PaperlayerView exact revision on arXiv

Source-linked AI summary

Principles and Algorithms for Forecasting Groups of Time Series: Locality and Globality

Pablo Montero-Manso, Rob J Hyndman

arXiv:2008.00444v3cs.LGstat.ML

TL;DR

Forecasting groups of time series raises questions about when to model series jointly rather than separately, especially because global methods also perform well on heterogeneous data. The paper formalizes local and global algorithms, derives generalization bounds and design principles, and empirically evaluates simple global models. It reports that global methods can match local forecasts, tolerate greater complexity and memory, and achieve competitive accuracy with far fewer parameters.

  • Problem

    Global methods perform well even on heterogeneous time-series datasets, but the general principles underlying this performance are not well understood.

  • Method

    The paper formalizes local and global forecasting, derives generalization bounds, proposes complexity-based design principles, and evaluates naive global models empirically.

  • Results

    Global and local methods can produce the same forecasts without assumptions about series similarity, while global complexity remains constant as local complexity grows with set size.

  • Takeaways & Limitations

    Global models can use greater complexity and longer memory, and global linear models can achieve comparable or superior accuracy with orders of magnitude fewer parameters than local alternatives.

  • Takeaways & Limitations

    Finite-memory local and global methods are not guaranteed to be equivalent, and data-driven partition selection can yield vacuous worst-case bounds.

Abstract

from arXiv · show

Forecasting groups of time series is of increasing practical importance, e.g. forecasting the demand for multiple products offered by a retailer or server loads within a data center. The local approach to this problem considers each time series separately and fits a function or model to each series. The global approach fits a single function to all series. For groups of similar time series, global methods outperform the more established local methods. However, recent results show good performance of global models even in heterogeneous datasets. This suggests a more general applicability of global methods, potentially leading to more accurate tools and new scenarios to study. Formalizing the setting of forecasting a set of time series with local and global methods, we provide the following contributions: 1) Global methods are not more restrictive than local methods, both can produce the same forecasts without any assumptions about similarity of the series. Global models can succeed in a wider range of problems than previously thought. 2) Basic generalization bounds for local and global algorithms. The complexity of local methods grows with the size of the set while it remains constant for global methods. In large datasets, a global algorithm can afford to be quite complex and still benefit from better generalization. These bounds serve to clarify and support recent experimental results in the field, and guide the design of new algorithms. For the class of autoregressive models, this implies that global models can have much larger memory than local methods. 3) In an extensive empirical study, purposely naive algorithms derived from these principles, such as global linear models or deep networks result in superior accuracy. In particular, global linear models can provide competitive accuracy with two orders of magnitude fewer parameters than local methods.

1 Introduction

The paper examines why global forecasting methods perform well even on heterogeneous groups of time series and formalizes principles for comparing global and local approaches. It derives complexity-based guidance and tests simple global models empirically.

  • Forecasting groups of time series matters in applications including retail demand, tourist arrivals, server loads, and ride shares.
  • Global methods pool series and fit one function, whereas local methods model each series separately.
  • Recent results show strong global-model performance even for heterogeneous datasets, but the principles behind that performance remain insufficiently understood.
  • The paper formalizes local and global forecasting, showing that their complexity can be compared within a common framework.
  • Generalization bounds show that local complexity grows with the number of series while global complexity remains constant, allowing complex global models to generalize better in large datasets.
  • Three design principles are highlighted: adding features or richer model classes, increasing autoregressive memory, and partitioning the series set.
  • A large benchmark study evaluates intentionally naive global linear, polynomial, and deep-network methods without feature engineering, tuning, preprocessing, or model selection.
  • The complexity analysis is presented as a theoretical explanation for recent empirical evidence on global forecasting methods.

2 Equivalence of global and local approaches for forecasting a set of time series

The paper formalizes local and global forecasting algorithms for finite sets of finite-length time series and studies when they can produce equivalent forecasts. Equivalence holds generally, but finite-memory restrictions create a boundary requiring longer global memory.

  • For time-series forecasting, every forecast expressible by a local algorithm can also be expressed by a global algorithm.
  • The equivalence relies on finite observed series and forecasting functions that may use the whole observed history.
  • A local algorithm returns a forecasting function for each series, while a global algorithm returns one function for the entire set.
  • Conversely, every global forecast can be reproduced by a local algorithm that applies the same function to each series.
  • If distinct series share the same recent M observations but differ earlier, finite-memory global methods may produce identical forecasts where local methods differ.
  • Guaranteeing equivalence for finite-memory models therefore requires global methods to use longer memory than local methods.

3 Generalization relationship between local and global approaches

The paper compares local and global forecasting through generalization bounds, showing that local complexity scales with the number of series while global complexity remains constant. These results motivate more complex global models, although the comparison depends on assumptions about sampling, independence, and fixed partitions.

  • 3.1 Generalization error bounds: Generalization error measures how far expected out-of-sample error may differ from in-sample error, using probabilistic bounds based on sample size and hypothesis-class complexity.The basic bound assumes bounded loss values and can apply to arbitrary finite hypothesis classes and data distributions under its stated conditions.
  • 3.1 Generalization error bounds: An effective sample size N is required for time-series observations, and it should generally increase with series length while remaining below T.This assumption abstracts important properties of the underlying time-series processes.
  • 3.3 Comparing local and global methods on groups of independent time series: For one isolated series with the same hypothesis class, the local approach has a better worst-case out-of-sample error guarantee than the global approach.The result relies on bounded errors and independence across series when applying the bound to the grouped setting.
  • 3.4 Relative complexity of local and global approaches hypothesis classes: Across a group of series, local complexity grows with set size, whereas global complexity remains constant, allowing a global model to match the complexity of the summed local models.With a state-of-the-art local baseline such as separate ARIMA fits, the global approach can be at least as complex as all individual ARIMAs combined.
  • 3.4 Relative complexity of local and global approaches hypothesis classes: The theory prescribes increasing global-model complexity through additional features, alternative model families, or longer autoregressive memory.For autoregressive models, global methods can afford substantially more memory than local alternatives.
  • 3.6 Relative Complexity of Partitioning S: Complexity bounds for partitions apply only when the partition is fixed before observing data; data-dependent clustering introduces combinatorial complexity and can make worst-case bounds vacuous.Averaging errors across series may also be inappropriate when series have different scales or measurement units.

4 Experimental Setup

The experiments compare established local forecasting benchmarks with pooled global autoregressive models across diverse datasets and error settings. The global methods vary complexity through linear features, neural networks, regression trees, or random partitioning.

  • Local methods: The benchmark includes auto.arima, ets, theta, TBATS, and STL-AR, with model selection or decomposition procedures specified for each method.The first three target monthly, quarterly, and annual data, while TBATS and STL-AR also handle multiple seasonalities.
  • Global methods: Global methods pool lag-embedded matrices from all series and fit model classes by minimizing loss in a standard regression problem.The final matrix column is the target, while preceding columns are predictors.
  • Global methods: Linear Autoregressive is the global baseline because least-squares linear models isolate globality while overlapping the ARIMA model class.This benchmark excludes advanced machine-learning techniques such as implicit regularization by SGD, bagging, and ensembling.
  • Global methods: Featurized Linear Autoregressive increases complexity with degree-2 and degree-3 polynomial features and subsumes the linear autoregressive model.It represents a deliberately naïve complexity increase through nonlinear features.
  • Global methods: The study also evaluates a five-layer, 32-unit-per-layer ReLU MLP, an XGBoost regression tree, and Partitioned Linear models applied after random dataset partitioning.Partitioned Linear tests partitioning as an alternative complexity-control strategy, while the neural network and tree use cross-validation early stopping.

5 Experiments

Across datasets, increasing memory and model complexity often improve global forecast accuracy, allowing global models to match or outperform local methods even for heterogeneous series. The experiments also show over-fitting, seasonal effects, and model-class-specific behavior that constrain these gains.

  • 5.1 Large memory: A global AR(25) fitted to 617 M1 Monthly series outperforms state-of-the-art local methods with 25 parameters versus approximately 1600 for auto.arima.This corresponds to a parameter reduction close to two orders of magnitude while achieving better accuracy.
  • 5.1 Large memory: Global linear models outperform local approaches for virtually all evaluated datasets, with exceptions in CIF2016 and FREDMD-Monthly.Even in those exceptions, the global linear model outperforms two of the three local methods.
  • 5.1 Large memory: Increasing memory generally improves accuracy, but very long memories can cause global models to over-fit and produce exploding error.The authors note that this behavior can be prevented in practice by simple methods, though the supplied passage does not specify them.
  • 5.1 Large memory: The benefit of greater complexity depends on the dataset: accuracy often improves with memory, but intermittent datasets are exceptions to the longer-memory pattern.The authors report no good explanation for the dominick and parts exceptions.
  • 5.1 Large memory: Long-memory coefficients are especially large at seasonal lags for quarterly and monthly data, indicating that longer memories contribute substantially to global predictions.The fitted coefficients are similar across the M1, M3, tourism, and M4 datasets.
  • 5.2 Nonlinear models: polynomials, deep networks and regression trees: On M3 Monthly data, degree-3 polynomials and deep networks improve over global linear models, while regression trees do not; the deep network approaches local-model accuracy.Recursive forecasting is competitive for the deep network but can be numerically unstable for polynomials and deep networks.

6 Discussion and related work

The discussion reframes global forecasting as broadly applicable beyond related series, linking its advantages to approximation capacity, complexity control, and generalization. It also connects global methods to partitioning, model-class choices, memory, preprocessing, seasonality, and multivariate forecasting.

  • Global and local: Global-and-complex methods more likely outperform local-and-simple methods through better generalization, even when local models achieve lower in-sample loss.The proposed explanation is overfitting control rather than an inability of local models to fit complex patterns.
  • Modern practices and model classes for global methods: Global models require more memory than local models for equal approximation capacity, including when complexity is increased through nonlinearities.The analysis suggests that longer memory can help global models capture long-memory patterns.
  • Global and local: Global methods need not assume related series: adding series-specific dummy variables and interactions can make a global linear model equivalent to a local linear model.This establishes that globality is not intrinsically more restrictive for approximation.
  • Global and local: Global models can support data augmentation, because a global algorithm can fit an augmented set as well as a local model fits the original set.The claim follows from Proposition 1 being valid for any practical set of time series.
  • Partitioning, relatedness and clustering: Clustering is not essential: it can be viewed as partitioning or a preprocessing layer, while global deep networks can forecast highly heterogeneous datasets without clustering.The paper contrasts clustering’s reduced flexibility with fully data-driven, end-to-end modeling.
  • Modern practices and model classes for global methods: The framework decouples model complexity from a particular architecture, extending its rationale to trees, featurization, kernel methods, nearest neighbors, and gradient-boosted decision trees.This broadens the available toolbox beyond prescribed deep-network architectures.
  • Modern practices and model classes for global methods: Global neural networks empirically do not appear unable to learn seasonal patterns, questioning seasonality removal as necessary preprocessing.The discussion presents this as an implication of the empirical results and theoretical justification.
  • Multivariate, Covariables and Sequence models: Globality and multivariate modeling are orthogonal: global models use groups to learn predictive functions, whereas multivariate models directly use interactions among series.Global models therefore require only a single series at forecasting time, unlike models requiring multiple contemporaneous series.

7 Conclusions

The paper argues that global and local forecasting methods are equally general, while global methods can support greater complexity and generalize better. Theory-derived principles and broad empirical evidence support using more complex global models, although practical issues can favor local methods when prior information is available.

  • Global and local methods are equally general: globality is neither restrictive nor dependent on series similarity or relatedness.
  • Global models’ better generalization does not guarantee better performance, but it can make failures easier to detect through cross-validation or in-sample loss.
  • When domain prior information is already available, it can usually be expressed only through local models, limiting the practical symmetry between approaches.
  • Basic generalization bounds show that local complexity grows with set size, whereas global complexity remains constant.
  • Increasing global model complexity, memory, and partitioning provides three principles for designing forecasting algorithms and connecting local and global approaches.
  • Naïve global methods are highly competitive, with global linear models matching or exceeding local state-of-the-art accuracy using orders of magnitude fewer parameters.

A Appendix: Extra results

Across many datasets, increasing global autoregressive memory generally improves accuracy and often lets simple global linear models outperform local alternatives. The appendix also documents frequency-specific saturation, exceptions, and instability where longer memory or recursive polynomial forecasting becomes harmful.

  • Global linear autoregressive models outperform local alternatives across all frequencies in M1, M3, tourism, and M4 when memory increases.
  • Global nonlinear models improve accuracy over global linear models on most yearly datasets as model complexity increases.
  • A global AR(14) model forecasts 24,000 M4 Quarterly series using only 14 numbers, illustrating extreme parameter parsimony.
  • Global linear models reach accuracy comparable to local state-of-the-art alternatives with orders of magnitude fewer parameters, while nonlinear models need less memory for good results.
  • A third-degree global polynomial becomes numerically unstable for M1 Monthly recursive forecasting, exploding after an 18-step horizon.
  • Longer memory can reduce accuracy in the Dominick and parts datasets, while FREDMD shows erratic but generally positive memory-related accuracy changes.
Loading 2008.00444v3…