electronics
Article Two-Dimensional Monte Carlo Filter for a Non-Gaussian Environment
Xingzi Qiang 1 , Rui Xue 1,* and Yanbo Zhu 2
1 School of Electrical and Information Engineering, Beihang University, Beijing 100191, China; [email protected] 2 Aviation Data Communication Corporation, Beijing 100191, China; [email protected] * Correspondence: [email protected]
Abstract: In a non-Gaussian environment, the accuracy of a Kalman filter might be reduced. In this paper, a two- dimensional Monte Carlo Filter is proposed to overcome the challenge of the non-Gaussian environment for filtering. The two-dimensional Monte Carlo (TMC) method is first proposed to improve the efficacy of the sampling. Then, the TMC filter (TMCF) algorithm is proposed to solve the non-Gaussian filter problem based on the TMC. In the TMCF, particles are deployed in the confidence interval uniformly in terms of the sampling interval, and their weights are calculated based on Bayesian inference. Then, the posterior distribution is described more accurately with less particles and their weights. Different from the PF, the TMCF completes the transfer of the distribution using a series of calculations of weights and uses particles to occupy the state space in the confidence interval. Numerical simulations demonstrated that, the accuracy of the TMCF approximates the Kalman filter (KF) (the error is about 10−6) in a two-dimensional linear/ Gaussian environment. In a two-dimensional linear/non-Gaussian system, the accuracy of the TMCF is improved by 0.01, and the computation time reduced to 0.067 s from 0.20 s, compared with the particle filter. Keywords: nonlinear filter; non-gaussian environment; particle filter; sequence monte carlo Citation: Qiang, X.; Xue, R.; Zhu, Y. Two-Dimensional Monte Carlo Filter for a Non-Gaussian Environment. Electronics 2021, 10, 1385. https:// 1. Introduction doi.org/10.3390/electronics10121385 Bayesian inference is one of the most popular theories in data fusion [1–5]. For a linear Gaussian dynamic system, the Bayesian filter can be achieved in terms of the well-known Academic Editor: Chiman Kwan updating equations of the Kalman Filter (KF) perfectly [6]. However, the analytical solution of the Bayesian filter is impossible to be obtained in a non-Gaussian scenario [7]. This Received: 18 May 2021 problem has attracted considerable attention for a few decades because of the wide appli- Accepted: 5 June 2021 Published: 9 June 2021 cation in signal processing [8,9], automatic control systems [10,11], biological information engineering [12], economic data analysis [13], and other subjects [14]. Approximation is one
Publisher’s Note: MDPI stays neutral of the most effective approaches for solving the nonlinear/non-Gaussian filter problem. with regard to jurisdictional claims in The linearization of the state model is an important strategy for solving the nonlinear/non- published maps and institutional affil- Gaussian filtering problem. The extended Kalman filter (EKF) was introduced to approxi- iations. mate the nonlinear model using the first-order term of the Taylor expansion of the state and observation equations in [15]. In [16], the unscented Kalman filter (UKF) was proposed to reduce the truncation error by introducing the unscented transformation (UT) [17]. The cubature Kalman filter based on the third-degree spherical–radial cubature rule was proposed in [18]. The third-degree cubature rule is a special form of the UT and has better Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. numerical stability in the application of filtering [19]. The Gauss–Hermite filter and central This article is an open access article difference filter (CDF) were proposed by Kazufumi Ito and Kaiqi Xiong in [20] and made distributed under the terms and the Gaussian assumption for the noise model. conditions of the Creative Commons Sequential Monte Carlo (SMC) provides another important strategy for the nonlinear/non- Attribution (CC BY) license (https:// Gaussian filtering problem and can approximate any probability density function (PDF) creativecommons.org/licenses/by/ conveniently using weighted particles. Particle Filter (PF) [21,22] is an algorithm de- 4.0/). rived from the recursive Bayesian filter based on the SMC approach that is used to solve
Electronics 2021, 10, 1385. https://doi.org/10.3390/electronics10121385 https://www.mdpi.com/journal/electronics Electronics 2021, 10, 1385 2 of 17
data/information fusion in a nonlinear/non-Gaussian environment [23]. The SMC ap- proach was introduced in filtering to tackle a nonlinear dynamic system that is analytically intractable. The core idea of PF is to describe the transformation of the state distribu- tion through the propagation of particles in a nonlinear dynamic system, and represent the posterior probability using weighted particles. As a flexible approach for avoiding solving complex integral problems, PF is widely used in the data/information fusion of nonlinear systems, such as fault detection [24], cooperative navigation and localization, visual tracking, and melody extraction. In [25] and [26], EKF and UKF were introduced, respectively, to optimize the proposal distribution for the PF framework. The feedback PF was designed based on an ensemble of controlled stochastic systems [27]. Additionally, because of the advantage of the resampling technique in solving the degeneracy problem, various resampling schemes were proposed in [28–30]. Both strategies are based on Bayesian theory, and approximation is also their main approach for solving the nonlinear filtering problem [31]. However, the perspectives of these two strategies are different, which results in different characteristics. For the first strategy, the Kalman filter (KF) is considered as the representative of Bayesian inference. Obtaining an analytical solution that is close to the real posterior distribution is the trick that the first strategy attempts to solve [32]. Researchers have attempted to approximate complex nonlinear non-Gaussian problems to linear Gaussian problems that can be di- rectly solved using the KF [33]. This approximation inevitably leads to a truncation error. Therefore, many improved algorithms based on the first strategy have been proposed, mainly to reduce the effect of the truncation error on nonlinear filtering [34]. However, it is difficult to accurately obtain the posterior distribution of a nonlinear system [35]. The first strategy is always accompanied by linearization errors. Reducing the influence of the Gaussian assumption of non-Gaussian noise on filtering performance is also a major problem to be considered in the first strategy [36]. For the second strategy, the SMC method is used to solve the difficult problem of integration in Bayesian filtering [37]. Theoretically, this strategy (PF and its improved algorithms) is not constrained by the model of non- linearity and non-Gaussian environment [38]. However, PF has been plagued by sample degeneracy and impoverishment since it was proposed. Many scholars have proposed several improvement methods to mitigate the two problems [39–41]. Increasing the particle number is an original approach to solve the problems, but it is not very effective because the particle number needs to increase exponentially to alleviate the two problems, which inevitably affects the efficiency of the filter [42]. The two main approaches for solving the two problems are improving the proposal distribution and resampling [43,44]. The improvement of the proposal distribution might greatly alleviate the impoverishment problem to improve the performance of the filter [45]. Hence, related improved algorithms have been widely used in engineering practice, such as EPF and UPF [46]. Resampling, as an important means to alleviate the sample degeneracy of PF has been widely studied by many scholars [47,48]. However, the model used to improve the proposal distribution must be based on some known noise model (such as the Gaussian model in EPF and Gaussian mixture model in UPF), which cannot fully solve the impoverishment problem [49]. The resampling step might alleviate the sample degeneracy, accompanied by the introduction of the resampling error [31]. The main problem of the second strategy is that the particle utilization efficiency is not high, which affects the effect and efficiency of the filtering [40]. Among the existing algorithms, the PF is the most flexible. The main procedure of PF can be roughly summed up as: (1) obtain particles according to a proposed distribution; (2) calculate the prior weights and the likelihood weights according to the process noise model and measurement noise model respectively and mix them; and (3) divide the mixed weights by the corresponding density of the proposed distribution and normalization. After that, the posterior distribution is reflected from these weighted particles. From this procedure, we can observe that the process of select particles is random, but the weights are calculated precisely according to the noise model. This phenomenon might cause random disturbances which could affect the filtering accuracy. Electronics 2021, 10, 1385 3 of 17
To overcome the aforementioned problem, the two-dimensional Monte Carlo (TMC) method is proposed to improve the efficiency of the sampled particles. Then, the TMC filter (TMCF) algorithm is proposed to solve the non-Gaussian filter problem based on the principle of the TMC method. The main contributions arising from this study are as follows: (1) The TMC method, as a deterministic sampling method, is proposed to improve the efficacy of particles. Particles are sampled in the confidence interval uniformly according to the sampling interval. Then, the posterior weight of each particle is calculated based on Bayesian inference. Subsequently, any probability distribution can be described by a small number of weighted particles. (2) A discrete solution to the problem of how to describe a known probability distribution transmitted in a linear or nonlinear state model is proposed. First, a small number of original weighted particles are obtained according to TMC method. Then, the confidence interval of the next time step for a fixed confidence is calculated according to the state model. Some new particles are then set in this confidence interval uni- formly in terms of the sampling interval. After that, the weights of these new particles are obtained using a series of calculations based on Bayesian inference. Then, the transferred probability distribution is described by these new weighted particles. (3) The TMCF algorithm is proposed based on the above two points. The proposed algorithm can be divided into four parts: initialization, particle deployment, weight mixing, and state estimation. The TMC method is used in the initialization step to generate the efficacy weighted particles. Particle deployment solves the problem of state space transfer for a certain degree of confidence and deploys particles in the confidence interval. The weight mixing step achieves the fusion of several arbitrary continuous probability densities in a discrete domain. Some invalid weighted particles are omitted in the particle choice step and the state is estimated using the remaining weighted particles. (4) The performance of TMCF was verified using a numerical simulation. The results demonstrated that the proposed algorithm with the approach of fewer particles and less computation estimated accuracy better than the PF in linear and Gaussian systems and performed better than the KF and PF in linear and Gaussian mixture noise model. The outline of this paper is as follows. In Section2, the problem statement and Bayesian filter are presented. The TMC method is introduced in Section3. In Section4, the TMCF algorithm is introduced in detail. The numerical simulation is described in Section5, and the validity of the proposed framework is demonstrated. In Section6, the conclusion of this study is presented.
2. Problem Statement and Bayesian Filter 2.1. Problem Statement For filtering algorithms introduced in this paper, the state space model is defined as: [37] xt = f(xt−1) + ut−1 (1)
zt = h(xt) + vt (2)
nx ny where xt ∈ < and yt ∈ < denote the state variable and observation at time step t, respectively; nx and ny denote the dimensions of the state vectors and observation, n ny respectively; ut ∈ < x and vt ∈ < denote the system noise and observation noise, respectively; and the mappings f : 2.2. Bayesian Estimation Recursive Bayesian filtering provides an effective guide for the real-time fusion of the state equation and observation. The procedure of the Bayesian filter framework can be divided into prediction and update steps as follows: Z p(xt|z1:t−1 ) = p(xt|xt−1 )p(xt−1|z1:t−1 )dxt−1 p(zt|xt )p(xt|z1:t−1 ) p(xt|z1:t ) = p(zt|z1:t−1 ) where p(xt|xt−1 ) denotes the state transition PDF, p(xt−1|z1:t−1 ) denotes the posterior PDF at time step t − 1, p(xt|z1:t−1 ) denotes the prior PDF at time step t, p(xt|z1:t ) denotes the likelihood PDF and Z p(zt|z1:t−1 ) = p(zt|xt )p(xt|z1:t−1 )dxt (3) For a linear and Gaussian environment, this procedure can be accurately operated by the celebrated KF as the integral problem of Equation (1), and the likelihood probability p(zk|xk ) can be solved conveniently. For a nonlinear and non-Gaussian environment, it is impossible to solve Equation (1) directly. 3. Two-Dimensional Monte Carlo Method The Monte Carlo approach provides a convenient track inference of the posterior PDF in a non-Gaussian environment. PF is a branch of the family of filter algorithms and is based on the Monte Carlo approach. It is used to process the nonlinear and non-Gaussian system filter problem. Several improved particle filter algorithms exist. The core of the PF approach is to sample particles according to the difference in the proposal distribution. Particles are used to describe the transition of the PDF in the system model. The integration of the observation depends on the likelihood weight. The concept of weight provides the possibility for the application of Monte Carlo to the filtering problem, which plays an important role. In the following, the TMC method is introduced to make full use of the weight and the noise model to enhance particle efficiency. Suppose p(x, y) is the PDF of a two-dimensional noise model. Its marginal PDF can be expressed as: Z +∞ p(x) = p(x, y)dy (4) −∞ Z +∞ p(y) = p(x, y)dx (5) −∞ where p(x) and p(y) denote the marginal PDF of x and y, respectively. The confidence interval c for confidence 1 − α can be defined as: cx = [ux1 , ux2 ] (6) cy = uy1 , uy2 (7) c c = x (8) cy where: u Z x1 α p(x)dx = (9) −∞ 2 Z +∞ α p(x)dx = (10) 2 ux2 u Z y1 α p(y)dy = (11) −∞ 2 Electronics 2021, 10, 1385 5 of 17 Z +∞ α p(y)dy = (12) 2 uy2 Particles X can be set according to sampling interval dT for c, as shown in Figure1. Additionally, Electronics 2021, 10, x FOR PEER REVIEW 6 of 18 T dT ≡ dtx, dty (13) x1 x2 xn Electronics 2021, 10, x FOR PEER REVIEW X ≡ ··· (14) 6 of 18 y1 y3 yn where n denotes the particle number. Figure 1. Sketch map of setting particles. w The weight of these particles is calculated as: Figure 1. Sketch map of setting particles. T Figure 1. Sketch map of setting particles. w w w， w， (15) 12 n The weight of these particles w is calculated as: whereThe: weight of these particles w is calculated as: T w ≡ [w1, w2, ··· wn] (15) T pxy ii, ww=w12 w， w， n (15) i n where: pxy , (16) p(xi, yi) ii where: = i1 wi n (16) ( ) ∑ p xipxy, yi , i=1 p x y, ii Xw， wi = n dT Then, is used to describe with the accuracy of in the confi-(16) Then, {X, w} is used to describe p(x, y) withpxy the , accuracy of dT in the confidence c 1 ii denceinterval intervalc for confidence for confidence1 − α discretely. The discretely. sketchi1 map ofThe the sketch particles map and of their the weights particles and theiris shown weights in Figure is shown2. in Figure 2. p x y, Then, Xw， is used to describe with the accuracy of dT in the confi- dence interval c for confidence 1 discretely. The sketch map of the particles and their weights is shown in Figure 2. Figure 2. Figure 2.Sketch Sketch map map of of particles particles and and their their weights. weights. 0 dT 0 n Theorem 1. When and ,then and Figure 2. Sketch map of particles and their weights. n lim, xii wxp x y dxdy (17) Theorem 1. When 0 and ndT 0 ,then n and i1 n n limyii w yp x , y dxdy (18) n lim,i1 xii wxp x y dxdy (17) n i1 n limyii w yp x , y dxdy (18) n i1 Electronics 2021, 10, 1385 6 of 17 Theorem 1. When α → 0 and dT → 0, then n → ∞ and n Z +∞ Z +∞ lim xiwi = xp(x, y)dxdy (17) n→∞∑ i=1 −∞ −∞ n Z +∞ Z +∞ lim yiwi = yp(x, y)dxdy (18) n→∞∑ i=1 −∞ −∞ Proof. Suppose the probability space of p(x, y) is divided into n small squares in terms of T dT, where dT ≡ dtx, dty . When dT → 0, n → ∞ and n Z +∞ Z +∞ lim p(xi, yi)dtxdty = p(x, y)dxdy = 1 n→+∞∑ i=1 −∞ −∞ As dtx and dty are independent of i, n 1 ∑ p(xi, yi) = (19) i=1 dtxdty Hence, n R +∞ R +∞ xp(x, y)dxdy = lim ∑ xi p(xi, yi)dtxdty −∞ −∞ n→+∞ i=1 n 1 = lim ∑ xi p(xi, yi) n n→+∞ i=1 ∑ p(xj,yj) j=1 n p(xi,yi) = lim ∑ xi n n→+∞ i=1 ∑ p(xj,yj) j=1 n = lim ∑ xiwi n→+∞i=1 n R +∞ R +∞ Thus, lim ∑ xiwi = −∞ −∞ xp(x, y)dxdy. n→∞i=1 Similarly, n R +∞ R +∞ lim ∑ yiwi = −∞ −∞ yp(x, y)dxdy. n→∞i=1 A simple sample is used to further demonstrate that TMC improves particle efficiency. Consider a one-dimensional gamma distribution: 1 X ∼ Γ 2, 3 The MC and TMC methods are used to generate particles from the gamma distribu- tion. Figure3 shows the sampling results from the two sampling methods. For the MC method, the selection of particles is random. Increasing the particle number allows for a better description of the gamma distribution, and this is the only way to mitigate the indeterminacy. For the TMC method, the position of particles is determined when the confidence and sampling interval are provided. The task of describing the probability distribution is transferred to the weights corresponding to the particles. The estimation results for the expectation errors of the Monte Carlo method for the gamma distribution Electronics 2021, 10, 1385 7 of 17 ElectronicsElectronics 20212021,, 1010,, xx FORFOR PEERPEER REVIEWREVIEW 88 ofof 1818 Electronics 2021, 10, x FOR PEER REVIEWare shown in Figure4. Considering the indeterminacy of MC, it is run 10,000 times and8 of the18 RMSE is used to reflect the size of the error: v m u monte !2 wherewhere m denotesdenotes particleparticle numbernumber u andandmonte montem denotesdenotes thethe MonteMonte CarloCarlo number.number. TheThe where m denotes particleRMSE number= tand montex denotes/m − 6 the/monte Monte Carlo number. 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Meanwhile, the particle number increases whereas the sampling interval is 0.4. slowlyTheThe asresultsresults confidence demonstratedemonstrate increases thatthat over particlesparticles a fixed generatedgenerated sampling byby interval. TMCTMC cancan The describedescribe absolute thethe expectation noisenoise dis-dis- The results demonstrate that particles generated by TMC can describe the noise dis- tributiontributionerror can moremore be reduced efficientlyefficiently to 0.015thanthan particlesparticles using only generatedgenerated 75 particles byby MC.MC. for the confidence of 0.999, whereas tribution more efficiently than particles generated by MC. the sampling interval is 0.4. Probability density/Weight Probability FigureFigure 3.3. SamplingSampling resultsresults fromfrom TMCTMC andand MCMC.. FigureFigure 3. 3.SamplingSampling results results from from TMC TMC and and MC. MC. FigureFigure 4.4. VariationVariation ofof thethe expectationexpectation RMSERMSE withwith thethe particleparticle numbernumber forfor MCMC.. FigureFigure 4. 4.VariationVariation of ofthe the expectation expectation RMSE RMSE with with the the particle particle number number for for MC. MC. FigureFigureFigure 5.5. 5. VariationVariationVariation ofof of thethe the particleparticle particle numbernumber number withwith with confidenceconfidence confidence forfor for TMCTMC TMC... Figure 5. Variation of the particle number with confidence for TMC. ElectronicsElectronics 20212021, ,1010, ,x 1385FOR PEER REVIEW 9 8of of 18 17 Electronics 2021, 10, x FOR PEER REVIEW 9 of 18 FigureFigure 6. 6. VariationVariation of of the the absolute absolute expectation expectation error error with with confidence confidence for for TMC TMC.. 4. ProposedThe results Filter demonstrateAlgorithm that particles generated by TMC can describe the noise distributionEach of the more efficient efficiently particles than from particles TMC generated is a possible by MC.state estimation. The weight of Figure 6. Variation of the absolute expectation error with confidence for TMC. each4. Proposed particle is Filter the probability Algorithm that the particle becomes the state estimation. The contin- uous probability distribution is discretized in terms of these particles and their weights. Each of the efficient particles from TMC is a possible state estimation. The weight The4. ProposedTMCF is further Filter Algorithmdesigned as shown in Figure 7. The entire filter system can be divided of each particle is the probability that the particle becomes the state estimation. The into fourEach parts: of the initialization, efficient particles particle from deployment, TMC is a weightpossible mixing state estimation. and state estimation. The weight In of continuous probability distribution is discretized in terms of these particles and their thiseach section, particle the is four the partsprobability are explained that the inparticle detail becomesand the TMCF the state algorithm estimation. is proposed. The contin- weights.uous probability The TMCF distribution is further designedis discretized as shown in terms in Figure of these7. Theparticles entire and filter their system weights. can be divided into four parts: initialization, particle deployment, weight mixing and state The TMCF is further designed as shown in Figure 7. The entire filter system can be divided estimation. In this section, theSystem four parts are explained in detail and the TMCF algorithm intoStart four parts:Initialization initialization, particle deployment, weight mixing and state estimation. In is proposed. model Particle deployment this section, the four parts are explained in detail and the TMCF algorithm is proposed. System Weight mix StateStart estimation InitializationParticle Choice model Particle deployment Likelihood funtion Observation State estimation Particle Choice Weight mix Figure 7. TMCF system block diagram. Likelihood funtion 4.1. Initialization Observation The target of initialization is to set several efficient particles to describe the initial probabilityFigureFigure 7.7. TMCFTMCF distribution systemsystem blockblock discretely diagram.diagram. to facilitate the subsequent filtering process. After the dT X co4.1.4.1.nfidence InitializationInitialization 1 and the sampling interval 0 are set, initial particles 0 and their w c weightsTheThe target0target can be ofof obtained initializationinitialization in the isis confidence toto setset severalseveral interval efficientefficient 0 using particlesparticles the TMC toto describedescribe method thethe accord- initialinitial probabilityprobability distributiondistribution discretelydiscretely toto facilitatefacilitate thethe subsequentsubsequent filteringfiltering process.process. AfterAfter thethe p0 x c ingconfidence to the known1 − α andinitial the probability sampling interval dT. Additionally,0 are set, initial the particles confidenceX0 and interval their weightsfor 1−α dT0 X 0 wconfidencecan be obtained and in the the confidence sampling interval interval c using are theset, TMCinitial method particles according and to their the 0 pu x 1 0 the system wnoise probability ( ) of can be obtainedc according to Equation (8), knownweights initial 0 can probability be obtainedp0 inx the. Additionally, confidenceε interval the confidence 0 using interval the TMCcε methodfor the systemaccord- and then the amplification( ) of −interval is defined as: noise probability pu x of 1 α can bep obtained()x according to Equation (8), and thenc the amplificationing to the known of interval initial εprobabilityis defined as:0 . Additionally, the confidence interval ε for ε=,columnEcxcx pcolumnE uu p (21) 12() pu x 1−α the system noise probability= [column (c ) − Eof(p (x)) cancolumn be obtained(c ) − E according(p (x))] to Equation (8), column c ε ε 1 u , ε 2 u Ep x (21) i ε u whereand then the amplification denotes ofthe interval ith column is ofdefined matrix/vector as: c and denotes where column(cε)i denotes the ith column of matrix/vector cε and E(pu(x)) denotes the pu x ()−−() () ()() () ε=,columncxεε E puu column cx E p (21) theexpectation expectation of ofpu (x). . 12 The real state xx now exists in the confidence interval c forc the probability 1 − α. The column real state()c realreal,0,0 now exists in the confidence interval0 0 forEp ()the() probabilityx p (x) is describedε byi {X , w } for the accuracy of dT . The probabilityc of eachu particle’s where0 denotes0 0 the i，th column of matrix/vector0 ε and denotes 1 p0 x Xw00 existence. is described is describedp by()x its by weight. for the accuracy of dT0 . The probability of each the expectation of u . particle’s existence is described by its weight. x c The real state real,0 now exists in the confidence interval 0 for the probability () { ， } 1−α p0 x Xw00 . is described by for the accuracy of dT0 . The probability of each particle’s existence is described by its weight. Electronics 2021, 10, 1385 9 of 17 4.2. Particle Deployment The target of this step is to analyze the transition of the confidence interval from time step t − 1 to time step t, and then deploy particles. At time step t − 1, the confidence interval ct−1 can be written as: ct−1 = [min(Xt−1), max(Xt−1)] (22) in terms of the principle of the TMC method. where min(Xt−1) denotes the minimum value of each row of matrix/vector Xt−1 and max(Xt−1) denotes the maximum value of each row of Xt−1. When the particles Xt−1 are transferred through the system model without system noise, the transferred particles can be expressed as: ^ Xt = f(Xt−1) (23) ^ Each particle in Xt then is considered to be a possible state estimation without system ^ noise at time step t, and the probability of each particle Xt(i) is the weight of Xt−1(i). ^ Considering system noise, the confidence interval of each particle Xt(i) for confidence 1 − α is ^ ^ ct(i) = Xt(i) + column(ε)1, Xt(i) + column(ε)2 (24) ^ where ct(i) denotes the confidence interval for confidence 1 − α corresponding to Xt(i). Then, the complete confidence interval for 1 − α can be obtained by calculating the union of all confidence intervals: ct = ct(1) ∪ ct(1) · · · ∪ ct(nt−1) (25) where nt−1 denotes the number of particles in set Xt−1. For simplicity, the complete confidence interval also can be estimated roughly by: ^ ¯ ct = c t + ε (26) ¯ ^ ^ where c t = min Xt , max(Xt) . ^ ^ As ct ⊂ ct. Hence, confidence corresponding to the confidence interval ct is greater than or equal to 1 − α. However, the amplification of the confidence interval might increase the number of deployed particles. Sometimes this phenomenon, particularly in the case of high dimensions, leads to too many particles, which might result in the failure of filtering. ¯ ^ Then, particles Xt can be deployed according to the confidence interval ct or ct and dTt at time step t. Generally, dTt is set to a constant vector: dTt = dT0 (t = 1, 2, 3 ···) (27) When the confidence interval is unstable (increases or decreases over time), a specific strategy corresponding to the specific system needs to be designed to change the size of the sampling interval. ¯ In this step, the deployed particles Xt are distributed in this confidence interval uniformly, which is preparation for the subsequent step. Electronics 2021, 10, x FOR PEER REVIEW 11 of 18 4.3. Weight Mix X X In the weight mix step, the relationship between t−1and t is analyzed to solve the X prior weight corresponding to t , the likelihood weight is calculated and the posterior weight is obtained. X As the distribution of the system noise is continuous, each particle in set t−1 might X X arrive at any particle in set t , in theory. The probability of each particle in set t−1 ar- X riving at each particle in set t can be expressed as: Electronics 2021 10 , , 1385 10 of 17 pj()XX()− () i ()= u tt wit ， j mt ()()− () pjXXtt i (28) 4.3. Weight Mix j=1 ()== ¯ injm1, 2,tt−1 ; 1, 2, In the weight mix step, the relationship between Xt−1 and Xt is analyzed to solve the ¯ wi()， j X priorwhere weight t corresponding denotes the probability to Xt, the likelihood of the ith particle weight in is calculatedt−1 being and transferred the posterior to the weight is obtained. Xt mt Xt jth particleAs the in distribution . denotes of thesystem the number noise of is particles continuous, in set each. particleAs shown in setin FigureXt−1 might 8, the prior weight is calculated ¯as: arrive at any particle in set Xt, in theory. The probability of each particle in set Xt−1 arriving ¯ nt at each particle in set X can be expressed ()=× as: () ( ) t wjtttww−1 i ij, (29) i=1 ¯ ^ The likelihood weight is written_ as: pu Xt(j)−Xt(i) wt(i, j) = m t ¯ ^ (28) wjpˆ ()= ∑()phX()Xt(j)()− jXt(−i) z tttj=v 1 (30) (i = 1, 2, ··· nt−1; j = 1, 2, ··· mt) Additionally, the posterior weight is mixed by: _ where w (i, j) denotes the probability of the ithww particle()j ˆ ()j in X − being transferred to the jth t ()= tt t 1 ¯ wjt n ¯ t (31) particle in Xt. mt denotes the number of particles () in setˆ ()Xt. As shown in Figure8, the prior wwttj j weight is calculated as: i=1 n t _ w (j) = w (i) × w (i, j) {Xwtt，(29)} Then, the posterior distributionet of∑ the tstate−1 at timet step t is described by i=1 discretely. Xt−1 Amplifying region Xt Xt 3 ()=× () ( ) ()⇐× ()ˆ () wwwttt11 −1 ii, wwwttt111 () i=1 wt−1 1 3 ()⇐× () () wwwttt222 ˆ ()=× () ( ) wwwttt22 −1 ii, = () i 1 wt−1 2 3 ()⇐× () () ()⇐ () wwwttt333 ˆ wwtt13 ()=× () ( ) Confidence wwwttt33 −1 ii, i=1 () 3 wt−1 3 www ()44=× ()ii (, ) ttt −1 ()⇐× () () i=1 wwwttt444 ˆ ()⇐ () wwtt24 3 ()=× () ( ) wwwttt55 −1 ii, www()555⇐× ()ˆ () ()⇐ () = ttt wwtt35 Amplifyingi 1 region Observation FigureFigure 8.8. Schematic diagram ofof particleparticle andand weightweight changes.changes. The likelihood weight is written as: ¯ wˆ t(j) = pv h Xt(j) − zt (30) Additionally, the posterior weight is mixed by: ~ ^ wt(j)wt(j) wt(j) = (31) nt ~ ^ ∑ wt(j)wt(j) i=1 ¯ ¯ Then, the posterior distribution of the state at time step t is described by Xt, wt discretely. Electronics 2021, 10, 1385 11 of 17 4.4. Particle Choice and State Estimation ¯ ¯ Xt, wt describes the posterior distribution after the fusion of the prior distribution and the likelihood distribution. All the distribution information is concentrated in the ¯ ¯ weight wt, and the role of the particles Xt is to only occupy the distribution space. Generally, many very low weighted particles emerge after fusion. Additionally, these particles with very low weights have very little effect on the accurate description of the distribution, so they can simply be omitted. nt particles are chosen in the order of largest to smallest so that the sum of the weights of the nt particles is 1 − α: ¯ ¯ 1−α ¯ ¯ Xt, wt ⇐ Xt, wt (32) mt nt Then, the weight is normalized: ¯ ¯ wt(i) wt(i) = (33) ¯ nt ¯ ∑ wt(i) i=1 The state estimation can be obtained by nt xt = ∑ Xt(i)wt(i) (34) i=1 In conclusion, the TMCF algorithm is summarized in Algorithm 1. Algorithm 1 1 Initialization: 2 Setting 1 − α and dT0 3 Generate {X0, w0} and ε according to TMC method and Equation (21) 4 //Over all time steps: 5 for t ← 1 to T do 6 Setting dTt = dT0, or other strategy is used to select dTt 7 Confidence interval choice according to Equation (25) or (26) 8 Particle deployment according to dTt 9 Weight fusion according to Equations (28)–(31) 10 Particles and their weights choice according to Equations (32) and (33) 11 State estimation according to Equation (34) 12 End 5. Numerical Simulation In this section, a two-dimensional linear system is used to assess the performance of the TMCF [43]: x (t) cos(θ) − sin(θ) x (t − 1) q (t − 1) 1 = 1 + 1 (35) x2(t) sin(θ) cos(θ) x2(t − 1) q2(t − 1) x (t) z(t) = 1 1 1 + r(t) (36) x2(t) where xi(t) and z(t) denote the state and observation at time step t, respectively; qi(t − 1) denotes the system noise sequence at time step t − 1; and r(t) denotes the observation noise sequence at time step t. In the two experiments, θ = π/18 and the initial state is Electronics 2021, 10, 1385 12 of 17 T [1, 1] . The initial probability satisfies p0(x) ∼ N(0, 0.1). It is well known that the KF is the optimal filter for a linear Gaussian system based on the Bayesian filter principle. Hence, the performance of the TMCF is first assessed in a linear and Gaussian system. The estimation results of the KF are used as a reference to evaluate the approximation degree of the TMCF algorithm and Bayesian filtering in the linear and Gaussian system. Because the TMCF algorithm is a filter based on the Monte Carlo principle, the performances of the TMCF algorithm and PF algorithm are compared in this experiment. Second, a heavy-tailed distribution (non-Gaussian environment) is considered in this linear system. The performance of the TMCF is compared with that of the KF and PF in this linear and non-Gaussian system. Four sets of parameters are selected for the TMCF algorithm, which are shown in Table1. Two forms of mean square errors (MSE) are used to evaluate the performance of the algorithms: T 2 MSE1 = ∑(xˆi − xi,real) /T (37) i=1 T 2 MSE2 = ∑(xˆi − xi,KF) /T (38) i=1 Table 1. Parameters of the TMCF. Parameter dT 1−α 1 1.2 0.999 2 0.8 0.999 3 1.2 0.9999 4 0.8 0.9999 In this section, MATLAB is used to build the simulation environment. The perfor- mance (including the filtering precision, the number of samples, filter time) of the TMCF, KF and PF are verified and compared by this simulation environment. All the data are generated by the simulated program. The configurations of the simulation computer can be seen in Table2. Table 2. Configuration environment. Basic Frequency Windows MATLAB CPU RAM (GB) (GHz) Version Version Intel(R) Core i5 1.70 16.0 Windows 10 R2018a 5.1. Gaussian Distribution System In this experiment, the Gaussian model is selected for both system noise and observed noise: q1(t) ∼ N(0, 1), q2(t) ∼ N(0, 1) and r(t) ∼ N(0, 1). PFs with 3000 and 5000 particles are used as the comparison algorithms of the TMCF. Figures9 and 10 show that the deviation between each filtering result of the different algorithms and the KF results for x1 and x2, respectively. The results show that it is difficult for the PF to approximate the performance of the KF in the linear and Gaussian system. Compared with the KF, although the number of particles is 3000, the results of the PF deviate by about 0.15 from that of the KF in each filtering process. Meanwhile, as the number of particles increases dramatically, this deviation declines very slowly. The deviation is about 0.1 when the number of particles is 5000. This is caused by the indeterminacy of the Monte Carlo method. The indeterminacy is greatly reduced when the TMC method is used to generate particles. Using the TMC method, the results of the TMCF are very close to those of the KF. The difference between the TMCF and KF is less than 0.01 for all four parameters selected. The deviation decreases as the confidence increases Electronics 2021, 10, x FOR PEER REVIEW 14 of 18 the sampling interval decreases. Particularly, the deviation is less than 0.001 when the Electronics 2021, 10, x FOR PEER REVIEW 14 of 18 confidence is 0.9999 and the sampling interval is 0.8. Figure 11 shows that only about 40 particles need to be transferred in each filtering process for parameter 1, and the number Electronics 2021, 10, 1385 13 of 17 of set particles is about 250. The number of particles required increases as the sampling the sampling interval decreases. Particularly, the deviation is less than 0.001 when the interval decreases and the confidence increases. For parameter 4, the number of trans- confidence is 0.9999 and the sampling interval is 0.8. Figure 11 shows that only about 40 ferred particles is about 130 and the number of set particles is only about 800. Figure 12 particles need to be transferred in each filtering process for parameter 1, and the number and the sampling interval decreases. Particularly, the deviation is less than 0.001 when ofshows set particles the time is consumedabout 250. The in each number filtering of particles process required by the increases different as algorithms the sampling on a com- puterthe confidenceusing the same is 0.9999 configuration. and the sampling The computation interval is 0.8. time Figure of 11 the shows TMCF that is onlymuch about less than interval40 particles decreases need and to be the transferred confidence in eachincrea filteringses. For process parameter for parameter 4, the number 1, and theof trans- number that of PF. Table 3 shows the filtering results of 5000 time steps processed by Equations ferredof set particles particles is isabout about 130 250. and The the number number of of particles set particles required is only increases about 800. as the Figure sampling 12 shows the time consumed in each filtering process by the different algorithms on a com- interval decreasesM andSE2 the confidence increases. For parameter 4, the number of transferred puter(37) and using (38). the The same configuration. is about The0.01 computation for PF with 5000time ofparticles, the TMCF and is muchthe computation less than time particles is about 130 and the number of set particlesMSE is only about 800. Figure 12 shows the thatis abouttime of PF. consumed 0.1 Table s for 3 inshowseach each filtering the filtering filtering process. process result by Thes of the 5000 different time2 reaches steps algorithms processed 10 on for a computerbythe Equations TMCF using with pa- MSE (37)rameterthe and same (38). 4, and configuration. The the computation2 is about The computation 0.01 time for PFis withonly time 5000about of the particles, 0.0035 TMCF and iss. muchThe the resultscomputation less than demonstrate that time of PF. that theTable TMCF3 shows can approximate the filtering results the KF of 5000algorithm timeMSE steps better processed with fewer by Equations particles (37) and and less (38). compu- is aboutMSE 0.1 s for each filtering process. The 2 reaches 10 for the TMCF with pa- tationThe in linear2 is aboutand Gaussian 0.01 for PF systems with 5000 compar particles,ed with and thePF. computationDifferent from time the is aboutKF, the 0.1 TMCF rameters for each4, and filtering the computation process. The timeMSE is onlyreaches about10 0.0035−6 for s. the The TMCF results with demonstrate parameter that 4, and does not use the propagation characteristics2 of the conditional means and covariances of thethe TMCF computation can approximate time is only the KF about algorithm 0.0035 s.better The with results fewer demonstrate particles and that less the TMCFcompu- can tationGaussianapproximate in linear noise and the in Gaussian KFlinear algorithm systems. systems better Therefore,compar withed fewer with this particles PF. method Different and is also lessfrom computationapplicable the KF, the toTMCF in non-Gauss- linear doesianand noise. not Gaussian use the propagation systems compared characteristics with PF. of Different the conditional from the means KF, the and TMCF covariances does not of use Gaussianthe propagation noise in linear characteristics systems. ofTherefore, the conditional this method means is and also covariances applicable ofto Gaussiannon-Gauss- noise ianin noise. linear systems. Therefore, this method is also applicable to non-Gaussian noise. State estimation deviation from that of Kalman Filter State estimation deviation from that of Kalman Filter Figure 9. Difference between the state estimation results of the different algorithms and those of FigureFigure 9.x Difference 9. Difference between between the thestate state estimation estimation results results of the of the different different algorithms algorithms and and those those of of KF KFfor forx x. 1 . KF for 1 1 . State estimation deviation from that of Kalman Filter Kalman of that from deviation estimation State State estimation deviation from that of Kalman Filter Kalman of that from deviation estimation State Figure 10. Difference between the state estimation results of different algorithms and those of KF FigureFigurex 10. 10. DifferenceDifference between between the the state state estimation results results of of different different algorithms algorithms and and those those of KF of KF for 2 . forxx2. for 2 . Electronics 2021, 10, x FOR PEER REVIEW 15 of 18 ElectronicsElectronics 20212021, 10,, 10x FOR, 1385 PEER REVIEW 15 of14 18 of 17 Particle number for TMCF in different parameters 900 n with parameter1 t m withParticle parameter1 number for TMCF in different parameters 900800 t n nwithwith parameter1 parameter2 t t m mwithwith parameter1 parameter2 800700 t t n nwithwith parameter2 parameter3 t t m with parameter2 tm with parameter3 700600 t n with parameter3 t n with parameter4 m witht parameter3 600 t 500 m with parameter4 n witht parameter4 t m with parameter4 500 t 400 400 Particle number Particle 300 Particle number Particle 300 200 200 100 100 0 0 0 102030405060708090100 0 102030405060708090100 Time steps Time steps FigureFigure 11. 11. 11.Number NumberNumber of ofparticles of particles particles required required for forthe for theTM the TMCFCF TM algorithmCF algorithm algorithm with with different with different different parameters. parameters. parameters. FilterFilter time time of different of different algorithms algorithms 0.180.18 KFKF TMCFTMCF with with parameter parameter 1 1 0.16 TMCF with parameter 2 0.16 TMCF with parameter 2 TMCF with parameter 3 TMCF with parameter 3 0.14 TMCF with parameter 4 0.14 PFTMCF with 3000 with particles parameter 4 PFPF with with 5000 3000 particles particles 0.12 PF with 5000 particles10-3 -3 0.12 6 10 6 0.1 4 0.1 2 4 0.08 0 2 0 102030405060708090100 0.08 0 0.06 0 102030405060708090100 0.06 0.04 0.04 0.02 0.020 0 102030405060708090100 0 Time steps 0 102030405060708090100 Figure 12. Computation time for theTime TMCF steps algorithm with different parameters. Table 3. Performance ofFigure theFigure different 12. 12. ComputationComputation filter algorithms time time withfor the different TMCFTMCF parameters algorithm algorithm with in with the different linear/Gaussian different parameters. parameters. system. x x TableTable 3. Performance 3. PerformanceParameters of ofthe the different different filter filter algorithms algorithms with1with different different parameters parameters in the in2 linear/Gaussianthe linear/GaussianComputation system. system. Algorithm −α dT 1 nmtt/ MSE1 MSE2 MSE1 MSE2 time (s) Parameters x x xx Parameters 1 1 22 ComputationComputation−5 AlgorithmAlgorithmKF - - - 3.421829 0 2.798550 0 3.069 × 10 dT 1−α nm¯/ ¯ Time (s) -dT 1− - α 3000ttnt/m t 3.456089MSE1MSE1 0.027105 MSE2 2.822853 MSE1MSE1 0.021010 MSE2MSE2 0.0608210time (s) PF −5 KF KF ------5000 - - 3.430909 3.421829 3.421829 0.015142 0 0 2.807029 2.7985502.798550 0.012138 0 0 0.13871773.069 3.069× 10 −×5 10 -1.2 0.999 - 42 3000 248 3.423760 3.456089 4.306 0.027105 × 10−4 2.799829 2.822853 3.155 × 0.02101010−4 6.282 × 10 0.0608210−5 PF - - 3000 3.456089 0.027105 2.822853 0.021010 0.0608210 PF -0.8 0.999 - 95 5000 560 3.422683 3.430909 4.238 0.015142 × 10−4 2.798926 2.807029 3.081 × 0.01213810−4 2.101 × 10 0.1387177−3 TMCF - - 5000 3.430909 0.015142 2.807029 0.012138 0.1387177 1.2 0.9999 57 343 3.421978 9.444 × 10−6 −2.7986104 7.098 × 10−6 9.293−4 × 10−4 −5 1.21.2 0.999 0.999 42 42 248 248 3.423760 3.423760 4.306 4.306× ×10 10−4 2.7998292.799829 3.1553.155× 10× −104 6.282 6.282× 10 −×5 10 −6 −6 −3 0.8 0.9999 130 782 3.421878 8.874 × 10 −2.7985394 6.484 × 10 3.511−4 × 10 −3 0.8 0.999 95 560 3.422683 4.238 × 10− 2.798926 3.081 × −10 2.101 −× 10 TMCF 0.8 0.999 95 560 3.422683 4.238 × 10 4 2.798926 3.081 × 10 4 2.101 × 10 3 TMCF 1.2 0.9999 57 343 3.421978 9.444 × 10−6 2.798610 7.098 × 10−6 9.293 × 10−4 1.2 0.99995.2. Gaussian 57 Mixture 343 Distribution 3.421978 System9.444 × 10−6 2.798610 7.098 × 10−6 9.293 × 10−4 0.8 0.9999 130 782 3.421878 8.874 × 10−6 2.798539 6.484 × 10−6 3.511 × 10−3 0.8 0.9999In 130 this experiment, 782 the 3.421878 Gaussian8.874 mixture× 10 −mo6 del2.798539 is selected 6.484for both× 10 system−6 3.511 noise× and10−3 qt() 0.6 N ( 0,1 )+ 0.4 N ( 0, 4 ) qt() 0.6 N ( 0,1 )+ 0.4 N ( 0, 4 ) observed5.2. Gaussian noise: Mixture 1 Distribution System , 2 and rt()5.2. 0.6 Gaussian N ( 0,1 ) Mixture+ 0.4 N ( 0,Distribution 4 ) System In this experiment, the. Gaussian mixture model is selected for both system noise and In this experiment, the Gaussian mixture model is selected for both system noise The noise model isqt shown() 0.6 in Figure N ( 0,1 )13.+ 0.4 Similar N ( 0, 4to ) Tableqt 3 for() the 0.6 previous N ( 0,1 ) experiment,+ 0.4 N ( 0, 4 ) and observed noise: 1q (t) ∼ 0.6N(0, 1) + 0.4N(0, 4),q (t2) ∼ 0.6N(0, 1) + 0.4N(0, 4) and Tableobserved 4 shows noise: the filtering1 results of 5000-time steps ,processed2 by Equation (37). The and r(t) ∼ 0.6N(0, 1) + 0.4N(0, 4). Mrt()SE 0.6 N ( 0,1 )+ 0.4 N ( 0, 4 ) 1 of the PF with 3000 particles. is greater than that of the KF. The performance of the The noise model is shown in Figure 13. Similar to Table 3 for the previous experiment, Table 4 shows the filtering results of 5000-time steps processed by Equation (37). The MSE 1 of the PF with 3000 particles is greater than that of the KF. The performance of the Electronics 2021, 10, 1385 15 of 17 Electronics 2021, 10, x FOR PEER REVIEW 16 of 18 The noise model is shown in Figure 13. Similar to Table3 for the previous experiment, Table4 shows the filtering results of 5000-time steps processed by Equation (37). The MSE1 of the PF with 3000 particles is greater than that of the KF. The performance of the TMCF TMCFwith with parameter parameter 1 is 1 better is better than than that that of PFof PF with with 5000 5000 particles particles and and KF. KF. Meanwhile, Meanwhile, the thenumber number of of transferred transferred particles particles is is only only 110 110 and and the the number number of of set set particles particles is is only only 650. 650. The Thecomputation computation time time is 0.006is 0.006 s, thats, that is, is, much much less less than than that that of theof the PF. PF. With With the decreasethe decrease of the of samplingthe sampling interval interval and and the increasethe increase of confidence, of confidence, the accuracy the accuracy of filter of isfilter improved, is improved, and the andcomputation the computation time istime increased. is increased. For the For parameters the parameters 4 of TMCF,4 of TMCF, the accuracy the accuracy of the of TMCF the TMCFis improved is improved by 0.01, by 0.01, and and the computationthe computation time time reduced reduced to 0.067 to 0.067 s from s from 0.20 0.20 s, comparing s, com- paringwith with the particle the particle filter filter (5000 (5000 particles). particles). Probability density Probability FigureFigure 13. 13.ProbabilityProbability model model of Gaussian of Gaussian mixture mixture noise. noise. Table 4. Performance of the different filter algorithms with different parameters in the linear/mixture Gaussian system. Table 4. Performance of the different filter algorithms with different parameters in the linear/mixture Gaussian system. ParametersParameters MSE1 MSE1 Computation −α ¯ ¯ ComputationTime Time (s) (s) dT dT 1 1− α nmtt/ nt /mt x1 x1 x2 x2 KFKF ------7.619301525 7.619301525 6.3982152 6.3982152 2.2242.224 × 10−×5 10−5 - - 3000 7.718597333 6.4887486 0.098728173 PF - - 3000 7.718597333 6.4887486 0.098728173 PF - - 5000 7.589158022 6.3695680 0.204914013 - - 5000 7.589158022 6.3695680 0.204914013 1.2 0.999 110 650 7.581596248 6.3654145 0.006746154 1.2 0.999 110 650 7.581596248 6.3654145 0.006746154 0.8 0.999 250 1479 7.582551190 6.3659966 0.028529454 TMCF 1.2 0.80.9999 0.999 157 250 961 14797.577101831 7.582551190 6.3616057 6.3659966 0.012136754 0.028529454 TMCF 0.8 1.20.9999 0.9999 358 157 2184 9617.576998566 7.577101831 6.3615759 6.3616057 0.067335938 0.012136754 0.8 0.9999 358 2184 7.576998566 6.3615759 0.067335938 6. Conclusions 6. ConclusionsThe TMCF algorithm was proposed to overcome the challenge of the non-Gaussian filteringThe in TMCFthis paper. algorithm First, wasthe TMC proposed method to overcome was proposed the challenge to sample of theparticles non-Gaussian in the confidencefiltering ininterval this paper. according First, to the the TMC sampling method interval. was proposed The performance to sample particlesof the TMC in the methodconfidence has been interval simulated according and the to theproperty sampling of the interval. TMC method The performance has been proved. of the Sec- TMC ond,method the TMCF has been algorithm simulated was and proposed the property by introducing of the TMC the method TMC method has been into proved. the PF Second, al- gorithm.the TMCF Different algorithm from was the proposedPF, the TMCF by introducing algorithm the completes TMC method the transfer into the of PF the algorithm. distri- butionDifferent using from a series the PF,of calculations the TMCF algorithm of weights completes and particles the transferwere used of theto occupy distribution the state using spacea series in the of confidence calculations interval. of weights Third, and Nume particlesrical simulations were used demons to occupytrated the that state the space MSE in of the confidenceTMCF was about interval. 10− Third,6 compared Numerical with that simulations of the Kalman demonstrated filter (KF) that in a the two-dimen- MSE of the sionalTMCF linear/Gaussian was about 10− 6system.compared In a with two-dimensional that of the Kalman linear/non-Gaussian filter (KF) in a two-dimensional system, the MSElinear/Gaussian of the TMCF for system. parameter In a two-dimensional 4 was 0.04 and linear/non-Gaussian0.01 less than that of system,the KF and the MSEPF with of the 5000TMCF particles, for parameter respectively. 4 was The 0.04 single and 0.01filter less times than of that the ofTMCF the KF and and PF PF with with 5000 5000 particles particles, wererespectively. 0.006 s and The 0.2 singles, respectively. filter times of the TMCF and PF with 5000 particles were 0.006 s andIn 0.2this s, paper, respectively. we have designed an improved PF algorithm, we called TMCF algo- rithm. In the non- Gaussian filter environment, it can not only improve the accuracy of filter, but also reduce the computation time. In the future development of new disciplines such as artificial intelligence, multi-sensor data fusion, and multi-target tracking, there are Electronics 2021, 10, 1385 16 of 17 In this paper, we have designed an improved PF algorithm, we called TMCF algorithm. In the non- Gaussian filter environment, it can not only improve the accuracy of filter, but also reduce the computation time. In the future development of new disciplines such as artificial intelligence, multi-sensor data fusion, and multi-target tracking, there are more and more types of data and more and more complex sources of data. the quality of nonlinear non-Gaussian filtering method becomes more and more important in data fusion. Our work lays a theoretical foundation for nonlinear/non-Gaussian filter and can be used to improve the filtering precision under the condition of reducing computation time in some non-Gaussian filter environments. In the future, we will try to apply the algorithm to the integrated navigation system to improve the positioning accuracy of satellite navigation. Author Contributions: Conceptualization, R.X. and X.Q.; methodology, R.X.; software, X.Q.; valida- tion, R.X., X.Q. and Y.Z.; formal analysis, X.Q.; writing—original draft preparation, X.Q.; writing— review and editing, R.X. and X.Q.; supervision, Y.Z.; funding acquisition, R.X., X.Q. and Y.Z. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported in part by the National Key Research and Development Program of China, grant number 2017YFB0503400, and in part by the National Natural Science Foundation of China, grant numbers U2033215, U1833125 and 61803037. 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