step?

• What source is telling you to whiten time series data? Jun 28, 2016 at 20:23
• @Maddenker Tutorial I am reading right now! Jun 29, 2016 at 12:26
• Can you post a link? Having some context to your question will help people answer it. Jun 30, 2016 at 14:38
• See some of this answer here: stats.stackexchange.com/questions/133155/… -- particularly the second reference Sep 30, 2017 at 0:43

The reason that you pre-whiten X is to identify a filter that can transform Y and X into y and x where x is white noise i.e. serially independent or free of autocorrelation in order to IDENTIFY an appropriate model. Note that one filter (ARMA developed on X ) is used on both the Y and X. Now with y and x you can form/identify a potential relationship which is then applied to the Y and X to construct/identify a polynomial distributed lag model (PDL/ADL/DGF . Fundamentally you are adjusting the Y and X ( transforming/filtering) so that the resultant cross-correlation between y and x (proxies) can be correctly/efficiently be interpreted and used on the observed series Y and X.

The single filter doesn't distort the causative structure. Note that differencing operators required for X and Y are not necessarily the same and are not necessarily part of the final model relating Y and X.

To further numerically illustrate this consider the GASX problem from the Box-Jenkins text where PINK reflects the predictor series . A simple filter (2,1,0) was used to prewhiten creating "adjusted cross-correlations or prewhitened cross-correlations" suggesting/identifying a three period delay culminating in this useful equation . Note clearly that Y is not CONDITIONALLY a function of X contemporarily (or lag 1 or lag 2) given the model form. In simpler terms X significantly affects Y after two periods and not before.

In contrast consider the simple (naive) cross-correlation between Y and X falsely suggesting structure (induced by the auto-correlation within the series ) .

It is interesting to me that most significant of these cross-correlations are at lags 3,4 and 5 illustrating that however flawed/contaminated they can still be directionally important.

geophysical time series are auto-correlated, which means the measurement value at 12:00 would be similar to 13:00, but more different to 19:00, like air temperature, this is just an example. Pre-whitening is used to detrend, and make the measurement "White", namely independent between each measurement.

• But that does not answer the question "why is prewhitening important?" Jun 2, 2017 at 17:37
• then I think you need to clarify your question. What's the E field? Would you mind provide some details about your question, or just give an example?
– Sam
Jun 2, 2017 at 20:25

The need for prewhiten your time series will depend on the model that you will use to analyze your data. For example, if you want to perform Pearson correlation analysis between two-time series, the pre-whitening will be needed because the autocorrelation in the time points (if it is the case) will violate the assumptions behind Pearson Correlation.

For example, suppose that you get a correlation of value C12 between time-series 1 and 2. Is C12 significant? You can infer the probability of getting the result C12 by chance given the number of points in each time series. However, if there is autocorrelation in any of the time series 1 and 2, you lose the real meaning of the calculated probability.

Another example, suppose that you want to apply some general linear analysis in a given times series (y) with a design matrix (x) such that y=x*beta + error. If y presents autocorrelation, it will introduce serially correlated errors, violating the Gauss Markov theorem.

• you mention "if you want to perform Pearson correlation analysis between two-time series, the pre-whitening will be needed" - I was under the impression that pearsons wasn't suitable for timeseries. Are you saying that if one pre-whitens the timeseries pearsons is suitable? Do you have any resources that support that ? Thanks
– baxx
Jun 6 at 10:34