In statistics, an<unk> <unk>  model is a generalisation of an<unk><unk>  average or <unk>  model.  These models are fitted to time series data either to better understand the data or to predict future points in the series.  The model is generally referred to as an<unk> (p,d,q) model where p, d, and q are integers greater than or equal to zero and refer to the order of the<unk> ,<unk> , and<unk>  average parts of the model respectively.
Given a time series of data 
X
t
{\displaystyle X_{t}}
 where 
t
{\displaystyle t}
 is an integer index and the 
X
t
{\displaystyle X_{t}}
 are real numbers, then an<unk> (p,q) model is given by
where 
L
{\displaystyle L}
 is the lag operator, the 
ϕ
i
{\displaystyle \phi _{i}}
 are the parameters of the<unk>  part of the model, the 
θ
i
{\displaystyle \theta _{i}}
 are the parameters of the<unk>  average part and the 
ε
t
{\displaystyle \varepsilon _{t}}
 are error terms.  The error terms 
ε
t
{\displaystyle \varepsilon _{t}}
 are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.
An<unk> (p,d,q) process is obtained by integrating an<unk> (p,q) process.  That is,
where d is a positive integer that controls the level of differencing (or, if 
d
=
0
{\displaystyle d=0}
, this model is equivalent to an<unk>  model).  Conversely, applying term-by-term differencing d times to an<unk> (p,q) process gives an<unk> (p,d,q) process.  Note that it is only necessary to difference the AR side of the<unk>  representation, because the MA component is always I(0).
It should be noted that not all choices of parameters produce well-behaved models.  In particular, if the model is required to be stationary then conditions on these parameters must be met.
Some well-known special cases arise naturally.  For example, an<unk> (0,1,0) model is given by:
which is simply a random walk.
A number of variations on the<unk>  model are commonly used.  For example, if multiple time series are used then the 
X
t
{\displaystyle X_{t}}
 can be thought of as vectors and a<unk>  model may be appropriate.  Sometimes a seasonal effect is suspected in the model.  For example, consider a model of daily road traffic volumes.  Weekends clearly exhibit different behaviour from weekdays.  In this case it is often considered better to use a<unk>  (seasonal<unk>  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
{\displaystyle d}
 parameter may be replaced by certain non-integer values in a Fractional<unk>  <unk>  also sometimes called<unk>  model.
Autoregressive integrated moving average
Receiver operating characteristic
Parts-per notation
Autoregressive moving average model
Triage
Ancillary statistic
Arithmetic mean
Kernel density estimation