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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 variaty of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc).


NOTATION


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 Number s. We also are accustomed to

:Y= \{Y_t : t \in T\ \}


ASSUMPTIONS


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}+\epsilon_t

where the term \epsilon_t is the source of randomness and is called White Noise . It is assumed to have the following characteristics:

1. E {Link without Title} =0

2. E {Link without Title} =\sigma^2

3. E {Link without Title} =0 orall t
ot=s

If it also has a Normal distribution, it is called Normal White-Noise:

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


RELATED TOOLS


Tools for investigating time-series data include:



INDUSTRY USAGE


Any associative array of times and numbers can be viewed as a time series. The times may not necessarily be of a regular interval length.

For example, the historical fluctuations in the price of a NYMEX Gold Contract can be said to be the time series for NYMEX Gold.

Analysts throughout the economy will use the tools outlined here to aid in the management of their corresponding businesses. Energy traders, for example, will often attempt to forecast power consumption based upon both weather normals and short term weather forecasts.


SEE ALSO




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