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# estimate

Class: ssm

Maximum likelihood parameter estimation of state-space models

## Syntax

• EstMdl = estimate(Mdl,Y,params0)
example
• EstMdl = estimate(Mdl,Y,params0,Name,Value)
example
• [EstMdl,estParams,EstParamCov,logL,Output] = estimate(___)
example

## Description

example

EstMdl = estimate(Mdl,Y,params0) returns an estimated state-space model from fitting the ssm model Mdl to the response data Y. params0 is the vector of initial values for the unknown parameters in Mdl.

example

EstMdl = estimate(Mdl,Y,params0,Name,Value) estimates the state-space model with additional options specified by one or more Name,Value pair arguments. For example, you can specify to deflate the observations by a linear regression using predictor data, control how the results appear in the Command Window, and indicate which estimation method to use for the parameter covariance matrix.

example

[EstMdl,estParams,EstParamCov,logL,Output] = estimate(___) additionally returns:

• estParams, a vector containing the estimated parameters

• EstParamCov, the estimated variance-covariance matrix of the estimated parameters

• logL, the optimized loglikelihood value

• Output, optimization diagnostic information structure

using any of the input arguments in the previous syntaxes.

## Tips

Constrained likelihood objective function maximization

• You can specify any combination of linear inequality, linear equality, and upper and lower bound constraints on the parameters.

• Good practice is to avoid equality and inequality constraints during optimization. For example, to constrain the parameter w to be positive, implicitly specify the state-space model using a parameter-to-matrix mapping function. Within the function, set w = exp(s) within the function. Then, use unconstrained optimization to estimate s. Consequently, s can assume any real value, but w must be positive.

Predictors and corresponding coefficients

• To include an overall mean to the observation model, include a column of 1s in Zt.

• To account for predictor effects when you simulate, you must deflate the observations manually. To deflate the observations, use ${W}_{t}={Y}_{t}-{Z}_{t}\stackrel{^}{\beta }.$

• If the regression model is complex, then consider implicitly defining the state space model. For example, define the parameter-to-matrix mapping function using the following syntax pattern.

function [A,B,C,D,Mean0,Cov0,StateType,DeflateY] = ParamMap(params,Y,Z)
...
DeflateY = Y - exp(params(9) + params(10)*Z);
...
end
In this example, Y is the matrix of observations and Z is the matrix of predictors. The function returns DeflateY, which is the matrix of deflated observations. Specify Y and Z in the MATLAB® Workspace before, and then pass ParamMap to ssm using the following syntax pattern.

Mdl = ssm(@(params)ParamMap(params,Y,Z))

This is also useful if each response series requires a distinct set of predictors.

• If the state equation requires predictors, then include the predictors as additional state variables. Since predictor data varies with time, a state-space model with predictors as states is time varying.

Additional Tips

• The software accommodates missing data. Indicate missing data using NaN values in the observed responses (Y).

• Good practice is to check the convergence status of the optimization routine by displaying Output.ExitFlag.

• If the optimization algorithm does not converge, then you can increase the number of iterations using the 'Options' name-value pair argument.

• If the optimization algorithm does not converge, then consider using refine, which might help you obtain better initial parameter values for optimization.

## Input Arguments

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Standard state-space model containing unknown parameters, specified as an ssm model object returned by ssm.

• For explicitly created state-space models, the software estimates all NaN values in the coefficient matrices (Mdl.A, Mdl.B, Mdl.C, and Mdl.D) and the initial state means and covariance matrix (Mdl.Mean0 and Mdl.Cov0). For details on explicit and implicit model creation, see ssm.

• For implicitly created state-space models, you specify the model structure and the location of the unknown parameters using the parameter-to-matrix mapping function. Implicitly create a state-space model to estimate complex models, impose parameter constraints, and estimate initial states. The parameter-to-mapping function can also accommodate additional output arguments.

 Note:   Mdl does not store observed responses or predictor data. Supply the data wherever necessary, using the appropriate input and name-value pair arguments.

Observed response data to which Mdl is fit, specified as a numeric matrix or a cell vector of numeric vectors.

• If Mdl is time invariant with respect to the observation equation, then Y is a T-by-n matrix. Each row of the matrix corresponds to a period and each column corresponds to a particular observation in the model. Therefore, T is the sample size and n is the number of observations per period. The last row of Y contains the latest observations.

• If Mdl is time varying with respect to the observation equation, then Y is a T-by-1 cell vector. Y{t} contains an nt-dimensional vector of observations for period t, where t = 1,...,T. The corresponding dimensions of the coefficient matrices in Mdl.C{t} and Mdl.D{t} must be consistent with the matrix in Y{t} for all periods. The last cell of Y contains the latest observations.

Suppose that you create Mdl implicitly by specifying a parameter-to-matrix mapping function, and the function has input arguments for the observed responses or predictors. Then, the mapping function establishes a link to observed responses and the predictor data in the MATLAB workspace, which overrides the value of Y.

NaN elements indicate missing observations. For details on how the Kalman filter accommodates missing observations, see Algorithms.

Data Types: double | cell

Initial values of unknown parameters for numeric maximum likelihood estimation, specified as a numeric vector.

The elements of params0 correspond to the unknown parameters in the state-space model matrices A, B, C, and D, and, optionally, the initial state mean Mean0 and covariance matrix Cov0.

• If you created Mdl explicitly (that is, by specifying the matrices without a parameter-to-matrix mapping function), then the software maps the elements of params to NaNs in the state-space model matrices and initial state values. The software searches for NaNs column-wise, following the order A, B, C, D, Mean0, Cov0.

• If you created Mdl implicitly (that is, by specifying the matrices with a parameter-to-matrix mapping function), then set initial parameter values for the state-space model matrices, initial state values, and state types within the parameter-to-matrix mapping function.

Data Types: double

### 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.

Example: 'CovMethod','hessian','Display','diagnostics','Predictors',Z specifies to estimate the asymptotic parameter covariance using the negative, inverted Hessian matrix, display optimization diagnostics at the Command Window, and to deflate the observations by a linear regression containing the predictor data Z.

#### Estimation Options

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Initial values of regression coefficients, specified as the comma-separated pair consisting of 'Beta0' and a d-by-n numeric matrix. d is the number of predictor variables (see Predictors) and n is the number of observed response series (see Y).

By default, Beta0 is the ordinary least-squares estimate of Y onto Predictors.

Data Types: double

Asymptotic covariance estimation method, specified as the comma-separated pair consisting of 'CovMethod' and a value in this table.

ValueDescription
'hessian'Negative, inverted Hessian matrix
'opg'Outer product of gradients (OPG)
'sandwich'Both Hessian and OPG

Example: 'CovMethod','sandwich'

Data Types: char

Command Window display option, specified as the comma-separated pair consisting of 'Display' and a character vector or cell vector of character vectors.

Set Display using any combination of values in this table.

Valueestimate Displays
'diagnostics'Optimization diagnostics
'full'Maximum likelihood parameter estimates, standard errors, t statistics, iterative optimization information, and optimization diagnostics
'iter'Iterative optimization information
'off'No display in the Command Window
'params'Maximum likelihood parameter estimates, standard errors, and t statistics

For example:

• To run a simulation where you are fitting many models, and therefore want to suppress all output, use 'Display','off'.

• To display all estimation results and the optimization diagnostics, use 'Display',{'params','diagnostics'}.

Data Types: char | cell

Optimization options, specified as the comma-separated pair consisting of 'Options' and an optimoptions optimization controller. Options replaces default optimization options of the optimizer. For details on altering default values of the optimizer, see the optimization controller optimoptions, the constrained optimization function fmincon, or the unconstrained optimization function fminunc in Optimization Toolbox™.

For example, to change the constraint tolerance to 1e-6, set Options = optimoptions(@fmincon,'ConstraintTolerance',1e-6,'Algorithm','sqp'). Then, pass Options into estimate using 'Options',Options.

By default:

• For constrained optimization, estimate maximizes the likelihood objective function using fmincon and its default options, but sets 'Algorithm','interior-point'.

• For unconstrained optimization, estimate maximizes the likelihood objective function using fminunc and its default options, but sets 'Algorithm','quasi-newton'.

Predictor data for the regression component in the observation equation, specified as the comma-separated pair consisting of 'Predictors' and a T-by-d numeric matrix. T is the number of periods and d is the number of predictor variables. Row t corresponds to the observed predictors at period t (Zt) in the expanded observation equation

${y}_{t}-{Z}_{t}\beta =C{x}_{t}+D{u}_{t}.$

In other words, the predictor series serve as observation deflators. β is a d-by-n time-invariant matrix of regression coefficients that the software estimates with all other parameters.

• For n observations per period, the software regresses all predictor series onto each observation.

• If you specify Predictors, then Mdl must be time invariant. Otherwise, the software returns an error.

• By default, the software excludes a regression component from the state-space model.

Data Types: double

Square root filter method flag, specified as the comma-separated pair consisting of 'SquareRoot' and true or false. If true, then estimate applies the square root filter method when implementing the Kalman filter.

If you suspect that the eigenvalues of the filtered state or forecasted observation covariance matrices are close to zero, then specify 'SquareRoot',true. The square root filter is robust to numerical issues arising from finite the precision of calculations, but requires more computational resources.

Example: 'SquareRoot',true

Data Types: logical

Forecast uncertainty threshold, specified as the comma-separated pair consisting of 'Tolerance' and a nonnegative scalar.

If the forecast uncertainty for a particular observation is less than Tolerance during numerical estimation, then the software removes the uncertainty corresponding to the observation from the forecast covariance matrix before its inversion.

It is best practice to set Tolerance to a small number, for example, le-15, to overcome numerical obstacles during estimation.

Example: 'Tolerance',le-15

Data Types: double

Univariate treatment of a multivariate series flag, specified as the comma-separated pair consisting of 'Univariate' and true or false. Univariate treatment of a multivariate series is also known as sequential filtering.

The univariate treatment can accelerate and improve numerical stability of the Kalman filter. However, all observation innovations must be uncorrelated. That is, DtDt' must be diagonal, where Dt, t = 1,...,T, is one of the following:

• The matrix D{t} in a time-varying state-space model

• The matrix D in a time-invariant state-space model

Example: 'Univariate',true

Data Types: logical

#### Constrained Optimization Options for fmincon

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Linear equality constraint parameter transformer for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'Aeq' and a matrix.

If you specify Aeq and beq, then estimate maximizes the likelihood objective function using the equality constraint $Aeq\theta =beq,$ where θ is a vector containing every Mdl parameter.

The number of rows of Aeq is the number of constraints, and the number of columns is the number of parameters that the software estimates. Order the columns of Aeq by Mdl.A, Mdl.B, Mdl.C, Mdl.D, Mdl.Mean0, Mdl.Cov0, and the regression coefficient (if the model has one).

Specify Aeq and beq together, otherwise estimate returns an error.

Aeq directly corresponds to the input argument Aeq of fmincon, not to the state-transition coefficient matrix Mdl.A.

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Linear inequality constraint parameter transformer for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'Aineq' and a matrix.

If you specify Aineq and bineq, then estimate maximizes the likelihood objective function using the inequality constraint $Aineq\theta \le bineq,$ where θ is a vector containing every Mdl parameter.

The number of rows of Aineq is the number of constraints, and the number of columns is the number of parameters that the software estimates. Order the columns of Aineq by Mdl.A, Mdl.B, Mdl.C, Mdl.D, Mdl.Mean0, Mdl.Cov0, and the regression coefficient (if the model has one).

Specify Aineq and bineq together, otherwise estimate returns an error.

Aineq directly corresponds to the input argument A of fmincon, not to the state-transition coefficient matrix Mdl.A.

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Data Types: double

Linear equality constraints of the transformed parameters for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'beq' and a numeric vector.

If you specify Aeq and beq, then estimate maximizes the likelihood objective function using the equality constraint $Aeq\theta =beq,$ where θ is a vector containing every Mdl parameter..

Specify Aeq and beq together, otherwise estimate returns an error.

beq directly corresponds to the input argument beq of fmincon, and is not associated with any component of Mdl.

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Data Types: double

Linear inequality constraint upper bounds of the transformed parameters for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'bineq' and a numeric vector.

If you specify Aineq and bineq, then estimate maximizes the likelihood objective function using the inequality constraint $Aineq\theta \le bineq,$ where θ is a vector containing every Mdl parameter.

Specify Aineq and bineq together, otherwise estimate returns an error.

bineq directly corresponds to the input argument b of fmincon, and is not associated with any component of Mdl.

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Data Types: double

Lower bounds of the parameters for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'lb' and a numeric vector.

If you specify lb and ub, then estimate maximizes the likelihood objective function subject to$lb\le \theta \le ub,$ where θ is a vector containing every Mdl parameter.

Order the elements of lb by Mdl.A, Mdl.B, Mdl.C, Mdl.D, Mdl.Mean0, Mdl.Cov0, and the regression coefficient (if the model has one).

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Data Types: double

Upper bounds of the parameters for constrained likelihood objective function maximization, specified as the comma-separated pair consisting of 'ub' and a numeric vector.

If you specify lb and ub, then estimate maximizes the likelihood objective function subject to$lb\le \theta \le ub,$ where θ is a vector every Mdl parameter.

Order the elements of ub by Mdl.A, Mdl.B, Mdl.C, Mdl.D, Mdl.Mean0, Mdl.Cov0, and the regression coefficient (if the model has one).

By default, if you did not specify any constraint (linear inequality, linear equality, or upper and lower bound), then estimate maximizes the likelihood objective function using unconstrained maximization.

Data Types: double

## Output Arguments

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State-space model containing the parameter estimates, returned as an ssm model object.

estimate uses the Kalman filter and maximum likelihood to estimate all unknown parameters.

Regardless of how you created Mdl, EstMdl stores:

• The parameter estimates of the coefficient matrices in the properties A, B, C, and D.

• The initial state means and covariance matrix in the properties Mean0 and Cov0.

For the estimated regression coefficient matrix, see estParams.

 Note:   EstMdl does not store observed responses or predictor data. If you plan to filter (using filter), forecast (using forecast), or smooth (using smooth) using EstMdl, then you might need to supply the appropriate data.

Maximum likelihood estimates of the model parameters known to the optimizer, returned as a numeric vector. estParams has the same dimensions as params0.

estimate arranges the estimates in estParams corresponding to unknown parameters in this order.

1. EstMdl.A(:), that is, estimates in EstMdl.A listed column-wise

2. EstMdl.B(:)

3. EstMdl.C(:)

4. EstMdl.D(:)

5. EstMdl.Mean0

6. EstMdl.Cov0(:)

7. In models with predictors, estimated regression coefficients listed column-wise

Variance-covariance matrix of maximum likelihood estimates of the model parameters known to the optimizer, returned as a numeric matrix.

The rows and columns contain the covariances of the parameter estimates. The standard errors of the parameter estimates are the square root of the entries along the main diagonal.

estimate arranges the estimates in the rows and columns of EstParamCov corresponding to unknown parameters in this order.

1. EstMdl.A(:), that is, estimates in EstMdl.A listed column-wise

2. EstMdl.B(:)

3. EstMdl.C(:)

4. EstMdl.D(:)

5. EstMdl.Mean0

6. EstMdl.Cov0(:)

7. In models with predictors, estimated regression coefficients listed column-wise

Optimized loglikelihood value, returned as a scalar.

Missing observations do not contribute to the loglikelihood.

Optimization information, returned as a structure array.

This table describes the fields of Output.

FieldDescription
ExitFlagOptimization exit flag that describes the exit condition. For details, see fmincon and fminunc.
OptionsOptimization options that the optimizer used for numerical estimation. For details, see optimoptions.

Data Types: struct

## Examples

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Generate data from a known model, and then fit a state-space model to the data.

Suppose that a latent process is this AR(1) process

where is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from , assuming that the series starts at 1.5.

T = 100;
ARMdl = arima('AR',0.5,'Constant',0,'Variance',1);
x0 = 1.5;
rng(1); % For reproducibility
x = simulate(ARMdl,T,'Y0',x0);

Suppose further that the latent process is subject to additive measurement error as indicated in the equation

where is Gaussian with mean 0 and standard deviation 0.1.

Use the random latent state process (x) and the observation equation to generate observations.

y = x + 0.1*randn(T,1);

Together, the latent process and observation equations compose a state-space model. Supposing that the coefficients and variances are unknown parameters, the state-space model is

Specify the state-transition matrix. Use NaN values for unknown parameters.

A = NaN;

Specify the state-disturbance-loading coefficient matrix.

B = NaN;

Specify the measurement-sensitivity coefficient matrix.

C = 1;

Specify the observation-innovation coefficient matrix

D = NaN;

Specify the state-space model using the coefficient matrices. Also, specify the initial state mean, variance, and distribution (which is stationary).

Mean0 = 0;
Cov0 = 10;
StateType = 0;
Mdl = ssm(A,B,C,D,'Mean0',Mean0,'Cov0',Cov0,'StateType',StateType);

Mdl is an ssm model. Verify that the model is correctly specified using the display in the Command Window.

Pass the observations to estimate to estimate the parameter. Set a starting value for the parameter to params0. and must be positive, so set the lower bound constraints using the 'lb' name-value pair argument. Specify that the lower bound of is -Inf.

params0 = [0.9; 0.5; 0.1];
EstMdl = estimate(Mdl,y,params0,'lb',[-Inf; 0; 0])
Method: Maximum likelihood (fmincon)
Sample size: 100
Logarithmic  likelihood:     -140.532
Akaike   info criterion:      287.064
Bayesian info criterion:      294.879
|     Coeff      Std Err   t Stat    Prob
-------------------------------------------------
c(1) | 0.45425       0.19870   2.28611  0.02225
c(2) | 0.89013       0.30359   2.93205  0.00337
c(3) | 0.38750       0.57858   0.66975  0.50302
|
|  Final State   Std Dev   t Stat    Prob
x(1) | 1.52989       0.35621   4.29498  0.00002

EstMdl =

State-space model type: <a href="matlab: doc ssm">ssm</a>

State vector length: 1
Observation vector length: 1
State disturbance vector length: 1
Observation innovation vector length: 1
Sample size supported by model: Unlimited

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...

State equation:
x1(t) = (0.45)x1(t-1) + (0.89)u1(t)

Observation equation:
y1(t) = x1(t) + (0.39)e1(t)

Initial state distribution:

Initial state means
x1
0

Initial state covariance matrix
x1
x1  10

State types
x1
Stationary

EstMdl is an ssm model. The results of the estimation appear in the Command Window, contain the fitted state-space equations, and contain a table of parameter estimates, their standard errors, t statistics, and p-values.

Use dot notation to use or display the fitted state-transition matrix.

EstMdl.A
ans =

0.4543

Pass EstMdl to forecast to forecast observations, or to simulate to conduct a Monte Carlo study.

Suppose that the linear relationship between the change in the unemployment rate and the nominal gross national product (nGNP) growth rate is of interest. Suppose further that the first difference of the unemployment rate is an ARMA(1,1) series. Symbolically, and in state-space form, the model is

where:

• is the change in the unemployment rate at time t.

• is a dummy state for the MA(1) effect.

• is the observed change in the unemployment being deflated by the growth rate of nGNP ( ).

• is the Gaussian series of state disturbances having mean 0 and standard deviation 1.

• is the Gaussian series of observation innovations having mean 0 and standard deviation .

Load the Nelson-Plosser data set, which contains the unemployment rate and nGNP series, among other things.

load Data_NelsonPlosser

Preprocess the data by taking the natural logarithm of the nGNP series and the first difference of each. Also, remove the starting NaN values from each series.

isNaN = any(ismissing(DataTable),2);       % Flag periods containing NaNs
gnpn = DataTable.GNPN(~isNaN);
u = DataTable.UR(~isNaN);
T = size(gnpn,1);                          % Sample size
Z = [ones(T-1,1) diff(log(gnpn))];
y = diff(u);

This example proceeds using series without NaN values. However, using the Kalman filter framework, the software can accommodate series containing missing values.

Specify the state-transition coefficient matrix.

A = [NaN NaN; 0 0];

Specify the state-disturbance-loading coefficient matrix.

B = [1; 1];

Specify the measurement-sensitivity coefficient matrix.

C = [1 0];

Specify the observation-innovation coefficient matrix.

D = NaN;

Specify the state-space model using ssm.

Mdl = ssm(A,B,C,D);

Estimate the model parameters. Specify the regression component and its initial value for optimization using the 'Predictors' and 'Beta0' name-value pair arguments, respectively. Display the estimates and all optimization diagnostic information. Restrict the estimate of to all positive, real numbers.

params0 = [0.3 0.2 0.1]; % Chosen arbitrarily
EstMdl = estimate(Mdl,y,params0,'Predictors',Z,'Display','full',...
'Beta0',[0.1 0.2],'lb',[-Inf,-Inf,0,-Inf,-Inf]);
____________________________________________________________
Diagnostic Information

Number of variables: 5

Functions
Objective:                            @(c)-fML(c,Mdl,Y,Predictors,unitFlag,sqrtFlag,mexFlag,mexTvFlag,tol,ind,switchTime,precaution)
Gradient:                             finite-differencing
Hessian:                              finite-differencing (or Quasi-Newton)

Constraints
Nonlinear constraints:                do not exist

Number of linear inequality constraints:    0
Number of linear equality constraints:      0
Number of lower bound constraints:          1
Number of upper bound constraints:          0

Algorithm selected
interior-point

____________________________________________________________
End diagnostic information
First-order      Norm of
Iter F-count            f(x)  Feasibility   optimality         step
0       6    2.579611e+02    0.000e+00    4.601e+01
1      20    2.556482e+02    0.000e+00    3.652e+01    1.392e-01
2      27    2.503349e+02    0.000e+00    4.319e+01    1.908e-01
3      35    2.379649e+02    0.000e+00    1.290e+01    1.083e+01
4      41    1.947847e+02    0.000e+00    1.946e+01    7.151e+00
5      47    1.604292e+02    0.000e+00    2.134e+02    1.186e+01
6      53    1.256701e+02    0.000e+00    8.822e+01    1.586e+00
7      59    1.116583e+02    0.000e+00    9.090e+00    2.379e+00
8      65    1.078927e+02    0.000e+00    1.784e+01    1.659e+00
9      71    1.028434e+02    0.000e+00    7.552e+00    1.054e+00
10      79    1.016331e+02    0.000e+00    4.613e+00    4.900e-01
11      85    1.009869e+02    0.000e+00    3.950e+00    3.493e-01
12      91    1.003250e+02    0.000e+00    5.110e+00    6.937e-01
13      97    9.995887e+01    0.000e+00    3.532e+00    2.484e-01
14     103    9.980272e+01    0.000e+00    2.733e+00    1.396e-01
15     109    9.973928e+01    0.000e+00    6.538e-01    1.195e-01
16     115    9.973060e+01    0.000e+00    5.589e-01    7.715e-02
17     121    9.972831e+01    0.000e+00    4.435e-01    6.038e-02
18     127    9.972820e+01    0.000e+00    4.306e-01    1.017e-02
19     133    9.972780e+01    0.000e+00    3.688e-01    3.495e-02
20     139    9.972729e+01    0.000e+00    2.828e-01    2.281e-02
21     145    9.972646e+01    0.000e+00    1.568e-01    1.055e-02
22     151    9.972579e+01    0.000e+00    1.000e-01    3.161e-02
23     157    9.972467e+01    0.000e+00    4.015e-02    3.117e-02
24     163    9.972457e+01    0.000e+00    2.000e-02    1.156e-02
25     169    9.972454e+01    0.000e+00    2.314e-03    3.019e-03
26     175    9.972454e+01    0.000e+00    2.000e-04    3.239e-04
27     181    9.972454e+01    0.000e+00    6.676e-06    3.839e-05
28     187    9.972454e+01    0.000e+00    2.861e-06    4.998e-07
29     193    9.972454e+01    0.000e+00    2.209e-06    3.381e-07
30     200    9.972454e+01    0.000e+00    3.633e-06    1.903e-07

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the default value of the step size tolerance and constraints are
satisfied to within the default value of the constraint tolerance.

Method: Maximum likelihood (fmincon)
Sample size: 61
Logarithmic  likelihood:     -99.7245
Akaike   info criterion:      209.449
Bayesian info criterion:      220.003
|      Coeff       Std Err    t Stat     Prob
----------------------------------------------------------
c(1)      |  -0.34098       0.29608    -1.15164  0.24948
c(2)      |   1.05003       0.41377     2.53771  0.01116
c(3)      |   0.48592       0.36790     1.32079  0.18657
y <- z(1) |   1.36121       0.22338     6.09358   0
y <- z(2) | -24.46711       1.60018   -15.29023   0
|
|    Final State   Std Dev     t Stat    Prob
x(1)      |   1.01264       0.44690     2.26592  0.02346
x(2)      |   0.77718       0.58917     1.31912  0.18713

Optimization information and a table of estimates and statistics output to the Command Window. EstMdl is an ssm model, and you can access its properties using dot notation.

The software implements the Kalman filter using the covariance filter by default, but you can specify to use the square-root filter instead. This example compares estimates from each method using simulated data.

Suppose that a latent process is an AR(1). Subsequently, the state equation is

where is Gaussian with mean 0 and standard deviation 0.3.

Generate a random series of 100 observations from , assuming that the series starts at 1.5.

T = 100;
ARMdl = arima('AR',0.5,'Constant',0,'Variance',0.3^2);
x0 = 1.5;
rng(1); % For reproducibility
x = simulate(ARMdl,T,'Y0',x0);

Suppose further that the latent process is subject to additive measurement error. Subsequently, the observation equation is

where is Gaussian with mean 0 and standard deviation 0.1.

Use the random latent state process (x) and the observation equation to generate observations.

y = x + 0.1*randn(T,1);

Together, the latent process and observation equations compose a state-space model. Supposing that the coefficients and variances are unknown parameter, the state-space model is

Specify the state-transition coefficient matrix. Use NaN values for unknown parameters.

A = NaN;

Specify the state-disturbance-loading coefficient matrix.

B = NaN;

Specify the measurement-sensitivity coefficient matrix.

C = 1;

Specify the observation-innovation coefficient matrix.

D = NaN;

Specify the state-space model using the coefficient matrices. Also, specify the initial state mean, variance, and distribution (which is stationary).

Mean0 = 0;
Cov0 = 10;
StateType = 0;
Mdl = ssm(A,B,C,D,'Mean0',Mean0,'Cov0',Cov0,'StateType',StateType);

Mdl is an ssm model.

Estimate the parameters using estimate two ways:

• Using the default, simple Kalman filter

• Using the square root filter variation

In both cases, specify that no output should be returned to the Command Window. This is good practice if you plan on running estimate multiple times (such as a Monte Carlo simulation).

params0 = [10,10,10];
[~,estParamsSKF,EstParamCovSKF,logLSKF,OutputSKF] = estimate(Mdl,y,params0,...
'Display','off');
[~,estParamsSR,EstParamCovSR,logLSR,OutputSR] = estimate(Mdl,y,params0,...
'Squareroot',true,'Display','off');

Check that the algorithms converged properly by printing the exit flag properties of OutputSKF and OutputSR.

exitFlagSKF = OutputSKF.ExitFlag
exitFlagSR = OutputSR.ExitFlag
exitFlagSKF =

1

exitFlagSR =

1

Both algorithms have an exit flag of 1, which indicates that the software met the convergence criteria.

Compare the estimates from each algorithm.

fprintf('\n Parameter Estimates\n')
table(estParamsSKF',estParamsSR','VariableNames',...
{'SimpleKalmanFilter','SquarerootFilter'})
fprintf('\nEstimated Parameter Covariance Matrix\n')
table(EstParamCovSKF,EstParamCovSR,'VariableNames',...
{'SimpleKalmanFilter','SquarerootFilter'})
Parameter Estimates

ans =

SimpleKalmanFilter    SquarerootFilter
__________________    ________________

0.51057               0.51057
0.23436               0.23436
-0.17904              -0.17904

Estimated Parameter Covariance Matrix

ans =

SimpleKalmanFilter                      SquarerootFilter
___________________________________    ___________________________________

0.036669    -0.013302    -0.014012     0.036669    -0.013302    -0.014012
-0.013302    0.0070187    0.0072533    -0.013302    0.0070187    0.0072533
-0.014012    0.0072533    0.0089019    -0.014012    0.0072533    0.0089019

In this case, the results are the same.

If you use the default, covariance filter method, and you run into numerical problems during estimation, filtering, or smoothing, try using the squareroot method.

## Algorithms

• The Kalman filter accommodates missing data by not updating filtered state estimates corresponding to missing observations. In other words, suppose there is a missing observation at period t. Then, the state forecast for period t based on the previous t – 1 observations and filtered state for period t are equivalent.

• For explicitly created state-space models, estimate applies all predictors to each response series. However, each response series has its own set of regression coefficients.

• If you do not specify optimization constraints, then estimate uses fminunc for unconstrained numerical estimation. If you specify any pair of optimization constraints, then estimate uses fmincon for constrained numerical estimation. For either type of optimization, optimization options you set using the name-value pair argument Options must be consistent with the options of the optimization algorithm.

• estimate passes the name-value pair arguments Options, Aineq, bineq, Aeq, beq, lb, and ub directly to the optimizer fmincon or fminunc.

• estimate fits regression coefficients along with all other state-space model parameters. The software is flexible enough to allow applying constraints to the regression coefficients using constrained optimization options. For more details, see the Name,Value pair arguments and fmincon.

• If you set 'Univariate',true then, during the filtering algorithm, the software sequentially updates rather then updating all at once. This practice might accelerate parameter estimation, especially for a low-dimensional, time-invariant model.

• Suppose that you want to create a state-space model using a parameter-to-matrix mapping function with this signature

[A,B,C,D,Mean0,Cov0,StateType,DeflateY] = paramMap(params,Y,Z)
and you specify the model using an anonymous function
Mdl = ssm(@(params)paramMap(params,Y,Z))
The observed responses Y and predictor data Z are not input arguments in the anonymous function. If Y and Z exist in the MATLAB Workspace before creating Mdl, then the software establishes a link to them. Otherwise, if you pass Mdl to estimate, the software throws an error.

The link to the data established by the anonymous function overrides all other corresponding input argument values of estimate. This distinction is important particularly when conducting a rolling window analysis. For details, see Rolling-Window Analysis of Time-Series Models.

## Limitations

If the model is time varying with respect the observed responses, then the software does not support including predictors. If the observation vectors among different periods vary in length, then the software cannot determine which coefficients to use to deflate the observed responses.

## References

[1] Durbin J., and S. J. Koopman. Time Series Analysis by State Space Methods. 2nd ed. Oxford: Oxford University Press, 2012.

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