As described in Data Exploration Methods(details), partial dependence functions (PDFs) summarize the effect of predictors on the probability of occurrence after accounting for the average effect of all other predictors. It is this "statistical control " that makes partial dependence functions well suited to explore and visualize the effects of predictors based on observational data.

In this section we use partial dependence functions to visualize the dynamics of wild bird populations. We do this by computing the partial dependence as a joint function of location and time, averaging out the effects of all other predictors. The resulting estimates of occurrence probability are plotted as surfaces and animated to visualize dynamic patterns of occurrence.

The main advantage of these model-based partial dependence surface-plots over simpler surface-plots of “raw” observations is that they control for many important sources of variation that would otherwise bias distributional estimates. For example, the bagged decision tree models used to predict the probability of species’ occurrence take into account the effects of varying observer effort, factors that attract birds to feeders, and varying human population density. These effects are then “averaged out” by the partial dependence plots producing improved estimates of species’ occurrence. Another advantage of model-based partial dependence surface plots is that they can be used to estimate the potential for species’ occurrence at unobserved locations.

Using all of our PFW analyses for American Goldfinch we generated two types of animations to highlight distinct dynamic patterns of the winter migration.  In Figure 3, we animate frames over each year at a common day within the Feederwatch season, December 1^st. This animation allows one to view between-season differences in winter migration. From this animation the following patterns can be seen:

• Alternating high and low occurrence probability in the Ontario - Maine regions characterize the irruptive winter migration of American Goldfinch.
• There is a relatively stable region of high occurrence probability from South of the great lakes to the Central Hardwoods of the Ozark Mountains.
• Little change in the western population.

 
The second animation in Figure 4 allows one to visualize within-season winter migration patterns. The animation below shows the estimated probability of occurrence from November 15 – April 1 for the 2003-04 Feederwatch season. From this animation the following patterns can be seen:

• The last part of the irruptive migration can be seen as the region of high occurrence probability in the North East moves southward in early winter.
  
• The region of high occurrence probability south of the great lakes moves southward until February and then it appears to start moving north as spring begins.