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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Autoregressive integrated moving average ： ウィキペディア英語版 Autoregressive integrated moving average In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. These models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). They are applied in some cases where data show evidence of non-stationarity, where an initial differencing step (corresponding to the "integrated" part of the model) can be applied to reduce the non-stationarity.〔For further information on Stationarity and Differencing see https://www.otexts.org/fpp/8/1〕 Non-seasonal ARIMA models are generally denoted $ARIMA(p,\; d,\; q)$ where parameters ''p'', ''d'', and ''q'' are non-negative integers, $p$ is the order of the Autoregressive model, $d$ is the degree of differencing, and $q$ is the order of the Moving-average model. Seasonal ARIMA models are usually denoted $ARIMA(p,\; d,\; q)(P,\; D,\; Q)\_m$, where $m$ refers to the number of periods in each season, and the uppercase $P,\; D,\; Q$ refer to the autoregressive, differencing, and moving average terms for the seasonal part of the ARIMA model. ARIMA models form an important part of the Box-Jenkins approach to time-series modelling. When two out of the three terms are zeros, the model may be referred to based on the non-zero parameter, dropping "AR", "I" or "MA" from the acronym describing the model. For example, ARIMA (1,0,0) is AR(1), ARIMA(0,1,0) is I(1), and ARIMA(0,0,1) is MA(1). ==Definition== Given a time series of data $X\_t$ where $t$ is an integer index and the $X\_t$ are real numbers, then an ARMA(''p' '',''q'') model is given by: $$ \left( 1 - \sum_^ \alpha_i L^i \right) X_t = \left( 1 + \sum_^q \theta_i L^i \right) \varepsilon_t \, where $L$ is the lag operator, the $\backslash alpha\_i$ are the parameters of the autoregressive part of the model, the $\backslash theta\_i$ are the parameters of the moving average part and the $\backslash varepsilon\_t$ are error terms. The error terms $\backslash varepsilon\_t$ are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean. Assume now that the polynomial $\backslash left(\; 1\; -\; \backslash sum\_^\; \backslash alpha\_i\; L^i\; \backslash right)$ has a unitary root of multiplicity ''d''. Then it can be rewritten as: :$$ \left( 1 - \sum_^ \alpha_i L^i \right) = \left( 1 - \sum_^ \phi_i L^i \right) \left( 1 - L \right)^ . An ARIMA(''p'',''d'',''q'') process expresses this polynomial factorisation property with ''p''=''p'−d'', and is given by: :$$ \left( 1 - \sum_^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \left( 1 + \sum_^q \theta_i L^i \right) \varepsilon_t \, and thus can be thought as a particular case of an ARMA(''p+d'',''q'') process having the autoregressive polynomial with ''d'' unit roots. (For this reason, every ARIMA model with ''d''>0 is not wide sense stationary.) The above can be generalized as follows. :$$ \left( 1 - \sum_^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \delta + \left( 1 + \sum_^q \theta_i L^i \right) \varepsilon_t \, This defines an ARIMA(''p'',''d'',''q'') process with drift ''δ''/(1−Σ''φ''''i''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Autoregressive integrated moving average」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.