Cognitive structure adaptive particle filter for radar manoeuvring target tracking

IET Radar, Sonar & NavigationVolume 13, Issue 1 p. 23-30 Research ArticleFree Access Cognitive structure adaptive particle filter for radar manoeuvring target tracking Shuliang Wang, Corresponding Author Shuliang Wang wangshuliang211@sina.com orcid.org/0000-0002-5504-6589 College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorDaping Bi, Daping Bi College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorHuailin Ruan, Huailin Ruan College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorShiwa Chen, Shiwa Chen College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this author Shuliang Wang, Corresponding Author Shuliang Wang wangshuliang211@sina.com orcid.org/0000-0002-5504-6589 College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorDaping Bi, Daping Bi College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorHuailin Ruan, Huailin Ruan College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this authorShiwa Chen, Shiwa Chen College of Electronic Engineering, National University of Defence Technology, Hefei, 230037 People's Republic of ChinaSearch for more papers by this author First published: 01 January 2019 https://doi.org/10.1049/iet-rsn.2018.5043Citations: 8AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The accuracy and efficiency of the particle filter algorithm partly depend on the number of particles used in the estimation. The number of particles is specified beforehand and kept fixed in the process of radar manoeuvring target tracking (MTT). In practice this may be highly inappropriate since it ignores the uncertainty in the models and the varying dynamics of the processes. A cognitive structure adaptive particle filter (CSAPF) algorithm is proposed to increase the accuracy and efficiency by adapting the number of particles online. The name CSAPF is due to the fact that the authors change the number of particles dynamically with radar's real-time perceptual information entropy for MTT. The authors cognitive structure approach chooses a transmitted waveform to improve tracking accuracy and chooses the lower bound of required particles to improve tracking efficiency. Monte–Carlo simulation results show that the approach has significant improvements over the particle filter with a fixed number of particles and over the fixed waveform particle filter algorithm. 1 Introduction The radar measurements for manoeuvring target tracking (MTT) contains range, range rate, and bearing, which are linear in polar coordinates but are nonlinear in Cartesian coordinates. In the tracking process, there are many interference factors, including target random manoeuver and random non-Gaussian noise, which make the target state estimation become a complicated nonlinear and non-Gaussian problem. Within a Bayesian framework, the particle filter (PF) is a useful tool to perform dynamic state estimation for MTT [1-5]. The main idea is to use samples, also called particles, to represent the posterior distribution of the state given a sequence of radar measurements. As new information arrives, these particles are constantly reallocated to update the estimation of the state of the system. Theoretically, the more number of particles, the higher the accuracy of state estimation, while the computational cost is larger. Some methods for online changing the number of particles are proposed [6-10]. In [6], the number of particles is M when the tracking is satisfactory, and the number of particles is transferred to 4M when it requires more particles. In [7], the number of particles is adaptively controlled depending on the position variance information of particles. In [8], the number of particles changes during the resampling process. When the distance between two neighbouring particles is below a predefined threshold, the algorithm keeps the one with the largest weight and rejects the other. Meanwhile, the proposed method needs that the practitioner sets a threshold to assess that the innovation error is acceptable. However, the lower bound of the particle number required is not mentioned in the above three methods. The Kullback–Leibler distance (KLD) sampling [9] is a method to adaptively estimate the number of particles under the restriction of estimated error. The error is measured by the Kullback–Leibler-divergence between the true posterior distribution and the empirical distribution. The bound given by KLD-sampling only uses information about the complexity of the true posterior, but it ignores any mismatch between the true and the proposed distribution [10]. By using the strong law of large numbers, it is possible to show that the estimation of the mean given by the PF is asymptotically unbiased at each iteration. Under the restriction of the desired error, the lower bound for the number of particles is given in [10]. From the equation of the lower bound, the number of particles required is proportional to the estimated error (uncertainty of state estimation) and is inversely proportional to the upper bound for the desired error. However, there is no further study about how to describe the uncertainty of multi-dimensional states (such as range, velocity, and acceleration in MTT), and the lower bound of required particle samples for MTT is not given explicitly. The uncertainty of state estimation is due to the predicted error and the measurement error. The measurement error covariance can be quantified by the Cramer–Rao lower bound (CRLB), which is derived from the curvature of the waveform ambiguity function at the origin in the delay-Doppler plane [11]. When the predicted uncertainty ellipsoid and measurement uncertainty ellipsoid are orthogonal, the uncertainty of state estimation is the smallest [12]. Furthermore, inspired by the human cognition mechanism of perception-action cycle (PAC), memory, attention, and intelligence, Haykin first explicitly proposed a cognitive radar [12, 13]. The intrinsic trait of the cognitive radar is to adaptively change the transmitter waveform to improve the tracking performance through the radar's real-time perception of the target and environment [14-21]. In [22], the determinant of the estimated error covariance matrix, called perceptual information entropy, is used to describe the radar's real-time perception for target's dynamics. It is practicable to introduce the PAC mechanism of human cognition into PF. We call it the cognitive structure adaptive PF (CSAPF) algorithm. This method attempts to give results on the following issues: (i) How to describe the uncertainty of multi-state estimation for MTT? (ii) With the restriction of the desired error, is there a lower bound of the required number of particles for MTT? Also, what factors can influence the lower bound? Dimension, desired error, uncertainty, and other factors? (iii) Can the number of particles be changed with the radar's perception of the environment and target? So is there an adaptive PF (APF) with a dynamic number of particles? (iv) Can waveform selection and APF algorithm be combined to improve the tracking performance based on PAC mechanism? Also, how to deal with the large computation cost of the combination? (v) Whether we can design a parallel structure with an extended Kalman filter (EKF) and APF? In the structure, the EKF is used to adaptively select the waveform to provide the best measurements for APF algorithm. Meanwhile, the extra computational cost is not added to the APF algorithm. This paper is organised as follows. Section 2 provides the target tracking model. Section 3 derives the lower bound of particle number for MTT and gives the process of APF algorithm. Section 4 presents the structure of the CSAPF algorithm. Section 5 shows the results of applying CSAPF to a radar MTT task. Finally, Section 6 presents the main conclusions of this work. 2 Target tracking model 2.1 Target state and measurement model 2.1.1 Target state model The target dynamics is modelled by a linear model by (1) where is the state transition matrix, is the acceleration input matrix, is the mean acceleration, and is the process noise. is the state vector of the target, which represents the range, velocity, and acceleration in the Cartesian coordinates, respectively. 2.1.2 Measurement model The measurement equation of the target is (2) where are the measurements of range, range rate, and bearing of the target. is the white Gaussian measurement noise with zero-mean and covariance matrix . Here, is related to radar transmitted waveform parameters (see Section 4). 2.2 Entropic state model Target motion and environmental change generate a series of uncertain problems. From the viewpoint of information theory, we can perceive the motion characteristics of the radar by means of information entropy [22] (3) where is the entropic state at time k in accordance with the state posterior probability in the Bayesian sense. As such, is the state of the preceptor and is a quantitative measure such as Shannon's information entropy by (4) It is important to note here that, in general, Shannon's entropy could assume the value zero; however, in cognitive control, the entropic state will always have non-zero, positive value due to the fact that the environment always involves uncertainty and we can never reach perfect target state reconstruction with 100% accuracy. Performance evaluation of the tracking model can be characterised by information entropy. Based on the PAC mechanism, the waveform in the radar transmitter can be selected adaptively according to the perception of the environment and target in the radar receiver. Thus, the perceptual information entropy can be used as a bridge between them. The equivalent expression of information entropy for short is given in [22] (5) where denotes the determinant of the operator, and the matrix is the estimated error covariance of the target at time k. 3 PF and dynamic number of particles 3.1 Sequential importance resampling (SIR) PF is a filter that estimates the state using a number of particles. Each particle has information about the dynamic system. In the PF through sequential importance sampling (SIS) [1, 2], the probability density function of the prior distribution of known states is assumed to be . is used to describe the posterior probability distribution of the target state at time k with N particles. is the particle set whose corresponding weight value set is . is the state set from time 0 to time k, and N is the number of particles. If the weights are normalised by , the posterior probability density distribution of the target state can be discretely weighed to (6) Weights are selected by important sampling (7) where is the importance distribution. is the prior distribution of the hidden Markov model. is the likelihood function. A simple choice is to use the prior distribution as importance function (8) The weights of importance can be simplified to (9) The degeneracy of the SIS algorithm is unavoidable. To reduce the degeneracy of the SIS algorithm, the SIR algorithm is proposed. In the SIR algorithm, the effective sample size is to measure the degeneracy. When it is below a fixed threshold , the resampling procedure is executed. The number of particles in the SIR is fixed in the whole estimation process. When the number of particles is large, the estimated accuracy can be improved well. However, the computation load becomes challenging to the hardware resources. In this study, based on the SIR, we deeply analyse the relationship among a number of particles, state dimension, and information entropy, in order to dynamically adjust the number of particles for the next time through the radar's real-time perception of the target. 3.2 APF algorithm If the number of particles in the state estimation is N, the discrete expression form of conditional probability density can be obtained. Considering the probability defined in space S, if , the mathematical expectation of is defined as (10) The expectation equation (10) is always recorded as the true target state. The analytic solution of (10) can be expressed in the form of a sum of discrete samples. If is the sample set from a posteriori probability, the empirical mean based on the function of sample is (11) With the assumption of that is the asymptotically unbiased estimation of . We record it as the estimated target state . Under the linear and Gaussian hypothesis condition, the conditional probability of the estimated state at time k can be expressed as (12) where () is the samples obtained from the upper equation of (12), which means the estimated state of every particle, is the dimension of estimated states, is the estimated error covariance matrix, and note also that the covariance matrix is assumed strictly positive definite, which also implies that its inverse exists. 3.2.1 Uncertainty ellipsoid Now, we further analyse the geometric structure of (12). The contours of any probability density function are defined by the curves of constant values of the density, given by the equation constant. For the multivariate Gaussian density, this is equivalent to requiring the exponent to be a constant . Also, all the effective particles to estimate the target state are included in the ellipsoid with the following form: (13) The main axis of this super ellipsoid is parallel to the eigenvector and its half axis length is . Fig. 1 is a two-dimensional state uncertainty ellipse. Fig. 1Open in figure viewerPowerPoint Two-dimensional state uncertainty ellipse The size of the ellipsoid (ellipse when ) reveals the uncertainty of state estimation. We call this ellipsoid the uncertainty ellipsoid. The relationship between the volume and the estimated error covariance matrix is given in [23] (14) Here, is a positive proportional coefficient associated with the state dimension. The probability of a real particle falling into -dimensional ellipsoid depends on the exponent constant (see Appendix 1). If is higher, the probability is higher too. Before proceeding further, the important remarks deserve particular attention: (a) Uncertainty ellipsoid can be used to describe the uncertainty of target state estimation. Also, its geometry is due to the estimated error covariance matrix. (b) The volume of uncertainty ellipsoid is proportional to the 1/2 power of information entropy. (c) If the state dimension and information entropy are determined, and the tracking error desired is low, it means is small, the probability of particle falling into the ellipsoid is small too. Thus, we need more particles to make sure that the random particles can fall into the uncertain ellipsoid to estimate the target state. 3.2.2 Dynamic number of particles The state estimated error of each particle can be expressed as (15) where . We define the square of the state estimated error as (16) So, the mean of the state estimated error square with N particles can be expressed as (17) where is a chi-square distribution with degrees of freedom. So, obeys distribution with degrees of freedom. When we use particles to estimate the state, the confidence region is , where is the confidence level of the distribution with degrees of freedom. Also, the corresponding confidence region of is (18) Fisher [24] has proved that when the degree of satisfy , there is an approximate expression (19) where denotes such a critical value of standard normal distribution with the quantile of . The following expression also can be obtained when (20) Combining (14), (18) and (20), the confidence region for defining the volume of the uncertainty ellipsoid is , so the confidence region volume can be expressed (see Appendix 2) as (21) From (21), we can see that under a specific state dimension , confidence level and estimated error covariance matrix , the confidence ellipsoid volume decreases with the increase of particles, so the lower bound of particles required can be expressed as (22) The APF algorithm with a dynamic number of particles is given as follows: Algorithm.APF for radar MTT(a) Parameter initialisationThe initial number of particles , the upper and lower bounds of the particle number , , the threshold of the resampling , the confidence covariance matrix and constant , which determine the tracking uncertainty required in engineering.The particle swarm is generated by a prior probability , and the weights of all particles are .(b) UpdateAt time k, using the prior distribution (8) as the importance distribution (23) (24)(c) ResamplingIf , do resampling [2].(d) OutputState estimation (25) (26)(e) Calculating the number of particles for the next timeAccording to formula (5), we calculate the information entropy with (26), then, calculate the number of particles with formula (22). The next time, we use the number of particles to estimate the target state for MTT. Before proceeding further, from the relationship expression (22), the following remarks deserve particular attention: (a) The number of particles is proportional to the information entropy , and the larger the estimated error, the more the particles needed. (b) The number of particles is inversely proportional to the square of confidence region volume , which means if the tracking accuracy desired is high ( is small), the particles needed are more. In engineering application, we often need to set the tracking accuracy threshold to judge whether the track work or fail. When the confidence covariance matrix and constant are determined, the confidence region volume can be achieved according to (14). (c) When the constant is determined, the corresponding confidence level is determined (see Appendix 1, the probability of particles falling into the ellipsoid). Also, can be obtained by the looking-up table. (d) The number of particles is proportional to , which means that if the dimension increases, the number of particles will increase sharply. 4 CSAPF algorithm A narrow-band signal model is given in [11]. A Gaussian pulse base band signal envelope is (27) with , where is the rise and fall time and is the duration of the Gaussian envelope. We use as the waveform parameter that will be optimised by the waveform-selection algorithm. For an optimum receiver, the measurement noise covariance of (range and range rate) can achieve the CRLB. Therefore, the CRLB can be obtained by inverting the Fisher information matrix through the following formula [11]: (28) where is the signal-to-noise ratio (SNR), c is the speed of light. In [15], the measurement error covariance of bearing is also considered, and its expression is (29) where denotes the 3 dB beamwidth of the radar antenna and is the monopulse error slope. Then it is possible to decouple into , i.e. (30) In the waveform selection tracking algorithm, it is important to select the next waveform from the existing waveform library with a certain cost function, so as to keep the tracking performance of the system the best. Here, we choose the information entropy described in Section 2.2 as the cost function (31) where represents the filter structure and is the range of . If we use the SIS algorithm to estimate the state error covariance matrix (from (26)) and to select the waveform parameter, the computation cost would be very large. In fact, some local approximation methods, such as the EKF algorithm, can satisfy the requirement of waveform selection. Based on the PAC mechanism of human cognition, the CSAPF algorithm is proposed. As shown in Fig. 2, in a radar receiver, a parallel structure with an EKF and APF is adopted, and the manoeuvring model is used to track the manoeuvring target. (i) At every tracking cycle, the filtering state is provided by APF, which avoids the divergence of the EKF, and the information entropy corresponding to different waveforms are stored and fed back to the radar transmitter by the EKF; (ii) The transmitter selects the best waveform from the waveform library to perceive the environment and the target according to a certain dynamic optimisation criterion; (iii) the measurements with selected waveform is used for APF at the next time to estimate the state of target. Meanwhile, the number of particles for APF at the next time is calculated by the perceptual information entropy with (22) and (26). Fig. 2Open in figure viewerPowerPoint Cognitive structure APF for MTT During the whole process, EKF and APF are parallel. EKF waveform selection works together with APF, which would not add the computation cost to APF. Also, the information entropy is kept least by the waveform selection, which makes the APF can always achieve the best tracking performance with the smallest number of particles. 5 Simulation results Here, the radar, located at (0, 0) m, provides the range, the range rate and the bearing measurements with the sampling interval . An aircraft, starting from m at time s, flies for 18 s with the initial velocity (−100, −50) m/s. Then, it performs left turn with the turn rate for 25 s. After the turn, the aircraft continues with current velocity for 10 s. Then, the aircraft performs right turn with the turn rate for 26 s. After the turn, the aircraft continues with current velocity for 21 s. The scenario is shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Considered sample tracking scenario (a) Target trajectory, (b) Variation of the target velocity In the PF, the resampling threshold is set to 1/3 times the current number of particles. The dimension of the target state is 6, containing range, velocity, and acceleration in x, y coordinates, respectively. If confidence covariance matrix and constant value are given, according to the volume formula (14), we can get the confidence region volume . The radar transmitted waveform is the X band, whose carrier frequency is 10.4 GHz, and the transmitted signal is followed as the type (27). The duration of the Gaussian envelope in the waveform library is chosen to be s, the interval is s. The half power beam width of the antenna is set to be 3°. There are ten kinds of optional waveforms. Each waveform has the same energy, but its precision of range and range rate are different because of the different pulse width. Under constantly transmitted energy constraint, the range dependent SNR is modelled according to (32) where r is the range between the target and radar. is the SNR for the target at the range and implicitly covers all the other factors except range in the SNR equation. Here, r0 is set to 50 km. To evaluate the tracking quality of the given algorithm, it is important to select proper criteria for tracking accuracy, efficiency, and the tracking loss. Here, (a) range and velocity estimation root mean square error (RMSE), and the average RMSE (ARMSE) are used to describe the accuracy; (b) the average computation time is used to describe the efficiency. (c) When the range estimation RMSE is greater than or equal to times confidence range, a track is considered to be lost in one scan. Also, the confidence range is determined by the pre-assumption of confidence covariance matrix . If the range error covariance in x, y coordinates are assumed to , , respectively, the confidence range is equal to . In the simulation, we use the current statistical model [25] for MTT. Its maximum acceleration is set to 50 m/s2, and the manoeuvring frequency constant is set to 1/60. 5.1 Not considering the waveform selection Here, the APF is an algorithm without considering the waveform selection. Also, the measurement accuracy of range, range rate, and bearing are fixed set to 150 m, 5 m/s, and 0.5°, respectively. The confidence covariance matrix is set to the constant is set to 9 ( can be calculated from Appendix 1, and can be obtained by using the looking-up table), so the confidence range is m, and the track loss threshold is 127.2 m. The upper and lower bounds of particles are set to 2000 and 400. The fixed particle number of the SIR-1/SIR-2 algorithm is 2000/500. The initial particle number is 500. With 50 times Monte–Carlo simulations, Table 1 gives the range and velocity tracking ARMSE, average computation time in one simulation, and the number of track loss, with fixed particles and adaptive particles. From Table 1, as expected, SIR-2 requires less computation time than SIR-1 and APF but has lower tracking accuracy and a larger number of track loss. The ARMSE given by SIR-1 is the lowest one, but with the more computation time. The APF has less computation time than SIR-1, and have better tracking accuracy than SIR-2. The number of track loss in APF is small, which can meet the requirement of engineering. Table 1. Performance comparison of three algorithms Algorithm Range ARMSE, m Velocity ARMSE, m/s Average computation time, s Number of track loss SIR-1 88.00 29.63 4.7255 0 SIR-2 108.72 34.84 1.1430 15 APF 94.86 33.99 2.7910 2 Furthermore, we choose different turn rate to analyse the influence of target manoeuver and choose different exponent constants to analyse the influence of the confidence region volume. In Fig. 4, (a) , the turn rates are set to /s, /s and /s, respectively; (b) /s, the exponent constants are set to , and , respectively. From Fig. 4, because of the insufficiency of the prior information, the number of particles needed is more. Fig. 4Open in figure viewerPowerPoint Number of particles in different manoeuver and exponent constant condition (a) Variation of particle number with target manoeuver intensity, (b) Variation of particle number with exponent constant Along with the continuous tracking and environment perception, the estimated performance gradually stabilises, the number of particles decreases and tends to be stable. As expected, the number of particles increases with the increase of the manoeuver intensity of the target. Also, it increases with the decrease of the exponent constant , which means that if the confidence region volume is smaller, the number of particles needed is large. From the comparison of the algorithms, we can easily get the following conclusion: (a) In the same condition, if the manoeuver of the target is strong, the number of particles needed is large. (b) If the high tracking accuracy is required, the number of particles needed is large. So, we should consider the requirement of accuracy and efficiency at the same time, thus obtain the performance compromise. 5.2 Considering the waveform selection The tracking performance comparison of the CSAPF algorithm and the APF with the fixed waveform algorithm is shown in Fig. 5 and Table 2. The fixed waveform parameter 1 is s, and the fixed waveform parameter 2 is s. The results of the dynamic waveform duration selection for the CSAPF is given in Fig. 6. The comparison of information entropy is shown in Fig. 7. Fig. 8 shows the dynamic change of the particles number. Table 2. Comparisons of performance metrics for CSAPF and APF with a fixed waveform tracking APF with waveform 1 APF with waveform 2 CSAPF with dynamic waveform EKF waveform selection APF Average computation time, s 6.87 7.19 0.47 4.77 Range ARMSE, m 104.26 67.35 — 53.98 Velocity ARMSE, m/s 21.87 29.10 — 23.78 Fig. 5Open in figure viewerPowerPoint Comparisons of range and velocity RMSE from tracks (a) Range RMSE comparison, (b) Velocity RMSE comparison Fig. 6Open in figure viewerPowerPoint Dynamic selection of pulse duration time Fig. 7Open in figure viewerPowerPoint Tracking uncertainty of three algorithms Fig. 8Open in figure viewerPowerPoint Dynamic number of particles for APF and CSAPF algorithm It is clear that the range RMSE of APF with waveform 1 is worse than that with waveform 2. Also, the velocity RMSE of the APF with waveform 1 is better than that with waveform 2. In the CSAPF algorithm, the waveform parameter is selected dynamically to balance the measurement error of range and range rate. From the comparison of information entropy in Fig. 7, we can find that the APF algorithm with waveform 2 is better than that with waveform 1, and the CSAPF algorithm is much better than that with the fixed waveform. Meanwhile, the tracking stability of the CSAPF algorithm is superior and robust. From Fig. 7 and Fig. 8, we can see that the CSAPF algorithm chooses the smallest information entropy waveform at any time to ensure that the number of particles is least. From the comparison of the average computation time in Table 2, the efficiency of the CSAPF is higher than that with the fixed waveform. The time consumption of the EKF waveform selection is lower than the APF, so it would not add extra time to the APF in the parallel structure. From the comparison of the algorithms, we can see that the waveform selection is an important part of the proposed algorithm. It has the following three advantages: (a) The measurement error of range and range rate can be balanced so that the performance of tracking can be improved. (b) Through the EKF waveform selection, the measurement is provided to APF, which ensures that the information entropy is minimum and the number of particles is least. (c) The number of particles can be changed dynamically according to the radar's real-time perception of the target. The better tracking performance can be achieved with affordable computation cost. 6 Conclusions In this study, we have investigated the information entropy, which is used to evaluate the radar's real-time perception of the target. Combined with the uncertainty ellipsoid theory, the lower bound of a number of particles is derived, and an APF algorithm is proposed. To further improve the accuracy and efficiency, we propose a CSAPF algorithm, which is a parallel structure algorithm with EKF and APF. According to the criterion of minimum information entropy, EKF is used to select the waveform for APF. Thus, APF can achieve best tracking accuracy with the least number of particles, by taking into account computational efficiency. 7 Acknowledgments The research was supported by the National Natural Science Foundation of China (61671453) and the Anhui Province Natural Science Fund Project (1608085MF123). 9 Appendices 9.1 Appendix 1 The volume of the ellipsoid with dimension is as follows: (33) where can be expressed as (34) The probability (or confidence level ) of a real particle falling into dimensional ellipsoid (depending on the exponent constant ; Table 3). Table 3. Probability of particles falling into the ellipsoid 1 2 3 4 5 6 where 9.2 Appendix 2 According to (18), the confidence region volume can be written as (35) According to (21), the upper equation of (35) can be written as (36) According to approximate (20), (36) can be approximated to the following expression: (37) , (37) can be written as (38) Owing to , we can obtain the approximate expression of (38): (39) Since the expression is a very small value, can be approximately equal to . Thus, (39) can be approximated to the following expression: (40) 8 References 1MacEachern, S.N., Clyde, M., Liu, J. S.: 'Sequential importance sampling for nonparametric Bayes models: the next generation', Can. J. Stat., 1999, 27, (2), pp. 251– 267 2Doucet, A., Godsill, S., Andrieu, C.: 'On sequential Monte Carlo sampling methods for Bayesian filtering', Stat. 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