Abstract. This paper proposes a new approach to detecting and describing differences in stationary processes. The approach is equivalent to comparing auto-covariance functions or power spectra. The basic idea is to fit an autoregressive model to each time series and then test whether the model parameters are equal. The likelihood ratio test for this hypothesis has appeared in the statistics literature, but the resulting test depends on maximum likelihood estimates, which are biased, neglect differences in noise parameters, and utilize sampling distributions that are valid only for large sample sizes. This paper derives a likelihood ratio test that corrects for bias, detects differences in noise parameters, and can be applied to small samples. Furthermore, if a significant difference is detected, we propose new methods to diagnose and visualize those differences. Specifically, the test statistic can be used to define a “distance” between two autoregressive processes, which in turn can be used for clustering analysis in multi-model comparisons. A multidimensional scaling technique is used to visualize the similarities and differences between time series. We also propose diagnosing differences in stationary processes by identifying initial conditions that optimally separate predictable responses. The procedure is illustrated by comparing simulations of an Atlantic Meridional Overturning Circulation (AMOC) index from 10 climate models in Phase 5 of the Coupled Model Intercomparison Project (CMIP5). Significant differences between most AMOC time series are detected. The main exceptions are time series from CMIP models from the same institution. Differences in stationary processes are explained primarily by differences in the mean square error of 1-year predictions and by differences in the predictability (i.e., R-square) of the associated autoregressive models.
The Laplace Likelihood Ratio Test for Heteroscedasticity
