How does one measure the fit of a model to data? Supppose data are (y_1,…,y_n), and the estimate from the model is (x_1,…,x_n). Then one can simply measure fit by the correlation of x and y, or by the root-mean-squared error (the square root of the average of the (y_i-x_i)^2’s).

When the n data points have structure, however, such simple pointwise error measures may miss the big picture. For example, suppose x and y are time series (that is, the n points are in a sequence), and x is a perfect predictor of y but just lagged by 2 time points (so that x_1=y_3, x_2=x_4, x_3=y_5, and so forth). Then we’d rather say that our error is “a lag of 2” rather than looking at the unlagged pointwise errors.

More generally, the lag need not be constant; thus, for example, there could be an error in the lag with standard deviation 1.3 time units, and an error in the prediction (after correcting for the lag) with standard deviation 0.4 units in the scale of y. Hence the title of this entry.

We have applied this idea to examples in time series and spatial statistics. Summarizing fitting error by a combination of distortion and additive error seems like a useful idea. It should be possible to do more by further decomposing fitting error.

For more, see the paper by Cavan Reilly, Phil Price, Scott Sandgathe, and myself (to appear in the journal Biometrics).