Asymmetric Bidirectional Predictive Dynamics in Google Search and COVID-19 Time Series: A Multi-Horizon Forecast-Error Analysis

Andrei Velichko; Imen Mahmoud

cs.LGOct 20, 2026

Asymmetric Bidirectional Predictive Dynamics in Google Search and COVID-19 Time Series: A Multi-Horizon Forecast-Error Analysis

Authors: Andrei Velichko, Imen Mahmoud

Organizations: Institute of Physics and Technology, Petrozavodsk State University, 33 Lenina Prospekt, Petrozavodsk 185910, Republic of Karelia, Russia · University of Mannouba, Tunisia · Laboratory of Research in Innovative and Management, Risk, Accounting and Finance

Abstract

Online search activity is widely used as a digital trace of information seeking during epidemics, but its predictive value is often reduced to correlation, a selected lag, or a single forecast-error score. We examine eight pre-specified Google Trends-COVID-19 pairs in both directions using matched LightGBM forecasts over rolling 400-day windows and horizons of 1-14 days. Log-RMSE gain is combined with forecast-error distribution entropy (FEDE), which captures horizon-dependent changes in the signed error distribution. The resulting directional predictive signature comprises response class, RMSE effect, FEDE endpoint and early-slope responses, persistence across rolling regimes, and horizon profile. For screening candidate directions, we also introduce a relative persistence-weighted directional predictive response strength (DPRS) on a within-study 0-100 scale; 100 denotes the strongest observed response in the comparison set, not an absolute coupling maximum. The pairs show concordant, RMSE-only, FEDE-only, and mixed responses. "loss of smell" -> reported cases is accuracy-dominant, with mean log-RMSE gain 0.104 (about 9.9% equivalent RMSE reduction) and DPRS 100, whereas the reverse direction is FEDE-dominant (D_FEDE=0.048; DPRS=42). "covid test" -> positive rate is FEDE-dominant (D_FEDE=0.039; positive in 72% of windows; DPRS=98) despite negative RMSE gain, while "lockdown" -> stringency is the clearest concordant case (DPRS=78). World-level symptom/case pairs are less persistent. Thus, forecast accuracy and error-distribution dynamics are complementary, predictive effects can be direction- and metric-dependent, and forecast horizon is part of the relationship. Aggregated regime-level patterns should not be extrapolated to individual time points or interpreted as evidence of causality.

Figures & tables

Rank	Direction	Class	RMSE effect	FEDE $D/E$	Positive windows $G/D/E$	Rel. DPRS	Dominant reading
1	B3 “loss of smell” $\rightarrow$ cases	II	$+9.9\%$	$-0.018/-0.003$	64/40/41%	100	Strong persistent accuracy response; gain increases with $h$
2	B2 “covid test” $\rightarrow$ positive rate	III	$-7.0\%$	$+0.039/+0.006$	18/72/69%	98	Strong persistent FEDE response while RMSE worsens
3	B7 “lockdown” $\rightarrow$ stringency	I	$+0.9\%$	$+0.040/+0.003$	73/72/62%	78	Clearest concordant response; FEDE grows with $h$
4	B1 “covid symptoms” $\rightarrow$ hospitalized	III	$-1.8\%$	$+0.019/+0.006$	57/57/67%	45	Persistent forward distributional tendency, strongest in early slope
5	B3 cases $\rightarrow$ “loss of smell”	III	$-1.8\%$	$+0.048/+0.001$	35/66/46%	42	Endpoint FEDE response without accuracy gain
6	B7 stringency $\rightarrow$ “lockdown”	III	$-22.7\%$	$+0.001/+0.006$	40/53/66%	38	Early FEDE response; no positive RMSE contribution

Table 1

Time Series Forecasting

Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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

Andrei Velichko

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

Time-Series Causal Discovery Multivariate Time Series Forecasting Forecasting Benchmarks

Open paper AskPDF arXiv

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.

Abstract

Time Series Forecasting

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