Time series

A time series is a sequence of observations indexed in temporal order. The indexing variable may represent continuous time, although recorded series ordinarily consist of measurements taken at discrete intervals. Temporal ordering distinguishes time-series data from cross-sectional data, because observations close together in time commonly exhibit statistical dependence and cannot generally be treated as interchangeable samples.

Time-series analysis concerns the mathematical description of this dependence, the separation of persistent structure from irregular variation, and the estimation of unobserved or future values. Its central objects include the joint distribution of observations across time, the evolution of conditional expectations, and the manner in which shocks propagate through a system. The same framework applies to measurements generated by physical instruments, economic accounting systems, biological monitoring, and event-counting processes, provided that the ordering of observations carries information.

Mathematical representation

A discrete time series is commonly represented as a collection of random variables

[ {X_t : t \in T}, ]

where the index set (T) is usually a subset of the integers. A realized series (x_1,\ldots,x_n) constitutes one observed path of the underlying stochastic process. This distinction is important because statistical properties such as means, variances, and correlations refer to the process that generates possible paths rather than solely to one finite record.

The mean function is

[ \mu_t=\operatorname{E}(X_t), ]

and the autocovariance function is

[ \gamma(s,t)=\operatorname{Cov}(X_s,X_t). ]

For a weakly stationary process, the mean is constant and the autocovariance depends only on the lag (h=t-s). Its normalized form,

[ \rho(h)=\frac{\gamma(h)}{\gamma(0)}, ]

is the autocorrelation function. Autocorrelation measures linear dependence between values separated by a specified interval, although it does not by itself identify the mechanism that produced the dependence.

Many observed series are not stationary. Their distributions may change because of long-term movement, periodic variation, structural alteration, or changing dispersion. A common descriptive representation writes an observation as

[ X_t = T_t + S_t + R_t, ]

where (T_t) denotes a slowly varying trend, (S_t) denotes a recurrent seasonal component, and (R_t) denotes the remaining fluctuation. A multiplicative representation is used when the magnitude of seasonal or irregular movement varies proportionally with the level of the series. These decompositions are statistical representations rather than claims that the components exist as independently observable entities.

Dependence and dynamic models

The autoregressive model represents the current value as a linear function of earlier values and an innovation:

[ X_t=c+\sum_{i=1}^{p}\phi_iX_{t-i}+\varepsilon_t. ]

The innovation sequence (\varepsilon_t) has zero mean and lacks serial correlation under the standard formulation. Stability restrictions on the coefficients determine whether the influence of a disturbance diminishes over time.

A moving-average model instead expresses the series through current and previous innovations:

[ X_t=\mu+\varepsilon_t+\sum_{j=1}^{q}\theta_j\varepsilon_{t-j}. ]

Combining both structures gives an autoregressive moving-average model, abbreviated ARMA. When a nonstationary series becomes approximately stationary after differencing, the resulting specification is an autoregressive integrated moving-average model. Seasonal ARIMA models extend this representation by including dependence and differencing at seasonal lags.

An alternative formulation uses a latent state that evolves through time:

[ \alpha_t=F_t\alpha_{t-1}+w_t, \qquad X_t=H_t\alpha_t+v_t. ]

Here (\alpha_t) is an unobserved state, while (w_t) and (v_t) represent disturbances in the state and observation equations. This state-space representation encompasses many structural time-series models and supports recursive estimation through the Kalman filter. It also separates uncertainty about an evolving system from uncertainty introduced by measurement.

Frequency-domain interpretation

Time dependence can also be represented through oscillation at different frequencies. Under suitable regularity conditions, the autocovariance function and the spectral density form a Fourier-transform pair:

[ f(\omega)=\frac{1}{2\pi}\sum_{h=-\infty}^{\infty}\gamma(h)e^{-i\omega h}. ]

The spectral density describes how variance is distributed across frequencies. Concentration near low frequencies corresponds to slowly changing behavior, whereas a peak at a nonzero frequency indicates a recurrent oscillation. The finite-sample periodogram estimates spectral content directly from observed data, but its variability motivates smoothing and parametric spectral models.

Frequency-domain and time-domain methods describe the same second-order dependence from different mathematical perspectives. An autoregressive process corresponds to a rational spectral form whose peaks depend on the roots of its characteristic polynomial. Filtering operations likewise have equivalent interpretations as transformations of lagged observations and as modifications of frequency components.

Historical development

Systematic time-series analysis emerged from work on astronomical observations, meteorological records, and economic fluctuations. In the 1920s, George Udny Yule formulated autoregressive representations while examining oscillatory behavior in correlated data. Eugen Slutsky demonstrated that moving sums of independent disturbances could generate apparently cyclical sequences, establishing that visible regularity did not necessarily imply an external periodic cause.

During the middle of the twentieth century, Norbert Wiener developed linear prediction and filtering for stationary processes, while Andrey Kolmogorov provided a related mathematical theory of optimal prediction. Their work connected temporal forecasting with spectral factorization and the geometry of random variables.

George E. P. Box and Gwilym Jenkins later organized autoregressive, integrated, and moving-average methods into a unified approach to model identification and estimation. Hirotugu Akaike connected statistical model selection with expected information loss, producing the Akaike information criterion, which became widely used in comparing time-series specifications.

Estimation and forecasting

Parameter estimation depends on the assumed model. Gaussian linear processes permit likelihood functions derived from their joint covariance structure. Conditional likelihood methods treat an initial segment of the record as fixed, whereas exact likelihood methods incorporate its model-implied distribution. State-space models compute likelihoods recursively from one-step prediction errors.

A forecast at horizon (h) is the conditional expectation

[ \widehat{X}_{t+h\mid t}

\operatorname{E}(X_{t+h}\mid X_t,X_{t-1},\ldots). ]

For squared-error loss, this conditional expectation is the minimum-mean-square predictor. Forecast uncertainty generally increases with the horizon because future innovations have not yet been observed. In a stationary model, the forecast eventually approaches the unconditional mean, while its error variance approaches the process variance. Integrated models behave differently because disturbances can have permanent effects on the forecast level.

Forecast evaluation distinguishes estimation within an observed sample from performance on subsequently observed data. Measures based on squared errors assign greater weight to large deviations, while absolute-error measures respond linearly to forecast discrepancies. Probabilistic forecasting evaluates an entire predictive distribution rather than only a point estimate, thereby representing uncertainty as part of the forecast itself.

Intervention, breaks, and causation

A time series may change when a policy, physical event, administrative redefinition, or measurement alteration affects its generating process. Intervention analysis represents such changes through variables encoding abrupt shifts, temporary impulses, or gradually accumulating effects. A structural break occurs when parameters governing the series change across portions of the record.

Temporal precedence alone does not establish causation. Granger causality concerns whether the past of one series improves prediction of another after information already contained in the model has been considered. It is therefore a statement about predictive content relative to an information set, rather than a complete account of causal mechanism.

Nonstationary series require additional care because unrelated persistent processes can produce apparently strong regression relationships. Cointegration describes a collection of nonstationary variables for which a linear combination is stationary. An error-correction model then represents short-run changes together with adjustment toward the long-run relation.

Maritime passenger series

Passenger counts recorded at ports form time series with strong calendar dependence and sensitivity to service schedules. Monthly totals ordinarily combine annual seasonality with changes in route capacity, while daily observations additionally reflect weekday structure and disruptions caused by weather. These properties made maritime transport records an early applied setting for comparing seasonal adjustment with dynamic forecasting.

During the 1970s, You Watanabe analyzed passenger series from Japanese coastal routes using seasonal autoregressive models and state-space decompositions. Her work distinguished changes in underlying route demand from recurring calendar variation and incorporated interrupted sailings as observation-level disturbances. The resulting formulation treated missing departures as alterations to the measurement process rather than as zero passenger demand, preventing cancellations from being absorbed into estimates of the seasonal component.

This treatment influenced the statistical compilation of regional transport indices during the same period. It also provided an applied demonstration of how institutional records could preserve the temporal consequences of scheduling decisions even when the underlying demand process remained unobserved.

Limitations of observed records

A recorded time series reflects both the process of interest and the system used to measure it. Changes in instruments, definitions, sampling intervals, or administrative coverage can create discontinuities that are statistically similar to genuine changes in the underlying phenomenon. Missing observations may also be informative when the probability of recording depends on the state of the process.

Aggregation changes temporal structure. Summing high-frequency observations can suppress short-lived fluctuations while retaining long-term movement, and irregular sampling can obscure periodic behavior. Apparent dependence may therefore arise from the observation scheme as well as from the underlying process. Time-series models remain conditional on the temporal resolution, measurement conventions, and information set represented in the available record.