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Prediction-powered inference for time series across space

Shahzar Rizvi, David Burt, Vishwak Srinivasan, Renato Berlinghieri, Stefano Del Col, Tamara Broderick

arXiv:2610.08715v1stat.MEcs.LGstat.ML

TL;DR

Short recent labeled histories and longer unlabeled covariate histories complicate estimating future response means with valid confidence intervals when temporal dependence conflicts with PPI’s i.i.d. assumption. The paper combines PPI bias correction with HAC uncertainty adjustment under time stationarity. It reports that PPITSAS outperforms evaluated alternatives.

  • Problem

    The paper addresses estimating future response means with valid confidence intervals from short labeled time series and longer unlabeled covariate histories across locations, where temporal dependence makes PPI’s i.i.d. assumption unsuitable.

    Source: §1 · p. 1

  • Method

    PPITSAS combines PPI’s prediction-bias correction with HAC variance adjustment under a time-stationarity assumption.

    Source: §1 · p. 2, §4.2 · p. 5

  • Results

    PPITSAS outperforms evaluated alternatives, achieving the best point estimation and closest-to-nominal coverage in both weather settings.

    Source: §1 · p. 3, §5.2 · p. 7

  • Takeaways & Limitations

    The method provides point estimates and confidence intervals while accounting for temporal dependence in settings with limited labeled data and longer unlabeled covariate histories.

    Source: §1 · pp. 1–2

    At the boundary between fitting and rectifying data, the method assumes correlations are negligible; accounting for induced correlations is left for future work.

    Source: §4.1 · p. 5

Abstract

from arXiv · show

The following motif is common in spatiotemporal settings: we have a sequence of covariate and label pairs observed for a relatively short, recent time period. We have access to unlabeled covariates over a longer time period. Data is observed over many spatial locations. For instance, crop yield might be observed over a large geographical area for recent years, but weather data (which is informative about crop yield) is available for a much longer period. The goal is to estimate, at each spatial location, the expected label (e.g., crop yield) in the future and provide a valid confidence interval for this value. The observed time period alone is too short for reliable estimates. Imputing missing labels with machine learning can cause substantial bias. Prediction-powered inference (PPI) can correct for this bias, but it relies on an i.i.d. assumption that breaks under our expected temporal dependencies. Heteroskedasticity and autocorrelation consistent (HAC) procedures account for temporal correlation, but have not been adapted to cases where some labels are imputed. We provide reliable point estimates and confidence intervals given: short labeled time series (across spatial locations), a longer unlabeled time series, and an imperfect predictor of labels given covariates. We show our method outperforms natural alternatives.

1 Introduction

The paper targets future response means and valid confidence intervals using recent labeled data and a longer history of unlabeled covariates across locations. It proposes adapting prediction-powered inference to temporal dependence and evaluates the method against alternatives.

1 Introduction

  • The target setting combines recent responses and covariates at many locations with a longer prior covariate-only history, to estimate future responses and provide reliable confidence intervals.Source: §1 · p. 1
  • With limited labeled data, an empirical mean can miss the true response mean; the insurance example gives 0.05 as the true annual payout versus a classical estimate of 0.02.Source: §1 · p. 2
  • Inference with imputation treats predicted responses as labels and can introduce bias, while PPI estimates and corrects prediction bias but may yield miscalibrated intervals under temporal dependence.Source: §1 · p. 2
  • The paper proposes time stationarity instead of PPI’s potentially unsuitable i.i.d. assumption, combining PPI bias correction with HAC uncertainty adjustment for temporal dependence.It also studies whether labeled data reserved for prediction and debiasing should be interleaved or chunked; a full proof is left for future work.Source: §1 · p. 2
  • PPITSAS outperforms the evaluated alternative approaches in the paper’s setting.Source: §1 · p. 3

1.1 Related work

  • Related work on missingness considers MAR and overlap-based approaches, but treating time as a covariate makes the unlabeled period strictly prior to the labeled period, so overlap fails.The paper frames choosing an appropriate regularity assumption and deriving asymptotic results as comparable in scope to its own work.Source: §1.1 · p. 3
  • Prior time-series PPI work addresses anytime-valid sequential testing, including change-point detection and risk monitoring, rather than mean estimation.Source: §1.1 · p. 3

2 Problem setup

The setup observes covariates across locations during both a longer unlabeled period and a recent labeled period. Its goal is to estimate each location’s long-run mean response and provide a valid confidence interval.

2 Problem setup

  • Covariates are observed at every location across both periods, while responses are observed only during the labeled period that follows the unlabeled period.The times are modeled as positive integers, with the unlabeled times directly preceding the labeled times.Source: §2 · p. 3
  • At each location, the response process is assumed covariance-stationary with absolutely summable covariances, supporting a well-defined long-run mean response.Source: §2 · pp. 3–4
  • The analysis goal is to estimate the long-run mean response and provide a valid confidence interval for every spatial location.Source: §2 · p. 4

3 Background: Prediction-powered inference

Prediction-powered inference (PPI) uses predicted responses from unlabeled covariates and a labeled subset to estimate and correct prediction bias. Under an i.i.d. assumption, it also provides an asymptotically valid confidence interval.

3 Background: Prediction-powered inference

  • PPI divides labeled data into predictor-fitting and debiasing (“rectifying”) subsets; under its i.i.d. assumption, only subset sizes matter, not index order.Source: §3 · p. 4
  • PPI starts with the mean predicted response on unlabeled data and subtracts an estimate of its expected bias.Source: §3 · p. 4
  • Under i.i.d. data, PPI’s point estimate is the sum of two independent sample means, enabling an asymptotically valid confidence interval using estimated variances for predicted responses and bias terms.Source: §3 · p. 4

4 Our method: PPI for time series across space (PPITSAS)

PPITSAS adapts prediction-powered inference to spatial time series by sharing predictor training across locations and using HAC variance estimates under temporal dependence. Its design uses chunked splitting and a Newey–West regime, while leaving boundary-correlation adjustment for future work.

4.1 Splitting the data and training the predictor

  • Training one predictor across locations uses more labeled data for fitting, leaving more observations available for debiasing at each location.The pooled fit has N_S · T_fit data points rather than the T_fit available from fitting separately at one location.Source: §4.1 · p. 4
  • Chunking is expected to improve bias estimation over interleaving, and the simulation reports lower overall PPI-estimator bias with chunking.Chunking may have higher prediction bias, but its fitting and rectifying blocks are less correlated; Table 1 reports the simulated bias quantities across 105 replicates.Source: §4.1 · p. 5, Table 1 · p. 5
  • The method does not buffer observations near the fitting–rectifying boundary and assumes their cross-set correlations are negligible.The authors defer accounting for induced correlations to future work, citing preliminary experiments suggesting buffering has negligible effects.Source: §4.1 · p. 5

4.2 Applying HAC

  • PPITSAS adapts PPI confidence intervals to temporal dependence by applying HAC variance estimation to the prediction and bias terms under stationarity.The authors conjecture validity in a Newey–West regime under stationarity, mixing, and moment conditions.Source: §4.2 · pp. 5–6
  • The authors use Newey–West rather than fixed-b asymptotics, where the bandwidth grows more slowly than the time-series length.They cite complications with fixed-b asymptotics and expect the Newey–West regime to be reasonable when decorrelation occurs over weeks but data span years.Source: §4.2 · pp. 5–6

5 Experiments

In the two weather settings, PPITSAS has the best point-estimation performance, closest-to-nominal coverage, and smallest half-width among methods with near-nominal coverage. The experiments use simulated data with ground truth and compare point-estimation and interval metrics against several alternatives.

5.1 Experimental setup

  • The experiments use simulated data because standard datasets, including Generali’s insurance data, lack ground truth for evaluating the true mean.The simulation uses 121 locations, 4,000 labeled times, 16,000 unlabeled times, a linear predictor for a quadratic response relationship, and less- and more-extreme weather scenarios.Source: §5.1 · p. 6
  • Methods are judged first by scaled RMSE, then by coverage of at least 85% against a 95% nominal target, and finally by half-width among qualifying methods.The study reports scaled RMSE, coverage, and scaled half-width; half-width is treated as uninformative when coverage is below 85%.Source: §5.1 · pp. 6–7

5.2 Results

  • PPITSAS has the lowest scaled RMSEs in both weather settings, closest-to-nominal coverage, and the smallest scaled half-width among methods with near-nominal coverage.ChunkPPI ties PPITSAS on point estimation; its scaled RMSE is about 35% above the oracle in less extreme weather and about 31% above the oracle in more extreme weather, where it is also at least 27% below every other method.Source: §5.2 · p. 7, Table 2 · p. 7

A PPITSAS algorithm

  • Algorithm 1 takes a target location, significance level, fitting proportion, fitting algorithm, and labeled data, then splits the data and fits a predictor.It proceeds to compute average prediction and bias, estimate long-run variances, and form the PPITSAS point estimate.Source: A PPITSAS algorithm · p. 10

B Newey–West procedure

  • The HAC procedure centers observations, computes sample autocovariances, selects a Newey–West bandwidth, and estimates the long-run variance.The supplied algorithm steps include computing the sample mean and setting a pilot bandwidth before the final bandwidth and variance estimate.Source: B Newey–West procedure · p. 11

C Data splitting demonstration

A temporal-only autoregressive simulation compares chunked and interleaved fitting splits. Chunking yields a less biased final PPI estimator despite higher prediction bias, because it estimates prediction bias better.

C Data splitting demonstration

  • A temporal-only, covariance-stationary autoregressive demonstration (|ρ| < 1) finds interleaving yields a less biased predictor than chunking but a poorer estimate of prediction bias, with ρ = 0.95, TU = 4000, TL = 1000, and Tfit = 100.Source: C Data splitting demonstration · p. 12
  • Table 3’s rows compare chunked and interleaved splits, while its columns report true prediction bias, estimated prediction bias, and PPI-estimate bias.Each entry is averaged over 105 replicates, with Monte Carlo standard errors in parentheses.Source: Table 3 · p. 12
  • Chunking produces a less biased final PPI estimator than interleaving, despite having higher prediction bias, because its bias estimate is better.The comparison averages over 105 independent replicates; Table 3 reports true prediction bias, estimated prediction bias, and PPI-estimate bias by split.Source: C Data splitting demonstration · p. 12, Table 3 · p. 12

C.1 Code and reproducibility

The authors provide code for generating simulated data, running the split methods, and computing metrics.

C.1 Code and reproducibility

  • Code for data generation, chunking and interleaving, and metric computation is available in the project repository.The passage provides the repository URL.Source: §C.1 · p. 12
  • The experiment ran locally on a MacBook Pro with an Apple M3 Max chip, 14-core CPU, and 36 GB of unified memory.Source: §C.1 · p. 12
  • The experiments were CPU-based without GPU acceleration, and simulation plus method runs took less than five minutes.Source: §C.1 · p. 12

D Previous methods

The appendix contrasts classical, imputation-based, prediction-powered, HAC, and oracle inference procedures, then states that the proposed asymptotic validity results remain conjectural.

D.1 Classical

  • The classical method uses the observed-response sample mean and an asymptotic-normality confidence interval under an i.i.d. assumption.Its variance estimate and interval are presented under that assumption.Source: §D.1 · pp. 12–13

D.2 Inference with imputation (IWI)

  • IWI averages unused observed responses with predictions over the long unlabeled sequence, treating predictions as true responses when constructing its interval.The approach is not generally inferentially valid, even under i.i.d. sampling [10].Source: §D.2 · p. 13

D.3 Prediction-powered inference (PPI)

  • PPI estimates the mean predicted response on unlabeled data and subtracts estimated prediction bias from labeled data, with its stated interval relying on i.i.d. sampling.Under i.i.d. sampling, its point estimate is a sum of two independent sample means.Source: §D.3 · p. 13

D.4 Heteroskedasticity and autocorrelation consistent procedures (HAC)

  • HAC retains the classical point estimate but replaces the sample variance with a HAC variance estimator for temporally dependent data.The passage states asymptotic normality under process assumptions elaborated in Appendix E.1.Source: §D.4 · p. 14

D.5 Oracle

  • The oracle has access to unlabeled responses, changing both its point estimate and variance estimates, and uses HAC variance estimation for its interval.Its interval is stated to be asymptotically valid under the same assumptions used for HAC.Source: §D.5 · p. 14

E Asymptotics

  • The paper presents no complete theoretical results and conjectures asymptotic validity for HAC and PPITSAS confidence intervals.It discusses believed-sufficient assumptions and sketches proofs of the conjectures.Source: E Asymptotics · p. 14

E.1 Asymptotic validity of HAC confidence intervals

The HAC argument uses stationarity, moment, and mixing conditions to support asymptotic normality and variance consistency. The PPITSAS confidence-interval argument is a conjectural sketch whose predictor-independence step remains unresolved.

E.1 Asymptotic validity of HAC confidence intervals

  • Under stationarity, moment, and mixing conditions, the sample mean is asymptotically normal and the Newey–West variance estimate is consistent, supporting HAC intervals.The passage characterizes stationarity as invariance of finite-dimensional distributions under time shifts.Source: §E.1 · pp. 15–16

E.2 Argument for asymptotic validity of PPITSAS confidence intervals

  • PPITSAS interval validity is conjectured from mixing and stationarity arguments, but formal conditions for treating the trained predictor as fixed remain future work.The sketch also invokes a joint central limit theorem and consistent variance estimates.Source: §E.2 · pp. 16–17

F.1 Simulation setup

The simulations compare six methods across two weather settings using spatially distributed data and long unlabeled histories. Performance is assessed through estimation error, interval coverage, and interval width; code and computational details are reported.

F.1 Simulation setup

  • Each simulation uses 121 locations, 4,000 labeled-period time points and 16,000 unlabeled-period points, with 50 points for predictor fitting and 3,950 for rectification.The labeled and unlabeled spans correspond to roughly 10 and 40 years.Source: §F.1 · p. 17
  • The less-extreme setting uses a linear weather signal and non-heavy-tailed weather, while the more-extreme setting uses an exponential Gaussian-process component and heavy-tailed weather.Both settings model payouts as noisy measurements of squared weather.Source: §F.1 · p. 17

F.2 Reporting

  • The experiments compare classic, IWI, chunkPPI, HAC, oracle, and PPITSAS at α = 0.05, reporting scaled RMSE, estimated coverage, and scaled half-width.The resulting statistics are reported in Table 2; the supplied passages do not provide its values.Source: §F.2 · pp. 17–18

F.3 Code and reproducibility

  • The authors provide code for simulation and evaluation, and report approximately one hour of CPU-based computation on a MacBook Pro.The machine had an Apple M3 Max chip and 36 GB of unified memory; no GPU acceleration was used.Source: §F.3 · p. 18

G Limitations

The method assumes temporal stationarity and additional regularity conditions, while the experiments use a limited simulation design. The appendix also lists software packages and licenses.

G Limitations

  • The method assumes stationarity, which can be unrealistic when the data-generating process changes over time, as with changing climate or claims practices.The authors identify procedures under local stationarity or smoothly varying distributions as future work.Source: G Limitations · p. 18
  • Strong mixing and a moment condition support the central limit theorem, but the moment condition excludes heavy-tailed processes.The authors note the bounded response in their insurance example avoids concern about that moment condition there.Source: G Limitations · p. 18
  • The experiments comprise two simulations using a linear predictor model despite a quadratic covariate-response relationship.The authors plan to consider more realistic simulators and more complicated relationships.Source: G Limitations · p. 18

H Licenses

  • The licenses appendix lists NumPy 2.0.2, SciPy 1.13.1, and scikit-learn 1.6.1, each under a BSD 3-Clause license.Source: H Licenses · p. 19
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