非高斯噪声下变分贝叶斯交互多模型状态估计

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本文探讨了在非高斯噪声环境中利用变分贝叶斯方法进行状态估计的交互多模型（IMM）算法。在现实应用中，许多系统中存在异常测量数据，即所谓的噪声中的异常值或"outliers"，这使得传统的基于高斯假设的IMM方法在处理这类问题时显得局限。作者针对这一挑战，提出了一个改进的IMM策略。该算法的核心是采用学生-τ分布来建模非高斯的测量噪声。相比于高斯分布，学生-τ分布因其尾部更厚，能够更好地适应含有异常值的数据。在交互步骤中，混合噪声参数的后验分布通过贝叶斯框架下的伽马近似和最近报道的矩匹配方法得到。这种方法有助于捕捉噪声的复杂性，尤其是在统计上难以处理的情况下。为了克服传统贝叶斯滤波中因噪声耦合导致的计算复杂性，文章引入了变分贝叶斯技术。变分贝叶斯是一种强大的无监督学习工具，它通过近似计算后验分布的最简化形式，将原本复杂的推理过程转化为易于处理的优化问题。这种方法允许作者递归地估计噪声和状态的后验分布，从而提高了算法的实时性和鲁棒性。作者通过一个机动目标跟踪的实例来验证新提出的算法。实验结果显示，该算法在处理含有噪声异常值的情况下表现出了良好的鲁棒性，能够在保持估计准确性的同时有效地抵抗这些干扰因素。因此，本文的研究对于那些需要在实际系统中处理非高斯噪声和异常值的状态估计任务具有重要的理论和实践价值。关键词包括：状态估计、鲁棒性、变分贝叶斯等，表明了这项工作对提高系统在复杂环境中的性能具有显著贡献。

, and H

are assumed constant over time. Rewriting the

non-Gaussian measurement model Eq. (2) yields:

Std z

ðÞx

; R

; υjðÞ¼∫

∞

N z

ðÞx

; u

1





G u

υ=2

; υ=2ðÞdu

(3)

The dependence on the mode s

makes it different

from the other Student-t measurement models in [13–15].

Since the heavy-tailed Student-t distribution does not

neglect the existence of outliers, the non-Gaussian ﬁltering

can proceed without setting other models for the outliers.

Unfortunately, there is no closed form solution to Eq. (3)

using conventional methods because u

has to be margi-

nalized out and the state and noise statistics are even

coupled. Therefore, direct use of Student-t in the Kalman

ﬁlter is intractable and we have to resort to approximation

techniques. In order to make this computation analytically

tractable, variational Bayesian methods are used to compute

the posterior distribution of the noise and the state. In

[6,7, 19,20], VB was proven to be able to handle such

coupled situations due to its properties, that is, factorizing

the coupled joint distribution into several conditionally inde-

pendent distributions. VB methods will be reviewed shortly.

III. VARIATIONAL BAYESIAN METHODS

By Bayes’ rule, the joint posterior distribution is

given as:

p ΘjYðÞ¼

p YjΘðÞp ΘðÞ

∫p YjΘðÞp ΘðÞdΘ

; (4)

where Y is the observed data set and Θ is a set of param-

eters. From the above, we ﬁnd that direct computation of

Eq. (4) can only be achieved for very limited models be-

cause of the intractability in computing the integral, that

is, the marginal likelihood p(Y). The key idea of VB is to

use a new free-form distribution q(Θ) to approximate the

true posterior distribution p(Θ|Y). The form of the new distri-

bution q(Θ) is selected freely, due to the conjugacy property

(see Chapter 2 in [9]). Furthermore, if the set Θ can be

partitioned into N parts as Θ ={Θ

, Θ

, ⋯, Θ

} and

each Θ

is assumed independent of each other, naturally,

q(Θ) factorizes into N independent q(Θ

) s, then the joint

approximate distribution becomes

q Θ

; Θ

⋯; Θ

; ⋯; Θ

ðÞ¼∏

i¼1

q Θ

ðÞ (5)

The VB method using this technique is also called

(mean ﬁeld) variational Bayes [6–8].

In VB framework, the goal is to ﬁnd a q(Θ) that is as

close as to p(Θ) (dependence on Y is omitted for concise-

ness). Fortunately, Kullback–Leibler (KL) divergence is a

non-negative dissimilarity function measuring the dis-

crepancy between two distributions q(Θ) and p(Θ). Hence,

we can obtain q(Θ) by minimizing the KL divergence

between p(Θ) and q(Θ). It has also been proven in [9] that

minimizing KL(q(Θ)‖ p(Θ)) is equivalent to maximizing

the lower bound of ln p(Y). The lower bound function can

be written as:

F ¼ ∫q ΘðÞln

p YjΘðÞp ΘðÞ

q ΘðÞ

dΘ (6)

Maximizing F tightens the lower bound and gives

rise to the optimal distribution [9]. The approximate dis-

tribution is optimal in the sense of KL divergence because

KL(q(Θ)‖ p(Θ)) equals zero if and only if q(Θ)=p(Θ).

Please note that this optimal sense differs from its coun-

ter part in optimal estimation theory, where the optimiza-

tion is measured by minimu m-variance rul es. Therefore,

q(Θ

) can be computed by differentiating F with respect

to q(Θ

j ≠ i

). General solutions of q(Θ

) s are given as [8,9]:

q Θ

ðÞ¼

exp E

j ≠ i

ln p Y; ΘðÞ½



∫exp E

j ≠i

ln p Y; ΘðÞ½



dΘ

; (7)

where E

j≠i

ln p Y; ΘðÞ½is the expectation of the logarithm

of the joint distribution over the entire set of parameters

except for the i th. Hereby, each factorized distribution

can be obtained after mean ﬁeld theory [9] is used. We

will see how this advantage of VB works out the state-

noise coupled problem in the following section.

It is worth noting that Eq. (7) is an implicit solution be-

cause of the circular dependencies. This forms an iterative

scheme of VB: given the appropriate initialization of the

hyper-parameters, the distribution of each parameter is in

turn estimated with the expectation over other distributions.

Each parameter’s distribution then is updated using Eq. (7)

for the next cycling step until the algorithm converges. The

convergence of VB has been guaranteed in [21].

IV. ROBUST IMM USING VB

In this section, an improved IMM approach, based on

the Student-t distribution, is derived. VB is used to recur-

sively estimate the noise statistics and the state of each

mode from the hybrid system. The state x

is augmented

as {x

, u

}. From Eqs (1) and (2), the prior forms of the state

and the latent variable are Gaussian and Gamma distribu-

tion, respectively. Hence, the distribution of the augmented

state of the j th mode conditioned on the measurements up

C. Shen et al.: A robust IMM Approach for State Estimation

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