Prediction of lncRNA-disease associations via an embedding learning HOPE in heterogeneous information networks

Abstract

Uncovering additional long non-coding RNA (lncRNA)-disease associations has become increasingly important for developing treatments for complex human diseases. Identification of lncRNA biomarkers and lncRNA-disease associations is central to diagnoses and treatment. However, traditional experimental methods are expensive and time-consuming. Enormous amounts of data present in public biological databases are available for computational methods used to predict lncRNA-disease associations. In this study, we propose a novel computational method to predict lncRNA-disease associations. More specifically, a heterogeneous network is first constructed by integrating the associations among microRNA (miRNA), lncRNA, protein, drug, and disease, Second, high-order proximity preserved embedding (HOPE) was used to embed nodes into a network. Finally, the rotation forest classifier was adopted to train the prediction model. In the 5-fold cross-validation experiment, the area under the curve (AUC) of our method achieved 0.8328 ± 0.0236. We compare it with the other four classifiers, in which the proposed method remarkably outperformed other comparison methods. Otherwise, we constructed three case studies for three excess death rate cancers, respectively. The results show that 9 (lung cancer, gastric cancer, and hepatocellular carcinomas) out of the top 15 predicted disease-related lncRNAs were confirmed by our method. In conclusion, our method could predict the unknown lncRNA-disease associations effectively.

Graphical Abstract

This article proposed a novel computational method to predict the potential associations between lncRNA and disease. This method extracts the information through the heterogeneous network, which is constructed by the information of serial biomolecular, drug, and disease. Then, the rotation forest was used as classifier to predict potential lncRNA-disease associations.

Introduction

RNA has been reported to play an intermediary role when the encoded protein is translated from DNA sequences.¹ Only 2% of the Human genome encodes protein, and the remaining of 98% is known as non-coding RNAs (ncRNAs).² Long non-coding RNA (lncRNA), which belongs to the heterogeneous class of ncRNAs, is an ncRNA with non-protein-coding transcripts longer than 200 nucleotides.³ Several previous works show that lncRNAs play a significant role in many biological processes, such as immune responses and chromosome, dynamic circuitry controlling pluripotency, and differentiation. Furthermore, lncRNA also play important role in many complex diseases, such as malignancies including lung cancer,⁴ hepatocellular cancer,⁵ and prostate cancer.⁶ For instance, the upregulation of DANCR is an essential factor in the development of lung cancer, especially in high-grade lung cancer tissues. The proliferation and colony formation of lung cancer is induced by ectopic DANCR expression. Thus, DANCR upregulation is an indicator of aggressive lung cancer. Another example is the HOTAIR, which is expressed about 100 to 2,000 times at the normal levels in breast cancer metastases.⁷ A similar association is found in other cancer types, including liver and gastric cancer. Therefore, finding an efficient way to predict additional associations among lncRNA and disease is a challenge for future progress.

In recent years, an increasing number of machine learning and data mining methods have been developed to take advantage of biological databases on large-scale lncRNA-disease associations. Blom et al.⁸ put forward a supervised machine learning for predicting gene-disease associations, in which a biased support vector machine is utilized. A random walk model with restart walking was proposed by Chen et al.⁹ to rank the microRNAs (miRNAs)-disease associations. However, these methods only recover a portion of all features contained in the databases. Currently, enormous datasets are populated using advanced technologies. A thorough understanding of the functions and mechanisms of lncRNAs will require a complete analysis of all this information.¹⁰ In addition, NONCODE provides a systematic platform including expression profiles and functions.¹¹ Thus, constructing a network among different molecular nodes simultaneously and systemically is beneficial to capture the complicated relationships between ncRNA and diseases.

Xiong et al.¹² constructed a heterogeneous biological network named HeteWalk, which includes miRNAs, genes, and diseases. You et al.¹³ developed a pathway method to construct the network through miRNA-miRNA, disease-disease similarities, and associations between miRNAs and diseases. Chen et al.¹⁴ proposed a network method, hyper-geometric distribution for lncRNA-disease association inference (HGLDA), in which the information of associations among miRNA and lncRNA, miRNA and disease is used to predict associations between lncRNA and miRNA. Sun et al.¹⁵ proposed a global network computational framework, RWRlncD, to identify possible associations between lncRNAs and diseases.

Previous efforts to capture attributes of proteins and ncRNAs and associations among these nodes form a substantive basis for additional work. The existing network graph method uses the associations among ncRNAs, proteins, and diseases. A more holistic and systematic heterogeneous network method would use a combination of known associations and attributes of all nodes in this existing network. There have several proposed methods focus on learning latent features from networks with multi-source features. He et al.¹⁶ proposed CCPMVFGC, which can discover clusters in graphs with multi-view vertex feature. For utilizing the constructed network, the network embedding method is introduced to our work to extract discriminative features from network topology. This method is popular for identifying potential relationships in social networks. This method allows for the preservation of vertex content, extra information, and topological structure, a low-dimensional feature space by mapping network data. In the low-dimensional vector space, each node in the network is mapped to a point, and connections between nodes are positively correlated to distances in the vector space.

The first challenge for constructing a heterogeneous network is choosing appropriate entities such as ncRNA and disease, along with other related items. Association among nodes can then be depicted concisely and clearly. A further challenge is identifying an efficient graph embedding algorithm. To address these issues, a heterogeneous network is constructed and analyzed by high-order proximity preserved embedding (HOPE) to identify potential associations among lncRNAs and diseases. First, a heterogeneous network is constructed by integrating the associations among miRNA, lncRNA, protein, disease, and drug. Second, each node in the network is represented as a vector by combining attribute feature of the node itself (e.g., sequences of ncRNAs and proteins, semantics of diseases and molecular fingerprints of drugs) and the behavior feature of the node in the complex network (associations with other nodes). Finally, the rotation forest model is chosen as the classifier for predicting new associations between lncRNA and disease. Furthermore, the proposed method was evaluated in lung, colorectal, and breast cancer. The experimental results demonstrate that the proposed method can quantitively identify the potential associations between lncRNA and disease.

Results

Performance evaluation measures

The k-fold cross-validation is a popular procedure in most algorithms designed for classifying two input categories of items or for comparing performance on a single dataset. The parameter k is set as 5 in our experiment. Specifically, a dataset is divided randomly into 5 disjointed folds of equal size. Each fold is then used in turns to test the model trained by the classifiers from the other 4 folds. The performance of classifiers is evaluated using the average of 5 values from 5-fold cross-validation. In practice, some nodes without association and nodes contribute simultaneously are isolated. Even though, the manual experiment of researchers can be simulated by this situation better. Manual assessment may address such a situation more efficiently.

For a comprehensive performance evaluation, a series of broader evaluation criteria are introduced to evaluate, including accuracy (Acc.), sensitivity (Sen.), specificity (Spec.), precision (Prec.), Matthews correlation coefficient (MCC), and area under the curve (AUC). The equations of accuracy (Acc.), sensitivity (Sen.), specificity (Spec.), precision (Prec.), and Matthews correlation coefficient (MCC) were listed as follows:

$$\begin{array}{l}Acc.=\frac{TP+TN}{TP+TN+FP+FN}\hfill \end{array}$$ (Equation 1)

$$\begin{array}{l}Sen.=\frac{TP}{TP+FP}\hfill \end{array}$$ (Equation 2)

$$\begin{array}{l}Spec.=\frac{TN}{TN+FN}\hfill \end{array}$$ (Equation 3)

$$\begin{array}{l}Prec.=\frac{TP}{TP+FP}\hfill \end{array}$$ (Equation 4)

$$\begin{array}{l}MCC=\frac{TP\times TN-FP\times FN}{\sqrt{\left(TP+FP\right)\left(TP-FN\right)\left(TN+FP\right)\left(TN+FN\right)}}\hfill \end{array}$$ (Equation 5)

While in these equations, TP, TN, FP, FN, mean true positive, true negative, false positive, and false negative, respectively. In addition, the proposed model is evaluated using two visualization methods. First method uses the AUC, which is the area surrounded by the Receiver characteristic curve (ROC) in a coordinate system whose abscissa is the false positive rate (FPR) and the ordinate is the true positive rate (TPR). For another perspective, the area under the Precision-Recall curve is also applied. The abscissa for the graph is recall and the ordinate is precision. Average AUC and AUPR for performance of the proposed model are both positive correlations. The experimental results shown in Figure 1 and Figure 2 illustrate that our model is acceptable.

Figure 1.

The ROC curves of our methods

The AUC is the area under the receiver operating characteristic curves (ROC).

Figure 2.

The PR curves of our methods

The AUPR is the area under the precision-recall (PR) curves.

Comparison with different types of features

In the constructed network, each node can be represented by both attribute and behavior features. These two kinds of information can be combined. A comparison experiment is designed for assessing attribute and behavior information separately and in combination. In Table 1, the results of the average of Acc., Sen., Spec., Prec., MCC, and AUC is 78.69%, 75.06%, 82.32%, 80.93%, 57.54%, and 83.28%, respectively. These average values represent the ability of the proposed model to predict random associations in the constructed network.

Table 1.

5-fold cross-validation results of our method

Fold	Acc. (%)	Sen. (%)	Spec. (%)	Prec. (%)	MCC (%)	AUC (%)
0	78.87	76.49	81.25	80.31	57.8	84.86
1	75.15	70.83	79.46	77.52	50.49	79.63
2	81.55	77.68	85.42	84.19	63.28	85.78
3	78.12	74.40	81.85	80.39	56.41	82.84
4	79.76	75.89	83.63	82.26	59.70	83.28
Average	78.69 ± 2.36	75.06 ± 2.64	82.32 ± 2.28	80.93 ± 2.48	57.54 ± 4.71	83.28 ± 2.36

The experimental results are shown in Table 2 and Figures 3 and 4. Both AUPR and AUC performance is improved when attribute and behavior information are combined. The AUC is 71.92% and 81.88% when attribute and behavior information are used separately, but it improves to 83.28% combined.

Table 2.

Comparison of different features, respectively and simultaneously

Feature	Acc. (%)	Sen. (%)	Spec. (%)	Prec. (%)	MCC (%)	AUC (%)
Attribute	71.85 ± 1.55	66.79 ± 3.37	76.90 ± 1.63	74.31 ± 1.27	43.94 ± 2.97	71.92 ± 2.09
Behavior	79.16 ± 2.88	71.37 ± 3.11	86.96 ± 2.97	84.57 ± 3.41	59.06 ± 5.81	81.88 ± 2.61
Both	78.69 ± 2.36	75.06 ± 2.64	82.32 ± 2.28	80.93 ± 2.48	57.54 ± 4.71	83.28 ± 2.36

Figure 3.

Comparison of different features under 5-fold cross-validation, respectively

Figure 4.

Comparison of different features under 5-fold cross-validation simultaneously

Comparison with other methods

The rotation forest model is chosen as the classifier for predicting new lncRNA-disease associations. To show the outstanding performance of this classifier, several classifiers, including Random Forest, Decision Tree, Gradient Boost, and Naive Bayes were initially compared. Parameters were set as the default unless otherwise indicated. The experimental results are shown in Table 3. It can be observed that the prediction performance of rotation forest model is better than Random Forest due to its use of PCA. It can also see that Bagging is more suitable than boosting in this situation and an ensemble structure of “trees “outperforms a series of “weak” classifiers.

Table 3.

Comparison of different classifiers

Classifier	Acc. (%)	Sen. (%)	Spec. (%)	Prec. (%)	MCC (%)	AUC (%)
Decision tree	73.57 ± 1.74	66.07 ± 2.69	81.07 ± 2.39	77.77 ± 2.23	47.71 ± 3.51	73.55 ± 1.79
GDBT	78.69 ± 1.66	68.21 ± 2.42	89.17 ± 1.24	86.29 ± 1.68	58.69 ± 3.24	81.27 ± 1.30
Naive bayes	68.69 ± 2.66	48.16 ± 3.33	89.23 ± 2.25	81.67 ± 4.05	40.99 ± 5.64	79.02 ± 2.45
Random forest	78.75 ± 2.43	68.75 ± 2.68	88.75 ± 2.47	85.95 ± 3.05	58.69 ± 4.93	81.35 ± 1.85
Rotation forest	78.69 ± 2.36	75.06 ± 2.64	82.32 ± 2.28	80.93 ± 2.48	57.54 ± 4.71	83.28 ± 2.36

Case study

After running the proposed method through 5-fold cross-validation, a case study was introduced to evaluate the overall performance. Three types of cancer with high mortality rates including lung, breast, and colorectal cancer were chosen for the test. This case study was designed as follows. Known associations were removed before datasets were embedded using HOPE. Before ranking results, rest lncRNA-disease associations were trained. As a result, 9 of the top 15 lung cancer associations, 9 of the top 15 gastric cancer associations, and 9 of the top 15 hepatocellular carcinoma associations were validated in lnc2cancer v2.0.

The three cancer types were chosen for their high mortality and incidence rates. More than 1.3 million patients are estimated to die of lung cancer annually,¹⁷ with high incidence in China.¹⁸ Even in Europe, the 5-year survival rate is only approximately 10%. The high rate of treatment failure correlates with metastatic disease at diagnosis. Earlier diagnosis during the stage where surgery is a viable treatment increases the survival rate to more than 70%. Several lncRNAs show obviously higher expression in tumor tissues in patients with small-cell lung cancer (SCLC) and non-small-cell lung cancer (NSCLC). The expression of lncRNA is both upregulation and downregulation in lung cancer cells. For example, the higher expression of some lncRNA is considered as an indicator for NSCLC. To be more specific, HOTAIR was indicated that there is an upregulated association between this lncRNA and NSCLC. Further, loss of imprinting of H19 can cause lung cancer. Expression of KCNQ1OT1, demonstrated by the knockdown experiment, is correlated to a series of behaviors observed for lung adenocarcinomas, such as positive lymphatic metastasis and larger tumor size. In the NSCLC cases, XIST is recognized as an oncogenic lncRNA in driving tumorigenesis is overexpressed in these cells.

Even though there still exists much geographical variation, especially in East Asia and China, gastric cancer is the second most frequent cause of cancer-related deaths worldwide. Most patients with gastric cancer are diagnosed at a late stage of the disease, and mortality can be reduced by early detection. Some specific risk profiles, including the presence of precursor lesions and gene polymorphisms, should be considered while deciding on the diagnosis and treatment. Overexpression of H19 is known to promote proliferation, invasion, and other features of gastric cancer.¹⁹ The expression of ANRIL, measured by qPCR, is correlated with a higher TNM stage and larger tumor size. These features are independent predictors for overall survival.²⁰ Downregulation associated with MEG3 is correlated with maternal expression of gene 3 and gastric cancer. Under-expression of MEG3 in gastric tissues is correlated with poor prognosis.²¹ Compared to patients with lower levels of NEAT1, patients with higher levels show poor survival. Associations between NEAT1 and gastric cancer include upregulation and overexpression of this lncRNA as an independent prognostic factor in these gastric cases.²²

Hepatocellular carcinoma has become the fifth most common cancer and the third most common cause of cancer mortality worldwide in recent years. Mortality from this cancer increased by 25% between 2002 and 2012 and reached an incidence of 3.1 per 100,000 for men. In Asia, the incidence of liver cancer is expected to decrease. In Japan, the incidence was predicted to decrease to 5.4 per 10,000 for men in 2020. Except in East Asia, younger individuals from most regions show a more encouraging trend in the incidence of liver cancer than middle-aged individuals. It was also common that the mortality among women is lower than for men by 3-fold to 5-fold. H19 and IGF2 are regulated in parallel in the hepatocellular carcinomas. A common feature of this disease is the disruption of IGF2 prompter regulation.²³ The tumor suppressor candidate 7 (TUSC7) suppresses epithelial-to-mesenchymal transformation through TUSC7-miR-10a-EphA4 axis, which is probably a potential target for treatment in hepatocellular carcinoma.²⁴ The correct prediction of case study was shown in Table 4.

Table 4.

Marked lncRNAs associations between lung cancer, gastric cancer, and hepatocellular carcinomas

lncRNA	Disease	Rank
HOTAIR	lung cancer	3.10
H19	lung cancer	6.7
KCNQ1OT1	lung cancer	8
MEG3	lung cancer	12
UCA1	lung cancer	13
XIST	lung cancer	14
linc-ROR	lung cancer	15
H19	gastric cancer	1
MEG3	gastric cancer	3
HOTAIR	gastric cancer	6
NEAT1	gastric cancer	7
XIST	gastric cancer	8
UCA1	gastric cancer	9
ANRIL	gastric cancer	10
CASC2	gastric cancer	12
link-ROR	gastric cancer	13
H19	hepatocellular carcinomas	1.7
HOTAIR	hepatocellular carcinomas	2.3
ANRIL	hepatocellular carcinomas	5
IGF2-AS	hepatocellular carcinomas	9
MEG3	hepatocellular carcinomas	12
TUSC7	hepatocellular carcinomas	13
NEAT1	hepatocellular carcinomas	15

Discussion

Recently, experimental technologies have proved plenty of known associations between lncRNA and disease. These associations play a crucial role in the prevention and treatment in severe disease, especially neoplasm. Due to the expensive and inefficient traditional experimental methods, we proposed this computational method based on known dataset and data mining method, for which the potential associations can be discovered.

This method has several advantages as listed. (1) The full use of biology information. The node representation was combined by not only the information of molecular function such as, lncRNA and miRNA molecular function, protein sequence, and semantic of disease, but also, the associations and interactions among the miRNA, lncRNA, protein, disease, and drug. (2) The HOPE was used to embed the network and present the node behavior. This method can preserve the feature information in graph network. (3) The rotation forest was used to train the model and predict the potential associations. The reason we chose these classifiers was for the following purposes: (1) for random forest classifier, principal-component analysis (PCA) was applied to each feature set to preserve variability information. The same number of “trees” was set at 45 for comparisons with rotation forest model. (2) Gradient boosting is a member of a family of classifiers; boosting uses an ensemble structure to boost some “weak” classifiers. (3) Decision tree model is a “weak” classifier chosen in the rotation forest because of its sensitivity to rotation of feature axes. (4) Naive Bayes is a popular “weak” classifier used when features are independent. The results of comparison with other classifiers shows that the process of PCA bring an outstanding performance.

Materials and methods

Construction of the molecular association network

Several databases that include known associations among miRNA, lncRNA, protein, disease, and drugs were used to construct the systematic and holistic molecular association network. The data used in the final model were selected by unifying identifiers, simplifying, eliminating redundancy, and deleting the extraneous items, which is shown in the Table 5. Divergent nodes were allocated separately from the Table 6 and concise amounts are provided in Table 6. The construction of the network is shown in Figure 5.

Table 5.

Nine associations involved in the heterogeneous network

Relationship type	Database	Number of associations
miRNA-lncRNA	lncRNASNP225	8,374
miRNA-disease	HMDD26	16,427
miRNA-protein	miRTarBase27	4,944
lncRNA-disease	lncRNAdisease28	1,680
	lncRNASNP225
	lnc2Cancer10
lncRNA-protein	lncRNA2Target29	690
protein-disease	DisGeNET30	25,087
drug-protein	DrugBank31	11,107
drug-disease	CTD32	18,416
protein-protein	STRING33	19,237
total	N/A	105,963

Table 6.

The number of kinds of nodes in the heterogeneous network

Node	Number of nodes
disease	2,062
lncRNA	769
miRNA	1,023
protein	1,649
drug	1,025
total	6,528

Figure 5.

The network constructed by the multiple associations among different biomolecules

Representing sequences of lncRNA

To obtain the discriminative attribute feature, we downloaded sequences of lncRNA from Database: NONCODE.¹¹ For simplifying our experiment, the sequences of lncRNA were encoded into a 64 (4 × 4 × 4) dimensional vector, where each dimension represents the normalized frequency of the corresponding 3-mer that was used in the lncRNA sequence (e.g., GAC, CUG, UGA).

MeSH for disease descriptions and directed acyclic graph

Medical Subject Headings (MeSH) are developed by the National Library of Medicine.³⁴ An effective “tree structure” was introduced to automatically retrieve these descriptions. The MeSH tree structures are poly-hierarchical, that is, one main heading can be found in more than one subcategory. For instance, for the heading, Gallbladder Neoplasms, Neoplasms is a disease listed under Neoplasms and under Digestive System Disease. Within this structure, the most search is accurate and objective. The tree can be expressed in another form as a directed acyclic graph (DAG). The method for describing the disease in the DAG is as follows:

$$\begin{array}{l}DAG\left(d\right)=\left(d,E\left(d\right),N\left(d\right)\right)\hfill \end{array}$$ (Equation 6)

where N(d) means the points in the set that include all the diseases in the DAG(d). E(d) denotes the edges in the set that include all connections between nodes in the DAG(d). Figure 6 illustrates the description of the disease Astrocytoma.

Figure 6.

The directed acyclic graph of a type of digestive system disease, gastrointestinal neoplasms

For defining semantic similarity among diseases, we used an approach that calculates means of DAG. The following formula reveals how disease $u$ in any ancestral disease contributes to disease $d$.

$$\begin{array}{l}\{\begin{array}{l}{D}_d\left(u\right)=1ifu=d\hfill \\ D_d\left(u\right)=max\left\{\Delta \text{∗}D{1}_d\left(u^{\text{′}}\right)|u^{\text{′}}\in childrenofu\right\}ift\ne d\hfill \end{array}\hfill \end{array}$$ (Equation 7)

where, $\Delta$, a factor that contributes semantically, is set at 0.5 according to previous works. The weight of disease $d$ for itself is 1. The contribution of other nodes to this disease is attenuated and assessed to yield $\Delta$. From Equation (1), the sum of contributions of all ancestral nodes in the graph to $d$ is estimated as follows:

$$\begin{array}{l}DV\left(d\right)=\sum _{u\in N\left(d\right)}{D}_d\left(u\right)\hfill \end{array}$$ (Equation 8)

Moreover, the semantic similarity between the two diseases is calculated using the Jaccard similarity coefficient:

$$\begin{array}{l}S1\left(p,q\right)=\frac{{\sum }_{t\in N\left(p\right)\cap N\left(q\right)}\left(D_p\left(u\right)+D_q\left(u\right)\right)}{DV\left(p\right)+DV\left(q\right)}\hfill \end{array}$$ (Equation 9)

In this equation, p and q are two diseases, which would be calculated to find the similarity. $\sum _{t\in N\left(p\right)\cap N\left(q\right)}\left(D_p\left(u\right)+D_q\left(u\right)\right)$ would calculate all same father node of these two diseases and $DV\left(p\right)+DV\left(q\right)$ would calculate the mean of these two diseases.

Stacked auto-encoder

To avoid the labor-intensive and handcraft feature design, we used Sparse Auto-Encoder (SAE) in our method, which is an unsupervised feature leaning method. The hidden layer is acted as a feature extraction of the input layer. The output and input are equal in SAE finally. More specifically, SAE is an unsupervised feedforward neural network. $K=\left\{n^{\left(1\right)},n^{\left(2\right)},\mathrm{...},n^{\left(m\right)},n^{\left(i\right)}\in R^d\right\}$ is set as unsupervised examples. Encoder function,$m=\text{σ}\left(Wn+p\right)$, maps the input layer n to hidden layer m, while decoder function, $y=\text{σ}\left(Wm+p\right)$, is used to recover y from m. W is the connected parameter between two layers, b is a bias, and $\text{σ}$ is a non-linear mapping. SAE can train a model that can approximate m with y, and the hidden layer is a new representation of data.

The encoder function and decoder are shown as below. The encoder function maps the input layer n to hidden layer m, while decoder function recover n from m. W is the connected parameters between two layers, p and q are bias. $f\left(n\right)$ and $g\left(m\right)$ denote non-linear mapping. SAE is for training a model that approximates x with y. The hidden layer can be considered as feature extraction of the input layer.

$$\begin{array}{l}m=f\left(n\right)S_f\left(Wn+p\right)\hfill \end{array}$$ (Equation 10)

$$\begin{array}{l}y=g\left(m\right)S_g\left(W^{\text{′}}n+q\right)\hfill \end{array}$$ (Equation 11)

Additional constraints are added for avoiding training a trivial identity mapping, which may lead to a compressive representation. It will also add noises to data, in which SAE would recover the data from a corrupted version, and the representation would also be robust. Here, the ReLU is chosen as the function for activation.

$$\begin{array}{l}{S}_f\left(u\right)=S_g\left(u\right)=max\left(0,Wu+b\right)\hfill \end{array}$$ (Equation 12)

Node representation

Two categories of information are used for representing nodes in the network. First, intrinsic attributes of nodes including sequences of miRNA, lncRNA, and protein, fingerprints of drugs, and semantic disease descriptions are used. Second, associations between nodes are defined as the behavior of nodes in the molecular network. Node behavior was represented using HOPE on the entire network.³⁵

Recently, undirected graphs have been targeted by graph embedding methods. However, the method used for undirected graphs cannot be used for the directed graphs because of asymmetric transitivity, which is a fundamental characteristic of these graphs. HOPE preserves asymmetric transitivity in directed graphs during embedding learning. In this study, the loss function is minimized as follows:

$$\begin{array}{l}\min {\Vert P-Q^M\cdot {Q^t}^{\top }\Vert }_F^2\hfill \end{array}$$ (Equation 13)

In this formula, asymmetric transitivity can be affected by several high-order proximities. However, the approximation of proximities is facilitated by the general formula as follows:

$$\begin{array}{l}Q=M_g^{-1}\cdot M_l\hfill \end{array}$$ (Equation 14)

Further, $K_g$ and $K_l$ are both polynomial matrices. A series of proximity measurements can be part of deduction in this equation, and all measurements can be classified into two categories. The first category is global proximity. Specifically, global asymmetric transitivity can be preserved with the Katz index and rooted PageRank, because these two measures are derived from a recurrent formula. The second category is local asymmetric transitivity. For instance, asymmetric transitivity only can be preserved in a local structure, because no recurrent structure exists in common neighbors and Adamic-Adar. Thus, an optimal rank-K approximation of the proximity matrix S can be determined by singular value decomposition (SVD). Embedding are constructed as follows:

$$\begin{array}{l}P=\sum _{i=1}^N{\tau }_1{U}_i^P{U}_i^{t\top }\hfill \end{array}$$ (Equation 15)

where $\left\{{\tau }_1,{\tau }_2,\mathrm{...},{\tau }_N\right\}$ are singular values sorted in decreasing order; $U_i^P$ and $U_i^t$ are corresponding singular vectors of ${\tau }_i$. Optimal embedding vectors can be expressed as follows:

$$\begin{array}{l}{V}^P=\left\{\sqrt{{\tau }_1}\cdot U_1^P,\dots ,\sqrt{{\tau }_K}\cdot U_K^P\right\}\hfill \end{array}$$ (Equation 16)

$$\begin{array}{l}{V}^t=\left\{\sqrt{{\tau }_1}\cdot U_1^t,\dots ,\sqrt{{\tau }_K}\cdot U_K^t\right\}\hfill \end{array}$$ (Equation 17)

To avoid calculating this matrix inversion in time complexity $O\left(N^3\right)$, it is necessary to use a previously proposed method to calculate ${\tau }_i$ by the equation below:

$$\begin{array}{l}{\tau }_i=\frac{{\tau }_i^l}{{\tau }_i^g}\hfill \end{array}$$ (Equation 18)

Finally, error bounds of this algorithm are produced as follows:

$$\begin{array}{l}\frac{{\Vert P-V^P\cdot V^t\Vert }_F^2}{{\Vert P\Vert }_F^2}=\frac{{\sum }_{i=K+1}^N{\tau }_i^2}{{\sum }_{i=1}^N{\tau }_i^2}\hfill \end{array}$$ (Equation 19)

Conclusions

An increasing number of associations between lncRNA and disease have been detected by advanced biological techniques. A computational method was are increasingly important for identifying potential associations because detection such associations using biological experiments is expensive. In this study, a novel computational method was developed by constructing and embedding a heterogeneous network using HOPE. The experimental results demonstrated that the proposed method achieved an outstanding performance for predicting lncRNA-disease associations. The proposed method was further validated using three case studies of high mortality cancer types (lung, breast, and colorectal cancer) to demonstrate its predictive power and reliability.