A variational particle filtering method based on KLD sampling

Abstract

This invention proposes a variational particle filter method based on KLD sampling, belonging to the field of signal processing. The proposed method is applicable to robust estimation of nonlinear systems. The algorithm uses a variational Bayesian method to estimate the time-varying measurement noise parameters implicit in the signal, while simultaneously employing a KLD sampling method to adjust the required number of particles in real time based on the estimated time-varying measurement noise parameters, thereby enhancing the robustness of the particle filter.

Description

Technical Field

[0001] This invention proposes a variational particle filtering method based on KLD sampling, which belongs to the field of signal processing. Background Technology

[0002] Nonlinear filtering problems exist in many fields, and numerous studies have shown that particle filters outperform traditional Gaussian filters (e.g., Kalman filters) for nonlinear systems. However, classic particle filters still rely on accurate modeling. For nonlinear systems with inaccurate measurement noise modeling or time-varying measurement noise (e.g., a limited number of valid satellite acquisitions in navigation systems), classic particle filters not only struggle to provide optimal estimates but also risk filter divergence. Therefore, this invention uses a variational Bayesian method to estimate the time-varying measurement noise parameters implicit in the signal, and employs the KLD sampling method to adjust the required number of particles in real time based on the estimated time-varying measurement noise parameters, thereby enhancing the robustness of the particle filter. Summary of the Invention

[0003] To address the aforementioned problems and shortcomings, this invention proposes a variational particle filtering method based on KLD sampling.

[0004] The specific steps of the method of the present invention are as follows:

[0005] The method of this invention is divided into a measurement update step and a status update step:

[0006] The variational measurement update step based on KLD sampling uses variational Bayesian estimation of the likelihood distribution and the KLD sampling method to estimate the posterior state distribution. It includes the following sub-steps:

[0007] Measurement update sub-step one: via formula Correct the variance parameter of the likelihood distribution, and then according to the formula... Obtain the initial posterior distribution of the state, and set the initial degrees of freedom parameters. Initial scaling matrix The measurement update sub-steps 2 and 3 are recursively executed M times to obtain the posterior distribution p(x) of the updated state. k，M |y 1：k ), posterior degrees of freedom parameters Posterior scale matrix in For the prior degrees of freedom parameters, Let p(x) be the prior scaling matrix. k，0 |y 1：k Let be the posterior distribution of the initial state at time k. Let be the initial likelihood distribution at time k, with the form and a mean of 0 and a variance of . Gaussian distribution, variance parameter For the latent variables of variational inference, p(x) k |y 1：k-1 Let be the prior distribution of the state at time k;

[0008] Measurement update sub-step two: Obtain the posterior distribution of the state particles using KLD sampling from the posterior distribution, and apply the Monte Carlo integration method according to the formula... Estimate the distribution parameters of the (j+1)th latent variable; where Let the degrees of freedom at time k be The scaling matrix is The (j+1)th corrected inverse Wissaud distribution For the j-th corrected degree of freedom parameter, Let h(·) be the scale matrix of the j-th correction, h(·) be the system measurement equation, and q(x) be the scale matrix of the j-th correction. k ,|y 1：k ) represents the sampling distribution of the state at time k, and N j The number of particles determined by the KLD sampling method;

[0009] Measurement update sub-step three: Through formula Correct the variance parameter of the likelihood distribution and then apply the formula Corrected state posterior distribution; where d is the state x k，j The dimension of p(x) k，j+1 |y 1：k Let be the posterior distribution of the state after the (j+1)th correction at time k. Let be the likelihood distribution of the (j+1)th correction at time k, with a mean of 0 and a variance of . Gaussian distribution, variance parameter These are latent variables for variational inference;

[0010] State updates employ a heuristic method for the degree-of-freedom parameters. Scale matrix The update process includes the following sub-steps:

[0011] State update sub-step one: The posterior distribution of the state at time k, updated by measurement, is determined according to the formula p(x) k+1 |y 1：k )=p(x k+1 |x k )p(x k，M |y 1：k ) to obtain the prior distribution of the state, where p(x k+1 |y 1：k Let p(x) be the prior distribution of the state at time k+1. k+1 |x k () represents the state transition distribution from time k to time k+1;

[0012] State update sub-step two: Update the degree of freedom parameters at time k based on the measurement. According to the formula Obtain the prior degrees of freedom at time k+1 The scale matrix at time k updated by measurement According to the formula Obtain the prior scaling matrix at time k+1 Where ρ is a constant and ρ∈(0,1], and B is a constant matrix whose determinant satisfies |B|∈(0,1).

[0013] The beneficial effects of this invention are as follows:

[0014] The method proposed in this invention is applicable to robust estimation of nonlinear systems. The algorithm uses a variational Bayesian method to estimate the time-varying measurement noise parameters implicit in the signal, and employs a KLD sampling method to adjust the required number of particles in real time based on the estimated time-varying measurement noise parameters, thereby enhancing the robustness of the particle filter. Attached Figure Description

[0015] Figure 1 This is a schematic diagram illustrating the principle of the method of the present invention;

[0016] Figure 2 The following are simulation verification diagrams to demonstrate the effectiveness of the method of the present invention: (a) is a diagram showing the change in the number of particles, (b) is a diagram showing the measurement noise estimation effect, and (c) is a diagram showing the state estimation effect.

[0017] Figure 3 The figures show a comparison between the method of this invention and the classical particle filtering method. In the figure, (a) is a comparison of the measurement noise estimation effect, and (b) is a comparison of the state estimation effect. VBCKF is the comparison method, and VBKLDPF is the method of this invention. Detailed Implementation

[0018] The embodiments of the present invention are described in detail below. These embodiments are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention. A variational particle filter based on KLD sampling according to the present invention will be described in detail below with reference to the accompanying drawings:

[0019] To demonstrate the specific steps of this invention, the following nonlinear system is constructed:

[0020]

[0021] Where k is the time index, u k The state noise follows a mean of zero and a standard deviation of σ. u Gaussian distribution v kThe measured noise follows a mean of zero and a standard deviation of σ. vk Gaussian distribution (Note that this variance is a time-varying parameter), and based on the above simulation experimental environment, according to Figure 1 The method's principle diagram and instruction manual steps are as follows:

[0022] Step 1: Initialize the filter parameters, assuming that the prior distribution at time zero follows a mean of zero and a standard deviation of σ. u If the distribution is Gaussian, then the initial particle points are sampled from this distribution, and the initial degree of freedom parameter is assumed to be d+1, where d is the state x. k The dimension is, and the initial scale matrix is The initial number of particles is N, and the initial KLD sampling parameters are: initial sampling interval size Δ, initial distribution error ε, lower quantile of the initial chi-square distribution δ, and minimum number of KLD samples N. min and the maximum number of samples N in KLD max .

[0023] Step 2: Upon receiving the observation data at time k, perform a variable measurement update step based on KLD sampling. According to the invention, this measurement update step can be implemented as a recursive execution of up to M measurement update sub-steps, where j represents the recursion index, and the sampling distribution at time k is selected as the prior state distribution at time k, i.e., q(x k |y 1：k )=p(x k |y 1：k-1 Then we have:

[0024] Measurement update sub-step one: via formula Correct the variance parameter of the likelihood distribution, and then according to the formula... Obtain the initial posterior distribution of the state, and set the initial degrees of freedom parameters. Initial scaling matrix The measurement update sub-steps 2 and 3 are recursively executed M times to obtain the posterior distribution p(x) of the updated state. k，M |y 1：k ), posterior degrees of freedom parameters Posterior scale matrix in For the prior degrees of freedom parameters, Let p(x) be the prior scaling matrix. k，0 |y 1：k Let be the posterior distribution of the initial state at time k. Let be the initial likelihood distribution at time k, with the form and a mean of 0 and a variance of . Gaussian distribution, variance parameter For the latent variables of variational inference, p(x) k |y1：k-1 Let be the prior distribution of the state at time k;

[0025] Measurement update sub-step two: using the posterior distribution Determine the required number of sampling points, obtain the posterior distribution of the state particles, and use the Monte Carlo integration method according to the formula... To estimate the distribution parameters of the (j+1)th latent variable, this formula can be implemented as: in Let the degrees of freedom at time k be The scaling matrix is The (j+1)th corrected inverse Wissaud distribution For the j-th corrected degree of freedom parameter, Let h(·) be the scale matrix of the j-th correction, h(·) be the system measurement equation, and q(x) be the scale matrix of the j-th correction. k |y 1：k ) represents the sampling distribution of the state at time k, and N j x represents the number of particles determined by the KLD sampling method. k，j，i For posterior distribution particles The i-th particle;

[0026] Measurement update sub-step three: Through formula Correct the variance parameter of the likelihood distribution and then apply the formula Corrected state posterior distribution; where d is the state x k，j The dimension of p(x) k，j+1 |y 1：k Let be the posterior distribution of the state after the (j+1)th correction at time k. Let be the likelihood distribution of the (j+1)th correction at time k, with a mean of 0 and a variance of . Gaussian distribution, variance parameter These are latent variables for variational inference;

[0027] Step 3:

[0028] State updates employ a heuristic method for the degree-of-freedom parameters. Scale matrix The update process includes the following sub-steps:

[0029] State update sub-step one: The posterior distribution of the state at time k, updated by measurement, is determined according to the formula p(x) k+1 |y 1：k )=p(x k+1 |x k )p(x k，M |y 1：k The prior distribution of the state is obtained, and this formula can be implemented as x k+1，j，i =f(x) k，j，i )+u k , where p(xk+1 |y 1：k Let p(x) be the prior distribution of the state at time k+1. k+1 |x k () represents the state transition distribution from time k to time k+1;

[0030] State update sub-step two: Update the degree of freedom parameters at time k based on the measurement. According to the formula Obtain the prior degrees of freedom at time k+1 The scale matrix at time k updated by measurement According to the formula Obtain the prior scaling matrix at time k+1 Where ρ is a constant and ρ∈(0,1], and B is a constant matrix whose determinant satisfies |B|∈(0,1).

Claims

1. A method of KLD sampling based variational particle filter applied in the field of signal processing, characterized in that, It includes variable measurement update steps and state update steps based on KLD sampling; The variational measurement update step based on KLD sampling uses variational Bayesian estimation of the likelihood distribution and the KLD sampling method to estimate the posterior state distribution. It includes the following sub-steps: Measurement update sub-step one: via formula Correct the variance parameter of the likelihood distribution, and then according to the formula... Obtain the initial posterior distribution of the state, and set the initial degrees of freedom parameters. Initial scaling matrix The measurement update sub-steps 2 and 3 are recursively executed M times to obtain the posterior distribution p(x) of the updated state. k，M |y 1：k ), posterior degrees of freedom parameters Posterior scale matrix in For the prior degrees of freedom parameters, Let p(x) be the prior scaling matrix. k，0 |y 1：k Let be the posterior distribution of the initial state at time k. Let be the initial likelihood distribution at time k, with the form and a mean of 0 and a variance of . Gaussian distribution, variance parameter p(xk|y1:k-1) is the latent variable for variational inference, and p(xk|y1:k-1) is the prior distribution of the state at time k. Measurement update sub-step two: Obtain the posterior distribution of the state particles using KLD sampling from the posterior distribution, and apply the Monte Carlo integration method according to the formula... Estimate the distribution parameters of the (j+1)th latent variable; where Let the degrees of freedom at time k be The scaling matrix is The (j+1)th corrected inverse Wissaud distribution For the j-th corrected degree of freedom parameter, Let h(·) be the scale matrix of the j-th correction, h(·) be the system measurement equation, and q(x) be the scale matrix of the j-th correction. k |y 1：k ) represents the sampling distribution of the state at time k, and N j The number of particles determined by the KLD sampling method; Measurement update sub-step three: Through formula Correct the variance parameter of the likelihood distribution and then apply the formula Corrected state posterior distribution; where d is the state x k，j The dimension of p(x) k，j+1 |y 1：k Let be the posterior distribution of the state after the (j+1)th correction at time k, and let be the likelihood distribution after the (j+1)th correction at time k. Let be a Gaussian distribution with mean 0 and variance , and let the variance parameter be a latent variable in variational inference. The state update uses a heuristic method to update the scaling matrix of the degree-of-freedom parameters, which includes the following sub-steps: State update sub-step one: The posterior distribution of the state at time k, updated by measurement, is determined according to the formula p(x) k+1 |y 1：k )=p(x k+1 |x k )p(x k ,M|y 1：k ) to obtain the prior distribution of the state, where p(x k+1 |y 1：k Let p(x) be the prior distribution of the state at time k+1. k+1 |x k () represents the state transition distribution from time k to time k+1; State update sub-step two: update the degree of freedom parameters at time k by the measurement According to the formula Get the prior degree of freedom at time k+1 Update the scale matrix at time k by the measurement According to the formula Get the prior scale matrix at time k+1 Where ρ is a constant and ρ∈(0,1], and B is a constant matrix whose determinant satisfies |B|∈(0,1]; The variational Bayesian method is used to estimate the time-varying measurement noise parameters hidden behind the signal. At the same time, the KLD sampling method is used to adjust the required number of particles in real time according to the estimated time-varying measurement noise parameters, thereby enhancing the robustness of the particle filter.

2. The method of claim 1, wherein, In the measurement update sub-step two, the posterior distribution particles of the state are obtained by KLD sampling from the posterior distribution and then processed using the Monte Carlo integration method according to the equation. Estimate the distribution parameters of the (j+1)th latent variable, and then use the formula Correct the variance parameter of the likelihood distribution.

3. The method of claim 1, wherein, In the measurement update sub-step three, the variance parameter of the modified likelihood distribution is updated according to the formula The state posterior distribution is modified.​

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