**Context**

Most introductory data mining texts include substantial coverage of model testing. Various methods of assessing true model performance (holdout testing, k-fold cross validation, etc.) are usually explained, perhaps with some important variants, such as stratification of the testing samples.

Generally, all of this exposition is aimed at in-time analysis: Model development data may span multiple time periods, but the testing is more or less blind to this: all periods are treated as fair game and mixed together. This is fine for model development. Once predictive models are deployed, however, it is desirable to continue testing to track model performance over time. Models which degrade over time need to be adjusted or replaced.

**Subtleties of Testing Over Time**

Nearly all production model evaluation is performed with new out-of-time data. As new periods of observed outcomes become available, they are used to calculate running performance measures. As far it goes, focusing on the actual performance metric makes sense. In my experience, though, some clients become distracted by movement in the independent variables or in the predicted or actual outcome distributions, in isolation. It is important to understand the dynamic of these changes to fully understand model performance over time.

For the sake of a thought experiment, consider a very simply problem with one independent variable, and one target variable, both real numbers. Historically, the distribution of each of these variables has been confined to specific ranges. A predictive model has been constructed as a linear regression which attempts to anticipate the target variable, using only the input of the single independent variable (and a constant). Assume that errors observed in the development data have been small and otherwise unremarkable (they are distributed normally, their magnitude is relatively constant across the range of the independent variable, there is no obvious pattern to them and so forth).

Once this model is deployed, it is executed on all future cases drawn from the relevant statistical universe, and predictions are saved for further analysis. Likewise, actual outcomes are recorded as they become available. At the conclusion of each future time period, model performance within that period is examined.

Consider the simplest change to well-developed model: the distribution of the independent variable remains the same, but the actual outcomes begin to depart the regression line. Any number of changes could be taking place in the output distribution, but the predicted distribution (the regression line) cannot move since it is entirely defined by the independent variable, which in this case is stable. By definition, model performance is degrading. This circumstance is easy to diagnose: the dynamic linking the target and independent variables is changing, hence a new model is necessary to restore performance.

What happens, though, when the independent variable begins to migrate? There are two possible effects (in reality, some combination of these extremes is likely): 1. The distribution of actual outcomes will either shift to appropriately match the change ("the dots march along the regression line"), or 2. The distribution of actual outcomes does not shift to match the change. In the first case, the model continues to correctly identify the relationship between the target and the independent variable, and model performance will more-or-less endure. In the second case, reality begins to wander from the model and performance deteriorates. Notice that, in the second case, the actual outcome distribution may or may not change noticeably- either way, the model no longer correctly anticipates reality and needs to be updated.

**Conclusion**

The example used here was deliberately chosen to be simple, for illustrations' sake. Qualitatively, though, the same basic behaviors are exhibited by much more complex models. Models featuring multiple independent variables or employing complex transformations (neural networks, decision trees, etc.) obey the same fundamental dynamic. Given the sensitivity of nonlinear models to each of their independent variables, a migration in even one of them may provoke the changes described above. Consideration of the components of this interplay in isolation only serves to confuse: Changes over time can only be understood as part of the larger whole.