CN104485921A - Distributed particle filter method based on information consistency - Google Patents

Abstract

The invention provides a sensor network distributed particle filter method based on information consistency in a nonlinear and non-Gaussian system. According to the method, based on centralized particle filter, network reliability is introduced to represent a global likelihood function, the interaction and the fusion of local reliability are performed through consistency iteration, the consistency of particle samples and weight value among nodes is realized, the precision is estimated to approach an centralized method. The distributed particle filter method based on information consistency is used for realizing consistency state estimation throughout of the network through a distributed mode, and has relatively excellent estimation performances in nonlinear and non-Gaussian environments of flicker noises and the like.

Description

Distributed particle filtering method based on information consistency

Technical Field

The invention belongs to the sensor network information fusion technology, relates to the problem of multi-sensor consistency state estimation, and provides a distributed particle filtering method based on information consistency under a nonlinear non-Gaussian condition.

Background

In order to meet the practical application requirements, the distributed sensor network is usually deployed in unknown or even dangerous environment to perform tasks, and the application environment of the distributed sensor network may have nonlinear and non-gaussian characteristics. For example, when a sonar sensor in a coastal anti-submarine detection system detects an offshore or underwater target, a received signal not only contains an echo signal of a moving target, but also has interference such as sea clutter noise, sea underwater sound environmental noise and the like, the noise has a significant spike characteristic at some time points, the noise variance is increased sharply, and the noise can be approximated to flicker noise. The above-mentioned maneuvering target tracking under the flicker noise and clutter environment is essentially a state estimation problem in a nonlinear non-gaussian system, and if the gaussian assumption is still adopted at this time, the system estimation performance will be degraded, or even the system cannot work.

For the problem of state filtering in a nonlinear non-gaussian system, the existing methods can be divided into three categories: (1) an analysis method based on system approximation and probability density function Gaussian mixture approximation; (2) solving a numerical method of an integral expression in a Bayes recursion relation in a numerical mode; (3) simulation based on Monte Carlo. The analysis method based on probability density function Gaussian mixture approximation integrates a series of local estimation methods in a weighting mode, specifically, non-Gaussian noise is fitted through weighted mixture Gaussian, a nonlinear system model is processed through a local nonlinear filter, complete conditional probability density function information can be provided, and high estimation quality can be obtained in a nonlinear system. However, this type of method does not consider the case where the observation of nodes in the sensor network is limited.

Disclosure of Invention

1. Technical problem to be solved

The invention aims to provide a consistency state estimation method under a nonlinear and non-Gaussian condition, namely a distributed particle filtering method IC-DPF based on information consistency. The method introduces a network reliability representation global likelihood function, realizes interaction and fusion of local reliability through consistency iteration, realizes consistency of particle samples and weights among nodes, and improves state estimation precision and robustness under complex environments such as flicker noise and the like.

2. Technical scheme

The distributed particle filtering method based on information consistency, which is disclosed by the invention, is based on a distributed solution method of a network global likelihood function and a particle filtering theory, and comprises the following technical measures: performing consistency iteration initialization based on standard reliability; embedding reliability transfer between nodes into a standard reliability process; and (5) consistency iteration based on credibility transfer.

3. Advantageous effects

Compared with the background technology, the invention has the following advantages:

(1) the method is suitable for complex environments with nonlinear non-Gaussian, limited network nodes and the like, and has high estimation precision and robustness;

(2) the invention realizes the consistency estimation of the whole network in a distributed mode, does not need the information collection of the whole network while ensuring the estimation performance, and can greatly reduce the network communication overhead;

drawings

FIG. 1: a flow chart of a distributed particle filtering method based on information consistency;

FIG. 2: a simulation scene graph;

FIG. 3: comparing the estimated performance;

Detailed Description

The invention is described in further detail below with reference to the drawings. Referring to the attached drawings, the single-cycle mode of the target state updating of the invention comprises the following steps:

1. description of the problem

Without loss of generality, consider the following discrete-time nonlinear non-Gaussian system

x_k＝f(x_k-1)+w_k-1 (1)

z_i,k＝h_i(x_k)+v_i,k,i＝1,2,...,N_S (2)

Wherein x_kAnd z_i,kRespectively representing the state vector of the system and the measurement vector of the ith sensor, N_SIndicating the number of sensor nodes, f and h_iRespectively representing the transfer function of the system and the measurement function of the ith sensor, w_kAnd v_i,kNon-gaussian process noise and non-gaussian metrology noise, respectively.

In practical applications, non-gaussian noise is more difficult to process than gaussian noise. Taking into account any profileThe rate density function can be approximated by weighting a finite number of Gaussian terms of the mixture, i.e., w_kAnd v_i,kThe estimation of the probability density function of (a) can be expressed in the following gaussian mixture:

wherein alpha is_pAnd beta_qThe sum of the mixing weights is required to be 1; g₁And G₂Is the number of gaussian mixture terms;the probability density function of gaussian random variables with mean σ and variance Σ is expressed. Without loss of generality, the mean of the Gaussian mixture term is set to zero, μ_k,p＝υ_i,k,q＝0。

2. Distributed particle filtering DPF

Theoretically, for distributed consistent particle filtering, all nodes in the network need to maintain a common particle sample and particle weight in each filtering period. Assuming that the particle samples and the particle weights for each node at time k-1 are identical, at time k, all nodes need to be first seeded with the same random number (i.e., their pseudo-random generators are guaranteed to be in the same state at any time) so that each node maintains a common particle sample. In addition, in order to ensure that each node has an equal particle weight, a Belief Consensus (BC) is introduced to represent a Joint Likelihood Function (JLF). BC, as shown in the following equation, functions by computing a series of real functions f with the same arguments, primarily in a distributed fashion_i(x) The product of (a):

<math> <mrow> <mi>BC</mi> <mo>[</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>f</mi> <msub> <mi>N</mi> <mi>s</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>=</mo> <munderover> <mi>&Pi;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>S</mi> </msub> </munderover> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </math>

it can be seen that if f is given_i(x) And the local likelihood function of the sensor i is represented, for the node i, the local likelihood functions of all nodes in the network can be obtained through the consistency information iteration among the neighbor nodes and based on the formula (5), so that the consistency of the local estimation to the global estimation is converged. However, in a sensor network with limited resources, most BC methods can only run a limited number of consistency iterations, and cannot collect complete local likelihood information, so that it is difficult to achieve accurate consistency of the whole network. At this time, to achieve consistency estimation of the network local area, a Max consistency rule (MC) is generally adopted, that is, the Max consistency rule

In this way, the first and second electrodes,nodes in a local area of the network can obtain accurate consistency in a limited number of iterations. More specifically, toRepresenting the function of node i at the ith iteration, MC may be implemented according to the following rules:

wherein,it should be noted that, in order to avoid estimation collisions in the network, a maximum consistency rule needs to be applied before the state estimates are calculated.

Compared with CF-PF, DPF has the following advantages: (1) the energy consumption of nodes in the network is relatively balanced; (2) communication costs in large sensor network applications are reduced; (3) each node in the network can calculate to obtain posterior distribution; (4) there is no need to predict the location, measurement and measurement model of any node. Table 1 shows the specific filtering process of node i at time k in the DPF method.

TABLE 1DPF method steps

Table 1The iterative process of DPF

3. Adaptive consistency rate factor based on dynamic topology

Step 1: standard-confidence-based consistency iteration initialization

To calculate uniform particle weights, based on Standard Belief Consistency (SBC), the following iterative form of information is defined:

wherein,indicating the state x at the l-th iteration_kζ ≈ 1/Δ of the global likelihood function_maxIs the update speed, depending on the maximum node degree in the networkInitialization of the consistency iteration is performed using the equation:

$M_i^{\left(1\right)}\left(x_k\right)=p\left(Z_{i,k}\right|x_k)---\left(9\right)$

step 2: embedding trust transfer between nodes into standard trust process

Due to the fact that

Thus, the local likelihood function estimates for all nodes in the network will converge to the global likelihood function estimate.

From above to above, forCan be calculated based on the formula (10)In practice, however, only a limited number of consistency iterations are typically performedThe result is an estimate of the true likelihood function.

In order to realize the calculation of the global likelihood function by the distributed consistency solution, the reliability transfer between the nodes needs to be embedded in the SBC process. Here, the following function is defined:

thus, for any node i, there is

<math> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mi>&Sigma;</mi> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>&NotEqual;</mo> <mi>i</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </munder> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </math>

The confidence of the first iteration is

Wherein the information transmitted from node j to node i is

<math> <mrow> <msubsup> <mi>m</mi> <mi>ji</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&Proportional;</mo> <msub> <mo>&Integral;</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mfrac> <mrow> <msubsup> <mi>M</mi> <mi>j</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>m</mi> <mi>ij</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mi>d</mi> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow> </math>

$m_{\mathrm{ji}}^{\left(l\right)}\left(x_{i,k}\right)=\frac{m_j^{(l-1)}\left(x_{i,k}\right)}{m_{\mathrm{ij}}^{(l-1)}\left(x_{i,k}\right)}---\left(16\right)$

And step 3: consistency iteration based on belief transfer

Since all arguments are consistent, they are not denoted as x_i,k＝x_j,k＝x_k. Then, there are

$m_{\mathrm{ji}}^{\left(l\right)}\left(x_k\right)=\frac{m_j^{(l-1)}\left(x_k\right)}{m_{\mathrm{ij}}^{(l-1)}\left(x_k\right)}---\left(17\right)$

And because of

$m_{\mathrm{ji}}^{(l-1)}\left(x_k\right)=\frac{m_j^{(l-2)}\left(x_k\right)}{m_{\mathrm{ij}}^{(l-2)}\left(x_k\right)}---\left(18\right)$

Combining formula (17) with formula (18) to obtain

$m_{\mathrm{ji}}^{\left(l\right)}\left(x_k\right)=\frac{m_j^{(l-1)}\left(x_k\right)}{m_{\mathrm{ij}}^{(l-2)}\left(x_k\right)}{m}_{\mathrm{ji}}^{(l-2)}\left(x_k\right)---\left(19\right)$

By recombining formula (19) and formula (15), the

The above equation characterizes the information consistency iterative process of the IC-DPF. In addition, it is necessary to provideCan the equation (20) be performed. Based on the expressions (15) and (16), and assume thatCan obtain

From the above, the IC-DPF can ensure the network state estimation convergenceWhere phi is a normalization constant independent of the variable. Furthermore, the IC-DPF enables the product of all local likelihood estimates to be kept consistent and does not require any other parameters in the network to be foreseen (such as the network maximum degree of node Δ)_maxAnd the number of network nodes N_S). Therefore, the IC-DPF is more robust to dynamic network topologies.

4. Simulation comparison and analysis

In this section, a typical numerical simulation scenario is constructed for verifying the validity of the method of the present invention. The distributed method involved in the simulation is IC-DPF, and the corresponding centralized optimal estimation method is also involved in the comparison as a reference (not referred to as CF-PF). For simplicity of labeling, it should be noted that the distributed particle filter for state estimation using only the measurement of the ith sensor is not labeled as Si-IC-DPF (the estimation curves for selecting two sensor nodes are shown in the figure and labeled as S1-IC-DPF and S2-IC-DPF respectively), and the particle filter for centralized state estimation using the whole-network measurement is labeled as CF-PF. 4.1 simulation setup

Consider a distributed sensor network target state estimation scenario in the presence of flicker noise, where flicker metric noise is modeled as non-gaussian noise. The distance unit of the relevant variable is hidden in the simulation experiment, and the environment setting and the simulation result in the simulation can be referred according to the specific application. Setting the target to move in a two-dimensional plane, and expressing the motion model as

$x_k=\left[\begin{array}{cccc}1& T_s& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T_s\\ 0& 0& 0& 1\end{array}\right]x_{k-1}+\left[\begin{array}{cc}{T_s}^2/2& 0\\ T_s& 0\\ 0& {T_s}^2/2\\ 0& T\end{array}\right]w_{k-1}---\left(22\right)$

Wherein T is the sampling period;is the target state vector, (x)_k,y_k) The target position at the time point k is,the target moving speed at the moment k is obtained; process noise is assumed to obey a zero mean variance of Q_k-1White gaussian noise. The sensor network is composed of Ns (total number of nodes) 12 isomorphic nodes, and the measurement equation is

$z_{i,k}=\left[\begin{array}{c}\sqrt{{(x_k-x_{i,0})}^2+{(y_k-y_{i,0})}^2}\\ \arctan [(y_k-y_{i,0})/(x_k-x_{i,0})]\end{array}\right]+v_{i,k},i=\mathrm{1,2},...,N_s---\left(23\right)$

Wherein (x)_i,0，y_i,0) The position of the ith radar; v. of_i,kNoise is measured for flicker. In the simulation, a two-term Gaussian mixture model is adopted to describe flicker noise v_i,kAnd there are the following two methods described.

Two zero mean Gaussian mixture terms are used to represent v_i,kAnd representing the outliers by a Gaussian term having a large variance, e.g.

Among these are the flicker probabilities of noise.

As shown in fig. 2, the coordinates of the 12 sensors are (0,30), (0,60), (0,90), (80,30), (80,60), (80,90), (160,30), (160,60), (160,90), (240,30), (240,60), (240,90), respectively. The lines between the sensors in fig. 2 represent possible communication links between the nodes, and it can be seen that the network is not fully connectedThe topology of (1). In the simulation, the method is initialized by adopting a uniform state prior initial value, namelyWherein x₀Is the true position of the target, x_bIs a random deviation (from a mean value of [ beta 0 ]]^TGaussian random variable generation). The target state is estimated by 4-term Gaussian weighted mixture, and the initial variances are all P_0|0＝diag{2²；0.1²；2²；0.1²}. Further, the initial value of the target state is set to x₀＝[-20 5 20 1.5]^T. The simulation period is K-50 time steps.

The number of consistent iterations of each method is set to be 7 in default, and the number of particles participating in estimation in particle filtering is set to be 500.

4.2 simulation results and analysis

Examining the estimated performance of each method when flicker measurement noise is described by method one, where the flicker probability is set to 0.1 and the process noise w_k-1Is set to Q_k-1Biag { 0.01; 0.01}. Variance matrix R of all sensors_i,1And R_i,2Are respectively set as R_i,1＝diag{0.2²；0.015²And R_i,2＝diag{2²；0.15²}. The parameter β for generating the initial state random offset is set to β ═ 1.

Fig. 3 shows the variation of the mean square error of the method with time step. It can be seen that for distributed consensus estimation, as can be seen from fig. 3, the selected single local filters (single sensors) all achieve an average consensus estimation and gradually converge to a centralized estimation result. Wherein, the mean value estimation precision of the IC-DPF is almost equal to that of a corresponding centralized method. 4.3 method overhead comparison

Without loss of generality, by N_packThe number of information packets broadcasted to the network by any node i at the moment k is shown, and P information packets are assumed to be contained in a single information packetA scalar value. In most hardware platforms, P > 1, and the energy required to deliver a packet does not depend primarily on the amount of data in the packet. It should be noted that determining Δ is omitted from the overhead analysis of the method of this section_maxΔ and N_SThe communication and computation costs required for the steps of equal parameters and clock synchronization.

(1)IC-DPF

In each consistency iteration (except for iteration 1), the node passes M_pWeight (using M)_pIndividual particles are filtered). Furthermore, the node must run the MC, which also requires the particle weight to be passed in each iteration. The iteration number of the SBC process in the IC-DPF isThe number of iterations of the MC procedure is equal to the diameter D of the network map_g. Thus, the average cost assigned to each node and each time interval in the IC-DPF is

<math> <mrow> <msubsup> <mi>N</mi> <mi>pack</mi> <mrow> <mi>IC</mi> <mo>-</mo> <mi>DPF</mi> </mrow> </msubsup> <mo>&ap;</mo> <mo>[</mo> <mfrac> <msub> <mi>M</mi> <mi>p</mi> </msub> <mi>P</mi> </mfrac> <mo>]</mo> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>g</mi> </msub> <mo>+</mo> <msubsup> <mi>L</mi> <mi>l</mi> <mi>SBC</mi> </msubsup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow> </math>

Furthermore, the IC-DPF is fully scalable, since increasing the number of nodes in a fixed deployment scenario does not significantly impact the network communication cost (communication is only performed between neighboring nodes).

(2)NCPF

Compared with the IC-DPF, the NCPF method does not need to transmit the particle weight, but needs to broadcast local data, and does not use N_dataRepresenting the number of these scalar values. Each node accumulates data during the iteration process, since the node must pass both local data and received data. Based on the number of iterations D_gThe communication cost of NCPF can be estimated by the following equation:

<math> <mrow> <msubsup> <mi>N</mi> <mi>pack</mi> <mi>NCPF</mi> </msubsup> <mo>&ap;</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <msub> <mi>D</mi> <mi>g</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>[</mo> <mfrac> <mrow> <msup> <mover> <mi>&Delta;</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msup> <msub> <mi>N</mi> <mi>data</mi> </msub> </mrow> <mi>P</mi> </mfrac> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein, the average value of the network node degrees is usedAs an approximation of a single degree of node, an

(3) IC-DPF and NCPF

Without loss of generality, assumeWhen in useWhen there is

<math> <mrow> <mo>[</mo> <mfrac> <msub> <mi>M</mi> <mi>p</mi> </msub> <mi>P</mi> </mfrac> <mo>]</mo> <mo>&lt;</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>D</mi> <mi>g</mi> </msub> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <msub> <mi>D</mi> <mi>g</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>[</mo> <mfrac> <mrow> <msup> <mover> <mi>&Delta;</mi> <mo>&OverBar;</mo> </mover> <mi>k</mi> </msup> <msub> <mi>N</mi> <mi>data</mi> </msub> </mrow> <mi>P</mi> </mfrac> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow> </math>

The formula represents the precondition that the IC-DPF is superior to NCPF, and can be used as a reference factor for reasonably selecting the method. For example, if the network is fully connected (D)_g1) or the capacity of the packet is large enough to convey the accumulated data of all sensors (i.e., the accumulated data of all sensors)) At this time, the NCPF method should be used; however, in a distributed sensor network with limited observation, the communication radius of the nodes is small (D)_gToo large), the information that a single node can communicate is limited, and should be taken at that timeWith the IC-DPF method, otherwise, flooding communication explosion may be caused during the whole network information collection.

In addition, for CF-PF, the communication cost of its method depends on many factors, including the routing protocol and the location of the fusion center, etc. Overall, the cost of CF-PF is much less than the cost of NCPF, since CF-PF has only one fusion center to which all sensor information needs to be routed, rather than N as NCPF does_SEach node needs to collect the full network information. However, since the energy consumption of the nodes in the CF-PF method is unbalanced and the non-fusion central node cannot calculate the posterior probability distribution of the target state, the CF-PF is not suitable for a distributed sensor network with limited node observation.

Regarding computational complexity, every node in the IC-DPF needs to be filtered within a single filtering instantThe number of times of operations that each node needs to perform in the NCPF method isHowever, considering that the energy consumption of the sensors comes mainly from the communication overhead, the computational overhead inside the nodes is relatively less influential. Therefore, IC-DPF is also superior to NCPF in terms of implementation cost.

Claims (1)

1. The distributed particle filtering method based on the information consistency is characterized by comprising the following steps of:

(1) standard-confidence-based consistency iteration initialization

$M_i^{\left(1\right)}\left(x_k\right)=p\left(Z_{i,k}\right|x_k);$

(2) Embedding trust transfer between nodes into standard trust process

For any node i, all have

<math> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mi>&Sigma;</mi> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>&NotEqual;</mo> <mi>i</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </munder> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>N</mi> <mi>S</mi> </msub> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </math>

The confidence of the first iteration is

Wherein the information transmitted from node j to node i is

<math> <mrow> <msubsup> <mi>m</mi> <mi>ji</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>&Proportional;</mo> <msub> <mo>&Integral;</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mfrac> <mrow> <msubsup> <mi>M</mi> <mi>j</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>m</mi> <mi>ij</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <msub> <mi>dx</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </math>

$m_{\mathrm{ji}}^{\left(l\right)}\left(x_{i,k}\right)=\frac{m_j^{(l-1)}\left(x_{i,k}\right)}{m_{\mathrm{ij}}^{(l-1)}\left(x_{i,k}\right)};$

(3) Consistency iteration based on belief transfer

The above equation characterizes the information consistency iterative process of the IC-DPF.