An AR or Auto-regressive model is a
model of a stochastic or random walk processes in which the future values of a data
series are determined based on the weighted sum of the past values.
The successive past values have
coefficients which are in fact the weights mentioned above and are the actual
correlations between the present and the particular past value. Thus
Autoregression model is also sometimes referred to as an autocorrelation model.
An auto-regression model
is expressed in the form:
xt
1 xt-1
+ …..αpxt-p + Zt
or
xt
+Zt
Where, the terms α1…αp are the auto correlation coefficients at lags
1, 2... and p and Zt
is the residual error term. This error term
relates specifically to the current time t.
Thus for a first order process, p=1 and
the model obtained is
xt = αxt-1
+ Zt……………………………….1
xt-1 = αxt-2
+ Zt-1 ……………………………..2
This expression implies that the
estimated value of x at time t is determined by the immediately
previous value of x or value at time = t-1 multiplied by a
correlation coefficient or the extent to which all pairs of the series
that are time period 1 lag distant are correlated (or autocorrelatred) plus a
residual term, which is an error term at time t.
This is also known as Markov process.
Thus a Markov process is a 1st order AR process and can be expressed
as AR(1) process.
Thus if α=1, the model states
that the next value of x is simply the previous value plus a random error term
and thus it is a simple 1 dimensional random walk.
However, if more terms
are introduced, the model predicts the value of x at time = t by a weighted sum
of the terms plus a random error component
In general an AR(p) process can be
expressed as
xt=
c +
ix
t-1+ Zt
Where α1….αp are
model parameters or the coefficients, c is a constant and Zt is
the error term or the white noise.
Applications
of AR models ( see
It can also be used to analyze audio/speech recognition
based on AR modelling of amplitude modulations see
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