Modelling the noise

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Modelling the noise The ARMAX model includes a
moving average model of noise, that is
Which can be represented as
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Now if we use LS estimate
by treating as noise, the model can be
wrong, because
Because
is correlated signal (AR
model). So it is very important to model the
noise.
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Recap Modelling the noise (Pseudo Linear
Regression PLR) but we do not usually have access
to the noise terms. However one approach is to
use the prediction errors as signal driving the
noise process. One way of modifying the recursive
least squares algorithm is to use the
following
Look at RLS again Notice that the data matrix
is now filled by prediction errors (5) from
previous lags.
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Instrument Variables (IVs) The least squares
algorithm assumes that the disturbance is not
correlated with the input. This is seldom the
case, and effects the quality of the model fit.
To reduce this problem generate an instrument
variable zn that is not correlated with the
disturbance. ,
the Instrumental least squares formula becomes
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Recursive Instrumental Least squares algorithm
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Choice of instrument variable Two possible
choices of instrument variables are
or
where is the data (i.e. the y values) that
would be outputted if a previous estimate of
model parameters were used. (sa and sb are the
number of a and b parameters in the model).
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That is where the and
coefficients are from a previous estimate. As
you imagine, a few iterations are needed per data
point.
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We now show that instrumental LS estimate is
unbiased, i.e.
Proof
because by definition if
then z is called instrumental variable.
Note that condition on noise model can be
relaxed, but the selection of IV is not trivial.