4.3. Example: Colored fox in Labrador (18391880)The time series comes from Elton (1942).
First step is to logtransform the data. When predicting population density, logtransformation is always better than no transformation.
The distribution of logtransformed data is more symmetric now. We will experiment with different factors trying to get the most accurate prediction of log population density.
1. One factor: previous year population counts
This means that our regression does not work any better than using average logdensity.
2. Two factors: population counts in two previous years
Effect of year t1 is not significant, but the effect of year t2 is significant. Nonsignificant effect can be ignored. Thus, we can reestimate regression using year t2 as the only predictor:
3. Plotting the regression
OK, there are no quadratic effects of year t2.
4. Adding year t3.
The model did not get significantly better after adding year t3 as a predictor. Thus, we cannot improve the model any further.
5. Plotting the residuals. This graph indicates no nonlinear effects of population counts in year t1
6. Prediction of population numbers
Predicted population counts follow the same pattern as observed values. However, predicted values have smaller variation because any regression has a "smoothing" effect. In the previous graph, we predicted population counts one year ahead at a time. Let's see what happen if we try to predict the entire time series from two initial values. In this case, the error will propagate because we will use predicted population counts as the base for further predictions.
Predicted population counts exhibit damped oscillations. After a few oscillations, they approach the equilibrium level of x=5.553. This model cannot be used to predict population numbers more than 13 years ahead.
