simsmooth
Bayesian nonlinear non-Gaussian state-space model simulation smoother
Since R2024a
Syntax
Description
simsmooth
provides random paths of states drawn from the
posterior smoothed state distribution, which is the distribution of the states conditioned on
model parameters Θ and the full-sample response data, of a Bayesian nonlinear non-Gaussian
state-space model (bnlssm
).
To draw state paths from the posterior smoothed state distribution,
simsmooth
uses the nonlinear forward-filtering,
backward-sampling method (FFBS), in which it implements sequential Monte Carlo
(SMC) to perform forward filtering, and then it resamples and reweights
particles (weighted random samples) generated by SMC to perform
backward sampling.
returns a randomly drawn path of states, simulated from the posterior smoothed state
distribution, by applying the simulation smoother to the input Bayesian nonlinear
non-Gaussian state-space model and responses data. X
= simsmooth(Mdl
,Y
,params
)simsmooth
uses
FFBS to obtain the random path from the posterior smoothed state distribution.
simsmooth
evaluates the parameter map
Mdl.ParamMap
by using the input vector of parameter values.
specifies additional options using one or more name-value arguments. For example,
X
= simsmooth(Mdl
,Y
,params
,Name=Value
)simsmooth(Mdl,Y,params,NumPaths=1e4,Resample="residual")
specifies
generating 1e4
random paths and to resample residuals.
[
additionally returns the following quantities using any of the input-argument combinations
in the previous syntaxes:X
,OutputFilter
,x0
] = simsmooth(___)
OutputFilter
— SMC forward filtering results by sampling time containing the following quantities:Approximate loglikelihood values associated with the input data, input parameters, particles, and posterior state filtering distribution
Filter estimate of state-distribution means
Filter estimate of state-distribution covariance
State particles and corresponding weights that approximate the filtering distribution
Effective sample size
Flags indicating which data the software used to filter
Flags indicating resampling
x0
— Simulation-smoothed initial state, computed only when you request the this output
Examples
Draw Path from Posterior Smoothed State Distribution
This example draws a random path from the approximate posterior smoothed state distribution of the Bayesian nonlinear state-space model in equation. The state-space model contains two independent, stationary, autoregressive states each with a model constant. The observations are a nonlinear function of the states with Gaussian noise. The prior distribution of the parameters is flat. Symbolically, the system of equations is
and are the unconditional means of the corresponding states. The initial distribution moments of each state are their unconditional mean and covariance.
Create a Bayesian nonlinear state-space model characterized by the system. The observation equation is in equation form, that is, the function composing the states is nonlinear and the innovation series is additive, linear, and Gaussian. The Local Functions section contains two functions required to specify the Bayesian nonlinear state-space model: the state-space model parameter mapping function and the prior distribution of the parameters. You can use the functions only within this script.
Mdl = bnlssm(@paramMap,@priorDistribution)
Mdl = bnlssm with properties: ParamMap: @paramMap ParamDistribution: @priorDistribution ObservationForm: "equation" Multipoint: [1x0 string]
Mdl
is a bnlssm
model specifying the state-space model structure and prior distribution of the state-space model parameters. Because Mdl
contains unknown values, it serves as a template for posterior analysis with observations.
Simulate a series of 100 observations from the following stationary 2-D VAR process.
where the disturbance series are standard Gaussian random variables.
rng(1,"twister") % For reproducibility T = 100; thetatrue = [0.9; 1; -0.75; -1; 0.3; 0.2; 0.1]; MdlSim = varm(AR={diag(thetatrue([1 3]))},Covariance=diag(thetatrue(5:6).^2), ... Constant=thetatrue([2 4])); XSim = simulate(MdlSim,T);
Compose simulated observations using the following equation.
where the innovation series is a standard Gaussian random variable.
ysim = log(sum(exp(XSim - mean(XSim)),2)) + thetatrue(7)*randn(T,1);
To draw from the approximate posterior smoothed state distribution, the simsmooth
function requires response data and a model with known state-space model parameters. Choose a random set with the following constraints:
and are within the unit circle. Use to generate values.
and are real numbers. Use the distribution to generate values.
, , and are positive real numbers. Use the distribution to generate values.
theta13 = (-1+(1-(-1)).*rand(2,1)); theta24 = 3*randn(2,1); theta567 = chi2rnd(1,3,1); theta = [theta13(1); theta24(1); theta13(2); theta24(2); theta567];
Draw a random path from the approximate smoothed state posterior distribution by passing the Bayesian nonlinear model, simulated data, and parameter values to simsmooth
.
SmoothX = simsmooth(Mdl,ysim,theta); size(SmoothX)
ans = 1×2
100 4
SmoothX
is a 100-by-4 matrix containing one path drawn from the approximate posterior smoothed state distribution, with rows corresponding to periods in the sample and columns corresponding to the state variables. The simsmooth
function uses the FFBS method (SMC and a bootstrap to resample particles and weights) to obtain draws from the posterior smoothed state distribution.
Draw another path, but specify used to simulate the data.
SmoothXSim = simsmooth(Mdl,ysim,thetatrue);
Plot the two paths with the true state values.
figure tiledlayout(2,1) nexttile plot([SmoothX(:,1) SmoothXSim(:,1) XSim(:,1)]) title("x_{t,1}") legend("Smoothed State, random \theta","Smoothed State, true \theta","XSim") nexttile plot([SmoothX(:,3) SmoothXSim(:,3) XSim(:,2)]) title("x_{t,3}") legend("Smoothed State, random \theta","Smoothed State, true \theta","XSim")
The paths using the true value of and the simulated state paths are close. The paths generated from the random value of is far from the simulated state paths.
Local Functions
These functions specify the state-space model parameter mappings, in equation form, and log prior distribution of the parameters.
function [A,B,C,D,Mean0,Cov0,StateType] = paramMap(theta) A = @(x)blkdiag([theta(1) theta(2); 0 1],[theta(3) theta(4); 0 1])*x; B = [theta(5) 0; 0 0; 0 theta(6); 0 0]; C = @(x)log(exp(x(1)-theta(2)/(1-theta(1))) + ... exp(x(3)-theta(4)/(1-theta(3)))); D = theta(7); Mean0 = [theta(2)/(1-theta(1)); 1; theta(4)/(1-theta(3)); 1]; Cov0 = diag([theta(5)^2/(1-theta(1)^2) 0 theta(6)^2/(1-theta(3)^2) 0]); StateType = [0; 1; 0; 1]; % Stationary state and constant 1 processes end function logprior = priorDistribution(theta) paramconstraints = [(abs(theta([1 3])) >= 1) (theta(5:7) <= 0)]; if(sum(paramconstraints)) logprior = -Inf; else logprior = 0; % Prior density is proportional to 1 for all values % in the parameter space. end end
Estimate Model Parameters Using Gibbs Sampler
This example shows how to draw from the posterior distribution of smoothed states and model parameters by using a Gibbs sampler. Consider this nonlinear state-space model
where the parameters in have the following priors:
, that is, a truncated normal distribution with .
, that is, an inverse gamma distribution with shape and scale .
, that is, a gamma distribution with shape and scale .
Simulate Series
Consider this data-generating process (DGP).
where the series is a standard Gaussian series of random variables.
Simulate a series of 200 observations from the process.
rng(500,"twister") % For reproducibility T = 200; thetaDGP = [0.7; 0.2; 3]; numparams = numel(thetaDGP); MdlXSim = arima(AR=thetaDGP(1),Variance=thetaDGP(2), ... Constant=0); xsim = simulate(MdlXSim,T); y = random("poisson",thetaDGP(3)*exp(xsim)); figure plot(y)
Create Bayesian Nonlinear Model
The Local Functions section contains the functions paramMap
and logPrior
required to specify the Bayesian nonlinear state-space model. The paramMap
function specifies the state-space model structure and initial state moments (chosen arbitrarily). The priorDistribution
function returns the log of the joint prior distribution of the state-space model parameters. You can use the functions only within this script.
Create a Bayesian nonlinear state-space model for the DGP. Indicate that the state-space model observation equation is expressed as a distribution. To speed up computations, the arguments A
and LogY
of the paramMap
function are written to enable simultaneous evaluation of the transition and observation densities of multiple particles. Specify this characteristic by using the Multipoint
name-value argument.
Mdl = bnlssm(@paramMap,@priorDistribution,ObservationForm="distribution", ... Multipoint=["A" "LogY"]);
Perform Gibbs Sampling
A Gibbs sampler is an Markov chain Monte Carlo method for drawing a sample from the joint posterior distribution of parameters. It successively draws from the full conditional distributions of the states and model parameters, one at a time; results of previous draws are substituted, which enables the Markov chains to explore the parameter space.
The full conditional distribution of the states is the posterior of the smoothed state distribution, which simsmooth
computes. The remaining full conditionals are:
, where and.
, where and .
, where and .
You can view the joint posterior distribution as the semiconjugate Bayesian linear regression model , where the response series , predictor series , and the error series . If you view the problem this way, you can speed up sampler. During sampling, you can reject any draws outside its support.
The Local Functions section contains the following functions:
phiFC
— Draws fromsigma2FC
— Draws fromlambdaFC
— Draws from
Conduct the Gibbs sampler. Draw a sample of 2000 from the full conditional distributions. Draw 1500 particles for each call of simsmooth
. Perform rejection sampling at a maximum of 50 iterations. This example chooses the initial conditions arbitrarily.
nGibbs = 2000; Gibbs = zeros(T+numparams,nGibbs); % Preallocate for state and parameter draws numparticles = 1500; maxiterations = 50; % Initial values % theta theta0 = [0.5; 0.1; 2]; % pi(phi,sigma2) hyperparameters m0 = 0; v02 = 1; a0 = 1; b0 = 1; MdlBLM = semiconjugateblm(1,Intercept=0,Mu=m0,V=v02, ... A=a0,B=b0); % lambda hyperparameters alpha0 = 3; beta0 = 1; hyperparams = [m0 v02 a0 b0 alpha0 beta0]; % Prepare wait bar dialog box wb = waitbar(0,"1",Name="Running Gibbs Sampler ...", ... CreateCancelBtn="setappdata(gcbf,Canceling=true)"); setappdata(wb,Canceling=false); % Gibbs sampler theta = theta0; for j = 1:nGibbs % Press Cancel in the dialog to break. if getappdata(wb,"Canceling") fprintf("Gibbs sampler canceled") break end waitbar(j/nGibbs,wb,sprintf("Draw %d of %d",j,nGibbs)); Gibbs(1:T,j) = simsmooth(Mdl,y,theta,NumParticles=numparticles, ... MaxIterations=maxiterations); [Gibbs(T+1,j),Gibbs(T+2,j)] = blmFC(Gibbs(1:T,j),MdlBLM,theta); Gibbs(T+3,j) = lambdaFC(Gibbs(1:T,j),y,hyperparams); theta = Gibbs(T+(1:3),j); end delete(wb)
Describe
To reduce the influence of initial conditions on the sample, remove the first 50 draws from the Gibbs sample. To reduce the influence of serial correlation in the sample, thin the sample by keeping every fourth draw.
burnin = 50; thin = 4; pphi = Gibbs(T+1,burnin:thin:end); psigma2 = Gibbs(T+2,burnin:thin:end); plambda = Gibbs(T+3,burnin:thin:end);
Plot trace plots of the Gibbs sampler.
figure h = tiledlayout(3,1); nexttile plot(pphi) title("\pi(\phi|y)") nexttile plot(psigma2) title("\pi(\sigma^2|y)") nexttile plot(plambda) title("\pi(\lambda|y)") title(h,"Trace Plots")
The trace plots show that the Markov chains are mixing adequately.
Summarize the posterior distributions by computing sample medians and 95% percentile intervals of the processed posterior draws. Plot histograms of the posterior distributions of the model parameters.
mphi = median(pphi); ciphi = quantile(pphi,[0.025 0.975]); msigma2 = median(psigma2); cisigma2 = quantile(psigma2,[0.025 0.975]); mlambda = median(plambda); cilambda = quantile(plambda,[0.025 0.975]); figure h = tiledlayout(3,1); nexttile histogram(pphi) title(sprintf("\\pi(\\phi|y), median=%.3f, ci=(%.3f, %.3f)",mphi,ciphi)) nexttile histogram(psigma2) title(sprintf("\\pi(\\sigma^2|y), median=%.3f, ci=(%.3f, %.3f)",msigma2,cisigma2)) nexttile histogram(plambda) title(sprintf("\\pi(\\lambda|y), median=%.3f, ci=(%.3f, %.3f)",mlambda,cilambda)) title(h,"Posterior Histograms")
The posterior medians are close to their DGP counterparts.
Local Functions
These functions specify the state-space model parameter mappings, in distribution form, the log prior distribution of the parameters, and random draws from full conditional distribution of each parameter.
function [A,B,LogY,Mean0,Cov0,StateType] = paramMap(theta) A = theta(1); B = sqrt(theta(2)); LogY = @(y,x)y.*x - exp(x).*theta(3); Mean0 = 0; Cov0 = 2; StateType = 0; % Stationary state process end function logprior = priorDistribution(theta,hyperparams) % Prior of phi m0 = hyperparams(1); v20 = hyperparams(2); pphi = makedist("normal",mu=m0,sigma=sqrt(v20)); pphi = truncate(pphi,-1,1); lpphi = log(pdf(pphi,theta(1))); % Prior of sigma2 a0 = hyperparams(3); b0 = hyperparams(4); lpsigma2 = -a0*log(b0) - log(gamma(a0)) + (-a0-1)*log(theta(2)) - ... 1./(b0*theta(2)); % Prior of lambda alpha0 = hyperparams(5); beta0 = hyperparams(6); plambda = makdist("gamma",alpha0,beta0); lplambda = log(pdf(plambda,theta(3))); logprior = lpphi + lpsigma2 + lplambda; end function [phi,sigma2] = blmFC(x,mdl,theta) % Reject sampled phi when it is outside the unit circle while true phi = simulate(mdl,x(1:end-1),x(2:end),Sigma2=theta(2)); if abs(phi) < 1 break end end [~,sigma2] = simulate(mdl,x(1:(end-1)),x(2:end),Beta=theta(1)); end function [lambda,alpha,beta] = lambdaFC(x,y,hyperparams) alpha0 = hyperparams(5); beta0 = hyperparams(6); alpha = sum(y) + alpha0; beta = beta0./(beta0*sum(exp(x))+1); lambda = gamrnd(alpha,beta,1); end
Monitor Underlying Sampling Procedures
simsmooth
runs SMC to forward filter the state-space model, which includes resampling particles. To assess the quality of the sample, including whether any posterior filtered state distribution is close to degenerate, you can monitor these algorithms by returning the third output of smooth
.
Consider this nonlinear state-space model.
and are the unconditional means of the corresponding states. The initial distribution moments of each state are their unconditional mean and covariance.
Simulate a series of 100 observations from the following stationary 2-D VAR process.
where the disturbance series and are standard Gaussian random variables.
rng(100,"twister") % For reproducibility T = 100; thetatrue = [0.9; 1; -0.75; -1; 0.3; 0.2; 0.1]; MdlSim = varm(AR={diag(thetatrue([1 3]))},Covariance=diag(thetatrue(5:6).^2), ... Constant=thetatrue([2 4])); XSim = simulate(MdlSim,T); y = log(sum(exp(XSim - mean(XSim)),2)) + thetatrue(7)*randn(T,1);
Create a Bayesian nonlinear state-space model. The Local Functions section contains the required functions specifying the Bayesian nonlinear state-space model structure and joint prior distribution.
Mdl = bnlssm(@paramMap,@priorDistribution);
Approximate the posterior smoothed state distribution of the state-space model. As in the Draw Path from Posterior Smoothed State Distribution example, choose a random set of initial parameter values. Specify the resampling residuals for the SMC. Return the forward-filtering results and the approximate initial smoothed state .
theta13 = (-1+(1-(-1)).*rand(2,1));
theta24 = 3*randn(2,1);
theta567 = chi2rnd(1,3,1);
theta = [theta13(1); theta24(1); theta13(2); theta24(2); theta567];
[~,OutputFilter,x0] = simsmooth(Mdl,y,theta,Resample="residual");
Output is a 100-by-1 structure array containing several fields, one set of fields for each observation, including:
FilteredStatesCov
— Approximate posterior filtered state distribution covariance for the states at each sampling timeDataUsed
— Whether the forward-filtering algorithm used an observation for posterior estimationResample
— Whether the forward-filtering algorithm resampled the particles associated with an observation
Plot the determinant of the approximate posterior filtered state covariance matrices for states that are not constant.
filteredstatecov = cellfun(@(x)det(x([1 3],[1 3])),{OutputFilter.FilteredStatesCov});
figure
plot(filteredstatecov)
title("Approx. Post. Filtered State Covariances")
Any covariance determinant that is close to 0 indicates a close-to-degenerate distribution. No covariance determinants in the analysis are close to 0.
Determine whether the forward-filtering algorithm omitted any observations from posterior estimation.
anyObsOmitted = sum([OutputFilter.DataUsed]) ~= T
anyObsOmitted = logical
0
anyObsOmitted = 0
indicates that the algorithm used all observations.
Determine whether filter
resampled any particles associated with observations.
whichResampled = numel(find([OutputFilter.Resampled] == true))
whichResampled = 75
The forward-filtering algorithm resampled particles associated with observations the 75 observations listed in whichResampled
.
Local Functions
These functions specify the state-space model parameter mappings, in equation form, and log prior distribution of the parameters.
function [A,B,C,D,Mean0,Cov0,StateType] = paramMap(theta) A = @(x)blkdiag([theta(1) theta(2); 0 1],[theta(3) theta(4); 0 1])*x; B = [theta(5) 0; 0 0; 0 theta(6); 0 0]; C = @(x)log(exp(x(1)-theta(2)/(1-theta(1))) + ... exp(x(3)-theta(4)/(1-theta(3)))); D = theta(7); Mean0 = [theta(2)/(1-theta(1)); 1; theta(4)/(1-theta(3)); 1]; Cov0 = diag([theta(5)^2/(1-theta(1)^2) 0 theta(6)^2/(1-theta(3)^2) 0]); StateType = [0; 1; 0; 1]; % Stationary state and constant 1 processes end function logprior = priorDistribution(theta) paramconstraints = [(abs(theta([1 3])) >= 1) (theta(5:7) <= 0)]; if(sum(paramconstraints)) logprior = -Inf; else logprior = 0; % Prior density is proportional to 1 for all values % in the parameter space. end end
Input Arguments
Mdl
— Bayesian nonlinear state-space model
bnlssm
model object
Y
— Observed response data
numeric matrix | cell vector of numeric vectors
Observed response data, specified as a numeric matrix or a cell vector of numeric vectors.
If
Mdl
is time invariant with respect to the observation equation,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. T is the sample size and n is the number of observations per period. The last row ofY
contains the latest observations.If
Mdl
is time varying with respect to the observation equation,Y
is a T-by-1 cell vector.Y{t}
contains an nt-dimensional vector of observations for period t, where t = 1, ..., T. For linear observation models, the corresponding dimensions of the coefficient matrices, outputs ofMdl.ParamMap
,C{t}
, andD{t}
must be consistent with the matrix inY{t}
for all periods. For nonlinear observation models, the dimensions of the inputs and outputs associated with the observations must be consistent. Regardless of model type, the last cell ofY
contains the latest observations.
NaN
elements indicate missing observations. For details on how the Kalman
filter accommodates missing observations, see Algorithms.
Data Types: double
| cell
params
— State-space model parameters Θ
numeric vector
State-space model parameters Θ to evaluate the parameter mapping Mdl.ParamMap
, specified as a numparams
-by-1 numeric vector. Elements of params0
must correspond to the elements of the first input arguments of Mdl.ParamMap
and Mdl.ParamDistribution
.
Data Types: double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: simsmooth(Mdl,Y,params,NumParticles=1e4,Resample="residual")
specifies generating 1e4
random paths and to use the residual-resampling
SMC method.
NumParticles
— Number of particles
1000
(default) | positive integer
Number of particles for SMC, specified as a positive integer.
Example: NumParticles=1e4
Data Types: double
NumPaths
— Number of sample state paths
1
(default) | positive integer
Number of sample state paths to draw from the posterior smoothed state distribution, specified as a positive integer.
Example: NumPaths=1e4
Data Types: double
MaxIterations
— Maximum number of rejection sampling iterations
10
(default) | nonnegative integer
Maximum number of rejection sampling iterations for posterior sampling using the simulation smoother, specified as a nonnegative integer. simsmooth
conducts rejection sampling before it conducts the computationally intensive importance sampling algorithm.
Example: MaxIterations=100
Data Types: double
Resample
— SMC resampling method
"multinomial"
(default) | "residual"
| "systematic"
SMC resampling method, specified as a value in this table.
Value | Description |
---|---|
"multinomial" | At time t, the set of previously generated particles (parent set) follows a standard multinomial distribution, with probabilities proportional to their weights. An offspring set is resampled with replacement from the parent set [1]. |
"residual" | Residual sampling, a modified version of multinomial resampling that can produce an estimator with lower variance than the multinomial resampling method [6]. |
"systematic" | Systematic sampling, which produces an estimator with lower variance than the multinomial resampling method [4]. |
Resampling methods downsample insignificant particles to achieve a smaller estimator variance than if no resampling is performed and to avoid sampling from a degenerate proposal [4].
Example: Resample="residual"
Data Types: char
| string
Cutoff
— Effective sample size threshold for resampling
0.5*NumParticles
(default) | nonnegative scalar
Effective sample size threshold, below which simsmooth
resamples particles, specified as a nonnegative scalar. For more details, see [4], Ch. 12.3.3.
Tip
To resample during every period, set
Cutoff=numparticles
, wherenumparticles
is the value of theNumParticles
name-value argument.To avoid resampling, set
Cutoff=0
.
Example: Cutoff=0.75*numparticles
Data Types: double
SortParticles
— Flag for sorting particles before resampling
false
(default) | true
Flag for sorting particles before resampling, specified as a value in this table.
Value | Description |
---|---|
true | simsmooth sorts the generated particles before resampling them. |
false | simsmooth does not sort the generated particles. |
When you set SortPartiles=true
, simsmooth
uses
Hilbert sorting during the SMC routine to sort the particles. This action can reduce
Monte Carlo variation, which is useful when you compare loglikelihoods resulting from
evaluating several params
arguments that are close to each other
[3]. However, the sorting routine
requires more computation resources, and can slow down computations, particularly in
problems with a high-dimensional state variable.
Example: SortParticles=true
Data Types: logical
RND
— Previously generated normal random numbers
structure array
Previously generated normal random numbers, as returned by
filter
, to reproduce simsmooth
results,
specified as the RND
output, a structure array, of previous
filter
call. Specify RND
to control the
random number generator.
The default is an empty structure array, which causes
simsmooth
to generate new random numbers.
Data Types: struct
Output Arguments
X
— Simulated paths of smoothed states
numeric matrix | 3-D numeric array | cell vector of numeric matrices
Simulated paths of smoothed states, drawn from the posterior smoothed state
distribution
p(xT,…,x0|yT,…,y1,Θ),
for t = 1,…,T, returned as a
T-by-m numeric matrix for one simulated path,
T-by-m-by-NumPaths
3-D
numeric array for NumPaths
simulated paths, or a
T-by-1 cell vector of numeric matrices.
Each row corresponds to a time point in the sample. The last row contains the latest simulated smoothed states.
If Mdl
is a time-invariant model with respect to the states,
each column of X
corresponds to a state in the model and each page
corresponds to a sample path.
If Mdl
is a time-varying model with respect to the states,
then, for each t
= 1,…,T, cell
X(
contains an
mt
)t
-by-NumPaths
matrix of simulated smoothed states. Each row corresponds to a state variable for the
corresponding time and each column corresponds to a simulated path. Each path across all
cells correspond.
OutputFilter
— SMC forward filtering results by sampling time
structure array
SMC forward filtering results by period, returned as a T-by-1 structure array with fields in this table, and where cell t corresponds to the filtering result for time t.
Field | Description | Estimate/Approximation of |
---|---|---|
LogLikelihood | Scalar approximate loglikelihood objective function value | log p(yt|y1,…,yt) |
FilteredStates | mt-by-1 vector of approximate filtered state estimates | |
FilteredStatesCov | mt-by-mt variance-covariance matrix of filtered states | |
CustomStatistics | (mt +
1)-by-NumParticles simulated particles and
corresponding weights that approximate the filtering distribution | N/A |
EffectiveSampleSize | Effective sample size for importance sampling, a scalar in
[0,NumParticles ] | N/A |
DataUsed | ht-by-1 flag indicating whether
the software filters using a particular observation. For example, if
observation j at time t is a
NaN , element j in
DataUsed at time t is
0 . | N/A |
Resampled | Flag indicating whether simsmooth resampled
particles | N/A |
x0
— Simulated paths of smoothed initial state vector
numeric vector | numeric matrix
Simulated paths of the smoothed initial state vector, drawn from the posterior
smoothed state distribution
p(xT,…,x0|yT,…,y1,Θ),
for t = 1,…,T, returned as an
m0-by-1 numeric vector for one simulated
path or an m0-by-NumPaths
numeric matrix for NumPaths
simulated paths.
Each row of x0
corresponds to a state in the model and each
column corresponds to a sample path.
If you do not request to return x0
,
simsmooth
does not compute it.
Tips
Smoothing has several advantages over filtering.
The smoothed state estimator is more accurate than the online filter state estimator because it is based on the full-sample data, rather than only observations up to the estimated sampling time.
A stable approximation to the gradient of the loglikelihood function, which is important for numerical optimization, is available from the smoothed state samples of the simulation smoother (finite differences of the approximated loglikelihood computed from the filter state estimates is numerically unstable).
You can use the simulation smoother to perform Bayesian estimation of the nonlinear state-space model via the Metropolis-within-Gibbs sampler.
Unless you set
Cutoff=0
,simsmooth
resamples particles according to the specified resampling methodResample
. Although resampling particles with high weights improves the results of the SMC, you should also allow the sampler traverse the proposal distribution to obtain novel, high-weight particles. To do this, experiment withCutoff
.
Algorithms
simsmooth
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.
References
[1] Andrieu, Christophe, Arnaud Doucet, and Roman Holenstein. "Particle Markov Chain Monte Carlo Methods." Journal of the Royal Statistical Society Series B: Statistical Methodology 72 (June 2010): 269–342. https://doi.org/10.1111/j.1467-9868.2009.00736.x.
[2] Andrieu, Christophe, and Gareth O. Roberts. "The Pseudo-Marginal Approach for Efficient Monte Carlo Computations." Ann. Statist. 37 (April 2009): 697–725. https://dx.doi.org/10.1214/07-AOS574.
[3] Deligiannidis, George, Arnaud Doucet, and Michael Pitt. "The Correlated Pseudo-Marginal Method." Journal of the Royal Statistical Society, Series B: Statistical Methodology 80 (June 2018): 839–870. https://doi.org/10.1111/rssb.12280.
[4] Durbin, J, and Siem Jan Koopman. Time Series Analysis by State Space Methods. 2nd ed. Oxford: Oxford University Press, 2012.
[5] Fernández-Villaverde, Jesús, and Juan F. Rubio-Ramírez. "Estimating Macroeconomic Models: A Likelihood Approach." Review of Economic Studies 70(October 2007): 1059–1087. https://doi.org/10.1111/j.1467-937X.2007.00437.x.
[6] Liu, Jun, and Rong Chen. "Sequential Monte Carlo Methods for Dynamic Systems." Journal of the American Statistical Association 93 (September 1998): 1032–1044. https://dx.doi.org/10.1080/01621459.1998.10473765.
Version History
Introduced in R2024a
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