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lag operator : ウィキペディア英語版
lag operator

In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series
:X= \\,
then
:\, L X_t = X_ for all \; t > 1\,
or equivalently
:\, X_t = L X_ for all \; t \geq 1\,
where ''L'' is the lag operator. Sometimes the symbol ''B'' for backshift is used instead. Note that the lag operator can be raised to arbitrary integer powers so that
:\, L^ X_ = X_\,
and
:\, L^k X_ = X_.\,
==Lag polynomials==
Also polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example,
: \varepsilon_t = X_t - \sum_^p \varphi_i X_ = \left(1 - \sum_^p \varphi_i L^i\right) X_t\,
specifies an AR(''p'') model.
A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as
: \varphi (L) X_t = \theta (L) \varepsilon_t\,
where \varphi (L) and \theta (L) respectively represent the lag polynomials
: \varphi (L) = 1 - \sum_^p \varphi_i L^i\,
and
: \theta (L)= 1 + \sum_^q \theta_i L^i.\,
Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,
: X_t = \frac\varepsilon_t,
means the same thing as
:\varphi (L) X_t = \theta (L) \varepsilon_t\, .
As with polynomials of variables, a polynomial in the lag operator can be divided by another one using polynomial long division. In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.
An annihilator operator, denoted ()_+, removes the entries of the polynomial with negative power (future values).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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