Particle Filters, otherwise known as Sequential Monte Carlo Methods, are a common numeri- cal approach to system identification. The basic formulation of the particle filter considers an approximate solution to the optimal recursive Bayesian filter.
The particle filter is largely based on the Monte Carlo simulation, proposed in the 1940’s by Ulam [94]. A Monte Carlo simulation is based on the concept of random samples of data in multiple dimensions. The sampling of this data is usually centered around an initial value with the sampling probability distribution encompassing random values around that point. Monte
Carlo simulations were initially conceived as a numerical approximation to solve difficult com- binatorics problems and have since seen applications in physics and math. The particle filter in contrast utilizes the random sampling as weighted particles to approximate a probability density of a state space dynamical model. This implementation was first proposed by Gordon et al. and was termed the ‘bootstrap filter’ [95]. The basic formulation of the particle filter considers an approximate solution to the optimal recursive Bayesian filter. Normally the prediction density function for this filter is given by Eq. 2.71.
p(xt+1|Yt) = Z
p(xt+1|xt)p(xt|Yt)dxt (2.71)
Here the filtering density, an update to the prior is then given by
p(xt|Yt) =
p(yt|xt)p(yt|Yt−1)
p(yt|Yt−1) (2.72)
The particle filter provides an estimate of the filtering density by using a set of random samples [xk−1(i) : i = 1, ..., N]. The distribution of these samples are taken from the Probability
Density Function (PDF) p(xk−1|Dk−1). Each sample represents a random estimate or particle
of the possible state. Using these particles, the discrete estimate is then given by:
p(xt|Yt) = N
∑
i=1 q(i)t δ (x (i) t − xt) (2.73)Here δ () represents the dirac delta function. The superscript (i) indicates a particular state particle which is an approximation to the actual state. Similarly q(i)t are the normalized weights
for each distinct particle. The weights allow the particles to approximate the PDF. These weights are updated each time step by Eq. 2.74.
q(i)t+1= p(yt+1|x(i)t+1)q (i)
t (2.74)
This update function means that particles with the largest likelihood will have larger weights. Then as time progresses, the particles and their associated weights will begin to approximate the PDF from the density function for the filter.
In order to implement this particle filter, first the state space model has to be specified (in the classic formulation). The state space model is assumed to follow some parametric functions
f(·) and h(·). The standard model is given by
zt+1= f (zt; θt) + vzt (2.75)
yt= h(zt; θt) + et (2.76)
These functions are dependent both on the state ztand the function parameters θt. Addition-
ally, initial estimates of the densities pz0, pθ0 and noise densities pvt must be specified. Finally
the particles are initialized as [x(i)0 : i = 1, ..., N] where x is distributed according to px0
The next step is measurement update from the current system output. In this step the weights are updated according to
qt(i)= q (i)
t−1p(yt|xt(i)) (2.77)
Here i = 1, ..., N. The next step is to re-sample from the previous particles according to their weights. Several methods for re-sampling exist, the most common is called Sampling Importance Re-Sampling (SIR). In this method N samples are picked from the previous set xt(i), θ
(i)
t . The probability of picking a particular sample i is defined by the weight q (i)
t . Hence
highly weighted particles are more likely to persist to the next time step, meaning that particle is more likely a true estimate of the state. Notice that re-sampling also propagates parameters estimates for the state space model. Therefore particles and weights can be used to gauge the likelihood of a particular parameter better representing the state space model.
The final step is the prediction step. Here the state is updated according to the state space model and the noise densities (Eq. 2.78). The parameters are also updated according to the parameter noise densities (Eq. 2.79).
x(i)t+1= f (x(i)t ; θ(i)) + vtz+ w z
t (2.78)
θt(i)+1= θt(i)+ vtθ+ wθt (2.79)
This algorithm continues for subsequent time steps until a sufficient condition is met. Usu- ally this condition is taken to be a certain weight threshold is achieved or a certain number of time steps have passed. The purpose of this particle filter formulation in terms of system
identification is for parameter identification. If the general model of a system is known, then by estimating the parameters that best represent the data, the specific underlying system can also be discovered. The use of the particle filters for state and parameter estimation has been proposed by several researcher groups [96–98].
More recently work has proposed explicit system identification via parameter estimation in a particle filter setting. Poyiadjis et al. proposed a more computationally efficient means to compute the score vector for the particle filter with explicit applications in parameter estima- tion [99]. With the proposed algorithm, model parameters for a stochastic volatility model were estimated with only 50 particles. Schon et al. also presented an explicit derivation of the use of particle filters for system identification [100]. This work highlighted the use of expectation maximization (EM) for parameter estimation in non-linear systems. Using this EM framework, convergence to the correct parameters for a non-linear, time varying system, was achieved with only 50 particles. Limetkai et al. proposed a Conditional Random Field (CRF) filter variation of the particle filter as a discriminative modeling technique [101]. The CRF filter is a discrim- inative undirected probabilistic model for use in continuous functions. This CRF method was implemented in a robot localization application with average errors of about 7cm. CRF meth- ods have certain negative traits, namely that the system requires a discrete number of states. For several applications, such as surgical skill evaluation, there is no deterministic method to define the number of necessary states. Another issue is that the training parameters cannot be subsequently used to inform the trainee about necessary improvements or errors. Finally the CRF model is not truly discriminant classifier in that optimal separation projections are not guaranteed.
In general the particle filtering method has the potential for use in a discriminant setting. The particle filter is favorable in situations with non-linear system models. However, in systems with high levels of noise relative to the inter-class separation, parameter estimation becomes increasing difficult. Additionally all PF based classification schemes are based on the closest parameter set to a known system. For situations where a known system model is not available, such as complex tissue models, the traditional particle filter approach will not work.