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Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments

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Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments Gerhard Kurz, Igor Gilitschenski, and Uwe D. Hanebeck Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics and Robotics, Karlsruhe Institute of Technology (KIT), Germany, [email protected], [email protected], [email protected] Abstract—Circular estimation problems arise in many ap- 0.5 plications and can be addressed with the help of circular WN distributions. In particular, the wrapped normal and von Mises WC distributions are widely used in the context of circular problems. 0.4 To facilitate the development of nonlinear filters, a deterministic VM sample-based approximation of these distributions with a so- called wrapped Dirac mixture distribution is beneficial. We pro- 0.3 pose a new closed-form solution to obtain a symmetric wrapped Dirac mixture with five components based on matching the first f(x) two circular moments. The proposed method is superior to state- 0.2 of-the-art methods, which only use the first circular moment to obtain three Dirac components, because a larger number of Dirac components results in a more accurate approximation. 0.1 Keywords—circular statistics, Dirac mixture, nonlinear filtering, moment matching 0 0 pi/2 pi 3pi/2 2pi x I. INTRODUCTION Figure 1: Probability density functions of WN, WC and VM Many estimation problems involve circular quantities, for distributions with identical first circular moment. example the orientation of a vehicle, the wind direction, or the angle of a robotic joint. Since conventional estimation algorithms perform poorly in these applications, particularly if the angular uncertainty is high, circular estimation methods samples. This is avoided in deterministic methods. However, such as [1], [2], [3], and [4] have been proposed. These methods deterministic methods for obtaining samples are typically more use circular probability distributions that stem from the field complicated and computationally demanding than nondeter- of directional statistics [5], [6]. ministic methods. To facilitate the development of circular filters, sample- In our previous publication [1], we have presented a based approaches are commonly used, because samples (i.e., deterministic approximation for von Mises and wrapped normal Dirac delta distributions) can easily be propagated through distributions with three components. This approximation is nonlinear functions. We distinguish deterministic and nondeter- based on matching the first circular moment. The first circular ministic approaches. In the noncircular case, typical examples moment is a complex number and both a measure of location for deterministic approaches include the unscented Kalman and dispersion. This approximation has already been applied to filter (UKF) [7], the cubature Kalman filter [8], and the smart constrained object tracking [15] as well as sensor scheduling sampling Kalman filter (S2KF) [9]. Nondeterministic filters for based on bearing-only measurements [16]. the noncircular case are the particle filter [10], the Gaussian particle filter [11], and the randomized UKF [12]. In this paper, we extend our previous approach [1] to match both the first and the second circular moment. This yields We focus on deterministic approaches because they have an approximation with five components. Even though this several distinct advantages. First of all, as a result of their approximation is somewhat more complicated, it can still be deterministic nature all results are reproducible. Second, the computed in closed form and does not require approximations. samples are placed according to a certain optimality criterion (i.e., moment matching [7] or shape approximation [13], [14]). II. PREREQUISITES Consequently, a much smaller number of samples is sufficient to achieve a good approximation. Third, nondeterministic In this section, we define the required probability distribu- approaches usually have a certain probability of causing the tions (see Fig. 1) by giving their probability density function filtering algorithm to fail just because of a poor choice of (pdf) and introduce the concept of circular moments. Definition 1 (Wrapped Normal Distribution). where L is the number of components, β1; : : : ; βL 2 [0; 2π) A wrapped normal (WN) distribution [5], [6] is given by the are the Dirac positions, w1; : : : ; wL > 0 are the weighting pdf coefficients and δ is the Dirac delta distribution [1], [16]. We PL 1 require wj = 1 to ensure that the WD distribution is 1 X (x − µ + 2kπ)2 j=1 f(x; µ, σ) = p exp − ; normalized. 2σ2 2πσ k=−∞ Unlike the continuous WN, WC and VM distributions, the where µ 2 [0; 2π) and σ > 0 are parameters for center and WD distribution is a discrete distribution consisting of a certain uncertainty respectively. number of Dirac delta components. These components can be seen as a set of samples and can be used to approximate a The WN distribution is obtained by wrapping a one- certain original density. WD densities are useful for nonlinear dimensional Gaussian density around the unit circle. It is of estimation because they can easily be propagated through particular interest because it appears as a limit distribution on nonlinear functions [1], just as Dirac mixture densities in Rn the circle, i.e., in a circular setting, it is reasonable to assume [9]. The WD distribution as defined above, does not contain an that noise is WN distributed. To see this, we consider i.i.d. infinite sum for wrapping, because wrapping a Dirac distribution random variables θi with E(θi) = 0 and finite variance. Then results in a single component according to the sum n 1 1 X X S = p θ δ(x + 2πk − β) = δ((x − β) mod 2π) : n n k k=1 k=−∞ converges to a normally distributed random variable if n ! 1. We still refer to the distribution as wrapped for consistency Consequently, the wrapped sum (Sn mod 2π) converges to a with the WN and WC distributions. WN distributed random variable. Definition 5 (Circular Moments). Definition 2 (Wrapped Cauchy Distribution). The n-th circular (or trigonometric) moment of a random The wrapped Cauchy (WC) distribution [5], [6] has the pdf variable x with pdf f(·) is given by [5], [6] 1 Z 2π 1 X γ n mn = (exp(ix) ) = exp(inx)f(x)dx ; f(x; µ, γ) = 2 2 ; E π γ + (x − µ + 2kπ) 0 k=−∞ where i is the imaginary unit. where µ 2 [0; 2π) and γ > 0. Circular moments are the circular analogon to the con- Similar to the WN distribution, a WC distribution is obtained n ventional real-valued moments E(x ). Note, however, that by wrapping a Cauchy distribution around the circle. Unlike mn 2 C is a complex number. For this reason, the first the WN distribution, here it is possible to simplify the infinite circular moment already describes both location and dispersion sum, yielding the closed-form expression of the distribution, similar to the first two conventional real- 1 sinh(γ) valued moments. The argument of the complex number is f(x; µ, γ) = : 2π cosh(γ) − cos(x − µ) analogous to the mean whereas the absolute value describes the concentration. Definition 3 (Von Mises Distribution). Lemma 1. The circular moments of WN, WC, VM, and WD A von Mises (VM) distribution [5], [6] is defined by the pdf distributions are given by 1 WN 2 2 f(x; µ, κ) = exp (κ cos(x − µ)) ; mn = exp(inµ − n σ =2) ; (1) 2πI0(κ) mWC = exp(inµ − jnjγ) ; (2) where µ 2 [0; 2π) and κ > 0 are parameters for location and n I (κ) concentration respectively, and I (·) is the modified Bessel VM jnj 0 mn = exp(inµ) ; (3) function of order 0. I0(κ) L WD X The modified Bessel function of integer order n is given mn = wj exp(inβj) : (4) by j=1 1 Z π In(z) = exp(z cos θ) cos(nθ)dθ π 0 Derivations can be found in [5], [6]. Here, In(·) is the modified Bessel function of order n [17]. The quotient of Bessel according to [17, eq. 9.6.19]. The von Mises distribution has a functions can be calculated numerically with the algorithm by similar shape as a WN distribution and is frequently used in [18]. Pseudo-code for this algorithm can be found in [1]. circular statistics. WN, WC and VM distributions are uniquely defined by their Definition 4 . (Wrapped Dirac Distribution) first circular moment. However, WN, WC and VM distributions A wrapped Dirac mixture (WD) distribution has the pdf with equal first moments significantly differ in their higher L moments. This is illustrated in Fig. 1 and Fig. 2. This difference X f(x; w1; : : : ; wL; β1; : : : ; βL) = wjδ(x − βj) ; motivates the use of second moments in deterministic Dirac j=1 mixture approximations. 1 1 β1 = −φ, β2 = φ, and equal weights w1 = w2 = 2 . For the WN first moment, we have WC 0.8 L 2 VM WD X m1 = wj exp(iβj) = cos(φ) : j=1 0.6 Solving for φ results in φ = arccos(m1). 0.4 2) Three Components: Now we extend the mixture with two components by adding an additional component at the second moment m mean. Consider the WD distribution with L = 3 components, 0.2 Dirac positions β1 = −φ, β2 = φ, β3 = 0, and equal weights 1 w1 = w2 = w3 = 3 . For the first moment, we have 0 L 0 0.2 0.4 0.6 0.8 1 X 1 first moment m mWD = w exp(iβ ) = (2 cos(φ) + 1) : 1 1 j j 3 j=1 Figure 2: First circular moment of wrapped normal, wrapped Cauchy, and von Mises distributions with mean zero plotted Notice that there is no imaginary part. Now, we match with m against their second circular moment. The moments are real- the first moment 1 of a WN or VM distribution and obtain valued in this case because µ = 0.
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