# smooth

Backward recursion of diffuse state-space models

## Description

returns smoothed states (`X`

= smooth(`Mdl`

,`Y`

)`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`

.

uses
additional options specified by one or more `X`

= smooth(`Mdl`

,`Y`

,`Name,Value`

)`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.

`[`

uses any of the input arguments
in the previous syntaxes to additionally return the loglikelihood
value (`X`

,`logL`

,`Output`

]
= smooth(___)`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, 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.

### 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.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**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 comma-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`

`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 _{t}*

*D*' must be diagonal, where

_{t}*D*,

_{t}*t*= 1,...,

*T*, is one of the following:

The matrix

`D{t}`

in a time-varying state-space modelThe 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`

*m*-by-1 vector of smoothed states_{t}$$E\left({x}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `SmoothedStatesCov`

*m*-by-_{t}*m*variance-covariance matrix of the smoothed states_{t}$$Var\left({x}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `SmoothedStateDisturb`

*k*-by-1 vector of smoothed, state disturbances_{t}$$E\left({u}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `SmoothedStateDisturbCov`

*k*-by-_{t}*k*variance-covariance matrix of the smoothed, state disturbances_{t}$$Var\left({u}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `SmoothedObsInnov`

*h*-by-1 vector of smoothed observation innovations_{t}$$E\left({\epsilon}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `SmoothedObsInnovCov`

*h*-by-_{t}*h*variance-covariance matrix of the smoothed, observation innovations_{t}$$Var\left({\epsilon}_{t}|{y}_{1},\mathrm{...},{y}_{T}\right)$$ `KalmanGain`

*m*-by-_{t}*n*adjusted Kalman gain matrix_{t}N/A `DataUsed`

*h*-by-1 logical vector indicating whether the software filters using a particular observation. For example, if observation_{t}*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. 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. 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*
{'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.

## Version History

**Introduced in R2015b**

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