**Class: **ssm

State-space model simulation smoother

`X = simsmooth(Mdl,Y)`

`X = simsmooth(Mdl,Y,Name,Value)`

returns
simulated states (`X`

= simsmooth(`Mdl`

,`Y`

)`X`

) by applying a simulation smoother to
the time-invariant or time-varying state-space model (`Mdl`

)
and responses (`Y`

). That is, the software uses forward
filtering and back sampling to obtain one random path from the posterior
distribution of the states.

returns
simulated states with additional options specified by one or more `X`

= simsmooth(`Mdl`

,`Y`

,`Name,Value`

)`Name,Value`

pair
arguments.

`Mdl`

— Standard state-space model`ssm`

model objectStandard state-space model, specified as an`ssm`

model
object returned by `ssm`

or `estimate`

. A
standard state-space model has finite initial state covariance matrix
elements. That is, `Mdl`

cannot be a `dssm`

model object.

If `Mdl`

is not fully specified (that is, `Mdl`

contains
unknown parameters), then specify values for the unknown parameters
using the `'`

`Params`

`'`

`Name,Value`

pair
argument. Otherwise, the software throws an error.

`Y`

— Observed response datanumeric matrix | cell vector of numeric vectors

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, where each row corresponds to a period and each column corresponds to a particular observation in the model.*T*is the sample size and*m*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. Each element of the cell vector corresponds to a period and contains an*n*-dimensional vector of observations for that period. The corresponding dimensions of the coefficient matrices in_{t}`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.

`NaN`

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

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'NumOut'`

— Number of output arguments of parameter-to-matrix mapping functionpositive integer

Number of output arguments of the parameter-to-matrix mapping
function for implicitly defined state-space models, specified as the
comma-separated pair consisting of `'NumOut'`

and
a positive integer.

If you implicitly define a state-space model and you do not
supply `NumOut`

, then the software automatically
detects the number of output arguments of the parameter-to-matrix
mapping function. Such detection consumes extra resources, and might
slow the simulation smoother.

For explicitly defined models, the software ignores `NumOut`

and
displays a warning message.

`'NumPaths'`

— Number of sample paths to generate variants`1`

(default) | positive integerNumber of sample paths to generate variants, specified as the
comma-separated pair consisting of `'NumPaths'`

and
a positive integer.

**Example: **`'NumPaths',1000`

**Data Types: **`double`

`'Params'`

— Values for unknown parametersnumeric vector

Values for unknown parameters in the state-space model, specified
as the column-separated pair consisting of `'Params'`

and
a numeric vector.

The elements of `Params`

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`NaN`

s in the state-space model matrices and initial state values. The software searches for`NaN`

s column-wise following the order`A`

,`B`

,`C`

,`D`

,`Mean0`

, and`Cov0`

.If you created

`Mdl`

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

If `Mdl`

contains unknown parameters,
then you must specify their values. Otherwise, the software ignores
the value of `Params`

.

**Data Types: **`double`

`'Tolerance'`

— Forecast uncertainty threshold`0`

(default) | nonnegative scalarForecast 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`

`X`

— Simulated statesnumeric matrix | cell matrix of numeric vectors

Simulated states, returned as a numeric matrix or cell matrix of vectors.

If `Mdl`

is a time-invariant model with respect
to the states, then `X`

is a `numObs`

-by-*m*-by-`numPaths`

array.
That is, each row corresponds to a period, each column corresponds
to a state in the model, and each page corresponds to a sample path.
The last row corresponds to the latest simulated states.

If `Mdl`

is a time-varying model with respect
to the states, then `X`

is a `numObs`

-by-`numPaths`

cell
matrix of vectors. `X{t,j}`

contains a vector of
length *m _{t}* of simulated states
for period

`X`

contains the latest set of simulated
states.Suppose that a latent process is an AR(1) model. Subsequently, the state equation is

$${x}_{t}=0.5{x}_{t-1}+{u}_{t},$$

where $${u}_{t}$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $${x}_{t}$$, 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. Subsequently, the observation equation is

$${y}_{t}={x}_{t}+{\epsilon}_{t},$$

where $${\epsilon}_{t}$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

Use the random latent state process (`x`

) and the observation equation to generate observations.

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5; B = 1; C = 1; D = 0.75;

Specify the state-space model using the coefficient matrices.

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

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) = (0.50)x1(t-1) + u1(t) Observation equation: y1(t) = x1(t) + (0.75)e1(t) Initial state distribution: Initial state means x1 0 Initial state covariance matrix x1 x1 1.33 State types x1 Stationary

`Mdl`

is an `ssm`

model. Verify that the model is correctly specified using the display in the Command Window. The software infers that the state process is stationary. Subsequently, the software sets the initial state mean and covariance to the mean and variance of the stationary distribution of an AR(1) model.

Simulate one path each of states and observations. Specify that the paths span 100 periods.

simX = simsmooth(Mdl,y);

`simX`

is a 100-by-1 vector of simulated states.

Plot the true state values with the simulated states.

figure; plot(1:T,x,'-k',1:T,simX,':r','LineWidth',2); title 'True State Values and Simulated States'; xlabel 'Period'; ylabel 'State'; legend({'True state values','Simulated state values'});

By default, `simulate`

simulates one path for each state in the state-space model. To conduct a Monte Carlo study, specify to simulate a large number of paths using the `'NumPaths'`

name-value pair argument.

The `simsmooth`

function draws random samples from the distribution of smoothed states, or the distribution of a state given all of the data and parameters. This is the definition of posterior distribution of a state. Suppose that a latent process is an AR(1). Subsequently, the state equation is

$${x}_{t}=0.5{x}_{t-1}+{u}_{t},$$

where $${u}_{t}$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $${x}_{t}$$, 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. Subsequently, the observation equation is

$${y}_{t}={x}_{t}+{\epsilon}_{t},$$

where $${\epsilon}_{t}$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

Use the random latent state process (`x`

) and the observation equation to generate observations.

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5; B = 1; C = 1; D = 0.75;

Specify the state-space model using the coefficient matrices.

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

Smooth the states of the state space model.

xsmooth = smooth(Mdl,y);

Draw 1000 paths from the posterior distribution of $${x}_{1}$$.

```
N = 1000;
SimX = simsmooth(Mdl,y,'NumPaths',N);
```

`SimX`

is a `100`

-by- `1`

-by- `1000`

array. Rows correspond to periods, columns correspond to individual states, and leaves correspond to separate paths.

Because `SimX`

has a singleton dimension, collapse it so that its leaves correspond to the columns using `squeeze`

.

SimX = squeeze(SimX);

Compute the mean, standard deviation, and 95% confidence intervals of the state at each period.

xbar = mean(SimX,2); xstd = std(SimX,[],2); ci = [xbar - 1.96*xstd, xbar + 1.96*xstd];

Plot the smoothed states, and the means and 95% confidence intervals of the draws at each period.

figure; plot(xsmooth,'k','LineWidth',2); hold on; plot(xbar,'--r','LineWidth',2); plot(1:T,ci(:,1),'--r',1:T,ci(:,2),'--r'); legend('Smoothed states','Simulation Mean','95% CIs'); title('Smooth States and Simulation Statistics'); xlabel('Period')

The *simulation smoother* is
an algorithm for drawing samples from the conditional, joint, posterior
distribution of the states given the complete observed response series.
You can use these random draws to conduct a simulation study of the
estimators.

For a univariate, time-invariant state-space model, the simulation smoother algorithm follows these steps.

Obtains the smoothed states ($${\widehat{x}}_{t};\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=1,\mathrm{..},T$$) using the Kalman filter.

Chooses initial state mean and variance values. Draw the initial random state $${x}_{0}^{\ast}$$ from the Gaussian distribution with the initial state mean and variance.

Randomly generates

*T*state disturbances and observation innovations from the standard normal distribution. Denote the random variants for period*t*$${\epsilon}_{t}^{\ast}$$ and $${u}_{t}^{\ast}$$, respectively.Creates random observations and states by plugging $${\epsilon}_{t}^{\ast}$$ and $${u}_{t}^{\ast}$$ into the state-space model

$$\begin{array}{c}{x}_{t}^{\ast}=A{x}_{t-1}^{\ast}+B{\epsilon}_{t}^{\ast}\\ {y}_{t}^{\ast}=C{x}_{t}^{\ast}+D{u}_{t}^{\ast}\end{array}.$$

Obtains smoothed states ($${\widehat{x}}_{t}^{\ast}$$) by applying the Kalman filter to the state-space model using the observation series $${y}_{t}^{\ast}$$.

Obtains the random path of smoothed states from the posterior distribution using

$${\tilde{x}}_{t}={\widehat{x}}_{t}-{\widehat{x}}_{t}^{\ast}+{x}_{t}^{\ast}.$$

For more details, see [1].

For increased speed in simulating states, the simulation smoother implements minimal dimensionality error checking. Therefore, for models with unknown parameter values, you should ensure that the dimensions of the data and the dimensions of the coefficient matrices are consistent.

[1] Durbin J., and S. J. Koopman. “A
Simple and Efficient Simulation Smoother for State Space Time Series
Analysis.” *Biometrika*. Vol 89., No.
3, 2002, pp. 603–615.

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

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