For the fast reader: links to DFA, MAFA and chronological clustering examples:

Two papers on DFA:

Background information

Underlying questions in time series studies

Common characteristics in environmental time series studies are that the series are (i) short, (ii) non-stationary, (iii) made up of many response variables which are interacting with each other, and (iv) have missing values. Common questions in these studies are:  

These questions can be summarised by one simple question: what's going on? Brodgar contains various different time series techniques to answer this question. Three important ones are: 

Each of these methods is discussed next.

Dynamic factor analysis

Biological time series are in general to short for techniques like spectral analysis, wavelet analysis, auto regressive (AR) models and auto regressive integrated moving average (ARIMA) models. Furthermore, aspects like missing values and the presence of many response variables are not handled well by these techniques.  To address the multivariate nature of the response variables, standard multivariate techniques such as canonical correspondence analysis, principal component analysis, multidimensional scaling, are sometimes used. However, these techniques do not handle missing data properly and furthermore they do note take dependencies over time into account. A more promising approach is structural time series analysis (Harvey 1989). Although this set of techniques originates from fields related to econometrics and psychology, it has several aspects that are of interest to biologists. Its main feature is that the time series are modelled in terms of a trend, seasonal effects, a cycle, explanatory variables and noise, each of which is allowed to be stochastic. This means that one might end up with  a seasonal component which changes slightly from year to year, a cyclic component which is not necessarily a cosine function, a trend which is not restricted to be a straight line or a polynomial, or explanatory variables which only have a significant influence in a certain period. 

One particular interesting approach is dynamic factor analysis. In this multivariate technique, underlying common components are identified, namely: common trends,  common seasonal patterns, common cycli and, effects of explanatory variables. If the time series are short, as in most environmental studies, cycli and seasonal patterns can be omitted, resulting in the estimation of common trends and effects of explanatory variables only. Traditionally, the parameter estimation process in dynamic factor analysis was carried out by direct optimisation of a maximum likelihood criterion (Harvey 1989). Due to numerical problems, this limits the number of time series that can be analysed. Zuur et al. (2003^a,b,c) however, addressed this limitation by using a different estimation procedure, the so-called EM algorithm. Furthermore, they extended the technique by including explanatory variables.

Links to dynamic factor analysis examples:

MAFA

MAFA stands for min/max autocorrelation factor analysis. MAFA can be described in various ways, e.g.:

MAFA is perhaps best explained with an example:

Chronological clustering

MAFA and dynamic factor analysis are techniques which can be used to estimate trends in multivariate time series. Application of these techniques on biological data assumes that the underlying  ecosystem is gradually changing over time. However, these techniques are less appropriate if the ecosystem changes rapidly from one state to another. Ordinary clustering techniques might be applied to identify sudden changes, but these methods are likely to result in groups of years that are difficult to interpret. For example, how does one explain a group containing 1970, 1976, 1992 and 2003? Chronological clustering, as the name already suggests, is especially designed for clustering of time series. The method is fully described in Legendre et al. (1985), Bell and Legendre (1987), and Legendre and Legendre (1998). The first two papers are downloadable from Legendre's website (search on "chronological clustering" in Google), and are easy to read for non-statisticians. Explaining chronological lustering is best done with examples:

To identify breakpoints in multivariate times series, Brodgar can also apply regime shift analysis, as explained in Hare & Mantua (2000), see the Brodgar manual for an example.

Besides dynamic factor analysis, MAFA and chronological clustering, Brodgar is capable of carrying out ‘standard’ multivariate techniques and multivariate time series techniques like principal component analysis, canonical correspondence analysis, discriminant analysis, redundancy analysis, multidimensional scaling, ARIMAX, spectral analysis, etc. For short time series data (say less than 15 points in time), some of the multivariate methods can be used. For example, partial RDA and partial CCA can be used to determine how much variation in the response variables is due to time. 

The  emphasis in Brodgar is on biological and environmental time series. However, Brodgar can be used in many other fields. For example, various Brodgar users work on economical time series data.

References