6.3.4 Sample analysis

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Definition 6.10 Let $({\overset{\leftarrow}{r}}_{i j}^{t}, {\overset{\leftarrow}{r}}_{i j}^{t}) = (({\overset{\leftarrow}{l}}_{i j}^{t}, {\vec{l}}_{i j}^{t}), ({\overset{\leftarrow}{m}}_{i j}^{t}, {\vec{m}}_{i j}^{t}), ({\overset{\leftarrow}{n}}_{i j}^{t}, {\vec{n}}_{i j}^{t}))$ $({\overset{\leftarrow}{r}}_{i j}^{t}, {\overset{\leftarrow}{r}}_{i j}^{t}) = (({\overset{\leftarrow}{l}}_{i j}^{t}, {\vec{l}}_{i j}^{t}), ({\overset{\leftarrow}{m}}_{i j}^{t}, {\vec{m}}_{i j}^{t}), ({\overset{\leftarrow}{n}}_{i j}^{t}, {\vec{n}}_{i j}^{t}))$ be the DFD given by the t th expert, where i, j = 1, 2, . . ., nk, t = 1, 2, . . .N. Then, the relationship matrix of all the elements in the k th level with respect to the h th element in the (k − 1)th level is given by the following formula:

$M_{i j}^{k} = \frac{1}{N} \otimes (({\overset{\leftarrow}{r}}_{i j}^{1}, {\vec{r}}_{i j}^{1}) + ({\overset{\leftarrow}{r}}_{i j}^{2}, {\vec{r}}_{i j}^{2}) + \dots + ({\overset{\leftarrow}{r}}_{i j}^{N}, {\vec{r}}_{i j}^{N})) . (6.13)$ $M_{i j}^{k} = \frac{1}{N} \otimes (({\overset{\leftarrow}{r}}_{i j}^{1}, {\vec{r}}_{i j}^{1}) + ({\overset{\leftarrow}{r}}_{i j}^{2}, {\vec{r}}_{i j}^{2}) + \dots + ({\overset{\leftarrow}{r}}_{i j}^{N}, {\vec{r}}_{i j}^{N})) . (6.13)$

The corresponding dynamic fuzzy comprehensive rate is

$S_{i}^{k} = \sum_{j = 1}^{n_{k}} M_{i j}^{k} \otimes \frac{1}{\sum_{i = 1}^{N_{k}} \sum_{j = 1}^{n_{k}} M_{i j}^{k}} . (6.14)$ $S_{i}^{k} = \sum_{j = 1}^{n_{k}} M_{i j}^{k} \otimes \frac{1}{\sum_{i = 1}^{N_{k}} \sum_{j = 1}^{n_{k}} M_{i j}^{k}} . (6.14)$

Definition 6.11 Assume ${(\overset{\leftarrow}{d}', \vec{d}')}_{({\overset{\leftarrow}{R}}_{t}, {\vec{R}}_{t})} = \land V (S_{i} \geq S_{k}) .$ ${(\overset{\leftarrow}{d}', \vec{d}')}_{({\overset{\leftarrow}{R}}_{t}, {\vec{R}}_{t})} = \land V (S_{i} \geq S_{k}) .$ The weight vector is written as

$W' = {({(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{1}}, \vec{R_{1}})}, {(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{1}}, \vec{R_{2}})}, \dots, {(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{n}}, \vec{R_{n}})})}^{T}, (6.15)$ $W' = {({(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{1}}, \vec{R_{1}})}, {(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{1}}, \vec{R_{2}})}, \dots, {(\overset{\leftarrow}{d}' \vec{d}')}_{(\overset{\leftarrow}{R_{n}}, \vec{R_{n}})})}^{T}, (6.15)$

where k, i = 1, 2, . . ., n; k ̸ i.

After normalization, we get the normalized weight vector

$W = {({(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{1}, {\vec{R}}_{1})}, {(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{2}, {\vec{R}}_{2})}, \dots, {(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{n}, {\vec{R}}_{n})})}^{T} . (6.16)$ $W = {({(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{1}, {\vec{R}}_{1})}, {(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{2}, {\vec{R}}_{2})}, \dots, {(\overset{\leftarrow}{d}, \vec{d})}_{({\overset{\leftarrow}{R}}_{n}, {\vec{R}}_{n})})}^{T} . (6.16)$

6.3.4Sample analysis

Consider a customer who wishes to buy an automobile based mainly on the fuel consumption; that is, the learning goal is mpg. There are three types of automobile to choose from, C1, C2, C3, and four related evaluation attributes, c1, c2, c3, c4. We wish to find the attribute that is most closely related to mpg, that is, the attribute that most affects city cycle fuel consumption. Attribute c1 is the number of cylinders, c2 is weight, c3 is displacement, and c4 is horsepower.

First, experts process the data and compare their opinions to set up the dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$ as given in Tab. 6.2.

To improve reliability and stability, we calculate the average of Tab. 6.2 and gain Tab. 6.3. Using (6.10)–(6.14) in a MATLAB program, we calculate

$W' = ((\overset{\leftarrow}{0.32}, \vec{0.32}), (\overset{\leftarrow}{0.96}, \vec{0.96}), \overset{\leftarrow}{(0.06)}, \vec{0.06}), {(\overset{\leftarrow}{1}, \vec{1})}^{T} .$ $W' = ((\overset{\leftarrow}{0.32}, \vec{0.32}), (\overset{\leftarrow}{0.96}, \vec{0.96}), \overset{\leftarrow}{(0.06)}, \vec{0.06}), {(\overset{\leftarrow}{1}, \vec{1})}^{T} .$

After normalization, we obtain the weight vector for c1, c2, c3, c4:

$W = {((\overset{\leftarrow}{013}, \vec{0.13}), (\overset{\leftarrow}{0.41}, \vec{0.41}), (\overset{\leftarrow}{0.03}, \vec{0.03}), (\overset{\leftarrow}{0.43}, \vec{0.43}))}^{T} .$ $W = {((\overset{\leftarrow}{013}, \vec{0.13}), (\overset{\leftarrow}{0.41}, \vec{0.41}), (\overset{\leftarrow}{0.03}, \vec{0.03}), (\overset{\leftarrow}{0.43}, \vec{0.43}))}^{T} .$

Tab. 6.2: Dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$

Tab. 6.3: Average value of dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$

Tab. 6.4: C value of dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$

Tab. 6.5: Final weight of dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$

Thus, the influence rate on mpg, from high to low, is horsepower, weight, cylinders, and displacement. This result is the same as reported in [56], where SVM was used to extract features from the same database.

Next, in the same case, the results were calculated according to C for each attribute c in Tab. 6.4.

Finally, we added the weight of this level and the multiple upper level weights. The final results are presented in Tab. 6.5.

Above all, the customer should choose car C1.

6.4Dynamic fuzzy tree hierarchical relation learning

The tree structure is an important kind of nonlinear data format that is widely observed in the real world, such as in the genealogy of human society and various social organization structures. Intuitively, the tree defines the hierarchical structure through the branch relation. Dynamic fuzzy theory is used to construct a dynamic fuzzy tree representation method, which is more suitable for processing uncertain data in the subjective and objective world.

6.4.1Dynamic fuzzy tree

The following uses the DF formulation to explain the basic concept of dynamic fuzzy trees.

There is a complicated relationship between different things or knowledge in the objective world. The causal relation is the most common and simple form. Therefore, production is a representation that is suitable for relationships. From a semantic perspective, this represents the causal inference “If A, then B”.

1) DF production definition [48]

The general form of a DF production is

$P \leftarrow Q, C D F, I,$ $P \leftarrow Q, C D F, I,$

where the right-hand side represents a group of conditions or premises and the left-hand side represents various premises Q and the conclusion P of the conclusion action that can belong to DF. CDF is called the rule confidence, and $(\overset{\leftarrow}{0}, \vec{0}) \leq C D F \leq$ $(\overset{\leftarrow}{0}, \vec{0}) \leq C D F \leq$ $(\overset{\leftarrow}{1}, \vec{1}) .$ $(\overset{\leftarrow}{1}, \vec{1}) .$ The above rule means that if premise Q can be satisfied, then we can induce whether conclusion P is true or execute action P with some degree of truth, where the confidence of the rule is CDF.

In the representation of production, premise Q is a template r, and “satisfied” means the template can be matched in some cases. Theoretically, we can use the form of a string to represent this. For example, production in formal language such as α BB ← α 1, AB means that, if there exists a substring in a string that matches α AB with r, we can exchange the substring with α AB. Thus, it is a more general expressive module. As a special string structure, the logical formula can be viewed as a special case of a module.

For example, the premise of implication $P (\overset{\leftarrow}{x}, \vec{x}) \leftarrow Q (\overset{\leftarrow}{x}, \vec{x})$ $P (\overset{\leftarrow}{x}, \vec{x}) \leftarrow Q (\overset{\leftarrow}{x}, \vec{x})$ can be viewed as a string $Q (\overset{\leftarrow}{x}, \vec{x})$ $Q (\overset{\leftarrow}{x}, \vec{x})$ made up of $Q'' ('','' (\overset{\leftarrow}{x}, \vec{x})'','')'' .$ $Q'' ('','' (\overset{\leftarrow}{x}, \vec{x})'','')'' .$ Only if there exists a string that matches r can the premise of the rule be satisfied, so that $P (\overset{\leftarrow}{x}, \vec{x})$ $P (\overset{\leftarrow}{x}, \vec{x})$ can be induced. Thus, we determine whether to use the rule by checking if the premise can be satisfied instead of calculating the truth value of the premise predicate in the production rule. This is the essential difference between production and implication. It is certain that if the appointment can be matched, the predicate is true; otherwise, it is false. In this sense, production contains implication.

Naturally, the match proposed above should define the DF match in some sense instead of giving completely the same meaning. Similar to using the semantic distance to measure the distance between DF data objects, we design a method to measure the degree of piecewise matching between any templates. When the matching degree reaches a certain level, it is considered that the match is successful and we can use a production rule.

In addition, the DF string sequence can be used as premises or conditions to make the premise and condition of production dynamic and fuzzy:

$Q = a_{1} ({\overset{\leftarrow}{u}}_{1}, {\vec{u}}_{1}) a_{2} ({\overset{\leftarrow}{u}}_{2}, {\vec{u}}_{2}) \dots a_{n} ({\overset{\leftarrow}{u}}_{n}, {\vec{u}}_{n}),$ $Q = a_{1} ({\overset{\leftarrow}{u}}_{1}, {\vec{u}}_{1}) a_{2} ({\overset{\leftarrow}{u}}_{2}, {\vec{u}}_{2}) \dots a_{n} ({\overset{\leftarrow}{u}}_{n}, {\vec{u}}_{n}),$

where ai is a character, $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i})$ $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i})$ is the membership rate of ai to string Q, and $(\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq (\overset{\leftarrow}{1}, \vec{1}) . When ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) = (\overset{\leftarrow}{1}, \vec{1}),$ $(\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq (\overset{\leftarrow}{1}, \vec{1}) . When ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) = (\overset{\leftarrow}{1}, \vec{1}),$ the i th character of string Q is ai. When $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) = (\overset{\leftarrow}{0}, \vec{0}),$ $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) = (\overset{\leftarrow}{0}, \vec{0}),$ the i th character of string Q is any character but ai. When $(\overset{\leftarrow}{0}, \vec{0}) < ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) < (\overset{\leftarrow}{1}, \vec{1}),$ $(\overset{\leftarrow}{0}, \vec{0}) < ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) < (\overset{\leftarrow}{1}, \vec{1}),$ the i th character of string Q is ai with probability $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) .$ $({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) .$ We use * to represent the wildcard that can match any characters.

2) DF match

Assume the production

$P \leftarrow Q, C D F, I .$ $P \leftarrow Q, C D F, I .$

When $Q = a_{1} ({\overset{\leftarrow}{u}}_{1}, {\overset{\leftarrow}{u}}_{2}) a_{2} ({\overset{\leftarrow}{u}}_{2}, {\overset{\leftarrow}{u}}_{2}) ... a_{n} ({\overset{\leftarrow}{u}}_{n}, {\overset{\leftarrow}{u}}_{n}), if Q' = a_{1}^{'} ({\overset{\leftarrow}{v}}_{1}, {\vec{v}}_{1}) a_{2}^{'} ({\overset{\leftarrow}{v}}_{2}, {\vec{v}}_{2}) ... a_{n}^{'} ({\overset{\leftarrow}{v}}_{n}, {\vec{v}}_{n}),$ $Q = a_{1} ({\overset{\leftarrow}{u}}_{1}, {\overset{\leftarrow}{u}}_{2}) a_{2} ({\overset{\leftarrow}{u}}_{2}, {\overset{\leftarrow}{u}}_{2}) ... a_{n} ({\overset{\leftarrow}{u}}_{n}, {\overset{\leftarrow}{u}}_{n}), if Q' = a_{1}^{'} ({\overset{\leftarrow}{v}}_{1}, {\vec{v}}_{1}) a_{2}^{'} ({\overset{\leftarrow}{v}}_{2}, {\vec{v}}_{2}) ... a_{n}^{'} ({\overset{\leftarrow}{v}}_{n}, {\vec{v}}_{n}),$ $(\overset{\leftarrow}{t'}, \vec{t'}), (\overset{\leftarrow}{0}, \vec{0}) < (\overset{\leftarrow}{t'}, \vec{t'}) \leq (\overset{\leftarrow}{1}, \vec{1}), where (\overset{\leftarrow}{t'}, \vec{t'})$ $(\overset{\leftarrow}{t'}, \vec{t'}), (\overset{\leftarrow}{0}, \vec{0}) < (\overset{\leftarrow}{t'}, \vec{t'}) \leq (\overset{\leftarrow}{1}, \vec{1}), where (\overset{\leftarrow}{t'}, \vec{t'})$ is the true value of knowledge $Q^{'} a_{i}^{'} = a_{i} or ★, (\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq ({\overset{\leftarrow}{v}}_{i}, {\vec{v}}_{i}) \leq (\overset{\leftarrow}{1}, \vec{1}) (or (\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{v}}_{i}, {\vec{v}}_{i}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq$ $Q^{'} a_{i}^{'} = a_{i} or ★, (\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq ({\overset{\leftarrow}{v}}_{i}, {\vec{v}}_{i}) \leq (\overset{\leftarrow}{1}, \vec{1}) (or (\overset{\leftarrow}{0}, \vec{0}) \leq ({\overset{\leftarrow}{v}}_{i}, {\vec{v}}_{i}) \leq ({\overset{\leftarrow}{u}}_{i}, {\vec{u}}_{i}) \leq$ $(\overset{\leftarrow}{1}, \vec{1})), i = 1, 2, ..., n,$ $(\overset{\leftarrow}{1}, \vec{1})), i = 1, 2, ..., n,$ , n, then Q can be matched.

The distance between Q and Q’ is

$(\overset{\leftarrow}{d}, \vec{d}) = \max {| ({\overset{\leftarrow}{v}}_{1}, {\vec{v}}_{1}) - ({\overset{\leftarrow}{u}}_{1}, {\vec{u}}_{1}) |, | (\overset{\leftarrow}{v_{2}}, \vec{v_{2}}) - (\overset{\leftarrow}{u_{2}}, \vec{u_{2}}) |, \dots, | ({\overset{\leftarrow}{v}}_{n}, \vec{v_{n}}) - ({\overset{\leftarrow}{u}}_{n}, {\vec{u}}_{n}) |} .$ $(\overset{\leftarrow}{d}, \vec{d}) = \max {| ({\overset{\leftarrow}{v}}_{1}, {\vec{v}}_{1}) - ({\overset{\leftarrow}{u}}_{1}, {\vec{u}}_{1}) |, | (\overset{\leftarrow}{v_{2}}, \vec{v_{2}}) - (\overset{\leftarrow}{u_{2}}, \vec{u_{2}}) |, \dots, | ({\overset{\leftarrow}{v}}_{n}, \vec{v_{n}}) - ({\overset{\leftarrow}{u}}_{n}, {\vec{u}}_{n}) |} .$

The definition of the matching rate of premise Q is

$(\overset{\leftarrow}{m}, \vec{m}) = \min {1 - (\overset{\leftarrow}{d}, \vec{d}), (\overset{\leftarrow}{t'}, \vec{t'})} (o r (\overset{\leftarrow}{m}, \vec{m}) = (\overset{\leftarrow}{t'}, \vec{t'}) * (1 - (\overset{\leftarrow}{d}, \vec{d}))) .$ $(\overset{\leftarrow}{m}, \vec{m}) = \min {1 - (\overset{\leftarrow}{d}, \vec{d}), (\overset{\leftarrow}{t'}, \vec{t'})} (o r (\overset{\leftarrow}{m}, \vec{m}) = (\overset{\leftarrow}{t'}, \vec{t'}) * (1 - (\overset{\leftarrow}{d}, \vec{d}))) .$

3) Representation and/or tree for the DF production rule

A group of production can be represented by an and/or tree. For example, the following production group [48] is known:

$\begin{aligned} (\overset{\leftarrow}{A}, \vec{A}) \leftarrow (\overset{\leftarrow}{B_{1}}, \vec{B_{1}}), (\overset{\leftarrow}{B_{2}}, \vec{B_{2}}) (C D F_{1}, I_{1}) \\ (\overset{\leftarrow}{A}, \vec{A}) \leftarrow (\overset{\leftarrow}{B_{3}}, \vec{B_{3}}), (\overset{\leftarrow}{B_{4}}, \vec{B_{4}}), (\overset{\leftarrow}{B_{5}}, \vec{B_{5}}) (C D F_{2}, I_{2}) \\ (\overset{\leftarrow}{B_{1}}, \vec{B_{1}}) \leftarrow (\overset{\leftarrow}{C_{1}}, \vec{C_{1}}), (\overset{\leftarrow}{C_{2}}, \vec{C_{2}}) (C D F_{3}, I_{3}) \\ (\overset{\leftarrow}{B_{2}}, \vec{B_{2}}) \leftarrow (\overset{\leftarrow}{C_{3}}, \vec{C_{3}}) (C D F_{4}, I_{4}) \\ (\overset{\leftarrow}{B_{3}}, \vec{B_{3}}) \leftarrow (\overset{\leftarrow}{C_{4}}, \vec{C_{4}}), (\overset{\leftarrow}{C_{5}}, \vec{C_{5}}), (\overset{\leftarrow}{C_{6}}, \vec{C_{6}}) (C D F_{5}, I_{5}) \\ (\overset{\leftarrow}{B_{4}}, \vec{B_{4}}) \leftarrow (\overset{\leftarrow}{C_{7}}, \vec{C_{7}}) (C D F_{6}, I_{6}) \\ (\overset{\leftarrow}{B_{5}}, \vec{B_{5}}) \leftarrow (\overset{\leftarrow}{C_{9}}, \vec{C_{9}}), (\overset{\leftarrow}{C_{10}}, \vec{C_{10}}) (C D F_{9}, I_{9}) \\ (\overset{\leftarrow}{C_{5}}, \vec{C_{5}}) \leftarrow (\overset{\leftarrow}{D_{1}}, \vec{D_{1}}), (\overset{\leftarrow}{D_{2}}, \vec{D_{2}}), (\overset{\leftarrow}{D_{3}}, \vec{D_{3}}) (C D F_{10}, I_{10}) \\ (\overset{\leftarrow}{C_{8}}, \vec{C_{8}}) \leftarrow (\overset{\leftarrow}{D_{4}}, \vec{D_{4}}), (\overset{\leftarrow}{D_{5}}, \vec{D_{5}}), (\overset{\leftarrow}{D_{6}}, \vec{D_{6}}) (C D F_{11}, I_{11}) \end{aligned}$ $\begin{aligned} (\overset{\leftarrow}{A}, \vec{A}) \leftarrow (\overset{\leftarrow}{B_{1}}, \vec{B_{1}}), (\overset{\leftarrow}{B_{2}}, \vec{B_{2}}) (C D F_{1}, I_{1}) \\ (\overset{\leftarrow}{A}, \vec{A}) \leftarrow (\overset{\leftarrow}{B_{3}}, \vec{B_{3}}), (\overset{\leftarrow}{B_{4}}, \vec{B_{4}}), (\overset{\leftarrow}{B_{5}}, \vec{B_{5}}) (C D F_{2}, I_{2}) \\ (\overset{\leftarrow}{B_{1}}, \vec{B_{1}}) \leftarrow (\overset{\leftarrow}{C_{1}}, \vec{C_{1}}), (\overset{\leftarrow}{C_{2}}, \vec{C_{2}}) (C D F_{3}, I_{3}) \\ (\overset{\leftarrow}{B_{2}}, \vec{B_{2}}) \leftarrow (\overset{\leftarrow}{C_{3}}, \vec{C_{3}}) (C D F_{4}, I_{4}) \\ (\overset{\leftarrow}{B_{3}}, \vec{B_{3}}) \leftarrow (\overset{\leftarrow}{C_{4}}, \vec{C_{4}}), (\overset{\leftarrow}{C_{5}}, \vec{C_{5}}), (\overset{\leftarrow}{C_{6}}, \vec{C_{6}}) (C D F_{5}, I_{5}) \\ (\overset{\leftarrow}{B_{4}}, \vec{B_{4}}) \leftarrow (\overset{\leftarrow}{C_{7}}, \vec{C_{7}}) (C D F_{6}, I_{6}) \\ (\overset{\leftarrow}{B_{5}}, \vec{B_{5}}) \leftarrow (\overset{\leftarrow}{C_{9}}, \vec{C_{9}}), (\overset{\leftarrow}{C_{10}}, \vec{C_{10}}) (C D F_{9}, I_{9}) \\ (\overset{\leftarrow}{C_{5}}, \vec{C_{5}}) \leftarrow (\overset{\leftarrow}{D_{1}}, \vec{D_{1}}), (\overset{\leftarrow}{D_{2}}, \vec{D_{2}}), (\overset{\leftarrow}{D_{3}}, \vec{D_{3}}) (C D F_{10}, I_{10}) \\ (\overset{\leftarrow}{C_{8}}, \vec{C_{8}}) \leftarrow (\overset{\leftarrow}{D_{4}}, \vec{D_{4}}), (\overset{\leftarrow}{D_{5}}, \vec{D_{5}}), (\overset{\leftarrow}{D_{6}}, \vec{D_{6}}) (C D F_{11}, I_{11}) \end{aligned}$

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6.3.4Sample analysis

6.4Dynamic fuzzy tree hierarchical relation learning

6.4.1Dynamic fuzzy tree

Table of Contents for
6.3.4 Sample analysis