ssm

Create standard linear Gaussian state-space model

Description

The ssm function returns an ssm object specifying the functional form and storing the parameter values of a standard linear Gaussian state-space model for a latent state process x_t possibly imperfectly observed through the variable y_t. The variables x_t and y_t can be univariate or multivariate and the model parameters or variable dimensions can be time-invariant or time-varying (see Decide on Model Structure).

The key components of an ssm object are the state-transition A, state-disturbance-loading B, measurement-sensitivity C, and observation-innovation D coefficient matrices because they completely specify the model structure. You can explicitly specify each matrix or supply a custom function that implicitly specifies them. Regardless, given the model structure, all coefficients are unknown and estimable unless you specify their values.

To estimate a model containing unknown parameter values, pass the model and data to estimate. To work with an estimated or fully specified ssm object, pass it to an object function.

Alternative state-space models include:

• The dssm model object — Linear Gaussian state-space model with diffuse initial state distribution

• The bssm model object — Bayesian linear state-space model

• The bnlssm model object — Bayesian nonlinear non-Gaussian state-space model

Creation

Syntax

Mdl = ssm(A,B,C)

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

Mdl = ssm(___,Name=Value)

Mdl = ssm(ParamMap)

Mdl = ssm(DSSMMdl)

Description

Explicitly Specify Coefficient Matrices

Mdl = ssm(A,B,C) returns the standard linear Gaussian state-space model Mdl with state-transition matrix A, state-disturbance-loading matrix B, and measurement-sensitivity matrix C. At each time t, the state combination y_t = Cx_t is observed without error. ssm sets the model properties A, B, and C from the corresponding inputs.

Mdl = ssm(A,B,C,D) additionally specifies the observation-innovation matrix D and sets the property D.

Mdl = ssm(___,Name=Value) sets properties that describe the initial state distribution using name-value arguments, and using any input-argument combination in the previous syntaxes. For example, ssm(A,B,C,StateType=[0; 1; 2]) specifies that the first state variable is initially stationary, the second state variable is initially the constant 1, and the third state variable is initially nonstationary.

Implicitly Specify Coefficient Matrices By Using Custom Function

Mdl = ssm(ParamMap) returns the state-space model Mdl whose structure is specified by the custom parameter-to-matrix mapping function ParamMap. The function maps a parameter vector θ to the matrices A, B, and C. Optionally, ParamMap can map parameters to D, Mean0, Cov0, or StateType. To accommodate a regression component in the observation equation, ParamMap can return deflated observation data.

Convert from Diffuse to Standard State-Space Model

Mdl = ssm(DSSMMdl) converts a diffuse state-space model object DSSMMdl to a state-space model object Mdl. ssm sets all initial variances of diffuse states in Mdl.Cov0 to 1e07.

Because Mdl is a standard state-space model, ssm object functions apply the standard Kalman filter, instead of the diffuse Kalman filter, for filtering, smoothing, and parameter estimation.

Input Arguments

DSSMMdl — Diffuse state-space model

dssm model object

Diffuse state-space model to convert to a state-space model, specified as a dssm model object.

ssm sets all Inf-valued initial state variances DSSMMdl.Cov0 to 1e7. Any states with variance other than Inf retain their values.

Properties

A — State-transition coefficient matrix

matrix | cell vector of matrices | empty array []

State-transition coefficient matrix for explicit state-space model creation, specified as a matrix or cell vector of matrices. When you implicitly create a model, ssm sets A to an empty array, and determines the state-transition coefficient matrix from ParamMap

The state-transition coefficient matrix, A_t, specifies how the states, x_t, are expected to transition from period t – 1 to t, for all t = 1,...,T. That is, the expected state-transition equation at period t is E(x_t|x_t–1) = A_tx_t–1.

For time-invariant state-space models, specify A as an m-by-m matrix, where m is the number of states per period.

For time-varying state-space models, specify A as a T-dimensional cell array, where A{t} contains an m_t-by-m_{t – 1} state-transition coefficient matrix. If the number of states changes from period t – 1 to t, then m_t ≠ m_{t – 1}.

NaN values in any coefficient matrix indicate unique, unknown parameters in the state-space model. A contributes:

• sum(isnan(A(:))) unknown parameters to time-invariant state-space models. In other words, if the state-space model is time invariant, then the software uses the same unknown parameters defined in A at each period.

• numParamsA unknown parameters to time-varying state-space models, where numParamsA = sum(cell2mat(cellfun(@(x)sum(sum(isnan(x))),A,'UniformOutput',0))). In other words, if the state-space model is time varying, then the software assigns a new set of parameters for each matrix in A.

You cannot specify A and ParamMap simultaneously.

Data Types: double | cell

B — State-disturbance-loading coefficient matrix

matrix | cell vector of matrices | empty array []

State-disturbance-loading coefficient matrix for explicit state-space model creation, specified as a matrix or cell vector of matrices. When you implicitly create a model, ssm sets B to an empty array, and determines the state-disturbance-loading coefficient matrix from ParamMap.

The state disturbances, u_t, are independent Gaussian random variables with mean 0 and standard deviation 1. The state-disturbance-loading coefficient matrix, B_t, specifies the additive error structure in the state-transition equation from period t – 1 to t, for all t = 1,...,T. That is, the state-transition equation at period t is x_t = A_tx_t–1 + B_tu_t.

For time-invariant state-space models, specify B as an m-by-k matrix, where m is the number of states and k is the number of state disturbances per period. B*B' is the state-disturbance covariance matrix for all periods.

For time-varying state-space models, specify B as a T-dimensional cell array, where B{t} contains an m_t-by-k_t state-disturbance-loading coefficient matrix. If the number of states or state disturbances changes at period t, then the matrix dimensions between B{t-1} and B{t} vary. B{t}*B{t}' is the state-disturbance covariance matrix for period t.

NaN values in any coefficient matrix indicate unique, unknown parameters in the state-space model. B contributes:

• sum(isnan(B(:))) unknown parameters to time-invariant state-space models. In other words, if the state-space model is time invariant, then the software uses the same unknown parameters defined in B at each period.

• numParamsB unknown parameters to time-varying state-space models, where numParamsB = sum(cell2mat(cellfun(@(x)sum(sum(isnan(x))),B,'UniformOutput',0))). In other words, if the state-space model is time varying, then the software assigns a new set of parameters for each matrix in B.

You cannot specify B and ParamMap simultaneously.

Data Types: double | cell

C — Measurement-sensitivity coefficient matrix

matrix | cell vector of matrices | empty array []

Measurement-sensitivity coefficient matrix for explicit state-space model creation, specified as a matrix or cell vector of matrices. When you implicitly create a model, ssm sets C to an empty array, and determines the measurement-sensitivity coefficient matrix from ParamMap.

The measurement-sensitivity coefficient matrix, C_t, specifies how the states are expected to linearly combine at period t to form the observations, y_t, for all t = 1,...,T. That is, the expected observation equation at period t is E(y_t|x_t) = C_tx_t.

For time-invariant state-space models, specify C as an n-by-m matrix, where n is the number of observations and m is the number of states per period.

For time-varying state-space models, specify C as a T-dimensional cell array, where C{t} contains an n_t-by-m_t measurement-sensitivity coefficient matrix. If the number of states or observations changes at period t, then the matrix dimensions between C{t-1} and C{t} vary.

NaN values in any coefficient matrix indicate unique, unknown parameters in the state-space model. C contributes:

• sum(isnan(C(:))) unknown parameters to time-invariant state-space models. In other words, if the state-space model is time invariant, then the software uses the same unknown parameters defined in C at each period.

• numParamsC unknown parameters to time-varying state-space models, where numParamsC = sum(cell2mat(cellfun(@(x)sum(sum(isnan(x))),C,'UniformOutput',0))). In other words, if the state-space model is time varying, then the software assigns a new set of parameters for each matrix in C.

You cannot specify C and ParamMap simultaneously.

Data Types: double | cell

D — Observation-innovation coefficient matrix

empty array [] (default) | matrix | cell vector of matrices

Observation-innovation coefficient matrix for explicit state-space model creation, specified as a matrix or cell vector of matrices.

The observation innovations, ε_t, are independent Gaussian random variables with mean 0 and standard deviation 1. The observation-innovation coefficient matrix, D_t, specifies the additive error structure in the observation equation at period t, for all t = 1,...,T. That is, the observation equation at period t is y_t = C_tx_t + D_tε_t.

For time-invariant state-space models, specify D as an n-by-h matrix, where n is the number of observations and h is the number of observation innovations per period. D*D' is the observation-innovation covariance matrix for all periods.

For time-varying state-space models, specify D as a T-dimensional cell array, where D{t} contains an n_t-by-h_t matrix. If the number of observations or observation innovations changes at period t, then the matrix dimensions between D{t-1} and D{t} vary. D{t}*D{t}' is the observation-innovation covariance matrix for period t.

NaN values in any coefficient matrix indicate unique, unknown parameters in the state-space model. D contributes:

• sum(isnan(D(:))) unknown parameters to time-invariant state-space models. In other words, if the state-space model is time invariant, then the software uses the same unknown parameters defined in D at each period.

• numParamsD unknown parameters to time-varying state-space models, where numParamsD = sum(cell2mat(cellfun(@(x)sum(sum(isnan(x))),D,'UniformOutput',0))). In other words, if the state-space model is time varying, then the software assigns a new set of parameters for each matrix in D.

By default, D is an empty matrix indicating no observation innovations in the state-space model. However, when you implicitly create a model, ssm sets D to [], and determines the observation-innovation coefficient matrix from ParamMap.

You cannot specify D and ParamMap simultaneously.

Data Types: double | cell

Mean0 — Initial state mean

numeric vector | empty array []

Initial state mean for explicit state-space model creation, specified as a numeric vector or an empty array ([]). As a numeric vector, Mean0 has length equal to the number of initial states (size(A,1) or size(A{1},1)). Mean0 is the mean of the Gaussian distribution of the states at period 0.

If you implicitly create a state-space model by specifying ParamMap, the following conditions apply:

• You cannot specify the Mean0 property by using name-value argument syntax. Instead, specify the initial state mean in the parameter-to-matrix mapping function.

• Before you estimate the model by using the estimate function, Mean0 is [] and read only. The estimate function specifies Mean0 after estimation.

For the default values, see Algorithms.

Data Types: double

Cov0 — Initial state covariance matrix

square matrix | empty array []

Initial state covariance matrix, specified as a square matrix or an empty array []. As a matrix, Cov0 has dimensions equal to the number of initial states (size(A,1) or size(A{1},1)). Cov0 is the covariance of the Gaussian distribution of the states at period 0.

If you implicitly create a state-space model by specifying ParamMap, the following conditions apply:

• You cannot specify the Cov0 property by using name-value argument syntax. Instead, specify the initial state covariance in the parameter-to-matrix mapping function.

• Before you estimate the model by using the estimate function, Cov0 is [] and read only. The estimate function specifies Cov0 after estimation.

For the default values, see Algorithms.

Data Types: double

StateType — Initial state distribution type

numeric vector | empty array []

Initial state distribution type, specified as a numeric vector or empty array []. As a numeric vector, StateType has length equal to the number of initial states (size(A,1) or size(A{1},1)).

This table summarizes the available types of initial state distributions.

Value	Initial State Distribution Type
0	Stationary (e.g., ARMA models)
1	The constant 1 (that is, the state is 1 with probability 1)
2	Nonstationary (e.g., random walk model, seasonal linear time series) or static state

Example: Suppose that the state equation has two state variables: The first state variable is an AR(1) process, and the second state variable is a random walk. Set StateType to [0; 2].

If you implicitly create a state-space model by specifying ParamMap, the following conditions apply:

• You cannot specify the StateType property by using name-value argument syntax. Instead, specify the initial state covariance in the parameter-to-matrix mapping function.

• Before you estimate the model by using the estimate function, StateType is [] and read only. The estimate function specifies StateType after estimation.

Data Types: double

ParamMap — Parameter-to-matrix mapping function

empty array [] (default) | function handle

Parameter-to-matrix mapping function for implicit state-space model creation, specified as a function handle.

The function, to which ParamMap is a function handle, must accept at least one input argument and return at least three output arguments. The requisite input argument is a vector of unknown state-space model parameters θ, and the requisite output arguments correspond to the coefficient matrices A, B, and C, respectively. If your parameter-to-mapping function requires the input θ only, then implicitly create a state-space model by entering

Mdl = ssm(@ParamMap)

In general, you can write an intermediate function, for example, ParamFun, using the syntax

function [A,B,C,D,Mean0,Cov0,StateType,DeflateY] = ...
    ParamFun(theta,...otherInputArgs...)

In this general case, create the state-space model by entering

Mdl = ssm(@(theta)ParamMap(theta,...otherInputArgs...))

However, the following conditions apply:

• Follow the order of the output arguments.

• theta is a vector, and each element corresponds to an unknown state-space model parameter.

• ParamFun must return A, B, and C, which correspond to the state-transition, state-disturbance-loading, and measurement-sensitivity coefficient matrices, respectively.

• For the optional output arguments D, Mean0, Cov0, StateType, and DeflateY:

  • The optional output arguments correspond to the observation-innovation coefficient matrix D and the properties Mean0, Cov0, and StateType.

  • To skip specifying an optional output argument, set the argument to [] in the function body. For example, to skip specifying D, then set D = []; in the function.

  • DeflateY is the deflated-observation data, which accommodates a regression component in the observation equation. For example, in this function, which has a linear regression component, Y is the vector of observed responses and Z is the vector of predictor data.

    function [A,B,C,D,Mean0,Cov0,StateType,DeflateY] = ParamFun(theta,Y,Z)
    	...
    	DeflateY = Y - theta(9) - theta(10)*Z;
    	...
    end

  • For the default values of Mean0, Cov0, and StateType, see Algorithms.

• It is best practice to:

  • Load the data to the MATLAB^® Workspace before specifying the model.

  • Create the parameter-to-matrix mapping function as its own file.

If you specify ParamMap, you cannot specify any other property or input argument.

If you explicitly create a state-space model, ParamMap is an empty array [].

Data Types: function_handle

Object Functions

Fit Model to Data

estimate	Maximum likelihood parameter estimation of state-space models
refine	Refine initial parameters to aid state-space model estimation
disp	Display summary information for state-space model

Estimate State Variables

filter	Forward recursion of state-space models
smooth	Backward recursion of state-space models
update	Real-time state update by state-space model Kalman filtering

Characterize Dynamic Behavior

irf	Impulse response function (IRF) of state-space model
irfplot	Plot impulse response function (IRF) of state-space model
fevd	Generate forecast error variance decomposition (FEVD) of state-space model
corr	Model-implied temporal correlations of state-space model

Generate Monte Carlo Simulations

simulate	Monte Carlo simulation of state-space models
simsmooth	State-space model simulation smoother

Generate Minimum Mean Square Error Forecasts

forecast

Forecast states and observations of state-space models

Examples

Explicitly Create Standard State-Space Model with Known and Unknown Parameters

Create a standard state-space model containing two independent, autoregressive states, and the observations are the deterministic sum of the two states. Symbolically, the system of equations is

$$\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\end{array}\right]=\left[\begin{array}{cc}{\varphi }_1& 0\\ 0& {\varphi }_2\end{array}\right]\left[\begin{array}{c}{x}_{t-1,1}\\ x_{t-1,2}\end{array}\right]+\left[\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_2\end{array}\right]\left[\begin{array}{c}{u}_{t,1}\\ u_{t,2}\end{array}\right]$$

$$y_t=\left[\begin{array}{cc}1& 1\end{array}\right]\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\end{array}\right].$$

Specify the state-transition matrix.

A = [NaN 0; 0 NaN];

Specify the state-disturbance-loading matrix.

B = [NaN 0; 0 NaN];

Specify the measurement-sensitivity matrix.

C = [1 1];

Define the state-space model using ssm.

Mdl = ssm(A,B,C)

Mdl = 
State-space model type: ssm

State vector length: 2
Observation vector length: 1
State disturbance vector length: 2
Observation innovation vector length: 0
Sample size supported by model: Unlimited
Unknown parameters for estimation: 4

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...
Unknown parameters: c1, c2,...

State equations:
x1(t) = (c1)x1(t-1) + (c3)u1(t)
x2(t) = (c2)x2(t-1) + (c4)u2(t)

Observation equation:
y1(t) = x1(t) + x2(t)

Initial state distribution:

Initial state means are not specified.
Initial state covariance matrix is not specified.
State types are not specified.

Mdl is an ssm model containing unknown parameters. A detailed summary of Mdl prints to the Command Window.

It is good practice to verify that the state and observation equations are correct. If the equations are not correct, then it might help to expand the state-space equation manually.

Explicitly Create State-Space Model with Observation Error

Create a state-space model containing two independent, autoregressive states, and the observations are the sum of the two states, plus Gaussian error. Symbolically, the equation is

$$\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\end{array}\right]=\left[\begin{array}{cc}{\varphi }_1& 0\\ 0& {\varphi }_2\end{array}\right]\left[\begin{array}{c}{x}_{t-1,1}\\ x_{t-1,2}\end{array}\right]+\left[\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_2\end{array}\right]\left[\begin{array}{c}{u}_{t,1}\\ u_{t,2}\end{array}\right]$$

$$y_t=\left[\begin{array}{cc}1& 1\end{array}\right]\left[\begin{array}{c}{x}_{t,1}\\ x_{t,2}\end{array}\right]+{\sigma }_3{\epsilon }_t.$$

Define the state-transition matrix.

A = [NaN 0; 0 NaN];

Define the state-disturbance-loading matrix.

B = [NaN 0; 0 NaN];

Define the measurement-sensitivity matrix.

C = [1 1];

Define the observation-innovation matrix.

D = NaN;

Create the state-space model using ssm.

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

Mdl = 
State-space model type: ssm

State vector length: 2
Observation vector length: 1
State disturbance vector length: 2
Observation innovation vector length: 1
Sample size supported by model: Unlimited
Unknown parameters for estimation: 5

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...
Unknown parameters: c1, c2,...

State equations:
x1(t) = (c1)x1(t-1) + (c3)u1(t)
x2(t) = (c2)x2(t-1) + (c4)u2(t)

Observation equation:
y1(t) = x1(t) + x2(t) + (c5)e1(t)

Initial state distribution:

Initial state means are not specified.
Initial state covariance matrix is not specified.
State types are not specified.

Mdl is an ssm model containing unknown parameters. A detailed summary of Mdl prints to the Command Window.

It is good practice to verify that the state and observations equations are correct. If the equations are not correct, then it might help to expand the state-space equation manually.

Pass the data and Mdl to estimate to estimate the parameters.

Create Known State-Space Model with Initial State Values

Create a state-space model, where the state equation is an AR(2) model. The state disturbances are mean zero Gaussian random variables with standard deviation of 0.3. The observation equation is the difference between the current and previous state plus a mean zero Gaussian observation innovation with a standard deviation of 0.1. Symbolically, the state-space model is

$$\left[\begin{array}{c}{x}_{1,t}\\ x_{2,t}\\ x_{3,t}\end{array}\right]=\left[\begin{array}{ccc}0.6& 0.2& 0.5\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ x_{2,t-1}\\ x_{3,t-1}\end{array}\right]+\left[\begin{array}{c}0.3\\ 0\\ 0\end{array}\right]u_{1,t}$$

$$y_t=\left[\begin{array}{ccc}1& -1& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t}\\ x_{2,t}\\ x_{3,t}\end{array}\right]+0.1{\epsilon }_t.$$

There are three states: $$x_{1,t}$$ is the AR(2) process, $$x_{2,t}$$ represents $$x_{1,t-1}$$, and $$x_{3,t}$$ is the AR(2) model constant.

Define the state-transition matrix.

A = [0.6 0.2 0.5; 1 0 0; 0 0 1];

Define the state-disturbance-loading matrix.

B = [0.3; 0; 0];

Define the measurement-sensitivity matrix.

C = [1 -1 0];

Define the observation-innovation matrix.

D = 0.1;

Use ssm to create the state-space model. Set the initial-state mean and covariance matrix. Identify the type of initial state distributions by noting the following:

• $$x_{1,t}$$ is a stationary, AR(2) process.

• $$x_{2,t}$$ is also a stationary, AR(2) process.

• $$x_{3,t}$$ is the constant 1 for all periods.

mean0 = [0; 0; 1];                                        % The mean of the AR(2)
varAR2 = 0.3*(1 - 0.2)/((1 + 0.2)*((1 - 0.2)^2 - 0.6^2)); % The variance of the AR(2)
Cov1AR2 = 0.6*0.3/((1 + 0.2)*((1 - 0.2)^2) - 0.6^2);      % The covariance of the AR(2)
Cov0Mat = zeros(3);
Cov0Mat(1:2,1:2) = varAR2*eye(2) + Cov1AR2*flip(eye(2));
stateType = [0; 0; 1];
Mdl = ssm(A,B,C,D,Mean0=mean0,Cov0=Cov0Mat,StateType=stateType)

Mdl = 
State-space model type: ssm

State vector length: 3
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 equations:
x1(t) = (0.60)x1(t-1) + (0.20)x2(t-1) + (0.50)x3(t-1) + (0.30)u1(t)
x2(t) = x1(t-1)
x3(t) = x3(t-1)

Observation equation:
y1(t) = x1(t) - x2(t) + (0.10)e1(t)

Initial state distribution:

Initial state means
 x1  x2  x3 
  0   0   1 

Initial state covariance matrix
     x1    x2    x3 
 x1  0.71  0.44   0 
 x2  0.44  0.71   0 
 x3   0     0     0 

State types
     x1          x2         x3    
 Stationary  Stationary  Constant 

Mdl is an ssm model object.

You can display properties of Mdl using dot notation. For example, display the initial state covariance matrix.

Mdl.Cov0

ans = 3×3

    0.7143    0.4412         0
    0.4412    0.7143         0
         0         0         0

Implicitly Create Time-Invariant State-Space Model

Use a parameter mapping function to create a time-invariant state-space model, where the state model is AR(1) model. The states are observed with bias, but without random error. Set the initial state mean and variance, and specify that the state is stationary.

Write a function that specifies how the parameters in params map to the state-space model matrices, the initial state values, and the type of state. Symbolically, the model is

% Copyright 2015 The MathWorks, Inc.

function [A,B,C,D,Mean0,Cov0,StateType] = timeInvariantParamMap(params)
% Time-invariant state-space model parameter mapping function example. This
% function maps the vector params to the state-space matrices (A, B, C, and
% D), the initial state value and the initial state variance (Mean0 and
% Cov0), and the type of state (StateType). The state model is AR(1)
% without observation error.
    varu1 = exp(params(2)); % Positive variance constraint
    A = params(1);
    B = sqrt(varu1);
    C = params(3);
    D = [];
    Mean0 = 0.5;
    Cov0 = 100;
    StateType = 0;
end

Save this code as a file named timeInvariantParamMap.m to a folder on your MATLAB® path.

Create the state-space model by passing the function timeInvariantParamMap as a function handle to ssm.

Mdl = ssm(@timeInvariantParamMap);

ssm implicitly creates the state-space model. Usually, you cannot verify implicitly defined state-space models.

Convert Diffuse to Standard State-Space Model

If you estimate, filter, or smooth a diffuse state-space model containing at least one diffuse state, then the software uses the diffuse Kalman filter. To use the standard Kalman filter instead, convert the diffuse state-space model to a standard state-space model. ssm attributes a large initial state variance (1e7) for diffuse states. A standard state-space model treatment results in an approximation to the results of the diffuse Kalman filter. However, estimate uses all of the data to fit the model, and filter and smooth return filtered and smoothed estimates for all periods, respectively.

Explicitly create a one-dimensional diffuse state-space model. Specify that the first state equation is $$x_t=x_{t-1}+u_t$$, and that the observation model is $$y_t=x_t+{\epsilon }_t$$.

A = 1;   
B = 1; 
C = 1; 
D = 1;
MdlDSSM = dssm(A,B,C,D)

MdlDSSM = 
State-space model type: dssm

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) = x1(t-1) + u1(t)

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

Initial state distribution:

Initial state means
 x1 
  0 

Initial state covariance matrix
     x1  
 x1  Inf 

State types
    x1   
 Diffuse 

MdlDSSM is a dssm model object. Because the model does not contain any unknown parameters, dssm infers the initial state distribution and its parameters. In particular, the initial state variance is Inf because the nonstationary state has a diffuse distribution by default.

Convert MdlDSSM to a standard state-space model.

Mdl = ssm(MdlDSSM)

Mdl = 
State-space model type: ssm

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) = x1(t-1) + u1(t)

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

Initial state distribution:

Initial state means
 x1 
  0 

Initial state covariance matrix
     x1    
 x1  1e+07 

State types
    x1   
 Diffuse 

Mdl is an ssm model object. The structures of Mdl and MdlDSSM are equivalent, except that the initial state variance of the state in Mdl is 1e7.

To see the difference between the two models, simulate 10 periods of data from a state-space model that is similar to Mdl, except it has known initial state mean of 5 and variance 2.

Mdl0 = ssm(A,B,C,D,Mean0=5,Cov0=2,StateType=2);
T = 10;
rng(1); % For reproducibility
y = simulate(Mdl0,T);

Obtain filtered and smoothed states from Mdl and MdlDSSM using the simulated data.

fY = filter(Mdl,y);
fDY = filter(MdlDSSM,y);
sY = smooth(Mdl,y);
sDY = smooth(MdlDSSM,y);

Plot the filtered and smoothed state estimates.

figure
plot(1:T,y,"-o",1:T,fY,"-d",1:T,fDY,"-*");
title("Filtered State Estimates")
legend("Simulated data","State estimates, Mdl","State estimates, MdlDSSM");

figure
plot(1:T,y,"-o",1:T,sY,"-d",1:T,sDY,"-*");
title("Smoothed State Estimates")
legend("Simulated data","State estimates, Mdl","State estimates, MdlDSSM");

Besides apparent transient behavior, the filtered and smoothed states between the standard and diffuse state-space models appear nearly equivalent. The slight difference occurs because filter and smooth set all diffuse state estimates in the diffuse state-space model to 0 while they implement the diffuse Kalman filter. Once the covariance matrices of the smoothed states attain full rank, filter and smooth switch to using the standard Kalman filter. In this case, the switching time occurs after the first period.

More About

Static State

A static state does not change in value throughout the sample, that is, $$P\left(x_{t+1}=x_t\right)=1$$ for all t = 1,...,T.

Tips

Specify ParamMap in a more general or complex setting, where, for example:

• The initial state values are parameters.

• In time-varying models, you want to use the same parameters for more than one period.

• You want to impose parameter constraints.

Algorithms

• Default values for Mean0 and Cov0:

  • If you explicitly specify the state-space model (that is, you provide the coefficient matrices A, B, C, and optionally D), then:

    • For stationary states, the software generates the initial value using the stationary distribution. If you provide all values in the coefficient matrices (that is, your model has no unknown parameters), then ssm generates the initial values. Otherwise, the software generates the initial values during estimation.

    • For states that are always the constant 1, ssm sets Mean0 to 1 and Cov0 to 0.

    • For diffuse states, the software sets Mean0 to 0 and Cov0 to 1e7 by default.

  • If you implicitly create the state-space model (that is, you provide the parameter vector to the coefficient-matrices-mapping function ParamMap), then the software generates any initial values during estimation.

  • For nonstationary states, ssm sets Cov0 to 1e7 by default. Subsequently, the software implements the Kalman filter for filtering, smoothing, and parameter estimation. This specification imposes relatively weak knowledge on the initial state values of diffuse states, and uses initial state covariance terms between all states.

• For static states that do not equal 1 throughout the sample, the software cannot assign a value to the degenerate, initial state distribution. Therefore, set static states to 2 using the property StateType. Subsequently, the software treats static states as nonstationary and assigns the static state a diffuse initial distribution.

• It is best practice to set StateType for each state. By default, the software generates StateType, but this behavior might not be accurate. For example, the software cannot distinguish between a constant 1 state and a static state.

• The software cannot infer StateType from data because the data theoretically comes from the observation equation. The realizations of the state equation are unobservable.

• ssm models do not store observed responses or predictor data. Supply the data wherever necessary using the appropriate inputs.

• 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 you create 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.

Alternatives

• If the states are observable, and the state equation resembles any of the following models, use the associated function instead.

  • ARIMA model, then you can specify an arima model instead.

  • Regression model with ARIMA errors, then you can specify a regARIMA model instead.

  • Conditional variance model, then you can specify a garch, egarch, or gjr model instead.

  • VAR model, then you can estimate such a model using varm and estimate.

• To impose no prior knowledge on the initial state values of diffuse states, and to implement the diffuse Kalman filter, create a dssm model object instead of an ssm model object.

• For a Bayesian view of a standard state-space model, use bssm.

References

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

Version History

Introduced in R2014a