A discrete time series consists of a set of observations {x₁, x₂, ... , x_t, ... } of some phenomenon, taken at equally spaced time intervals. (We assume that x_t is real.) The main purpose of time series analysis is to understand the underlying mechanism that generates the observed data and, in turn, to forecast future values. We assume that the generating mechanism is probabilistic and that the observed series {x₁, x₂, ... , x_t, ... } is a realization of a stochastic process {X₁, X₂, ... , X_t, ... }. Typically, the process is assumed to be stationary and described by a class of linear models called autoregressive moving average (ARMA) models.

An ARMA model of orders p and q (ARMA(p, q)) is defined by

where {_i} and {_i} are the coefficients of the autoregressive (AR) and moving average (MA) parts, respectively, and {Z_t} is white noise with mean zero and variance ². (We assume Z_t is normally distributed.) Using the backward shift operator B defined by B^jX_t=X_t-j, the ARMA(p, q) model above can be written as

where (B)=1-₁B-₂B²-... -_pB^p and (B)=1+₁B+₂B²+... +_qB^q. We assume that the polynomials (x) and (x) have no common zeros. When X_t is a vector, we have a multivariate or vector ARMA model; in this case, the AR and MA coefficients are matrices and the noise covariance is also a matrix denoted by . When all the zeros of the polynomial (x) (or its determinant in the multivariate case) are outside the unit circle, the model is said to be stationary and the ARMA(p, q) model can be expanded formally as an MA() model (X_t=_iZ_t-i). Similarly, if the zeros of (x) are all outside the unit circle, the ARMA(p, q) model is said to be invertible and can be expanded as an AR() model.

Autoregressive integrated moving average (ARIMA) models are used to model a special class of nonstationary series. An ARIMA(p, d, q) (d non-negative integer) model is defined by

Seasonal models are used to incorporate cyclic components in models. A class of commonly encountered seasonal models is that of seasonal ARIMA (SARIMA) models. A SARIMA(p, d, q)(P, D, Q)_s model (d and D are non-negative integers and s is the period) is defined by

where (x) and (x) are polynomials that describe the seasonal part of the process.

The commonly used time series models are represented in this package by objects of the generic form model[param₁, param₂,...]. Each of these objects serves to specify a particular model and does not itself evaluate to anything. They can be entered as arguments of time series functions as well as generated as output.

Here, when model is used as a Mathematica function argument it means the model object defined above. The notation philist denotes a list of AR coefficients {₁, ₂, ... , _p}, thetalist specifies a list of MA coefficients {₁, ₂, ... , _q}, and so on. The noise is zero-mean Gaussian white noise, and its variance or covariance matrix will be called the noise parameter. d (or D) is the order of differencing, and s is the seasonal period. To extract any of these arguments from a model, we can use the function Part or one of the following functions.

All of the functions in this package with the exception of AsymptoticCovariance and the functions analyzing ARCH models work for both univariate and multivariate cases, although some are illustrated using examples of univariate series only.

Given a model, its covariance function and correlation function can be calculated. For a univariate zero-mean, stationary process, the covariance function at lag h is defined by (h)=E(X_t+hX_t) (E denotes the expectation); the correlation function, by (h)=(h)/(0). For a multivariate process, the matrix covariance function is defined by (X denotes the transpose of X), and the matrix correlation function at lag h, R(h), is given by where _ij(h)=((h))_ij. Another useful function is the partial correlation function. The partial correlation function at lag k, _{k, k}, is defined to be the correlation between X_t+k and X_t with all intervening variables fixed. The power spectrum of a univariate ARMA(p, q) process is given by , and for a multivariate process the power spectrum is .

Both RandomSequence and TimeSeries uses the current default random number generator. Sequences generated prior to Version 6.0 of Mathematica can be obtained by including the option LegacySequence→True.