The correlation matrix Ciis specified by assuming that the correlation coefficientbetween two residual errors eij and eij’, corresponding to two observations from thesame group i, is given byCor(eij, eij’) = hd(tij,tij’), ?)where ? is a vector of correlation parameters, d(tij,tij’) is a distance function of vectorsof position or serial variables tij and tij’ corresponding to respectively, eij andeij’, and h(·) is a continuous function with respect to ?, such that it takes values between-1 and 1, and h(0, ?) ? 1, that is, if two observations have identical position1.4.
Linear Mixed Models 3vectors, they are the same observation and therefore have a correlation 1 (Pinheiroand Bates, 2000; Ga?ecki and Burzykowski, 2013).In general there can be two sources of correlation: spatial and temporal. Spatialcorrelation refers to the fact that sites closer together will generally be more similar.The same effect can also appear when responses are recorded over time, so that observationscollected closer together in time are likely to be more similar than thosefurther apart temporally. The latter is the definition for temporal/serial correlation.When the response variable is influenced by underlying spatial or temporal processes,then the data are auto-correlated – the closer the observations are in spaceor time, the more highly correlated they are. For time-series data it is assumed thatthe serial correlation model depends on the one-dimensional positions tij, tij’, onlythrough their absolute difference.
The general serial correlation model is then definedas:Cor(eij, eij’) = h(|tij ? tij’|, ?)In the context of time-series data, the correlation function h(·) is referred to as theautocorrelation function (Pinheiro and Bates, 2000). For a more detailed descriptionof the autocorrelation function that is used in this report, see section