# recursiveARMA

Create System object for online parameter estimation of ARMA model

## Syntax

obj = recursiveARMA
obj = recursiveARMA(Orders)
obj = recursiveARMA(Orders,A0,C0)
obj = recursiveARMA(___,Name,Value)

## Description

Use recursiveARMA command for parameter estimation with real-time data. If all data necessary for estimation is available at once, and you are estimating a time-invariant model, use the offline estimation command, armax.

obj = recursiveARMA creates a System object™ for online parameter estimation of a default single output ARMA model structure. The default model structure has polynomials of order 1 and initial polynomial coefficient values eps.

After creating the object, use the step command to update model parameter estimates using recursive estimation algorithms and real-time data.

obj = recursiveARMA(Orders) specifies the polynomial orders of the ARMA model to be estimated.

obj = recursiveARMA(Orders,A0,C0) specifies the polynomial orders and initial values of the polynomial coefficients. Specify initial values to potentially avoid local minima during estimation. If the initial values are small compared to the default InitialParameterCovariance property value, and you have confidence in your initial values, also specify a smaller InitialParameterCovariance.

obj = recursiveARMA(___,Name,Value) specifies additional attributes of the ARMA model structure and recursive estimation algorithm using one or more Name,Value pair arguments.

## Object Description

recursiveARMA creates a System object for online parameter estimation of single output ARMA models using a recursive estimation algorithm.

A System object is a specialized MATLAB® object designed specifically for implementing and simulating dynamic systems with inputs that change over time. System objects use internal states to store past behavior, which is used in the next computational step.

After you create a System object, you use commands to process data or obtain information from or about the object. System objects use a minimum of two commands to process data — a constructor to create the object and the step command to update object parameters using real-time data. This separation of declaration from execution lets you create multiple, persistent, reusable objects, each with different settings.

You can use the following commands with the online estimation System objects in System Identification Toolbox™:

CommandDescription
step

Update model parameter estimates using recursive estimation algorithms and real-time data.

step puts the object into a locked state. In a locked state, you cannot change any nontunable properties or input specifications, such as model order, data type, or estimation algorithm. During execution, you can only change tunable properties.

release

Unlock the System object. Use this command to enable setting of nontunable parameters.

reset

Reset the internal states of a locked System object to the initial values, and leave the object locked.

clone

Create another System object with the same object property values.

Do not create additional objects using syntax obj2 = obj. Any changes made to the properties of the new object created this way (obj2) also change the properties of the original object (obj).

isLocked

Query locked status for input attributes and nontunable properties of the System object.

Use the recursiveARMA command to create an online estimation System object. Then estimate the ARMA model parameters (A and C) and output using the step command with output data y.

[A,C,EstimatedOutput] = step(obj,y)

For recursiveARMA object properties, see Properties.

## Examples

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Create a System object for online parameter estimation of an ARMA model.

obj = recursiveARMA;

The ARMA model has a default structure with polynomials of order 1 and initial polynomial coefficient values, eps.

Load the time-series estimation data. In this example, use a static data set for illustration.

output = z9.y;

Estimate ARMA model parameters online using step.

for i = 1:numel(output)
[A,C,EstimatedOutput] = step(obj,output(i));
end

View the current estimated values of polynomial C coefficients.

obj.C
ans = 1×2

1.0000    0.2315

View the current covariance estimate of the parameters.

obj.ParameterCovariance
ans = 2×2
10-3 ×

0.6372   -0.0257
-0.0257    0.0017

View the current estimated output.

EstimatedOutput
EstimatedOutput = 11.8121

Specify ARMA model orders.

na = 2;
nc = 1;

Create a System object for online estimation of an ARMA model with the specified orders.

obj = recursiveARMA([na nc]);

Specify ARMA model orders.

na = 2;
nc = 1;

Create a System object for online estimation of ARMA model with known initial polynomial coefficients.

A0 = [1 0.5 0.3];
C0 = [1 0.7];
obj = recursiveARMA([na nc],A0,C0);

Specify the initial parameter covariance.

obj.InitialParameterCovariance = 0.1;

InitialParameterCovariance represents the uncertainty in your guess for the initial parameters. Typically, the default InitialParameterCovariance (10000) is too large relative to the parameter values. This results in initial guesses being given less importance during estimation. If you have confidence in the initial parameter guesses, specify a smaller initial parameter covariance.

Create a System object that uses the unnormalized gradient algorithm for online parameter estimation of an ARMA model.

## Input Arguments

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Model orders of an ARMA model, specified as a 1-by-2 vector of integers, [na nc].

• na — Order of the polynomial A(q), specified as a nonnegative integer.

• nc — Order of the polynomial C(q), specified as a nonnegative integer.

Initial value of polynomial coefficients, specified as row vectors of real values with elements in order of ascending powers of q-1.

• A0 — Initial guess for the coefficients of the polynomial A(q), specified as a 1-by-(na+1) vector with 1 as the first element.

• C0 — Initial guess for the coefficients of the polynomial C(q), specified as a 1-by-(nc+1) vector with 1 as the first element.

The coefficients in C0 must define a stable discrete-time polynomial with roots within a unit disk. For example,

C0 = [1 0.5 0.5];
all(abs(roots(C0))<1)
ans =

1

Specifying as [], uses the default value of eps for the polynomial coefficients.

Note

If the initial guesses are much smaller than the default InitialParameterCovariance, 10000, the initial guesses are given less importance during estimation. In that case, specify a smaller initial parameter covariance.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Use Name,Value arguments to specify writable properties of recursiveARMA System object during object creation. For example, obj = recursiveARMA([2 1],'EstimationMethod','Gradient') creates a System object to estimate an ARMA model using the 'Gradient' recursive estimation algorithm.

## Properties

recursiveARMA System object properties consist of read-only and writable properties. The writable properties are tunable and nontunable properties. The nontunable properties cannot be changed when the object is locked, that is, after you use the step command.

Use Name,Value arguments to specify writable properties of recursiveARMA objects during object creation. After object creation, use dot notation to modify the tunable properties.

obj = recursiveARMA;
obj.ForgettingFactor = 0.99;

 A Estimated coefficients of polynomial A(q), returned as a row vector of real values specified in order of ascending powers of q-1. A is a read-only property and is initially empty after you create the object. It is populated after you use the step command for online parameter estimation. C Estimated coefficients of polynomial C(q), returned as a vector of real values specified in order of ascending powers of q-1. C is a read-only property and is initially empty after you create the object. It is populated after you use the step command for online parameter estimation. InitialA Initial values for the coefficients of polynomial A(q) of order na, specified as a row vector of length na+1, with 1 as the first element. Specify the coefficients in order of ascending powers of q-1. If the initial guesses are much smaller than the default InitialParameterCovariance, 10000, the initial guesses are given less importance during estimation. In that case, specify a smaller initial parameter covariance. InitialA is a tunable property. You can change it when the object is in a locked state. Default: [1 eps] InitialC Initial values for the coefficients of polynomial C(q) of order nc, specified as a row vector of length nc+1, with 1 as the first element. Specify the coefficients in order of ascending powers of q-1. The coefficients in InitialC must define a stable discrete-time polynomial with roots within a unit circle. For example,InitialC = [1 0.5 0.5]; all(abs(roots(InitialC))<1)ans = 1 If the initial guesses are much smaller than the default InitialParameterCovariance, 10000, the initial guesses are given less importance during estimation. In that case, specify a smaller initial parameter covariance. InitialC is a tunable property. You can change it when the object is in a locked state. Default: [1 eps] ParameterCovariance Estimated covariance P of the parameters, returned as an N-by-N symmetric positive-definite matrix. N is the number of parameters to be estimated. The software computes P assuming that the residuals (difference between estimated and measured outputs) are white noise, and the variance of these residuals is 1. ParameterCovariance is applicable only when EstimationMethod is 'ForgettingFactor' or 'KalmanFilter'. The interpretation of P depends on the estimation method: 'ForgettingFactor' — R2/2 * P is approximately equal to the covariance matrix of the estimated parameters, where R2 is the true variance of the residuals.'KalmanFilter' — R2* P is the covariance matrix of the estimated parameters, and R1 /R2 is the covariance matrix of the parameter changes. Where, R1 is the covariance matrix that you specify in ProcessNoiseCovariance. ParameterCovariance is a read-only property and is initially empty after you create the object. It is populated after you use the step command for online parameter estimation. InitialParameterCovariance Covariance of the initial parameter estimates, specified as one of the following: Real positive scalar, α — Covariance matrix is an N-by-N diagonal matrix, with α as the diagonal elements. N is the number of parameters to be estimated.Vector of real positive scalars, [α1,...,αN] — Covariance matrix is an N-by-N diagonal matrix, with [α1,...,αN] as the diagonal elements. N-by-N symmetric positive-definite matrix. InitialParameterCovariance represents the uncertainty in the initial parameter estimates. For large values of InitialParameterCovariance, less importance is placed on the initial parameter values and more on the measured data during beginning of estimation using step. Use only when EstimationMethod is 'ForgettingFactor' or 'KalmanFilter'. InitialParameterCovariance is a tunable property. You can change it when the object is in a locked state. Default: 10000 EstimationMethod Recursive estimation algorithm used for online estimation of model parameters, specified as one of the following values: 'ForgettingFactor' — Algorithm used for parameter estimation'KalmanFilter' — Algorithm used for parameter estimation'NormalizedGradient' — Algorithm used for parameter estimation'Gradient' — Unnormalized gradient algorithm used for parameter estimation Forgetting factor and Kalman filter algorithms are more computationally intensive than gradient and unnormalized gradient methods. However, they have better convergence properties. For information about these algorithms, see Recursive Algorithms for Online Parameter Estimation. EstimationMethod is a nontunable property. You cannot change it during execution, that is after the object is locked using the step command. If you want to deploy code using MATLAB Coder™, EstimationMethod can only be assigned once. Default: 'ForgettingFactor' ForgettingFactor Forgetting factor, λ, relevant for parameter estimation, specified as a scalar in the range (0,1]. Suppose that the system remains approximately constant over T0 samples. You can choose λ such that: ${T}_{0}=\frac{1}{1-\lambda }$ Setting λ = 1 corresponds to “no forgetting” and estimating constant coefficients. Setting λ < 1 implies that past measurements are less significant for parameter estimation and can be “forgotten”. Set λ < 1 to estimate time-varying coefficients. Typical choices of λ are in the range [0.98 0.995]. Use only when EstimationMethod is 'ForgettingFactor'. ForgettingFactor is a tunable property. You can change it when the object is in a locked state. Default: 1 EnableAdapation Enable or disable parameter estimation, specified as one of the following: true or 1— The step command estimates the parameter values for that time step and updates the parameter values.false or 0 — The step command does not update the parameters for that time step and instead outputs the last estimated value. You can use this option when your system enters a mode where the parameter values do not vary with time.NoteIf you set EnableAdapation to false, you must still execute the step command. Do not skip step to keep parameter values constant, because parameter estimation depends on current and past I/O measurements. step ensures past I/O data is stored, even when it does not update the parameters. EnableAdapation is a tunable property. You can change it when the object is in a locked state. Default: true DataType Floating point precision of parameters, specified as one of the following values: 'double' — Double-precision floating point'single' — Single-precision floating point Setting DataType to 'single' saves memory, but leads to loss of precision. Specify DataType based on the precision required by the target processor where you will deploy generated code. DataType is a nontunable property. It can only be set during object construction using Name,Value arguments and cannot be changed afterward. Default: 'double' ProcessNoiseCovariance Covariance matrix of parameter variations, specified as one of the following: Real nonnegative scalar, α — Covariance matrix is an N-by-N diagonal matrix, with α as the diagonal elements.Vector of real nonnegative scalars, [α1,...,αN] — Covariance matrix is an N-by-N diagonal matrix, with [α1,...,αN] as the diagonal elements. N-by-N symmetric positive semidefinite matrix. N is the number of parameters to be estimated. ProcessNoiseCovariance is applicable when EstimationMethod is 'KalmanFilter'. Kalman filter algorithm treats the parameters as states of a dynamic system and estimates these parameters using a Kalman filter. ProcessNoiseCovariance is the covariance of the process noise acting on these parameters. Zero values in the noise covariance matrix correspond to estimating constant coefficients. Values larger than 0 correspond to time-varying parameters. Use large values for rapidly changing parameters. However, the larger values result in noisier parameter estimates. ProcessNoiseCovariance is a tunable property. You can change it when the object is in a locked state. Default: 0.1 AdaptationGain Adaptation gain, γ, used in gradient recursive estimation algorithms, specified as a positive scalar. AdaptationGain is applicable when EstimationMethod is 'Gradient' or 'NormalizedGradient'. Specify a large value for AdaptationGain when your measurements have a high signal-to-noise ratio. AdaptationGain is a tunable property. You can change it when the object is in a locked state. Default: 1 NormalizationBias Bias in adaptation gain scaling used in the 'NormalizedGradient' method, specified as a nonnegative scalar. NormalizationBias is applicable when EstimationMethod is 'NormalizedGradient'. The normalized gradient algorithm divides the adaptation gain at each step by the square of the two-norm of the gradient vector. If the gradient is close to zero, this can cause jumps in the estimated parameters. NormalizationBias is the term introduced in the denominator to prevent these jumps. Increase NormalizationBias if you observe jumps in estimated parameters. NormalizationBias is a tunable property. You can change it when the object is in a locked state. Default: eps

## Output Arguments

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System object for online parameter estimation of ARMA model, returned as a recursiveARMA System object. This object is created using the specified model orders and properties. Use step command to estimate the coefficients of the ARMA model polynomials. You can then access the estimated coefficients and parameter covariance using dot notation. For example, type obj.A to view the estimated A polynomial coefficients.

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### ARMA Model Structure

The ARMA model structure is:

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

where,

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

$C\left(q\right)=1+{c}_{1}{q}^{-1}+\dots +{c}_{{n}_{c}}{q}^{-{n}_{c}}$

Here,

• y(t)— Output at time t. Data is a time series that has no input channels and one output channel.

• na — Number of A polynomial coefficients

• nc — Number of C polynomial coefficients

• e(t) — White-noise disturbance value at time t

• q-1 — Time-shift operator

## Tips

• Starting in R2016b, instead of using the step command to update model parameter estimates, you can call the System object with input arguments, as if it were a function. For example, [A,C,EstimatedOutput] = step(obj,y) and [A,C,EstimatedOutput] = obj(y) perform equivalent operations.

## Version History

Introduced in R2015b