Ksmooth function

Quick Kalman Smoother

Returns the smoother values for various linear state space models. The predicted and filtered values and the likelihood at the given parameter values are also returned (via Kfilter). This script replaces Ksmooth0, Ksmooth1, and Ksmooth2.

Ksmooth(y, A, mu0, Sigma0, Phi, sQ, sR, Ups = NULL, Gam = NULL, 
         input = NULL, S = NULL, version = 1)

Arguments

• y: data matrix (n x q), vector or time series, n = number of observations. Use NA or zero (0) for missing data.
• A: can be constant or an array with dimension dim=c(q,p,n) if time varying (see details). Use NA or zero (0) for missing data.
• mu0: initial state mean vector (p x 1)
• Sigma0: initial state covariance matrix (p x p)
• Phi: state transition matrix (p x p)
• sQ: state error pre-matrix (see details)
• sR: observation error pre-matrix (see details)
• Ups: state input matrix (p x r); leave as NULL (default) if not needed
• Gam: observation input matrix (q x r); leave as NULL (default) if not needed
• input: NULL (default) if not needed or a matrix (n x r) of inputs having the same row dimension (n) as y
• S: covariance matrix between state and observation errors; not necessary to specify if not needed and only used if version=2; see details
• version: either 1 (default) or 2; version 2 allows for correlated errors

Details

This script replaces Ksmooth0, Ksmooth1, and Ksmooth2 by combining all cases. The major difference is how to specify the covariance matrices; in particular, sQ = t(cQ) and sR = t(cR) where cQ and cR were used in Kfilter0-1-2 scripts.

The states $x_t$ are p-dimensional, the data $y_t$ are q-dimensional, and the inputs $u_t$ are r-dimensional for $t=1, \dots, n$. The initial state is $x_0 \sim N(\mu_0, \Sigma_0)$.

The measurement matrices $A_t$ can be constant or time varying. If time varying, they should be entered as an array of dimension dim = c(q,p,n). Otherwise, just enter the constant value making sure it has the appropriate $q \times p$ dimension.

Version 1 (default): The general model is

$$x_t = \Phi x_{t-1} + \Upsilon u_{t} + sQ\, w_t \quad w_t \sim iid\ N(0,I)$$ $$y_t = A_t x_{t-1} + \Gamma u_{t} + sR\, v_t \quad v_t \sim iid\ N(0,I)$$

where $w_t \perp v_t$. Consequently the state noise covariance matrix is $Q = sQ\, sQ'$ and the observation noise covariance matrix is $R = sR\, sR'$ and $sQ, sR$ do not have to be square as long as everything is conformable. Notice the specification of the state and observation covariances has changed from the original scripts.

NOTE: If it is easier to model in terms of $Q$ and $R$, simply input the square root matrices sQ = Q %^% .5 and sR = R %^% .5.

Version 2 (correlated errors): The general model is

$$x_{t+1} = \Phi x_{t} + \Upsilon u_{t+1} + sQ\, w_t \quad w_t \sim iid\ N(0,I)$$ $$y_t = A_t x_{t-1} + \Gamma u_{t} + sR\, v_t \quad v_t \sim iid\ N(0,I)$$

where $S = {\rm Cov}(w_t, v_t)$, and NOT ${\rm Cov}(sQ\, w_t, sR\, v_t)$.

NOTE: If it is easier to model in terms of $Q$ and $R$, simply input the square root matrices sQ = Q %^% .5 and sR = R %^% .5.

Note that in either version, $sQ\, w_t$ has to be p-dimensional, but $w_t$ does not, and $sR\, v_t$ has to be q-dimensional, but $v_t$ does not.

Returns

Time varying values are returned as arrays. - Xs: state smoothers

• Ps: smoother mean square error

• X0n: initial mean smoother

• P0n: initial smoother covariance

• J0: initial value of the J matrix

• J: the J matrices

• Xp: state predictors

• Pp: mean square prediction error

• Xf: state filters

• Pf: mean square filter error

• like: negative of the log likelihood

• innov: innovation series

• sig: innovation covariances

• Kn: the value of the last Gain

References

You can find demonstrations of astsa capabilities at FUN WITH ASTSA.

The most recent version of the package can be found at https://github.com/nickpoison/astsa/.

In addition, the News and ChangeLog files are at https://github.com/nickpoison/astsa/blob/master/NEWS.md.

The webpages for the texts and some help on using R for time series analysis can be found at https://nickpoison.github.io/.

Note

Note that Ksmooth is similar to Ksmooth-0-1-2 except that only the essential values need to be entered (and come first in the statement); the optional values such as input are set to NULL by default if they are not needed. This version is faster than the older versions. The biggest change was to how the covarainces are specified. For example, if you have code that used Ksmooth1, just use sQ = t(cQ) and sR = t(cR) here.

Author(s)

D.S. Stoffer

See Also

Kfilter

Examples

# generate some data
 set.seed(1)
 sQ  = 1; sR = 3; n = 100  
 mu0 = 0; Sigma0 = 10; x0 = rnorm(1,mu0,Sigma0)
 w = rnorm(n); v = rnorm(n)
 x = c(x0 + sQ*w[1]);  y = c(x[1] + sR*v[1])   # initialize
for (t in 2:n){
  x[t] = x[t-1] + sQ*w[t]
  y[t] = x[t] + sR*v[t]   
  }
# run and plot the smoother  
run = Ksmooth(y, A=1, mu0, Sigma0, Phi=1, sQ, sR)
tsplot(cbind(y,run$Xs), spaghetti=TRUE, type='o', col=c(4,6), pch=c(1,NA), margins=1)
# CRAN tests need extra white space :( so margins=1 above is not necessary otherwise
legend('topleft', legend=c("y(t)","Xs(t)"), lty=1, col=c(4,6), bty="n", pch=c(1,NA))