Inspect what a model leaves unexplained

A residual — open definition can be defined as observed value minus the model prediction. Positive residuals then mean the observation is above the prediction, and negative ones mean it is below. Always state the convention so readers do not reverse the interpretation.

For fictional predictions 10, 12 and 14 with observations 11, 11 and 16, the residuals are +1, −1 and +2. Their mean is about +0.67, but that summary alone does not show their timing, spread or relationship with the predicted value.

Residuals can reveal a trend that is hard to see beside the original signal. A repeated drift with time or a curved pattern with a predictor suggests that something systematic remains unexplained. A model can have an impressive overall fit statistic while still missing behavior relevant to its intended use.

Imagine a fictional orbit model whose position residual grows steadily after a maneuver. Possible explanations include an incorrect initial state, a missing force or a measurement-timing issue. The residual identifies a problem to investigate; it does not uniquely choose among these explanations.

Residual = observed − predicted

Adjusting parameters to match a dataset is fitting. Testing a prediction on suitable observations not used for that adjustment provides a different check. A very flexible model can reproduce its fitting data while predicting poorly elsewhere, especially outside the conditions it has encountered.

A useful review states the data used for fitting, the independent comparison, uncertainty assumptions and intended range — open definition of use. If a model is changed after inspecting the test set, that set has helped shape the model and is no longer a fully untouched test. Preserve that history when reporting performance.