Retrodictive Forecasting: A Proof-of-Concept for Exploiting Temporal Asymmetry in Time Series Prediction

Cedric Damour

cs.LGFeb 28, 2026

Retrodictive Forecasting: A Proof-of-Concept for Exploiting Temporal Asymmetry in Time Series Prediction

Authors: Cedric Damour

Organizations: ENERGY-Lab, Université de La Réunion, Saint-Denis, La Réunion, France

Abstract

We propose a retrodictive forecasting paradigm for time series: instead of predicting the future from the past, we identify the future that best explains the observed present via inverse MAP optimization over a Conditional Variational Autoencoder (CVAE). This conditioning is a statistical modeling choice for Bayesian inversion; it does not assert that future events cause past observations. The approach is theoretically grounded in an information-theoretic arrow-of-time measure: the symmetrized Kullback-Leibler divergence between forward and time-reversed trajectory ensembles provides both the conceptual rationale and an operational GO/NO-GO diagnostic for applicability. We implement the paradigm as MAP inference over an inverse CVAE with a learned RealNVP normalizing-flow prior and evaluate it on six time series cases: four synthetic processes with controlled temporal asymmetry and two ERA5 reanalysis datasets (wind speed and solar irradiance). The work makes four contributions: (i) a formal retrodictive inference formulation; (ii) an inverse CVAE architecture; (iii) a model-free irreversibility diagnostic; and (iv) a falsifiable validation protocol with four pre-specified predictions. All pre-specified predictions are empirically supported: the diagnostic correctly classifies all six cases; the learned flow prior improves over an isotropic Gaussian baseline on GO cases; the inverse MAP yields no spurious advantage on time-reversible dynamics; and on irreversible GO cases, it achieves competitive or superior RMSE relative to forward baselines, with a statistically significant 17.7% reduction over a forward MLP on ERA5 solar irradiance. These results provide a structured proof-of-concept that retrodictive forecasting can constitute a viable alternative to conventional forward prediction when statistical time-irreversibility is present and exploitable.

Figures & tables

Case	Verdict	Process	Noise structure	Irreversibility mechanism	Irreversibility strength ( $\Delta_{\mathrm{arrow}}$ )	Key parameters
A	GO	NLAR dissipative	$\sigma(s){=}\sigma_{0}{+}$ $\sigma_{1}\|s\|$	tanh contraction + cubic dissipation + multiplicative noise	Strong (LEVEL)	$\alpha{=}0.7$ , $\gamma_{q}{=}0.05$ , $\gamma_{c}{=}{-}0.08$ , $\sigma_{0}{=}0.3$ , $\sigma_{1}{=}0.35$
B	NO-GO	Symmetric RW	Gaussian increments	None	${\approx}\,0$	$T{=}20$ k, $\sigma{=}0.50$
C	GO	Shot-noise relaxation	Excitation $p{=}0.04$ , exp. relax.	Causal excitation-relax.	Very strong (LEVEL+DIFF)	$T{=}20$ k, decay $=0.95$ , $p_{\mathrm{shot}}{=}0.04$ , shot_scale $=1.5$ , $\sigma_{\mathrm{obs}}{=}0.05$
D	NO-GO	Noisy sinusoid	Symmetric noise on period	None (symmetric)	${\approx}\,0$	$T{=}20$ k, $A{=}1.0$ , $P{=}40$ , $\sigma{=}0.50$
ERA5	GO	North Sea wind (10m)	ERA5 reanalysis 2023	Turbulence cascade	Moderate (DIFF)	$T{=}8{,}736$ h, $56^{\circ}$ N $3^{\circ}$ E
ERA_ssrd	GO	North Sea solar (SSRD)	ERA5 reanalysis 2023	Cloud attenuation	Strong (DIFF)	$T{\approx}5{,}110$ h (daylight), $56^{\circ}$ N $5^{\circ}$ E

Table 1: Process suite summary. All six cases with generative mechanism, noise structure, expected irreversibility, and key parameters.

ID	Name	Training obj.	Inference	Description
M1	Naïve mean	—	$\hat{\bm{y}}=\bar{\bm{y}}_{\mathrm{train}}$	Unconditional mean of training futures. Zero-information baseline.
M2	Forward MLP	MSE	$\hat{\bm{y}}=f(\bm{x})$	2-hidden-layer MLP (128 units, ReLU), trained on $(\bm{x},\bm{y})$ pairs.
M3	Forward CVAE	ELBO (forward)	$\hat{\bm{y}}\sim p_{\theta}(\bm{y}\mid\bm{x})$	Standard CVAE with prior $p(\bm{z}\mid\bm{x})$ , decoder $p_{\theta}(\bm{y}\mid\bm{x},\bm{z})$ . Samples averaged.
M4	Inverse MAP (flow prior)	ELBO (inverse)	$(\hat{\bm{y}},\hat{\bm{z}})=\mathrm{MAP}$	Inverted CVAE with RealNVP flow prior. MAP over $(\bm{y},\bm{z})$ . Primary method.
M5	Inverse MAP (N(0,I) prior)	ELBO (inverse)	$(\hat{\bm{y}},\hat{\bm{z}})=\mathrm{MAP}$	Same as M4 but with isotropic Gaussian prior $p(\bm{y})=\mathcal{N}(\mathbf{0},\mathbf{I})$ . Ablation for P2.

Table 2: Methods compared. Five evaluation methods enabling ablation of retrodictive paradigm components.

ID	Statement	Cases	Metric	Pass criterion	Observed
P1	Arrow-of-time diagnostic matches expected verdicts	A–D, ERA5, ERA_ssrd	GO / NO-GO verdict	GO : A,C,ERA5, ERA_ssrd; NO-GO : B,D	A,C,ERA5,ERA_ssrd $\to$ GO ; B,D $\to$ NO-GO
P2	Flow prior improves inverse inference vs $\mathcal{N}(0,I)$ on GO cases	A, C	RMSE	RMSE ${}_{\mathrm{flow}}<$ RMSE N(0,I)	A: $1.038<1.074$ ; C: $0.782<0.872$
P3	No meaningful directional advantage on NO-GO cases	B, D	Ratio Inv/MLP	$\geq 0.95$ ( $\pm 5\%$ )	B: 1.870; D: 0.984
P4	Inverse MAP competitive with or beats MLP on GO cases	A, C, ERA5, ERA_ssrd	Ratio Inv/MLP	$\leq 1.05$ ( $\pm 5\%$ )	A: 0.897; C: 1.014; ERA5: 0.991; ERA_ssrd: 0.823

Table 3: Falsifiable predictions. Four pre-specified predictions with pass criteria and observed outcomes (Section 9 ).

Figure 1: Training curves (ELBO loss) for all six cases. Solid: training loss; dashed: validation loss. Stable convergence without overfitting across all cases confirms correct CVAE training.

Figure 2: Cross-case RMSE comparison (all six cases, five methods). The inverse MAP (flow, green) is competitive or best on all GO cases, while the forward MLP (orange) clearly dominates on Case B ( NO-GO ). Case D shows near-perfect equivalence across methods.

Figure 3: RMSE per forecast horizon $h\in\{1,\ldots,16\}$ for all six cases and five methods. GO cases (A, C, ERA5, ERA_ssrd): the inverse MAP with flow prior (dark green) is competitive with or below the forward MLP (orange) across most horizons, with the advantage widening at longer horizons on ERA_ssrd. NO-GO cases (B, D): the forward MLP dominates on Case B; Case D shows near-equivalence consistent with time-reversibility. The N(0,I) prior (pink) uniformly underperforms the flow prior on GO cases. Compare with Fig. 10 ( GO cases only, overlay).

Figure 4: Falsifiable predictions scorecard. All 4/4 predictions PASS. Summary panel produced by the automated verification module.

Case	Verdict	Naïve	MLP	CVAE	Inv(flow)	Ratio	DM stat	DM $p$ -value
A	GO	1.030	1.156	1.048	1.038	0.897	$-22.9$	<0.001 ✓
B	NO-GO	1.728	0.066	0.087	0.124	1.870	$+19.3$	$<0.001$
C	GO	0.883	0.772	0.749	0.782	1.014	$+1.7$	$0.092$ (ns)
D	NO-GO	1.001	0.622	0.616	0.612	0.984	$-8.1$	$<0.001$
ERA5	GO	1.384	1.034	0.931	1.025	0.991	$-0.5$	$0.642$ (ns)
ERA_ssrd	GO	0.963	0.195	0.168	0.160	0.823	$-10.4$	<0.001 ✓

Table 4: Main results. RMSE in standardized space for all six cases and five methods. Ratio

=

RMSE Inv(flow) / RMSE MLP . Green ratios satisfy the paradigm prediction; DM stat and p-value from two-sided Diebold–Mariano test vs. MLP. ✓

=

paradigm-confirming significant result.

Figure 5: Arrow-of-time diagnostic (block permutation test) for all six cases. Top panels: $J_{\mathrm{obs}}(w)$ at $w=2,4,8$ for LEVEL (left) and DIFF (right) representations. Bottom panels: $-\log_{10}(p)$ with the $\alpha=0.05$ threshold (red dashed). GO cases (A, C, ERA5, ERA_ssrd) exceed the threshold; NO-GO cases (B, D) remain below. Case C dominates with $J_{\mathrm{obs}}$ up to 22.9 in LEVEL; ERA_ssrd leads in DIFF with $J_{\mathrm{obs}}=5.3$ .

Figure 6: Arrow-of-time strength per case ( $J_{\mathrm{obs}}$ median across all windows and scales). GO cases (green) are clearly separated from NO-GO cases (orange). Case C dominates ( $J_{\mathrm{obs}}=12.84$ ); ERA_ssrd is the strongest real-world case (5.30). Cases B and D score $\approx 0$ .

Figure 7: Prior ablation study (P2). RMSE comparison between Inverse MAP with flow prior (green) and N(0,I) prior (pink) for all six cases. The flow prior is systematically better on GO cases; improvements range from $3.3\%$ (Case A) to $37\%$ (ERA_ssrd).

Figure 8: MAP loss distributions (flow prior vs. N(0,I) prior) for all six cases. Lower MAP loss indicates better objective convergence. On GO cases (A, C, ERA5, ERA_ssrd), the flow prior distribution (green) is shifted toward lower loss values. The separation is largest for ERA_ssrd (flow median $-121.6$ vs. N(0,I) median $-80.6$ ). NO-GO cases (B, D) show overlapping distributions.

Figure 9: Forward-Inverse Chaining (FIC) landscape diagnostic for GO cases (A, C, ERA5, ERA_ssrd). Left panel: scatter of multi-start standard deviation (optimization roughness) vs. best-of-K MAP loss; right panel: cumulative distribution of MAP loss (flow prior). Low dispersion and a concentrated CDF indicate a tractable landscape. ERA_ssrd (P10–P90 range: 14.7 units, $\sigma\approx 0.05$ ) and Case A ( $\sigma<0.15$ ) exhibit well-conditioned landscapes; Case C ( $\sigma=0.35$ ) and ERA5 ( $\sigma=0.23$ ) show greater roughness.

Figure 10: RMSE per forecast horizon $h$ for GO cases only (forward CVAE vs. inverse MAP; shaded $=$ inverse advantage). ERA_ssrd shows expanding advantage at $h>8$ ; ERA5 shows crossover at $h\approx 6$ ; Cases A and C show moderate inverse advantage at mid-range horizons.

Figure 11: Example reconstructions for Case ERA_ssrd (representative). Left: best sample (RMSE $=0.029$ ); centre: median (0.130); right: worst (0.357). Inverse MAP (green) tracks the true future (dashed black) closely in the best and median cases; forward MLP (orange) degrades at longer horizons.

Figure 12: Multi-start dispersion (mean std of best- $K$ MAP solutions) per case. Higher dispersion indicates a rugged optimization landscape. Case C (0.35) and ERA5 (0.23) are most challenging; ERA_ssrd (0.05) and Case A (0.08) are best-conditioned—consistent with their superior inverse MAP performance.

Figure 13: RetroNLL ( $-\log p_{\theta}(\bm{x}_{\mathrm{obs}}\mid\hat{\bm{y}},\hat{\bm{z}})$ ) vs. per-sample RMSE for GO cases. Theoretically expected: a negative correlation (low RetroNLL $\Leftrightarrow$ accurate prediction). Case A: $r=-0.037$ , $p=0.100$ (ns)—no significant correlation, consistent with correct decoder coupling. Case C: $r=-0.070$ , $p=0.002$ ✓—expected negative correlation confirmed. Cases ERA5 ( $r=+0.172$ , $p<0.001$ ) and ERA_ssrd ( $r=+0.088$ , $p=0.012$ ) show positive correlations, indicating partial decoder likelihood–RMSE decoupling attributable to the homogeneous Gaussian reconstruction assumption.

ID	Statement	Cases	Pass criterion	Verdict Observed values
P1	Arrow-of-time diagnostic matches expected verdicts	A–D, ERA5, ERA_ssrd	GO / NO-GO verdict	PASS all 6 cases correctly classified
P2	Flow prior improves over $\mathcal{N}(0,I)$ on GO cases	A, C	RMSE ${}_{\mathrm{flow}}<$ RMSE N(0,I)	PASS A: $-3.3\%$ ; C: $-10.3\%$
P3	No meaningful directional advantage on NO-GO cases	B, D	Ratio Inv/MLP $\geq 0.95$	PASS B: 1.870; D: 0.984
P4	Inverse MAP competitive on GO cases	A, C, ERA5, ERA_ssrd	Ratio Inv/MLP $\leq 1.05$	PASS A: 0.897; C: 1.014; ERA5: 0.991; ERA_ssrd: 0.823

Table 5: Falsifiable predictions scorecard. All four predictions pass. Observed values from experimental results (Sections 9 .2– 9 .5).

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