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# arx

Estimate parameters of ARX or AR model using least squares

## Syntax

```sys = arx(data,[na nb nk])sys = arx(data,[na nb nk],Name,Value)sys = arx(data,[na nb nk],___,opt)```

## Description

 Note:   `arx` does not support continuous-time estimations. Use `tfest` instead.

```sys = arx(data,[na nb nk])``` returns an ARX structure polynomial model, `sys`, with estimated parameters and covariances (parameter uncertainties) using the least-squares method and specified `orders`.

```sys = arx(data,[na nb nk],Name,Value)``` estimates a polynomial model with additional options specified by one or more `Name,Value` pair arguments.

```sys = arx(data,[na nb nk],___,opt)``` specifies estimation options that configure the estimation objective, initial conditions and handle input/output data offsets.

## Input Arguments

 `data` Estimation data. Specify `data` as an `iddata` object, an `frd` object, or an `idfrd` frequency-response-data object. `[na nb nk]` Polynomial orders. `[na nb nk]` define the polynomial orders of an ARX model. `na` — Order of the polynomial A(q). Specify `na` as an Ny-by-Ny matrix of nonnegative integers. Ny is the number of outputs.`nb` — Order of the polynomial B(q) + 1.`nb` is an Ny-by-Nu matrix of nonnegative integers. Ny is the number of outputs and Nu is the number of inputs.`nk` — Input-output delay expressed as fixed leading zeros of the B polynomial. Specify `nk` as an Ny-by-Nu matrix of nonnegative integers. Ny is the number of outputs and Nu is the number of inputs. `opt` Estimation options. `opt` is an options set that specifies estimation options, including: input/output data offsetsoutput weight Use `arxOptions` to create the options set.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

 `'InputDelay'` Input delays. `InputDelay` is a numeric vector specifying a time delay for each input channel. Specify input delays in integer multiples of the sample time `Ts`. For example, `InputDelay = 3` means a delay of three sampling periods. For a system with `Nu` inputs, set `InputDelay` to an `Nu`-by-1 vector, where each entry is a numerical value representing the input delay for the corresponding input channel. You can also set `InputDelay` to a scalar value to apply the same delay to all channels. Default: 0 for all input channels `'IODelay'` Transport delays. `IODelay` is a numeric array specifying a separate transport delay for each input/output pair. Specify transport delays as integers denoting delay of a multiple of the sample time, `Ts`. For a MIMO system with `Ny` outputs and `Nu` inputs, set `IODelay` to a `Ny`-by-`Nu` array, where each entry is a numerical value representing the transport delay for the corresponding input/output pair. You can also set `IODelay` to a scalar value to apply the same delay to all input/output pairs. Useful as a replacement for the `nk` order, you can factor out `max(nk-1,0)` lags as the `IODelay` value. Default: 0 for all input/output pairs `'IntegrateNoise'` Specify integrators in the noise channels. Adding an integrator creates an ARIX model represented by: `$A\left(q\right)y\left(t\right)=B\left(q\right)u\left(t-nk\right)+\frac{1}{1-{q}^{-1}}e\left(t\right)$` where,$\frac{1}{1-{q}^{-1}}$ is the integrator in the noise channel, e(t). `IntegrateNoise` is a logical vector of length `Ny`, where `Ny` is the number of outputs. Default: `false(Ny,1)`, where `Ny` is the number of outputs

## Output Arguments

`sys`

ARX model that fits the estimation data, returned as a discrete-time `idpoly` object. This model is created using the specified model orders, delays, and estimation options.

Information about the estimation results and options used is stored in the `Report` property of the model. `Report` has the following fields:

Report FieldDescription
`Status`

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.

`Method`

Estimation command used.

`InitialCondition`

Handling of initial conditions during model estimation, returned as one of the following values:

• `'zero'` — The initial conditions were set to zero.

• `'estimate'` — The initial conditions were treated as independent estimation parameters.

This field is especially useful to view how the initial conditions were handled when the `InitialCondition` option in the estimation option set is `'auto'`.

`Fit`

Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has the following fields:

FieldDescription
`FitPercent`

Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as a percentage.

`LossFcn`

Value of the loss function when the estimation completes.

`MSE`

Mean squared error (MSE) measure of how well the response of the model fits the estimation data.

`FPE`

Final prediction error for the model.

`AIC`

Raw Akaike Information Criteria (AIC) measure of model quality.

`AICc`

Small sample-size corrected AIC.

`nAIC`

Normalized AIC.

`BIC`

Bayesian Information Criteria (BIC).

`Parameters`

Estimated values of model parameters.

`OptionsUsed`

Option set used for estimation. If no custom options were configured, this is a set of default options. See `arxOptions` for more information.

`RandState`

State of the random number stream at the start of estimation. Empty, `[]`, if randomization was not used during estimation. For more information, see `rng` in the MATLAB® documentation.

`DataUsed`

Attributes of the data used for estimation, returned as a structure with the following fields:

FieldDescription
`Name`

Name of the data set.

`Type`

Data type.

`Length`

Number of data samples.

`Ts`

Sample time.

`InterSample`

Input intersample behavior, returned as one of the following values:

• `'zoh'` — Zero-order hold maintains a piecewise-constant input signal between samples.

• `'foh'` — First-order hold maintains a piecewise-linear input signal between samples.

• `'bl'` — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

`InputOffset`

Offset removed from time-domain input data during estimation. For nonlinear models, it is `[]`.

`OutputOffset`

Offset removed from time-domain output data during estimation. For nonlinear models, it is `[]`.

For more information on using `Report`, see Estimation Report.

## Examples

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Generate input data based on a specified ARX model, and then use this data to estimate an ARX model.

```A = [1 -1.5 0.7]; B = [0 1 0.5]; m0 = idpoly(A,B); u = iddata([],idinput(300,'rbs')); e = iddata([],randn(300,1)); y = sim(m0,[u e]); z = [y,u]; m = arx(z,[2 2 1]); ```

Use `arxRegul` to automatically determine regularization constants and use the values for estimating an FIR model of order 50.

Obtain `L` and `R` values.

```load regularizationExampleData eData; orders = [0 50 0]; [L,R] = arxRegul(eData,orders); ```

By default, the `TC` kernel is used.

Use the returned `Lambda` and `R` values for regularized ARX model estimation.

```opt = arxOptions; opt.Regularization.Lambda = L; opt.Regularization.R = R; model = arx(eData,orders,opt); ```

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### ARX structure

The ARX model structure is :

`$\begin{array}{l}y\left(t\right)+{a}_{1}y\left(t-1\right)+...+{a}_{na}y\left(t-na\right)=\\ {b}_{1}u\left(t-nk\right)+...+{b}_{nb}u\left(t-nb-nk+1\right)+e\left(t\right)\end{array}$`

The parameters `na` and `nb` are the orders of the ARX model, and `nk` is the delay.

• $y\left(t\right)$— Output at time $t$.

• ${n}_{a}$ — Number of poles.

• ${n}_{b}$ — Number of zeroes plus 1.

• ${n}_{k}$ — Number of input samples that occur before the input affects the output, also called the dead time in the system.

• $y\left(t-1\right)\dots y\left(t-{n}_{a}\right)$ — Previous outputs on which the current output depends.

• $u\left(t-{n}_{k}\right)\dots u\left(t-{n}_{k}-{n}_{b}+1\right)$ — Previous and delayed inputs on which the current output depends.

• $e\left(t\right)$ — White-noise disturbance value.

A more compact way to write the difference equation is

`$A\left(q\right)y\left(t\right)=B\left(q\right)u\left(t-{n}_{k}\right)+e\left(t\right)$`

q is the delay operator. Specifically,

`$A\left(q\right)=1+{a}_{1}{q}^{-1}+\dots +{a}_{{n}_{a}}{q}^{-{n}_{a}}$`
`$B\left(q\right)={b}_{1}+{b}_{2}{q}^{-1}+\dots +{b}_{{n}_{b}}{q}^{-{n}_{b}+1}$`

### Time Series Models

For time-series data that contains no inputs, one output and ```orders = na```, the model has AR structure of order `na`.

The AR model structure is

`$A\left(q\right)y\left(t\right)=e\left(t\right)$`

### Multiple Inputs and Single-Output Models

For multiple-input systems, `nb` and `nk` are row vectors where the `i`th element corresponds to the order and delay associated with the `i`th input.

### Multi-Output Models

For models with multiple inputs and multiple outputs, `na`, `nb`, and `nk` contain one row for each output signal.

In the multiple-output case, `arx` minimizes the trace of the prediction error covariance matrix, or the norm

`$\sum _{t=1}^{N}{e}^{T}\left(t\right)e\left(t\right)$`

To transform this to an arbitrary quadratic norm using a weighting matrix `Lambda`

`$\sum _{t=1}^{N}{e}^{T}\left(t\right){\Lambda }^{-1}e\left(t\right)$`

use the syntax

```opt = arxOptions('OutputWeight',inv(lambda)) m = arx(data,orders,opt) ```

### Estimating Initial Conditions

For time-domain data, the signals are shifted such that unmeasured signals are never required in the predictors. Therefore, there is no need to estimate initial conditions.

For frequency-domain data, it might be necessary to adjust the data by initial conditions that support circular convolution.

Set the `InitialCondition` estimation option (see `arxOptions`) to one the following values:

• `'zero'` — No adjustment.

• `'estimate'` — Perform adjustment to the data by initial conditions that support circular convolution.

• `'auto'` — Automatically choose between `'zero'` and `'estimate'` based on the data.

### Algorithms

QR factorization solves the overdetermined set of linear equations that constitutes the least-squares estimation problem.

The regression matrix is formed so that only measured quantities are used (no fill-out with zeros). When the regression matrix is larger than `MaxSize`, data is segmented and QR factorization is performed iteratively on these data segments.

Without regularization, the ARX model parameters vector θ is estimated by solving the normal equation:

$\left({J}^{T}J\right)\theta ={J}^{T}y$

where J is the regressor matrix and y is the measured output. Therefore,

$\theta ={\left({J}^{T}J\right)}^{-1}{J}^{T}y$.

Using regularization adds a regularization term:

$\theta ={\left({J}^{T}J+\lambda R\right)}^{-1}{J}^{T}y$

where, λ and R are the regularization constants. See `arxOptions` for more information on the regularization constants.