Estimating common trends in multivariate time series using dynamic factor analysis

Abstract

This paper discusses dynamic factor analysis, a technique for estimating common trends in multivariate time series. Unlike more common time series techniques such as spectral analysis and ARIMA models, dynamic factor analysis can analyse short, non-stationary time series containing missing values. Typically, the parameters in dynamic factor analysis are estimated by direct optimisation, which means that only small data sets can be analysed if computing time is not to become prohibitively long and the chances of obtaining sub-optimal estimates are to be avoided. This paper shows how the parameters of dynamic factor analysis can be estimated using the EM algorithm, allowing larger data sets to be analysed. The technique is illustrated on a marine environmental data set.

Analysis of trends and sudden changes in long-term environmental data from King George Island (Antarctica): relationships between global climatic oscillations and local system response

“…To confirm whether the present environmental time series bears any general patterns over time, and whether the measured variables are related to climatic cycles such as SAM or ENSO, a dynamic factor analysis (DFA) after (Zuur et al 2003) was performed using the software Brodgar (v. 2.6.6). DFA is a method to estimate "trends" that are common to all series, as well as the effects of explanatory variables and interactions in a multivariate time series data set that may even contain missing values (Zuur et al 2003(Zuur et al , 2007.…”

“…DFA is a method to estimate "trends" that are common to all series, as well as the effects of explanatory variables and interactions in a multivariate time series data set that may even contain missing values (Zuur et al 2003(Zuur et al , 2007. Dynamic factor analysis is based on structural time-series models (Jalles 2009).…”

“…The 'explanatory variable' component is modeled as linear regression (Zuur and Pierce 2004;Zuur et al 2003). Dynamic factor analysis estimates a linear combination of M common trends of all N time series and can be written in matrix notation as: where the N x M matrix A contains the factor loadings, z t the trend, B the regression parameters, x t the value of the explanatory variable, and e t the noise components (Zuur et al 2003;Zuur and Pierce 2004). Magnitude and sign of factor loadings determine how the common trends are related to the original time series.…”

“…All plots indicated that the original data are almost normally distributed and that transformations do not considerably improve normality of the data set. While normality of data is generally beneficial in DFA models, it is not a prerequisite (Zuur et al 2003). Table 1 shows the four dynamic factor models that were used in our analysis.…”

“…Regarding the modeling of the error covariance matrix R, the simplest approach is to use a diagonal matrix. Alternatively, a symmetric, non-diagonal matrix for R can be used (Zuur et al 2003) Off-diagonal elements of R represent the information in the response variables which cannot be explained with the other terms (Zuur et al 2003;Zuur and Pierce 2004). We applied different models which i) yield one, two or three "model-specific" common trends (see column M in Table 3); ii) models with and without explanatory variables (SAM, ENSO).…”