RMSEszer

RMSEszer is a scale-adjusted variant of the root mean squared error used to evaluate predictive models. It is designed to enable fair comparisons when different datasets or models yield residuals with varying variances, by normalizing the standard RMSE with a dispersion-based scale.

The core idea behind RMSEszer is to divide the RMSE by a scale estimate that reflects the spread of the residuals. A common choice for the scale is sigma_hat, an estimate of the residual standard deviation, derived from the model or from the data. Robust alternatives include using the median absolute deviation (MAD) of the residuals, scaled to approximate a standard deviation. The resulting RMSEszer is computed as RMSE divided by the chosen scale s.

Calculation typically follows these steps: fit the model and compute residuals e_i = y_i - y_hat_i; compute RMSE = sqrt(mean(e_i^2)); determine a scale s from the residuals or an accompanying uncertainty model; then calculate RMSEszer = RMSE / s. In multi-target settings, RMSEszer can be averaged across targets or aggregated using macro or micro schemes, depending on the benchmarking protocol.

RMSEszer is used in contexts where cross-dataset or cross-model comparisons are needed under differing noise levels or error distributions. It is common in benchmarking suites and in forecasting tasks involving heteroskedastic data, where a scale-adjusted metric helps highlight relative predictive accuracy independent of residual variance.

Advantages of RMSEszer include improved comparability across heterogeneous problems and robustness when a robust scale estimator is used. Limitations include dependence on the chosen scale estimator, potential interpretability challenges, and the need for reliable estimation of residual dispersion to avoid biased conclusions. Related metrics include RMSE, normalized RMSE, and scale-invariant error measures.

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