Time Series

A time series is a sequence of observations recorded in time order, often at regular intervals: daily retail sales, hourly electricity demand, per-minute server latency, quarterly GDP. Time series analysis is the set of statistical and machine learning methods built for data of that shape, where ordering can carry information and observations need not be independent. That possible temporal dependence separates the field from ordinary regression or classification on tabular data, and it is why some supervised learning routines, such as shuffling rows into random train and test splits, do not transfer without modification.

This article focuses on time series as a data structure and gives a broad overview of tasks, model families, historical development, competitions, and benchmarks. The companion Time Series Analysis article covers model specification and estimation, diagnostics, time-aware validation, statistical inference and intervention analysis, and structural change.

Several distinct tasks sit under the same heading. Forecasting predicts future values from past ones. Classification assigns a label to a whole series or window. Anomaly detection flags points or subsequences that do not fit the normal pattern. Imputation fills gaps. A survey first released in 2024 and revised in May 2026 reports that its Time Series Library implements 41 small- and large-scale models, covers 30 datasets, and supports five analysis tasks. Its authors found that models with specific structures suit distinct tasks, so an architecture that forecasts well is not automatically good at classification or anomaly detection ^[1].

Classical time series methods predate modern machine learning; exponential smoothing and ARIMA became standard model families during the twentieth century ^[4][7]. Neural networks trained jointly across large collections of related series arrived in the late 2010s ^[12]. Since 2023, pretrained foundation models for time series have been released that generate predictions for datasets not seen during training, importing the pretrain-once-and-reuse pattern from language modeling ^[21].

Data structure and sampling

Time distinguishes a time series from cross-sectional data. Cross-sectional observations describe many units at one point or period; a time series follows one quantity or a vector of quantities through successive times. Forecasting texts commonly focus on equally spaced hourly, daily, monthly, or annual observations, but transaction logs, clinical records, astronomy measurements, and other event-driven data can arrive at unequal intervals ^[47].

A series is univariate when each time index has one measured value and multivariate when each index has a vector of values. A forecasting data set may also contain static attributes, past-observed covariates, and future-known covariates such as calendar variables ^[15][31]. These distinctions matter because a method that accepts one regularly sampled target cannot automatically consume irregular timestamps, varying channel counts, or future covariates.

Irregular sampling is not equivalent to an evenly spaced series with occasional blank cells. The observation times themselves can carry information, and different variables may be measured on different schedules. A survey of machine learning for irregular series groups the main responses into temporal discretization, interpolation, recurrent models, attention, and methods designed to be invariant to the observation structure ^[48]. Resampling to a regular grid can make standard tools available, but the interpolation rule and a missingness indicator then become part of the modeling assumptions.

Series characteristics

A series can contain a long-run trend, patterns that repeat at a fixed period, cycles without a fixed period, and irregular variation. Decomposition methods such as STL estimate selected components so that they can be inspected or modeled separately ^[2]. Stationarity describes a series whose statistical properties do not depend on the observation time. Some classical model families work with a stationary representation, but stationarity is not a defining property of every time series ^[3].

Model families

Exponential smoothing, developed through work by Holt, Brown, and Winters in the late 1950s and 1960, represents a series through combinations of level, trend, and seasonality. Modern taxonomies distinguish nine smoothing methods, while ETS adds probabilistic error models and an innovations state-space interpretation ^[4][5][6].

ARIMA represents a different classical family built around autoregression, differencing, and moving-average errors. Seasonal ARIMA adds terms at the seasonal period, while SARIMAX can include exogenous regressors ^{[7][8][44][45]}. The companion analysis article treats model specification, estimation, residual diagnostics, and transformations in depth.

Machine learning and deep learning

Prophet, released as open source by Facebook's Core Data Science team, sits between the two worlds. It fits an additive model in which non-linear trends are combined with yearly, weekly, and daily seasonality plus holiday effects, and it ships as R and Python packages sharing the same underlying Stan code ^[9]. The accompanying paper, published as "Forecasting at scale" in The American Statistician, frames the design around analyst-in-the-loop use: a modular regression model with parameters a domain expert can adjust, plus tooling that flags forecasts for manual review ^[10][11]. The stated goal is "completely automated forecasts that can be tuned by hand by data scientists and analysts" ^[9].

Deep learning entered forecasting through models trained across many related series at once rather than one model per series. DeepAR trains an autoregressive recurrent neural network on a large collection of related series and outputs a full predictive distribution instead of a point forecast; the original paper reported accuracy improvements of around 15 percent over the methods it compared against ^[12]. It was later published in the International Journal of Forecasting and ships in GluonTS, Amazon Web Services' open source library for probabilistic time series modeling, alongside N-BEATS, the Temporal Fusion Transformer, WaveNet, and others ^[13].

N-BEATS, by Boris Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio, dropped recurrence entirely for a deep stack of fully connected layers linked by backward and forward residual connections. On the M3, M4, and TOURISM datasets the authors reported forecast accuracy improvements of 11 percent over a statistical benchmark and 3 percent over the winner of the M4 competition, and argued that generic residual blocks with no time-series-specific components were enough ^[14].

The Temporal Fusion Transformer (TFT), by Bryan Lim, Sercan Arik, Nicolas Loeff, and Tomas Pfister, targets multi-horizon problems that mix static covariates, known future inputs such as scheduled promotions, and historically observed exogenous series. It uses recurrent layers for local processing and interpretable self-attention for long-range structure, plus variable selection and gating components ^[15].

Whether the transformer helps at all became contested. In 2022 Ailing Zeng and co-authors introduced LTSF-Linear, a set of "embarrassingly simple one-layer linear models," and reported that it outperformed the existing transformer-based long-term forecasting models on nine real-life datasets "in all cases, and often by a large margin," arguing that permutation-invariant attention discards ordering information ^[16]. PatchTST answered by changing what the transformer sees: it segments the series into subseries-level patches used as input tokens, and adopts channel independence, where each variate is a univariate series sharing one embedding and one set of transformer weights ^[17].

What the forecasting competitions showed

The Makridakis competitions provide large comparative studies of forecasting methods. M4, organized by Spyros Makridakis, Evangelos Spiliotis, and Vassilios Assimakopoulos, evaluated 61 forecasting methods on 100,000 series and added prediction intervals to the scoring ^[18]. It was won by Slawek Smyl, then a data scientist at Uber, whose submission mixed a standard exponential smoothing model with long short-term memory networks in a single hybrid, hierarchical method ^[40][41]. The organizers singled out that hybrid as the competition's biggest surprise, and reported that the six pure machine learning entries performed poorly: none was more accurate than the combination benchmark, and only one beat the Naive2 baseline ^[19].

M5 produced a different outcome in a single retail-hierarchy setting. It ran on the Kaggle platform from 2 March to 30 June 2020 over 42,840 hierarchical series of Walmart unit sales, with participants forecasting 28 days ahead ^[43][20]. Gradient boosting dominated the result. A retrospective by members of the Amazon forecasting team called the prevalence of gradient boosted trees among the top contestants "potentially the most eye-catching result" of M5, found that "tree-based methods out-shone other solutions, in particular deep learning-based solutions," and noted that the winners in both tracks relied heavily on them ^[42]. Read together, the two competitions bracket the field: pure machine learning methods lost on M4's pool of 100,000 series and won on M5's single retail hierarchy ^[18][19][42].

Time series foundation models

Starting in late 2023, several groups pretrained a single model on very large time series corpora and released it for zero-shot use. TimeGPT, announced in October 2023, described itself as "the first foundation model for time series," able to generate predictions for datasets not seen during training ^[21]. Its developer Nixtla states that it was trained on over 100 billion data points and keeps the model closed source behind an API, distributing an Apache-licensed SDK that also exposes anomaly detection, exogenous variables, and fine-tuning ^[22].

Google Research's TimesFM is a patched decoder-style attention model, described by its authors as reaching zero-shot accuracy close to supervised models trained per dataset ^[23]. Google stated it has 200M parameters and was pretrained on a corpus of 100 billion real-world time points including Google Trends and Wikipedia pageviews, and that it was accepted at ICML 2024 ^[24]. TimesFM 2.5, released on 15 September 2025, cut the parameter count back to 200M from the 500M of TimesFM 2.0 while raising the maximum context from 2,048 to 16k points ^[25].

Amazon's Chronos took a different route, treating forecasting as language modeling: it scales and quantizes values into a fixed vocabulary, then trains T5 architectures on the resulting tokens with a cross-entropy loss, with model sizes from 20M to 710M parameters evaluated across 42 datasets ^[26]. Chronos-Bolt followed in November 2024 with models of 9M to 205M parameters that Amazon reports are up to 250 times faster and 20 times more memory efficient than the original Chronos models of the same size, while achieving 5 percent lower error ^[27].

Salesforce AI Research's Moirai is a masked encoder trained on LOTSA, an archive of over 27 billion observations across nine domains, and was published at ICML 2024 ^[28]. Its successor Moirai-MoE replaced Moirai's per-frequency projection layers with a sparse mixture of experts inside the transformer, on the argument that frequency is not a reliable indicator of the patterns underlying a series, since series of different frequencies can display similar patterns and series of the same frequency can display different ones ^[29].

TiRex, from a team including Sepp Hochreiter, breaks with the transformer default that its paper says most zero-shot forecasters rely on. It is built on xLSTM and argues that recurrent state tracking, which transformers, state space models, and parallelizable RNNs such as RWKV give up, matters for long-horizon forecasting; the authors reported state-of-the-art zero-shot results on the GiftEval and Chronos-ZS Hugging Face benchmarks against TabPFN-TS, Chronos-Bolt, TimesFM, and Moirai ^[30].

Scroll sideways for more →

The 2025-2026 releases moved past univariate point forecasting. Chronos-2 handles univariate, multivariate, and covariate-informed forecasting zero-shot through a group attention mechanism that shares information across related series, which the authors describe as in-context learning; they report the best results among pretrained models on the fev-bench, GIFT-Eval, and Chronos Benchmark II suites. On fev-bench under scaled quantile loss, the 120M-parameter model records an average win rate of 90.7 percent, against 80.8 for the next best model (TiRex) and 60.3 for its own predecessor Chronos-Bolt ^[31]. TiRex-2, posted in July 2026, extends the xLSTM approach to multivariate data with past and future covariates and to streaming at arbitrary context lengths with constant inference cost per patch, using 38.4M active parameters in univariate mode and 44.1M more for multivariate forecasting ^[32].

Classification, anomaly detection, and other tasks

Time series classification has its own benchmark tradition built on the UCR archive. A 2024 experimental review extended an earlier comparison from 18 algorithms on 85 datasets to 112 UCR datasets plus 30 newly introduced problems, and widened the taxonomy from five algorithm categories to eight to take in convolution-based, feature-based, and deep learning approaches. It found that Hydra+MultiROCKET and HIVE-COTEv2 "perform significantly better than other approaches on both the current and new TSC problems" ^[33].

Anomaly detection is the task where evaluation has been most heavily criticized. In work introducing the UCR Time Series Anomaly Archive, Renjie Wu and Eamonn Keogh argued that most examples in the popular Yahoo, Numenta, and NASA benchmarks suffer from one or more of four flaws, that published comparisons built on them may be unreliable, and that "much of the apparent progress in recent years may be illusionary" ^[34]. A 2024 survey covering a decade of the field organizes the methods under a process-centric taxonomy and traces the shift from statistical measures toward machine learning; it names cyber security, financial markets, law enforcement, and health care among the application areas ^[35].

Benchmark design

Time series benchmarks must state their forecast horizons, aggregation rules, and accuracy measures. Scale-dependent errors such as MAE and RMSE cannot be compared directly across series measured in different units. MAPE is undefined or infinite when an actual value is zero and can become extreme near zero. MASE scales errors by an in-sample naive forecast and is comparable across series ^[36].

Pretrained models create an additional data-leakage problem because public evaluation series may have appeared in a pretraining corpus. GIFT-Eval covers 23 datasets with more than 144,000 time series and 177 million data points across seven domains and 10 frequencies, and provides a separate non-leaking pretraining corpus of about 230 billion points ^[37]. fev-bench contains 100 forecasting tasks across seven domains, including 46 tasks with covariates, and reports bootstrapped uncertainty for win rates and skill scores ^[38].

Limitations

The gap between benchmark results and deployment decisions remains wide. A 2026 break-even analysis compared zero-shot and LoRA-fine-tuned foundation models (Chronos, Moirai, Lag-Llama) against Naive, ETS, ARIMA, and XGBoost baselines across 30 datasets at six training-set sizes. Foundation models won at every training fraction on 15 of the 30 datasets, but on 6 the classical methods overtook zero-shot foundation models with as little as 2 percent of the training data, and on the remaining 9 the break-even point ranged from 24 to 8,361 samples. The authors also found that LoRA fine-tuning can actively degrade performance on short series ^[39]. Running a pretrained model on GPUs is therefore not automatically the right call, and the answer depends on how much history exists.

Task boundaries also matter. Some applications are spatiotemporal rather than purely sequential and form a separate problem class. In medium-range weather prediction, for example, GraphCast predicts hundreds of weather variables 10 days ahead at 0.25 degree resolution globally in under a minute ^[46]. The anomaly-detection critique and the uncertainty analysis in fev-bench also show why a result on one archive, metric, or aggregation rule should not be generalized automatically to a different task ^[34][38].

References

1. ^Wang, Y., Wu, H., Dong, J., Liu, Y., Wang, C., Long, M., Wang, J. "Deep Time Series Models: A Comprehensive Survey and Benchmark." arXiv:2407.13278 (first submitted 18 July 2024; version 3 revised 4 May 2026). arxiv.org/...2407.13278
2. ^Hyndman, R. J., Athanasopoulos, G. "STL decomposition," in Forecasting: Principles and Practice, 3rd ed., OTexts, 2021. otexts.com/...stl
3. ^Hyndman, R. J., Athanasopoulos, G. "Stationarity and differencing," Forecasting: Principles and Practice, 3rd ed. otexts.com/...stationarity
4. ^Hyndman, R. J., Athanasopoulos, G. "Exponential smoothing," Forecasting: Principles and Practice, 3rd ed. otexts.com/...expsmooth
5. ^Hyndman, R. J., Athanasopoulos, G. "A taxonomy of exponential smoothing methods," Forecasting: Principles and Practice, 3rd ed. otexts.com/...taxonomy
6. ^Hyndman, R. J., Athanasopoulos, G. "Innovations state space models for exponential smoothing," Forecasting: Principles and Practice, 3rd ed. otexts.com/...ets
7. ^Hyndman, R. J., Athanasopoulos, G. "ARIMA models," Forecasting: Principles and Practice, 3rd ed. otexts.com/...arima
8. ^Hyndman, R. J., Athanasopoulos, G. "Non-seasonal ARIMA models" (including Table 9.1), Forecasting: Principles and Practice, 3rd ed. otexts.com/...non-seasonal-arima
9. ^Prophet project site, Facebook Core Data Science. facebook.github.io/prophet
10. ^Taylor, S. J., Letham, B. "Forecasting at Scale." Meta Research publication page. research.facebook.com/...forecasting-at-scale
11. ^facebook/prophet repository (citation: Taylor, S. J., Letham, B., 2018, "Forecasting at scale," The American Statistician 72(1):37-45). github.com/...prophet
12. ^Salinas, D., Flunkert, V., Gasthaus, J. "DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks." arXiv:1704.04110 (13 April 2017). Published as Salinas, Flunkert, Gasthaus, Januschowski, International Journal of Forecasting 36(3):1181-1191. arxiv.org/...1704.04110
13. ^GluonTS documentation, AWS Labs. ts.gluon.ai/stable
14. ^Oreshkin, B. N., Carpov, D., Chapados, N., Bengio, Y. "N-BEATS: Neural basis expansion analysis for interpretable time series forecasting." arXiv:1905.10437 (24 May 2019; revised 20 February 2020). arxiv.org/...1905.10437
15. ^Lim, B., Arik, S. O., Loeff, N., Pfister, T. "Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting." arXiv:1912.09363 (19 December 2019). Published in International Journal of Forecasting 37(4):1748-1764 (2021). arxiv.org/...1912.09363
16. ^Zeng, A., Chen, M., Zhang, L., Xu, Q. "Are Transformers Effective for Time Series Forecasting?" arXiv:2205.13504 (26 May 2022). arxiv.org/...2205.13504
17. ^Nie, Y., Nguyen, N. H., Sinthong, P., Kalagnanam, J. "A Time Series is Worth 64 Words: Long-term Forecasting with Transformers." arXiv:2211.14730, ICLR 2023. arxiv.org/...2211.14730
18. ^Makridakis, S., Spiliotis, E., Assimakopoulos, V. "The M4 Competition: 100,000 time series and 61 forecasting methods." International Journal of Forecasting 36(1):54-74 (2020). doi.org/...j.ijforecast.2019.04.014
19. ^Makridakis, S., Spiliotis, E., Assimakopoulos, V. "The M4 Competition: Results, findings, conclusion and way forward." International Journal of Forecasting 34(4):802-808 (2018). doi.org/...j.ijforecast.2018.06.001
20. ^Makridakis, S., Spiliotis, E., Assimakopoulos, V. "M5 accuracy competition: Results, findings, and conclusions." International Journal of Forecasting 38(4):1346-1364 (2022). doi.org/...j.ijforecast.2021.11.013
21. ^Garza, A., Challu, C., Mergenthaler-Canseco, M. "TimeGPT-1." arXiv:2310.03589 (5 October 2023). arxiv.org/...2310.03589
22. ^Nixtla/nixtla repository (TimeGPT SDK). github.com/...nixtla
23. ^Das, A., Kong, W., Sen, R., Zhou, Y. "A decoder-only foundation model for time-series forecasting." arXiv:2310.10688 (14 October 2023). arxiv.org/...2310.10688
24. ^Sen, R., Zhou, Y. "A decoder-only foundation model for time-series forecasting." Google Research blog, 2 February 2024. research.google/...del-for-time-series-forecasting
25. ^google-research/timesfm repository (TimesFM 2.5 release notes, 15 September 2025). github.com/...timesfm
26. ^Ansari, A. F., Stella, L., Turkmen, C., et al. "Chronos: Learning the Language of Time Series." arXiv:2403.07815 (12 March 2024); Transactions on Machine Learning Research, 2024. arxiv.org/...2403.07815
27. ^amazon-science/chronos-forecasting repository (Chronos, Chronos-Bolt, and Chronos-2 release notes). github.com/...chronos-forecasting
28. ^Woo, G., Liu, C., Kumar, A., Xiong, C., Savarese, S., Sahoo, D. "Unified Training of Universal Time Series Forecasting Transformers." Proceedings of the 41st International Conference on Machine Learning, PMLR 235:53140-53164 (2024). proceedings.mlr.press/...woo24a
29. ^Liu, X., Liu, J., Woo, G., et al. "Moirai-MoE: Empowering Time Series Foundation Models with Sparse Mixture of Experts." arXiv:2410.10469 (14 October 2024). arxiv.org/...2410.10469
30. ^Auer, A., Podest, P., Klotz, D., Böck, S., Klambauer, G., Hochreiter, S. "TiRex: Zero-Shot Forecasting Across Long and Short Horizons with Enhanced In-Context Learning." arXiv:2505.23719 (29 May 2025). arxiv.org/...2505.23719
31. ^Ansari, A. F., Shchur, O., Küken, J., et al. "Chronos-2: From Univariate to Universal Forecasting." arXiv:2510.15821 (17 October 2025). arxiv.org/...2510.15821
32. ^Podest, P., Pichler, M., Bürger, E., et al. "TiRex-2: Generalizing TiRex to Multivariate Data and Streaming." arXiv:2607.01204 (1 July 2026). arxiv.org/...2607.01204
33. ^Middlehurst, M., Schäfer, P., Bagnall, A. "Bake off redux: a review and experimental evaluation of recent time series classification algorithms." arXiv:2304.13029 (25 April 2023; revised 8 May 2024); Data Mining and Knowledge Discovery, DOI 10.1007/s10618-024-01022-1. arxiv.org/...2304.13029
34. ^Wu, R., Keogh, E. J. "Current Time Series Anomaly Detection Benchmarks are Flawed and are Creating the Illusion of Progress." arXiv:2009.13807 (29 September 2020); IEEE ICDE 2022, pp. 1479-1480. arxiv.org/...2009.13807
35. ^Boniol, P., Liu, Q., Huang, M., Palpanas, T., Paparrizos, J. "Dive into Time-Series Anomaly Detection: A Decade Review." arXiv:2412.20512 (29 December 2024). arxiv.org/...2412.20512
36. ^Hyndman, R. J., Athanasopoulos, G. "Evaluating point forecast accuracy," Forecasting: Principles and Practice, 3rd ed. otexts.com/...accuracy
37. ^Aksu, T., Woo, G., Liu, J., Liu, X., Liu, C., Savarese, S., Xiong, C., Sahoo, D. "GIFT-Eval: A Benchmark For General Time Series Forecasting Model Evaluation." arXiv:2410.10393 (14 October 2024). arxiv.org/...2410.10393
38. ^Shchur, O., Ansari, A. F., Turkmen, C., Stella, L., Erickson, N., Guerron, P., Bohlke-Schneider, M., Wang, Y. "fev-bench: A Realistic Benchmark for Time Series Forecasting." arXiv:2509.26468 (30 September 2025). arxiv.org/...2509.26468
39. ^Tan Jerome, N., Simon, F. "When Do Foundation Models Pay Off? A Break-Even Analysis of Pretrained Time Series Forecasters." arXiv:2607.04919 (6 July 2026). arxiv.org/...2607.04919
40. ^Smyl, S. "A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting" (the winning M4 submission). International Journal of Forecasting 36(1):75-85 (2020). doi.org/...j.ijforecast.2019.03.017
41. ^Smyl, S., Ranganathan, J., Pasqua, A. "M4 Forecasting Competition: Introducing a New Hybrid ES-RNN Model." Uber Engineering blog, 25 June 2018. uber.com/...m4-forecasting-competition
42. ^Januschowski, T., Wang, Y., Torkkola, K., Erkkilä, T., Hasson, H., Gasthaus, J. "Forecasting with trees." International Journal of Forecasting 38(4):1473-1481 (2022). doi.org/...j.ijforecast.2021.10.004
43. ^"M5 Competition." Institute For the Future, University of Nicosia. unic.ac.cy/...m5-competition
44. ^Hyndman, R. J., Athanasopoulos, G. "Seasonal ARIMA models," Forecasting: Principles and Practice, 3rd ed. otexts.com/...seasonal-arima
45. ^"Time Series Analysis by State Space Methods (statespace)." statsmodels documentation. statsmodels.org/...statespace
46. ^Lam, R., Sanchez-Gonzalez, A., Willson, M., et al. "Learning skillful medium-range global weather forecasting." Science 382(6677):1416-1421 (2023). doi.org/...science.adi2336
47. ^Hyndman, R. J., Athanasopoulos, G. "Forecasting data and methods," Forecasting: Principles and Practice, 3rd ed. otexts.com/...data-methods
48. ^Shukla, S. N., Marlin, B. M. "A Survey on Principles, Models and Methods for Learning from Irregularly Sampled Time Series." arXiv:2012.00168 (30 November 2020; revised 5 January 2021). arxiv.org/...2012.00168