smooth

Class: dssm

Backward recursion of diffuse state-space models

Syntax

X = smooth(Mdl,Y)

X = smooth(Mdl,Y,Name,Value)

[X,logL,Output] = smooth(___)

Description

X = smooth(Mdl,Y) returns smoothed states (X) by performing backward recursion of the fully-specified diffuse state-space model Mdl. That is, smooth applies the diffuse Kalman filter using Mdl and the observed responses Y.

X = smooth(Mdl,Y,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, specify the regression coefficients and predictor data to deflate the observations, or specify to use the univariate treatment of a multivariate model.

If Mdl is not fully specified, then you must specify the unknown parameters as known scalars using the 'Params' Name,Value pair argument.

[X,logL,Output] = smooth(___) uses any of the input arguments in the previous syntaxes to additionally return the loglikelihood value (logL) and an output structure array (Output) using any of the input arguments in the previous syntaxes. The fields of Output include:

• Smoothed states and their estimated covariance matrix

• Smoothed state disturbances and their estimated covariance matrix

• Smoothed observation innovations and their estimated covariance matrix

• The loglikelihood value

• The adjusted Kalman gain

• And a vector indicating which data the software used to filter

Input Arguments

Mdl — Diffuse state-space model

dssm model object

Diffuse state-space model, specified as an dssm model object returned by dssm or estimate.

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 issues an error. estimate returns fully-specified state-space models.

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

Y — Observed response data

numeric 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_t-dimensional vector of observations for that period. 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.

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

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

Example: 'Beta',beta,'Predictors',Z specifies to deflate the observations by the regression component composed of the predictor data Z and the coefficient matrix beta.

'Beta' — Regression coefficients

[] (default) | numeric matrix

Regression coefficients corresponding to predictor variables, specified as the comma-separated pair consisting of 'Beta' 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).

If Mdl is an estimated state-space model, then specify the estimated regression coefficients stored in estParams.

'Params' — Values for unknown parameters

numeric 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 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, 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

'Predictors' — Predictor variables in state-space model observation equation

[] (default) | numeric matrix

Predictor variables in the state-space model 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 (Z_t). The expanded observation equation is

$$y_t-Z_t\beta =C{x}_t+D{u}_t.$$

That is, the software deflates the observations using the regression component. β is the time-invariant vector of regression coefficients that the software estimates with all other parameters.

If there are n observations per period, then 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

'SwitchTime' — Final period for diffuse state initialization

positive integer

Final period for diffuse state initialization, specified as the comma-separated pair consisting of 'SwitchTime' and a positive integer. That is, estimate uses the observations from period 1 to period SwitchTime as a presample to implement the exact initial Kalman filter (see Diffuse Kalman Filter and [1]). After initializing the diffuse states, estimate applies the standard Kalman filter to the observations from periods SwitchTime + 1 to T.

The default value for SwitchTime is the last period in which the estimated smoothed state precision matrix is singular (i.e., the inverse of the covariance matrix). This specification represents the fewest number of observations required to initialize the diffuse states. Therefore, it is a best practice to use the default value.

If you set SwitchTime to a value greater than the default, then the effective sample size decreases. If you set SwitchTime to a value that is fewer than the default, then estimate might not have enough observations to initialize the diffuse states, which can result in an error or improper values.

In general, estimating, filtering, and smoothing state-space models with at least one diffuse state requires SwitchTime to be at least one. The default estimation display contains the effective sample size.

Data Types: double

'Tolerance' — Forecast uncertainty threshold

0 (default) | nonnegative scalar

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' — Univariate treatment of multivariate series flag

false (default) | true

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, D_tD_t' must be diagonal, where D_t, 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

Output Arguments

X — Smoothed states

numeric matrix | cell vector of vectors

Smoothed states, returned as a numeric matrix or a cell vector of matrices.

If Mdl is time invariant, then the number of rows of X is the sample size, and the number of columns of X is the number of states. The last row of X contains the latest smoothed states.

If Mdl is time varying, then X is a cell vector with length equal to the sample size. Cell t of X contains a vector of smoothed states with length equal to the number of states in period t. The last cell of X contains the latest smoothed states.

smooth pads the first SwitchTime periods of X with zeros or empty cells. The zeros or empty cells represent the periods required to initialize the diffuse states.

logL — Loglikelihood function value

scalar

Loglikelihood function value, returned as a scalar.

Missing observations and observations before SwitchTime do not contribute to the loglikelihood.

Output — Smoothing results by period

structure array

Smoothing results by period, returned as a structure array.

Output is a T-by-1 structure, where element t corresponds to the smoothing recursion at time t.

• If Univariate is false (it is by default), then the following table describes the fields of Output.

  Field	Description	Estimate
  LogLikelihood	Scalar loglikelihood objective function value	N/A
  SmoothedStates	mt-by-1 vector of smoothed states	$$E\left(x_t|y_1,\mathrm{...},y_T\right)$$
  SmoothedStatesCov	mt-by-mt variance-covariance matrix of the smoothed states	$$Var\left(x_t|y_1,\mathrm{...},y_T\right)$$
  SmoothedStateDisturb	kt-by-1 vector of smoothed, state disturbances	$$E\left(u_t|y_1,\mathrm{...},y_T\right)$$
  SmoothedStateDisturbCov	kt-by-kt variance-covariance matrix of the smoothed, state disturbances	$$Var\left(u_t|y_1,\mathrm{...},y_T\right)$$
  SmoothedObsInnov	ht-by-1 vector of smoothed observation innovations	$$E\left({\epsilon }_t|y_1,\mathrm{...},y_T\right)$$
  SmoothedObsInnovCov	ht-by-ht variance-covariance matrix of the smoothed, observation innovations	$$Var\left({\epsilon }_t|y_1,\mathrm{...},y_T\right)$$
  KalmanGain	mt-by-nt adjusted Kalman gain matrix	N/A
  DataUsed	ht-by-1 logical vector indicating whether the software filters using a particular observation. For example, if observation i at time t is a NaN, then element i in DataUsed at time t is 0.	N/A

• If Univarite is true, then the fields of Output are the same as in the previous table, but the values in KalmanGain might vary.

smooth pads the first SwitchTime periods of the fields of Output with empty cells. These empty cells represent the periods required to initialize the diffuse states.

Data Types: struct

Examples

Smooth States of Time-Invariant Diffuse State-Space Model

Suppose that a latent process is a random walk. Subsequently, the state equation is

$$x_t=x_{t-1}+u_t,$$

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;
x0 = 1.5;
rng(1); % For reproducibility
u = randn(T,1);
x = cumsum([x0;u]);
x = x(2:end);

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 = 1;
B = 1;
C = 1;
D = 0.75;

Create the diffuse state-space model using the coefficient matrices. Specify that the initial state distribution is diffuse.

Mdl = dssm(A,B,C,D,'StateType',2)

Mdl = 
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) + (0.75)e1(t)

Initial state distribution:

Initial state means
 x1 
  0 

Initial state covariance matrix
     x1  
 x1  Inf 

State types
    x1   
 Diffuse 

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

Smooth states for periods 1 through 100. Plot the true state values and the smoothed state estimates.

SmoothedX = smooth(Mdl,y);

figure
plot(1:T,x,'-k',1:T,SmoothedX,':r','LineWidth',2)
title({'State Values'})
xlabel('Period')
ylabel('State')
legend({'True state values','Smoothed state values'})

The true values and smoothed estimates are approximately the same.

Smooth States of Diffuse State-Space Model Containing Regression Component

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

$$\begin{array}{l}{x}_t=\varphi x_{t-1}+\sigma u_t\\ y_t-\beta Z_t=x_t,\end{array}$$

where:

• $$x_t$$ is the unemployment rate at time t.

• $$y_t$$ is the observed change in the unemployment rate being deflated by the return of nGNP ($$Z_t$$).

• $$u_t$$ is the Gaussian series of state disturbances having mean 0 and unknown standard deviation $$\sigma$$.

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 removing the starting NaN values from each series.

isNaN = any(ismissing(DataTable),2);       % Flag periods containing NaNs
gnpn = DataTable.GNPN(~isNaN);
y = diff(DataTable.UR(~isNaN));
T = size(gnpn,1);                          % The sample size
Z = price2ret(gnpn);

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

Specify the coefficient matrices.

A = NaN;
B = NaN;
C = 1;

Create the state-space model using dssm by supplying the coefficient matrices and specifying that the state values come from a diffuse distribution. The diffuse specification indicates complete ignorance about the moments of the initial distribution.

StateType = 2;
Mdl = dssm(A,B,C,'StateType',StateType);

Estimate the 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 $$\sigma$$ to all positive, real numbers.

params0 = [0.3 0.2]; % Initial values chosen arbitrarily
Beta0 = 0.1;
[EstMdl,estParams] = estimate(Mdl,y,params0,'Predictors',Z,'Beta0',Beta0,...
    'lb',[-Inf 0 -Inf]);

Method: Maximum likelihood (fmincon)
Effective Sample size:             60
Logarithmic  likelihood:     -110.477
Akaike   info criterion:      226.954
Bayesian info criterion:      233.287
           |      Coeff       Std Err    t Stat    Prob 
--------------------------------------------------------
 c(1)      |   0.59436       0.09408     6.31738  0     
 c(2)      |   1.52554       0.10758    14.17991  0     
 y <- z(1) | -24.26161       1.55730   -15.57930  0     
           |                                            
           |    Final State   Std Dev     t Stat   Prob 
 x(1)      |   2.54764        0           Inf     0     

EstMdl is a dssm model, and you can access its properties using dot notation.

Smooth the estimated diffuse state-space model. EstMdl does not store the data or the regression coefficients, so you must pass in them in using the name-value pair arguments 'Predictors' and 'Beta', respectively. Plot the smoothed states.

SmoothedX = smooth(EstMdl,y,'Predictors',Z,'Beta',estParams(end));

figure
plot(dates(end-(T-1)+1:end),SmoothedX);
xlabel('Period')
ylabel('Change in the unemployment rate')
title('Smoothed Change in the Unemployment Rate')
axis tight

Extract Other Estimates from Output

Estimate a diffuse state-space model, smooth the states, and then extract other estimates from the Output output argument.

Consider the diffuse state-space model

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

The state variable $$x_{1,t}$$ is an AR(1) model with autoregressive coefficient $$\varphi$$. $$x_{2,t}$$ is a random walk. The disturbances $$u_{1,t}$$ and $$u_{2,t}$$ are independent Gaussian random variables with mean 0 and standard deviations $${\sigma }_1$$ and $${\sigma }_2$$, respectively. The observation $$y_t$$ is the error-free sum of $$x_{1,t}$$ and $$x_{2,t}$$.

Generate data from the state-space model. To simulate the data, suppose that the sample size $$T=100$$, $$\varphi =0.6$$, $${\sigma }_1=0.2$$, $${\sigma }_2=0.1$$, and $$x_{1,0}=x_{2,0}=2$$.

rng(1); % For reproducibility
T = 100;
ARMdl = arima('AR',0.6,'Constant',0,'Variance',0.2^2);
x1 = simulate(ARMdl,T,'Y0',2);
u3 = 0.1*randn(T,1);
x3 = cumsum([2;u3]);
x3 = x3(2:end);
y = x1 + x3;

Specify the coefficient matrices of the state-space model. To indicate unknown parameters, use NaN values.

A = [NaN 0; 0 1];
B = [NaN 0; 0 NaN];
C = [1 1];

Create a diffuse state-space model that describes the model above. Specify that $$x_{1,t}$$ and $$x_{2,t}$$ have diffuse initial state distributions.

StateType = [2 2];
Mdl = dssm(A,B,C,'StateType',StateType);

Estimate the unknown parameters of Mdl. Choose initial parameter values for optimization. Specify that the standard deviations are constrained to be positive, but all other parameters are unconstrained using the 'lb' name-value pair argument.

params0 = [0.01 0.1 0.01]; % Initial values chosen arbitrarily
EstMdl = estimate(Mdl,y,params0,'lb',[-Inf 0 0]);

Method: Maximum likelihood (fmincon)
Effective Sample size:             98
Logarithmic  likelihood:      3.44283
Akaike   info criterion:    -0.885655
Bayesian info criterion:      6.92986
      |     Coeff      Std Err   t Stat     Prob  
--------------------------------------------------
 c(1) | 0.54134       0.20494    2.64145  0.00826 
 c(2) | 0.18439       0.03305    5.57897   0      
 c(3) | 0.11783       0.04347    2.71039  0.00672 
      |                                           
      |  Final State   Std Dev    t Stat    Prob  
 x(1) | 0.24884       0.17168    1.44943  0.14722 
 x(2) | 1.73762       0.17168   10.12121   0      

The parameters are close to their true values.

Smooth the states of EstMdl, and request all other available output.

[X,logL,Output] = smooth(EstMdl,y);

X is a T-by-2 matrix of smoothed states, logL is the final optimized log-likelihood value, and Output is a structure array containing various estimates that the Kalman filter requires. List the fields of output using fields.

fields(Output)

ans = 9x1 cell array
    {'LogLikelihood'          }
    {'SmoothedStates'         }
    {'SmoothedStatesCov'      }
    {'SmoothedStateDisturb'   }
    {'SmoothedStateDisturbCov'}
    {'SmoothedObsInnov'       }
    {'SmoothedObsInnovCov'    }
    {'KalmanGain'             }
    {'DataUsed'               }

Convert Output to a table.

OutputTbl = struct2table(Output);
OutputTbl(1:10,1:4) % Display first ten rows of first four variables

ans=10×4 table
    LogLikelihood    SmoothedStates    SmoothedStatesCov    SmoothedStateDisturb
    _____________    ______________    _________________    ____________________

    {0x0 double}      {0x0 double}       {0x0 double}           {0x0 double}    
    {0x0 double}      {0x0 double}       {0x0 double}           {0x0 double}    
    {[  0.1827]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.0972]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.4472]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.2073]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.5167]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.2389]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[  0.5064]}      {2x1 double}       {2x2 double}           {2x1 double}    
    {[ -0.0105]}      {2x1 double}       {2x2 double}           {2x1 double}    

The first two rows of the table contain empty cells or zeros. These correspond to the observations required to initialize the diffuse Kalman filter. That is, SwitchTime is 2.

SwitchTime = 2;

Plot the smoothed states and their individual 95% Wald-type confidence intervals.

CI = nan(T,2,2);

for j = (SwitchTime + 1):T
    CovX = OutputTbl.SmoothedStatesCov{j};
    CI(j,:,1) = X(j,1) + 1.96*sqrt(CovX(1,1))*[-1 1];
    CI(j,:,2) = X(j,2) + 1.96*sqrt(CovX(2,2))*[-1 1];    
end

figure;
plot(1:T,X(:,1),'k',1:T,CI(:,:,1),'--r');
xlabel('Period');
ylabel('Smoothed states');
title('State 1 Estimates')
legend('Smoothed','95% Individual CIs');
grid on;

figure;
plot(1:T,X(:,2),'k',1:T,CI(:,:,2),'--r');
xlabel('Period');
ylabel('Smoothed states');
title('State 2 Estimates')
legend('Smoothed','95% Individual CIs');
grid on;

Tips

• Mdl does not store the response data, predictor data, and the regression coefficients. Supply the data wherever necessary using the appropriate input or name-value pair arguments.

• It is a best practice to allow smooth to determine the value of SwitchTime. However, in rare cases, you might experience numerical issues during estimation, filtering, or smoothing diffuse state-space models. For such cases, try experimenting with various SwitchTime specifications, or consider a different model structure (e.g., simplify or reverify the model). For example, convert the diffuse state-space model to a standard state-space model using ssm.

• To accelerate estimation for low-dimensional, time-invariant models, set 'Univariate',true. Using this specification, the software sequentially updates rather then updating all at once during the filtering process.

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 defined state-space models, filter applies all predictors to each response series. However, each response series has its own set of regression coefficients.

• The diffuse Kalman filter requires presample data. If missing observations begin the time series, then the diffuse Kalman filter must gather enough nonmissing observations to initialize the diffuse states.

• For diffuse state-space models, filter usually switches from the diffuse Kalman filter to the standard Kalman filter when the number of cumulative observations and the number of diffuse states are equal. However, if a diffuse state-space model has identifiability issues (e.g., the model is too complex to fit to the data), then filter might require more observations to initialize the diffuse states. In extreme cases, filter requires the entire sample.

References

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