Autoregressive Integrated Moving Average (ARIMA) Algorithm
An Autoregressive Integrated Moving Average (ARIMA) Algorithm is a parametric univariate timeseries modeling algorithm with non-negative integers (p,d,q) which refer to the order of the autoregressive, integrated, and moving average algorithm parameters
- Context:
- parameters: ARIMA(p,d,q), where p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts.
- It can be implemented by an ARIMA-based system (to solve an ARIMA-based task that produces an trained ARIMA model structure).
- …
- Example(s):
- an ARMA Model (is a specialization?)
- a Seasonal ARIMA Algorithm.
- …
- Counter-Example(s):
- a Naive Forecasting Algorithm, such as a moving average model, or an exponential forecasting algorithm.
- a Kalman Filter.
- a Vector Autoregressive Algorithm.
- See: Box-Jenkins Algorithm, Multi-Relational Data Mining, ARCH Model, Timeseries Analysis Method.
References
2014
- (Gupta et al., 2014a) ⇒ Madhu Gupta, Jing Gao, Charu C Aggarwal, and Jiawei Han. (2014). “Outlier Detection for Temporal Data: A Survey.” In: Knowledge and Data Engineering, IEEE Transactions on, 26(9).
- QUOTE: … Parametric models for time series outliers [15] represents the first work on outlier detection for time series data. Several models were subsequently proposed in the statistics literature, including autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), vector autoregression (VARMA), CUmulative SUM Statistics (CUSUM), exponentially weighted moving average, etc.
2013
- http://en.wikipedia.org/wiki/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 remove the non-stationarity.
The model is generally referred to as an ARIMA(p,d,q) model where p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively. ARIMA models form an important part of the Box-Jenkins approach to time-series modelling.
When one of the three terms is zero, it's usual to drop "AR", "I" or "MA". For example, ARIMA(0,1,0) is I(1), and ARIMA(0,0,1) is MA(1).
- 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 remove the non-stationarity.
2013
- http://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average#Definition
- Given a time series of data [math]\displaystyle{ X_t }[/math] where [math]\displaystyle{ t }[/math] is an integer index and the [math]\displaystyle{ X_t }[/math] are real numbers, then an ARMA(p' ,q) model is given by: :[math]\displaystyle{ \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 \, }[/math] where [math]\displaystyle{ L }[/math] is the lag operator, the [math]\displaystyle{ \alpha_i }[/math] are the parameters of the autoregressive part of the model, the [math]\displaystyle{ \theta_i }[/math] are the parameters of the moving average part and the [math]\displaystyle{ \varepsilon_t }[/math] are error terms. The error terms [math]\displaystyle{ \varepsilon_t }[/math] are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.
Assume now that the polynomial [math]\displaystyle{ \left( 1 - \sum_{i=1}^{p'} \alpha_i L^i \right) }[/math] has a unitary root of multiplicity d. Then it can be rewritten as: :[math]\displaystyle{ \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} . }[/math] An ARIMA(p,d,q) process expresses this polynomial factorisation property with p=p'−d, and is given by: :[math]\displaystyle{ \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 \, }[/math] 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. :[math]\displaystyle{ \left( 1 - \sum_{i=1}^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \delta + \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \, }[/math]
This defines an ARIMA(p,d,q) process with drift δ.
- Given a time series of data [math]\displaystyle{ X_t }[/math] where [math]\displaystyle{ t }[/math] is an integer index and the [math]\displaystyle{ X_t }[/math] are real numbers, then an ARMA(p' ,q) model is given by: :[math]\displaystyle{ \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 \, }[/math] where [math]\displaystyle{ L }[/math] is the lag operator, the [math]\displaystyle{ \alpha_i }[/math] are the parameters of the autoregressive part of the model, the [math]\displaystyle{ \theta_i }[/math] are the parameters of the moving average part and the [math]\displaystyle{ \varepsilon_t }[/math] are error terms. The error terms [math]\displaystyle{ \varepsilon_t }[/math] are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.
2008
- (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
- QUOTE: Models for time series which resemble ARMA models except in that it is presumed the time series has a steady underlying trend. The models therefore work with the differences between the successive observed values, instead of the values themselves. To retrieve the original data from the differences requires a form of integration and the models are therefore called autoregressive integrated moving average models. Models incorporating additional space-time variation are called STARIMA models.