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Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
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2 times per year
Languages
English
access type Open Access

Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 18 Jan 2022
Accepted: 27 Mar 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

We use the Legendre wavelet method to study nonlinear fractional differential equations. Based on the in-depth study of the characteristics of various fractional-order dynamic system models, this paper designs a system for solving fractional-order differential equations, and we apply them to the anomaly analysis of big computer data. This method can improve the efficiency of big data classification. The results of computer numerical simulation show that the designed algorithm for solving fractional differential equations has high accuracy. At the same time, the algorithm can avoid misclassification and omission in big data analysis.

Keywords

MSC 2010

Introduction

At present, the database has penetrated the data processing of various industries in society. And the development of data technology and the growth of data volume have brought modern people into the era of information and data explosion. The main research direction of modern related researchers is how to realize the effective processing of a large amount of information and data to find the knowledge contained in it [1]. Only based on database query and retrieval technology has been unable to meet people's needs for data information processing. Data mining technology automatically and intelligently transforms anonymous data and confidential information in the data into useful knowledge and technology. This technology helps people extract the knowledge that people are interested in from the database and then analyze the data. This can make full use of the value of large amounts of data. In the process of continuous data mining, we cannot only grasp the process of traditional data development, but we can also realize the prediction of future data development trends. Data mining is a brand-new discipline that integrates multiple technologies. Association rules are a relatively active component in the knowledge model, and they play an important role in data mining. Association rules belong to the research direction of data mining technology, which is widely used in the industry.

The process of association mining

The goal of data association mining is to use many noisy and incomplete data sets to find useful knowledge and information processing procedures. It mainly includes three steps: data preparation, data mining, and knowledge evaluation [2]. Figure 1 shows the basic model of association rule mining.

Figure 1

Basic model of association rule mining

Data preparation

This process mainly includes the process of preparing data and sorting it out. For example, employment data has a large number of attributes [3]. In researching the data model, students’ employment information mainly includes gender, ethnicity, and major. We can use this description database model to get student employment information (Table 1).

Employment information data

Gender Boys Boys Girl
Nationality Han nationality Han nationality Han nationality
Political status member member member
Department name computer science and technology computer science and technology computer science and technology
Professional computer science computer science computer science
Family Engineering Engineering Engineering
Graduate category Rural area Rural area Rural area
Employment unit Private enterprise Private enterprise Private enterprise

Because this research object originated from three colleges and universities, where there are more girls, the attributes of students mainly include gender and major. Table 2 shows the employment data after processing.

Employment data after processing

Serial number gender Professional Nature of Employment Unit
1 male computer science and technology Private enterprise
2 male computer science and technology Private enterprise
3 Female computer science and technology Private enterprise
4 Female applied mathematics Private enterprise
5 Female applied mathematics Private enterprise
6 Female applied mathematics Private enterprise
Algorithm flow

The Association rule algorithm is the main analysis method in the data mining algorithm. It can realize the key mining of data association and help find the multiple domain dependencies that meet the conditions [4]. This algorithm is widely used in the industry. The idea of the association algorithm is to find frequent itemsets whose support degree is greater than the minimum support degree. In implementing association mining, we must first scan the transaction set records to find frequent candidate sets. In this way, association rules that are of interest to users are generated. Figure 2 shows the algorithm flow of association rules.

Figure 2

Algorithm flow of association rules

Improved technology of association mining based on partial differential classification mathematical model
Theorem 1

In the process of adjusting the weight vector αc of the partial differential classification mathematical model, the convex polyhedron group Ss of a matrix WZ11WT WZ_1^{- 1}{W^T} is not empty [5]. The time-delay term of the dual-boundary stability convergence of the partial differential equations with time-delay under the constraints of dual-boundary conditions satisfies: αTQα=i=1nj=1nαiαjQij0 {\alpha ^T}Q\alpha= \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{\alpha _i}{\alpha _j}{Q_{ij}} \ge 0}}

In updating the solution space matrix Q, the marginal integral term matrix of the differential equation is invertible. In the sample set αc of data classification, the set Ss is not empty. At this time, the semi-definite eigendecomposition matrix is: Q×R=Q×RttQRtt(R*t×R*t)\tt=Q×Rtt+(R*t×R*t)\tt=I {Q^ -} \times {R^ -} = {Q^ -} \times {R_{tt}} - {{{Q^ -}} \over {{R_{tt}}}}{({R_{*t}} \times {R_{*t}})_{\backslash tt}} = {Q^ -} \times {R_{tt}} + {({R_{*t}} \times {R_{*t}})_{\backslash tt}} = I

Furthermore, the stability of the dual-boundary convergence control of the partial differential classification mathematical model is proved as follows.

Proof

According to the semi-definite constraint, the second moment of a second-order delay partial differential equations class is shown in equation (2). In the finite field YH (q) A (0), there is a n order square matrix A that satisfies the convergence condition in the Bochner-Riesz space. When Ψ(d1(t), d2(t)) < 0, there are: V˙(t)ξT(t)Ψ(d1(t),d2(t))ξ(t)<0 \dot V(t) \le {\xi ^T}(t)\Psi ({d_1}(t),{d_2}(t))\xi (t) < 0

The partial differential classification mathematical model has a stable boundary equilibrium point for any matrix with suitable dimensions. When λ1, λ2 has an inequality greater than 0, the inequality 1+3T213<0 \sum\nolimits_1 {}+ \sum\nolimits_3^T {} \sum\nolimits_2^{- 1} {\sum\nolimits_3 {}< 0} holds, if and only if: [13T32]<0,or[233T1]<0 \left[{\matrix{{\sum\nolimits_1 {}} & {\sum\nolimits_3^T {}}\cr{\sum\nolimits_3 {}} & {- \sum\nolimits_2 {}}\cr}} \right] < 0,\,\,{\rm{or}}\left[{\matrix{{- \sum\nolimits_2 {}} & {\sum\nolimits_3 {}}\cr{\sum\nolimits_3^T {}} & {\sum\nolimits_1 {}}\cr}} \right] < 0

The partial differential classification mathematical model has two equilibrium points with continuous-time delays. It can be seen that using it for association mining has dual-boundary convergence [6]. The control process of data classification is stable and convergent. The proposition is proved.

Introduce the partial differential classification mathematical model into the association rule data set Ss. According to the change of training speed, we use an increase-decrease support vector machine for fuzzy control of data classification. There are ∀iSs, βic± \beta _i^c \ne\pm \infty and ∀iSSs, γic± \gamma _i^c \ne\pm \infty . The sub-sequences generated by combining constrained bundled clustering method under the control of partial differential classification mathematical model for associated data mining are: X1=(αc[1],αc[2],αc[3],) {X_1} = (\alpha _c^{[1]},\alpha _c^{[2]},\alpha _c^{[3]}, \cdots) X2=(gc[1],gc[2],gc[3],) {X_2} = (g_c^{[1]},g_c^{[2]},g_c^{[3]}, \cdots)

Adjust the weight vector ac to generate sequences X1 and X2 as finite sequences. Under the convergence condition of meeting the minimum iteration, the associated data sequence of mining is regulated and controlled by measurement information. We use Gaussian kernel function for finite step adjustment. The expressions of the Gaussian kernel function are: K(xi,xj)=exp(xixj2/(2σ2)) K({x_i},{x_j}) = \exp \left({{{\left\| {{x_i} - {x_j}} \right\|}^2}/(2{\sigma ^2})} \right)

In the formula σ = 0.707. The maximum adjustment required Δαcmax>0 \Delta \alpha _c^{\max} > 0 . In the mathematical model of partial differential classification, we deal with ∀iSs to obtain: βic=kSsRikQkcRi1yc=1det(Q)(kSs(1)i+kdet(Q\ki)Qkc+yc(1)i+1det(Qi1)) \matrix{{\beta _i^c =- \sum\limits_{k \in {S_s}} {{R_{ik}}{Q_{kc}} - {R_{i1}}{y_c}}= {1 \over {\det ({Q^{'}})}}} \hfill\cr{\left({\sum\limits_{k \in {S_s}} {{{(- 1)}^{i + k}}\det ({Q_{\backslash ki}}'){Q_{kc}} + {y_c}{{(- 1)}^{i + 1}}\det ({Q_{i1}}')}} \right)} \hfill\cr}

According to the stability principle of Lyapunov, the association mining model designed in this paper converges gradually.

Improved technology of association mining

A genetic algorithm is an efficient global search method. It has certain robustness, randomness, and implicit parallelism [7]. This algorithm can effectively realize the global optimization search. Genetic algorithms and partial differential classification mathematical models in the optimization process of association mining can shorten the search time for large itemsets. Figure 3 shows the model structure of improved association mining.

Figure 3

Improved model structure for association mining

The main problem in the improvement of association mining is coding. This article implements data encoding with a transaction database because the encoding of real numbers is relatively simple and easy to implement [8]. Table 3 is the decision information table.

Decision information table

Condition attribute 1 Condition attribute 2 ..... Conditional attribute N Decision attributes
1 value 1 value ..... 1 value 1 value
2 value 2 value ..... 2 value 2 value
..... ..... ..... ..... .....
H value 1 value ..... P-value Q value

Although the reduction in the improvement of association rule mining is an innovation, the effectiveness and importance of association rules cannot achieve accurate data mining. Therefore, it is necessary to propose an improved attribute reduction method to achieve attribute reduction and delete attributes that do not affect the conclusion [9]. After that, we can realize data association rule mining.

The fitness function belongs to the interface in the improvement process of association mining. It is designed for application problems. It realizes the selection of different fitness functions according to different problem-solving. Because the support degree is the main measurement index in the association rules, it represents the symbolic meaning of all things in the rules, so the association rule support degree realizes the definition of its fitness function.

Improve results

Table 4 shows the results of various mining algorithms. Table 4 shows that the improved association mining technology studied in this paper can solve the slower efficiency of traditional algorithms [10]. And in the process of increasing the minimum support threshold, the number of rules is continuously decreasing.

Results of various mining algorithms

Algorithm Minimum support Minimum confidence The average number of rules Correct rate Average running time
Apriori 0.08 0.35 41 100% 32.06
GAs 0.08 0.35 32.7 89% 19.23
RS 0.08 0.35 34.3 91% 11.76
Apriori 0.12 0.4 31 100% 14.35
GAs 0.12 0.4 28 91% 9.23
RS 0.12 0.4 30.9 93% 7.3
Apriori 0.18 0.5 20 100% 12.1
GAs 0.18 0.5 18.7 87% 8.47
RS 0.18 0.5 18.4 91% 5.23
Apriori 0.23 0.6 8 100% 8.45
GAs 0.23 0.6 7.2 93% 7.07
RS 0.23 0.6 7.69 90% 5.14

Use Quest to realize the generation of a large-scale comprehensive database. After sampling from it, the sampling of the distinguished database is realized. To reduce the dependence of different experimental processes, the scale of the sampling database is smaller than that of the original database. It is necessary to realize the scanning of the min freq value. The hypothetical support number is encoded as 4 bytes, and the item set number is encoded as 2 bytes. Figures 4 to 6 show the communication load of different databases [11]. We use this to express the comparison of the three algorithms. Two of them use relatively little communication. DDM and PDDM in the load database behave the same, and DDDM is the best.

Figure 4

The relationship between the number of transmitted bytes, the partition book, the minimum support, and the communication load

Figure 5

The relationship between support, number of nodes, partition book, and sent bytes

Figure 6

The number of messages sent, the number of bytes, and the rate of message usage are related to changes in the buffer

First, check the buffer size change. The result shows that it is not much different from the ideal network environment and the network result in the buffer. The algorithm is good in the number of bytes sent. Figure 6 shows the relationship between the size of the buffer and the number of bytes sent. For the buffer, if there are many candidate bases, the algorithm sends bytes and information lower than FDM. If the candidate base set is small, the sent information is half empty. FDM will have a certain degree of competitiveness. The experimental results show that the improved technology of association mining based on the partial differential classification mathematical model proposed in this paper can solve communication complexity. Compared with other algorithms, this algorithm can guarantee the same growth rate.

Concluding remarks

In increasing current information, the demand for network data domain database creation is also increasing. This will expand the scale of data information processing. So how to achieve efficient and fast data mining is a problem that needs to be solved in the modern field. The association rule mining optimization designed in this paper can improve the algorithm's efficiency while reducing the co-workload of the object scanning data set. This algorithm can be used in the evaluation of enterprise selection.

Figure 1

Basic model of association rule mining
Basic model of association rule mining

Figure 2

Algorithm flow of association rules
Algorithm flow of association rules

Figure 3

Improved model structure for association mining
Improved model structure for association mining

Figure 4

The relationship between the number of transmitted bytes, the partition book, the minimum support, and the communication load
The relationship between the number of transmitted bytes, the partition book, the minimum support, and the communication load

Figure 5

The relationship between support, number of nodes, partition book, and sent bytes
The relationship between support, number of nodes, partition book, and sent bytes

Figure 6

The number of messages sent, the number of bytes, and the rate of message usage are related to changes in the buffer
The number of messages sent, the number of bytes, and the rate of message usage are related to changes in the buffer

Results of various mining algorithms

Algorithm Minimum support Minimum confidence The average number of rules Correct rate Average running time
Apriori 0.08 0.35 41 100% 32.06
GAs 0.08 0.35 32.7 89% 19.23
RS 0.08 0.35 34.3 91% 11.76
Apriori 0.12 0.4 31 100% 14.35
GAs 0.12 0.4 28 91% 9.23
RS 0.12 0.4 30.9 93% 7.3
Apriori 0.18 0.5 20 100% 12.1
GAs 0.18 0.5 18.7 87% 8.47
RS 0.18 0.5 18.4 91% 5.23
Apriori 0.23 0.6 8 100% 8.45
GAs 0.23 0.6 7.2 93% 7.07
RS 0.23 0.6 7.69 90% 5.14

Employment data after processing

Serial number gender Professional Nature of Employment Unit
1 male computer science and technology Private enterprise
2 male computer science and technology Private enterprise
3 Female computer science and technology Private enterprise
4 Female applied mathematics Private enterprise
5 Female applied mathematics Private enterprise
6 Female applied mathematics Private enterprise

Decision information table

Condition attribute 1 Condition attribute 2 ..... Conditional attribute N Decision attributes
1 value 1 value ..... 1 value 1 value
2 value 2 value ..... 2 value 2 value
..... ..... ..... ..... .....
H value 1 value ..... P-value Q value

Employment information data

Gender Boys Boys Girl
Nationality Han nationality Han nationality Han nationality
Political status member member member
Department name computer science and technology computer science and technology computer science and technology
Professional computer science computer science computer science
Family Engineering Engineering Engineering
Graduate category Rural area Rural area Rural area
Employment unit Private enterprise Private enterprise Private enterprise

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