Forecast coupling beyond RMSE: multi-horizon error entropy holds its sign across four hidden nonlinearities

The object of predictive-causality assessment shifts from error magnitude to how the error distribution evolves across forecast horizons. Whether an auxiliary history adds predictive information is answered not only by whether average loss falls, but by how the entropy profile of the signed forecast-error distribution changes between short and longer horizons. Prior practice compressed each horizon's error distribution into a single magnitude number, such as RMSE, MAE or out-of-sample R-squared, and compared that number with and without the auxiliary series; such measures reflect average error size but not how the error distribution's shape changes with horizon. Across four equal-power hidden nonlinear channels the noise-averaged response-sensitivity statistic is 29.00, 27.47, 23.22 and 28.67, while mean log-RMSE gain from the same LightGBM forecasts is 14.09, 10.68, 12.67 and 12.71; the two quantities share identical forecasts, so the difference isolates the summary.

A hidden nonlinear predictive channel can be extracted from the raw auxiliary series by a nonlinear predictor even when raw linear association is near zero. In the linear-reference benchmarks the injected auxiliary contribution is itself linear and Ridge regression is the natural reference; in the Gaussian AR(1) nonlinear benchmarks the source enters through a normalized even transform orthogonal to the raw first-order component, so raw Pearson correlation cannot expose the channel. At the strongest coupling the mean one-step raw Pearson correlation is only 0.0079 and 0.0082, while the synthetic oracle rises monotonically with coupling to 0.792, 0.784 and 0.761; LightGBM entropy responses rise with coupling whereas Ridge stays near zero or anti-aligned.

The entropy response points to the source and forecast direction that actually carry the predictive information, not to any structured series that is added. If only error magnitude is inspected, adding an unrelated but temporally structured auxiliary series can produce numerical fluctuations that are hard to separate from a genuine channel. The true channel's noise-averaged response-sensitivity statistic ranges from 24.50 to 29.06 across four noise levels with mean 26.71; reverse direction, sinusoid, quasiperiodic, chaotic and independent AR(1) controls have means between 0.39 and 1.08, and the independent AR(1) control matching the true source's first-order memory coefficient has mean only 0.39.

When the hidden interaction form is unknown the entropy response is more stable than fixed-basis comparators, while a correctly specified parametric model is more sensitive when its basis matches. Linear out-of-sample Granger-style controls are negative for all four hidden nonlinear channels, ranging from -7.94 to -3.08; the fixed degree-2 polynomial control is strong only when the true channel is quadratic. The degree-2 polynomial control reaches 74.21 on the matched quadratic channel but gives -6.77, 3.53 and -4.01 on the other three; the entropy response gives noise-averaged values of 29.00, 27.47, 23.22 and 28.67 across the four channels, a max/min ratio of 1.25.

The results rest on controlled synthetic benchmarks: the linear-reference family uses logistic-map drivers, the nonlinear family uses Gaussian AR(1) source and target, the hidden channels are four normalized nonlinear mappings, the predictors are Ridge and LightGBM, and convergent cross mapping is a basic implementation without lag scanning, surrogate testing or multivariate conditioning. Parameter robustness covers a predefined grid of window length, history length and maximum forecast horizon, 108 combinations in total, and conclusions are limited to those tested ranges. The authors propose next validating on observational time series, with candidate domains including financial and economic systems, climate and hydrology, epidemiological surveillance, physiology and neuroscience, and ecological interaction networks, compared against stronger domain-appropriate baselines.

The entropy descriptor is model-relative: if the prediction engine cannot extract a real interaction, the descriptor cannot recover information absent from its forecast errors, and a highly flexible predictor can exploit spurious structure unless the validation design, source controls and temporal split are chosen carefully. The synthetic benchmarks use one stochastic driver family, one target architecture, a finite set of hidden nonlinear mappings, one nonlinear prediction engine, and the parameter study establishes robustness only over the tested ranges rather than universal invariance. Real observational data add common drivers, delayed feedback, regime changes, missing observations, unequal sampling, multiple candidate sources and limited sample sizes, none of which are tested here. The authors also note that when the interaction form is known, a correctly specified low-dimensional parametric model can be both faster and more sensitive.