Valid Prediction Time (VPT)

• Valid Prediction Time (VPT) is a concept defining calibrated prediction horizons that satisfy explicit validity criteria in various settings.
• Different formulations—such as anytime-valid conformal risk control, vessel trajectory prediction, chaotic forecasting, and survival analysis—demonstrate its versatility and domain-specific applications.
• Practical implementations of VPT leverage methods like concentration bounds, regression-based confidence intervals, and reservoir computing to ensure controlled risk, error, or survival guarantees.

Searching arXiv for papers on Valid Prediction Time and related conformal risk-control formulations. Valid Prediction Time (VPT) denotes a prediction horizon or lower bound certified by an explicit validity criterion. In recent arXiv literature, the term appears in several technically distinct forms: an anytime-valid guarantee that a sequentially updated prediction set never exceeds a target risk level with probability at least $1-\delta$ (Hultberg et al., 4 Feb 2026); a vessel-trajectory reliability horizon $a_{90/95}$, defined as the largest forecast horizon whose $95\%$ lower-confidence bound on the probability of accurate prediction remains at least $0.90$ (Rastin et al., 19 Aug 2025); the first time at which normalized forecast error in a chaotic system crosses a fixed threshold (Hurley et al., 8 Aug 2025); and a covariate-specific lower bound on a right-censored survival time satisfying training-set conditional validity (Si et al., 4 May 2025). This suggests that VPT is best viewed as a family of validity-qualified horizon concepts rather than a single universal metric.

1. Terminological scope and formal variants

The literature summarized here uses the same term for several non-equivalent mathematical objects. What unifies them is not a common formula but a common role: each VPT is a horizon up to which a prediction remains valid under a stated criterion.

Setting	Formal object called VPT	Validity criterion
Anytime-valid conformal risk control	Implicitly, an infinite valid prediction time with high probability	$$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$
Vessel trajectory prediction	$a_{90/95}$	Largest $h$ such that the $95\%$ lower bound on $\mathrm{POAP}(h)$ is at least $0.90$
Reservoir computing for Lorenz	Threshold-crossing time	$a_{90/95}$0
Right-censored survival	Lower prediction bound $a_{90/95}$1	$a_{90/95}$2

In the sequential conformal setting, VPT is best understood as the duration over which a calibrated predictor continues to satisfy a risk constraint at every sample size. In vessel trajectory prediction, VPT is a reliability horizon attached to a detection-style confidence analysis. In chaotic forecasting, it is the first failure time of a normalized trajectory error. In right-censored survival analysis, VPT is not a time of future breakdown but a lower predictive bound for an event time, tailored to covariates and endowed with conditional validity.

2. Anytime-valid conformal risk control

In the sequential conformal formulation, calibration points $a_{90/95}$3 are observed i.i.d. from $a_{90/95}$4, possibly with an importance-weight $a_{90/95}$5 for test-time shift. One fixes a loss $a_{90/95}$6 that is monotone in the size of the prediction set $a_{90/95}$7, a target risk level $a_{90/95}$8, and a nested family of sets $a_{90/95}$9. Calibration produces a data-dependent $95\%$0, and the output is $95\%$1. The defining guarantee is

$95\%$2

where $95\%$3. The associated empirical risk is

$95\%$4

The key distinction from standard conformal formulations is that error is not merely controlled on average over many possible calibration datasets of fixed size; it remains valid with high probability over a cumulatively growing calibration dataset at any time point (Hultberg et al., 4 Feb 2026).

The main theorem assumes that $95\%$5 is monotone and right-continuous in $95\%$6, and introduces the stitched correction term

$95\%$7

With

$95\%$8

the update rule is

$95\%$9

The resulting sets satisfy

$0.90$0

For split conformal miscoverage, the loss becomes the $0.90$1–$0.90$2 loss $0.90$3, so $0.90$4, and the guarantee specializes to

$0.90$5

The formulation therefore turns VPT into a statement about indefinite sequential validity: with probability at least $0.90$6, the valid prediction time is infinite.

3. Concentration, tightness, and online operation

The proof architecture is quantile-based and handles discontinuities by a randomized mixture at the threshold $0.90$7. For each $0.90$8, the losses

$0.90$9

are right-continuous and nonincreasing in $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$0. Let $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$1 be the infimum of $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$2 with $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$3, and define a randomized mixture at any discontinuity so that

$$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$4

This yields the zero-mean process

$$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$5

whose increments lie in $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$6. The variance proxy is $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$7 and the scale parameter is $$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$8. A time-uniform stitched boundary gives

$$\Pr[\forall n:\;E(\ell(C_n(X),Y)\mid \mathcal Z_n)\le \alpha]\ge 1-\delta$$9

Rearranging the boundary and using the definition of $a_{90/95}$0 shows $a_{90/95}$1 on the same event, hence $a_{90/95}$2 for all $a_{90/95}$3 (Hultberg et al., 4 Feb 2026).

The same work proves a matching lower bound in the sense of asymptotic tightness. If the random leaps of $a_{90/95}$4 vanish,

$a_{90/95}$5

then with probability $a_{90/95}$6,

$a_{90/95}$7

with

$a_{90/95}$8

In particular,

$a_{90/95}$9

No faster time-uniform rate than $h$0 can be achieved, so the rates are asymptotically optimal.

The online implementation is correspondingly simple. On each new calibration point, one updates the variance proxy $h$1, computes $h$2, and selects the smallest $h$3 on a discrete grid $h$4 such that the empirical risk is at most $h$5, with an additional monotonicity step $h$6. For continuous $h$7, the infimum can be implemented by sorting scores, as in split conformal, or by binary search.

A synthetic regression experiment illustrates the operational meaning of VPT. The setup is $h$8, $h$9, $95\%$0, with target miscoverage $95\%$1, error probability $95\%$2, score $95\%$3, and prediction sets $95\%$4. Over $95\%$5 independent runs, approximately $95\%$6 of runs remain green, matching the nominal $95\%$7, whereas standard split conformal has almost every run exceed $95\%$8 at some $95\%$9, despite controlling marginal miscoverage on average. The realized miscoverage curves remain below $\mathrm{POAP}(h)$0 and approach it from below as $\mathrm{POAP}(h)$1 grows.

4. Reliability-based VPT in vessel trajectory prediction

In vessel trajectory prediction, the paper does not introduce a separate symbol “VPT” but instead uses the quantity $\mathrm{POAP}(h)$2 as a time-horizon reliability metric. It is defined as the largest prediction horizon $\mathrm{POAP}(h)$3, in minutes, for which the $\mathrm{POAP}(h)$4 lower-confidence bound on the probability of making an error below a chosen threshold still exceeds $\mathrm{POAP}(h)$5. If $\mathrm{POAP}(h)$6 denotes the probability that the displacement error at horizon $\mathrm{POAP}(h)$7 stays below a decision threshold $\mathrm{POAP}(h)$8, the construction is

$\mathrm{POAP}(h)$9

$0.90$0

and

$0.90$1

The $0.90$2 lower-confidence bound is obtained by the Wald method (Rastin et al., 19 Aug 2025).

The step-by-step procedure is explicit. Predicted trajectories are interpolated to $0.90$3 steps, displacement error is computed at each minute-level horizon $0.90$4, mean error versus horizon is fitted with the linear regression above by maximum likelihood, a decision threshold $0.90$5 is chosen, $0.90$6 and its $0.90$7 lower bound are computed, and $0.90$8 is read off as the largest admissible horizon. The reported VPT is therefore threshold-specific and confidence-specific.

The numerical results show clear scenario dependence. For the overall dataset of $0.90$9 samples, the VPTs are $a_{90/95}$00 for STT-R-CSCT, $a_{90/95}$01 for N-CSCT, and $a_{90/95}$02 for GMM-Trans-GRU. In Encounter-3, with $a_{90/95}$03 samples, the values are $a_{90/95}$04, $a_{90/95}$05, and $a_{90/95}$06, respectively. In Encounter-2, with $a_{90/95}$07 samples, they are $a_{90/95}$08, $a_{90/95}$09, and $a_{90/95}$10. The only row in which GMM-Trans-GRU is not best is Encounter-1 & Overtaken-1, where STT-R-CSCT achieves $a_{90/95}$11, N-CSCT $a_{90/95}$12, and GMM-Trans-GRU $a_{90/95}$13.

These values are intended for operational use. A navigation-decision module can restrict itself to predictions with horizon $a_{90/95}$14, thereby guaranteeing, with $a_{90/95}$15 confidence, an error below the chosen threshold. Stratifying by traffic complexity permits adaptive horizon selection for collision avoidance, route planning, or pilot-assist systems.

5. Chaotic systems and reservoir computing

For chaotic dynamical systems, VPT is defined as a threshold-crossing time of a normalized trajectory error. Let $a_{90/95}$16 be the ground-truth state and $a_{90/95}$17 the predicted state. The normalized mean-squared error is

$a_{90/95}$18

with

$a_{90/95}$19

The VPT is

$a_{90/95}$20

measured in units of the largest Lyapunov time $a_{90/95}$21. The threshold $a_{90/95}$22 is chosen because once $a_{90/95}$23 exceeds it, the trajectory has typically jumped onto the wrong lobe of the Lorenz attractor (Hurley et al., 8 Aug 2025).

The relation to chaos is explicit. If $a_{90/95}$24 is the maximal Lyapunov exponent, nearby trajectories diverge on average like $a_{90/95}$25, so $a_{90/95}$26 scales approximately as $a_{90/95}$27. In practice, this means that VPT is controlled by the initial prediction error and the known $a_{90/95}$28 of the Lorenz system. The same argument motivates an efficient shortcut: VPT can be estimated from the error after only a small number of autonomous prediction steps, avoiding full rollout during hyperparameter search.

The paper also introduces Valid Ground Truth Time (VGTT), defined as the time up to which independent ODE solvers agree to within the same error threshold. Using the 5th-order Adams–Bashforth–Moulton method with $a_{90/95}$29 tolerances and $a_{90/95}$30, VGTT is approximately $a_{90/95}$31 Lyapunov times for the Lorenz system. A reported VPT exceeding VGTT is not meaningful, because beyond VGTT the reference trajectory itself becomes solver-dependent.

The reservoir computer is an echo-state architecture with state update

$a_{90/95}$32

readout

$a_{90/95}$33

and ridge objective

$a_{90/95}$34

whose closed-form solution is

$a_{90/95}$35

As $a_{90/95}$36, both the reservoir computer’s VPT and the benchmark VPT increase and asymptote to a limit; for $a_{90/95}$37, the reservoir computer achieves approximately $a_{90/95}$38 of the benchmark VPT. Maximum VPT values exceed $a_{90/95}$39 Lyapunov times.

A two-regime structure appears in the spectral radius $a_{90/95}$40. One regime is near zero, $a_{90/95}$41, essentially independent of reservoir size $a_{90/95}$42, where the reservoir behaves like an extreme learning machine: stable, with little temporal memory, but recoverable by sufficiently large $a_{90/95}$43. The other is an edge-of-chaos regime at $a_{90/95}$44, which retains more memory, slightly outperforms the near-zero regime, and requires more delicate tuning in $a_{90/95}$45.

6. Right-censored survival outcomes

In right-censored survival analysis, VPT is a covariate-specific lower prediction bound for the event time. The observed data are i.i.d. copies of

$a_{90/95}$46

where $a_{90/95}$47 are covariates, $a_{90/95}$48 is the event time, and $a_{90/95}$49 is the censoring time. A Valid Prediction Time is a function $a_{90/95}$50 such that, for a new subject with covariates $a_{90/95}$51,

$a_{90/95}$52

The paper studies asymptotic training-set conditional validity, also called APAC, after splitting the sample into a training set of size $a_{90/95}$53 and a calibration set of size $a_{90/95}$54 (Si et al., 4 May 2025).

The target coverage functional is

$a_{90/95}$55

identified under two conditions: $a_{90/95}$56 on $a_{90/95}$57, and $a_{90/95}$58. The method uses a semiparametric one-step estimator. First, nuisance estimators $a_{90/95}$59 are fitted on the training data. Next, a candidate VPT is defined by

$a_{90/95}$60

for a small $a_{90/95}$61. On the calibration set, one computes the plug-in coverage and corrects it by the average efficient influence function: $a_{90/95}$62

The final lower bound is selected by a confidence-based thresholding rule. For each $a_{90/95}$63 in a grid $a_{90/95}$64, an estimated standard error $a_{90/95}$65 is formed, and the Wald lower confidence bound is

$a_{90/95}$66

One then chooses the largest $a_{90/95}$67 such that, for all $a_{90/95}$68, $a_{90/95}$69, and outputs $a_{90/95}$70. The main efficiency theorem gives the asymptotic linear expansion

$a_{90/95}$71

uniformly over $a_{90/95}$72, implying asymptotic normality. The threshold selection then yields

$a_{90/95}$73

The empirical study reports six synthetic settings spanning univariate versus multivariate covariates, independent versus covariate-dependent censoring, and moderate versus high dimension. In Setting 3 with $a_{90/95}$74, TCsurv has coverage $a_{90/95}$75, average VPT $a_{90/95}$76, and standard deviation $a_{90/95}$77; DFT-fixed has $a_{90/95}$78, $a_{90/95}$79, $a_{90/95}$80; DFT-adaptive-T has $a_{90/95}$81, $a_{90/95}$82, $a_{90/95}$83; and DFT-adaptive-CT has $a_{90/95}$84, $a_{90/95}$85, $a_{90/95}$86. The real-data application tracks users’ active times on a mobile application, with $a_{90/95}$87, covariates including gender, age, and number of children, and outcome defined as time to the ninth active day with artificial right-censoring at days $a_{90/95}$88–$a_{90/95}$89. The implementation cost is dominated by nuisance fitting on the training set, while calibration corrections scale as $a_{90/95}$90 with typically $a_{90/95}$91. The method is available as an R package and uses survSuperLearner for $a_{90/95}$92.

7. Comparative interpretation and limitations

The four formulations reveal that VPT is domain-specific both in its mathematical definition and in its operational meaning. In sequential conformal risk control, it is a high-probability statement that a data-conditional risk constraint holds for every calibration size; in vessel trajectory prediction, it is the largest horizon supported by a lower confidence bound on $a_{90/95}$93; in chaotic prediction, it is the first threshold-crossing time of normalized error; and in right-censored survival analysis, it is a predictive lower bound on the event time itself (Hultberg et al., 4 Feb 2026, Rastin et al., 19 Aug 2025, Hurley et al., 8 Aug 2025, Si et al., 4 May 2025).

This suggests several common misconceptions should be avoided. A longer VPT is not directly comparable across these literatures, because each construction is tied to a distinct threshold or validity convention: $a_{90/95}$94 and $a_{90/95}$95 in anytime-valid conformal risk control, $a_{90/95}$96 together with the $a_{90/95}$97 rule in vessel prediction, the error threshold $a_{90/95}$98 and Lyapunov-time normalization in chaotic forecasting, and the pair $a_{90/95}$99 together with censoring assumptions in survival analysis. Likewise, “infinite” VPT in the conformal setting does not mean zero error; it means that the data-conditional risk remains below a target level at all times with probability at least $95\%$00. In chaotic systems, a VPT larger than VGTT is explicitly not meaningful. In survival analysis, APAC validity depends on $95\%$01, positivity, and sufficiently fast nuisance-rate conditions, whereas the marginal guarantee follows from a simpler threshold rule.

Taken together, these works show that VPT functions as a calibrated horizon concept. Its substantive interpretation depends on what is being guaranteed: bounded risk, bounded displacement error, bounded trajectory divergence, or lower-bounded survival time.