\section{Simulation Studies}
In this section, we present a brief simulation study to show the superiority
of the risk-sensitive filtering to standard $L_2$ filtering of HMMs with
finite-discrete states. These simulations have been run on a set of
$N_d = 40000$
data points and a number of computer-generated noise realisations. The
readers will notice that we have deliberately chosen high noise environments
to demonstrate the applicability of risk-sensitive filtering in case
of noise uncertainty to obtain improved performance. It was found that the
risk-sensitive filtering does not do appreciably better than risk-neutral
filtering when there is no uncertainty in the plant or in the noise, or even
when the plant model is perturbed a little. But it definitely improves the
performance substantially in case of noise uncertainties. Different types of
uncertainties are taken up in the following examples.
In all the examples to follow, we take a $3$-state HMM of the form
(\ref{eq:ssphmm}) with $v_k \sim N(0,\sigma^2)$ and the performance index $\frac{1}{N_d} \sum_{k=1}^{N_d} ({\cal X}_k - \hat {\cal X}_k)^{\p}
Q_k ({\cal X}_k - \hat {\cal X}_k)$.
We take
$$
Q_k \equiv Q = \left[ \begin{array}{ccc}
6.0 & -2.0 & -6.0 \\
-2.0 & 3.0 & 2.0 \\
-6.0 & 2.0 & 6.0 \end{array} \right] ,\;\;
A = \left[ \begin{array}{ccc}
0.5 & 0.3 & 0.2 \\
0.2 & 0.5 & 0.3 \\
0.3 & 0.2 & 0.5 \end{array} \right] ,\;\; C = (-1\; 0\; 1)$$
\subsection{Examples}
{\bf Robustness Against High Noise}
This simulation has been carried out on a set of measurements corrupted by
computer-generated Gaussian noise with $\sigma = 2$. The model underestimates
the noise variance and runs the algorithm with the assumption that the
noise variance is lower than what it actually is. The cost is plotted against
$\theta$ for two different values of the model variance ($\sigma_m = 0.5,\;
\sigma_m = 1.0$). Figure \ref{fig:theta1} shows that when $\theta \rightarrow
0$, the cost is higher than when (for example) $5.0 \leq \theta \leq 50.0$.
As $\theta$ increases higher and higher, the cost increases making the
risk-sensitive filter not useful anymore. The reasoning behind this is that
when $\theta$ is very high, the state distribution becomes more and more
uniform and thus makes the risk-sensitive filter more prone to errors.
This proves that the amount of risk involved depends on the nature and extent
of the underlying uncertainty and therefore, to exploit the superiority
of the risk-sensitive filter, it is wise not to choose $\theta$ very high.
\begin{remark}
It has been shown in \cite{Speyer} and \cite{SM1} that $\theta$ should
be ``sufficiently small'' for a risk-sensitive filter to exist for linear and
nonlinear state-space signal models with continuous-range states. This condition manifests itself by requiring the existence of the
solution of a certain Riccati equation in the linear case,
and the integrability of a certain integral in the nonlinear case. For details,
see \cite{SM1}. It is not surprising therefore that we lose the superiority
of the risk-sensitive filter in the case of HMMs as well, when $\theta$ is
very high, although, we do not have any theoretical condition that limits
the maximum value of $\theta$.
\end{remark}
{\bf Robustness Against Coloured Noise}
It is a well known fact that the standard filtering techniques for HMMs do
not perform very well when the noise is coloured instead of being white
as assumed by the algorithm. This drawback has, for example, led to a new
blind equalisation technique for IIR channels based on HMMs and Extended
Least Squares algorithms \cite{Svik}. It is of interest, therefore, to
study the effect of noise colouring on the cost for different values of
$\theta$. In our simulations, the noise colouring polynomial is chosen to
be $D(q^{-1}) = 1.0 + d_1 q^{-1}$, where $q^{-1}$
is the delay operator. Also $\sigma = \sigma_m = 2.0$.
Figure \ref{fig:theta2} shows the cost plotted against different values
of $\theta$ for high colouring ($d_1 = 0.9$) and comparatively low
colouring ($d_1 = 0.6$). It is seen that the risk-sensitive filter performs
better than the risk-neutral filter in a certain range of values for $\theta$.
Here also, the cost goes up when $\theta$ exceeds a certain value. This
part of the curve is not shown in the plot, because it is similar to the
corresponding part of Figure \ref{fig:theta1} and is of no interest to us.
It is also seen that the performance improvement is more when noise
colouring is high, thus proving the robustness of the risk-sensitive filter
against coloured noise.
{\bf Robustness Against High Bursty Noise}
It is also of interest to see how robust the risk-sensitive filter is
by subjecting it to occasional burst of high noise. This was implemented
by incorporating state-dependent noise, i.e, the noise variance is different
for different states. In our simulations, we choose $\sigma[1] = 2.0,\;
\sigma[2] = 1.0,\; \sigma[3] = 2.0$ whereas the algorithm assumes that
$\sigma_m[i] = 1.0,\; \forall i \in \{1,2,3\}$. This
implies that whenever there evolves a sequence of state
$e_1$ or state $e_3$, high noise occurs in bursts. Figure \ref{fig:theta3}
shows that again, the risk-sensitive filter performs better than the risk-neutral filter for a certain range of $\theta$.
Of course, state-dependent noise can be taken care of by standard HMM filtering techniques or risk-sensitive filtering techniques (see Remark \ref{stnoise}).
Here, it is used as a tool to generate high bursty noise
to illustrate the
improvement obtained by implementing a risk-sensitive filter.