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:

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)