# smooth

Backward recursion of state-space models

## Description

returns smoothed states (`X`

= smooth(`Mdl`

,`Y`

)`X`

)
by performing backward recursion of the fully-specified state-space model `Mdl`

.
That is, `smooth`

applies the standard 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.

If `Mdl`

is not fully specified, then you must
set the unknown parameters to 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`

)
containing:

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

## Examples

### Smooth States of Time-Invariant State-Space Model

Suppose that a latent process is an AR(1). The state equation is

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

where $${u}_{t}$$ is Gaussian with mean 0 and standard deviation 0.5.

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',0.5^2); 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. The observation equation is

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

where $${\epsilon}_{t}$$ is Gaussian with mean 0 and standard deviation 0.05. 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 + 0.05*randn(T,1);

Specify the four coefficient matrices.

A = 0.5; B = 0.5; C = 1; D = 0.05;

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) + (0.50)u1(t) Observation equation: y1(t) = x1(t) + (0.05)e1(t) Initial state distribution: Initial state means x1 0 Initial state covariance matrix x1 x1 0.33 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.

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

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'})

### Smooth States of State-Space Model Containing Regression Component

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

$$\begin{array}{l}\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\end{array}\right]=\left[\begin{array}{cc}\varphi & \theta \\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]{u}_{1,t}\\ {y}_{t}-\beta {Z}_{t}={x}_{1,t}+\sigma {\epsilon}_{t},\end{array}$$

where:

$${x}_{1,t}$$ is the change in the unemployment rate at time

*t*.$${x}_{2,t}$$ is a dummy state for the MA(1) effect.

$${y}_{1,t}$$ is the observed unemployment rate being deflated by the growth rate of nGNP ($${Z}_{t}$$).

$${u}_{1,t}$$ is the Gaussian series of state disturbances having mean 0 and standard deviation 1.

$${\epsilon}_{t}$$ is the Gaussian series of observation innovations having mean 0 and 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 the first difference of each series. Also, remove the starting `NaN`

values from each series.

isNaN = any(ismissing(DataTable),2); % Flag periods containing NaNs gnpn = DataTable.GNPN(~isNaN); u = DataTable.UR(~isNaN); T = size(gnpn,1); % Sample size Z = [ones(T-1,1) diff(log(gnpn))]; y = diff(u);

Though this example removes missing values, the software can accommodate series containing missing values in the Kalman filter framework.

Specify the coefficient matrices.

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

Specify the state-space model using `ssm`

.

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

Estimate the model parameters. Specify the regression component and its initial value for optimization using the `'Predictors'`

and `'Beta0'`

name-value pair arguments, respectively. Restrict the estimate of $$\sigma $$ to all positive, real numbers.

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

Method: Maximum likelihood (fmincon) Sample size: 61 Logarithmic likelihood: -99.7245 Akaike info criterion: 209.449 Bayesian info criterion: 220.003 | Coeff Std Err t Stat Prob ---------------------------------------------------------- c(1) | -0.34098 0.29608 -1.15164 0.24948 c(2) | 1.05003 0.41377 2.53771 0.01116 c(3) | 0.48592 0.36790 1.32079 0.18657 y <- z(1) | 1.36121 0.22338 6.09358 0 y <- z(2) | -24.46711 1.60018 -15.29024 0 | | Final State Std Dev t Stat Prob x(1) | 1.01264 0.44690 2.26592 0.02346 x(2) | 0.77718 0.58917 1.31912 0.18713

`EstMdl`

is an `ssm`

model, and you can access its properties using dot notation.

Smooth the states. `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. Recall that the first state is the change in the unemployment rate, and the second state helps build the first.

SmoothedX = smooth(EstMdl,y,'Predictors',Z,'Beta',estParams(end-1:end)); figure plot(dates(end-(T-1)+1:end),SmoothedX(:,1)); xlabel('Period') ylabel('Change in the unemployment rate') title('Smoothed Change in the Unemployment Rate')

## Input Arguments

`Mdl`

— Standard state-space model

`ssm`

model object

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

model
object returned by `ssm`

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
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 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`

`SquareRoot`

— Square root filter method flag

`false`

(default) | `true`

Square root filter method flag, specified as the comma-separated pair consisting of
`'SquareRoot'`

and `true`

or
`false`

. If `true`

, then
`smooth`

applies the square root filter method when
implementing the Kalman filter.

If you suspect that the eigenvalues of the filtered state or
forecasted observation covariance matrices are close to zero, then
specify `'SquareRoot',true`

. The square root filter
is robust to numerical issues arising from finite the precision of
calculations, but requires more computational resources.

**Example: **`'SquareRoot',true`

**Data Types: **`logical`

`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

matrix | cell vector of vectors

Smoothed states, returned as a 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.

**Data Types: **`cell`

| `double`

`logL`

— Loglikelihood function value

scalar

Loglikelihood function value, returned as a scalar.

Missing observations 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.

## 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 arguments.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,

`smooth`

applies all predictors to each response series. However, each response series has its own set of regression coefficients.

## 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 R2014a**

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