6.3.2Sample analysis

1. Sample description

Consider a database that describes a town of 20 houses. First, there are some dynamic fuzzy relational predicates, such as $\left(close\left(x,y\right),\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\right)$ to describe that house x is close to house y with membership rate $\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)$ (0 represents strangers, 1 represents friends). Dynamic fuzzy predicates, some of which are similar to propositions such as $price\left(x,\overleftarrow{expensive},\overrightarrow{expensive}\right)\text{or}size\left(x,\left(\overleftarrow{small},\overrightarrow{small}\right)\right),$ can be used to describe the houses.

Thus, in this case, we can attain good confidence values for each type of dynamic fuzzy rule. For instance, consider the flexible rule

$close\left(x,y\right),price\left(y,\left(\overleftarrow{expensive},\overrightarrow{expensive}\right)\right)\to price\left(x,\left(\overleftarrow{expensive},\overrightarrow{expensive}\right)\right).$

This has a confidence value of 0.81, so it is reasonable to predict that if a house is close to the price of an expensive house, the price of this house is also expensive (because expensive houses are usually located in the same area). A typical gradual rule is

$size\left(x,\left(\overleftarrow{large},\overrightarrow{large}\right)\right)\to price\left(x,\overleftarrow{expensive},\overrightarrow{expensive}\right)).$

That is, the bigger the house, the more expensive it is, which explains the fact that price increases with size. The confidence of this rule is 0.80. A good example of certainty is

$close\left(x,y\right)\to know\left(x,y\right).$

If this is type-one certainty rule, the confidence level is 0.95. If it is a type-two certainty rule, the confidence level is 0.88. This rule means “the closer the two houses, the more likely it is that the owners know each other”. Obviously, the owners of houses close to each other have a greater probability of knowing one another. As the distance between the houses increases, the possibility of the owners knowing each other will drop. At the end of the study, we can observe that all of these rules are obviously affected by exceptions, and regardless of what they concern, they cannot be obtained by the classical ILP machine.

2. Results analysis and comparison

The data used in this chapter are taken from the auto-mpg database in the UCI repository (http://www.ics.uci.edu/~mlearn/MLRepository.html). The database consists of a car and a concept to learn. The concept to learn is the fuel consumption per mile of city driving. There are 398 car samples in the database. They are described by nine attributes, five of which are continuous, including the concept of miles per gallon (mpg). We conducted an experiment using three multi-valued discrete attributes and five continuous attributes to predict the future mpg properties, that is, the city cycle fuel consumption/mile. The database cannot be represented by propositional logic, but it is sufficient to illustrate the interesting areas of the method. First, the database can be “discretized” into dynamic fuzzy sets. Moreover, we can construct three distinct discrete correspondences:

1.A clear part of the property domain is most closely connected to the dynamic fuzzy part;

2.The support of the dynamic fuzzy sets; and

3.The core of the dynamic fuzzy sets.

Second, we learn the city cycle fuel consumption category for all types of dynamic fuzzy rules, according to clear and dynamic fuzzy sets. Table 6.1 presents the results of predicting the mpg using different rules.

The following describes an example for each type of rule combined into an algorithm. (For the clear part of the domain of the property that corresponds to the nearest dynamic fuzzy part, the classical rule is obtained by discretization.) According to different rules, the algorithm automatically generates the following example:

Classic rule: cylinders(A, 8) → mpg(A, low).

The classic rule predicted the attribute value of mpg to be lower than the low field when the automobile cylinder number is less than eight. The data are coincident with the known data in Fig. 6.1.

Tab. 6.1: Prediction result of different rules.

Flexible rule:

$displacement\left(A,low\right),weight\left(A,medium\right)\to mpg\left(A,medium\right).$

The rule explains that we can induce that mpg is in the low field when the number of cylinders is six, the weight is high, origin = 1, automobile acceleration is low, and horsepower is low.

Gradual rule: weight(A, high) → mpg(A, low).

The rule explains that we can induce that attribute mpg is in the low field when weight is in the high field.

Type one certainty rule: cylinders(A, 6), weight(A, high) → mpg(A, low).

The rule explains that we can induce that mpg is in the low field when the number of cylinders of automobile A is six and its weight is in the high field.

Type two certainty rule:

$\begin{array}{l}cylinders\left(A,6\right),weight\left(A,high\right),orign\left(A,1\right),acceleration\left(A,low\right),\\ horespower\left(A,low\right)\to mpg\left(A,low\right).\end{array}$

The rule explains that we can induce that mpg is in the low field when the number of cylinders of automobile is six, its weight is in the high field, origin = 1, automobile acceleration is low, and horsepower is low.

As expected, the coverage rate of the classical rules is between that of the classical rules with a dynamic fuzzy set kernel and the classical rules supported by dynamic fuzzy sets. If this is due to the dynamic fuzzy rules kernel, samples on the clear category boundaries will not be processed. In contrast, with the support of dynamic fuzzy sets, it is clear that samples on the class boundary will belong to two categories. Relative to the coverage, because the dynamic fuzzy rules are more difficult to find than the classical rules, the kernel of the dynamic fuzzy rules is smaller. This result is to be expected because the estimated value of each predicate must be considered, and the dynamic fuzzy rules will be subject to more conditions. In fact, the confidence level of the classical rule does not depend on the distance between the data on the discrete set boundary. For example, consider the classic rule F with good confidence, as well as its dynamic fuzzy flexible copy F′. Relative to the dynamic fuzzy sets, if many examples of F belong to the boundary line, the confidence of F′ will be lower than the confidence of F. On the contrary, if many counterexamples of F lie in the border region, the confidence of F′ will be higher than that of F. Thus, compared to the small change in the set boundary, the confidence of the dynamic fuzzy flexible copy is a good indicator of the robustness of the classical rule.

Type-one certainty rules concern the degree of membership of the rule condition. Gradual and certainty rules show how the conditions and conclusions evolve together. The meanings of these rules are far different from those of the classical rules and need not have a clear copy or approximation. Dealing with certainty in dynamic fuzzy rules, it tends to be the non-dynamic fuzzy predicates that favour the conditional part. This is because the non-dynamic fuzzy predicate allows more freedom about the conformity of the sample to be covered. Note that there are some rules that can be described by propositional logic, but the examples are automatically generated by the algorithm. As some of the rules have shown, this algorithm can mix dynamic fuzzy predicates and non-dynamic fuzzy predicates.

6.3.3Dynamic fuzzy matrix HRL algorithm

1. Dynamic fuzzy matrix

Definition 6.5 If $\left(\overleftarrow{R},\overrightarrow{R}\right)\in D{F}_L\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),$ we say that $\left(\overleftarrow{R},\overrightarrow{R}\right)$ is the type L relationship from $\left(\overleftarrow{X},\overrightarrow{X}\right)\text{to}\left(\overleftarrow{Y},\overrightarrow{Y}\right).$ If

$L=\left[\left(\overleftarrow{0},\overrightarrow{0}\right),\left(\overleftarrow{1},\overrightarrow{1}\right)\right],\left(\overleftarrow{R},\overrightarrow{R}\right)\in D{F}_r\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),\left(\overleftarrow{R},\overrightarrow{R}\right)$ is the DF relationship from $\left(\overleftarrow{X},\overrightarrow{X}\right)\text{to}\left(\overleftarrow{Y},\overrightarrow{Y}\right).$

Definition 6.6 If $\left({\overleftarrow{R}}_1,{\overrightarrow{R}}_1\right),\left({\overleftarrow{R}}_2,{\overrightarrow{R}}_2\right)\in D{F}_L\left(\left(\overleftarrow{X},\overrightarrow{X}\right)\times \left(\overleftarrow{Y},\overrightarrow{Y}\right)\right),$ write

$\begin{array}{rl}& (1)({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)\subset <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)\iff <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))\le <!--\; =\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y})),\\ & \text{where(}(\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y})\mathrm{ϵ<!--\; ?\; -->}((\stackrel{\leftarrow <!--\; ?\; -->}{X},\stackrel{\to <!--\; ?\; -->}{X})\times (\stackrel{\leftarrow <!--\; ?\; -->}{Y},\stackrel{\to <!--\; ?\; -->}{Y})))\\ & (2)({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)\bigcup <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))=({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))\\ & \vee <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))\\ & ({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)\bigcap <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))=({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_1,{\stackrel{\to <!--\; ?\; -->}{R}}_1)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))\\ & \wedge <!--\; ?\; -->({\stackrel{\leftarrow <!--\; ?\; -->}{R}}_2,{\stackrel{\to <!--\; ?\; -->}{R}}_2)((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))\\ & (3){(\stackrel{\leftarrow <!--\; ?\; -->}{R},\stackrel{\to <!--\; ?\; -->}{R})}^{-<!--\; -\; -->1}((\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}),(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}))=(\stackrel{\leftarrow <!--\; ?\; -->}{R},\stackrel{\to <!--\; ?\; -->}{R})(\stackrel{\leftarrow <!--\; ?\; -->}{y},\stackrel{\to <!--\; ?\; -->}{y}),(\stackrel{\leftarrow <!--\; ?\; -->}{x},\stackrel{\to <!--\; ?\; -->}{x}))\end{array}$