Predictability-Stratified Time Series (PreSTS)

• Predictability-Stratified Time Series (PreSTS) is a framework that organizes time series into tiers based on intrinsic forecastability for better benchmarking and model selection.
• It employs diverse estimation methods, including entropy bounds, spectral analysis, and loss ranking, to capture multiple dimensions of predictability.
• Practical applications include enhanced data curation and pretraining, as demonstrated by the Kairos corpus with its five predictability tiers and weighted sampling.

Searching arXiv for recent and foundational papers related to Predictability-Stratified Time Series (PreSTS). Predictability-Stratified Time Series (PreSTS) denotes, in the literature considered here, a family of workflows that organize time series, datasets, samples, or latent components into strata according to intrinsic or operational predictability. In its most specific recent usage, "PreSTS" names the large-scale pretraining corpus used by Kairos, where real-world datasets are stratified into five predictability tiers and sampled non-uniformly during training (Feng et al., 30 Sep 2025). In the broader methodological sense, PreSTS rests on the premise that predictability is an intrinsic property or upper bound of a process, distinct from the realized performance of any single forecasting model, and that estimating this property is useful for benchmarking, model selection, data curation, and training-time weighting (Xu et al., 18 Oct 2025, Xu et al., 2022).

1. Scope and levels of stratification

PreSTS is not a single estimator or architecture. The literature instead presents several distinct but compatible levels at which predictability can be used to organize temporal data. At one extreme, predictability is treated as a corpus-level property used to bias sampling during foundation-model pretraining; at another, it is treated as a sample-level or state-dependent quantity used to reweight losses during optimization; elsewhere it is pushed into latent or periodic components that are expected to be more forecastable than the raw observations.

Level	Mechanism	Representative source
Dataset level	Five predictability tiers with non-uniform sampling	Kairos (Feng et al., 30 Sep 2025)
Sample level	Within-batch loss ranking into ordered predictability buckets	APTF (Zhang et al., 18 Feb 2026)
Latent-factor level	Multiple predictable and independent signal components	DPLF (Hou et al., 2023)
Periodicity/component level	Reference-series alignment to dominant frequencies and harmonics	MFRS (Yu et al., 11 Mar 2025)

A closely related component-level interpretation appears in predictability-aware multichannel compression, where a compressed representation is designed to preserve shared predictable periodic structure while a residual decoder reconstructs harder channel-specific content. This suggests that PreSTS can stratify not only entire sequences, but also substructures such as channels, components, windows, or local states when predictability is heterogeneous within the observed process (Liu et al., 31 May 2025).

2. Formal notions of predictability

The most classical formalization in this literature is next-state predictability for a symbolic process. For a finite state space $\Omega=\{z_1,\dots,z_C\}$ and history $h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$, local one-step predictability is defined as

$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$

and average predictability over histories is

$\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$

with asymptotic overall predictability

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$

Within information-theoretic approaches, the same quantity is connected to an entropy-based upper bound through the Fano-style relation

$$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$

which yields a scalar ceiling $\Pi^{\max}$ for next-step accuracy (Xu et al., 18 Oct 2025).

A second formalization identifies predictability with Bayes-optimal classification accuracy. If the next state is treated as the class and the history as the feature, then time-series predictability is exactly

$\Pi = 1-\mathcal{R}_B,$

where $\mathcal{R}_B$ is the Bayes error rate. This equivalence matters because it replaces an indirect entropy surrogate with a direct decision-theoretic quantity and makes predictability interpretable as irreducible next-state classification difficulty (Xu et al., 2022).

A third formalization is local and state-dependent rather than global. For the conditionally heteroscedastic nonlinear autoregression

$x_{t+s}=f(X_t)+g^{1/2}(X_t)\epsilon_t,$

predictability is measured by the local sensitivity of the conditional law of $h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$0 to perturbations of the delay vector $h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$1. The symmetric Kullback-Leibler divergence between nearby conditional distributions admits the second-order expansion

$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$2

with Fisher information matrix

$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$3

In this formulation, predictability is local, state-dependent, and jointly determined by conditional-mean sensitivity, conditional-variance sensitivity, and noise magnitude (Gianetto et al., 2012).

These definitions are not equivalent in all settings, but they share a common theme: predictability is treated as structure already present in the process, not as a property of a particular forecasting architecture. This suggests that PreSTS is best viewed as a methodology for organizing data by latent forecasting difficulty rather than by raw task labels alone.

3. Estimation, proxies, and predictability descriptors

Operational PreSTS depends on estimators or proxies that can be computed before, during, or alongside forecasting. The literature provides several families of such measures, each emphasizing a different aspect of forecastability, including symbolic uncertainty, spectral regularity, chaos, local persistence, and tail-event dependence (Wang et al., 17 Jul 2025, Garland et al., 2014, Dierick et al., 2017, Han et al., 2014).

Measure	Quantity	Typical role
Entropy/Fano	$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$4 from $h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$5	Symbolic upper bound
Bayes-error route	$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$6	Direct next-state predictability
Spectral predictability	$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$7	Ex ante forecastability
Largest Lyapunov exponent	$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$8	Chaos and stability
Weighted permutation entropy	$h_{t-1}=\{Z_{t-1},Z_{t-2},\dots,Z_1\}$9	Model-free complexity proxy
Hurst exponent	$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$0	Persistence-based predictability
Cross-quantilogram	$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$1	Quantile-specific directional predictability

The spectral and dynamical route is explicitly ex ante. For a de-trended series $$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$2, spectral entropy

$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$3

and spectral predictability

$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$4

quantify concentration of power in frequency space, while the largest Lyapunov exponent

$$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$5

measures exponential divergence in reconstructed phase space. The paper proposing these measures states that spectral predictability below $$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$6 or Lyapunov exponent above $$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$7 are indicative of low forecastability, and reports strong correlations with downstream WAPE on M5, with $$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$8 for both daily and weekly frequencies (Wang et al., 17 Jul 2025).

For real-valued series, weighted permutation entropy provides a model-free complexity proxy. Given ordinal-pattern probabilities $$\pi(h_{t-1})=\sup_{z\in\Omega}Pr[Z_t=z\mid h_{t-1}],$$9, weighted permutation entropy is

$\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$0

Low WPE corresponds to concentrated ordinal structure and higher apparent predictability; high WPE corresponds to weak reusable temporal structure. The empirical study on 120 series links WPE to best-achievable forecast error and uses it as a diagnostic of whether a forecasting method is mismatched to the available predictive structure (Garland et al., 2014).

In physiological series, the Hurst exponent $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$1 is used as a predictability axis and the Minkowski fractal dimension $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$2 as a complexity axis. The gait study reports that backward walking increases predictability through a more stereotyped stride-interval pattern, while forward walking shows maximal complexity, and finds $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$3 and $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$4 empirically independent with Pearson $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$5, $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$6 (Dierick et al., 2017). This is important because it shows that a PreSTS system may need at least two axes—predictability and complexity—rather than a single ranking.

Quantile-specific predictability can be measured by the cross-quantilogram

$\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$7

which measures directional predictability at lag $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$8 between quantile-hit processes. This is especially relevant when dependence is sparse, asymmetric, heavy-tailed, or visible only in lower or upper tails (Han et al., 2014).

A central caution is that measurement itself can be unstable. In the Lempel-Ziv/Fano pipeline, inconsistent logarithm bases and ambiguous implementation of the match length $\Pi(h_{t-1})=\sum_{h_{t-1}}P(h_{t-1})\pi(h_{t-1}),$9 systematically inflate predictability estimates. The corrected estimator uses

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$0

with

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$1

and the paper reports that, on an empirical interaction dataset, the average predictability falls from $$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$2 under unmatched bases to $$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$3 under matched bases. It further concludes that the LZ estimator itself fails when the time series is highly random (Xu et al., 2018).

4. Algorithmic forms of PreSTS

One operational form of PreSTS appears during optimization. In APTF, a dataset

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$4

is stratified within each batch by current per-sample loss. Samples are sorted by ascending loss and divided into $$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$5 buckets $$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$6, with lower-loss buckets interpreted as higher predictability and assigned weights

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$7

The method then applies a Hierarchical Predictability-aware Loss, progressively reducing bucket count across stages so that broader low-predictability regions are penalized, and uses a second model to mitigate predictability-estimation bias caused by the source model’s own errors (Zhang et al., 18 Feb 2026). This is a sample-stratified PreSTS implementation in which predictability is an online training quantity rather than a precomputed corpus label.

A second form is latent-factor stratification. In "Discovering Predictable Latent Factors for Time Series Forecasting," the observed history

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$8

is mapped to latent components

$$\Pi=\lim_{t\to\infty}\frac{1}{t}\sum_{i=1}^t \Pi(h_{i-1}).$$9

with each component required to satisfy predictability, sufficiency, and identifiability. Prediction proceeds through component-wise latent transitions

$$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$0

followed by decoding and weighted aggregation. The paper reports that the learned latent factors are more stationary than the original stock-price signals under the ADF test, which supports the idea that PreSTS can be implemented by moving from raw series to more predictable latent sub-series before forecasting (Hou et al., 2023).

A third form is periodicity-based stratification. MFRS assumes that time series predictability is derived from periodic characteristics at different frequencies and performs long-term FFT analysis to identify dominant spectral components and harmonics. After converting amplitudes to a period-domain spectrum

$$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$1

it extracts primary base-patterns and harmonic base-patterns, constructs reference series, and computes cross-attention between observed series and these references. The paper reports especially strong gains on Traffic, which it describes as having strong periodicity, and shows on synthetic data that long-period deterministic structure remains forecastable even when the lookback window is shorter than the dominant period (Yu et al., 11 Mar 2025). This suggests a PreSTS stratum of globally periodic but locally under-observed series.

A fourth form is finite-sample model-class stratification. The stationary-versus-locally-stationary model-choice procedure compares empirical MSPEs

$$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$2

through

$$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$3

Locally stationary models are chosen only if they beat stationary models by at least a threshold $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$4, that is, when $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$5. The paper’s main practical conclusion is that locally stationary forecasting outperforms stationary forecasting only if the sequence is long, or the coefficient function exhibits considerable variation, or the tangent processes are close to the unit root; otherwise the stationary approach can be chosen without a large loss (Kley et al., 2016). In PreSTS terms, this yields strata such as effectively stationary, locally predictable, and borderline.

5. PreSTS as the Kairos pretraining corpus

In Kairos, PreSTS is a concrete corpus rather than a general framework. The paper states that the Predictability-Stratified Time Series corpus contains over $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$6 billion real-world time series observations from Chronos and Moirai in conjunction with $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$7 billion synthetic time points, and that the training loader samples $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$8 real data and $$S=-\Pi^{\max}\log_2\Pi^{\max}-(1-\Pi^{\max})\log_2(1-\Pi^{\max})+(1-\Pi^{\max})\log_2(C-1),$$9 synthetic data (Feng et al., 30 Sep 2025). Real-world datasets are stratified into five tiers based on their predictability, and datasets with higher predictability are assigned a greater sampling probability during training.

The tiering criterion is intentionally qualitative rather than formulaic. Tier 1 contains datasets characterized by pronounced periodicity and trends with low noise; Tier 2 contains datasets with similarly distinct patterns but high noise; Tier 3 contains those with subtle trends and considerable noise; Tiers 4 and 5 are classified based on a composite assessment of size and pattern regularity. The paper does not provide an explicit predictability score, exact thresholds, or a formal entropy-based assignment rule. This matters because PreSTS, in this corpus-specific sense, is a curated sampling policy rather than a published estimator.

The synthetic part of the corpus is also stratification-oriented. Each synthetic series has length $\Pi^{\max}$0 and is drawn from two families: composite time series of the form

$\Pi^{\max}$1

where seasonal, trend, and Gaussian-noise components are combined, and idealized industrial signals with regular machine-like cycles. These synthetic sequences are meant to provide broad coverage of predictable structures, especially periodic and low-noise regimes (Feng et al., 30 Sep 2025).

In Kairos, PreSTS affects sampling probability during pretraining, not the model objective. Kairos is trained with a weighted quantile loss

$\Pi^{\max}$2

where

$\Pi^{\max}$3

The paper does not present a dedicated ablation isolating PreSTS from non-stratified corpus construction, so the independent effect of predictability-tiered sampling remains an open empirical question.

6. Limitations, controversies, and open questions

A first recurrent controversy is measurement fidelity. Entropy-based predictability has been widely used, but the methodological clarification in (Xu et al., 2018) shows that common implementations can overestimate predictability through mismatched logarithm bases and incorrect handling of $\Pi^{\max}$4, and that low-predictability regimes are intrinsically hard to estimate even after these corrections. Any PreSTS system that uses a scalar predictability score inherits these estimator-level failure modes.

A second limitation is that predictability is not unidimensional. The gait literature shows that a series can become more predictable yet less complex, and that healthy dynamics may occupy a regime of intermediate predictability and maximal complexity rather than maximal predictability (Dierick et al., 2017). This directly contradicts the simplistic interpretation that "more predictable" is always "better." It suggests that PreSTS should often separate at least predictability, complexity, and possibly adaptability.

A third issue is finite-sample efficacy. The locally stationary model-choice literature argues that even if a more complex locally stationary model is true, a simpler stationary model may still be preferable in finite samples because parameter uncertainty can dominate bias reduction (Kley et al., 2016). Thus, a time series can be structurally nonstationary yet belong to a stratum where stationary forecasting is operationally preferable. PreSTS therefore depends not only on intrinsic process properties but also on sample size, forecast horizon, and estimator variance.

A fourth limitation concerns under-specified stratification criteria. The Kairos corpus uses a clearly stated five-tier hierarchy, but the paper does not provide exact numerical rules for tier assignment, exact tier-level sampling probabilities, or an ablation isolating the effect of the stratification policy (Feng et al., 30 Sep 2025). This suggests that current large-scale PreSTS corpora remain partly heuristic.

A fifth issue is context dependence. Review evidence indicates that predictability can change substantially with aggregation level, contextual covariates, candidate state space, sparsity, and analysis window (Xu et al., 18 Oct 2025). This suggests that a robust PreSTS system should report not just a stratum label, but also the assumptions under which that label was assigned: symbolic or continuous representation, conditional or unconditional measure, local or global window, and estimator uncertainty.

Taken together, the literature suggests that PreSTS is best understood not as a single model or metric, but as a design principle for organizing time-series analysis around estimated forecasting difficulty. In that principle, stratification may be dataset-level, sample-level, latent, periodic, local-state-specific, or tail-specific; predictability may be estimated via entropy, Bayes error, spectral regularity, Lyapunov instability, permutation structure, local Fisher information, or quantile-hit dependence; and the practical value of the framework lies in aligning benchmarking, data curation, model selection, and training dynamics with the heterogeneous forecastability structure already present in time-series data.