The h(·) is referred to as the autocorrelation function

The correlation matrix Ci
is specified by assuming that the correlation coefficient
between two residual errors eij and eij’
, corresponding to two observations from the
same group i, is given by
Cor(eij, eij’) = hd(tij,tij’), ?)
where ? is a vector of correlation parameters, d(tij,tij’) is a distance function of vectors
of position or serial variables tij and tij’ corresponding to respectively, eij and
eij’
, 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 position
1.4. Linear Mixed Models 3
vectors, they are the same observation and therefore have a correlation 1 (Pinheiro
and Bates, 2000; Ga?ecki and Burzykowski, 2013).
In general there can be two sources of correlation: spatial and temporal. Spatial
correlation 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 observations
collected closer together in time are likely to be more similar than those
further 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 space
or time, the more highly correlated they are. For time-series data it is assumed that
the serial correlation model depends on the one-dimensional positions tij, tij’
, only
through their absolute difference. The general serial correlation model is then defined
as:
Cor(eij, eij’) = h(|tij ? tij’
|, ?)
In the context of time-series data, the correlation function h(·) is referred to as the
autocorrelation function (Pinheiro and Bates, 2000). For a more detailed description
of the autocorrelation function that is used in this report, see section