Applied

Packet sets and inverse packet sets are two kinds of novel mathematical tools to analyze dynamic information systems. With advances in inverse packet sets, random inverse packet information is proposed by introducing random characteristics into inverse packet sets. Hence, random inverse packet information has dynamic and random characteristics, and is an extended form of inverse packet sets. Furthermore, random feature, dynamic feature, and identiﬁcation relation about the random inverse packet information are discussed. Finally, based on the above theory, an instance is used to illustrate the applications of intelligent acquisition of investment information


Introduction
With regard to finite common set theory with static feature, research on some dynamic systems often faced problems because change always exists. It became necessary to construct a new kind of set model with dynamic characteristics. Hence, Refs. [1][2][3][4] proposed two types of dynamic set models-packet sets and inverse packet sets (IPSs), by replacing "static" with "dynamic" to improve the finite common set. These dynamic set models provide a better theory foundation for dealing with dynamic applied systems. Later, the mathematical characteristics of the new sets such as quantitative characteristics, algebraic characteristics, geometrical characteristics, genetic characteristics, random characteristics, and theory applications are discussed by more and more scholars [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Especially, literatures [5][6][7][8][9][10][11][12][13][14][15][16][17] developed the latter model by taking information instead of sets to obtain the inverse packet information (IPI) model and provide some applications for information fusionseparation, hidden information discovery, intelligent data digging, and big decomposition-fusion acquisition. However current research on random inverse packet information (RIPI) is rare. Hence, we consider the probabilities of information element migration in IPI and present some concepts about the RIPI and their structures. Furthermore, the random feature, dynamic characteristics, and identification relations on RIPI are discussed and applied to intelligent acquisition-separation of investment information. Convention: (x) = {x 1 , x 2 , · · · , x s } ⊂ Uis a nonempty finite ordinary information and α ⊂ V is its nonempty attribute set; F, F are information transition function families in which f ∈ F, f ∈ F are transition functions, whose detailed characteristics and occurrence probabilities can be found in Hao et al. [23]. The occurrence probabilities of two events and pF (f ) in order.

RIPI and its construction
The theory model of IPSs [3,4] with the inner IPSX F and exterior IPSXF combined, has the following dynamic characteristics: given a finite common element set X = {x 1 , x 2 , ..., x r } with α = {α 1 , α 2 , ..., α r }. I. If some added attributes are transferred by f to α and to get α F such thatα ⊆ α F , then some extra elements are accordingly removed to X to generate a new element set called inner IPSX F , X ⊆X F . II. If some attributes are transferred by f from α to generate αF such that αF ⊆ α, then some elements in X are accordingly deleted to generate a new element set called exterior IPSXF ,XF ⊆ X. III. If it happens in the same time that some extra attributes are moved into α and some other attributes in α are migrated out, that is, α becomes α F , and meanwhile α does αF , αF ⊆ α ⊆ α F , then X becomes an IPS(X F ,XF ), which fulfillsXF ⊆ X ⊆X F and has dynamic characteristics. All of the IPSs generated by setXconstitute a set family called the IPS family {(XF i , X F j )|i ∈ I, j ∈ J} [3]. Especially, if the above process occurs continuously, X would dynamically generate a linked IPS ( .., s.Let us treat the setsXF , X,X F as information indicated orderly by (x) F , (x), and (x)F . Then we obtain IPI((x) F , (x)F ) with all the characteristics of IPS [18][19][20][21][22].
For inner IPI (x) F , the dynamic process is shown by adding information elements under the condition that some attributes are migrated into α, as For exterior IPI (x)F , the dynamic process is done by some elements in (x) migrated out under the condition that some attributes in α are removed out, as Obviously, all of the inner IPI and exterior IPI generated by (x) can, respectively, form an inner IPI family and an exterior IPI family expressed as {(x) F i |i ∈ I},{(x) F j | j ∈ J}. Certainly, all of the IPI generated by(x) can also form an IPI family as i is obtained in the case of the event occurrence probability equal to 1, namely, (x) F is obtained with the fact that {x i |u i∈ (x), f (u i ) = x i ∈ (x)}is bound to happen. As we know, it is stochastic thatw i / ∈ (x) is transferred in(x) by f . The same goes for exterior IPI and IPI [23]. Definition 1 is called random inner IPI generated by (x) depending on information element migration probability σ , generally written as random inner IPI, such that where σ ∈ (0, 1) and α F are also thought to be the attribute sets of (x) Fσ . Definition 2 (x)F σ is called the random exterior IPI obtained by (x) depending on information element migration probability σ , briefly written as random exterior IPI, such that where the nonempty attribute setαF of (x)F is also that of(x)F σ = / 0 and σ ∈ (0, 1). Definition 3 The information pair formed by the random inner IPI and the random exterior IPI generated by (x) is called a random IPI generated by (x) depending on information element migration probability σ , also called RIPI as where (α F , αF ) is also the attribute set of ((x) Fσ , (x)F σ ). Considering Definitions 1-3 and the above assumption, it is easily noted that Formulas (1) and (4) can, respectively, be represented with other forms as the following: Formulas (8) and (9) show RIPI to be as one information pair generated by not only the corresponding IPI, but also the ordinary information(x), as shown in Fig. 1. All of the RIPI generated by information(x)constitute an RIPI family as According to Definitions 1-3, Propositions 1-4 are simply derived as follows.
. Under certain conditions, RIPI could restore to homologous IPI, and to information (x). Theorem 1 (Relation theorem between RIPI and IPI) Assume where Formula (11) represents the relation of Fig. 1.

RIPI characteristics
Supplementing some additional attributes into α, some information elements would be migrated into information (x) depending on certain probability in succession and form a chain of random inner IPI showing the following dynamic process: Deleting some attributes out from α continuously, some information elements in (x) are migrated successively, depending on certain probability and form a chain of random exterior IPI showing the dynamic process as follows: If the above change processes take place at the same time, we obtain a chain of the RIPI implying the dynamic process as According to Formulas (12)- (14), we get the dynamic characteristics depending on the attribute sets indicated by Theorems 3-5. Theorem 3 (Depending attribute theorem of RIPI) Let (x) Fρ i , (x) Fρ j be the random inner IPI and α F i , α F j expressing their attribute sets in order. Then (x) Fσ i ⊆ (x) Fσ j iff α F i ⊆ α F j . Theorem 4 (Depending attribute theorem of RIPI) Let (x)F ρ i , (x)F ρ j be random exterior IPI and α F i , α F j expressing their attribute sets in order. According to the dynamic characteristics of RIPI, the measurement of dynamic change degree is proposed in Definitions 4-6. Definition 4 Let (x) Fσ be a random inner IPI derived by (x). Then call the real number γ(x) Fσ to be F−measure degree of (x) Fσ relative to(x)as where (x) = {x 1 , x 2 , ..x s },(x) Fσ = {x 1 , x 2 , ..x s , x s+1 , ..., x s+t }; the sequence of information value is expressed as .., x im ), i = 1, 2, ..., s + t, x ik ∈ [0, 1], and Definition 5 Let (x)F σ be a random exterior IPI derived by (x). Then call γ(x)F σF − measure degree of (x)F σ relative to(x)as where x (0) and x (0) are the same as Definition 4, and the sequence of information value is written as .., s, and

Definition 6
Let ((x) Fσ , (x)F σ ) be an RIPI generated by (x). Then call the real number pair composed by Formulas (15)and (16) to be (F,F)− measure degree of ((x) Fσ , (x)F σ ) relative to (x), and Because Formula (15) notes the change measurement between (x) F p and (x) caused by attribute supplementing set ∆α; the same goes for Formulas (16) and (17). Thus Propositions 5-7 can be obtained.
Accordingly, we have the following result. Table 1 The profit discrete distributions x (0) , x

Applications of RIPI model in intelligent acquisition-separation of investment information
For convenience, call x (0) the information value and x (Fσ ) the inner IPI value in Definition 4; call x (Fρ) the exterior IPI value based on which Definition 7 is given.
in which the attribute sets of (x)F i , (x)F i+1 are αF i , αF i+1 , respectively, and they satisfy αF i+1 = α F i − {α k |α k ∈ α,f (α k ) = δ k / ∈ α}. Assumption For simplicity, this section only proposes the applications of random inner IPI in intelligent separationacquisition of investment information. Suppose that W is a group company that produces petroleum and chemical products W = {W 1 ,W 2 ,W 3 ,W 4 ,W 5 }, where W i ∈ W , i = 1, 2, 3, 4, 5 are subsidiary corporations of W . α = {α 1 , α 2 , α 3 , α 4 , α 5 , α 6 } is the attribute set of W (product market characteristics of W ). Information form of W is (x) = {x 1 , x 2 , x 3 , x 4 , x 5 }. Due to trade secret, the group company and its subsidiary corporations and attributes (market characteristics) are expressed as W,W i , and α 1 , α 2 , α 3 , α 4 , α 5 , α 6 , respectively. x (0) , x (0) i are profit discrete value distributions of W,W i from January to June in 2019 as Values in x (0) , x  Table 1. By profit discrete distribution in Table 1, the profit information value of W is  A global disease COVID-19 broke out during preliminary stage in 2020 and caused a series of economic changes such as some manufacturing industry profits reduced in different probabilities. In contrast, products relating to protective apparatus, therapeutic apparatus, their appurtenance, and so on, have great market potential and earn better profit in big probabilities. This random dynamic change suited the RIPI model in this paper. For simplicity, this section only considers the latter.
The detailed profit discrete distribution of W F0.8 is shown in Table 2. By profit discrete distribution in Table 2, the profit information value of W F0.8 is Analysis on intelligent acquisition of RIPI(x) Fρ On the condition that α F is generated by supplementing attributes α 7 into α, one can obtain the random inner inverse packet information (x) F0.8 = {x 1 , x 2 , x 3 , ..., x 6 } based on information (x) = {x 1 , x 2 , x 3 , x 4 } through using Definition 1 and fulfill Formula (18).
x (F0.8) is intelligently separated out and acquired. If α 7 does not occur, x (F0.8) would never have been gained, or(x) Fσ would never have been known depending on the probability 0.8. The example simply tells us that the following: I. When α andα F satisfy α ⊆ α F , information (x) Fσ is intelligently discovered randomly out of information (x) by using the random inner inverse packet information generation model. While W 5 ,W 6 are found out of W due to (x) Fσ . II. When α 7 is thought to be a chance attribute and it invades the attribute se tα. Random inner IPI (x) Fσ is generated by (x) in Definition 1.
III. When the chance attributes α 7 invades α, the profit discrete distribution data x (0) of group company is turned to x (Fσ ) , which makes the profit of W increase. This conclusion has been proved in the financial statement published by W .
IV. Formula (23) means that W 5 ,W 6 will bring the extra profit 85.7% with a probability of 0.8 or greater.

Discussion
In Refs. [3,4], dynamic feature was brought into common set X and proposed the structure of IPS. Based on IPS, IPI, and its applications in resolving practical problems with dynamic characteristics and heredity are discussed in [8,13,17,20, and 21]. The randomness of element transfer is considered in this paper according to the dynamic characteristics of IPI [23]. By integrating the possibility knowledge into IPI, this paper proposes the concepts and structures of RIPI and their applications. RIPI theory enriches IPI and enlarges its application category. It also provides a new theory tool for studying the information system. T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k