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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under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model