Understanding fluctuations through Multivariate Circulant Singular Spectrum Analysis

The decomposition of multichannel signals into their underlying components is a problem of key interest in many disciplines. It involves the analysis of nonlinear and nonstationary time series and the need to identify the most relevant frequencies of oscillation that characterize the extracted underlying components, disentangling their sources of variation and assessing their relative phase or cyclical position. Many successful multivariate methods for signal extraction identify the frequencies after the decomposition problem has been solved. They may not be able to easily recognize the full information contained in each frequency though. Within the context of Multivariate Singular Spectrum Analysis, this paper develops Multivariate Circulant Singular Spectrum Analysis (MCiSSA), a self-identifying procedure for the frequencies without the need of further processing the data. Seeing MSSA procedures as a generalization of principal component analysis that includes series and their lags, the key for the self-identification of the frequencies is the use of block circulant matrices of second moments, instead of the usual variance–covariance matrix, that allows matching eigenvalues with frequencies. Additionally, MCiSSA performs a double diagonalization of that block circulant matrix that uncovers cross section relations per frequency. We show the performance of MCiSSA through several synthetic examples that include latent signals modulated both in amplitude and frequency. To do so, we compare the Root Mean Squared Error (RMSE) of the adjusted series to the simulated ones across several multivariate alternatives being MCiSSA the best option with RMSE 40% lower than the second best option in the first example and 24% in the second example. Finally, we apply MCiSSA to two real data applications: energy commodity prices and temperatures. The first application shows how MCiSSA helps to understand the main drivers and co-movements of primary energy commodity prices at various frequencies that are key to assess energy policy at different time horizons. We find that prices are dominated by the European gas at all frequencies. More specifically, the volatility of the price of the European gas accounts for almost half of the cyclical variability in the system. The second application shows the growing trend of temperatures in 12 South European cities (on average, trends have increased 1.77 Celsius degrees between 1950 to 2022) and their changing pattern in seasonality.