4.4. Autocorrelation of factors and model validation

   The variable is called autocorrelated if its value in specific place and time is
   correlated with its values in other places and/or time. Spatial autocorrelation
   is a particular case of autocorrelation. Temporal autocorrelation is also a very
   common phenomenon in ecology. For example, weather conditions are highly
   autocorrelated within one year due to seasonality. A weaker correlation exists
   between weather variables in consecutive years. Examples of autocorrelated biotic
   factors are: periodicity in food supply, in predator or prey density, etc.

   Autocorrelation of factors may create problems with stochastic modeling. In
   standard regression analysis, which is designed for fitting the equation to a
   given set of data, the autocorrelation of factors does not create any problems .
   The effect of the factor is considered significant if it sufficiently improves
   the fit of the equation to this data set. This approach works well if we are
   interested in one particular data set.
   For example, when geologists predict the concentration of specific compounds in a
   particular area they can use any factors that can improve prediction. And they
   usually don't care if the effect of these factors will be different in another
   area.

   However, ecologists are mostly interested in proving that some factor helps to
   predict population density in all data sets within a specific class of data sets.
   It appeared that models may work well with the data to which they were fit, but
   show no fit to other data sets obtained an different time or in a different
   geographical points.To solve this problem, the concept of validation was
   developed.

   Model Validation is testing the model on another independent data set.

   Example 1. In 60-s an 70-s it was very popular to relate population dynamics to
   the cycles of solar activity. Solar activity exhibits 11-yr cycles which seemed
   to coincide with the cycles of insect population outbreaks and population
   dynamics of rodents. Most analyses were done using from 20 to 40-yr time series.
   However, 2 independent cyclic processes with similar periods may coincide very
   well in short time intervals. When larger time series became available, it
   appeared that periods of population oscillations were usually smaller or greater
   than the period of the solar cycle. As a result, the relationship between
   population density and solar activity may change its sign in a larger time scale.

   Example 2. Our fox model (section 4.3.) was developed by fitting the equation to
   the time series. Thus, it is not surprising that it fits these data rather well.
   The question is, will this model work if tested on an independent data set which
   was not used for fitting the equation. We can separate the data into two
   portions, one of which is used for model fitting and the other portion is used
   for model validation.

   In our example, we select first 22 years and used them for estimating the
   regression:

                                    [eqseqex8.gif]

   t-ratio = 1.03; P=0.317; [eqrsquar.gif] = 5.6%. Effect is non-significant! There
   is nothing to validate. We can stop at this point and say that there is not
   enough data for validation. However we can make another try and separate the data
   in a different way. In our case, population numbers in year t depends on
   population numbers in tear t-2. Thus, we can estimate the regression using uneven
   years only and then test it using even years.

   Regression obtained from uneven years:

                                    [eqseqex9.gif]

   t-ratio = 13.14; P