Time Series

A time series is a sequence of observations ordered in time, most often recorded at regular intervals: a stock's daily price, hourly temperature, a city's electricity demand, or the periodic reading of a sensor. What sets it apart from any other dataset is that order matters: each value is related to the ones before it, and shuffling the observations destroys that information.

The most common task is prediction, or forecasting: anticipating future values from past ones. Time series are also used to flag anomalies, monitor a process, or pull patterns apart. When the observations are spread across space as well —sensor networks, satellite imagery— the problem moves into spatio-temporal analysis, which has its own entry in this glossary.

Trend, seasonality, and noise

A series is usually read as the sum of three components. The trend is the long-run direction, such as demand that grows year after year. Seasonality is the set of patterns that repeat over a fixed period: more electricity in winter, more web traffic on Mondays. The noise, or residual, is what remains once the first two are removed, the irregular part no model should try to memorize. Pulling these pieces apart is called decomposition. Hyndman and Athanasopoulos present it as a tool for understanding a series and, in some cases, improving forecasts; it is useful, but not a universal prerequisite for every model.

Stationarity and autocorrelation

Many classical methods start from a stationary series: its relevant statistical properties do not change over time. A trend often calls for ordinary differencing; seasonality may call for differencing at the seasonal period, and no transformation guarantees stationarity by itself. The other central idea is autocorrelation: the correlation of the series with lagged copies of itself. The shape of the ACF provides clues: a slow decay appears with trend, while peaks at seasonal lags signal seasonality. It guides diagnosis and model choice, but is not by itself a conclusive test.

From ARIMA to neural methods

The classical toolkit begins with George Box and Gwilym Jenkins, whose 1970 book popularized ARIMA models (autoregressive, integrated, moving average) and a three-stage method: identify, estimate, and diagnose. Alongside them sit exponential smoothing (Holt-Winters) and state-space models. Neural methods provide alternatives: LSTMs introduced memory for long dependencies; N-BEATS uses residual blocks to produce forecasts; and models such as the Temporal Fusion Transformer combine recurrence and attention across several horizons. The Prophet paper, published in 2017, describes an automated tool for business series. None of these papers establishes a universal winner: results depend on the data, horizon, and metric. ARIMA remains a reference that a new model must beat through time-ordered validation on the actual series. Additional documentation: the NIST handbook.

Validating without temporal leakage

Evaluating a time-series model requires preserving time order. A random split can place future observations in training and yield an optimistic estimate. Time-series cross-validation creates several forecast origins: at each one, the model is fitted on the past and forecasts the next stretch. That better reproduces real use, but it does not guarantee the future. Hirano and Wright show that structural instability changes forecast risk and even the preferred training window. A backtest metric describes the regimes represented in its folds; a new break remains outside the observed sample.