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University Library Lending System Model Based on Fractional Differential Equations

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 13 Apr 2022
Accepté: 23 Jun 2022
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Introduction

Two contradictions faced by mobile digital libraries have become increasingly prominent: one is the contradiction between the explosive growth of knowledge and the limitations of library users’ ability to choose. Chinese digital library promotion project provides users with more than 140TB of shared digital resources. This makes it difficult for readers’ learning ability to match the explosive growth of information. The second is how readers can discover their interests from the prosperous digital resource information. There is a contradiction between the extreme abundance of information and the limitation of information that users are interested in. The fundamental problem of the above two contradictions is the personalized recommendation problem of the mobile digital library. Mobile digital libraries’ personalized recommendation is to process information such as readers’ interests and knowledge fields into knowledge elements that can vividly describe readers’ preferences [1]. This supports various recommendation services of the digital library. Ultimately, the system can provide users with knowledge resources that meet their individual needs. The primary user groups of mobile digital libraries access resources through mobile handheld devices. The environments and scenarios in which such users are located are changing. Studies have shown that changes in the user's context will have varying degrees of impact on their personalized needs. But at present, most mobile digital libraries have insufficient ability to perceive situational factors. Therefore, it is difficult for these systems to provide library users with accurate and personalized services that best match their contexts.

Extensive data-oriented mobile digital library contextualized recommendation system
System Architecture

The mobile digital library contextualized service recommendation system is deployed in the Hadoop distributed environment (Figure 1). With the continuous increase in the scale of library user service requests, the pressure of big data processing faced by the recommendation system is also gradually increasing. At this time, the computing advantages of Hadoop distributed processing can be fully utilized: we deploy library user service requests to Hadoop clusters for processing [2]. The system increases the parallel computing performance of the cluster by horizontally expanding the number of Hadoop nodes. This can alleviate the pressure of big data processing faced by the recommendation system.

Figure 1

Architecture of extensive data-oriented mobile digital library contextualized recommendation system

Fractional differential equation denoising model

In this paper, the minimized energy functional is first given as follows: E(I)=infuBV(Ω){ 12Ω(gI)2dΩ+μΩ| DpI |dΩ } E\left( I \right) = \mathop {\inf }\limits_{u \in BV\left( \Omega \right)} \left\{ {{1 \over 2}\int {\int_\Omega {{{\left( {g - I} \right)}^2}d\Omega + \mu \int {\int_\Omega {\left| {{D^p}I} \right|} d\Omega } } } } \right\}

We use the variational method to derive the Euler-Lagrange equation for this functional: μDp(DpI| DpI |)+(gI)=0 \mu {D^p}\left( {{{{D^p}I} \over {\left| {{D^p}I} \right|}}} \right) + \left( {g - I} \right) = 0 Where Dp=(pxp,pyp) {D^p} = \left( {{{{\partial ^p}} \over {\partial {x^p}}},{{{\partial ^p}} \over {\partial {y^p}}}} \right) . The corresponding diffusion equation is as follows: { It=(gI)+μDp(DpI| DpI |)I(0,x,y)=g(x,y)In|Ω=0 \left\{ {\matrix{ {{{\partial I} \over {\partial t}} = \left( {g - I} \right) + \mu {D^p}\left( {{{{D^p}I} \over {\left| {{D^p}I} \right|}}} \right)} \hfill \cr {I\left( {0,\,x,\,y} \right) = g\left( {x,\,y} \right)} \hfill \cr {{{\partial I} \over {\partial n}}\left| {_{\partial \Omega } = 0} \right.} \hfill \cr } } \right.

When p = 1 the model is the ROF model. When p = 2, the model is a fourth-order PDE denoising model. When 1 < p < 2 is the model is an interpolation of the second-order and fourth-order partial differential denoising models [3]. The author believes that 1 ≤ p ≤ 2 is a suitable audio smooth interval. We need to choose an appropriate value of p to suppress the “staircase effect.”

Fractional derivatives are a reasonable generalization of integer derivatives. Although fractional derivatives have been used in many fields, the definition of fractional derivatives is not unique. Generally, it is only required to be compatible with integer order derivatives. The commonly used definitions of fractional derivatives include the Riemann-Liouville (R-L) fractional derivative, the Cauchy integral fractional derivative, the Fourier domain fractional derivative, and the Grunwald-Letnikov (G-L) fractional derivative. In this paper, an audio regularization model and its numerical solution are given based on the definition of G-L fractional derivative. This paper first gives the G-L fractional derivative: Dpf(x)=limh0k0n(1)k(pk)f(xkh)hp {D^p}f\left( x \right) = \mathop {\lim }\limits_{h \to 0} {{\sum\limits_{k \ge 0}^n {{{\left( { - 1} \right)}^k}\left( {\matrix{ p \hfill \cr k \hfill \cr } } \right)f\left( {x - kh} \right)} } \over {{h^p}}} Where p is the order of the fractional order. (pk)=Γ(p+1)Γ(k+1)Γ(pk+1) \left( {\matrix{ p \hfill \cr k \hfill \cr } } \right) = {{\Gamma \left( {p + 1} \right)} \over {\Gamma \left( {k + 1} \right)\Gamma \left( {p - k + 1} \right)}} .

Here Γ(k)=0+xk1exdx \Gamma \left( k \right) = \int_0^{ + \infty } {{x^{k - 1}}{e^{ - x}}dx} is the Gamma function, which satisfies Γ (k) = k−1 Γ (k + 1). If k is a positive integer, then Γ (k + 1) = k!. When p = 1, the above formula is the first derivative in the usual sense.

Fixed p, then limk(pk)=0 \mathop {\lim }\limits_{k \to \infty } \left( {\matrix{ p \hfill \cr k \hfill \cr } } \right) = 0 . Therefore, when h = 1, the GL type fractional derivative can be approximated by the fractional difference of finite terms (take the first k terms): Dpf(x)k=0k1(1)k(pk)f(xk) {D^p}f\left( x \right) \approx \sum\limits_{k = 0}^{k - 1} {{{\left( { - 1} \right)}^k}\left( {\matrix{ p \hfill \cr k \hfill \cr } } \right)f\left( {x - k} \right)}

We generalize the above definition to the two-dimensional case to obtain the definition of fractional partial differential of I (x, y): pIxp=limhk0n(1)k(pk)I(xkh,y)hp {{{\partial ^p}I} \over {\partial {x^p}}} = \mathop {\lim }\limits_{h \to \infty } {{\sum\limits_{k \ge 0}^n {{{\left( { - 1} \right)}^k}\left( {\matrix{ p \hfill \cr k \hfill \cr } } \right)I\left( {x - kh,y} \right)} } \over {{h^p}}} pIyp=limhk0n(1)k(pk)I(x,ykh)hp {{{\partial ^p}I} \over {\partial {y^p}}} = \mathop {\lim }\limits_{h \to \infty } {{\sum\limits_{k \ge 0}^n {{{\left( { - 1} \right)}^k}\left( {\matrix{ p \hfill \cr k \hfill \cr } } \right)I\left( {x,y - kh} \right)} } \over {{h^p}}}

Since the algorithm in this paper only involves the case of 1 ≤ p ≤ 2, we only need to take the fractional difference of the first three terms to approximate the fractional partial differential: pI(x,y)xpI(x,y)+(p)I(x1,y)+p(p+1)2I(x2,y) {{{\partial ^p}I\left( {x,\,y} \right)} \over {\partial {x^p}}} \approx I\left( {x,\,y} \right) + \left( { - p} \right)I\left( {x - 1,\,y} \right) + {{ - p\left( { - p + 1} \right)} \over 2}I\left( {x - 2,\,y} \right) pI(x,y)ypI(x,y)+(p)I(x,y1)+p(p+1)2I(x,y2) {{{\partial ^p}I\left( {x,\,y} \right)} \over {\partial {y^p}}} \approx I\left( {x,\,y} \right) + \left( { - p} \right)I\left( {x,\,\,y - 1} \right) + {{ - p\left( { - p + 1} \right)} \over 2}I\left( {x,\,\,y - 2} \right)

Contextual interest modeling

Contextual interest modeling is the core part of contextualized recommendation in the mobile digital library. The user group is denoted as M. The project group participating in the evaluation is denoted as S. The external context information faced by the user is represented as Context [4]. The target user studied in this paper is set as ui (uiU). This article sets the ungraded item set ui as Sj. ui to Sj scores were predicted by the CF recommendation method during the study. Then we recommend the top-ranked items to ui according to the Top-N rule. In general, the model in this paper is described in detail.

Mobile digital library users are in different situations such as geographic location, ambient temperature, and service time. This article uses the vector calculation formula to label it Contexty = (C1, C2, ⋯, Cn. where Ck(k = 1, 2, ⋯, n) represents the user context under a particular type. Contextx and Contexty represent two different situations. Its similarity is denoted as Sim (Contextx, Contexty) in this paper. Simk (Contextx, Contexty) indicates how similar situations Contextx and Contexty are compared in the context of the class k.

This paper introduces the contextual information of library users in the scoring process. We extend the “user-item” scoring matrix of traditional CF. This forms a “user-item-context” scoring matrix. We incorporate contextual information Contextk into each item's score rsui, sj in the original mobile digital library user. In this way, the personalized recommendation of a mobile digital library that integrates contextual interests can be realized.

Step 1: First, we obtain the “user-item” scoring matrix and the context in which the user is placed when scoring. When building a library user contextual recommendation model, a traditional user rating matrix RSM×N is established first [5]. The matrix reflects the user's rating of the item. The recommendation relation is shown in formula (10). RSM×N=[ rsu1s1rsu1s2rsu1sNrsu2s1rsu2s2rsu2sNrsuMs1rsuMs2rsuMsN ] R{S_{M \times N}} = \left[ {\matrix{ {r{s_{{u_1}{s_1}}}} & {r{s_{{u_1}{s_2}}}} & \cdots & {r{s_{{u_1}{s_N}}}} \cr {r{s_{{u_2}{s_1}}}} & {r{s_{{u_2}{s_2}}}} & \cdots & {r{s_{{u_2}{s_N}}}} \cr \cdots & \cdots & \cdots & \cdots \cr {r{s_{{u_M}{s_1}}}} & {r{s_{{u_M}{s_2}}}} & \cdots & {r{s_{{u_M}{s_N}}}} \cr } } \right]

User ui ‘s rating for item sj is represented by rsui, sj. The average rating of user ui is denoted by rsui¯ \overline {r{s_{{u_i}}}} . The user's average rating is denoted by rsuj¯ \overline {r{s_{{u_j}}}} .

Step 2: Calculate the similarity sim (ui, uj) between the target user and other users. In this paper, the Parson correlation coefficient measurement formula is used in the calculation. From this we can calculate the similarity between the target user ui and user uj: sim(ui,uj)=skUSui,uj(rsui,skrsui¯)(rsuj,skrsuj¯)skUSui,uj(rsui,skrsui¯)2skUSui,uj(rsuj,skrsuj¯)2 sim\left( {{u_i},{u_j}} \right) = {{\sum\limits_{{s_k} \in U{S_{{u_i},{u_j}}}} {\left( {r{s_{{u_i},{s_k}}} - \overline {r{s_{{u_i}}}} } \right)\,\left( {r{s_{{u_j},{s_k}}} - \overline {r{s_{{u_j}}}} } \right)} } \over {\sqrt {\sum\limits_{{s_k} \in U{S_{{u_i},{u_j}}}} {\left( {r{s_{{u_i},{s_k}}} - \overline {r{s_{{u_i}}}} } \right){\,^2}} \sqrt {\sum\limits_{{s_k} \in U{S_{{u_i},{u_j}}}} {\left( {r{s_{{u_j},{s_k}}} - \overline {r{s_{{u_j}}}} } \right){\,^2}} } } }}

The similarity between target user ui and user uj is expressed by sim(ui, uj) in this paper. Items scored by both ui and uj are denoted by USui, uj in this paper. The average ratings of ui and uj are denoted by rsui¯ \overline {r{s_{{u_i}}}} and rsuj¯ \overline {r{s_{{u_j}}}} in this paper.

The size of USui, uj is significant here. The larger the modulus of the item set USui, uj, that both ui and uj have jointly rated, the higher the similarity between ui and uj is. If sim(ui, uj) = sim(ui, uk), |USui, uj| > |USui, uk|, the similarity between the target user ui and user uk is lower than the similarity between target user ui and user uj. The traditional Pearson measure formula (11) fails to consider the influence of these factors when calculating the similarity between target user ui and user uk. This paper uses the contextual information recommendation mechanism to introduce |USui, uj| to modify: sim(ui,uj)=11+e| USui,uj |2×sim(ui,uj) sim^\prime\left( {{u_i},{u_j}} \right) = {1 \over {1 + {e^{{{\left| {U{S_{{u_i},{u_j}}}} \right|} \over 2}}}}} \times sim\left( {{u_i},{u_j}} \right)

11+e| USui,uj |2 {1 \over {1 + {e^{{{\left| {U{S_{{u_i},{u_j}}}} \right|} \over 2}}}}} is a variant of Sigmoid function 11+et {1 \over {1 + {e^{ - t}}}} . The value of 11+e| USui,uj |2 {1 \over {1 + {e^{{{\left| {U{S_{{u_i},{u_j}}}} \right|} \over 2}}}}} is infinitely close to 1 when the value of |USui, uk| is infinite.

Step 3: Revise the similarity measurement formula according to the situation. When the context information of the target user is more similar to the scoring context Simi (ItemContext, ItemRatingContext) of the rated user, the rated user obtains a higher recommendation weight [6]. We’ll label it k = Sim (ItemContext, ItemRatingContext). We refer to the similarity measurement formula to modify formula (12): Sim(ui,uj)=c×sim(ui,uj)+(1c)×k×sim(ui,uj) Sim''\left( {{u_i},{u_j}} \right) = c \times sim'\left( {{u_i},{u_j}} \right) + \left( {1 - c} \right) \times k \times sim'\left( {{u_i},{u_j}} \right)

Among them, in formula (13) is the adjustable coefficient. Its size belongs to between 0–1. At that time, the calculation result of Equation (13) is equivalent to not considering the user's contextual factors.

Big data-parallel recommendation

The contextualized recommendation for mobile digital library users proposed in this paper is oriented to massive data. It isn’t easy to carry out practical library user extensive data mining through the recommendation method that only runs on a single machine. We introduced big data parallel processing technology [7]. In this paper, the library contextualized recommendation system constructed in this paper is deployed in a distributed environment. We parallelize the recommendation methods to improve the mining performance and scalability of recommendation systems in massive data environments.

The similarity calculation of the “user-item” rating matrix and the rating prediction of unrated items are two critical recommendations links. These two links can be considered sequential steps in a logical relationship [8]. This is consistent with the principle of distributed parallel processing in the environment. The specific calculation process is summarized as follows: The first step uses the effective data processing method to calculate the similarity of the “user-item” scoring matrix. The second step makes rating predictions for unrated items by similarity. The key-value pairs input in this paper is expressed in the form of <null, (User, Item, Score)> when calculating the similarity of the “user-item” scoring matrix of the library. The output key value is represented by < (Item1, Item2), Sim>. Two MapReduce tasks can realize parallel computing in this stage. The first MapReduce mainly summarizes the rating information of library users on items and ranks them. The function converts the input library user information and item rating information into corresponding key-value pairs. The function merges rating items with the same user. The second is devoted to calculating the similarity between items. We convert key-value pairs between library users and projects into key-value pairs between projects and projects. We use the Map function to obtain the scores of the same User among each Item. At the same time, we use the Reduce function to calculate the similarity between items. After two times of MapReduce processing, we can get the calculation result of similarity and the similarity list of each Item [9]. Finally, we calculate the similarity list of recommendation scores of library users based on MapReduce. In this paper, the Map function is used for CF recommendation, and the function Reduce is used to output the recommendation results.

Experiment and Verification
Dataset and Evaluation Criteria

The experimental evaluation criteria in this paper mainly include two aspects: First, we test whether the parallel mining of big data based on Hadoop can improve the performance of the model calculation. The second is to test situational recommendation mitigation to reduce recommendation accuracy caused by data sparsity [10]. The speedup ratio is mainly used to compare the time it takes for a specific algorithm to run in two different environments, single-machine, and parallel computing. The calculation method is the ratio between the running time of a single machine and the running time of a parallel machine: S=T(1)/T(N). Among them, T (1) is the algorithm's running time in a stand-alone environment. T(N) is the time for multi-machine parallel processing. This paper introduces the most common mean absolute deviation MAE in recommendation performance testing. The calculation formula is: MAE=i=1N| piqi |N MAE = {{\sum\nolimits_{i = 1}^N {\left| {pi - qi} \right|} } \over N}

{p1, p2, ⋯⋯, pN} is the set of user ratings predicted by the algorithm. {q1, q2, ⋯⋯, qN} is a collection of actual user ratings.

Analysis of experimental results

This part adopts the speedup ratio S to analyze the performance of the contextualized recommendation algorithm. Data distribution is first performed on the dataset. We define them as datasets D1, D2, D3, and D4, respectively. It contains data sets of 1000, 2000, 3000, and 4000 user ratings [11]. This paper tests the parallel running time of the above datasets in the Hadoop environment. We selected 2, 4, 6, and 8 cases with different numbers of running nodes, respectively. At the same time, we recorded the running time of the D1–D4 dataset in the above four cases (Fig. 2).

Figure 2

Speedup of algorithm parallel recommendation in Hadoop environment

When using the D1 test data set, the speedup curve of the recommended model in the Hadoop environment is less than 1. It will first drop and then rise as the number of nodes increases. When the test data set is D2–D4, the speedup ratio S obtained by the test will increase rapidly with the amount of data. In the case of the same amount of data, the speedup ratio obtained by the test will continue to rise with the increase of the number of nodes [12]. From the comparison results in Figure 2, it can also be seen that in the test case of small data volume such as D1, the operating efficiency of the recommendation model in the Hadoop environment is low. It does not reflect the advantages of parallel computing very well. The main reason is that the job startup time and interaction time of the recommendation system in the environment are relatively long.

In contrast, the time used for parallel computing is relatively short. Therefore, the combined final run time is longer. It isn’t easy to reflect the advantages of Hadoop cluster parallel computing. In the Hadoop environment, the proportion of time between start-up and interaction of the recommender system will gradually decrease with the increase of the amount of test data. Therefore, the larger the test data set, the more pronounced the speed advantage of parallel computing of the algorithm. The efficiency of parallel processing of the recommended model in this paper also tends to be more stable.

Conclusion

In the big data environment, the sparseness of user member ratings in mobile digital libraries will lead to many users having no or very few items in standard ratings. It isn’t easy to perform efficient mining when using standard CF recommendations. The accuracy of its recommendations is significantly reduced. The model in this paper better integrates user contextual interests based on fractional differential equations into the recommendation process. It improves the nearest neighbor search problem due to sparse scoring data. At the same time, it effectively alleviates the cold start problem caused by the mobile digital library's sparse user rating data in the big data environment.

Figure 1

Architecture of extensive data-oriented mobile digital library contextualized recommendation system
Architecture of extensive data-oriented mobile digital library contextualized recommendation system

Figure 2

Speedup of algorithm parallel recommendation in Hadoop environment
Speedup of algorithm parallel recommendation in Hadoop environment

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