Уақыт қатарларын талдау үшін ARIMA моделі

Statistical model used in time series analysis

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. To better comprehend the data or to forecast upcoming series points, both of these models are fitted to time series data. ARIMA models are applied in some cases where data show evidence of non stationarity in the sense of mean (but not variance/autocovariance), where an initial differencing step (corresponding to the "integrated" part of the model) can be applied one or more times to eliminate the non stationarity of the mean function (i. e., the trend). When the seasonality shows in a time series, the seasonal differencing is theoretically sufficient to describe a regular (a. k. a. purely nondeterministic Note that if the time series contains a predictable sub process (a. k. a. pure sine or complex valued exponential process The (for "integrated") indicates that the data values have been replaced with the difference between their values and the previous values (and this differencing process may have been performed more than once). The purpose of each of these features is to make the model fit the data as well as possible. 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 (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), 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. When two out of the three terms are zeros, the model may be referred to based on the non zero parameter, dropping "", "" or "" from the acronym describing the model. For example, \text{ARIMA} (1,0,0) is AR(1), \text{ARIMA}(0,1,0) is I(1), and \text{ARIMA}(0,0,1) is MA(1). ARIMA models can be estimated following the Box–Jenkins approach.

The explicit identification of the factorization of the autoregression polynomial into factors as above can be extended to other cases, firstly to apply to the moving average polynomial and secondly to include other special factors. For example, having a factor in a model is one way of including a non stationary seasonality of period s into the model; this factor has the effect of re expressing the data as changes from s periods ago. Another example is the factor , which includes a (non stationary) seasonality of period 2. The effect of the first type of factor is to allow each season's value to drift separately over time, whereas with the second type values for adjacent seasons move together. Identification and specification of appropriate factors in an ARIMA model can be an important step in modeling as it can allow a reduction in the overall number of parameters to be estimated while allowing the imposition on the model of types of behavior that logic and experience suggest should be there.

A stationary time series's properties do not depend on the time at which the series is observed. Specifically, for a wide sense stationary time series, the mean and the variance/autocovariance keep constant over time. Differencing in statistics is a transformation applied to a non stationary time series in order to make it stationary in the mean sense (viz., to remove the non constant trend), but having nothing to do with the non stationarity of the variance or autocovariance. Likewise, the seasonal differencing is applied to a seasonal time series to remove the seasonal component. From the perspective of signal processing, especially the Fourier spectral analysis theory, the trend is the low frequency part in the spectrum of a non stationary time series, while the season is the periodic frequency part in the spectrum of it. Therefore, the differencing works as a high pass (i. e., low stop) filter and the seasonal differencing as a comb filter to suppress the low frequency trend and the periodic frequency season in the spectrum domain (rather than directly in the time domain), respectively. An ARIMA(0, 2, 2) model is given by — which is equivalent to Holt's linear method with additive errors, or double exponential smoothing. Other alternative methods include AIC, BIC, etc. If the time series is suspected to exhibit long range dependence, then the d parameter may be allowed to have non integer values in an autoregressive fractionally integrated moving average model, which is also called a Fractional ARIMA (FARIMA or ARFIMA) model.