Autoregressive Integrated Moving Average - Definition

Definition

Given a time series of data where is an integer index and the are real numbers, then an ARMA(p,q) model is given by:

\left( 1 - \sum_{i=1}^{p} \alpha_i L^i \right) X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,

where is the lag operator, the are the parameters of the autoregressive part of the model, the are the parameters of the moving average part and the are error terms. The error terms are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.

Assume now that the polynomial has a unitary root of multiplicity d. Then it can be rewritten as:

\left( 1 - \sum_{i=1}^p \alpha_i L^i \right) = \left( 1 - \sum_{i=1}^{p-d} \phi_i L^i \right) \left( 1 - L \right)^{d} .


An ARIMA(p,d,q) process expresses this polynomial factorisation property, and is given by:

\left( 1 - \sum_{i=1}^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,

and thus can be thought as a particular case of an ARMA(p+d,q) process having the auto-regressive polynomial with some roots in the unity. For this reason every ARIMA model with d>0 is not wide sense stationary.

Read more about this topic:  Autoregressive Integrated Moving Average

Famous quotes containing the word definition:

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)