ar
Estimate parameters when identifying AR model or ARI model for scalar time series
Syntax
Description
specifies additional options using one or more namevalue pair arguments. For instance,
using the namevalue pair argument sys
= ar(y
,n
,___,Name,Value
)'IntegrateNoise',1
estimates an ARI model, which is
useful for systems with nonstationary disturbances. Specify Name,Value
after any of the input argument combinations in the previous syntaxes.
Examples
AR Model
Estimate an AR model and compare its response with the measured output.
Load the data, which contains the time series tt9
with noise.
load sdata9 tt9
Estimate a fourthorder AR model.
sys = ar(tt9,4)
sys = Discretetime AR model: A(z)y(t) = e(t) A(z) = 1  0.8369 z^1  0.4744 z^2  0.06621 z^3 + 0.4857 z^4 Sample time: 0.0039062 seconds Parameterization: Polynomial orders: na=4 Number of free coefficients: 4 Use "polydata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using AR ('fb/now') on time domain data "tt9". Fit to estimation data: 79.38% FPE: 0.5189, MSE: 0.5108
The output displays the polynomial containing the estimated parameters alongside other estimation details. Under Status
, Fit to estimation data
shows that the estimated model has 1stepahead prediction accuracy above 75%.
You can find additional information about the estimation results by exploring the estimation report, sys.Report
. For instance, you can retrieve the parameter covariance.
covar = sys.Report.Parameters.FreeParCovariance
covar = 4×4
0.0015 0.0015 0.0005 0.0007
0.0015 0.0027 0.0008 0.0004
0.0005 0.0008 0.0028 0.0015
0.0007 0.0004 0.0015 0.0014
For more information on viewing the estimation report, see Estimation Report.
Compare Burg's Method with ForwardBackward Approach
Given a sinusoidal signal with noise, compare the spectral estimates of Burg's method with those found using the forwardbackward approach.
Generate an output signal and convert it into an iddata
object.
y = sin([1:300]') + 0.5*randn(300,1); y = iddata(y);
Estimate fourthorder AR models using Burg's method and using the default forwardbackward approach. Plot the model spectra together.
sys_b = ar(y,4,'burg'); sys_fb = ar(y,4); spectrum(sys_b,sys_fb) legend('Burg','ForwardBackward')
The two responses match closely throughout most of the frequency range.
ARI Model
Estimate an ARI model, which includes an integrator in the noise source.
Load the data, which contains the time series ymat9
with noise. Ts
contains the sample time.
load sdata9 ymat9 Ts
Integrate the output signal.
y = cumsum(ymat9);
Estimate an AR model with 'IntegrateNoise'
set to true
. Use the leastsquares method 'ls'
.
sys = ar(y,4,'ls','Ts',Ts,'IntegrateNoise',true);
Predict the model output using 5step prediction and compare the result with the integrated output signal y
.
compare(y,sys,5)
Modify Default Options
Modify the default options for the AR
function.
Load the data, which contains a time series z9
with noise.
load iddata9 z9
Modify the default options so that the function uses the 'ls'
approach and does not estimate covariance.
opt = arOptions('Approach','ls','EstimateCovariance',false)
opt = Option set for the ar command: Approach: 'ls' Window: 'now' DataOffset: 0 EstimateCovariance: 0 MaxSize: 250000
Estimate a fourthorder AR model using the updated options.
sys = ar(z9,4,opt);
Retrieve Reflection Coefficients for Burg's Method
Retrieve reflection coefficients and loss functions when using Burg's method.
Latticebased approaches such, as Burg's method 'burg'
and geometric lattice 'gl'
, compute reflection coefficients and corresponding loss function values as part of the estimation process. Use a second output argument to retrieve these values.
Generate an output signal and convert it into an iddata
object.
y = sin([1:300]') + 0.5*randn(300,1); y = iddata(y);
Estimate a fourthorder AR model using Burg's method and include an output argument for the reflection coefficients.
[sys,refl] = ar(y,4,'burg');
refl
refl = 2×5
0 0.3562 0.4430 0.5528 0.2385
0.8494 0.7416 0.5960 0.4139 0.3904
Input Arguments
y
— Timeseries data
iddata
object  numeric vector  timetable
Timeseries data, specified as one of the following:
For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.
n
— Model order
positive integer
Model order, specified as a positive integer. The value of n
determines the number of A parameters in the AR model.
Example: ar(idy,2)
computes a secondorder AR model from the
singlechannel iddata
object idy
approach
— Algorithm for computing AR model
'fb'
(default)  'burg'
 'gl'
 'ls'
 'yw'
Algorithm for computing the AR model, specified as one of the following values:
'burg'
: Burg's latticebased method. Solves the lattice filter equations using the harmonic mean of forward and backward squared prediction errors.'fb'
: (Default) Forwardbackward approach. Minimizes the sum of a leastsquares criterion for a forward model, and the analogous criterion for a timereversed model.'gl'
: Geometric lattice approach. Similar to Burg's method, but uses the geometric mean instead of the harmonic mean during minimization.'ls'
: Leastsquares approach. Minimizes the standard sum of squared forwardprediction errors.'yw'
: YuleWalker approach. Solves the YuleWalker equations, formed from sample covariances.
All of these algorithms are variants of the leastsquares method. For more information, see Algorithms .
Example: ar(idy,2,'ls')
computes an AR model using the
leastsquares approach
window
— Prewindowing and postwindowing
'now'
 'pow'
 'ppw'
 'prw
Prewindowing and postwindowing outside the measured time interval (past and future values), specified as one of the following values:
'now'
: No windowing. This value is the default except when you setapproach
to'yw'
. Only measured data is used to form regression vectors. The summation in the criteria starts at the sample index equal ton+1
.'pow'
: Postwindowing. Missing end values are replaced with zeros and the summation is extended to timeN+n
(N
is the number of observations).'ppw'
: Prewindowing and postwindowing. The software uses this value whenever you select the YuleWalker approach'yw'
, regardless of yourwindow
specification.'prw'
: Prewindowing. Missing past values are replaced with zeros so that the summation in the criteria can start at time equal to zero.
Example: ar(idy,2,'yw','ppw')
computes an AR model using the
YuleWalker approach with prewindowing and postwindowing.
opt
— Estimation options
arOptions
option set
Estimation options for AR model identification, specified as an
arOptions
option set. opt
specifies the
following options:
Estimation approach
Data windowing technique
Data offset
Maximum number of elements in a segment of data
For more information, see arOptions
. For an example, see Modify Default Options.
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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.
Example: 'IntegrateNoise',true
adds an integrator in the noise source
OutputName
— Output signal names
" "
(default)  character vector  string
Output channel names for timetable data, specified as a string or a character
vector. By default, the software interprets the last variable in
tt
as the sole output channel. When you want to select a
different timetable variable for the output channel, use
'OutputName'
to identify it. For example, sys =
ar(tt,__,'OutputName',"y3")
selects the variable
y3
as the output channel for the estimation.
Ts
— Sample time
1
(default)  positive scalar
Sample time, specified as the commaseparated pair consisting of
'Ts'
and the sample time in seconds. If y
is
a numeric vector, then you must specify 'Ts'
.
Example: ar(y_signal,2,'Ts',0.08)
computes a secondorder AR
model with sample time of 0.08 seconds
IntegrateNoise
— Add integrator to noise channel
false
(default)  logical vector
Noisechannel integration option for estimating ARI models,
specified as the commaseparated pair consisting of
'IntegrateNoise'
and a logical. Noise integration is useful in
cases where the disturbance is nonstationary.
When using 'IntegrateNoise'
, you must also integrate the
outputchannel data. For an example, see ARI Model.
Output Arguments
sys
— AR or ARI model
idpoly
model object
AR or ARI model that
fits the given estimation data, returned as a discretetime idpoly
model 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 Field  Description  

Status  Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.  
Method  Estimation command used.  
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:
 
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  
RandState  State of the random number stream at the start of estimation. Empty,
 
DataUsed  Attributes of the data used for estimation, returned as a structure with the following fields.

For more information on using Report
, see Estimation Report.
refl
— Reflection coefficients and loss functions
array
Reflection coefficients and loss functions, returned as a 2by2 array. For the two
latticebased approaches 'burg'
and 'gl'
,
refl
stores the reflection coefficients in the first row and the
corresponding loss function values in the second row. The first column of
refl
is the zerothorder model, and the (2,1)
element of refl
is the norm of the time series itself. For an
example, see Retrieve Reflection Coefficients for Burg's Method.
More About
AR (Autoregressive) Model
The AR model structure has no input, and is given by the following equation:
$$A(q)y(t)=e(t)$$
This model structure accommodates estimation for scalar timeseries data, which have no input channel. The structure is a special case of the ARX structure.
ARI (Autoregressive Integrated) Model
The ARI model is an AR model with an integrator in the noise channel. The ARI model structure is given by the following equation:
$$A(q)y(t)=\frac{1}{1{q}^{1}}e(t)$$
Algorithms
AR and ARI model parameters are estimated using variants of the leastsquares method. The
following table summarizes the common names for methods with a specific combination of
approach
and window
argument values.
Method  Approach and Windowing 

Modified covariance method  (Default) Forwardbackward approach with no windowing 
Correlation method  YuleWalker approach with prewindowing and postwindowing 
Covariance method  Least squares approach with no windowing. arx uses this
routine 
References
[1] Marple, S. L., Jr. Chapter 8. Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice Hall, 1987.
Version History
Introduced in R2006aR2022b: Timedomain estimation data is accepted in the form of timetables and matrices
Most estimation, validation, analysis, and utility functions now accept timedomain
input/output data in the form of a single timetable that contains both input and output data
or a pair of matrices that contain the input and output data separately. These functions
continue to accept iddata
objects as a data source as well, for
both timedomain and frequencydomain data.
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