Lecture 22: Dynamic Filtering

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1 ECE 830 Fall 2011 Statistical Signal Processing instructor: R. Nowak Lecture 22: Dynamic Filtering 1 Dynamic Filtering In many applications we want to track a time-varying (dynamic) phenomenon. Example 1 Tracking temperature or humidity in a museum room with an inaccurate device. Key: Temperature changes slowly with time so we should be able to average across time to obtain better estimates. How to do this? Model dynamics of temperature changes and noise/uncertainties in measurement. 2 Dynamical State Equation (Prior) Let x 1, x 2,... denotes quantity ( state ) of interest. The state is changing over time and we will model this variation stochastically as follows. The state at time n depends causally on the past. Let p(x n x n 1, x n 2,..., x 1 ) denote the conditional distribution of the state at time n given all the past states. This distribution is a n-variate function, and as n grows it becomes more and more complex (to specifiy, to compute, etc). A reasonable simplifying assumption is to assume that the probability distribution of the state at time n depends only on value of the state at time n 1, a so-called Markovian assumption, p(x n x n 1,..., x 1 ) = p(x n x n 1 ). Note that p(x n x n 1 ) is bivariate and therefore much simpler than the general causal model. To define the state process we must to specify (a) p(x 1 ), the initial state distribution (b) p(x n x n 1 ), n = 2, 3,..., the state transition probability density functions This is illustrated in the following example. Example 2 Santa Tracker On December 25th legend has it that Santa Claus makes his way around the globe, delivering toys to all the good girls and boys. Tracking Santa s delivery trip has attracted considerable 1

2 Lecture 22: Dynamic Filtering 2 interest by the signal processing research community in recent years, see Here is a simple approach to the problem. x(t) = Santa s position at time t on Christmas Eve x(t) t = v(t), velocity We can sample Santa s position once every second, producing a sequence of position values x 1, x 2,.... His velocity is also represented by a discrete-time process v 1, v 2,... We use the following model for Santa s dynamics: [ ] xn+1 v n+1 = [ ] [ xn v n ] + [ 0 σ 2 ] u n, u n N (0, 1), small Also, Santa s initial position is the North Pole, denoted by x 0. So we take p(x 1 ) = δ(x 1 x 0 ). In words, Santa s position at time x n+1 is equal to his position at time n plus a small step proportional to his velocity. His velocity is modeled as a Gaussian white noise process, representing the fact that he randomly speeds up and slows down as he makes his stops around the world. 3 Observation Model (Likelihood) Usually we cannot observe x n directly. Instead we observe z 1, z 2,..., which are noisy and/or indirect measurements related to the states. Example 3 Here are a few examples of observation processes. z n = x n + w n, w n N (0, σ 2 ), simple signal+noise model z n = Ax n + w n, where A is a matrix representing, for example, a blur z n = f(x n ) + w n, f is a non-linear function Let p(z n x n ) denote the likelihood of x n based on observation z n. We can combine the likelihoods and the priors p(x n x n 1 ) to compute the posterior distribution of x = (x 1,..., x n ) given z = (z 1,..., z n ) p(x z) p(z x)p(x) = n p(z i x i )p(x i x i 1 ). The posterior can be computed efficiently in an incremental fasion by exploiting Markovian structure of state transitions (prior). This incremental procedure is called Density Propagation. 4 Density Propagation Density Propagation is an incremental procedure for efficiently computing p(x n z 1,..., z n ). First let s establish some notation. i=1 Prior: S n (x n x n 1 ) := p(x n x n 1 ), P 1 (x 1 ) = p(x 1 ) Likelihood: L n (z n x n ) := p(z n x n ) Posterior: F n (x n ) := p(x n z 1,..., z n )

3 Lecture 22: Dynamic Filtering 3 Prediction: P n (x n ) := p(x n z 1,..., z n 1 ) P n (x n ) is the prediction of the value of x n using only observations up to time n 1, and this will play a key role in the Density Propagation algorithm. 4.1 Density Propogation Algorithm n = 1: predict x 1 : observe z 1 and x 1 p 1 (x 1 ) F 1 (x 1 ) = p(x 1 z 1 ) = p(z 1 x 1 )p(x 1 ) L 1 (z 1 x 1 )p 1 (x 1 ) n = 2: predict x 2 : p(x 1, x 2 z 1 ) = p(x 1, x 2, z 1 ) = p(x 2 x 1, z 1 )p(x 1 z 1 ) = p(x 2 x 1 )F 1 (x 1 ) = S 2 (x 2 x 1 )F 1 (x 1 ) p(x 2 z 1 ) = S 2 (x 2 x 1 )F 1 (x 1 )dx =: P 2 (x 2 ) observe z 2 and at time step n: predict x n : F 2 (x 2 ) = p(x 2 z 1, z 2 ) = p(x 2, z 1, z 2 ) p(z 1, z 2 ) = p(z 2 x 2 )p(x 2 z 1 ) p(z 1, z 2 ) L 2 (z 2 x 2 )P 2 (x 2 ) P n (x n ) = p(x n z 1,..., z n 1 ) = S n (x n x n 1 )F n 1 (x n 1 )dx n 1 observe z n and F n (x n ) = p(x n z 1,..., z n ) L n (z n x n )P n (x n )

4 Lecture 22: Dynamic Filtering Block Diagram Figure 1: Block diagram of dynamic filtering Filtering F n (x n ) = L n (z n x n )P n (x n ) Figure 2: The filtering or focus portion of the dynamical filtering block diagram Prediction P n+1 (x n+1 ) = S n (x n+1 x n )F n (x n )dx n

5 Lecture 22: Dynamic Filtering 5 Figure 3: The prediction or diffusion portion of the dynamical filtering block diagram. 5 Estimating x n We have many possibilities. Given, F n (x n ) = p(x n z 1,..., z n ) We can minimize various risk functions based on a loss and the posterior distribution F n. l 2 : x n = arg min E Fn [(x n x) 2 ] ex = x n F n (x n )dx n l 1 : l 0/1 : x n = arg min E Fn [ x n x ] ex x n = arg max x F n(x n )

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