Time series

From Wikipedia, the free encyclopedia

(Redirected from Time series prediction)
Jump to: navigation, search

In statistics, signal processing, and many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at (often 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.

As shown by Box and Jenkins in their book, models for time series data can have many forms and represent different stochastic processes. When modeling the mean of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models (the MA process is related but not to be confused with the concept of moving average ). These three classes depend linearly on previous data points and are treated in more detail in the articles autoregressive moving average models (ARMA) and autoregressive integrated moving average (ARIMA). The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Non-linear dependence on previous data points is of interest because of the possibility of producing a chaotic time series.

Among non-linear time series, there are models to represent the changes of variance along time (heteroskedasticity). These models are called autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc). Recently, wavelet transform based methods (for example locally stationary wavelets and wavelet decomposed neural networks) have gained favor. Multiscale (often referred to as multiresolution) techniques decompose a given time series, attempting to illustrate time dependence at multiple scales.

Contents

A number of different notations are in use for time-series analysis:

X= \{X_1, X_2, \dots \}

is a common notation which specifies a time series X which is indexed by the natural numbers. We also are accustomed to

Y= \{Y_t : t \in T\ \}

There are only two assumptions from which the theory is built:

The general representation of an autoregressive model well-known as AR(p) is

Y_t =\alpha_0+\alpha_1 Y_{t-1}+\alpha_2 Y_{t-2}+\cdots+\alpha_p Y_{t-p}+\varepsilon_t\,

where the term εt is the source of randomness and is called white noise. It is assumed to have the following characteristics:

1. E[\varepsilon_t]=0 \,

2. E[\varepsilon^2_t]=\sigma^2 \,

3. E[\varepsilon_t\varepsilon_s]=0 \quad\forall t\not=s \,

If it also has a normal distribution, it is called normal white noise:

\{\varepsilon_t\}_{(t \in T)} : \mbox{Normal-WN}

Tools for investigating time-series data include:

Time series analysis is exercised in numerous applied fields, from astrophysics to geology. Model selection is often based on the underlying assumption on the data generating process. Take, for example, traffic flow, here we would fully expect periodic behavior (with bursts at peak travel times). In such a situation one may consider applying Dynamic Harmonic Regression (this is highly similar to airline data which is frequently analyzed in the statistics literature).

More recently there has been increased use of time series methods in geophysics (the analysis of rain fall and climate change for example). Within industry, almost every sector will in some way perform time series analysis. With retail, for example, tracking and predicting sales. Analysts will typically load their data into a statistics package ( R and S-Plus are examples of such programs). The most important step is the review of the Autocorrelation function (ACF) which indicates the number of lagged observations to be included in any time series model (one should always analyze the partial autocorrelation function as well).

In general financial series often require non-linear models (such as ARCH) as the application of autoregressive models often results in a model suggesting that to predict the value of tomorrows, lets say share price here, depends almost entirely on yesterday's share price:

Y_t =\alpha_0+\alpha_1 Y_{t-1}+\varepsilon_t \,

(where α1 is close to 1).

Robert Engle recognized the importance of including lagged values of the series' variance. In general time series can be considered in the time domain and/or the frequency domain. This duality has led to many of the recent developments in time series analysis. Wavelet-based methods are an attempt to model series in both domains. Wavelets are compactly supported "small waves", which when convolved with the series itself (when scaled and dilated) gives a scale by scale analysis of the temporal dependence of a series. Such wavelet based methods are frequently applied for climate change problems.

One other (and less researched) area of time series analysis considers the "mining" of series to retrospectively extract knowledge. In the literature this is referred to as time series data mining (TSDM). Techniques in this area often depend on "feature detection". In essence this is an attempt to find the "characteristic" behavior of the series, and use this to find areas of the series which do not adhere to this behavior. Current efforts are led by the computer science department at the University of California (Riverside).

Retrieved from "http://en.wikipedia.org/wiki/Time_series"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.