White Noise Test: detecting autocorrelation and nonstationarities in long time series after ARIMA modeling

Abstract

Time series analysis has been a dominant technique for assessing relations within datasets collected over time and is becoming increasingly prevalent in the scientific community; for example, assessing brain networks by calculating pairwise correlations of time series generated from different areas of the brain. The assessment of these relations relies, in turn, on the proper calculation of interactions between time series, which is achieved by rendering each individual series stationary and nonautocorrelated (i.e., white noise, or to prewhiten the series). This ensures that the relations computed subsequently are due to the interactions between the series and do not reflect internal dependencies of the series themselves. An established method for prewhitening time series is to apply an Autoregressive (AR, p) Integrative (I, d) Moving Average (MA, q) model (ARIMA) and retain the residuals. To diagnostically check whether the model orders (p, d, q) are sufficient, both visualization and statistical tests (e.g., Ljung-Box test) of the residuals are performed. However, these tests are not robust for high-order models in long time series. Additionally, as dataset size increases (i.e., number of time series to model) it is not feasible to visually inspect each series independently. As a result, there is a need for robust alternatives to diagnostic evaluations of ARIMA modeling. Here, we demonstrate how to perform ARIMA modeling of long time series using Statsmodels, a library for statistical analysis in Python. Then, we present a comprehensive procedure (White Noise Test) to detect autocorrelation and nonstationarities in prewhitened time series, thereby establishing that the series does not differ significantly from white noise. This test was validated using time series collected from magnetoencephalography recordings. Overall, our White Noise Test provides a robust alternative to diagnostic checks of ARIMA modeling for long time series.