Research on Forecastability and the Limits of Prediction
Forecastability is a property of the data, not the model. This page presents research on measuring forecastability using auto-mutual information (AMI) and entropy. It addresses a prior problem, before model selection or benchmarking: determining how much of the future is knowable from the past.
Rather than benchmarking models against each other, these papers examine the structure of the data itself and the limits that structure imposes on achievable accuracy. Together, they develop forecastability as a measurable property of time series, show how it varies across forecast horizons, and link it directly to forecast error, model choice, and economic value.
Learning Predictive Information in Temporal Systems
We present a self-supervised framework for learning and quantifying predictive information in temporal systems. Temporal prediction is often evaluated through realised error alone, despite the fact that temporal processes may contain latent predictive structure that determines how much information about the future is encoded in the past and how efficiently predictive models can extract it. We define predictive information as the mutual information between a future observation and a learned representation of past context, estimated across horizons with a contrastive objective based on InfoNCE. Building on this formulation, we introduce the exploitation ratio, a horizon-resolved measure of the fraction of available predictive information converted into predictive performance under logarithmic loss. We evaluate the framework across four classes of temporal systems spanning heterogeneous human-organisational series from M4 Monthly, physiological signals from electrocardiography and heart-rate variability, natural physical systems represented by daily temperature series, and controlled nonlinear dynamical systems including Mackey–Glass and Lorenz attractors. Predictive-information geometry differs substantially across system classes, with physiological and seasonal physical systems retaining local structure over longer horizons, whereas chaotic systems display rapid information decay. Across all domains, exploitation efficiency varies systematically with both system class and forecast horizon: some predictive models recover much of the available predictive information at short horizons, whereas substantial residual predictive information remains unexploited at longer horizons even for high-capacity neural architectures. These results recast temporal prediction as predictive information extraction and position predictive information as a general representation-learning object governing temporal predictability across heterogeneous systems.
Manuscript in preparation
Horizon-resolved Forecastability of Time Series via Auto Mutual Information
In many social, business, economic, and physical systems the true data-generating process is rarely known, so a series’ forecastability cannot be assumed; it must be assessed from the observed history. This paper evaluates a horizon-specific, pre-modelling measure for that assessment: auto-mutual information (AMI), a training-only measure of past-future dependence at each forecast horizon; higher AMI indicates more recoverable temporal structure. It is evaluated on the M4 Monthly dataset (47,992 of 48,000 series, 18-month horizon) with seasonal naive, ETS, and N-BEATS as probes and MASE as the error measure. The series-level association with realised skill is modest but systematic (mean Spearman ρ of 0.11 to 0.13 for ETS and N-BEATS), accumulating into a pronounced descriptive gradient: median MASE is 29% (ETS) to 37% (N-BEATS) lower in the top within-horizon AMI decile than in the bottom. AMI outperforms absolute autocorrelation as a horizon-specific diagnostic by a small, consistent margin; paired bootstrap intervals exclude zero for all three probes, and synthetic benchmarks with analytically exact AMI locate the advantage on nonlinear dependence. Decile assignments are sample-relative and require recalibration on a new portfolio. The contribution is a pre-modelling diagnostic, not a forecasting model.
APA
Catt, P. M. (2026). Horizon-resolved Forecastability of Time Series via Auto Mutual Information (under review).
BibTeX
@article{catt2026diagnostic,
author = {Catt, Peter M.},
title = {Horizon-resolved Forecastability of Time Series via Auto Mutual Information},
year = {2026},
note = {Manuscript under review}
}
Forecastability as an Information-Theoretic Limit on Prediction
Abstract: Forecasting is usually framed as a problem of model choice. This paper starts earlier, asking how much predictive information is available at each horizon. Under logarithmic loss, the answer is exact: the mutual information between the future observation and the declared information set equals the maximum achievable reduction in expected loss. This paper develops the consequences of that identity. Forecastability, defined as this mutual information evaluated across horizons, forms a profile whose shape reflects the dependence structure of the process and need not be monotone. Three structural properties are derived: compression of the information set can only reduce forecastability; the gap between the profile under a finite lag window and the full history gives an exact truncation error budget; and for processes with periodic dependence, the profile inherits the periodicity. Predictive loss decomposes into an irreducible component fixed by the information structure and an approximation component attributable to the method; their ratio defines the exploitation ratio, a normalised diagnostic for method adequacy. The exact equality is specific to log loss, but when forecastability is near zero, classical inequalities imply that no method under any loss can materially improve on the unconditional baseline. The framework provides a theoretical foundation for assessing, prior to any modelling, whether the declared information set contains sufficient predictive information at the horizon of interest.
APA
Catt, P. M. (2026). Forecastability as an Information-Theoretic Limit on Prediction. arXiv. https://doi.org/10.48550/arXiv.2603.27074
BibTeX
@article{catt2026forecastability,
author = {Catt, Peter M.},
title = {Forecastability as an Information-Theoretic Limit on Prediction},
year = {2026},
journal = {arXiv preprint arXiv:2603.27074},
doi = {10.48550/arXiv.2603.27074},
url = {https://doi.org/10.48550/arXiv.2603.27074}
note = {Manuscript under review}
}
Entropy as an A Priori Indicator of Forecastability
Abstract: The ability to accurately determine the a priori forecastability of a time series is an important endeavour for forecasting practitioners as it provides guidance on the potential for accurate forecasts and the associated degree of effort that is warranted. Measures of entropy, such as sample entropy, provide an assessment of the regularity or similarity within a time series. We posit that series with low in-sample entropy, i.e. high regularity, will positively correlate with low out-of-sample forecast error, as measured by the mean absolute scaled error (MASE). To assess this we adopt the 3003 time series used in the M3 forecasting competition spanning micro, industry, macro, finance, and demographic series. We calculate the in-sample sample entropy and out-of-sample MASE for all 3003 series using a common univariate forecasting method. We demonstrate that the a priori sample entropy is indeed a useful predictor of out-of-sample forecast performance, subject to the well-researched problem of structural breaks. We recommend that forecasting practitioners adopt such entropy measures, alongside well-established tests for seasonality and trend, to better understand the likelihood of successful forecasting outcomes.
APA
Catt, P. M. (2014). Entropy as an a priori indicator of forecastability. (SSRN Working Paper No. 6235738). SSRN. https://dx.doi.org/10.2139/ssrn.6235738
BibTeX
@unpublished{catt2014entropy,
author = {Catt, Peter M.},
title = {Entropy as an A Priori Indicator of Forecastability},
year = {2014},
month = {November},
note = {SSRN Working Paper No. 6235738},
url = {https://dx.doi.org/10.2139/ssrn.6235738}
}
Forecastability: Insights from Physics, Graphical Decomposition, and Information Theory
Abstract: This paper explores the concept of forecastability from multiple theoretical perspectives, drawing on physics, graphical decomposition methods, and information theory. Beginning with foundational concepts from classical determinism, quantum mechanics, and chaos theory, the work establishes a classification of data-generating processes along a continuum from deterministic to chaotic to complex to random. Six illustrative time series, spanning a pure sine wave, the Box-Jenkins airline passengers series, the Hénon map, polystyrene shipments, IBM stock price changes, and pseudo-random numbers, are analysed using time plots, classical decomposition, and lag plots to reveal their underlying structural components. The paper evaluates the coefficient of variation as a forecastability metric and identifies critical limitations, including its sensitivity to near-zero means and its inability to detect patterns beyond trend and seasonality. As an alternative, the paper proposes normalised approximate entropy (ApEn), an information-theoretic measure of series regularity that captures a broader range of deterministic structure. Empirical results across the six series demonstrate that ApEn provides a more reliable ordering of relative forecastability than the coefficient of variation, offering forecasting practitioners a principled basis for assessing how predictable a given time series is and, by extension, the degree to which investment in sophisticated modelling methods may be justified.
APA
Catt, P. M. (2009). Forecastability: Insights from physics, graphical decomposition, and information theory. Foresight: The International Journal of Applied Forecasting, 13, 24–33.
BibTeX
@article{catt2009forecastability,
author = {Catt, Peter M.},
title = {Forecastability: Insights from Physics, Graphical Decomposition, and Information Theory},
journal = {Foresight: The International Journal of Applied Forecasting},
year = {2009},
volume = {13},
pages = {24--33}
}