Automated Generation of Reduced Stochastic Weather Models I: Simultaneous Dimension and Model Reduction for Time Series Analysis

We present a method for simultaneous dimension reduction, model fitting, and metastability analysis of high-dimensional time series. The approach is based on the combination of hidden Markov models (HMMs) with localized principal component analysis (PCA) (which is used to identify the essential dimensions in the form of empirical orthogonal functions (EOFs) for each of the hidden states) and fitting of multidimensional stochastic differential equations (SDEs). This means that the analyzed data is clustered according to differences in essential dimensions and SDE models specific to each of the hidden states. We derive explicit estimators for PCA-SDE model parameters in the case of fixed sequences of HMM states and employ the expectation-maximization algorithm for numerical optimization of HMM-PCA-SDE parameters. We demonstrate the performance of the method by application to historical temperature data in Europe during 1976–2002. In a comparison with the standard SARMA (seasonal autoregressive moving average model) technique for time series analysis, the HMM-PCA-SDE method exhibits better numerical performance and efficiency, especially on high-dimensional data sets and for a 20-dimensional reduced state space. We also compare the results of both models w.r.t. errors of one-day temperature predictions.