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Figure 1 - from "The Unscented Particle Filter"

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arrow_forwardFigure 1: In this example, a particle filter starts at time ¢ — 1 with an unweighted measure (x2, N71}, which provides an approximation of p(xz-1|y1.:-2). For each particle we compute the importance weights using the information at time ¢ — 1. This results in the weighted measure (x, OO, , which yields an approximation p(X:-1|¥1:2-1)- Subsequently, a resampling step selects only the “fittest” particles to obtain the unweighted measure {x,, N-1}, which is still an approximation of p(Xt-1|¥1:t-1) - Finally, the sampling (prediction) step introduces variety, resulting in the measure {x , N-}. where “*$ denotes almost sure convergence and f; : R’= > R’% is some func- tion of interest. For example, it could be the conditional mean, in which case fi (Xo) = Xo, or the conditional covariance of x; with f;(xo%) = xix}; —

Figure 1: In this example, a particle filter starts at time ¢ — 1 with an unweighted measure (x2, N71}, which provides an approximation of p(xz-1|y1.:-2). For each particle we compute the importance weights using the information at time ¢ — 1. This results in the weighted measure (x, OO, , which yields an approximation p(X:-1|¥1:2-1)- Subsequently, a resampling step selects only the “fittest” particles to obtain the unweighted measure {x,, N-1}, which is still an approximation of p(Xt-1|¥1:t-1) - Finally, the sampling (prediction) step introduces variety, resulting in the measure {x , N-}. where “*$ denotes almost sure convergence and f; : R’= > R’% is some func- tion of interest. For example, it could be the conditional mean, in which case fi (Xo) = Xo, or the conditional covariance of x; with f;(xo%) = xix}; —

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Figure 1: In this example, a particle filter starts at time ¢ — 1 with an unweighted measure (x2, N71}, which provides an approximation of p(xz-1|y1.:-2). For each particle we compute the importance weights using the information at time ¢ — 1. This results in the weighted measure (x, OO, , which yields an approximation p(X:-1|¥1:2-1)- Subsequently, a resampling step selects only the “fittest” particles to obtain the unweighted measure {x,, N-1}, which is still an approximation of p(Xt-1|¥1:t-1) - Finally, the sampling (prediction) step introduces variety, resulting in the measure {x , N-}. where “*$ denotes almost sure convergence and f; : R’= > R’% is some func- tion of interest. For example, it could be the conditional mean, in which case fi (Xo) = Xo, or the conditional covariance of x; with f;(xo%) = xix}; —
As mentioned earlier, using the transition prior as proposal distribution can be inefficient. As illustrated in Figure 2, if we fail to use the latest available informa- tion to propose new values for the states, only a few particles might survive. I[¢ is therefore of paramount importance to move the particles towards the regions of high likelihood. To achieve this, we propose to use the unscented filter as proposal distribution. This simply requires that we propagate the sufficient statistics of the UKF for each particle. For exact details, please refer to our technical report (van der Merwe et al. 2000). Figure 2: The UKF proposal distribution allows us to move the samples in the prior to regions of high likelihood. This is of paramount importance if the likelihood happens to lie in one of the tails of the prior distribution, or if it is too narrow (low measurement error).
Table 1: Mean and variance of the MSE calculated over 100 independent runs. is used. The observation noise, nz, is drawn from a zero-mean Gaussian distribution. Given only the noisy observations, y:, a few different filters were used to estimate the underlying clean state sequence x; for t = 1...60. The experiment was repeated 100 times with random re-initialization for each run. All of the particle filters used 200 particles. Table 1 summarizes the performance of the different filters. The
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Figure 3: Plot of the state estimates generated by different filters. table shows the means and variances of the mean-square-error (MSE) of the state estimates. Note that MCMC could improve results in other situations. Figure 3 compares the estimates generated from a single run of the different particle filters. The superior performance of the unscented particle filter is clearly evident. Figure
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Figure 4: EKF and UKF estimates of state covariance. 7 Conclusions
Eric Wan
Rudolph van der Merwe, Eric Wan
The Unscented Particle Filter
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