Research Article  Open Access
A Novel Method for Predicting DiseaseAssociated LncRNAMiRNA Pairs Based on the HigherOrder Orthogonal Iteration
Abstract
A lot of research studies have shown that many complex human diseases are associated not only with microRNAs (miRNAs) but also with long noncoding RNAs (lncRNAs). However, most of the current existing studies focus on the prediction of diseaserelated miRNAs or lncRNAs, and to our knowledge, until now, there are few literature studies reported to pay attention to the study of impact of miRNAlncRNA pairs on diseases, although more and more studies have shown that both lncRNAs and miRNAs play important roles in cell proliferation and differentiation during the recent years. The identification of diseaserelated genes provides great insight into the underlying pathogenesis of diseases at a system level. In this study, a novel model called PADLMHOOI was proposed to predict potential associations between diseases and lncRNAmiRNA pairs based on the higherorder orthogonal iteration, and in order to evaluate its prediction performance, the global and local LOOCV were implemented, respectively, and simulation results demonstrated that PADLMHOOI could achieve reliable AUCs of 0.9545 and 0.8874 in global and local LOOCV separately. Moreover, case studies further demonstrated the effectiveness of PADLMHOOI to infer unknown diseaserelated lncRNAmiRNA pairs.
1. Introduction
Noncoding RNA, according to its size, can be divided into small and long noncoding RNAs approximately. Generally, small RNAs include tRNAs, miRNAs, piRNAs, and snoRNAs [1–4], and miRNAs are widely present in the cytoplasm of eukaryotic cells and are approximately 18–22 nucleotides in length, which can bind to 3′untranslated region of mRNA (3′UTR) to inhibit the translation process of mRNA or to degrade mRNA, thereby affecting the expression of related genes [5–7]. miRNAs play important roles in a series of life activities such as cell differentiation of living body [8], growth and development [9], and apoptosis [10]. Compared to smallmolecule ncRNA, lncRNA has a longer nucleotide chain with more than 200 nucleotides and has a specific and complex secondary space structure inside the molecule and can provide multiple sites for protein binding [11]. In addition, both lncRNAs and miRNAs are key members of noncoding RNAs and play important roles in coding and regulation of many complex human diseases [12–16].
Up to now, there have been many studies on relationships between diseases and miRNAs. For example, some important methods proposed by Xing Chen et al. [17–20] and Zou et al. [21–24]. In terms of prediction of potential associations between lncRNAs and diseases, Yu et al. [25] and Xing et al. [26] proposed two kinds of computational models called NBCLAD and LRLSLDA, respectively. Moreover, studies have also shown that there exist relationships between lncRNAs and miRNAs. For example, Gernapudi et al. demonstrated that miRNA 140 can induce the expression of lncRNA NEAT1 [27]. Dey et al. showed that the silencing of lncRNA H19 and knockout of H19 gene in myoblasts significantly decreased skeletal muscle differentiation [28]. Yilong et al. discovered that, after low XIST expression in gliomas, XIST could regulate miR152 glioma stem cells to inhibit cell proliferation, migration, and invasion [29]. Xinyu et al. demonstrated that lncRNA MALAT1 could achieve posttranscriptional regulation of esophageal squamous cell carcinoma cells through miR101 and miR217 [30]. Erbao et al. proposed that lncRNA ANRIL interacted with miR99a/miR449a to regulate cell proliferation during gastric cancer formation [31]. You et al. found that the expression of miR449a and the expression of lncRNA NEAT1 in lung cancer cell L9981 inhibited each other. When miR449a was overexpressed, NEAT1 expression was decreased, cell proliferation was inhibited, and apoptosis was increased, and vice versa [32]. Emmrich et al. found that the expression of lncRNA MONC and MIR100HG was closely related to the miRNA groups of miR99a∼125b2 and miR100∼125b1. After silencing of lncRNA MONC and MIR100HG, acute megakaryocytes in the early stage of the disease, the tumor cells of leukemia patients, were severely inhibited [33]. Amy et al. found that lncRNA Ang362 was the host transcriptor of miR211 and miR222, and their interactions regulated Ang II and induced proliferation of vascular smooth muscle cells [34]. Miaojun et al. found that the interactions between lncRNA H19 and miRNA675 play an important role in the metastasis of prostate cancer [35]. Obviously, the exploration of these relationships was conducive to the construction of gene regulatory networks and the identification of the mechanisms of complex human diseases [36–38].
From the above description, it is easy to see that more and more studies have shown that lncRNAmiRNA interactions are involved in the development of complex diseases. However, to the best of our knowledge, so far, in addition to the model of PADLMP proposed by Zhou et al. [39], few models have been proposed for largescale prediction of potential associations between diseases and lncRNAmiRNA interactions. Hence, inspired by stateoftheart methods [40–44], which show that the miRNAmiRNA pairs can work cooperatively to regulate a single gene or gene clusters being involved in similar processes [45], and simultaneously, based on the reasonable assumption that functionally similar lncRNAmiRNA pairs tend to be associated with similar diseases, in this paper, a new prediction model called PADLMHOOI was proposed to infer potential associations between diseases and the lncRNAmiRNA pairs. And, as illustrated in Figure 1, our newly proposed prediction model PADLMHOOI consists of the following four major steps: Step 1 (Data Integration and Network Construction). In this step, first of all, we downloaded known diseaselncRNA associations from three different diseaselncRNA databases such as diseaselncRNA [46], MNDR [47, 48], and lnc2cancer [49], respectively, and then, based on these datasets, we constructed a bipartite network of diseaselncRNA. Next, we downloaded known diseasemiRNA associations from three different databases such as miR2Disease [50], HMDD [51], and miRCancer [52] separately, and then, based on these datasets, we constructed a bipartite network of diseasemiRNA. Moreover, we downloaded the 2015 and 2017 versions of known lncRNAmiRNA associations from the starBasev2.0 database [53] (http://starbase.sysu.edu.cn/) on Feb 2, 2017, and based on these datasets, we constructed a bipartite network of lncRNAmiRNA. Finally, based on these three kinds of bipartite networks, we constructed an integrated tripartite network of diseaselncRNAmiRNA, which could be denoted as a tensor T. Step 2 (Similarity Calculation). In this step, we would integrate the disease semantic similarity and Gaussian Interaction Profile Kernel similarity firstly to measure the similarity of diseases. Next, we would integrate the lncRNA functional similarity and miRNA functional similarity in three different ways to measure the functional similarity of lncRNAmiRNA pairs. Step 3 (Weighted KNearest Neighbor Profile). Considering that there may be diseases that are unrelated to all lncRNAmiRNA pairs, which may lead to unsatisfactory prediction results while implementing PADLMHOOI to infer potential associations between diseases and lncRNAmiRNA pairs. Hence, in this step, we would introduce the weighted Knearest neighbor profile (WKNNP) to add more interaction information between diseases, lncRNAs, and miRNAs to improve the prediction performance of PADLMHOOI. Step 4 (Tensor Decomposition). In this step, we would perform tensor decomposition on the newly constructed diseaselncRNAmiRNA tensor T. Since the results of tensor decomposition include a core tensor and three matrices, we can define the final predicted association tensor as the modal product between the core tensor and these three matrices. Thereafter, we would sort scores of the lncRNAmiRNA pairs associated with each disease in the descending order in the final predicted association tensor, and it is obvious that the higher the ranking of the score, the bigger the possibility that there may exist potential association between the disease and the lncRNAmiRNA pair would be.
2. Materials and Methods
2.1. Construction of the Bipartite Network of DiseaselncRNA
In order to construct the bipartite network of diseaselncRNA, firstly, known associations between diseases and lncRNAs were downloaded from three different databases such as the LncRNADisease, MNDR, and Lnc2Cancer, respectively, and then, after feature processing (including feature cleaning and data imbalance processing etc.), 2048 different diseaselncRNA associations were finally obtained (Supplementary Table 1). Thereafter, based on these newly obtained 2048 known diseaselncRNA associations, we can construct a diseaselncRNA bipartite network G_{1} = (V_{1}, E_{1}) according to the following steps: Step 1. Let be the set of all different lncRNAs in these 2048 known diseaselncRNA associations and be the set of all different diseases in these 2048 known diseaselncRNA associations, then we define as the vertex set in G_{1}. Step 2. , if (l_{i}, d_{j}) belongs to these 2048 downloaded known diseaselncRNA associations, then we define that there is an edge between l_{i} and d_{j} in G_{1}; thereafter, we can obtain the edge set E_{1} in G_{1}.
2.2. Construction of the Bipartite Network of DiseasemiRNA
In order to construct the bipartite network of diseasemiRNA, at first, known diseasemiRNA associations were downloaded from three different databases such as the miR2Disease, HMDD, and miRCancer separately, and then, after these newly acquired miRNAs and diseases being mapped to the database miRBase v21 [54] and disease ontology (DO) [55], respectively, 4041 different diseasemiRNA associations were finally obtained (Supplementary Table 2). Hence, based on these newly obtained 4041 known diseasemiRNA associations, we can construct a diseasemiRNA bipartite network G_{2} = (V_{2}, E_{2}) according to the following steps: Step 1. Let be the set of all different miRNAs in these 4041 known diseasemiRNA associations and be the set of all different diseases in these 4041 known diseasemiRNA associations, then we define as the vertex set in G_{2}. Step 2. , if (m_{i}, d_{j}) belongs to these 4041 known diseasemiRNA associations, then we define that there is an edge between m_{i} and d_{j} in G_{2}; thereafter, we can obtain the edge set E_{2} in G_{2}.
2.3. Construction of the Bipartite Network of lncRNAmiRNA
In order to construct the bipartite network of lncRNAmiRNA, at first, two different versions (2015 and 2017) of lncRNAmiRNA dataset were downloaded from the starBasev2.0 database separately, and then, after feature processing (including feature cleaning and data imbalance processing), 20324 different lncRNAmiRNA interactions were finally obtained (Supplementary Table 3). Thereafter, based on these newly obtained 20324 known lncRNAmiRNA associations, we can construct a lncRNAmiRNA bipartite network G_{3} = (V_{3}, E_{3}) according to the following steps: Step 1. Let denote the set of all different lncRNAs in these 20324 known lncRNAmiRNA associations and denote the set of all different miRNAs in these 20324 known lncRNAmiRNA associations, then we define as the vertex set in G_{3}. Step 2. , if (l_{i}, m_{j}) belongs to these 20324 known lncRNAmiRNA associations, then we define that there is an edge between l_{i} and m_{j} in G_{3}; thereafter, we can obtain the edge set E_{3} in G_{3}.
2.4. Construction of the Tripartite Network of DiseaselncRNAmiRNA
Based on the above newly obtained networks such as G_{1}, G_{2}, and G_{3}, we can construct a tripartite network G_{4} = (V_{4}, E_{4}) according to the following steps: Step 1. Let , , and . Step 2. While V_{d} is not null, Repeat: , If and satisfyies the following three kinds of conditions simultaneously:(a)(b)(c) Then (d_{i}, l_{j}), (d_{i}, m_{k}), and (l_{j}, m_{k}) will be added into E_{4} firstly, and then, d_{i} will be added into and removed from V_{d}. Finally, l_{j} and m_{k} will be added into V_{4} if l_{j} and m_{k} are not inV_{4}. Else, d_{i} will be removed from V_{d}. Step 3. Let .
According to above steps, a tripartite diseaselncRNAmiRNA association network can be obtained finally. And, it is obvious that, in the tripartite network, there are three kinds of different nodes such as disease nodes, lncRNA nodes, and miRNA nodes; moreover, the number of disease nodes, lncRNA nodes, and miRNA nodes is 68, 44, and 211, respectively, and the number of associations between diseases and lncRNAmiRNA pairs is 3,047.
2.5. Construction of the DiseaselncRNAmiRNA Tensor
Based on the newly constructed tripartite network, for any given disease node d_{i}, lncRNA node l_{j}, and miRNA node m_{k} in G_{4}, we can define a tensor T as follows:
2.6. Calculation of the Similarity of Disease Pairs
2.6.1. Calculation of the Disease Semantic Similarity (DisSemSim)
In order to estimate the semantic similarity between diseases, we first downloaded the MeSH descriptor from the National Medical Library (http://www.nlm.nih.gov/) and selected the standard MeSH disease terminology. And then, for each disease d, we can construct a Directed Acyclic Graph (DAG) such as , where denotes the set of nodes containing the node d itself and its ancestors and denotes the set of edges of the respective direct links from parent to child nodes [56]. Thereafter, based on the newly constructed directed acyclic graph , the semantic contribution of an ancestor node d_{s} to the disease d can be calculated as follows:where is the semantic contribution decay factor with value between 0 and 1. And, in addition, according to the experimental results of some previous stateoftheart methods [57, 58], the most appropriate value for will be 0.5. Hence, based on the assumption that two diseases with more common ancestor nodes in their DAGs shall have higher semantic similarity, the semantic similarity between two diseases d_{i} and d_{j} can be defined as follows:
2.6.2. Calculation of the Gaussian Interaction Profile Kernel Similarity for Diseases (GIPSim)
Based on the hypothesis that functionally similar genes are often associated with similar diseases, in this section, we will adopt the Gaussian Interaction Profile Kernel to calculate the similarity of diseases according to the following steps:
Firstly, based on the networks G_{1} and G_{2} constructed above, for any given lncRNA l_{i} and disease d_{j}, we define that
Next, for any given miRNA m_{i} and disease d_{j}, we define that
Hence, let denote the ith column of the matrix Y_{1}, then we can calculate the Gaussian Kernel Similarity between diseases d_{i} and d_{j} based on their interaction profiles as follows:where the parameter denotes the number of different diseases in G_{1}.
In a similar way, let denote the ith column of matrix Y_{2}, then we can calculate the Gaussian Kernel Similarity between diseases d_{i} and d_{j} based on their interaction profiles as follows:Here, the parameter denotes the number of different diseases in G_{2}.
Thereafter, based on these above formulas, we can calculate the Gaussian Interaction Profile Kernel Similarity between diseases d_{i} and d_{j} as follows:
2.7. Calculation of the Similarity of lncRNA Pairs
2.7.1. Calculation of the lncRNA Functional Similarity
For any two given lncRNAs such as l_{i} and l_{j}, let be all the diseases related to l_{i} in G_{1} and be all the diseases related to l_{j} in G_{1}, then we can define the functional similarity between l_{i} and l_{j} as follows:where
2.7.2. Calculation of the Gaussian Interaction Profile Kernel Similarity for lncRNAs
For any two given lncRNAs such as l_{i} and l_{j}, similar to the definition of formula (6), let and denote the ith and the jth row of the matrix Y_{1}, respectively, then we can calculate the Gaussian Kernel Similarity between diseases l_{i} and l_{j} based on their interaction profiles as follows:where denotes the number of different lncRNAs in G_{1}.
Hence, based on these formulas given above, we can finally define the similarity measurement between lncRNAs l_{i} and l_{j} as follows:
2.8. Calculation of the Similarity between miRNAs (miRSim)
2.8.1. Calculation of the miRNA Function Similarity (miRfunSim)
For any two given miRNAs, such as m_{i} and m_{j}, let be all the diseases related to m_{i} in G_{2} and be all the diseases related to m_{j} in G_{2}, then we can define the functional similarity between m_{i} and m_{j} as follows:
2.8.2. Calculation of the Gaussian Interaction Profile Kernel Similarity for miRNAs
For any two given miRNAs, such as m_{i} and m_{j}, in a similar way, let and represent the ith and jth row in matrix Y_{2}, respectively, then we can calculate the Gaussian Kernel Similarity between diseases m_{i} and m_{j} based on their interaction profiles as follows:where denotes the number of miRNAs in G_{2.}
Hence, based on these formulas presented above, we can finally define the similarity measurement between miRNAs m_{i} and m_{j} as follows:
2.9. Weighted K Nearest Neighbor Profiles for Diseases, lncRNAs, and miRNAs (WKNNP)
Let , , and denote the set of diseases, lncRNAs, and miRNAs, respectively. Let denote the horizontal slice matrix in disease axis of the tensor T, hence, also represents the interaction profile for the disease d_{i}. Let denote the jth lateral slice matrix in lncRNA axis of the tensor T, hence, also represents the interaction profile for lncRNA l_{j}. Let denote the frontal slice matrix in miRNA axis of the tensor T, hence, also denotes the interaction profile for miRNA m_{p}. Then, it is obvious that the values in these three kinds of interaction profiles of any novel diseases, lncRNAs, or miRNAs are all zeros, which may lead to unsatisfactory prediction performance during inferring potential associations between diseases and lncRNAmiRNA pairs. Hence, in this section, we will perform a procedure for the construction of new interaction profiles to address the problem mentioned above. And, in this procedure, for each disease d_{i}, its association with other K nearest known diseases (including at least one experimentally verified association) and corresponding K interaction profiles will be utilized to obtain the following interaction profile:where, are the diseases sorted in descending order based on their similarity to d_{i}, is the weight coefficient, and , which means that a higher weight will be assigned if d_{t} is more similar to d_{i}. The parameter is a decay term with values between 0 and 1. The parameter is a normalization term, and there is .
In the same manner, the new interaction profile for each can be determined as follows:where are the lncRNAs sorted in the descending order based on their similarity to is the weight coefficient, and , which means that a higher weight will be assigned if is more similar to . The parameter is a normalization term, and there is .
Similarly, the new interaction profile for each can be determined as follows:where are the miRNAs sorted in the descending order based on their similarity to is the weight coefficient, and , which means that a higher weight is assigned if m_{t} is more similar to m_{p}. The parameter is a normalization term, and there is .
Thereafter, after combining the above three kinds of tensors , and obtained from different data spaces and replacing with an associated likelihood score, we can update the original adjacency matrix T as follows:where .
2.10. PADLMHOOI
Inspired by the successful application of tensor decomposition in the field of link prediction and the application of nonnegative matrix decomposition methods in inferring diseasemiRNA associations, in this section, we proposed a novel model called PADLMHOOI to predict new associations between diseases and miRNAlncRNA pairs. From above descriptions, it is easy to know that a tensor is a multidimensional array. Currently, the most commonly used tensor decomposition techniques include Tucker decomposition [59], HOSVD [60], and HOOI [61]. In this section, we will perform Tucker decomposition on above constructed tensor T. Assuming , the tucker decomposition aims at finding and core tensor that can solve the following optimization problem:
Hence, based on formula (21), we can further transform this equation to following simple form:, and are the factor matrices, which are usually orthogonal and can be considered as the main component of each mode. R_{1}, R_{2}, and R_{3} are the number of columns () in the factor matrices Z_{1}, Z_{2}, and Z_{3} respectively. The notation denotes nmode product; is the shorthand introduced by Kolda and Gibson [62] (Supplementary File A).
Based on equation (22), the above optimization problem can be solved according to the following steps:
Considering that the derivation forms of Z_{1}, Z_{2}, and Z_{3} are similar, we will only derive the iterative formula of Z_{1} as an example. Firstly, as illustrated in formula (23), the objective function given in formula (22) can be rewritten as a matrix form of T along the first dimension:where is the unfolding of T along the first dimension (Supplementary File A). Assuming that the optimal solution Z_{1} satisfies all the constraints in equation (22), we havewhere denotes the Kronecker product, and moreover, we have
Hence, formula (24) can be regarded as a nonnegative matrix factorization (NMF) form [63]. Then, we can finally obtain the solution of Z_{1} by updating NMF as follows:
Hence, we can finally obtain the factor matrices Z_{2} and Z_{3} in a similar way. Thereafter, while fixing the factor matrices Z_{1}, Z_{2}, and Z_{3}, the objective function in formula (22) can be converted to the following form:where denotes the vectorization of the tensor. And moreover, based on formula (27), the following linear equation can be obtained:
Let , then obviously, formula (28) can also be regarded as a NMF, and thereafter, the core tensor in formula (28) can be obtained as follows [63]:
Based on above formulas, the pseudocode of our prediction model PADLMHOOI based on tensor decomposition can be described as follows: Step 1. Input: T, R_{1}, R_{2}, R_{3}, Z_{1}, Z_{2}, Z_{3}, G, and the convergence threshold . Step 2. Repeat For i = 1 to 3: Update Z_{i} according to formula (26) End For Update G according to formula (30) Until Step 3. Return Z_{1}, Z_{2}, Z_{3}, G
According to above steps, we can obtain the final predicted diseaselncRNAmiRNA association tensor , and after prioritizing the diseaserelated lncRNAmiRNA pairs based on the entities in the tensor , obviously, the topranked lncRNAmiRNA pairs can be regarded as more likely to be related to the corresponding disease.
3. Results and Analysis
3.1. LeaveOneOut CrossValidation (LOOCV)
In order to estimate the prediction performance of our newly proposed prediction model, the global leaveoneout crossvalidation (LOOCV), 2fold crossvalidation (2fold CV), and 10fold crossvalidation (10fold CV) were implemented on PADLMHOOI, respectively. In the Kfold crossvalidation, the initial sample will be divided into K subsample sets, and a single subsample set is retained as the data for the validation model, while the other K − 1 samples are used to train the model. During simulation, the crossvalidation will be performed K times, and each subsampling set will be verified once, and the average results of K times will be utilized to obtain a single estimation. Moreover, in order to reduce the performance deviation caused by the random sample partitioning, we divide the partition 100 times and then obtain the ROC curve and the AUC value in the same way as the LOOCV. And, as a result, from the following Table 1, it is easy to see that PADLMHOOI can achieve reliable AUCs of 0.9545, 0.9730 ± 0.0119, and 0.9626 ± 0.0150 in the frameworks of global LOOCV, 2fold CV, and 10fold CV, respectively. Additionally, in order to further estimate the prediction performance of PADLMHOOI, we implemented it under the framework of local LOOCV, and the simulation results of 50 predicted related diseases were illustrated in Supplementary Table 4.

3.2. Performance Comparison with Other Methods
To the best of our knowledge, up to now, PADLMP [39] is the unique model having been proposed for predicting potential associations between disease and lncRNAmiRNA pairs, in which, these three kinds of nodes such as disease nodes, lncRNA nodes, and miRNA nodes are considered simultaneously to construct a triple network. And, the major difference between PADLMP and our model PADLMHOOI is that PADLMP is based on the method of link prediction. Therefore, in order to compare PADLMP with our model PADLMHOOI, we implemented LOOCV to verify the prediction performance of these two models based on the 3047 known diseaselncRNAmiRNA associations downloaded above. In the first experiment, we set the parameters in PADLMP to their best values; specifically, the step size K is set to 2 and the attenuation coefficient is set to 0.01. Meanwhile, for convenience, we set the parameters in PADLMHOOI as follows: the parameters a_{1}, a_{2}, and a_{3} in formula (20) are all set to 1, the parameters r_{1}, r_{2}, and r_{3} in formula (21) are all set to 5, and the parameters K and in formulas (17)–(19) are all set to 3 and 0.1 separately. And, as illustrated in Figure 2, it is easy to see that PADLMHOOI and PADLMP can achieve the AUCs of 0.9545 and 0.9318 separately, which demonstrate that the prediction performance of PADLMHOOI is superior to that of PADLMP.
As time went by, we found that some databases have been updated. Hence, in order to further demonstrate the advancement of PADLMHOOI, we once again collected the latest diseaselncRNA correlations from the databases lnc2cancer v2.0, lncRNADisease 2.0 [64], and MNDR v2.0 [48], collected the latest diseasemiRNA associations from the database HMDD v3.0, and collected the latest lncRNAmiRNA associations from the database RAID v2.0 [65] separately. And thereafter, we reconstructed the triple network based on these newly collected latest datasets. In the newly constructed triple network, the numbers of disease nodes, lncRNA nodes, and miRNA nodes are 42, 234, and 251 respectively; the number of known associations between diseases and lncRNAmiRNA pairs is 3,768; the number of known associations between diseases and lncRNAs is 733; and the number of known associations between diseases and miRNAs is 674. Then, based on the new triple network, we compared our model PADLMHOOI with PADLMP once more. And, in this second experiment, we set the parameters K and to 10 and 0.5, respectively, in PADLMHOOI and kept other parameters unchanged as in the first experiment. And, as illustrated in Figure 3, simulation results show that PADLMHOOI and PADLMP can achieve AUCs of 0.9026 and 0.9013, respectively, which demonstrate that the prediction performance of PADLMHOOI outperforms that of PADLMP markedly.
Additionally, the interesting point is that our model can infer potential diseaselncRNA associations and diseasemiRNA associations incidentally, while predicting potential associations between diseases and lncRNAmiRNA pairs. Hence, it is reasonable as well to compare our model PADLMHOOI with prediction models for inferring potential diseaselnRNA or diseasemiRNA associations. Therefore, in this section, we would compare PADLMHOOI with some stateoftheart computational prediction models such as the LRLSLDA [26], NBCLAD [25], WBSMDA [66], and RLSMDA [67]. Among them, LRLSLDA is a semisupervised learningbased prediction model for inferring potential lncRNAdisease associations; NBCLAD is a probabilistic model for predicting potential associations between diseases and lncRNAs; WBSMDA is a prediction model for predicting potential associations between diseases and miRNAs; and RLSMDA is a prediction model for predicting diseaserelated miRNAs based on the framework of regularized least squares. In addition, while comparing with LRSLDA, known diseaselncRNA associations were obtained from the triple diseaselncRNAmiRNA network; however, the parameters in LRSLDA are set to the same values given in the literature. Moreover, while comparing with NBCLDA, considering that there are four kinds of nodes such as diseases, lncRNAs, miRNAs, and genes included in NBCLDA, there are three kinds of nodes such as diseases, lncRNAs, and miRNAs in our model PADLMHOOI. Hence, for the sake of fairness, we only compared PADLMHOOI with the submethod NBCLDAGN1SD. And, as illustrated in Figure 4, simulation results show that PADLMHOOI, NBCLDAG1SD, and LRSLDA can achieve AUCs of 0.9568, 0.7928, and 0.5924 separately, which demonstrate that PADLMHOOI thoroughly defeats both NBCLDAG1SD and LRSLDA. In addition, while comparing with WBSMDA and RLSMDA, 674 known diseasemiRNA associations were obtained from the triple diseaselncRNAmiRNA network; however, the parameters in both WBSMDA and RLSMDA are set to the same values given in the literatures. And, as illustrated in Figure 5, simulation results show that PADLMHOOI, WBSMDA, and RLSMDA can achieve AUCs of 0.9157, 0.8544, and 0.8991, respectively, which demonstrate that PADLMHOOI outperforms both WBSMDA and RLSMDA thoroughly as well.
3.3. Recall Ratio Analysis
In this section, in order to further evaluate the prediction performance of PADLMHOOI, we compared the recall value of PADLMHOOI and other stateoftheart models. It is well known that the higher recall ratio of all selected diseases in a top k ranking list means that the more positive testing samples (real diseaserelated lncRNAmiRNA pairs) have been identified successfully. And, as a result, Figure 6 illustrates the recall rate of all selected diseases in different top k ranking lists. Moreover, we further listed the recall rate of some given diseases associated with at least 80 verified lncRNAmiRNA associations in Supplementary Table 5.
3.4. Case Studies
In this section, case studies of breast neoplasms, colon neoplasms, and prostate neoplasms were conducted to further verify the capability of PADLMHOOI to detect novel associations between diseases and lncRNAmiRNA pairs separately. And, among these three kinds of case studies, breast cancer is the second leading cause of female cancer death and comprises 22% of all cancers in women [68, 69]. The related literature has suggested that lncRNAs and miRNAs play an important role in the formation of many diseases, and the formation of breast cancer may be more relevant to them [70, 71]. Predicting breast cancerassociated lncRNAmiRNA pairs and identifying lncRNAs and miRNAs as biomarkers may make a significant contribution to better diagnosis and treatment of breast cancer [71]. In Supplementary Table 6, we have listed the top 30 candidate lncRNAmiRNA pairs related to breast cancer. And, in Supplementary Table 6, the column of lncRID and miRID denotes lncRNA ID and miRNA ID, respectively. Evi1 and Evi2 denote some authority database or published literature containing verified diseaselncRNA or diseasemiRNA associations separately. “#” and “∗” stand for databases of lncRNADisease and MNDR v2.0, respectively, which consist of known diseaselncRNA associations or contain published literatures to support the association between predicted lncRNAs and breast cancer. “!,” “&,” and “+” stand for databases of HMDD, miR2Disease, and miRCancer, respectively, which consist of known diseasemiRNA associations or contain published literature to support the association between predicted miRNAs and breast cancer. Particularly, “Nan” indicates that there is no database or no published literature to support the predicted results. From Supplementary Table 6, it is easy to see that all candidate diseaselncRNA associations have been verified in databases of the lncRNADisease and MNDR v2.0 or published papers containing these databases. And, in addition, there are 42 out of 50 candidate diseasemiRNA associations having been reported by HMDD, miR2Disease, and miRCancer or published paper containing these databases. Moreover, we discovered that those novel miRNAs with miRID 35, 51, 73, 164, and 186 are related to some important factors affecting the development of breast neoplasms. Hence, it is obvious that we infer that these lncRNAmiRNA pairs may be associated with breast cancer.
In addition, colonic tumors are a type of malignancy that is common in the rectum and sigmoid borders [72]. Early colon cancer is difficult to detect because of its insignificant symptoms [73]. Unfortunately, the related literature reports that its incidence has been on the rise in recent years [74]. Therefore, predicting potential miRNAs and lncRNAs associated with colon tumors is of great significance for the diagnosis of early colon cancer. In Supplementary Table 7, we have listed the top 30 candidate lncRNAmiRNA pairs predicted to be associated with colon tumors. Moreover, all of these candidate lncRNAs and most of these candidate miRNAs have been verified by lncRNADisease database and MNDR v2.0, respectively.
Moreover, prostate neoplasm is one of the most common cancers in white and AfricanAmerican men, and it is reported that there are about one in six white men and one in five AfricanAmerican men having prostate cancer in their lifetime. Recent researches have shown that prostate neoplasm is caused by the malignancy of prostate epithelial cells [75], its formation includes many factors such as age, family history, and race [76], and particularly, some miRNAs such as haslet7a5p and lncRNAs such as XIST have been found to be involved in the formation of prostate neoplasms successively. Hence, it is interesting to infer potential miRNAs and lncRNAs associated with prostate neoplasms. In Supplementary Table 8, we have listed the top 30 prostate neoplasmrelated candidate lncRNAmiRNA pairs. Moreover, all of these candidate lncRNAs and most of these candidate miRNAs have been verified by lncRNADisease and MNDR v2.0, respectively.
3.5. Parameter Sensitivity Analysis
Considering that there are some key parameters such as K and , which may be significant to the performance of our prediction model PADLMHOOI, in this section, we will further estimate the effects of these key parameters to the prediction performance of PADLMHOOI. Firstly, we varied K from 1 to 10 during simulation. And, as a result, Table 2 illustrates the impacts of parameter K on the performance of PADLMHOOI. By observing Table 2, it is obvious that PADLMHOOI can achieve the maximum AUC value of 0.9708 while K = 8. And additionally, as for the impacts of the parameter , considering the time costs, we set K = 3 and varied from 0.1 to 0.9 during simulation. And as a result, Table 3 illustrates the impacts of parameter on the performance of PADLMHOOI. By observing Table 3, it is obvious that PADLMHOOI can achieve the maximum AUC value of 0.9591 while = 0.7.


4. Discussion and Conclusion
Researches on prediction of potential associations between lncRNAmiRNA pairs and diseases not only are helpful in understanding the disease mechanisms on lncRNA and miRNA levels but also play an important role in the detection of disease biomarkers, diagnosis, prognosis, and prevention. However, to our knowledge, although there are many researches having demonstrated that lncRNAmiRNA interactions are associated with the development of complex diseases, up to now, there are few models having been proposed for largescale forecasting potential associations between diseases and lncRNAmiRNA pairs. Since traditional biological experiments are quite expensive and timeconsuming, in this paper, based on the existing diseasemiRNA associations, diseaselncRNA associations, lncRNAmiRNA interactions, and the assumption that genes with similar functions are often associated with similar diseases; we firstly constructed a threeorder tensor T by adopting the method of WKNNP, and then, based on the method of tensor factorization, we further proposed a prediction model called PADLMHOOI to infer potential relations between diseases and lncRNAmiRNA pairs. And thereafter, simulation results under the frameworks of global and local LOOCV, 2fold CV, and 10fold CV, all confirmed the superiority of PADLMHOOI. Moreover, case studies of breast neoplasms, colon neoplasms, and prostate neoplasms further demonstrate that our model PADLMHOOI is an effective method for predicting potential diseaseassociated lncRNAmiRNA pairs. Certainly, there are still some limitations in PADLMHOOI. For example, although a large number of datasets have been integrated in PADLMHOOI, the amount of data available is still not enough; it is obvious that the prediction performance of PADLMHOOI will be better if more datasets can be collected. And in addition, in this paper, we only predicted the association between disease and a single lncRNAmiRNA pair. In the future, we will further modify PADLMHOOI to predict potential associations between diseases and multiple lncRNAmiRNA pairs.
Abbreviations
PADLMHOOI:  Prediction of potential associations between diseases and lncRNAmiRNA pairs based on the higherorder orthogonal iteration. 
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The project is partly sponsored by the National Natural Science Foundation of China (Nos. 61873221 and 61672447), the Natural Science Foundation of Hunan Province (Nos. 2018JJ4058 and 2017JJ5036), and the CERNET Next Generation Internet Technology Innovation Project (Nos. NGII20160305 and NGII20170109).
Supplementary Materials
Supplementary 1. File 1: introduction to tensor and optimization of objective function and update of factor matrix and core tensor.
Supplementary 2. Table 1: 2048 known diseaselncRNA associations.
Supplementary 3. Table 2: 4041 known diseasemiRNA associations.
Supplementary 4. Table 3: 20324 known lncRNAmiRNA associations.
Supplementary 5. Table 4: AUC score for 50 diseases in the framework of local LOOCV.
Supplementary 6. Table 5: recall ratio of some important diseases.
Supplementary 7. Table 6: the candidate lncRNAmiRNA pairs associated with breast cancer. In addition, the LncRNADisease and MNDR v2.0 databases have confirmed that these lncRNAs or miRNAs are associated with breast cancer.
Supplementary 8. Table 7: the candidate lncRNAmiRNA pairs associated with colon cancer. In addition, the LncRNADisease and MNDR v2.0 databases have confirmed that these lncRNAs or miRNAs are associated with colon cancer.
Supplementary 9. Table 8: the candidate lncRNAmiRNA pairs associated with pprostate cancer. In addition, the LncRNADisease and MNDR v2.0 databases have confirmed that these lncRNAs or miRNAs are associated with colon cancer.
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