observed: y(t) y(t + dt) y(t + 2dt) y(t + 3dt)
hidden: x(t) x(t + dt) x(t + 2dt) x(t + 3dt) . . .
Figure 1.9: Continuous-time filtering can be considered as limiting case of discrete-time filtering, where the time interval between the successive states and the corresponding measurements goes to zero.
In analog communication systems the measured signals are typically con- tinuous-time processes and analog receivers are devices, which demodulate or estimate the actual transmitted continuous-time signals from the noisy measured signals. Also in many analog (electrical or mechanical) control systems operating without digital computers the measured signals are continuous-time, not discrete- time signals.
Optimal continuous-time filtering (Bucy and Joseph, 1968; Jazwinski, 1970;
Gelb, 1974; Liptser and Shiryayev, 1977; Kallianpur, 1980) considers stochastic inference problems, where the system model consists of a continuous-time state process x(t) and a continuous-time measurement process z(t). Thus, continu-
ous-time filtering considers the limiting filtering problem, where the time interval between the states and measurements goes to zero, as illustrated in the Figure 1.9. In the limitdt → 0 both the state and measurement sequences become random functions, which can be modeled as stochastic differential equations of the form2
dx(t)
dt = f (x(t), t) + L(t) w(t) z(t) = h(x(t), t) + V(t) e(t),
(1.8)
where w(t) and e(t) are white noise processes.
The purpose of the continuous-time optimal (Bayesian) filter is to compute the posterior distribution (or the filtering distribution) of the process x(t) given
the observed process{z(τ) : 0 ≤ τ ≤ t}
p(x(t) | {z(τ) : 0 ≤ τ ≤ t}). (1.9)
2
As in the continuous-discrete filtering case, formally, the continuous-time filtering problem is defined in terms of Brownian motion driven Itô processes. Due to this the formal measurement process is actually the integral of measurement process y(t) =Rt
0z(t) dt. In this thesis we only
consider non-linear continuous-time optimal filtering problems with additive noise, which is also the reason why we can safely use the white noise interpretation of the equations.
An example of continuous-time filtering is shown in the Figure 1.10. The under- lying signal is the same noisy resonator as in the continuous-discrete filtering case in the previous section. The measurement signalz(t) is a continuous-time process
and it has been truncated a bit for visualization purposes such that it does not take values from minus infinity to plus infinity as it formally should.
0 2 4 6 8 10 12 14 16 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 Time Signal Measurement Filter Estimate 95% Quantiles
Figure 1.10: An example of continuous-time filtering result.
The optimal continuous-time smoothing equations (Leondes et al., 1970) can
be used for computing the (smoothed) posterior distribution
p(x(t) | {z(τ) : 0 ≤ τ ≤ T }), (1.10) of the state at timet given the history of the measurements up to the time instance T > t.
The Kalman filter equations can be generalized to continuous-time measure- ment processes and the resulting filter is called the Kalman-Bucy filter (Kalman and Bucy, 1961). In this filter both the state process and measurements are mod- eled as linear stochastic differential equations. The continuous-time linear filter- ing equations consist of a vectorial linear differential equation for the mean and a non-linear matrix Riccati differential equation for the covariance.
The non-linear Bayesian filtering equations can also be generalized to contin- uous time and the resulting equation is called the Kushner-Stratonovich equation (Kushner, 1964; Stratonovich, 1968; Bucy, 1965), which is a measure valued stochastic partial differential equation. The unnormalized version of the equation
is called Zakai equation (Zakai, 1969) (see also Kallianpur, 1980). However, the problem in these equations is that they only give the formal solution, and the actual computation of the distribution or its expectations would require an infinite amount of computational resources. For this reason, in general, approximations must be used. In certain cases, the equations do have finite dimensional solutions, which lead to Kalman-Bucy filters (Kalman and Bucy, 1961) and Bene˘s filters (Bene˘s, 1981).
The extended Kalman-Bucy filter (EKBF) (see, e.g., Gelb, 1974) approxi- mates the exact solution by replacing the non-linear model with a suitably lin- earized approximate model, which can be solved by the Kalman-Bucy filter. The EKBF can be interpreted as a method for forming a Gaussian process approxi- mation to the optimal filtering solution (i.e., the posterior process). In Section 3.3 the equations of the new unscented Kalman-Bucy filter are derived, in which the continuous-time optimal filtering solution is approximated by using the con- tinuous-time form of the unscented transform. Another general way of forming approximations is Monte Carlo sampling (Crisan and Lyons, 1997, 1999), where a set of weighted particles is used for approximating the posterior probability measure.
More information on practical continuous-time Kalman filtering and lineariza- tion based non-linear filtering can be found in books of Gelb (1974), Stengel (1994), Bar-Shalom et al. (2001), and Grewal and Andrews (2001). A bit more theoretical material and also material on more general non-linear filtering can be found in books of Bucy and Joseph (1968) and Jazwinski (1970). Analysis of continuous-time filtering problems in Itô calculus point of view can be found, for example, in books of Liptser and Shiryayev (1977), Kallianpur (1980) and Øksendal (2003). Good introduction to this point of view is given in the lecture notes of Karatzas (1988).
For general treatment of the probabilistic continuous-time smoothing problem and the related stochastic partial differential equations, see (Liptser and Shiryayev, 1977). There also exists so called Zakai forms of these equations (Antonelli and Elliott, 1986; Krishnamurthy and Elliott, 2002). Leondes et al. (1970) presents partial differential equations, which can be used for computing the smoothing solution once the filtering solution has been computed.
As in the filtering case, the optimal continuous smoothing problem can be solved exactly only if the model is linear Gaussian (Rauch et al., 1965; Gelb, 1974). Approximate solutions to more general non-linear continuous optimal smoothing problems can be computed by using the extended continuous-time Kalman-Bucy (Rauch-Tung-Striebel) smoother (see, e.g., Leondes et al., 1970; Sage and Melsa, 1971; Gelb, 1974), which uses a linear or quadratic approxi- mation of the non-linear continuous-time model. The novel continuous-discrete unscented Kalman smoother derived in Section 3.2 can also be used for approxi- mating the smoothing solutions of continuous-time filtering models.
Probabilistic Inference and
Stochastic Processes
This chapter briefly presents the principles of Bayesian inference and the theory of stochastic differential equations. The mathematical complexity is kept to the minimum but still at the level, which should be enough for understanding the theory of optimal filtering in the next chapter.
2.1
Bayesian Inference
This section provides a brief presentation of the philosophical and mathematical foundations of Bayesian inference. The connections to the classical statistical inference are also briefly discussed.