## Special Cases of Regression Based Methods Correlated data models

Fitting and applying regression models depends on the assumptions made regarding the dependence structure between the predictors and the errors, as well as between the errors themselves. The standard assumptions of independence are usually violated when one is modeling ecological processes that occur over space and time. In such cases, it is natural to assume correlated data, in which the correlations themselves are unknown parameters in the model. Fortunately, such situations can still be formulated to fit into the framework of multi-variate regression.

The simplest case is the problem of trend estimation. If the response variable y is to be predicted as a function of time t as the single predictor, then a simple linear or nonlinear regression of y on t is appropriate. This can then easily be combined with other predictor variables x into a larger linear or GLM.

A more sophisticated time-series model involves modeling correlations between measurements, rather than a systematic trend with time. Usually these correlations are modeled as a function of the time separation between measurements. Such correlation models can be combined with trend models by fitting correlations to data that have been corrected for the effects of time or other predictor variables.

'Autoregressive models' express the conditional distribution of measured variables at each point in time as a function only of the model parameters and the previous k measurements. When k — 1, these are referred to as 'Markov models'. If the observation time t itself is added as a predictor, then the model can also account for trends.

Correlation in time can also be modeled as a mixture of sinusoidal curves with different frequencies. This approach may make sense for capturing the seasonal or diurnal fluctuations common in ecological data. If the frequencies can be considered fixed, the resulting spectral model simply states that the response variable is a linear function of sinusoidal transformations of the predictor variable, time. Thus, linear regression techniques apply.

Spatial models can often be developed using many of the techniques of temporal modeling. However, unlike time, there is no natural ordering in space. Thus, one-way, autoregressive models are not appropriate. 