# [R] Dynamic Linear Models for Times Series - Implemented?

Thomas Lumley tlumley at u.washington.edu
Wed Feb 12 17:04:44 CET 2003

```On Wed, 12 Feb 2003, Gavin Simpson wrote:

> Hi,
>
> Following an off-list reply to my original post, I realised that I hadn't
> really provided very much information for you to work with.  So here's a
> second attempt:
>
> Following West & Harrison (1989) and Pole et al. (1994) a DLM is defined as:
>
> Y[t] = F'[t]theta[t] + v[t],	v[t] ~ N[0,V] #Observation equation
> theta[t] = G[t]theta[t-1] + w[t],  w[t] ~ N[0,W] #system equation
>
> The system equation is a first order Markov process, where G[t] is a matrix
> of known coefficients that defines the systematic evolution of the state
> vector (theta[t]) across time, and w[t] is an unobservable stochastic error
> term having a normal distribution with zero mean and covariance matrix.
>
> Y[t] denotes the observation series at time t
> F[t] is a vector of known constants (the regression vector)
> theta[t] denotes the vector of model state parameters
> v[t] is a stochastic error term having zero mean and variance V[t]
>
> If I have understood Brockwell and Davis (1991) correctly, the DLM can be
> considered from the point of view of State-space models (although I am
> venturing some way out of my statistical depth here, all the papers I have
> collected are applied examples and they all refer to dynamic Linear Models,
> not State-space models).
>

There are some models of this form in the ts package, see eg StructTS. It
may be possible to use some of the underlying Kalman filter machinery to
fit more models.

I'm trying to fit a model of this sort where F[t] contains a linear
regression term, and having some difficulty with optimisation. (Brian
Ripley has noted that parameter estimation is more difficult in this
models than is generally realised).

-thomas

```