Ranking based ontology scheming using eigenpair computation

Abstract

Ontology, which is regularly used in computer information retrieval and other computer applications, plays an essential role in effectively retrieving the concepts that have highly semantic similarity with the original query concept. Meanwhile, ontology also returns the results to the user. Ontology mapping is used to connect the relationship between different ontologies, and similarity computation is the essence of such applications. In this article, we present a ranking based ontology optimizing algorithm to get the ontology score function which map each ontology vertex to a real number, and the similarity between ontology vertices is determined according to the difference of the scores. The ontology framework is designed relied on the eigenpair computation, and the solution is obtained by means of operator calculation. The result data of simulation experiment implies that our new ontology method has high efficiency and accuracy in ontology similarity measure and ontology mapping in multiple disciplines.

Keywords

1 Introduction

The term “ontology” has its origin in the field of philosophy. It has been used to describe the natural connection of things and the inherently hidden connections of their components. In information and computer science, ontology is a model for knowledge storing and representation. This term has been widely applied in knowledge management, machine learning, information systems, image retrieval, information retrieval search extension, collaboration and intelligent information integration. The past decade has witnessed how ontology, as an effective concept semantic model and a powerful analysis tool has been widely applied in pharmacology science, biology science, medical science, geographic information system and social sciences (for instance, see [1 –5]).

A simple graph is usually used to represent the structure of ontology. Each concept, object or element in ontology is corresponding to a vertex. Each (directed or undirected) edge on an ontology graph represents a relationship (or potential link) between two concepts (objects or elements). Let O be an ontology. Let G be a simple graph corresponding to O. The nature of ontology engineer application falls onto getting the similarity calculating function. Such a function is to compute the similarities between ontology vertices. The intrinsic link between vertices in ontology graph can be represented by these similarities. Ontology mapping aims to get the ontology similarity measuring function through measuring the similarity between vertices from different ontologies. A mapping like this is a bridge between different ontologies. It gets a potential association between the objects or elements from different ontologies. Specifically speaking, the ontology similarity function is a semi-positive score function and it maps each pair of vertices to a non-negative real number.

In recent years, ontology technologies can be found in a variety of applications. A graph derivation representation based technology for stable semantic measurement was presented by Ma et al. [6]. An ontology representation method for online shopping customers knowledge in enterprise information was raised by Li et al. [7]. An innovative ontology matching system that finds complex correspondences by processing expert knowledge from external domain ontologies and in terms of using novel matching tricks was proposed by Santodomingo et al. [8]. Description of the main features of the food ontology and some examples of application for traceability purposes was given by Pizzuti et al. [9]. That ontologies can be used in designing an architecture for monitoring patients at home was aroused in argument by Lasierra et al. [10].

Using ontology learning algorithm which gets an ontology function is an advanced idea in dealing with the ontology similarity computation. The ontology graph is mapped into a line which consists of real numbers through ontology function. Comparing the difference between their corresponding real numbers is used to measure the similarity between two concepts. Dimensionality reduction is the essence of this idea. To associate the ontology function with ontology application, a vector is used to express all its information for vertex v. We slightly confuse the notations and v is used to denote both the ontology vertex and its corresponding vector because we want to facilitate the representation. We map the vector to a real number by ontology function. The ontology function is a dimensionality reduction operator and it maps multi-dimensional vectors into one dimensional vectors.

Several effective methods for getting efficient ontology similarity measure or ontology mapping algorithm in terms of ontology function can be used. The ontology similarity calculation in terms of ranking learning technology was considered by Wang et al. [11]. The fast ontology algorithm in order to cut the time complexity for ontology application was raised by Huang et al. [12]. An ontology optimizing model such that the ontology function is determined by virtue of NDCG measure was presented by Gao and Liang [13]. Meanwhile, it is successfully applied in physics education. Gao and Xu [14] presented ontology similarity measuring and ontology mapping algorithm via MEE (minimum error entropy) criterion. An ontology algorithm by virtue of diffusion and harmonic analysis on hyper-graph framework was inferred by Gao et al. [15]. By using special loss functions, Gao et al. [16] proposed new multi-dividing ontology learning algorithms for similarity measure and ontology mapping construction.

Several articles make contributions to the theoretical analysis of ontology algorithms. Ontology algorithm from a theoretical perspective was studied by Gao et al. [17] based on the regression. Considering the strong and weak stability of k-partite ranking, ontology algorithm was studied by Gao et al. [18]. Gao et al. [19] continued to discuss the ontology algorithm in multi-dividing setting. They pointed out that conditional linear statistical can be used to represent the empirical multi-dividing ontology framework. An approximation result is obtained from the projection method. A gradient learning model for ontology similarity measuring and ontology mapping in multi-dividing setting was raised by Gao and Zhu [20] based on their cooperation. Furthermore, the hypothesis space and the ontology dividing operator determines the sample error in this setting. Based on the assumptions of low noise, the upper bound and lower bound minimax learning rate was obtained by Gao et al. [21]. Other theoretical results can be found in Gao et al. [22].

In recent years, ranking learning algorithm and its applications have raised high attention among scholars. A large number of ranking tricks are introduced for a variety of applications (see [23 –28]). In this paper, we present a new ontology optimization model based on ranking technologies. The solution is calculated in terms of eigenpair computation. At last, the proposed ontology algorithm will be applied in plant science, physical education, biology science and humanoid robotics. The effectiveness of ontology algorithm in these fields proved to be high according to experiment results.

2 Setting

Let V be a compact metric input space (or ontology instance space) and Y = [0, M] be an output space, where M is a fixed positive number. Suppose that ρ is a Borel probability measure on Z = V × Y. Set ρ_V as the marginal distribution on V and its conditional distribution on Y for fixed ontology vertex v is denoted by ρ (· |v). The aim of ranking based ontology algorithm is to obtain a ontology function $f_{z} : V \to ℝ$ in terms of given ontology data (sample) set $z = {(v_{i}, y_{i})}_{i = 1}^{n} \in Z^{n}$ which is drawn independently and identically according to unknown ρ. This ontology function can be regarded as a score function which values future vertices with larger scores higher than those with smaller labels, that is to say, vertex v is to be assigned with a larger value than v′ if f_z (v) > f_z (v′), and a smaller value than v′ if f_z (v) < f_z (v′). For convenience, we say that f_z (v) = f_z (v′) implies there is no difference in ranking order between vertices v and v′. Ontology learning algorithms were used aiming to obtain an optimal ontology function f. In this way, the similarity between two vertices v_i and v_j is determined by the value of |f (v_i) - f (v_j) |.

The punishment of such ranking based ontology function f on a pair of ontology vertex pair (v, v′) with corresponding labels y and y′ can be expressed as the least squares ontology loss $\begin{matrix} l (f, (v, y), (v^{'}, y^{'})) \\ = ((y - y^{'}) - (f (v) - f (v^{'})))^{2}, \end{matrix}$ and we measure the quality of ontology function f by means of its ontology expected error raised by $\begin{matrix} R (f) \\ = \int_{Z} \int_{Z} ((y - y^{'}) - (f (v) - f (v^{'})))^{2} d ρ (v^{'}, y^{'}) . \end{matrix}$

Let $H = {f : f = \underset{f \in L_{ρ_{V}}^{2}}{arg min} R (f)}$ be the set of target ontology functions which are defined to be the real score functions minimizing the error R (f) over $L_{ρ_{V}}^{2}$ . It is not hard to check that any ontology function with the form f + c has the same ontology expected error as that of ontology function f, where $c \in ℝ$ is a fixed constant. From this point of view, different from the target ontology function in regression setting, the ranking based target ontology function is not unique. It can be verified that the regression ontology function of ρ denoted as $f_{ρ} (v) = \int_{Y} yd ρ (y | v), v \in V$ is precisely a minimizer of the ontology expected error R (f). This fact reveals that any ontology function with the form f_ρ + c can be a target ontology function. Furthermore, by virtue of Lemma 11 in [29], we see that any target function in $L_{ρ_{V}}^{2}$ has the form f_ρ + c. This implies that the ontology function set $H = {f_{ρ} (v) + c : c \in ℝ}$ contains ontology functions with the expression f_ρ (v) + c for any constant $c \in ℝ$ .

A Mercer kernel (see [30 –35] for more details) on V is defined as a symmetrically continuous function $K : V \times V \to ℝ$ satisfying that the n × n matrix K with its (i, j)-element K (v_i, v_j) is a positive semi-definite matrix for any ontology data set ${v_{i}}_{i = 1}^{n}$ . Set span{K_v : v ∈ V} as the space spanned by the collection {K_v = K (· , v) : v ∈ V}. Therefore, the inner product in the space s span {K_v : v ∈ V} can be expressed by ${〈 \sum_{i = 1}^{s} α_{i} K_{v_{i}}, \sum_{j = 1}^{t} α_{j} K_{v_{j}} 〉}_{K} = \sum_{i = 1}^{s} \sum_{j = 1}^{t} α_{i} β_{j} K (v_{i}, v_{j}) .$

The reproducing kernel Hilbert space (RKHS) H_K associated with kernel function K was introduced as the completion of span{K_v : v ∈ V} with the norm ∥ · ∥ _K induced by the inner product 〈 · , · 〉 _K. The reproducing property of RKHS H_K can be formulated by f (v) = 〈 f, K_v 〉 _K for any v ∈ V and any f ∈ H_K. By virtue of reproducing property and set $κ = \sqrt{sup_{v, v^{'} \in V} | K (v, v^{'}) |}$ , we infer that ∥f ∥ _∞ ≤ κ ∥ f ∥ _K for all f ∈ H_K.

A kernel associated ontology learning algorithm in ranking setting with the least squares ontology loss function can be stated as follows:

$\begin{matrix} f_{γ}^{z} & = & \underset{f \in H_{K}}{arg min} \frac{1}{n^{2}} \sum_{i, j = 1}^{n} ((y_{i} - y_{j}) - (f (v_{i}) - f (v_{j})))^{2} \\ + γ {∥ f ∥}_{K}^{2}, \end{matrix}$ (1) where γ is a positive balance parameter.

3 Main algorithm

3.1 Framework of ontology computation

Let K be a Mercer kernel on ontology input space V. For any ontology function f ∈ H_K, L_K is the operator associated with K, which is defined by $L_{K} (f) = \int_{V} \int_{V} f (v) (K_{v} - K_{v^{'}}) d ρ_{V} (v) d ρ_{V} (v^{'}) .$

It is easy to deduce that L_K has at most countable eigenvalues. All of these eigenvalues are nonnegative since they are self-adjoint, compact, and positive operators. The eigenvalues {λ_l} (with multiplicities) of L_K can be ranked as a non-increasing sequence with lower bound 0 and associated with the sequence of eigenfunctions {φ_l} which forms the orthonormal basis of reproducing kernel Hilbert space H_K. The unlabeled part of the ontology samples $z = {(v_{i}, y_{i})}_{i = 1}^{n}$ is expressed as $v = {v_{i}}_{i = 1}^{n}$ . For any f ∈ H_K, the operator $L_{K}^{v}$ defined on H_K is denoted by $L_{K}^{v} = \frac{1}{n (n - 1)} \sum_{i, j = 1}^{n} f (v_{i}) (K_{v_{i}} - K_{v_{j}}) .$

The relationship between $L_{K}^{v}$ and L_K can be formulated as $E (L_{K}^{v}) = L_{K}$ , which implies that $L_{K}^{v}$ can be regarded as the empirical version of the operator L_K with respect to v. Clearly, the operator $L_{K}^{v}$ is a self-adjoint, positive operator with $r (L_{K}^{v}) \leq n$ . The eigensystem of $L_{K}^{v}$ can be called as an empirical eigensystem, which is denoted by ${(λ_{l}^{v}, φ_{l}^{v})}$ . It is emphasized here that the eigenvalues ${λ_{l}^{v}}$ are ranked in the non-increasing order, and all the empirical eigenfunctions ${φ_{l}^{v}}$ exactly form an orthonormal basis of reproducing kernel Hilbert space H_K. Furthermore, at most n eigenvalues are nonzero. We can formulate as $λ_{l}^{v} = 0$ for all l > n.

In this paper, our ontology framework is a modified version of (1) based on the computation of the first n empirical eigenfunctions ${φ_{l}^{v}}_{l = 1}^{n}$ in ranking setting which can be stated as

$\begin{matrix} c_{γ}^{z} & = & \underset{c \in ℝ^{n}}{arg min} {\frac{1}{n^{2}} \sum_{i, j = 1}^{n} ((y_{i} - y_{j}) - (\sum_{l = 1}^{n} c_{l} φ_{l}^{v} (v_{i}) \\ - \sum_{l = 1}^{n} c_{l} φ_{l}^{v} (v_{j})))^{2} + γ {∥ c ∥}_{1}} . \end{matrix}$ (2)

According to the representation theory in learning theory, the output ontology function of computation model (2) is denoted by $f_{γ}^{z} = \sum_{l = 1}^{n} c_{γ, l}^{z} φ_{l}^{v}$ .

We should determine the suitable target ontology function approximated by the ontology function $f_{γ}^{z}$ that forms the set of infinitely number of target ontology functions. Set $L_{K}^{r}$ as the rth power of L_K. Our selection strategy is described as follows: suppose that we can search an ontology function ${\tilde{f}}_{ρ} \in H$ to meet that for certain r > 0 and ${\tilde{g}}_{ρ} \in H_{K}$ , ${\tilde{f}}_{ρ} = L_{K}^{r} ({\tilde{g}}_{ρ})$ .

If there exists another ontology function ${\hat{f}}_{ρ} \in H$ satisfying ${\hat{f}}_{ρ} = L_{K}^{r} ({\hat{g}}_{ρ})$ for certain ${\hat{g}}_{ρ} \in H_{K}$ . Then, we get ${\tilde{f}}_{ρ} - {\hat{f}}_{ρ} = L_{K}^{r} ({\tilde{g}}_{ρ} - {\hat{g}}_{ρ}) = c$ for some constant $c \in ℝ$ in view of $H = {f_{ρ} (v) + c : c \in ℝ}$ . We rewrote ${\tilde{g}}_{ρ} - {\hat{g}}_{ρ}$ as ${\tilde{g}}_{ρ} - {\hat{g}}_{ρ} = \sum_{l = 1}^{\infty} c_{l} φ_{l}$ by means of the fact that $L_{K}^{r}$ can be formulated by using {λ_l} and {φ_l} as $L_{K}^{r} (\sum_{l = 1}^{\infty} c_{l} φ_{l}) = \sum_{l = 1}^{\infty} λ_{l}^{r} c_{l} φ_{l}$ . Set $Λ = {l \in ℕ : λ_{l} > 0}$ . Then, we deduce $c = L_{K}^{r} (\sum_{l = 1}^{\infty} c_{l} φ_{l}) = \sum_{l \in Λ} λ_{l}^{r} c_{l} φ_{l}$ and $L_{K} c = \sum_{l \in Λ} λ_{l}^{r + 1} c_{l} φ_{l}$ . Obviously, c_l = 0 for all l ∈ Λ by means of L_K (c) =0. It implies that ${\tilde{g}}_{ρ} - {\hat{g}}_{ρ} = \sum_{l \notin Λ} c_{l} φ_{l}$ and $c = L_{K}^{r} (\sum_{l \notin Λ} c_{l} φ_{l}) = \sum_{l \notin Λ} λ_{l}^{r} c_{l} φ_{l} = 0$ . This reveals the uniqueness of ${\tilde{f}}_{ρ}$ .

It is verified above that there exists only one ontology score function in H satisfying the condition above. Thus, it is reasonable to take ${\tilde{f}}_{ρ}$ as the target ontology function for our ontology framework (2).

3.2 Solution of ontology optimization algorithm

Now, we present the detailed technologies to calculate the empirical eigenpairs and then obtain the solution of our ontology problem (2).

Let I be the n-th identity matrix and $1 = (1, \dots, 1)^{T} \in ℝ^{n}$ . Set $P = I - \frac{1}{n} 11^{T}$ , and we can verify that P is an orthogonal projection matrix such that P ^T = P = P ². Set the sampling operator (this operator has been widely used in learning theory and algorithms, see [36 –40] for more details) $S_{v} : H_{K} \to ℝ^{n}$ as $S_{v} (f) = (f (v_{1}), \dots, f (v_{n}))^{T} .$

Then, $K = S_{v} S_{v}^{T}$ and $L_{K}^{v} = \frac{1}{n - 1} S_{v}^{T} {PS}_{v}$ .

Set B = PKP . It is not hard to check that B is a positive semi-definite matrix. We denote all its eigenvalues as ${\tilde{λ}}_{1}^{v} \geq \dots \geq {\tilde{λ}}_{n}^{v}$ , and the corresponding orthonormal eigenvectors as ${\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{m}$ . Set $r (L_{K}^{v}) = d^{v}$ (the rank of $L_{K}^{v}$ ). Then the empirical eigenpairs ${(λ_{l}^{v}, φ_{l}^{v})}_{l = 1}^{d^{v}}$ of $L_{K}^{v}$ can be computed by means of the eigenpairs of B stated as follows: $λ_{l}^{v} = \frac{{\tilde{λ}}_{l}^{v}}{n - 1}$ and $φ_{l}^{v} = \frac{S_{v}^{T} P {\tilde{x}}_{l}}{\sqrt{{\tilde{λ}}_{l}^{v}}}$ , 1 ≤ l ≤ d^v.

Furthermore, for 1 ≤ l ≤ n, we set $\begin{matrix} S_{l}^{z} \\ = {\begin{matrix} \frac{1}{n^{2} λ_{l}^{v}} \sum_{i, j = 1}^{n} (y_{i} - y_{j}) (φ_{l}^{v} (v_{i}) - φ_{l}^{v} (v_{j})), & if λ_{l}^{v} > 0 \\ 0, & otherwise \end{matrix} . \end{matrix}$

Then, the solution to ontology problem (2) is determined by $\begin{matrix} c_{γ, l}^{z} \\ = {\begin{matrix} 0, & if 2 λ_{l}^{v} | S_{l}^{z} | \leq γ \\ \frac{n}{2 (n - 1)} (S_{l}^{z} - \frac{γ}{2 λ_{l}^{v}}), & if 2 λ_{l}^{v} | S_{l}^{z} | > γ and S_{l}^{z} > \frac{γ}{2 λ_{l}^{v}}, \\ \frac{n}{2 (n - 1)} (S_{l}^{z} + \frac{γ}{2 λ_{l}^{v}}), & if 2 λ_{l}^{v} | S_{l}^{z} | > γ and S_{l}^{z} < - \frac{γ}{2 λ_{l}^{v}}, \end{matrix} . \end{matrix}$

3.3 Description of ontology algorithms

The new ontology learning algorithm can be used in ontology concepts similarity measurement and ontology mapping. The basic idea is: via the computation of ontology framework (2), the ontology graph is mapped into a line consisting of real numbers. The similarity between two concepts then can be measured by comparing the difference between their corresponding real numbers.

Algorithm 1. Ranking based ontology similarity measuring with eigenpair computation:

Step 1. Mathematizing ontology information.

Step 2. By obtaining the solution of ontology algorithm (2), we map the ontology graph to the real line and map the vertices of ontology graph to real numbers.

Step 3. For each v ∈ V (G), we obtain the similar vertices and return the outcomes to the users.

Algorithm 2. Ranking based ontology mapping with eigenpair computation:

Let G₁, G₂, …, G_m be ontology graphs corresponding to ontologies O₁, O₂, …, O_m.

Step 1. Mathematizing ontology information.

Step 2. By obtaining the solution of ontology algorithm (2), we map the ontology graph to the real line and map the vertices of ontology graph to real numbers.

Step 3. For v ∈ V (G_i), where 1 ≤ i ≤ m, we obtain the mapping vertices and return the outcome to the users.

4 Experiments

In this section, four simulation experiments relevance ontology similarity measure and ontology mapping below are designed.

4.1 Experiment on biology data

We use “Go” ontology O₁ which was constructed in http://www.geneontology.org. (Figure 1 shows the basic structure of O₁) in our experiment. P@N (Precision Ratio, see [41] for more detail) is used to measure the equality of the experiment. First of all, we give the closest N concepts for every vertex on the ontology graph by expert, and then we obtain the first N concepts for every vertex on ontology graph by the algorithm and compute the precision ratio. Huang et al. [12], Gao and Liang [13] and Gao and Gao [42] employed ontology algorithms to “Go” ontology. We compare the precision ratio which we got from four methods. Several experiment results can are shown in Table 1.

When N = 3, 5, 10 or 20, the precision ratio by virtue of our algorithm 1 is higher than the precision ratio determined by algorithms as [12 , 42] proposed in their research. Particularly, when N increases, such precision ratios will apparently increase. Conclusion can be drawn from the fact that the algorithm described in our paper is superior to the method proposed by [12 , 42].

4.2 Experiment on physical education data

We use physical education ontologies O₂ and O₃ (the structures of O₂ and O₃ are presented in Figs. 2 and 3 respectively) for our second experiment. This experiment aims at determining the ontology mapping between O₂ and O₃ via Algorithm 2. P@N criterion is applied to measure the equality of the experiment. We gave the closest N concepts for each vertex on the ontology graph with the help of experts. Then, we obtained the first N concepts for every vertex on ontology graph by the algorithm and compute the precision ratio. Ontology algorithms to “physical education” ontology was also employed in [12, 13, 15]. We made a comparison among the precision ratios which we get from four methods. Several experiment results can be found in Table 2.

Our Algorithm 2 is more efficiently than algorithms raised in [12 , 15]. It is especially the case when N is sufficiently large. This is revealed by the experiment results in Table 2.

4.3 Experiment on plant data

In this subsection, “PO” ontology O₄ which was constructed in http://www.plantontology.org. (Figure 4 shows the basic structure of O₄) is used to test the efficiency of our new algorithm for ontology similarity measuring. The P@N standard is used again for this experiment. Furthermore, we apply ontology method in [11 –13] to the “PO” ontology in our experiment. We calculate the accuracy by these three algorithms, and also compare the result to algorithm rose in our paper. Part of the data can be referred to Table 3.

When N = 3, 5, or 10, the precision ratio determined by algorithms that [11 –13] proposed in their research is lower than the precision ratio in terms of our algorithm. Such precision ratio apparently increase as N increases. That the algorithm described in our paper is superior to the method that [11 –13] proposed in their research can be drawn from the paper as a conclusion.

4.4 Experiment on humanoid robotics data

We used humanoid robotics ontologies O₅ and O₆ (constructed by [42], and the structures of O₅ and O₆ are presented in Figs. 5 and 6 respectively) for our last experiment. This experiment aims at determining ontology mapping between O₅ and O₆ via Algorithm 2. P@N criterion we applied to measure the equality of the experiment. Ontology algorithms in [12 , 43] were employed to humanoid robotics ontologies. We compare the precision ratio that we get from four methods. Several experiment results are shown in Table 4.

That our algorithm 2 works with more efficiency than algorithms [12 , 43] raised in their research especially when N is sufficiently large is revealed by the experiment results in Table 4.

5 Conclusions

In our article, a ranking based computation technology is manifested for ontology similarity measure and ontology mapping application. The detailed optimization scheming relies on the eigenpair computation, and the solution has uniqueness in view of our method of selection. Simulation data in the last part show that new ontology trick has high efficiency in biology, physics education, plant science and humanoid robotics. The ranking based ontology algorithm raised in this paper illustrates promising application prospects in multiple disciplines.

Footnotes

Acknowledgments

We thank all the reviewers for their constructive suggestions on how to improve the quality of this paper. We wish to acknowledge the National Natural Science Foundation of China (11401519).

References

Przydzial

J.M.

, Bhhatarai

and Koleti

, GPCR ontology: Development and application of a G protein-coupled receptor pharmacology knowledge framework, Bioinformatics29(24) (2013), 3211–3219.

Koehler

, Doelken

S.C.

and Mungall

C.J.

, The human phenotype ontology project: Linking molecular biology and disease through phenotype data, Nucleic Acids Research42 (2014), 966–974.

Ivanovic

and Budimac

, An overview of ontologies and data resources in medical domains, Expert Systerms and Applications41(11) (2014), 5158–5166.

Hristoskova

, Sakkalis

and Zacharioudakis

, Ontology-driven monitoring of patient’s vital signs enabling personalized medical detection and alert, Sensors14(1) (2014), 1598–1628.

Kabir

M.A.

, Han

and Yu

, User-centric social context information management: An ontology-based approach and platform, Personal and Ubiquitous Computing18(5) (2014), 1061–1083.

Y.L.

, Liu

, Lu

, Jin

B.H.

and Liu

X.J.

, A graph derivation based approach for measuring and comparing structural semantics of ontologies, IEEE Transactions on Knowledge and Data Engineering26(5) (2014), 1039–1052.

, Guo

H.S.

, Yuan

Y.S.

and Sun

L.B.

, Ontology representation of online shopping customers knowledge in enterprise information, Applied Mechanics and Materials483 (2014), 603–606.

Santodomingo

, Rohjans

, Uslar

, Rodriguez-Mondejar

J.A.

and Sanz-Bobi

M.A.

, Ontology matching system for future energy smart grids, Engineering Applications of Artificial Intelligence32 (2014), 242–257.

Pizzuti

, Mirabelli

, Sanz-Bobi

M.A.

and Gomez-Gonzalez

, Food track & trace ontology for helping the food traceability control, Journal of Food Engineering120(1) (2014), 17–30.

10.

Lasierra

, Alesanco

and Garcia

, Designing an architecture for monitoring patients at home: Ontologies and web services for clinical and technical management integration, IEEE Journal of Biomedical and Health Informatics18(3) (2014), 896–906.

11.

Wang

Y.Y.

, Gao

, Zhang

Y.G.

and Gao

, Ontology similarity computation use ranking learning method, The 3rd International Conference on Computational Intelligence and Industrial Application, Wuhan, China, 2010, pp. 20–22.

12.

Huang

, Xu

T.W.

, Gao

and Jia

Z.Y.

, Ontology similarity measure and ontology mapping via fast ranking method, International Journal of Applied Physics and Mathematics1(1) (2011), 54–59.

13.

Gao

and Liang

, Ontology similarity measure by optimizing NDCG measure and application in physics education, Future Communication, Computing, Control and Management142 (2011), 415–421.

14.

Gao

and Xu

T.W.

, Ontology similarity measuring and ontology mapping algorithm based on MEE criterion, Energy Education Science and Technology Part A: Energy Science and Research32(5) (2014), 3793–3806.

15.

Gao

, Gao

and Liang

, Diffusion and harmonic analysis on hypergraph and application in ontology similarity measure and ontology mapping, Journal of Chemical and Pharmaceutical Research5(9) (2013), 592–598.

16.

Gao

, Xu

T.W.

and Liang

, New multi-dividing ontology learning algorithm using special loss functions, The Open Cybernetics and Systemics Journal8 (2014), 259–268.

17.

Gao

, Zhang

Y.G.

, Xu

T.W.

and Zhou

J.X.

, Analysis for learning a similarity function with ontology applications, Journal of Information & Computational Science9(17) (2012), 5311–5318.

18.

Gao

, Gao

and Zhang

Y.G.

, Strong and weak stability of k-partite ranking algorithm, Information15(11(A)) (2012), 4585–4590.

19.

Gao

, Xu

T.W.

, Gan

J.H.

and Zhou

J.X.

, Linear statistical analysis of multi-dividing ontology algorithm, Journal of Information & Computational Science11(1) (2014), 151–159.

20.

Gao

and Zhu

L.L.

, Gradient learning algorithms for ontology computing, Computational Intelligence and NeuroscienceVolume 2014, Article ID 438291. https://dx-doi-org.web.bisu.edu.cn/10.1155/2014-438291

21.

Gao

, Gao

, Zhang

Y.G.

and Liang

, Minimax learning rate for multi-dividing ontology algorithm, Journal of Information & Computational Science11(6) (2014), 1853–1860.

22.

Gao

, Farahani

M.R.

and Gao

, Ontology optimization tactics via distance calculating, Applied Mathematics and Nonlinear Sciences1(1) (2016), 154–169.

23.

Rossello

, Ranking of investment funds: Acceptability versus robustness, European Journal of Operational Research245(3) (2015), 828–836.

24.

Nikolakopoulos

A.N.

, Kouneli

M.A.

and Garofalakis

J.D.

, Hierarchical itemspace rank: Exploiting hierarchy to alleviate sparsity in ranking-based recommendation, Neurocomputing163 (2015), 126–136.

25.

Lee

and Kim

, Fast multi-label feature selection based on information-theoretic feature ranking, Pattern Recognition48(9) (2015), 2761–2771.

26.

Beikkhakhian

, Javanmardi

, Karbasian

and Khayambashi

, The application of ISM model in evaluating agile suppliers selection criteria and ranking suppliers using fuzzy TOPSIS-AHP methods, Expert Systems with Applications42(15-16) (2015), 6224–6236.

27.

Ebrahimnejad

, A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems, International Journal of Systems Science46(11) (2015), 2048–2060.

28.

Anderson

, A monte carlo comparison of alternative methods of maximum likelihood ranking in racing sports, Journal of Applied Statistics42(8) (2015), 1740–1756.

29.

, Fan

, Wu

and Zhou

D.X.

, Learning theory approach to minimum error entropy criterion, Journal of Machine Learning Research14 (2013), 377–397.

30.

Castro

M.H.

, Menegatto

V.A.

and Peron

A.P.

, Integral operators generated by mercer-like kernels on topological spaces, Colloquium Mathematicum126(1) (2012), 125–138.

31.

Jordao

and Menegatto

V.A.

, Reproducing properties of differentiable mercer-like kernels on the sphere, Numerical Functional Analysis and Optimization33(10) (2012), 1221–1243.

32.

Steinwart

and Scovel

, Mercer’s theorem on general domains: On the interaction between measures, kernels, and RKHSs, Constructive Approximation35(3) (2012), 363–417.

33.

Ferreira

J.C.

and Menegatto

V.A.

, Reproducing properties of differentiable Mercer-like kernels, Mathematische Nachrichten285(8-9) (2012), 959–973.

34.

Figuera

, Barquero-Perez

, Rojo-Alvarez

J.L.

, Martinez-Ramon

, Guerrero-Curieses

and Caamano

A.J.

, Spectrally adapted Mercer kernels for support vector nonuniform interpolation, Signal Processing94 (2014), 421–433.

35.

Bourrier

, Perronnin

, Gribonval

, Perez

and Jegou

, Explicit embeddings for nearest neighbor search with Mercer kernels, Journal of Mathematical Imaging and Vision52(3) (2015), 459–468.

36.

Cluni

, Costarelli

, Minotti

A.M.

and Vinti

, Applications of sampling Kantorovich operators to thermographic images for seismic engineering, Journal of Computational Analysis and Applications19(4) (2015), 602–617.

37.

Hong

Y.M.

and Pfander

G.E.

, Irregular and multi-channel sampling of operators, Applied and Computational Harmonic Analysis29(2) (2010), 214–231.

38.

Syed

and Ogawa

, Optimal sampling operator for signal restoration in the presence of signal space and observation space noises, IEICE Transactions on Information and SystemsE88D(12) (2005), 2828–2838.

39.

Zizler

, The spectral radius of a multivariate sampling operator, Linear Algebra and its Applications385 (2004), 463–474.

40.

Costarelli

and Vinti

, Order of approximation for sampling Kantorovich operators, Journal of Integral Equations and Applications26(3) (2014), 345–368.

41.

Craswell

and Hawking

, Overview of the TREC web track, pp, Proceeding of the Twelfth Text Retrieval Conference, Gaithersburg, Maryland, NIST Special Publication (2003), 78–92.

42.

Gao

and Gao

, Ontology similarity measure and ontology mapping via learning optimization similarity function, International Journal of Machine Learning and Computing2(2) (2012), 107–112.

43.

Gao

and Lan

M.H.

, Ontology mapping algorithm based on ranking learning method, Microelectronics and Computer28(9) (2011), 59–61.