# bj

Estimate Box-Jenkins polynomial model using time domain data

## Syntax

```sys = bj(data, [nb nc nd nf nk]) sys = bj(data,[nb nc nd nf nk], Name,Value) sys = bj(data, init_sys) sys = bj(data, ___, opt) [sys,ic] = bj(___) ```

## Description

```sys = bj(data, [nb nc nd nf nk])``` estimates a Box-Jenkins polynomial model, `sys`, using the time-domain data, `data`. ```[nb nc nd nf nk]``` define the orders of the polynomials used for estimation.

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

`sys = bj(data, init_sys)` estimates a Box-Jenkins polynomial using the polynomial model `init_sys` to configure the initial parameterization of `sys`.

`sys = bj(data, ___, opt)` estimates a Box-Jenkins polynomial using the option set, `opt`, to specify estimation behavior.

`[sys,ic] = bj(___)` returns the estimated initial conditions as an `initialCondition` object. Use this syntax if you plan to simulate or predict the model response using the same estimation input data and then compare the response with the same estimation output data. Incorporating the initial conditions yields a better match during the first part of the simulation.

## Input Arguments

 `data` Estimation data. `data` is an `iddata` object that contains time-domain input and output signal values. You cannot use frequency-domain data for estimating Box-Jenkins models. `[nb nc nd nf nk]` A vector of matrices containing the orders and delays of the Box-Jenkins model. Matrices must contain nonnegative integers. `nb` is the order of the B polynomial plus 1 (Ny-by-Nu matrix)`nc` is the order of the C polynomial plus 1 (Ny-by–1 matrix)`nd` is the order of the D polynomial plus 1 (Ny-by-1 matrix)`nf` is the order of the F polynomial plus 1 (Ny-by-Nu matrix)`nk` is the input delay (in number of samples, Ny-by-Nu matrix) where Nu is the number of inputs and Ny is the number of outputs. `opt` Estimation options. `opt` is an options set that configures, among others, the following: estimation objectiveinitial conditionsnumerical search method to be used in estimation Use `bjOptions` to create the options set. `init_sys` Polynomial model that configures the initial parameterization of `sys`. `init_sys` must be an `idpoly` model with the Box-Jenkins structure that has only B, C, D and F polynomials active. `bj` uses the parameters and constraints defined in `init_sys` as the initial guess for estimating `sys`. Use the `Structure` property of `init_sys` to configure initial guesses and constraints for B(q), F(q), C(q) and D(q). To specify an initial guess for, say, the C(q) term of `init_sys`, set `init_sys.Structure.C.Value` as the initial guess. To specify constraints for, say, the B(q) term of `init_sys`: set `init_sys.Structure.B.Minimum` to the minimum B(q) coefficient valuesset `init_sys.Structure.B.Maximum` to the maximum B(q) coefficient valuesset `init_sys.Structure.B.Free` to indicate which B(q) coefficients are free for estimation You can similarly specify the initial guess and constraints for the other polynomials.

### Name-Value 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 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. Default: 0 for all input/output pairs `IntegrateNoise` Logical specifying integrators in the noise channel. `IntegrateNoise` is a logical vector of length Ny, where Ny is the number of outputs. Setting `IntegrateNoise` to `true` for a particular output results in the model: `$y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t-nk\right)+\frac{C\left(q\right)}{D\left(q\right)}\frac{e\left(t\right)}{1-{q}^{-1}}$` Where, $\frac{1}{1-{q}^{-1}}$ is the integrator in the noise channel,e(t). Default: `false(Ny,1)`(Ny is the number of outputs)

## Output Arguments

`sys`

BJ 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.

• `'backcast'` — The initial conditions were estimated using the best least squares fit.

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 the percentage fitpercent = 100(1-NRMSE).

`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 `bjOptions` 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`.

`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 `[]`.

`Termination`

Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields:

FieldDescription
`WhyStop`

Reason for terminating the numerical search.

`Iterations`

Number of search iterations performed by the estimation algorithm.

`FirstOrderOptimality`

$\infty$-norm of the gradient search vector when the search algorithm terminates.

`FcnCount`

Number of times the objective function was called.

`UpdateNorm`

Norm of the gradient search vector in the last iteration. Omitted when the search method is `'lsqnonlin'` or `'fmincon'`.

`LastImprovement`

Criterion improvement in the last iteration, expressed as a percentage. Omitted when the search method is `'lsqnonlin'` or `'fmincon'`.

`Algorithm`

Algorithm used by `'lsqnonlin'` or `'fmincon'` search method. Omitted when other search methods are used.

For estimation methods that do not require numerical search optimization, the `Termination` field is omitted.

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

`[nb nc nd nf nk]`

A vector of matrices containing the orders and delays of the Box-Jenkins model. Matrices must contain nonnegative integers.

• `nb` is the order of the B polynomial plus 1 (Ny-by-Nu matrix)

• `nc` is the order of the C polynomial plus 1 (Ny-by–1 matrix)

• `nd` is the order of the D polynomial plus 1 (Ny-by-1 matrix)

• `nf` is the order of the F polynomial plus 1 (Ny-by-Nu matrix)

• `nk` is the input delay (in number of samples, Ny-by-Nu matrix) where Nu is the number of inputs and Ny is the number of outputs.

`ic`

Estimated initial conditions, returned as an `initialCondition` object or an object array of `initialCondition` values.

• For a single-experiment data set, `ic` represents, in state-space form, the free response of the transfer function model (A and C matrices) to the estimated initial states (x0).

• For a multiple-experiment data set with Ne experiments, `ic` is an object array of length Ne that contains one set of `initialCondition` values for each experiment.

If `bj` returns `ic` values of `0` and the you know that you have non-zero initial conditions, set the `'InitialCondition'` option in `bjOptions` to `'estimate'` and pass the updated option set to `bj`. For example:

```opt = bjOptions('InitialCondition','estimate') [sys,ic] = bj(data,[nb nc nd nf nk],opt)```
The default `'auto'` setting of `'InitialCondition'` uses the `'zero'` method when the initial conditions have a negligible effect on the overall estimation-error minimization process. Specifying `'estimate'` ensures that the software estimates values for `ic`.

For more information, see `initialCondition`. For an example of using this argument, see Obtain Initial Conditions.

## Examples

collapse all

Estimate the parameters of a single-input, single-output Box-Jenkins model from measured data.

```load iddata1 z1; nb = 2; nc = 2; nd = 2; nf = 2; nk = 1; sys = bj(z1,[nb nc nd nf nk]);```

`sys` is a discrete-time `idpoly` model with estimated coefficients. The order of sys is as described by `nb`, `nc`, `nd`, `nf`, and `nk`.

Use `getpvec` to obtain the estimated parameters and `getcov` to obtain the covariance associated with the estimated parameters.

Estimate the parameters of a multi-input, single-output Box-Jenkins model from measured data.

```load iddata8 nb = [2 1 1]; nc = 1; nd = 1; nf = [2 1 2]; nk = [5 10 15]; sys = bj(z8,[nb nc nd nf nk]);```

`sys` estimates the parameters of a model with three inputs and one output. Each of the inputs has a delay associated with it.

Estimate a regularized BJ model by converting a regularized ARX model.

`load regularizationExampleData.mat m0simdata;`

Estimate an unregularized BJ model of order 30.

`m1 = bj(m0simdata(1:150),[15 15 15 15 1]);`

Estimate a regularized BJ model by determining Lambda value by trial and error.

```opt = bjOptions; opt.Regularization.Lambda = 1; m2 = bj(m0simdata(1:150),[15 15 15 15 1],opt);```

Obtain a lower-order BJ model by converting a regularized ARX model followed by order reduction.

```opt1 = arxOptions; [L,R] = arxRegul(m0simdata(1:150),[30 30 1]); opt1.Regularization.Lambda = L; opt1.Regularization.R = R; m0 = arx(m0simdata(1:150),[30 30 1],opt1); mr = idpoly(balred(idss(m0),7));```

Compare the model outputs against data.

```opt2 = compareOptions('InitialCondition','z'); compare(m0simdata(150:end),m1,m2,mr,opt2);```

Estimate the parameters of a single-input, single-output Box-Jenkins model while configuring some estimation options.

Generate estimation data.

```B = [0 1 0.5]; C = [1 -1 0.2]; D = [1 1.5 0.7]; F = [1 -1.5 0.7]; sys0 = idpoly(1,B,C,D,F,0.1); e = iddata([],randn(200,1)); u = iddata([],idinput(200)); y = sim(sys0,[u e]); data = [y u];```

`data` is a single-input, single-output data set created by simulating a known model.

Estimate initial Box-Jenkins model.

```nb = 2; nc = 2; nd = 2; nf = 2; nk = 1; init_sys = bj(data,[2 2 2 2 1]);```

Create an estimation option set to refine the parameters of the estimated model.

```opt = bjOptions; opt.Display = 'on'; opt.SearchOptions.MaxIterations = 50;```

`opt` is an estimation option set that configures the estimation to iterate 50 times at most and display the estimation progress.

Reestimate the model parameters using the estimation option set.

`sys = bj(data,init_sys,opt);`

`sys` is estimated using `init_sys` for the initial parameterization for the polynomial coefficients.

To view the estimation result, enter `sys.Report`.

Estimate a multi-input, multi-output Box-Jenkins model from estimated data.

```load iddata1 z1 load iddata2 z2 data = [z1 z2(1:300)];```

`data` contains the measured data for two inputs and two outputs.

Estimate the model.

``` nb = [2 2; 3 4]; nc = [2;2]; nd = [2;2]; nf = [1 0; 2 2]; nk = [1 1; 0 0]; sys = bj(data,[nb nc nd nf nk]);```

The polynomial order coefficients contain one row for each output.

`sys` is a discrete-time `idpoly` model with two inputs and two outputs.

`load iddata1ic z1i`

Estimate a second-order Box-Jenkins model `sys` and return the initial conditions in `ic`.

```nb = 2; nc = 2; nd = 2; nf = 2; nk = 1; [sys,ic] = bj(z1i,[nb nc nd nf nk]); ic```
```ic = initialCondition with properties: A: [4x4 double] X0: [4x1 double] C: [0.8744 0.5426 0.4647 -0.5285] Ts: 0.1000 ```

`ic` is an `initialCondition` object that encapsulates the free response of `sys`, in state-space form, to the initial state vector in `X0`. You can incorporate `ic` when you simulate `sys` with the `z1i` input signal and compare the response with the `z1i` output signal.

collapse all

### Box-Jenkins Model Structure

The general Box-Jenkins model structure is:

`$y\left(t\right)=\sum _{i=1}^{nu}\frac{{B}_{i}\left(q\right)}{{F}_{i}\left(q\right)}{u}_{i}\left(t-n{k}_{i}\right)+\frac{C\left(q\right)}{D\left(q\right)}e\left(t\right)$`

where nu is the number of input channels.

The orders of Box-Jenkins model are defined as follows:

## Alternatives

To estimate a continuous-time model, use:

## References

[1] Ljung, L. System Identification: Theory for the User, Upper Saddle River, NJ, Prentice-Hall PTR, 1999.