1. Exam SRM Guide
  2. Time Series Models
  3. Overview

Overview

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language[1]).

Methods for analysis

Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.

Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.

Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate.

Models

Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models. These three classes depend linearly on previous data points.[2] Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for vector autoregression. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".

Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, as for example in nonlinear autoregressive exogenous models. Further references on nonlinear time series analysis: (Kantz and Schreiber),[3] and (Abarbanel)[4]

Among other types of non-linear time series models, there are models to represent the changes of variance over time (heteroskedasticity). These models represent autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc.). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modelled as being driven by a separate time-varying process, as in a doubly stochastic model.

Notation

A number of different notations are in use for time-series analysis. A common notation specifying a time series [math]X[/math] that is indexed by the natural numbers is written

[[math]]X = (X_1, X_2, \ldots).[[/math]]

Another common notation is

[[math]]Y = (Y_t: t\in T) [[/math]]

where [math]T[/math] is the [y[index set]].

Conditions

There are two sets of conditions under which much of the theory is built:

Ergodicity implies stationarity, but the converse is not necessarily the case. Stationarity is usually classified into strict stationarity and wide-sense or second-order stationarity. Both models and applications can be developed under each of these conditions, although the models in the latter case might be considered as only partly specified.

In addition, time-series analysis can be applied where the series are seasonally stationary or non-stationary. Situations where the amplitudes of frequency components change with time can be dealt with in time-frequency analysis which makes use of a time–frequency representation of a time-series or signal.[5]

References

  1. Lin, Jessica; Keogh, Eamonn; Lonardi, Stefano; Chiu, Bill (2003). "A symbolic representation of time series, with implications for streaming algorithms". Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery. New York: ACM Press. pp. 2–11. CiteSeerX 10.1.1.14.5597. doi:10.1145/882082.882086. S2CID 6084733.
  2. Gershenfeld, N. (1999). The Nature of Mathematical Modeling. New York: Cambridge University Press. pp. 205–208. ISBN 978-0521570954.
  3. Kantz, Holger; Thomas, Schreiber (2004). Nonlinear Time Series Analysis. London: Cambridge University Press. ISBN 978-0521529020.
  4. Abarbanel, Henry (Nov 25, 1997). Analysis of Observed Chaotic Data. New York: Springer. ISBN 978-0387983721.
  5. Boashash, B. (ed.), (2003) Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003 ISBN 0-08-044335-4

Wikipedia References

  • Wikipedia contributors. "Time series". Wikipedia. Wikipedia. Retrieved 17 August 2022.