Time Series · aiwiki.page

A time series is a sequence of observations arranged according to when they occur. Examples include monthly atmospheric carbon dioxide concentrations, daily sales, and measurements collected during an industrial process. In statistics, time-series analysis examines patterns and dependence within such records. Unlike methods that treat observations as interchangeable, it preserves their temporal order, because earlier and later values may be related through persistence, trends, or recurring fluctuations. (itl.nist.gov)

Representation and temporal structure

A discrete-time series is commonly written as $y_1,y_2,\ldots,y_T$ , where the index identifies successive observation times. Many standard methods assume equally spaced observations, such as hourly readings or monthly totals. Under that assumption, timestamps can be represented implicitly by the index, although the actual sampling interval remains important for interpretation. Irregularly spaced observations require methods that account for unequal elapsed times. (itl.nist.gov)

A univariate series records one quantity at each time; multivariate analysis considers several quantities together. Statistical modeling distinguishes the observed record from the underlying stochastic process that generates it. In this formulation, observations are realizations of time-indexed random variables, and a model describes their temporal dependence rather than only their individual distributions. (itl.nist.gov)

The sampling interval determines which patterns are visible. Monthly observations, for example, can reveal annual seasonality but cannot directly resolve variation within a day. A time plot is therefore an important starting point for exploratory data analysis: it can expose changes in level, variability, unusual observations, and recurring patterns before a model is fitted. (itl.nist.gov)

Trend, seasonality, and decomposition

A series may contain a long-term trend, recurring seasonality, and an irregular remainder. Seasonality is associated with a known repeating period, such as a week or year. Cyclical movements need not have a fixed duration; decomposition methods often combine them with the trend into a trend-cycle component. (otexts.com)

An additive decomposition represents the observations as

y_t=T_t+S_t+R_t,

where $T_t$ is the trend-cycle, $S_t$ the seasonal component, and $R_t$ the remainder. This form is appropriate when seasonal fluctuations have approximately constant magnitude. A multiplicative form, $y_t=T_tS_tR_t$ , describes fluctuations that scale with the series level. For positive data, logarithmic transformation can convert a multiplicative relationship into an additive one. (otexts.com)

Removing the estimated seasonal component produces seasonally adjusted data. These still contain irregular variation and should not be confused with the underlying trend. Decomposition describes patterns in the observations; it does not by itself explain the mechanisms producing them. (otexts.com)

Dependence and stationarity

Autocorrelation measures the relationship between a series and its own lagged values. Lag $k$ compares observations separated by $k$ sampling intervals. For equally spaced observations, a common sample estimate is

r_k= \frac{\sum_{t=1}^{T-k}(y_t-\bar y)(y_{t+k}-\bar y)} {\sum_{t=1}^{T}(y_t-\bar y)^2}.

An autocorrelation plot displays these estimates across lags. It can help identify persistence, oscillation, and candidate model structures. Positive autocorrelation indicates that observations separated by the specified lag tend to deviate from their mean in the same direction. (itl.nist.gov)

Many models assume stationarity. In the weak, or covariance, sense, a process has a constant expected value, finite constant variance, and covariance depending on the lag rather than the absolute observation time. Stationarity does not imply independence: a stationary process can retain substantial temporal dependence. Trends, changing variability, and deterministic seasonal patterns can violate these assumptions. (itl.nist.gov)

Differencing replaces levels with changes, for example $\Delta y_t=y_t-y_{t-1}$ . Seasonal differencing compares observations one seasonal period apart. These transformations can remove particular forms of nonstationarity; logarithmic or other transformations may stabilize variability. Their suitability depends on the series and the model being considered. (itl.nist.gov)

Statistical and machine-learning models

An autoregressive model predicts a value from earlier values of the same series. A moving-average model instead expresses it through current and previous random innovations. Here “moving average” denotes a stochastic model, not simply a rolling arithmetic average. Combining autoregressive and moving-average terms gives an ARMA model; adding differencing gives an ARIMA model. (itl.nist.gov)

ARIMA models emphasize autocorrelation structure. Exponential smoothing provides a complementary approach based on evolving level, trend, and seasonal components. Model identification can use time plots, autocorrelation, and partial autocorrelation; information criteria compare fit while penalizing complexity. Diagnostics remain necessary after model selection. (otexts.com)

Machine learning offers additional forecasting formulations. Lagged observations can become predictors in supervised learning, while an artificial neural network learns nonlinear relationships between those inputs and later values. A network without hidden layers can reproduce linear regression; hidden layers permit more complex mappings. (otexts.com)

Forecast evaluation

Forecast evaluation separates fitting from prediction. In rolling-origin cross-validation, each forecast uses only observations available before its target time. The origin then moves forward, producing multiple evaluations. The procedure can assess one-step or multi-step forecasts, depending on the intended forecasting horizon. (otexts.com)

Chronological separation of training data and the test set prevents future observations from influencing earlier forecasts—a form of data leakage. Accuracy can be measured using mean squared error, its square root, or absolute-error measures. Errors computed from fitted residuals are not equivalent to genuine out-of-sample forecast errors, because the fitted model has already used those observations. (otexts.com)