Time Series

Definitions

"time series, n. (Statistics) a sequence of data indexed by time, often comprising uniformly spaced observations."
Borowski and Borwein (Collins Dictionary of Mathematics, 1989)

"time series n. Statistics. a series of values of a variable taken in successive periods of time."
Collins English Dictionary (1991)

"time-series analysis Techniques used in statistics to study data, often economic, collected over a period of time, e.g. quarterly sales figures for cars over a number of years. Time series are used mainly for prediction purposes. To make useful predictions, allowance must be made for seasonal variation; this gives seasonally adjusted data. Interest then centres on any trend in the series. Typically, for car sales there may be a steady, or accelerating, increase or decrease in demand. Properties such as a five-year periodicity in sales may also be of interest."
Daintith and Nelson (The Penguin Dictionary of Mathematics, 1989)

"A time series is a set of data collected at successive points in time or over successive periods of time. A sequence of monthly data on new housing starts and a sequence of weekly data on product sales are examples of time series. Usually the data in a time series are collected at equally spaced periods of time, such as hour, day, week, month, or year."
Encyclopædia Britannica (2006)

"Time series: Values of a variable recorded, usually at a regular interval, over a long period of time. The observed movement and fluctuations of many such series are composed of four different components, secular trend, seasonal variation, cyclical variation, and irregular variation. An example from medicine is the incidence of a disease recorded yearly over several decades (see Fig. 125). Such data usually require special methods for their analysis because of the presence of serial correlation between the separate observations."
Everitt (The Cambridge Dictionary of Statistics, 1998)

"Time series analysis deals with records that are collected over time."
Fan and Yao (2003)

"A time series is a set of ordered observations on a quantitative characteristic of an individual or collective phenomenon taken at different points of time. Although it is not essential, it is common for these points to be equidistant in time. The essential quality of the series is the order of the observations according to the time variable, as distinct from those which are not ordered at all, e.g. in a random sample chosen simultaneously or are ordered according to their internal properties e.g. a sat arranged in order of magnitude."
Marriott (A Dictionary of Statistical Terms, 1990)

"time series noun Statistics a series of values of a quantity obtained at successive times, often with equal intervals between them."
The New Oxford Dictionary of English (1998)

"A time series is a series of observations taken sequentially over time. In a standard regression model the order in which observations are included in the data set is irrelevant: any ordering is equally satisfactory as far as the analysis is concerned. It is the order property that is crucial to time series and that distinguishes time series from non-time-series data. Actions taken at some time have consequences and effects that are experienced at some later time. Time itself, through the mechanism of causality, imparts structure into a time series."
Pole, West and Harrison (1994)

"time series A series of measurements over time, usually at regular intervals, of a random variable. A prime concern is the forecasting of future values using methods such as exponential smoothing, Holt-Winters forecasting, or Box-Jenkins methods. Models fitted include autoregressive models and moving average models. It is often necessary to deseasonalize the data and to remove any underlying trend before undertaking the analysis."
Upton and Cook (Oxford Dictionary of Statistics, 2002)

"In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying theory of the data points (where did they come from? what generated them?), or to make forecasts (predictions). Time series prediction is the use of a model to predict future events based on known past events: to predict future data points before they are measured. The standard example is the opening price of a share of stock based on its past performance.
Models for time series data can have many forms. Two broad classes of practical importance are the moving average (MA) models, and the autoregressive (AR) models. These two classes depend linearly on previous data points and are treated in more detail in the article on autoregressive moving average models (ARMA). Non-linear dependence on previous data points is of interest because of the possibility of producing a chaotic time series."
Wikipedia (2006)

Books

For more information on time series analysis, Box, Jenkins and Reinsel (1994) is a complete revision of the classic, whilst Hamilton (1994)'s tomb is the bible. To search Amazon.com for books on time series, click here.

Glossary

AR (autoregressive) model
Models for a time series where the next point is dependent on the previous n points: AR(n). (AR(1) is a Markov chain.)
ARCH (autoregressive conditional heteroskedasticity)
In econometrics, ARCH (Engle, 1982) is a model used for forecasting volatility which captures the conditional heteroscedasticity (serial correlation of volatility) of financial returns. Today's conditional variance is a weighted average of past squared unexpected returns. ARCH is an AR process for the variance.
ARIMA (autoregressive integrated moving average) models
Models for time series which resemble ARMA models except in that it is presumed the time series has a steady underlying trend. The models therefore work with the differences between the successive observed values, instead of the values themselves. To retrieve the original data from the differences requires a form of integration and the models are therefore called autoregressive integrated moving average models.
ARMA (autoregressive moving average) models
Models for a time series with no trend (the constant mean is taken as 0). They incorporate the terms in both an autoregressive model and a moving average model.
autocorrelation
A measure of the linear relationship between two separate instances of the same random variable.
Box-Jenkins procedure
A general strategy for the analysis of time series based on the use of ARIMA models or, for seasonal data, SARIMA models. The procedure was set out by Box and Jenkins in their 1970 book Time Series Analysis: Forecasting and Control. The first stage consists of removing trends or cycles from the data. An appropriate type of model must then be identified and its parameters estimated. The estimated model is then compared with the original data and adjustments are made if necessary.
deseasonalize
To remove regular seasonal fluctuations from a time series for the purposes of analysis (for example, to estimate an underlying trend).
GARCH (generalized autoregressive conditional heteroskedasticity)
GARCH (Bollerslev, 1986) generalizes the ARCH model. Today's conditional variance is a function of past squared unexpected returns and its own past values. The model is an infinite weighted average of all past squared forecast errors, with weights that are constrained to be geometrically declining. GARCH is an ARMA(p,q) process in the variance.
Holt-Winters forecasting
An application of exponential smoothing to a time series that displays a trend and seasonality.
MA (moving average) models
Models for a time series with constant mean (taken as 0) where the next point is dependent on the previous n errors: MA(n).
STARIMA models, STARMA models
The space-time equivalents of ARIMA models and ARMA models.
trend
If the mean of a time series changes steadily over time then it is said to exhibit a trend.
unit root
In autoregressive models in econometrics, a unit root is present if yt = yt-1 + other terms.

Mailing Lists

Links