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Segment data and estimate models for each segment

`segment`

is not compatible with MATLAB^{®}
Coder™ or MATLAB
Compiler™.

segm = segment(z,nn) [segm,V,thm,R2e] = segment(z,nn,R2,q,R1,M,th0,P0,ll,mu)

`segment`

builds models of AR, ARX, or ARMAX/ARMA
type,

$$A(q)y(t)=B(q)u(t-nk)+C(q)e(t)$$

assuming that the model parameters are piecewise constant over time. It results in a model that has split the data record into segments over which the model remains constant. The function models signals and systems that might undergo abrupt changes.

The input-output data is contained in `z`

,
which is either an `iddata`

object or a matrix ```
z
= [y u]
```

where `y`

and `u`

are
column vectors. If the system has several inputs, `u`

has
the corresponding number of columns.

The argument `nn`

defines the model order.
For the ARMAX model

nn = [na nb nc nk];

where `na`

, `nb`

, and `nc`

are
the orders of the corresponding polynomials. See What Are Polynomial Models?. Moreover, `nk`

is
the delay. If the model has several inputs, `nb`

and `nk`

are
row vectors, giving the orders and delays for each input.

For an ARX model (`nc = 0`

) enter

nn = [na nb nk];

For an ARMA model of a time series

z = y; nn = [na nc];

and for an AR model

nn = na;

The output argument `segm`

is a matrix, where
the `k`

th row contains the parameters corresponding
to time `k`

. This is analogous to output estimates
returned by the `recursiveARX`

and `recursiveARMAX`

estimators. The output
argument `thm`

of `segment`

contains
the corresponding model parameters that have not yet been segmented.
Each row of `thm`

contains the parameter estimates
at the corresponding time instant. These estimates are formed by weighting
together the parameters of `M`

(default:
5) different time-varying models, with the participating models changing
at every time step. Consider `segment`

as
an alternative to the online estimation commands when you are not
interested in continuously tracking the changes in parameters of a
single model, but need to detect abrupt changes in the system dynamics.

The output argument `V`

contains the sum of
the squared prediction errors of the segmented model. It is a measure
of how successful the segmentation has been.

The input argument `R2`

is the assumed variance
of the innovations *e*(*t*)
in the model. The default value of `R2`

, ```
R2
= []
```

, is that it is estimated. Then the output argument `R2e`

is
a vector whose `k`

th element contains the estimate
of `R2`

at time `k`

.

The argument `q`

is the probability that the
model exhibits an abrupt change at any given time. The default value
is `0.01`

.

`R1`

is the assumed covariance matrix of the
parameter jumps when they occur. The default value is the identity
matrix with dimension equal to the number of estimated parameters.

`M`

is the number of parallel models used in
the algorithm (see below). Its default value is `5`

.

`th0`

is the initial value of the parameters.
Its default is zero. `P0`

is the initial covariance
matrix of the parameters. The default is 10 times the identity matrix.

`ll`

is the guaranteed life of each of the
models. That is, any created candidate model is not abolished until
after at least `ll`

time steps. The default is ```
ll
= 1
```

. `Mu`

is a forgetting parameter that
is used in the scheme that estimates `R2`

. The default
is `0.97`

.

The most critical parameter for you to choose is `R2`

.
It is usually more robust to have a reasonable guess of `R2`

than
to estimate it. Typically, you need to try different values of `R2`

and
evaluate the results. (See the example below.) `sqrt(R2)`

corresponds
to a change in the value *y*(*t*)
that is normal, giving no indication that the system or the input
might have changed.

The algorithm is based on `M`

parallel models,
each recursively estimated by an algorithm of Kalman filter type.
Each model is updated independently, and its posterior probability
is computed. The time-varying estimate `thm`

is formed
by weighting together the `M`

different models with
weights equal to their posterior probability. At each time step the
model (among those that have lived at least `ll`

samples)
that has the lowest posterior probability is abolished. A new model
is started, assuming that the system parameters have changed, with
probability `q`

, a random jump from the most likely
among the models. The covariance matrix of the parameter change is
set to `R1`

.

After all the data are examined, the surviving model with the
highest posterior probability is tracked back and the time instances
where it jumped are marked. This defines the different segments of
the data. (If no models had been abolished in the algorithm, this
would have been the maximum likelihood estimates of the jump instances.)
The segmented model `segm`

is then formed by smoothing
the parameter estimate, assuming that the jump instances are correct.
In other words, the last estimate over a segment is chosen to represent
the whole segment.

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