## 1 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 fusion–separation, 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}*} ⊂

*U*is a nonempty finite ordinary information and

*α*⊂

*V*is its nonempty attribute set;

*F*,

*F̅*are information transition function families in which

*f*∈

*F*,

*f̅*∈

*F̅*are transition functions, whose detailed characteristics and occurrence probabilities can be found in Hao et al. [23]. The occurrence probabilities of two events

*p*(

_{F}*f*) and

*p*(

_{F̅}*f̅*) in order.

## 2 RIPI and its construction

The theory model of IPSs [3, 4] with the inner IPS *X̅ ^{F}* and exterior IPS

*X̅*combined, has the following dynamic characteristics: given a finite common element set

^{F̅}*X*= {

*x*

_{1},

*x*

_{2}, ...,

*x*} with

_{r}*α*= {

*α*

_{1},

*α*

_{2},...,

*α*}. I. If some added attributes are transferred by

_{r′}*f*to

*α*and to get

*α*such that

^{F}*α*⊆

*α*, then some extra elements are accordingly removed to

^{F}*X*to generate a new element set called inner IPS

*X̅*,

^{F}*X*⊆

*X̅*. II. If some attributes are transferred by

^{F}*f̅*from

*α*to generate

*α*such that

^{F̅}*α*⊆

^{F̅}*α*, then some elements in

*X*are accordingly deleted to generate a new element set called exterior IPS

*X̅*,

^{F̅}*X̅*⊆

^{F̅}*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*and meanwhile

^{F},*α*does

*α*,

^{F̅}*α*⊆

^{F̅}*α*⊆

*α*, then

^{F}*X*becomes an IPS(

*X̅*,

^{F}*X̅*), which fulfills

^{F̅}*X̅*⊆

^{F̅}*X*⊆

*X̅*and has dynamic characteristics. All of the IPSs generated by set

^{F}*X*constitute a set family called the IPS family

*X*would dynamically generate a linked IPS

*i*= 1, 2, ...,

*s*. Let us treat the sets

*X̅*,

^{F̅}, X*X̅*as information indicated orderly by (

^{F}*x̅*)

^{F},(

*x*), and (

*x̅*)

^{F̅}. Then we obtain IPI((

*x̅*)

^{F}, (

*x̅*)

^{F̅}) with all the characteristics of IPS [18,19,20,21,22].

For inner IPI (*x̅*)^{F}, the dynamic process is shown by adding information elements under the condition that some attributes are migrated into *α*, as ∃*w _{i}* ∉ (

*x*),

*f*(

*w*) =

_{i}*x*∈ (

_{i}*x*), where (

*x̅*)

^{F}indicates (

*x*) ∪ {

*x*|

_{i}*w*∉ (

_{i}*x*),

*f*(

*w*) =

_{i}*x*∈ (

_{i}*x*)}. For exterior IPI (

*x̅*)

^{F̅}, the dynamic process is done by some elements in (

*x*) migrated out under the condition that some attributes in

*α*are removed out, as ∃

*x*∈ (

_{i}*x*),

*f̅*(

*x*) =

_{i}*w*∉ (

_{i}*x*), where (

*x̅*)

^{F̅}indicates (

*x*) − {

*x*∈ (

_{i}*x*)|

*f̅*(

*x*) =

_{i}*w*∉ (

_{i}*x*)}. 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*) can also form an IPI family as

*x̅*)

^{F}is obtained with the fact that

*w*∉ (

_{i}*x*) is transferred in(

*x*) by

*f*. The same goes for exterior IPI and IPI [23].

*x*) depending on information element migration probability

*σ*, generally written as random inner IPI, such that

*α*fulfills

^{F}*σ*∈ (0, 1) and

*α*are also thought to be the attribute sets of (

^{F}*x̅*)

^{Fσ}.

*x̅*)

^{F̅σ}is called the random exterior IPI obtained by (

*x*) depending on information element migration probability

*σ*, briefly written as random exterior IPI, such that

*α*fulfills

^{F̅}*α*of (

^{F̅}*x̅*)

^{F̅}is also that of (

*x̅*)

^{F̅σ}≠ ∅ and

*σ*∈ (0, 1).

*x*) is called a random IPI generated by (

*x*) depending on information element migration probability

*σ*, also called RIPI as

*α*,

^{F}*α*) is also the attribute set of ((

^{F̅}*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ}).

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.

*x*) constitute an RIPI family as

Let *p _{F}*(

*f*) ≡ 0 for ∀

*f*∈

*F*, then (

*x̅*)

^{F}, (

*x̅*)

^{Fσ}are not indentified expressed as UNI((

*x̅*)

^{F}, (

*x̅*)

^{Fσ}).

*Let p _{F̅}*(

*f̅*) ≡ 0

*for*∀

*f̅*∈

*F̅*,

*then*UNI((

*x̅*)

^{F̅}, (

*x̅*)

^{F̅σ}).

*Let p _{F̅}*(

*f̅*) =

*p*(

_{F}*f*) ≡ 0

*for*∀

*f̅*∈

*F̅ and*∀

*f*∈

*F*,

*then*UNI(((

*x̅*)

^{F}, (

*x̅*)

^{F̅}), ((

*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ})).

*For* ∀*σ* ∈ (0, 1), *there is* UNI(Φ(*x*), *R*Φ(*x*)).

Propositions 1–4 state that RIPI ((*x̅*)^{Fσ}, (*x̅*)^{F̅σ}) is the extension of ((*x̅*)^{F}, (*x̅*)^{F̅}), and ((*x̅*)^{F}, (*x̅*)^{F̅}) is the particular case of ((*x̅*)^{Fσ}, (*x̅*)^{F̅σ}). Under certain conditions, RIPI could restore to homologous IPI, and to information (*x*).

*Assume*((

*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ}) ∈

*R*Φ(

*x*)

*and*((

*x̅*)

^{F}, (

*x̅*)

^{F̅}) ∈ Φ(

*x*),

*then for*∀

*σ*∈ (0, 1)

*we have*

*where Formula (11) represents the relation of Fig. 1*.

The assumption condition and Formulas (1) and (4) guarantee that (*x̅*)^{F} ⊆ (*x̅*)^{Fσ} and (*x̅*)^{F̅σ} ⊆ (*x̅*)^{F̅} are fulfilled. According to the definition of IPI derived by common information (*x*) in Refs [3, 4, and 18], we can obtain (*x̅*)^{F̅} ⊆ (*x̅*)^{F}. Hence, we get Formula (11) by set hereditary property.

(Generation theorem of RIPI) *Assume that* (*α ^{F}*,

*α*)

^{F̅}*is the attribute sets of RIPI*((

*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ}).

*There exists the nonempty pair*(Δ

*α*, ∇

*α*) ≠ ∅ such that

*α*− (

^{F}*α*∪ Δ

*α*) = ∅ and

*α*− (

^{F̅}*α*− ∇

*α*) = ∅,

*where*(Δ

*α*, ∇

*α*) ≠ ∅

*is equal to*Δ

*α*≠ ∅ and ∇

*α*≠ ∅.

Let ((*x̅*)^{Fσ}, (*x̅*)^{F̅σ}) be different from (*x*), namely, (*x̅*)^{Fσ} = (*x̅*)^{F̅σ} = (*x*) fails the Theorem 2 condition, then there exists one working between (*x̅*)^{Fρ} ≠ (*x*) and (*x̅*)^{F̅ρ} ≠ (*x*) at least. Let us assume that (*x̅*)^{Fσ} ≠ (*x*), the generating process of random inner IPI (*x̅*)^{Fρ} depending on its attribute set *α ^{F}* points out that

*α*meets

^{F}*α*⊂

*α*. Assuming Δ

^{F}*α*=

*α*−

^{F}*α*, we obtained Δ

*α*≠ ∅ and

*α*− (

^{F}*α*∪ Δ

*α*) = ∅ according to Definition 1. In the same way, it is proved that there exists ∇

*α*≠ ∅ such that

*α*− (

^{F̅}*α*− ∇

*α*) = ∅.

## 3 RIPI characteristics

*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:

*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:

According to Formulas (12)–(14), we get the dynamic characteristics depending on the attribute sets indicated by Theorems 3–5.

(Depending attribute theorem of RIPI) *Let**be the random inner IPI and**expressing their attribute sets in order. Then*

(Depending attribute theorem of RIPI) *Let**be random exterior IPI and**expressing their attribute sets in order. Then*

(Depending attribute theorem of RIPI) *Let**and**be their attribute sets, respectively. Then*

**Inference 1** Let card(*V − α*) = *t*. Then information (*x*) can generate *t*! dynamic chains of RIPI.

**Inference 2** Let card(*α*) = *m*. Then information (*x*) can generate *m*! dynamic chains of random exterior IPI.

**Inference 3** Let card(*α*) = *m* and card(*V* − *α*) = *t*. Then information (*x*) can generate *t*! × *m*! dynamic chains of RIPI.

According to the dynamic characteristics of RIPI, the measurement of dynamic change degree is proposed in Definitions 4–6.

*x̅*)

^{Fσ}be a random inner IPI derived by (

*x*). Then call the real number

*γ*(

*x̅*)

^{Fσ}to be

*F−*measure degree of (

*x̅*)

^{Fσ}relative to (

*x*) as

*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

*x̅*)

^{F̅σ}be a random exterior IPI derived by (

*x*). Then call

*γ*(

*x̅*)

^{F̅σ}

*F̅*− measure degree of (

*x̅*)

^{F̅σ}relative to (

*x*) as

*x*) = {

*x*

_{1},

*x*

_{2},..

*x*},(

_{s}*x*)

^{F̅σ}= {

*x*

_{1},

*x*

_{2},..

*x*}, 0 ≤

_{s}−_{p}*p*<

*s*,

*p*∈

*Z*

^{+};

*x*

^{(0)}and ‖

*x*

^{(0)}‖ are the same as Definition 4, and the sequence of information value is written as

*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ}) be an RIPI generated by (

*x*). Then call the real number pair composed by Formulas (15) and (16) to be (

*F*,

*F̅*)

*−*measure degree of ((

*x̅*)

^{Fσ}, (

*x̅*)

^{F̅σ}) relative to (

*x*), and

Because ((*x̅*)^{F},(*x̅*)^{F̅}) is a special case of ((*x̅*)^{Fσ}, (*x̅*)^{F̅σ}), (*γ*(*x̅*)^{F}, *γ*(*x̅*)^{F̅}) is chosen to express (*F*, *F̅*)*−* measure degree of ((*x̅*)^{F},(*x̅*)^{F̅}) when UNI((*x̅*)^{F}, (*x̅*)^{Fσ}), UNI((*x̅*)^{F̅}, (*x̅*)^{F̅σ}) in Definitions 4 and 5.

Formula (15) notes the change measurement between (*x̅*)^{Fp} and (*x*) caused by attribute supplementing set Δ*α*; the same goes for Formulas (16) and (17). Thus Propositions 5–7 can be obtained.

*Given F − measure degree γ*(*x̅*)^{Fσ}, *γ*(*x̅*)^{Fσ} ≠ 0 *iff* IDE((*x̅*)^{Fσ}, (*x*)) *or* IDE(*α ^{F},α*).

*Given F̅ − measure degree γ*(*x̅*)^{F̅σ}, *γ*(*x̅*)^{F̅σ} ≠ 0 *iff* IDE((*x̅*)^{F̅σ}, (*x*)) *or* IDE(*α ^{F̅}*,

*α*).

*Given*(*F*, *F̅*)*−measure degree* (*γ*(*x̅*)^{Fσ}, *γ*(*x̅*)^{F̅σ}), (*γ*(*x̅*)^{Fσ}, *γ*(*x̅*)^{F̅σ}) ≠ 0 *iff* IDE(((*x̅*)^{Fσ}, (*x̅*)^{F̅σ}),(*x*)) *or* IDE((*α ^{F}*,

*α*),

^{F̅}*α*).

*Let*(*x̅*)_{i}^{Fσ}, (*x̅*)_{j}^{Fσ}*be random inner IPI. Then*
(*x̅*)_{i}^{Fσ} ⊆ (*x̅*)_{j}^{Fσ}*iff γ _{i}*(

*x*)

^{Fσ}≤

*γ*(

_{j}*x*)

^{Fσ}.

*x̅*)

*= {*

_{j}^{Fσ}*x*

_{1},

*x*

_{2},

*x*

_{3},...,

*x*} and (

_{q}*x̅*)

*= {*

_{j}^{Fσ}*x*

_{1},

*x*

_{2},

*x*

_{3},...,

*x*,

_{q}*x*

_{q+1},...

*x*

_{q+t}}, (

*x̅*)

*⊆ (*

_{i}^{Fσ}*x̅*)

*implies that (*

_{j}^{Fσ}*x̅*)

*is a sub-information of (*

_{j}^{Fσ}*x̅*)

*, so*

_{i}^{Fσ}*Suppose that* (*x̅*)* _{i}^{F̅σ} and* (

*x̅*)

*(*

_{j}^{F̅σ}are random exterior IPI. Then*x̅*)

*≤ (*

_{i}^{F̅σ}*x̅*)

*(*

_{j}^{F̅σ}iff γ_{i}*x*)

*≤*

^{F̅σ}*γ*(

_{j}*x*)

^{F̅σ}.**Inference 4**Suppose ((

*x̅*)

*, (*

_{i}^{Fσ}*x̅*)

*), ((*

_{i}^{F̅σ}*x̅*)

*(*

_{j}^{Fσ},*x̅*)

*) are RIPI.*

_{j}^{F̅σ}*γ*(

_{i}*x̅*)

^{Fσ}≤

*γ*(

_{j}*x̅*)

^{Fσ}and

*γ*(

_{i}*x̅*)

^{F̅σ}≥

*γ*(

_{j}*x̅*)

^{F̅σ.}

## 4 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*^{(F̅ρ)} the exterior IPI value based on which Definition 7 is given.

*α*⇒

*α*,

^{F}*x*

^{(0)}⇒

*x*

^{(Fσ)}are equivalent to

*α*⊆

*α*,

^{F}*x*

^{(0)}⊆

*x*

^{(Fσ)}, respectively; call

*α*⇒

^{F̅}*α*and

*x*

^{(F̅σ)}⇒

*x*

^{(0)}are equivalent so that

*α*,

^{F̅}*x*

^{(F̅σ)}are subsets of

*α,x*

^{(0)}, respectively.

*If**then**in which the attribute sets of**are**respectively*, *and they satisfy**δ* ∉ *α*, *f* (*δ*) = *α*′ ∈ *α*}.

*If**then**in which the attribute sets of**are**respectively, and they satisfy*

*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*, and

_{i}*α*

_{1},

*α*

_{2},

*α*

_{3},

*α*

_{4},

*α*

_{5},

*α*

_{6}, respectively.

*x*

^{(0)},

*W,W*from January to June in 2019 as

_{i}*x*

^{(0)},

*x*

^{(0)},

*W,W*are listed in Table 1.

_{i}The profit discrete distributions *x*^{(0)},
*W,W _{i},i* = 1,2,3,4 from June to December in 2019

k | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

0.51 | 0.30 | 0.52 | 0.22 | 0.50 | 0.44 | |

0.21 | 0.28 | 0.45 | 0.36 | 0.66 | 0.53 | |

0.43 | 0.29 | 0.60 | 0.40 | 0.28 | 0.24 | |

0.19 | 0.35 | 0.48 | 0.66 | 0.67 | 0.26 |

*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.

*α*

_{7}= outbreakofCOVID

*−*19 and the attribute set of (

*x*) is

*α*= {

*α*

_{1},

*α*

_{2},

*α*

_{3},

*α*

_{4},

*α*

_{5},

*α*

_{6}}, then

*α*is derived through transferring

^{F}*α*

_{7}into

*α*as

*W*

_{6},

*W*

_{7},

*W*

_{8},

_{5},W

_{6},W

_{7},W

_{8}omit) would bring much profit differently with probabilities 1,0.8,0.75,0.67 after overall consideration. If probability

*ρ*= 0.8 is chosen, then

*W*

_{5},

*W*

_{6}turn into

*W*to form

*W*

^{F}^{0.8}whose information is (

*x*)

^{Fσ}= {

*x*

_{1},

*x*

_{2},

*x*

_{3},...,

*x*

_{6}},

The detailed profit discrete distribution of *W ^{F}*

^{0.8}is shown in Table 2.

The profit discrete distributions *x*^{(F0.8)},
*W*^{F0.8} and sub-company *W*_{i},*i* = 1,2,3,4,5,6 from January to June in 2020

k | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

0.51 | 0.30 | 0.52 | 0.22 | 0.50 | 0.44 | |

0.21 | 0.28 | 0.45 | 0.36 | 0.66 | 0.53 | |

0.43 | 0.28 | 0.60 | 0.40 | 0.28 | 0.24 | |

0.19 | 0.35 | 0.48 | 0.66 | 0.67 | 0.26 | |

0.61 | 0.60 | 0.72 | 0.69 | 0.65 | 0.70 | |

0.71 | 0.69 | 0.82 | 0.72 | 0.66 | 0.69 |

*W*

^{F}^{0.8}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:

- When
*α*and*α*satisfy^{F}*α*⊆*α*, information (^{F}*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}. - When
*α*_{7}is thought to be a chance attribute and it invades the attribute se t*α*. Random inner IPI (*x*)^{Fσ}is generated by (*x*) in Definition 1. - 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*. - Formula (23) means that
*W*_{5},*W*_{6}will bring the extra profit 85.7% with a probability of 0.8 or greater.

## 5 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.

The authors acknowledge the National Statistical Science Research Project (Grant No: 2018LY14).

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