P-sets (P stand for Packet) and P-matrix are novel and effective mathematical tools for studying dynamic information systems. In this paper, the concept of P-information mining is given by using the dynamic characteristics of P-sets and P-matrix structure. In addition, the reasoning theorem of P-matrix and the reasoning structure are given. Moreover, the information intelligent mining method under the condition of P-matrix reasoning is obtained. As the application, intelligent recognition of information image was shown.

#### Keywords

- P-sets
- P-matrix reasoning
- reasoning theorem
- information intelligent mining

#### MSC 2010

- F18996

Information (data) mining and applications are the important branch of theoretical and application research in information (data) system and information (data) engineering. Information (data) mining is a dynamic process. For a given source information (_{1},_{2} ⋯ ,_{q}_{i}

Deleting some redundant information elements _{i}^{F̅} = {_{1},_{2} ⋯ ,_{p}}, ^{F̅} is mined-discovered in (

Supplementing some information elements _{j}^{F} = {_{1},_{2} ⋯ ,_{r}^{F} is mined, i.e. discovered outside (

Deleting some redundant information elements _{i}_{j}^{F̅}, and (^{F}, (^{F̅} ⊆ (^{F}; ((^{F̅},(^{F}) is mined, i.e. discovered inside-outside (

I-III are the three different concepts of information (data) mining. A number of authors are studying the dynamic characteristics of I–III to find a mathematical method. In Ref. [1], the concept of P-sets with dynamic characteristics is proposed by improving the finite common element set (Conter set). In Refs. [2,3,4,5,6,7,8,9,10], the application of P-sets in dynamic information system is shown. In Ref. [11], P-matrix (P-augmented matrix) is obtained by using the structure and dynamic characteristics of P-sets. P-matrix reasoning (p-augmented matrix reasoning) is a new reasoning structure with dynamic characteristics generated by P-sets. In this paper, the basic theory and application of information (data) intelligence mining are discussed using the structure and dynamic characteristics of P-sets and P-matrix reasoning.

Conter set

Supplementing some attributes in ^{F}^{F}^{F̅}, ^{F̅} ⊆

Deleting some attributes in ^{F̅}, ^{F̅} ⊆ ^{F}^{F}

Supplementing some attributes in ^{F}^{F̅}, ^{F̅} ⊆ ^{F}^{F̅}, and ^{F}^{F̅} ⊆ ^{F}^{F̅},^{F}^{F̅},^{F}

The dynamic characteristics (1–3) of P-sets are the same as those of the information mining concepts I–III. If ^{F̅},^{F},^{F̅},^{F}^{F̅}, outer P-information (^{F}, P-information ((^{F̅},(^{F}); or (^{F̅} = ^{F̅}, (^{F} = ^{F}^{F̅},(^{F}) = (^{F̅},^{F}

This paper gives the structures and characteristics of P-sets and P-matrix (P-augmented matrix), gives three new concepts of information mining, proposes P-matrix reasoning (P-augmented matrix reasoning) and reasoning theorems, presents the information (data) intelligent mining method under the condition of P-matrix reasoning and gives the simple application of information (data) intelligent recursive mining. All the results given in this paper are new.

In Ref. [1], P-sets and its structure are presented as follows: Conter set _{1},_{2},⋯ ,_{q}_{1},_{2},⋯ ,_{k}^{F̅} is referred to as internal packet set generated by ^{F̅} is internal packet set for short,
^{−} is referred to as ^{F}^{F̅} meets

In Eq. (3), _{i}_{i}

In Eq. (1), ^{F̅} ≠ ∅, ^{F̅} = {_{1},_{2},⋯ ,_{p}^{+}.

Conter set _{1},_{2},⋯ ,_{q}_{1},_{2},⋯ ,_{k}^{F}^{F}^{+} is referred to as ^{F̅} of ^{F}

In Eq. (6), _{i}_{i}^{F̅} ≠ ∅;

In Eq. (4), ^{F}_{1},_{2},...,_{r}^{+}.

The set pair which is composed of internal packet set ^{F̅} and outer packet set ^{F}

From Eq. (3),

From Eqs (1)–(7) and (12), we get the following proposition.

^{F̅}, ^{F}

_{1}, _{2},⋯ , _{n}_{1}, _{2},⋯ , _{n}} are the family of element (attribute) transfer. The characteristic of _{i}_{i}_{i}_{i}_{i}_{i}′_{i}_{i}_{i} ∈ α, _{i} into

The dynamic characteristic of Eq. (1) are the same as the dynamic characteristics of down-counter

The dynamic characteristic of Eq. (4) are the same as the dynamic characteristics of accumulator _{1}^{F} = _{1}^{+}; let _{1}^{F} = _{2}^{F} = _{1}^{F} ∪ _{2}^{+} = (_{1}^{+}) ∪ _{2}^{+},⋯, and so on.

Take an example: _{1},_{2},_{3},_{4},_{5}} is a finite commodity element set of five apples, and _{1},_{2},_{3}} is the attribute set confined in _{1} denotes red colour, _{2} denotes sweet taste, _{3} denotes produced in Henan province of China. Obviously, ∀_{i}_{i}_{1},_{2} and _{3}; the attribute _{i}_{i}_{4} denotes weight is 150 g, supplementing attribute _{4} in ^{F} = _{4}} = {_{1},_{2},_{3}, _{4}}, and ^{F̅} = {_{1},_{2},_{5}}. Obviously, ∀_{i}^{F̅}, _{i}_{1},_{2},_{3} and _{4}. The attribute _{i}_{i}_{3} in ^{F̅} = _{3}} = {_{1},_{2}}, and ^{F}_{1},_{2},_{3},_{4},_{5},_{6},_{7}}. Obviously, ∀_{i}^{F}, i_{i}_{1},_{2}. The attribute _{i}_{i}

In Ref. [11], the structure and generation of p-augmented matrix are proposed as follows: Conter set _{1},_{2},⋯ ,_{q}_{i}_{1,i},_{2,i},⋯ ,_{m,i}; _{i}_{1,i},_{2,i},⋯ ,_{m,i})^{T} is a vector generated by element values _{1,i},_{2,i},⋯ ,_{m,i} of _{i}

Internal P-sets ^{F̅} = {_{1},_{2},⋯ ,_{p}_{i}^{F̅} has element values _{1,i},_{2,i},⋯ ,_{m,i}; _{i}_{1,i},_{2,i},⋯, _{m,i}^{T} is a vector generated of _{1,i},_{2,i},⋯ ,_{m,i}, ^{F̅}, and is called internal P-augmented matrix of

Outer P-sets ^{F}_{1},_{2},⋯ ,_{r}_{i}^{F}_{1,i},_{2,i},⋯ ,_{m,i}; _{i}_{1,i},_{2,i},⋯, _{m,i}^{T} is a vector generated of _{1,i},_{2,i},⋯ ,_{m,i}, ^{F}

Here: in Eqs (15)–(17), ^{+}.

The matrix pair composed of ^{F̅} and ^{F}^{F̅},^{F}

It should be noted that: In Ref. [11], by using the structure and dynamic characteristic of P-sets, they improved the definition of augmented matrix in classical mathematics, proposed internal P-augmented matrix ^{F̅}, outer P-augmented matrix ^{F}^{F̅},^{F}^{F}^{F̅} and P-augmented matrix (^{F̅},^{F}^{F̅},^{F}^{F̅},^{F}

The internal P-augmented matrix ^{F̅}, outer P-augmented matrix ^{F}^{F̅},^{F}^{F̅}, outer P-matrix ^{F}^{F̅},^{F}^{F̅},^{F}^{F̅},^{F}^{F̅},(^{F} and ((^{F̅},(^{F}), or (^{F̅} = ^{F̅},(^{F} = ^{F}^{F̅},(^{F}) = (^{F̅},^{F}^{F̅},(^{F} and ((^{F̅},(^{F}) are called information, internal P-information, outer P-information and P-information; ∀_{i}_{i}^{F̅}, or ∀_{i}^{F}) is called information element, its numerical value _{i}_{i}

If internal P-matrix

If outer P-matrix

If P-matrix

By Eqs (20)–(22), we get:

By Eqs (15) and (16) in Section 3, we get:

Here:

Since the proof of Theorem 2 is similar to the proof of Theorem 1, the proof is omitted. By Theorems 1 and 2, we get Theorem 3.

Using the concepts in Sections 2 and 3 and results in Section 4, Section 5 is given.

Assumption: For simple and without losing generality, in this section only the application of internal P-information intelligent recursive mining under the condition of internal P-matrix reasoning is given. The application of outer P-information intelligent recursive mining under the condition of outer P-matrix reasoning and the application of P-information intelligent recursive mining under the condition of P-matrix reasoning are omitted.

Source of information (

Here: column _{i}_{1,j},_{2,j},_{3,j},_{4,j},_{5,j})^{T} composed of the information values _{1,j},_{2,j},_{3,j},_{4,j},_{5,j} of information element _{j}

Under the condition of internal P-matrix reasoning:

By using internal P-matrix reasoning family Eq. (23),

Special note: Examples in this section are taken from the computer information image intelligent recognition system. Matrix

The cross and penetration of P-sets and P-matrix reasoning is a new research field of information intelligent dynamic mining. In this paper, the basic theory and application of information (data) intelligent mining are given by using the intersection of P-sets and P-matrix reasoning. This paper discusses information intelligent recursive mining and mining methods, and gives its application in information image intelligent mining and recognition. Several new concepts are discussed, and a new reasoning method (intelligent method) is proposed. Ref. [12] gives the dual form of inverse P-sets of P-sets, and Ref. [13] presented inverse P-matrix, which is the dual form of P-matrix. In Refs. [14,15,16,17,18], inverse P-sets and inverse P-matrix have been applied in the dynamic discovery of unknown information and the intelligent fusion and separation of information laws. P-sets, inverse P-sets, P-matrix and inverse P-matrix provide new mathematical theories and methods for information intelligent mining, and become a new tool to study dynamic information system.

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