This chapter is divided into six sections. Section 1.1 proposes the problem, Section 1.2 introduces the dynamic fuzzy machine learning (DFML) model, and Section 1.3 describes the dynamic fuzzy machine learning system (DFMLS)-related algorithms. Section 1.4 discusses a process control model for DFML, and Section 1.5 presents a dynamic fuzzy relation learning algorithm. Finally, Section 1.6 gives a summary of this chapter.
Although machine learning theory has been widely used in various learning paradigms, the currently accepted concept of machine learning is as follows: “If a system can improve its performance by performing a certain process, it is learning”. The point of this argument is, first, that learning is a process; second, that learning considers the whole system; and third, that learning can change the performance of the system. “Process”, “system” and “performance change” are the three main points of learning. Obviously, if the system is considered human, the same argument can be established. According to this statement, existing machine learning methods could be considered somewhat limited. The reason is that these three points have a dynamic fuzzy character: the learning process is essentially dynamic and fuzzy; changes in the system (i.e. whether a system is “good” or “bad” and so on) are essentially dynamic and fuzzy; changes in system performance, results, and so on are all dynamic and fuzzy. These dynamic fuzzy characteristics are ubiquitous in the whole of machine learning and machine learning systems. Therefore, we believe that to make progress and meet the above definition of machine learning, the key question is to be able to effectively solve the dynamic fuzzy problems arising from learning activities.
However, existing machine learning methods are not sufficient for dynamic fuzzy problems. For example, Rough Set theory is based on fuzzy sets and can solve fuzziness problems, but it cannot handle dynamic problems; statistical machine learning is based on small-sample statistics and can solve static problems but not dynamic problems; Reinforcement learning is based on Markov processes and can solve dynamic problems but not fuzzy problems. Therefore, choosing dynamic fuzzy sets (DFSs) to study machine learning is a kind of inevitable choice. For the basic concepts of DFSs, see Appendix 8.1 and References [1, 2–4].
To effectively deal with the dynamic fuzzy problems that arise in machine learning activities, a coordination machine learning model based on dynamic fuzzy sets (DFS) and related algorithms has been proposed [9, 10]. Further work on this foundation has put forward the method of DFML and described a DFMLM and related algorithms in terms of their algebra and geometry [5–8, 11]. This section introduces the basic concepts of DFML, DFML algorithms and their stability analysis, and the geometry of DFML.
The process of system learning can be considered as self-adjusting, reflecting a series of changes in the system structure or parameters. Using mathematical language, learning can be defined as a mapping from one set to another set.
Definition 1.1 Dynamic fuzzy machine learning space: The space used to describe the learning process, which consists of all the DFML elements, is called the DFML space. It consists of five elements: {learning sample, learning algorithm, input data, output data, representation theory}, and can be expressed as (↼S
Definition 1.2 Dynamic fuzzy machine learning: DFML (↼l
Definition 1.3 Dynamic fuzzy machine learning system (DFMLS): The five elements of the DFMLS (, ) can be combined with a certain learning mechanism to form a computer system with learning ability. This is called a Dynamic Fuzzy Machine Learning System.
Definition 1.4 Dynamic fuzzy machine learning model (DFMLM): DFMLM = {(, ), (←L,→L)
If we discretize the processing system, (←S,→S)and(←L,→L)
Definition 1.5 DFMLM can be described as
where (←x,→x)(k)
According to the definition, the following three propositions can be obtained:
Proposition 1.1 DFMLS is a random system.
Proposition 1.2 DFMLS is an open system.
Proposition 1.3 DFMLS is a nonlinear system.
Definition 1.6 The model of DFML process can be described as
where:
(←So←So)
(←Y,→Y)
(←Op,→Op)
(←V,→V)
ER is an execution algorithm, which is the implementation of the algorithm to verify the source field elements and provide executive information.
(←G,→G)
This definition leads to the following proposition:
Proposition 1.4 DFML is an orderly process controlled by an incentive mechanism. Its general procedure is as follows:
(1)For a relevant subset (←S0+,←S0+)
(2)Consider the elements (←y0,←y0)
(3)For a subset (←So+,←So+)
(4)Take the above target as a new source field.
(5)Repeat the above steps until the learning process meets the accuracy requirements. A schematic diagram is shown in Fig. 1.2.
In DFML, the curse of dimensionality can be a serious problem. In machine learning, many of the data are nonlinear and high-dimensional, which brings further difficulties to data processing in machine learning. Therefore, we need to reduce the dimension (i.e. dimensionality reduction) of the high-dimensional data.
LLE is a well-known nonlinear dimension reduction method proposed by Saul and Roweis in 2000 [12]. Its basic idea is to transform the global nonlinearity into local linearity, with the overlapping local neighbourhoods providing the global structure of the information. In accordance with certain rules, each part of the linear dimension reduction is then combined with the results to give a low-dimensional global coordinate representation.
In the dynamic fuzzy high-dimensional space (←RD,←RD),
Definition 1.7 The model for the dynamic fuzzy dimensionality reduction problem is ((←X,→X),(←F,→F)),
Definition 1.8 The mapping (←f,→f):(←Y,→Y)→(←T,→T)⊂(←RD,←RD)
Consider the arbitrary point (←xi,←xi)∈(←X,→X),Let{(←xli,←xli),l=1,2,...,k}
Theorem 1.1 If (←xi,←xi)=k∑l=1(←wli,←wli)(←xli,←xli).
where (←wli,←wli)(i=1,2,...,n;l=1,2,...,k)
Proof: Because (→Fi,→Fi)
We have
This suggests that the local linear dimension reduction is a constant. (←xi,←xi)
The core of the LLE method is to find the k-dimensional dynamic fuzzy vector (←Wi,←Wi)=((←W1i,←W1i),(←W1i,←W2i),...,(←Wki,←Wki))
where k∑l−1(←w1i,←w1i)=(←1,←1),(←w1i,←w1i)=(←0,→0),
because (, ) is solved by the dynamic fuzzy dataset (, , and the computation is very sensitive to noise, especially when the eigenvalues of (←X,→X)T(←X,→X) are small. In this case, the desired result may not be obtained.
Let (←x1l,←x1l)'=(←x1l,←x1l)+(←ε1i,←ε1i),(i=1,2,...,k) denote the corresponding noise-affected point and (←xi,←xi)'=k∑l=1(←w1i,←w1i)'(←xli,←xli)',k∑l=1(←w1i,←w1i)'=(←1,→1);(←U(←xi)),→U(→xi))=((←x1i,←x1i),(←x2i,←x2i),...,(←xki,←xki)) is the k-neighbourhood DFS of (←xi,←xi),(←U(←xi),←U(←xi))'=((←x1i,←x1i)', (←x2i,←x2i)',...,(←xki,←xki)') is the k-neighbourhood DFS of (←xi,←xi)',, and