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.
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.
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).
ARIMA models can be estimated following the Box-Jenkins approach.
Given a time series of data Xt where t is an integer index and the Xt are real numbers, then an ARMA(p,q) model is given by:
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:
An ARIMA(p,d,q) process expresses this polynomial factorisation property with p=p'−d, and is given by:
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.
This defines an ARIMA(p,d,q) process with drift δ/(1−Σφi).
Other special forms
The explicit identification of the factorisation 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.[clarification needed] 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.[clarification needed]
Identification and specification of appropriate factors in an ARIMA model can be an important step in modelling 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 behaviour that logic and experience suggest should be there.
Differencing in statistics refers to a transformation applied to time-series data in order to make it stationary. A stationary time series' properties do not depend on the time at which the series is observed.
In order to difference the data - the difference between consecutive observations is computed. Mathematically, this is shown as:
Differencing removes the changes in the level of a time series, eliminating trend and seasonality and consequently stabilizing the mean of the time series.
Sometimes it may be necessary to difference the data a second time to obtain a stationary time series, which is referred to as second order differencing
Another method of differencing data is seasonal differencing, which involves computing the difference between an observation and the corresponding observation in the previous year. This is shown as:
The differenced data is then used as a parameter in the ARIMA model.
Forecasts using ARIMA models
The ARIMA model can be viewed as a "cascade" of two models. The first is non-stationary:
while the second is wide-sense stationary:
Now forecasts can be made for the process , using a generalization of the method of autoregressive forecasting.
The forecast intervals for ARIMA models are based on assumptions that the residuals are uncorrelated and normally distributed. If either of these are assumptions do not hold, then the forecast intervals may be incorrect. For this reason, always plot the ACF and histogram of the residuals to check the assumptions before producing forecast intervals.
95% forecast interval: , where is the variance of .
For , for all ARIMA models regardless of parameters and orders.
For ARIMA(0,0,q), , , for h=2,3,....
In general, forecast intervals from ARIMA models will increase as the forecast horizon increases.
Some well-known special cases arise naturally or are mathematically equivalent to other popular forecasting models. For example:
- An ARIMA(0,1,0) model (or I(1) model) is given by — which is simply a random walk.
- An ARIMA(0,1,0) with a constant, given by — which is a random walk with drift.
- An ARIMA(0,0,0) model is a white noise model.
- An ARIMA(0,1,2) model is a Damped Holt's model.
- An ARIMA(0,1,1) model without constant is a Basic Exponential Smoothing.
- An ARIMA(0,2,2) model is given by — which is equivalent to Holt's linear method with additive errors, or Double exponential smoothing.
To determine the order of a non-seasonal ARIMA model, a useful criteria is Akaike information criterion (AIC) . It is written as:
AIC= −2log(L)+2(p+q+k+1), where L is the likelihood of the data, p is the order of the autoregressive part and q is the order of the moving average part. The k term in these criterion are defined as the number of parameters in the model being fitted to the data. For AIC, if k = 1 then c ≠ 0 and if k = 0 then c = 0.
The corrected AIC for ARIMA models can be written as:
AICc= AIC+ (2(p+q+k+1)(p+q+k+2))/(T−p−q−k−2)
The Bayesian Information Criterion can be written as:
The objective is to minimize the AIC, AICc or BIC values for a good model. The lower the value of one of these criterion for a range of models being investigated, the more better the model will suit the data. It should be noted however that the AIC and the BIC are used for two completely different purposes. Whilst the AIC tries to approximate models towards the reality of the situation, the BIC attempts to find the perfect fit. The BIC approach is often criticized as there never is a perfect fit to real-life complex data, however it is still a useful method for selection as it penalizes models more heavily for having more parameters than the AIC would.
AICc can only be used to compare ARIMA model with the same orders of differencing. For ARIMA with different order of differencing, RMSE can be used for model comparison.
Variations and extensions
A number of variations on the ARIMA model are commonly employed. If multiple time series are used then the can be thought of as vectors and a VARIMA model may be appropriate. Sometimes a seasonal effect is suspected in the model; in that case, it is generally better to use a SARIMA (seasonal ARIMA) model than to increase the order of the AR or MA parts of the model. 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.
- For further information on Stationarity and Differencing see 
- "Notation for ARIMA Models". Time Series Forecasting System. SAS Institute. Retrieved 19 May 2015.
- Hyndman, Rob J; Athanasopoulos, George. "8.9 Seasonal ARIMA models". Forecasting: principles and practice. oTexts. Retrieved 19 May 2015.
- "Introduction to ARIMA models". people.duke.edu. Retrieved 2016-06-05.
- TimeModels.jl www.github.com
- ARIMA in NCSS,
- Automatic ARMA in NCSS,
- Autocorrelations and Partial Autocorrelations in NCSS