Multi-source information fusion model in rule-based Gaussian-shaped fuzzy control inference system incorporating Gaussian density function

2.1Definitions

Definitions for our information structure and kernel computing method were established as follows.

Definition 1. Assume object Ω is described by the multi-source information set I = {I _k |k = 1, 2, ⋯ , m} and that measure set V = {v _k |k = 1, 2, ⋯ , m} is a set of information structures corresponding to set I. For ∀v _k  ∈ V, we define v _k  =< P _k ,  d _k ,  ρ _k  >, where P _k is a typical point as the kernel of I _k . Moreover, d _k is a metric of the information structure v _k related to the population or scale of information and will be used for boundary computing. Lastly, ρ _k is a density function on the threshold of v _k .

Definition 2. Let the fusion operator be ⊕ so that Ω can be formalized as:

where ⊕ is a minimum operator.

Definition 3. ∀P _k ,  Q _k in an n-dimensional Euclidean space R ⁿ , P _k  = [P _k1,  P _k2, ⋯ ,  P _kn ], and Q _k  = [Q _k1,  Q _k2, ⋯ Q _kn ]. Moreover, let d =∥  ∥, and it has the following properties:

We have d (αP _k ± βQ _k ) = ∥ αP _k  ± βQ _k  ∥; in addition,

Definition 4. For sample points {Pkl|l=1,2,⋯}, the statistics-based kernel point computation is calculated as:

Pkil indicates the value of the i-th dimension of the l-th sample point in the k-th information source.

The boundary of v _k gives the scale of the neighborhood of all elements in this special information structure. This is defined below.

Definition 5. For ∀P _k  ∈ R ⁿ , there exists a neighborhood,

Definition 6. For calculating the boundary of I, two sets were defined as:

- The Upper Approximation Boundary (UAB)

- The Lower Approximation Boundary (LAB)

Therefore, the boundary is P _B  = UP _B  ∖ LP _B  = B (u, t). Thus, we have P _B  = P _K  + λ (P _B  - P _K ), λ ∈ [0, 1], which exhibits fuzziness attributes at the boundary.

2.2Probability density function

–Single Gaussian Model for single-source information

The Gaussian distribution is a continuous probability distribution with a bell-shaped PDF in one-dimensional space:

The parameter μ is the mean or expectation, and σ ² is the variance. The SGM is applied to induct the density function of the proposed information structure I, and we define:

where X is a vector in n-dimensional space, Φ is the covariance matrix, and μ is the mean value of the density function. The density function’s properties are determined by (Φ,  μ), so this is a parameter estimation problem [29]. For any point P _i  ∈ R ⁿ , its probability density function is δ (P _i ,  μ,  Φ), and if, for any information structure v _k , each P _i in v _k is regarded as an independent event, then the PDF of v _k is:

The maximum likelihood estimation can be used to estimate the parameters (Φ,  μ) under (8). Taking the logarithm of (8), we have:

Taking the partial derivative w.r.t. μ of O (μ, Φ) and setting it to 0, we obtain the following:

This gives μˆ=12∑iPi. Similarly, for Φ, we can obtain Φˆ=1n-1∑i(Pi-μˆ)(Pi-μˆ)T. Thus, if the density of each point in v _k is δ(P, μˆ, Φˆ), then our estimation of the parameter μ is:

where e _li is the coordinate of P _i in R ⁿ .

The covariance Φˆ is converted to

- Gaussian Mixed Model and parameter estimation

For multi-source information fusion, we need to calculate all of I _k ’s density functions as well as calculate the new density function. For m multi-source information structures, let Ifusion=∑i=1lαiδ(P, μi, Φi) for a normalized weight parameter α: i.e., ∑ _i α _i  = 1. To calculate and simplify the covariance matrix Φ, let

From the SGM, we have that

Calculate:

and

Then, for Δ=cI→, c ∈ R, GMM is defined as G (P) = ∑ _i α _i δ (P, μ _i , σ _i ), i = 1, 2, ⋯ , lf. The number of parameters for estimation is 3l. If we let θ=[α1,α2,⋯,αl,μ1,μ2,⋯μl,σ12,σ22,⋯,σl2], the object is that:

which can be differentiated w.r.t. μ _j and σ _j . Thus, we have that:

Let φj(Pi)=αjδ(Pi,μj,σj2)∑j=1lαjδ(Pi,μj,σj2), so that:

Similarly, we can find:

Setting the above two equations equal to 0, we have

For α _j , under the constraint ∑ _j α _j  = 1, we use Lagrange multipliers to re-define the object as:

Differentiating this new object w.r.t. α _j , we have that:

Furthermore, we know λ = l, so: where φ is also a function of parameters, and we can resolve this using the following iteration:

Step 1: Let

θ=[α1,α2,⋯αl,μ1,μ2,⋯μl,σ12,σ22,⋯σl2]

Given an initial value and in order to achieve convergence, μ ₁, μ ₂, ⋯ μ _m may be calculated by the cluster method.

Step 2: Calculate φ _j  (P _i ).

Step 3: Calculate μ˜j=∑iφj(Pi)Pi∑iφj(Pi).

Step 4: Calculate

Step 5: Calculate αj=1lf∑iφj(Pi).

Step 6: Let

θˆ=[αˆ1,αˆ2,⋯αˆl,μˆ1,μˆ2,⋯μˆl,σˆ12,σˆ22,⋯σˆl2] If ∥θ-θˆ∥<δ for a given threshold δ, then stop the process; otherwise, proceed to Step 2.

In actuality, the density function of information fusion under this special structure is a product of the fusion of SGMs. For all information structures v _k and their SGM densities δ (v _k ), δ(Ifusion)=∏kδ(vk); therefore, we have that:

where C is an undetermined constant.

In particular, in a one-dimensional space with σ = 1, we have that:

This is a linear transformation of the basic Gaussian function. Thus, for any two information structures v _i ,  v _j , the fusion result is v _ij  =< αP _ij ,  βd _ij ,  γδ _ij  > where α,  β, and γ are undetermined coefficients.

3.1Fuzzy implications of information structures under IF-THEN rules

In fuzzy sets, the rule “IF x is A¯, THEN y is B¯” indicates a fuzzy implication between A¯ and B¯ as denoted by A¯→B¯. If we let x, y ∈ [0, 1] be the memberships of A¯ and B¯, respectively, we list the Mamdani model for the membership computing as t ∀x, y ∈ [0, 1], F (x, y) = Min {x, y}.

We construct a fuzzy membership based on a new fuzzy implication and inference system. We also derive a similarity relationship and apply this to the Gaussian density function-based fuzzy rule inference system. For δ _k in a rule-based IF-THEN inference system, suppose that the rule set is:

IF I ₁ is v ₁, then I _o is v _o , ω ₁, I ₂ is v ₂, then I _o is v _o , ω ₂.

We can integrate these rules as:

IF I ₁ is v ₁ and I ₂ is v ₂, THEN I _o is v _o ,

The Mamdani model for δ _k in two-dimensional space (x,  y) will be:

In particular, if σ ₁₁ = σ ₁₂ = σ ₂₁ = σ ₂₁ = 1 and μ ₁₁ = μ ₁₂ = μ ₂₁ = μ ₂₂ = 0, we have that:

Thus, for any other implication operators, the function of rules will have the form:

4.1Mamdani model-based fuzzy control inference system using nonlinear conjugate gradient

In the previous section, information was formalized as v _k  =< P _k ,  d _k ,  ρ _k  > where P _k is a central point in R ⁿ , d _k  ∈ R, and ρ _k is a Gaussian density function. P _k and d _k operate as fuzzy numbers using the fuzzy logical operation in Section 2. In our fuzzy rule-based inference system if different information (multi-source information) implies the same conclusion, then this information is integrated. Supposing that the multisource information structure v _k will conclude with a particular assertion at the ρ _k level, we have that:

This can be simplified to,

and

Now, we only discuss the density function δ _k (here ρ _k ) in our FIS. Letting Θ = [δ ₁, δ ₂, ⋯ , δ _n ], we have that:

From the previous section, we know that ϖ (Δ) is a Gaussian density function, so the rule is re-labeled as IF X THEN f (X). For this rule set, we have

However, as f (X) is a nonlinear function, it is difficult to find its minimum point under the Mamdani model, so we need to linearize f (X) and use the nonlinear conjugate gradient algorithm to optimize the parameters of f (X).

If we suppose that f (X) = [Ax - b]  ^T  [Ax - b], then the gradient is ∇ _x f (x) =2A ^T  (Ax - b), and the objective is to find x subject to ∇ _x f (x) =0. The nonlinear conjugate gradient requires f being twice differentiable, but as f is a Gaussian function, it is infinitely differentiable. Starting from the opposite direction as Δx ₀ = - ∇  _x f (x ₀) with step size α, we have that:

This is the first iteration in the direction of Δx ₀, and by setting the initial conjugate direction s ₀ = Δx ₀, the following steps will calculate Δx _n :

Step 1: Calculate Δx _n  = - ∇  _x f (x _n ).

Step 2: Calculate βn:βn=ΔxnT(Δxn-Δxn-1)Δxn-1TΔxn-1. (Polak–Ribière)

Step 3: Update the conjugate direction s _n  = Δx _n  + β _n s _n-1.

Step 4: Calculate αn=argminαf(xn+αsn).

Step 5: Update x _n+1 = x _n  + α _n s _n .

The algorithm is based on the quadratic function that we use to normalize the Gaussian function f (x) in order to speed up the iterations. Considering a simplified Mamdani model and from formula (20), we know that:

Using the nonlinear conjugate gradient, we obtain the results given in Fig. 1 and Table 1 by comparing with other special functions. From Table 1, we know that the Gaussian density function will be approximated in just a few steps by the nonlinear conjugate gradient algorithm, which is the reason we selected the Gaussian distribution as the density function of this special structure. We also compare other forms of density function, which appear to require more steps under the nonlinear conjugate gradient algorithm.

4.2Takagi–Sugeno model in RGS-FIS

Takagi and Sugeno [18] proposed a fuzzy IF-THEN rules system as the local input–output relations of a nonlinear system to scale the population of rules under a multi-dimensional fuzzy inference system, known as the T-S model [21]. The normal rules for the T-S model under the special information structure proposed for our information fusion method are:

The T-S model outputs a linear, non-constant function that will reduce the population of rules.

From rule set R _T-S , we can simplify If=∑i=1naiδi+bidi, in which a _i and b _i are undetermined constants. Let the standard deviation in Equation (6) σ=I→, μ = 0, and thus, If=12π∑i=1naiei-x22+bix. The first part of I _f is a GMM model that can be estimated by Section 2.2-(2), and the second part of I _f is a linear function (see Fig. 2).

Furthermore, for the nonlinear conjugate gradient proposed in Section 4.1, we obtain 100 steps and 301 gradients to find the minimum point (the Mamdani model). As a result, we can simplify this in RGS-FIS under the T-S model to output three linear membership functions. Suppose that the inputs are Gaussian-shaped rules, and the outputs are linear functions. Let the membership function of INPUT 1 and INPUT 2 be a Gaussian function, and the OUTPUT is composed of three linear functions [33]. We have this RGS-FIS system under the T-S model (see Fig. 3).