ffbs function

Forward Filtering Backward Sampling

FFBS algorithm for state space models

ffbs(y, A, mu0, Sigma0, Phi, sQ, sR, Ups = NULL, Gam = NULL, input = NULL)

Arguments

• y: Data matrix, vector or time series.
• A: Observation matrix. Can be constant or an array with dim=c(q,p,n) if time varying.
• mu0: Initial state mean.
• Sigma0: Initial state covariance matrix.
• Phi: State transition matrix.
• sQ: State error covariance matrix is Q = sQ%*%t(sQ) -- see details below. In the univariate case, it is the standard deviation.
• sR: Observation error covariance matrix is R = sR%*%t(sR) -- see details below. In the univariate case, it is the standard deviation.
• Ups: State input matrix.
• Gam: Observation input matrix.
• input: matrix or vector of inputs having the same row dimension as y.

Details

For a linear state space model, the FFBS algorithm provides a way to sample a state sequence $x_{0:n}$

from the posterior $\pi(x_{0:n} \mid \Theta, y_{1:n})$

with parameters $\Theta$ and data $y_{1:n}$.

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.

$x_t$ is p-dimensional, $y_t$ is q-dimensional, and $u_t$ is r-dimensional. Note that $sQ\, w_t$ 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

• Xs: An array of sampled states

• X0n: The sampled initial state (because R is 1-based)

Source

Chapter 6 of the Shumway and Stoffer Springer text.

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/.

Author(s)

D.S. Stoffer

Note

The script uses Kfilter. If $A_t$ is constant wrt time, it is not necessary to input an array; see the example. The example below is just one pass of the algorithm; see the example at FUN WITH ASTSA for the real fun.

Examples

## Not run:

## -- this is just one pass --##
# 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 one pass of FFBS, plot data, states and sampled states  
run = ffbs(y, A=1, mu0=0, Sigma0=10, Phi=1, sQ=1, sR=3)
tsplot(cbind(y,run$Xs), spaghetti=TRUE, type='o', col=c(8,4), pch=c(1,NA))
legend('topleft', legend=c("y(t)","xs(t)"), lty=1, col=c(8,4), bty="n", pch=c(1,NA))
## End(Not run)