considering autocorrelation in predictive models - Department of ...

28 Definition of the Problem
• Non-linear relationships may exist between species and environments, but these relationships may
be incorrectly modeled as linear;
• Classical statistical modeling may fail in the identification of the relationships between different
kinds of data without taking into account their spatial arrangement (Besag, 1974);
• The spatial resolution of data should be taken into account: Coarser grains lead to spatial smoothing
of data.
Spatial autocorrelation has many interpretations. We present some of them (Dubin, 1998):
• As a nuisance parameter, inserted in a model specification because its presence is necessary for a
good description but it is not the topic of interest.
• As a self-correlation, arising from the geographical context within which attribute values occur.
• As a map pattern, viewed in terms of trends, gradients or mosaics across a map.
• As a diagnostic tool for proper sampling design, model misspecification, nonconstant variance or
outliers.
• As a redundant (duplicate) information in geographical data, connected to the missing values estimation,
as well as to notation of effective sample size and degrees of freedom.
• As a missing variables indicator/surrogate, popular in spatial econometrics (Anselin, 1988).
• As an outcome of areal unit demarcation in statistical analysis.
Consequently, when analyzing spatial data, it is important to check for autocorrelation. If there is no
evidence of spatial autocorrelation, then proceeding with a standard approach is acceptable. However, if
there is evidence of spatial autocorrelation, then one of the underlying assumptions of your analysis may
be violated and your results may not be valid.
Spatial autocorrelation is more complicated than temporal autocorrelation as it can occur in any
direction. Moreover, the phenomenon of existing spatial and temporal autocorrelation at the same time
has even higher complexity.
Spatio-Temporal Autocorrelation
In nature, it is a common case that data is not only affected by the phenomenon of spatial autocorrelation,
but also by the phenomenon of temporal autocorrelation at the same time. In that case, we consider
spatio-temporal autocorrelation, as a special case of autocorrelation.
Spatio-temporal autocorrelation is a property of a random variable taking values, at pairs location
a certain distance apart in space and time, that a more similar or less similar than expected for pairs
of observations at random selected locations and times. It is the actual correlation among values of a
variable strictly due to their relative location proximity and time proximity.
For a continuous variable X, measured at locations at a distance (∆a, ∆b), the spatio-temporal autocorrelation
can be expressed as:
SpatioTemporal_AC(∆a,∆b,τ) = E[(Xa,b,k − µx)(Xa+∆a,b+∆b,k+τ − µx)]
E[(Xa,b,k − µx)]
(2.22)
where d = (∆a) 2 + (∆b) 2 is the spatial distance between the values of the variable X at the two locations,
τ is the temporal lag, E is the expected value operator and µx is the mean of the variable X.