Solved – LogLikelihood Parameter Estimation for Linear Gaussian Kalman Filter

algorithmskalman filterlikelihoodmaximum likelihood

I have written some code that can do Kalman filtering (using a number of different Kalman-type filters [Information Filter et al.]) for Linear Gaussian State Space Analysis for an n-dimensional state vector. The filters work great and I am getting some nice output. However, the parameter estimation via loglikelihood estimation is confusing me. I am not a statistician but a physicist, so please be kind.

Let us consider the linear Gaussian State Space model

$$y_t = \mathbf{Z}_{t}\alpha_{t} + \epsilon_{t},$$
$$\alpha_{t + 1} = \mathbf{T}_{t}\alpha_{t} + \mathbf{R}_{t}\eta_{t},$$

where $y_{t}$ is our observation vector, $\alpha_{t}$ our state vector at time step $t$. The quantities in bold are the transformation matrices of the state space model which are set according to the characteristics of the system under consideration. We also have

$$\epsilon_{t} \sim NID(0, \mathbf{H}_{t}),$$
$$\eta_{t} \sim NID(0, \mathbf{Q}_{t}),$$
$$\alpha_{1} \sim NID(a_{1}, \mathbf{P}_{1}).$$

where $t = 1,\ldots, n$. Now, I have derived and implemented the recursion for the Kalman Filter for this generic state space model by guessing the initial parameters and variance matrices $\mathbf{H}_{1}$ and $\mathbf{Q}_{1}$ I can produce plots like

Kalman Filter

where the points are the Nile River water levels for Jan over 100 years, the line is the Kalamn Estimated state, and the dashed lines are the 90% confidence levels.

Now, for this 1D data set the matrices $\mathbf{H}_{t}$ and $\mathbf{Q}_{t}$ are just scalars $\sigma_{\epsilon}$ and $\sigma_{\eta}$ respectively. So now I want to get the correct parameters for these scalars using the output from the Kalman Filter and the loglikelihood function

$$\log L(Y_{n}) = -\frac{np}{2}\log(2\pi) – \frac{1}{2}\sum^{n}_{t = 1}(log|\mathbf{F}_{t}| + v^{\mathsf{T}}_{t}\mathbf{F}_{t}^{-1}v_{t})$$

Where $v_{t}$ is the state error and $\mathbf{F}_{t}$ is the state error variance. Now, here's where I am confused. From the Kalman filter, I have all the information I need to work out $L$, but this seems to get me no closer to being able to calculate the maximum likelihood of $\sigma_{\epsilon}$ and $\sigma_{\eta}$. My question is how can I calculate the maximum likelihood of $\sigma_{\epsilon}$ and $\sigma_{\eta}$ using the loglikelihood approach and the equation above? An algorithmic break down would be like a cold beer to me right now…

Thanks for your time.

Note. For the 1D case $\mathbf{H}_{t} = \sigma^{2}_{\epsilon}$ and $\mathbf{H}_{t} = \sigma^{2}_{\eta}$. This is the univariate local level model.

Best Answer

When you run the Kalman filter as you have, with given values of $\sigma_\epsilon^2$ and $\sigma^2_\eta$, you get a sequence of innovations $\nu_t$ and their covariances $\boldsymbol{F_t}$, hence you can calculate the value of $\log L(Y_n)$ using the formula you give.

In other words, you can regard the Kalman filter as a way to compute an implicit function of $\sigma_\epsilon^2$ and $\sigma^2_\eta$. The only thing that you need to do then is to package this computation into a function or subroutine and handle that function to an optimization routine --like optim in R. That function should accept as inputs $\sigma_\epsilon^2$ and $\sigma^2_\eta$ and return $\log L(Y_n)$.

Some packages in R (e.g. dlm) do this for you (see for instance function dlmMLE).

Edit: Link in one of my comments below, which I cannot edit, is now here.

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Solved – Kalman Filtering, Smoothing and Parameter Estimation for State Space Models in R and C#

Im not sure why KFS does not return V_eta or V_eps, but why not use modelNile$H and modelNile$Q? They are also kept in the return object from KFS, out$model$H and out$model$Q. You will find what the objects contains by using the names() function as in names(modelNile) or names(out$model).

Solved – Maximum likelihood estimation for state space models using BFGS

I will give you an answer from my experience developing the R package stsm that is introduced in this document. By default ithe package uses the function optimize, which is a combination of a golden section search and successive parabolic interpolation. Other line search algorithms, such as the zoom line search algorithm will do the job similarly well. The algorithm in function optimize is convenient because it does not require the gradient of the function to be minimized. Here is some pseudo code to compute the optimal step size:

fcn.step <- function(step, model, direction)
{
  trial.pars <- model$pars + step * direction
  -logLik(model, pars = trial.pars)
}
s <- line.search(f = fcn.step, interval = c(0, 1))

fcn.step is the function to be minimized, the only parameter is the step size; the parameters of the model are fixed to the values from the last iteration of the BFGS optimization algorithm. The negative of the log-likelihood function is minimized with respect to the step size; direction is the optimal direction vector computed at the current iteration of the BFGS algorithm, $\tilde{\pi(\psi)}$ in your notation.

It is also important the choice of the upper limit of the interval where the optimal value of the step size s is searched. By default, the package stsm calculates the maximum value of s that is compatible with positive values for the variance parameters updated in equation (3). If this value is lower than $1$ then it is taken as the upper bound passed to the line search algorithm, otherwise this bound is set equal to $1$. You may look at function step.maxsize in the package or the function calc.constraint in this implementation of the BFGS algorithm.

With this approach you can apply the BFGS algorithm without the risk of reaching a local optimum with negative variance parameters. Otherwise, I would recommend you either the L-BFGS-B algorithm, which allows setting a lower bound equal to zero in the parameter space where the algorithm searches for the optimum; or a reparameterization of the model, e.g. variance = theta^2 or variance = exp(theta) and maximize the likelihood with respect to theta.

As regards the EM algorithm, as you mention, it is slow to converge. Strangely enough, it converges slower as it approaches the local optimum. For some insight into this issue and a modified EM algorithm you may look at this document. I am the author of this document, you may send me an e-mail if you have any questions about it or if you want a copy (the link is not always available).

Optimization of the likelihood function of state space models can be hard in practice. Some enhancements to the general-purpose optimization algorithm are recommended. For example, one of the parameters of the model can be concentrated out of the likelihood function; maximum likelihood in the frequency domain enjoys some advantages from a computational point of view and may provide good values to be used as starting values in the optimization of the time domain likelihood function.

Best Answer

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Solved – Kalman Filtering, Smoothing and Parameter Estimation for State Space Models in R and C#

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