5.3.3 DFSSMTL model

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DFSSL seeks to obtain a function $(\overset{\leftarrow}{f}, \vec{f}) : (\overset{\leftarrow}{X}, \vec{X}) \to (\overset{\leftarrow}{Y}, \vec{Y})$ $(\overset{\leftarrow}{f}, \vec{f}) : (\overset{\leftarrow}{X}, \vec{X}) \to (\overset{\leftarrow}{Y}, \vec{Y})$ that can accurately predict the label $(\overset{\leftarrow}{y}, \vec{y})$ $(\overset{\leftarrow}{y}, \vec{y})$ of sample $(\overset{\leftarrow}{x}, \vec{x})$ $(\overset{\leftarrow}{x}, \vec{x})$ and classify it into class $({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y})$ $({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y})$ with membership degree $({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), {\vec{C}}_{y} (\vec{x})) .$ $({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), {\vec{C}}_{y} (\vec{x})) .$ Here, $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}) and ({\overset{\leftarrow}{x}}_{i}, \vec{x}'_{i})$ $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}) and ({\overset{\leftarrow}{x}}_{i}, \vec{x}'_{i})$ are d-dimensional vectors, $({\overset{\leftarrow}{y}}_{i}, {\vec{y}}_{i}) \in (\overset{\leftarrow}{Y}, \vec{Y})$ $({\overset{\leftarrow}{y}}_{i}, {\vec{y}}_{i}) \in (\overset{\leftarrow}{Y}, \vec{Y})$ is the label of sample $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}), and | (\overset{\leftarrow}{L}, \vec{L}) |, | \overset{\leftarrow}{U}, \vec{U} |$ $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}), and | (\overset{\leftarrow}{L}, \vec{L}) |, | \overset{\leftarrow}{U}, \vec{U} |$ are the sizes of $(\overset{\leftarrow}{L}, \vec{L}) and (\overset{\leftarrow}{U}, \vec{U}),$ $(\overset{\leftarrow}{L}, \vec{L}) and (\overset{\leftarrow}{U}, \vec{U}),$ respectively (that is, the number of samples contained in the labelled and unlabelled sample sets).

Compared with the previous definitions, it is easy to find that DFSSL is a subclass of dynamic fuzzy machine learning. If the learning sample in the dynamic fuzzy machine learning space is refined into unlabelled and tagged data, dynamic fuzzy machine learning can be called DFSSL.

DFSSL is an attempt to find an optimal learner for the sample set $(\overset{\leftarrow}{X}, \vec{X}) .$ $(\overset{\leftarrow}{X}, \vec{X}) .$ If $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{L}, \vec{L}),$ $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{L}, \vec{L}),$ then the problem is transformed into dynamic fuzzy supervised learning; if $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{U}, \vec{U}),$ $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{U}, \vec{U}),$ then the problem becomes dynamic fuzzy unsupervised learning. However, how to use the labelled samples and the unmarked samples effectively is one area of study for DFSSL. It is worth mentioning that the three hypotheses in traditional semi-supervised learning still exist in DFSSL.

5.3.3DFSSMTL model

Definition 5.6 Dynamic fuzzy task (DFT): A problem or task with dynamic ambiguity is called a dynamic fuzzy task.

Obviously, descriptions of people’s age, weight, and height are dynamic and fuzzy, so the description of these three tasks is a DFT. Figure 5.4 describes three ambiguous dynamic tasks. We assume that Task 1 describes the age of a person and is composed of three categories: old, young, and juvenile. Task 2 describes a person’s weight and consists of obese, average, and slim. Task 3 describes the height of the person as either tall, medium, or short. Obviously, these three tasks are dynamic fuzzy characteristics because the age, weight, and height of the standard itself have some dynamic ambiguity.

Definition 5.7 Dynamic fuzzy multi-task learning (DFMTL): The learning model of dynamic fuzzy multi-task learning considers the sequential or simultaneous learning of multiple related DFTs.

Fig. 5.4: Three examples of dynamic ambiguous tasks that consist of three classes.

It is easy to see that the different descriptive tasks describing different aspects of the same person from different angles are relevant; that is, the three dynamic ambiguity tasks are related. If we study these DFTs at the same time or sequentially, we are performing DFMTL.

Definition 5.8 Dynamic fuzzy semi-supervised multi-task learning: Given a set of M related DFTs, the sample set of each task is from an unknown distribution. The sample of the t-th task set is labelled as

${(\overset{\leftarrow}{X}, \vec{X})}_{t} = {(\overset{\leftarrow}{L}, \vec{L})}_{t} {(\overset{\leftarrow}{U,} \vec{U})}_{t}$ ${(\overset{\leftarrow}{X}, \vec{X})}_{t} = {(\overset{\leftarrow}{L}, \vec{L})}_{t} {(\overset{\leftarrow}{U,} \vec{U})}_{t}$

where ${(\overset{\leftarrow}{L}, \vec{L})}_{t} = {({({\overset{\leftarrow}{x}}_{1}, {\vec{x}}_{1})}_{t}, {({\overset{\leftarrow}{y}}_{1}, {\vec{y}}_{1})}_{t}), ({({\overset{\leftarrow}{x}}_{2}, {\vec{x}}_{2})}_{t} {({\overset{\leftarrow}{y}}_{2}, {\vec{y}}_{2})}_{t}), ..., ({({\overset{\leftarrow}{x}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}, {\vec{x}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |})}_{t}),$ ${(\overset{\leftarrow}{L}, \vec{L})}_{t} = {({({\overset{\leftarrow}{x}}_{1}, {\vec{x}}_{1})}_{t}, {({\overset{\leftarrow}{y}}_{1}, {\vec{y}}_{1})}_{t}), ({({\overset{\leftarrow}{x}}_{2}, {\vec{x}}_{2})}_{t} {({\overset{\leftarrow}{y}}_{2}, {\vec{y}}_{2})}_{t}), ..., ({({\overset{\leftarrow}{x}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}, {\vec{x}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |})}_{t}),$ $({\overset{\leftarrow}{y}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}, {\vec{y}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}))}$ $({\overset{\leftarrow}{y}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}, {\vec{y}}_{| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |}))}$ is the tagged sample set for the t th task and ${(\overset{\leftarrow}{U}, \vec{U})}_{t} =$ ${(\overset{\leftarrow}{U}, \vec{U})}_{t} =$ ${{(\overset{\leftarrow}{x}'_{1}, \vec{x}'_{1})}_{t}, {(\overset{\leftarrow}{x}'_{2}, \vec{x}'_{2})}_{t}, ..., {(\overset{\leftarrow}{x}'_{| {(\overset{\leftarrow}{U}, \vec{U})}_{t} |}, \vec{x}'_{| {(\overset{\leftarrow}{U}, \vec{U})}_{t} |})}_{t}}$ ${{(\overset{\leftarrow}{x}'_{1}, \vec{x}'_{1})}_{t}, {(\overset{\leftarrow}{x}'_{2}, \vec{x}'_{2})}_{t}, ..., {(\overset{\leftarrow}{x}'_{| {(\overset{\leftarrow}{U}, \vec{U})}_{t} |}, \vec{x}'_{| {(\overset{\leftarrow}{U}, \vec{U})}_{t} |})}_{t}}$ is the unlabelled sample set for task t.

DFSS-MTL seeks to learn multiple DFTs simultaneously or sequentially and aims to learn a function for each task. The functions of the t th task are written as ${(\overset{\leftarrow}{f}, \vec{f})}_{t} : {(\overset{\leftarrow}{X}, \vec{X})}_{t} \to {(\overset{\leftarrow}{Y}, \vec{Y})}_{t},$ ${(\overset{\leftarrow}{f}, \vec{f})}_{t} : {(\overset{\leftarrow}{X}, \vec{X})}_{t} \to {(\overset{\leftarrow}{Y}, \vec{Y})}_{t},$ so that we can accurately predict the tag ${(\overset{\leftarrow}{y}, \vec{y})}_{t}$ ${(\overset{\leftarrow}{y}, \vec{y})}_{t}$ of the samples ${(\overset{\leftarrow}{x}, \vec{x})}_{t}$ ${(\overset{\leftarrow}{x}, \vec{x})}_{t}$ of the t th task and classify ${(\overset{\leftarrow}{x}, \vec{x})}_{t}$ ${(\overset{\leftarrow}{x}, \vec{x})}_{t}$ into classes ${({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y})}_{t}$ ${({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y})}_{t}$ whose membership degree is denoted as ${({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), {\vec{C}}_{y} (\vec{x}))}_{t} .$ ${({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), {\vec{C}}_{y} (\vec{x}))}_{t} .$ Here, ${({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})}_{t} and {(\overset{\leftarrow}{x_{i}^{'}}, \vec{x_{i}^{'}})}_{t}$ ${({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})}_{t} and {(\overset{\leftarrow}{x_{i}^{'}}, \vec{x_{i}^{'}})}_{t}$ are d-dimensional vectors, ${({\overset{\leftarrow}{y}}_{i}, {\vec{y}}_{i})}_{t} \in {(\overset{\leftarrow}{Y}, \vec{Y})}_{t}$ ${({\overset{\leftarrow}{y}}_{i}, {\vec{y}}_{i})}_{t} \in {(\overset{\leftarrow}{Y}, \vec{Y})}_{t}$ is the tag for the t th task samples ${({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})}_{t},$ ${({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})}_{t},$ and $| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |, | {(\overset{\leftarrow}{U}, \vec{U})}_{t} |$ $| {(\overset{\leftarrow}{L}, \vec{L})}_{t} |, | {(\overset{\leftarrow}{U}, \vec{U})}_{t} |$ represent the size of the untagged sample set ${(\overset{\leftarrow}{L}, \vec{L})}_{t}$ ${(\overset{\leftarrow}{L}, \vec{L})}_{t}$ of the t th task and the unlabelled sample set ${(\overset{\leftarrow}{U}, \vec{U})}_{t}$ ${(\overset{\leftarrow}{U}, \vec{U})}_{t}$ of the t th task (that is, the number of samples in the labelled sample set and unlabelled sample set of the t th task).

For the former definition, it is easy to find that DFSS-MTL is also a subclass of DFML. If the learning sample in the dynamic fuzzy machine learning space is refined into unlabelled data and tagged data in a single DFT and multiple related DFTs are learned in sequence or simultaneously, it can be said that the dynamic fuzzy machine learning is DFSSMTL.

DFSS-MTL is mainly considered from two aspects. First, when the data in the task to be learned are extremely rare, the DFT can be learned using sample data from similar or related DFTs learned in the past, that is, learning in the context of DFMTL. Second, in the case of DFMTL, there are unlabelled sample data in addition to the labelled sample data, which belong to the DFSSL domain.

Figure 5.5 depicts a simple DFSS-MTL model. The learning sample sets ${{(\overset{\leftarrow}{X}, \vec{X})}_{1}, {(\overset{\leftarrow}{X}, \vec{X})}_{M}}$ ${{(\overset{\leftarrow}{X}, \vec{X})}_{1}, {(\overset{\leftarrow}{X}, \vec{X})}_{M}}$ of M tasks are obtained under this model. The class label sets ${{(\overset{\leftarrow}{X}, \vec{X})}_{1}, {(\overset{\leftarrow}{X}, \vec{X})}_{M}}$ ${{(\overset{\leftarrow}{X}, \vec{X})}_{1}, {(\overset{\leftarrow}{X}, \vec{X})}_{M}}$ of each task are obtained and the membership degree ${({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), \vec{C} (\vec{x}))}_{t}$ ${({\overset{\leftarrow}{C}}_{y} (\overset{\leftarrow}{x}), \vec{C} (\vec{x}))}_{t}$ of a sample $(\overset{\leftarrow}{x}, \vec{x})$ $(\overset{\leftarrow}{x}, \vec{x})$ in task t is represented by $({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y}) .$ $({\overset{\leftarrow}{C}}_{y}, {\vec{C}}_{y}) .$

5.4Dynamic fuzzy semi-supervised multi-task matching algorithm

We can regard the performance of the dynamic fuzzy system as different operating states in different time periods or moments, and each state in the pattern recognition method represents a dynamic fuzzy set of similar patterns, which together form finite regions in the feature space. We call this the class. Thus, the membership function can be used to identify a state or class of the new model. In this chapter, we will introduce a DFSS-MTL algorithm for dynamic fuzzy semi-supervised multi-task matching. The main purpose of this algorithm is to learn the membership functions from finite initial datasets and update the membership functions of the corresponding categories with the addition of a new model. This can effectively identify each model and then solve the corresponding dynamic fuzzy problem.

Fig. 5.5: A simple dynamic fuzzy semi-supervised multi-task learning model.

5.4.1Dynamic fuzzy random probability

In this algorithm, dynamic fuzzy stochastic probabilities will be used as the main way to calculate the membership degree of a certain model. Therefore, the relevant dynamic fuzzy random probability is briefly introduced [36, 37].

Definition 5.9 Dynamic fuzzy data: $(\overset{\leftarrow}{A}, \vec{A}) \in$ $(\overset{\leftarrow}{A}, \vec{A}) \in$ is DFD if

(1) $(\overset{\leftarrow}{A}, \vec{A})$ $(\overset{\leftarrow}{A}, \vec{A})$ is normal; that is, there exists some $(\overset{\leftarrow}{x}, \vec{x}) \in R$ $(\overset{\leftarrow}{x}, \vec{x}) \in R$ such that $(\overset{\leftarrow}{A}, \vec{A}) (\overset{\leftarrow}{x}, \vec{x}) =$ $(\overset{\leftarrow}{A}, \vec{A}) (\overset{\leftarrow}{x}, \vec{x}) =$ $(\overset{↽}{1}, \overset{⇁}{1}) .$ $(\overset{↽}{1}, \overset{⇁}{1}) .$

(2) $\forall (\overset{\leftarrow}{α}, \vec{α}) \in [(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})], {(\overset{\leftarrow}{A}, \vec{A})}_{(\overset{\leftarrow}{α}, \vec{α})}$ $\forall (\overset{\leftarrow}{α}, \vec{α}) \in [(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})], {(\overset{\leftarrow}{A}, \vec{A})}_{(\overset{\leftarrow}{α}, \vec{α})}$ is bounded by a closed interval.

Definition 5.10 Distribution number: The map $D F : (\overset{\leftarrow}{R}, \vec{R}) \to [(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})]$ $D F : (\overset{\leftarrow}{R}, \vec{R}) \to [(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})]$ is a distribution number if

(1)DF is monotone decreasing;

(2)DF is left continuous;

(3) $D F (- \infty) = (\overset{\leftarrow}{0}, \vec{0}), D F (+ \infty) = (\overset{\leftarrow}{1}, \vec{1}) .$ $D F (- \infty) = (\overset{\leftarrow}{0}, \vec{0}), D F (+ \infty) = (\overset{\leftarrow}{1}, \vec{1}) .$

If the distribution number $D F (\overset{\leftarrow}{0}, \vec{0}) = (\overset{\leftarrow}{0}, \vec{0}),$ $D F (\overset{\leftarrow}{0}, \vec{0}) = (\overset{\leftarrow}{0}, \vec{0}),$ it is called a positive distribution function.

If DF is a positive distribution number, when $(\overset{\leftarrow}{x}, \vec{x}) \leq (\overset{\leftarrow}{0}, \vec{0}), D F (\overset{\leftarrow}{x}, \vec{x}) = (\overset{\leftarrow}{0}, \vec{0}) .$ $(\overset{\leftarrow}{x}, \vec{x}) \leq (\overset{\leftarrow}{0}, \vec{0}), D F (\overset{\leftarrow}{x}, \vec{x}) = (\overset{\leftarrow}{0}, \vec{0}) .$

Definition 5.11 Dynamic fuzzy variable: For some domain $(\overset{\leftarrow}{U}, \vec{U}), if (\overset{\leftarrow}{x}, \vec{x})$ $(\overset{\leftarrow}{U}, \vec{U}), if (\overset{\leftarrow}{x}, \vec{x})$ is a dynamic fuzzy variable, then its value is in the domain, that is, $(\overset{\leftarrow}{x}, \vec{x}) =$ $(\overset{\leftarrow}{x}, \vec{x}) =$ $(\overset{\leftarrow}{u}, \vec{u}), (\overset{\leftarrow}{u}, \vec{u}) \in (\overset{\leftarrow}{U}, \vec{U}) .$ $(\overset{\leftarrow}{u}, \vec{u}), (\overset{\leftarrow}{u}, \vec{u}) \in (\overset{\leftarrow}{U}, \vec{U}) .$

As $(\overset{\leftarrow}{F}, \vec{F})$ $(\overset{\leftarrow}{F}, \vec{F})$ is a dynamic fuzzy set on $(\overset{\leftarrow}{U}, \vec{U}), (\overset{\leftarrow}{F} (\overset{\leftarrow}{u}), \vec{F} (\vec{u}))$ $(\overset{\leftarrow}{U}, \vec{U}), (\overset{\leftarrow}{F} (\overset{\leftarrow}{u}), \vec{F} (\vec{u}))$ is the membership degree of $(\overset{\leftarrow}{F}, \vec{F}) .$ $(\overset{\leftarrow}{F}, \vec{F}) .$

Definition 5.12 Dynamic fuzzy distribution: As $(\overset{\leftarrow}{x}, \vec{x})$ $(\overset{\leftarrow}{x}, \vec{x})$ is a dynamic fuzzy variable, $(\overset{\leftarrow}{F}, \vec{F})$ $(\overset{\leftarrow}{F}, \vec{F})$ is the dynamic fuzzy limit of $(\overset{\leftarrow}{x}, \vec{x}),$ $(\overset{\leftarrow}{x}, \vec{x}),$ that is, $D R (\overset{\leftarrow}{x}, \vec{x}) .$ $D R (\overset{\leftarrow}{x}, \vec{x}) .$ Then, the proposition that $(\overset{\leftarrow}{x}, \vec{x}) is (\overset{\leftarrow}{F}, \vec{F})$ $(\overset{\leftarrow}{x}, \vec{x}) is (\overset{\leftarrow}{F}, \vec{F})$ can be expressed as $D R (\overset{\leftarrow}{x}, \vec{x}) = (\overset{\leftarrow}{F}, \vec{F}) .$ $D R (\overset{\leftarrow}{x}, \vec{x}) = (\overset{\leftarrow}{F}, \vec{F}) .$ This is associated with a dynamic fuzzy distribution $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ that makes $D Π_{(\overset{\leftarrow}{x}, \vec{x})} = D R (\overset{\leftarrow}{x}, \vec{x}) .$ $D Π_{(\overset{\leftarrow}{x}, \vec{x})} = D R (\overset{\leftarrow}{x}, \vec{x}) .$

If $D Π_{(\overset{\leftarrow}{x}, \vec{x})}$ $D Π_{(\overset{\leftarrow}{x}, \vec{x})}$ is a set $(\overset{\leftarrow}{X}, \vec{X})$ $(\overset{\leftarrow}{X}, \vec{X})$ that is composed of a dynamic ambiguity variable $(\overset{\leftarrow}{x}, \vec{x})$ $(\overset{\leftarrow}{x}, \vec{x})$ and is mapped to an interval $[(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})],$ $[(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})],$ we set $D Π_{(\overset{\leftarrow}{X}, \vec{X})} : (\overset{\leftarrow}{X}, \vec{X}) \to$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})} : (\overset{\leftarrow}{X}, \vec{X}) \to$ $[(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})]$ $[(\overset{\leftarrow}{0}, \vec{0}), (\overset{\leftarrow}{1}, \vec{1})]$ so as to satisfy the following:

(1) $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ is monotone decreasing;

(2) $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ is left continuous; and

(3) $D Π_{(\overset{\leftarrow}{X}, \vec{X})} (- \infty) = (\overset{\leftarrow}{0}, \vec{0}), D Π_{(\overset{\leftarrow}{X}, \vec{X})} (+ \infty) = (\overset{\leftarrow}{1}, \vec{1}) .$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})} (- \infty) = (\overset{\leftarrow}{0}, \vec{0}), D Π_{(\overset{\leftarrow}{X}, \vec{X})} (+ \infty) = (\overset{\leftarrow}{1}, \vec{1}) .$

Then, it can be said that the dynamic fuzzy distribution is the distribution number defined above.

In fact, if the distribution function of the dynamic fuzzy distribution $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ $D Π_{(\overset{\leftarrow}{X}, \vec{X})}$ is denoted by $d π_{(\overset{\leftarrow}{X}, \vec{X})},$ $d π_{(\overset{\leftarrow}{X}, \vec{X})},$ then the dynamic fuzzy distribution function associated with $(\overset{\leftarrow}{X}, \vec{X})$ $(\overset{\leftarrow}{X}, \vec{X})$ is numerically equal to the membership degree of $(\overset{\leftarrow}{F}, \vec{F}) .$ $(\overset{\leftarrow}{F}, \vec{F}) .$ That is, the dynamic fuzzy probability of $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{u}, \vec{u})$ $(\overset{\leftarrow}{X}, \vec{X}) = (\overset{\leftarrow}{u}, \vec{u})$ can be approximated as $d π_{(\overset{\leftarrow}{X}, \vec{X})} ((\overset{\leftarrow}{u}, \vec{u})) \overset{Δ}{=}$ $d π_{(\overset{\leftarrow}{X}, \vec{X})} ((\overset{\leftarrow}{u}, \vec{u})) \overset{Δ}{=}$ $(\overset{\leftarrow}{F} (\overset{\leftarrow}{u}), \vec{F} (\vec{u})), where d π_{(\overset{\leftarrow}{X}, \vec{X})} ((\overset{\leftarrow}{u}, \vec{u}))$ $(\overset{\leftarrow}{F} (\overset{\leftarrow}{u}), \vec{F} (\vec{u})), where d π_{(\overset{\leftarrow}{X}, \vec{X})} ((\overset{\leftarrow}{u}, \vec{u}))$ is abbreviated as $d π ((\overset{\leftarrow}{u}, \vec{u})) .$ $d π ((\overset{\leftarrow}{u}, \vec{u})) .$

In [49], fuzzy sets form the basic theory of fuzzy stochastic probability, which solves fuzzy problems well. However, it cannot deal with dynamic fuzzy problems. Therefore, we can say that dynamic fuzzy variables correspond to classical random probability, fuzzy random probability, and dynamic fuzzy probability. In [50], the transition theory between classical stochastic probability and fuzzy stochastic probability is discussed. Similarly, to better deal with dynamic fuzzy problems, we transform the classical stochastic probability to dynamic fuzzy stochastic probability. If there are n dynamic fuzzy variables $(\overset{\leftarrow}{x}, \vec{x}), and p (({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})), p (({\overset{\leftarrow}{x}}_{j}, {\vec{x}}_{j}))$ $(\overset{\leftarrow}{x}, \vec{x}), and p (({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})), p (({\overset{\leftarrow}{x}}_{j}, {\vec{x}}_{j}))$ are the classical random probabilities of the dynamic fuzzy variables $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})$ $({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i})$ and $({\overset{\leftarrow}{x}}_{j}, {\vec{x}}_{j}),$ $({\overset{\leftarrow}{x}}_{j}, {\vec{x}}_{j}),$ then the dynamic fuzzy random probability $d π (({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}))$ $d π (({\overset{\leftarrow}{x}}_{i}, {\vec{x}}_{i}))$ can be expressed as

$d π (({\overset{\leftarrow}{x}}_{i}, {\overset{⇀}{x}}_{i})) = \sum_{j = 1, \dots, n} \min (p (({\overset{\leftarrow}{x}}_{i}, {\overset{⇀}{x}}_{i})), p (({\overset{\leftarrow}{x}}_{j}, {\overset{⇀}{x}}_{j}))) (5.1)$ $d π (({\overset{\leftarrow}{x}}_{i}, {\overset{⇀}{x}}_{i})) = \sum_{j = 1, \dots, n} \min (p (({\overset{\leftarrow}{x}}_{i}, {\overset{⇀}{x}}_{i})), p (({\overset{\leftarrow}{x}}_{j}, {\overset{⇀}{x}}_{j}))) (5.1)$

5.4.2Dynamic fuzzy semi-supervised multi-task matching algorithm

1. Dynamic fuzzy semi-supervised matching algorithm

Firstly, the dynamic fuzzy semi-supervised matching algorithm is described in the context of single tasks, which is an improvement on semi-supervised fuzzy pattern matching [51]. Without knowing the number of classes in advance, a probabilistic histogram is used to estimate the edge probability density of each attribute in the class. As patterns are added, the dynamic fuzzy probability (or membership) for each class is gradually formed.

${( {\overset{\leftarrow}{C}}_{1}, {\vec{C}}_{1}), ({\overset{\leftarrow}{C}}_{2}, {\vec{C}}_{2}), ..., ({\overset{\leftarrow}{C}}_{c}, {\vec{C}}_{c})}$ ${( {\overset{\leftarrow}{C}}_{1}, {\vec{C}}_{1}), ({\overset{\leftarrow}{C}}_{2}, {\vec{C}}_{2}), ..., ({\overset{\leftarrow}{C}}_{c}, {\vec{C}}_{c})}$ represents c categories, each of which can be regarded as a dynamic fuzzy set. Each model of each class consists of d attributes; $(\overset{\leftarrow}{x}, \vec{x}) = (({\overset{\leftarrow}{x}}^{1}, {\vec{x}}^{2}), ({\overset{\leftarrow}{x}}^{2}, {\vec{x}}^{2}), ..., ({\overset{\leftarrow}{x}}^{d}, {\vec{x}}^{d}))$ $(\overset{\leftarrow}{x}, \vec{x}) = (({\overset{\leftarrow}{x}}^{1}, {\vec{x}}^{2}), ({\overset{\leftarrow}{x}}^{2}, {\vec{x}}^{2}), ..., ({\overset{\leftarrow}{x}}^{d}, {\vec{x}}^{d}))$ denotes a dynamic fuzzy pattern consisting of d attributes in the feature space; $({\overset{\leftarrow}{X}}_{\max}^{j}, {\vec{X}}_{\max}^{j}) and ({\overset{\leftarrow}{X}}_{\max}^{j}, {\vec{X}}_{\max}^{j})$ $({\overset{\leftarrow}{X}}_{\max}^{j}, {\vec{X}}_{\max}^{j}) and ({\overset{\leftarrow}{X}}_{\max}^{j}, {\vec{X}}_{\max}^{j})$ represent the property j histogram of the maximum and minimum values, respectively, and determine the histogram’s size according to the actual situation; H represents the number of intervals in the histogram; and 1 denotes the k th interval of attribute j. When an attribute value of some new pattern 1 exceeds the maximum and minimum values of the attribute, the current histogram of the attribute needs to be updated. Let 1, 2, and h denote the maximum and minimum values of the updated attribute j and the number of intervals. These are updated by the following formula:

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5.3.3DFSSMTL model

5.4Dynamic fuzzy semi-supervised multi-task matching algorithm

5.4.1Dynamic fuzzy random probability

5.4.2Dynamic fuzzy semi-supervised multi-task matching algorithm

Table of Contents for
5.3.3 DFSSMTL model