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Forecasting Process Details |

A dependent time series that is modeled as a linear combination of its own past values and past values of an error series is known as a (pure) ARIMA model.

*p*- is the order of the autoregressive part
*d*- is the order of the differencing (rarely should
*d*> 2 be needed) *q*- is the order of the moving-average process

Given a dependent time series ,mathematically the ARIMA model is written as

where

*t*- indexes time
- is the mean term
*B*- is the backshift operator; that is,
*B**X*_{t}=*X*_{t-1} - is the autoregressive operator,
represented as a polynomial in the back shift operator:
- is the moving-average operator,
represented as a polynomial in the back shift operator:
*a*_{t}- is the independent disturbance, also called the random error

For example, the mathematical form of the ARIMA(1,1,2) model is

*P*- is the order of the seasonal autoregressive part
*D*- is the order of the seasonal differencing (rarely should
*D*> 1 be needed) *Q*- is the order of the seasonal moving-average process
*s*- is the length of the seasonal cycle

Given a dependent time series ,mathematically the ARIMA seasonal model is written as

where

- is the seasonal autoregressive operator,
represented as a polynomial in the back shift operator:

- is the seasonal moving-average operator, represented as a polynomial in the back shift operator:

- is the seasonal autoregressive operator,
represented as a polynomial in the back shift operator:

For example, the mathematical form of the ARIMA(1,0,1)(1,1,2)

- I(
*d*)(*D*)_{s} - integrated model or ARIMA(
*0,d,0*)(*0,D,0*) - AR(
*p*)(*P*)_{s} - autoregressive model or ARIMA(
*p,0,0*)(*P,0,0*) - IAR(
*p,d*)(*P,D*)_{s} - integrated autoregressive model or ARIMA(
*p,d,0*)(*P,D,0*)_{s} - MA(
*q*)(*Q*)_{s} - moving average model or ARIMA(
*0,0,q*)(*0,0,Q*)_{s} - IMA(
*d,q*)(*D,Q*)_{s} - integrated moving average model or ARIMA(
*0,d,q*)(*0,D,Q*)_{s} - ARMA(
*p,q*)(*P,Q*)_{s} - autoregressive moving-average model
or ARIMA(
*p,0,q*)(*P,0,Q*)_{s}

- I(

Let

This model is called a

Given the

*d*_{i}- is the simple order of the differencing for the
*i*th predictor time series, (1-*B*)^{di}*X*_{i,t}(rarely should*d*_{i}> 2 be needed) *k*_{i}- is the pure time delay (lag) for the effect of the
*i*th predictor time series,*X*_{i,t}*B*^{ki}=*X*_{i,t-ki} *p*_{i}- is the simple order of the denominator for the
*i*th predictor time series *q*_{i}- is the simple order of the numerator for the
*i*th predictor time series

The mathematical notation used to describe a transfer function is

where

*B*- is the backshift operator; that is,
*B**X*_{t}=*X*_{t-1} - is the denominator polynomial of the transfer function
for the
*i*th predictor time series: - is the numerator polynomial of the transfer function
for the
*i*th predictor time series:

The numerator factors for a transfer function for a predictor series are like the MA part of the ARMA model for the noise series. The denominator factors for a transfer function for a predictor series are like the AR part of the ARMA model for the noise series. Denominator factors introduce exponentially weighted, infinite distributed lags into the transfer function.

For example, the transfer function for the

*k*_{i}=3- time lag is 3
*d*_{i}=1- simple order of differencing is one
*p*_{i}=1- simple order of the denominator is one
*q*_{i}=2- simple order of the numerator is two

would be written as [Dif(1)Lag(3)N(2)/D(1)]. The mathematical notation for the transfer function in this example is

The general transfer function notation for the

*D*_{i}- is the seasonal order of the differencing for the
*i*th predictor time series (rarely should*D*> 1 be needed)_{i} *P*_{i}- is the seasonal order of the denominator for the
*i*th predictor time series (rarely should*P*> 2 be needed)_{i} *Q*_{i}- is the seasonal order of the numerator for the
*i*th predictor time series, (rarely should*Q*> 2 be needed)_{i} *s*- is the length of the seasonal cycle

The mathematical notation used to describe a seasonal transfer function is

where

- is the denominator seasonal polynomial of the transfer function
for the
*i*th predictor time series:

- is the numerator seasonal polynomial of the transfer function
for the
*i*th predictor time series:

- is the denominator seasonal polynomial of the transfer function
for the

For example, the transfer function for the

*d*_{i}=2- simple order of differencing is two
*D*_{i}= 1- seasonal order of differencing is one
*q*_{i}=2- simple order of the numerator is two
*Q*_{i}= 1- seasonal order of the numerator is one

would be written as [Dif(2)(1)

Note: In this case,
[Dif(2)(1)_{s} N(2)(1)_{s}]
= [Dif(2)(1)_{s}Lag(0)N(2)(1)_{s}/D(0)(0)_{s}].

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