Beyond RMSE and Granger Gain: Multi-Horizon Forecast-Error Entropy for Nonlinear Predictive Coupling

Andrei Velichko

nlin.CDOct 13, 2026

Beyond RMSE and Granger Gain: Multi-Horizon Forecast-Error Entropy for Nonlinear Predictive Coupling

Authors: Andrei Velichko

Organizations: Institute of Physics and Technology, Petrozavodsk State University, Petrozavodsk, Russia

Abstract

We investigate whether the past of an auxiliary time series provides predictive information about a target beyond that contained in the target's own history. We compare matched out-of-sample target-only and target-plus-source forecasts and introduce the forecast-error distribution entropy (FEDE) predictive-causality framework, where predictive causality is understood in the Granger sense: auxiliary history is informative when it improves prediction beyond the target's own past. FEDE quantifies how the full signed forecast-error distribution changes across prediction horizons. Controlled nonlinear benchmarks hide source-target dependence behind nonlinear transformations not supplied to the predictors. LightGBM recovers these hidden channels, whereas Ridge does not. Across four equal-power nonlinearities and four noise levels, the mean response-sensitivity statistic for FEDE remains 29.00, 27.47, 23.22, and 28.67, compared with 14.09, 10.68, 12.67, and 12.71 for mean log-RMSE gain. Directional and source controls separate the true source-to-target channel from the reverse direction and from independent periodic, quasiperiodic, chaotic, and autoregressive auxiliaries. Chronological Granger-style comparisons show strong model-form dependence: a degree-2 polynomial model is exceptionally sensitive to the matched quadratic channel but becomes weak or oppositely aligned for other nonlinearities, while basic convergent cross mapping is weak or directionally inconsistent in this stochastic, partly non-invertible setting. FEDE remains positively aligned across all 108 tested combinations of window length, history length, and maximum forecast horizon. Together, these results establish FEDE as a sensitive and robust horizon-resolved tool that extracts predictive information beyond conventional error-magnitude measures when the nonlinear interaction form is unknown. Code available in Zenodo.

Figures & tables

Method family	Core question	Typical output	Role of forecast horizon	Main limitation for the present problem	Representative references
Linear Granger causality	Does past $y$ reduce the prediction-error variance of $x$ beyond past $x$ in a linear model?	$F$ /Wald statistic, $p$ value, GC index, variance ratio	Usually one-step or lag-model based	Misses nonlinear interactions when the linear basis is inadequate; sensitive to lag/model specification	[ 9 , 15 ]
Nonlinear / ML Granger	Same restricted-versus-full question with a nonlinear predictor or feature basis	Nonlinear GC score, generalized variance, variable importance, test statistic	Usually one-step/few-step; model dependent	Sensitivity depends on architecture or chosen basis; greater tuning burden	[ 10 , 11 , 12 , 16 ]
Out-of-sample forecast gain	How much does adding $y$ improve held-out prediction of future $x$ ?	$\Delta$ RMSE, $\Delta$ MAE, loss ratio, OOS $R^{2}$	Chosen explicitly by analyst; may be single- or multi-horizon	Collapses forecast behavior to an accuracy summary and does not describe how the error distribution changes with $h$	[ 17 , 18 ]
Diebold–Mariano-type forecast comparison	Is the mean loss difference between two forecast systems distinguishable from zero?	DM statistic and $p$ value	Horizon-specific or extended to multiple horizons	Primarily a significance test, not a coupling-magnitude or error-structure descriptor	[ 17 , 18 ]
CCM / EDM	Is information about one variable recoverable from the reconstructed state of another, and does reconstruction converge?	Cross-map skill $\rho(L)$ , convergence, nonlinear forecast skill, local coefficients	Library size and embedding are central; forecast horizon enters EDM variants	Reconstruction and predictive contribution are not equivalent; noise, synchronization, embedding, and common forcing matter	[ 13 , 20 , 21 , 14 ]
FEDE predictive-causality framework (this work)	How does adding $y$ change the distributional evolution of out-of-sample forecast errors for $x$ across horizons?	Directed FEDE $\Delta$ response and early-horizon entropy-slope response	Explicitly uses $h=1,\ldots,H$ as the analysis axis	Model-relative; requires multi-horizon forecasts and an adequate predictive engine	Present study

Table 1: Representative method families for directional or incremental dependence between time series. The table emphasizes the object each method measures rather than treating the methods as interchangeable causal estimators.

Panel of Figure 1 — Figure 1: Core construction for auxiliary predictive contribution. The two branches forecast exactly the same future target values. The only information difference is the availability of the aligned auxiliary history $y$ . Conventional accuracy gain and FEG/FEDE descriptors are all computed from this same matched pair of out-of-sample forecasts.

Method	Within-window object	Scalar response	Native axis
FEDE $\Delta$ + LightGBM	signed-error entropy profile	$D_{\mathrm{FEDE}}$	$h=1,\ldots,14$
Mean log-RMSE + LightGBM	forecast-error magnitude	$G_{\log\mathrm{RMSE}}$	$h=1,\ldots,14$
Linear Granger-style OOS	restricted vs full OLS	Eq. ( 27 )	one step
Polynomial Granger-style OOS	restricted vs degree-2 full OLS	Eq. ( 27 )	one step
CCM	source reconstruction from target manifold	$\rho(90)-\rho(30)$	library size

Table 2: Scalar quantities used in the method-comparison benchmark. Each method is evaluated on the same

W=100

windows and the same controlled

\gamma

scan, but its native within-window construction is preserved.

Benchmark	Generating family	Main purpose	Results
L1	logistic-map $y,q$	calibration to known direct linear coupling	Sec. 4.1
L2	logistic-map $y,q$	weak coupling, process noise, and null-relative detection	Secs. 4.2 – 4.4
N1	Gaussian AR(1) $y,q$	hidden nonlinear extraction and Ridge–LightGBM comparison	Secs. 4.5 – 4.6
N2	Gaussian AR(1) $y,q$	time-resolved tracking of intermittent coupling	Sec. 4.9
N3	Gaussian AR(1) $y,q$	direction and source specificity	Sec. 4.7
N4	Gaussian AR(1) $y,q$	robustness to hidden functional form; Granger/CCM comparison	Secs. 4.8 , 4.10

Table 3: Roadmap of the controlled synthetic benchmarks. L1–L2 and N1–N5 are separate generating families; they share the same matched forecasting logic but not the same source/background process.

Setting	Benchmark L1	Benchmark L2
Purpose	calibration	weak coupling + noise
Driver family	logistic map	logistic map
Realizations	10	30
Retained samples	5000	3000
Window length $W$	100	100
Forecast horizon $H$	14	14

Table 4: Main settings of the linear-reference benchmarks L1 and L2.

Setting	N1 stationary	N2 intermittent
Tested source	Gaussian AR(1), $\varphi_{y}=0.60$	same
Background forcing	Gaussian AR(1), $\varphi_{q}=0.35$	same
Hidden transform	$\phi_{2}(y)=(y^{2}-1)/\sqrt{2}$	same
Predictor input	raw $y$ only	raw $y$ only
Realizations	20	10
Retained samples	3000	8000

Table 5: Settings of the stationary and intermittent hidden nonlinear benchmarks N1 and N2.

Figure 2: Stationary linear-coupling calibration at $\sigma=0$ . Each metric is endpoint-normalized to emphasize response shape rather than numerical scale. The dashed line is the ideal linear response.

Predictor	Metric	$\sigma=0$	$\sigma=0.20$	$\sigma=0.80$
Ridge	mean log-RMSE gain	0.952	0.941	0.920
Ridge	FEDE delta change	0.995	0.910	0.832
Ridge	FEDE early-slope change	0.994	0.934	0.862
LightGBM	mean log-RMSE gain	0.964	0.933	0.887
LightGBM	FEDE delta change	0.984	0.904	0.545
LightGBM	FEDE early-slope change	0.986	0.923	0.365

Table 6: Linearity of the metric-versus-coupling calibration, measured by

R^{2}

of a linear fit to the condition means. Selected history-

1

metrics are shown.

Figure 3: Weak-coupling zoom at zero process noise. Lines show condition means and shaded bands show between-realization standard deviations. The Ridge FEDE descriptors respond to weak auxiliary coupling before a stable conventional log-RMSE gain appears.

Figure 4: Linearity of metric-versus- $\gamma$ calibration as process noise increases. The dashed line marks $R^{2}=0.95$ .

Figure 5: Minimum coupling strength required for sustained 80% null-relative detection using predictive metrics only. Missing points indicate that the criterion was not reached within the tested range $\gamma\leq 0.30$ .

Predictor	Metric	0	0.05	0.10	0.20	0.40	0.80
Ridge	mean log-RMSE gain	0.150	0.150	0.150	0.150	0.300	$>0.30$
Ridge	FEDE delta change	0.050	0.075	0.150	0.200	0.300	$>0.30$
Ridge	FEDE early-slope change	0.050	0.075	0.150	0.200	0.300	$>0.30$
LightGBM	mean log-RMSE gain	0.150	0.150	0.150	0.200	0.300	$>0.30$
LightGBM	FEDE early-slope change	0.100	0.150	0.200	0.300	$>0.30$	$>0.30$

Table 7: Sustained 80% detection limit

\gamma_{\min}

for the main history-

1

predictive metrics.

>0.30

means that the criterion was not reached on the tested grid.

Figure 6: Validation of the hidden nonlinear channel. Lines are means over 20 realizations and shaded bands are 95% confidence intervals of the mean. (a) The raw one-step Pearson association between $y_{t}$ and $x_{t+1}$ remains near zero. (b) The synthetic oracle based on the known hidden transform $\phi(y_{t})$ increases strongly with the imposed coupling $\gamma$ .

Figure 7: Nonlinear specificity at $\sigma=0.10$ and history $m=1$ . Each point is the mean of 20 realization-level values; each realization value is itself averaged over 30 non-overlapping $W=100$ windows. Shaded bands are 95% confidence intervals of the mean across realizations. LightGBM develops a systematic coupling-dependent response, whereas Ridge remains close to its null FEDE response.

Metric	$\bar{\beta}$ at $\sigma=0.10$ [95% CI]	$T_{0}$	$T_{0.05}$	$T_{0.10}$	$T_{0.20}$	mean $T$
Mean log-RMSE gain	0.0379 [0.0321, 0.0437]	17.84	15.58	13.74	12.52	14.92
FEDE $\Delta$ change	0.0981 [0.0902, 0.1059]	27.04	24.50	26.24	29.06	26.71
FEDE early-slope change	0.0230 [0.0210, 0.0250]	25.87	29.80	24.24	26.21	26.53

Table 8: Noise-normalized response sensitivity

T

for LightGBM, history

m=1

, in the stationary hidden nonlinear benchmark. For each realization, a linear slope is fitted across the full 13-point

\gamma

grid;

T

is then computed from Eq. ( 22 ). The second column gives the mean slope and its 95% confidence interval at

\sigma=0.10

Metric	$T_{0}$	$T_{0.05}$	$T_{0.10}$	$T_{0.20}$	mean $T$
Mean log-RMSE gain	-3.37	-3.56	-3.65	-3.32	-3.47
FEDE $\Delta$ change	-0.07	-1.20	-0.97	-0.86	-0.77
FEDE early-slope change	-1.73	-1.34	-1.19	-0.29	-1.14

Table 9: Signed response sensitivity

T

for Ridge, history

m=1

, in the stationary hidden nonlinear benchmark. Negative

T

denotes a dose response opposite to the predefined positive orientation of the metric. A large

|T|

would still indicate reproducible sensitivity, but in the inverse direction; values near zero indicate little reproducible response.

Figure 8: Minimum hidden nonlinear coupling required for sustained 80% and 90% null-relative detection with LightGBM, history $m=1$ . The FEDE descriptors reduce the strict 90% threshold relative to mean log-RMSE gain in several low-to-moderate noise cases.

Candidate source / direction	$T_{0}$	$T_{0.05}$	$T_{0.10}$	$T_{0.20}$	mean $\|T\|$
$Y\rightarrow X$ (true nonlinear driver)	27.04	24.50	26.24	29.06	26.71
$X\rightarrow Y$ (reverse)	0.03	0.99	1.23	0.15	0.60
Sinusoid $\rightarrow X$	0.34	0.50	0.98	2.49	1.08
Quasiperiodic $\rightarrow X$	1.01	-0.07	0.26	0.59	0.48
Logistic chaos $\rightarrow X$	-0.55	1.21	0.42	0.29	0.62
Independent AR(1) $\rightarrow X$	0.20	-0.30	0.96	0.11	0.39

Table 10: Directional and source specificity of the LightGBM FEDE

\Delta

response.

T

is the signed response-sensitivity statistic obtained from 20 realization-wise slopes across the complete coupling grid. The final column reports mean

|T|

across noise levels so that positive and negative control responses cannot cancel.

Method	$H_{2}$	$H_{3}$	$H_{4}$	$\sin(y^{2})$
FEDE $\Delta$ + LightGBM	29.00	27.47	23.22	28.67
Mean log-RMSE + LightGBM	14.09	10.68	12.67	12.71
Linear Granger-style OOS	-7.94	-3.08	-6.70	-3.56
Polynomial Granger-style OOS, degree 2	74.21	-6.77	3.53	-4.01
CCM convergence $\Delta\rho$	1.85	9.07	0.36	0.02

Table 11: Signed response-sensitivity statistic

T

, averaged over

\sigma\in\{0,0.05,0.10,0.20\}

, for four equal-power hidden nonlinear channels. Positive values are aligned with increasing useful predictive contribution. The polynomial Granger basis is fixed at degree two for all channels.

Figure 9: Benchmark N2: time-resolved hidden nonlinear coupling at $\sigma=0.10$ , history $m=1$ , and rolling-window length $W=400$ . (a) Exact imposed coupling schedule with 1000 samples per block. (b) Rolling raw linear correlation $r(y_{t},x_{t+1})$ and nonlinear-oracle correlation $r(\phi_{2}(y_{t}),x_{t+1})$ . (c) FEDE $\Delta$ change and (d) FEDE early-slope change for LightGBM and Ridge. Curves are unsmoothed ensemble means over 10 realizations with rolling step 20 and $H=14$ . Negative FEDE values are retained numerically and shown with lower opacity rather than clipped.

Figure 10: Benchmark N2 tracking quality against the known window-mean imposed coupling $\bar{\gamma}(\tau)$ . (a) Pearson and (b) Spearman correlations for mean log-RMSE gain, FEDE $\Delta$ , and FEDE early-slope change. Solid curves use $W=400$ and dashed curves use $W=200$ ; circles denote LightGBM and squares denote Ridge. Correlations are computed from the unsmoothed ensemble-mean rolling responses.

Method	seconds / 100 windows	relative to FEDE
FEDE $\Delta$ + LightGBM	29.451	1.000
Mean log-RMSE + LightGBM	28.820	0.979
CCM convergence $\Delta\rho$	1.864	0.063
Linear Granger-style OOS	0.0158	0.00054
Polynomial Granger-style OOS, degree 2	0.0207	0.00070

Table 12: Approximate computation time for 100 true-direction window evaluations. FEDE and mean log-RMSE share the same LightGBM forecast core. OOS Granger timings come from the dedicated chronological control run.

Hidden channel	Primary $T$	Grid mean $T$	Grid min $T$	Grid max $T$
$H_{2}$	32.95	41.34	16.49	55.35
$H_{3}$	24.31	28.32	16.90	42.88
$H_{4}$	20.35	22.30	18.02	32.13
$\sin(y^{2})$	29.71	45.39	17.01	73.78

Table 13: FEDE parameter robustness at

\sigma=0.10

. “Primary” is the configuration

W=100

m=1

H=14

used in the main nonlinear experiments. Grid statistics summarize all 27 combinations in Eq. ( 48 ) for each hidden channel.

Panel of Figure 11 — Figure 11: Marginal FEDE response sensitivity over the predefined parameter grid at $\sigma=0.10$ . (a) Window length $W$ ; (b) history length $m$ ; (c) maximum forecast horizon $H$ . Curves show the four hidden nonlinear channels and their overall mean. For each channel, the marker is the mean $T$ after averaging over the other two grid parameters; vertical bars span the minimum and maximum over those remaining configurations and are not confidence intervals.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 12: Stationary hidden nonlinear coupling response for LightGBM across the complete noise grid. Bands show 95% confidence intervals of the mean across 20 realizations.

Figure 13: Benchmark N2 supplementary time-domain comparison at $W=200$ , $\sigma=0.10$ , history $m=1$ , rolling step 20, and $H=14$ . The imposed schedule and all stochastic realizations are the same as in the $W=400$ main-text analysis. The shorter window preserves sharper temporal localization around switching boundaries but yields a visibly less stable FEDE response. Negative FEDE values are retained and displayed with lower opacity rather than clipped.

Figure 14: Linearity diagnostic as the fitted coupling range is expanded. The main text therefore emphasizes the complete controlled calibration and the noise-dependent linearity analysis rather than any single very narrow range.

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