Jun Minagawa

Hiromi Ban

Factors to Affect Descriptions on Intra-Concept Relation in Introductory Concept Mapping

Keywords: Concept Map, association task, Introductory Concept Map, relation of concepts

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Concept Mapping is an externalizing strategy for a cognitive structure developed by Novak & Gowin (1984) based on a learning theory of [START_REF] Ausubel | Educational Psychology :A Cognitive view[END_REF] as well as a learning strategy for forming more reasonable cognitive structure.

With a focus put on a strict systematicity between concepts, in particular, hierarchical systematicity reaching from subordinate to superordinate concept among others, Novak et al. included links between concepts as well as explanation (linking label) described on the links as elements of concept map using a material of science education. Concepts to be written, way of linking, and linking label is strictly determined in the corresponding unit. However, it was revealed in the subsequent development that non-hierarchical map could exist, and Novak himself proposed to perform association task in the course of introduction of concept map to learning. Further subsequently, even such opinion has also appeared in Japan that association task itself should be taken as a concept map resulting in diffusing the very meaning of the concept itself in a form of concept map. There have been pros and cons regarding association task. For example, [START_REF] Cachapuz | Detecting changes with learning in the organization of Knowledge: Using association tests to follow the learning of collision theory[END_REF] performed association task in order to examine intraconcept relation held by subjects. As the result, it has been proved that there is a large difference between experts and beginners using a diagram with less cognitive load and an easily viewable method. At the same time, Cachapuz et al. have criticized such strategy that describes intra-concept relation in detail is a very time-consuming method. On the other hand, [START_REF] Stewart | Content and Cognitive Structure: Critique of Assessment and Representation Techniques Used by Science Education Re-searchers[END_REF][START_REF] Stewart | Techniques for Assessing and Representing Information in Cognitive Structure[END_REF] and [START_REF] Shinkai | Of a cognitive construct check by a related description way during a concept-about junior high school creature teaching materials[END_REF] have criticized regarding association task that how subjects thought to link between concepts is unclear. In the author's series of experiments, showing association task graphically really has an advantage to keep cognitive load lower and it is possible to assume how subjects though to link between concepts as long as they are able to endure strict experiment, but it has been proved that such strategy as to memorize the whole things of typical maps is used by subjects who have in-adequate previous knowledge or who are judged to lack of learning motivation required for understanding intra-concept relation. Therefore, it is desired to describe link-labels in detain from a perspective of learning support which is the true purpose of Concept Mapping. However, it has been revealed that when making subjects with less knowledge describe link-labels on the map during the experiment they get confused by the time consumed to think how to describe them resulting in great decrease in number of concepts to be de-scribed. Amid such trend, [START_REF] Minagawa | The Relation between Elements of Introductory Concept Map and the Score of Multiple-Choice Test[END_REF] figured out a way to positively utilize association task by revising it. With appearance of various factors observed in association task similarly to a case of concept map, such as hierarchical property, concept bifurcation phenomenon (a few subordinate concepts to be bifurcated from specific concept and described), and cross-linking (a new relationship between subordinate concepts to be recognized after completing hierarchical upper and lower structures), it has been proved to be possible to obtain clues for recognizing cognitive structure of learners by rather considering these observation results and association task as such meaning has be-come to be called as "Introductory Concept Mapping." Practically in this method, learners are handed over with a sheet of white paper and asked to write original concept for association on the center of it and additionally write down concepts associated from the original one in order. This method has characteristics such as ① to describe related concepts consecutively without caring for hierarchical intra-concept relation, ② to note re-called order by numbers, ③ to continuously describe concepts group in a direction free from the original concept for association at the center, and ④ to describe link-labels too similarly to a case of Concept Mapping, and it also demands efforts to find out cross links asking to describe link-labels too in principle in such a case. However, as it often becomes to be extremely complicated, such link-label and numbers to indicate order are often omitted. In this study, Minagawa has demonstrated in a multi-regression analysis using a test score of the corresponding field as the dependent variable that numbers of concept bifurcation and cross-linking are useful as independent variables. Then, why hierarchical property was not necessarily demanded between associated words groups in the Introductory Concept Mapping is based on a fact that recalled order of concept groups in association is random to some extent. That is, for example, in a concept of Ausubel's meaningful learning, subordinate concepts are believed to consist of systematization with several concepts and refining for which meaningfulness, understanding with profound contents and correlation are considered.

What is important is that association is not necessarily developed from superordinate concepts in association task even if subjects systematized them by Concept Mapping. For example, first response word for stimulus word, meaningful learning, may be "meaningfulness." Further, it may be often developed in order from correlation, refining to understanding with profound con-tents. Since these examples are often observed in association, any association should be determined to be correct as long as it is within the same super-ordinate concept.

Then, there is a problem of link-label. Link label should be described in de-tail as much as possible from a perspective of learning support, but it is inappropriate at early stage of unit learning due to tight time constraint. Therefore, adopting intra-concept relation (including a link-label to directly and indirectly link between concepts) as dependent variables in the present study, independent variable and dependent variables were considered from among various possible variables using examples from [START_REF] Minagawa | The Relation between Elements of Introductory Concept Map and the Score of Multiple-Choice Test[END_REF][START_REF] Minagawa | The effect of the Introductory Concept Map creation in shorttime[END_REF].

Variables ever considered in the previous studies include posttest score, number of correct associations, number of erroneous associations, total number of associations, number of cross-links, number of bifurcated concepts, related previous knowledge score, score of intra-concept relation explanation, and mean number of expressed concepts per series (total number of associated concepts/number of direct connection) Of those items, number of erroneous associations and total number of associations were excluded from the study because the former had less importance for interpreting the corresponding experiment and the latter was difficult to be interpreted due to the concurrently included correct and erroneous associations. As the result, previous knowledge score, score of intra-concept relation explanation, mean number of expressed concepts, number of bifurcated concepts, number of cross-links, and number of correct responses were adopted as variables, among which score of intra-concept relation explanation was used as a dependent variable because it corresponded to a link-label in a broad sense. Our research hypotheses are as follows: 1. Subjects of the present study may be difficult to function as an independent variable because neither were they able to find out even a few cross-links nor were they instructed how to find out them in a concept mapping practice of previous lectures. 2. Mean number of expressed concepts per series may be difficult to function as an effective independent variable because number of concept maps is originally determined by the types of subordinate concepts. 3. Previous knowledge may work as an effective independent variable this time too because it was able to effectively function every time up to now. 4. Numbers of correct responses and bifurcated concepts may function as effective independent variables because they are indexes whether the whole concept groups is well understood or not.

Experiments are performed and the results are considered in order to verify these hypotheses.

1. Experiment participants: Totally 26 sophomores including 2 men and 24 women of B department in a private A university in Okayama prefecture 2. Date of experiment: December in 2014] 3. Venue of experiment: Class room within A university 4. Procedures: We performed a previous knowledge survey (scale of one to ten) a week before the experiment and a relation explanation test (scale of one to ten) a week after them without preliminary notice for both of them. The experiment was performed in a lecture of "Developmental psychology" as a part of class. First, students were handed with a sheet of A4 size white paper and instructed as follows: "We perform an association experiment now. Fill in your affiliation, student ID number and name at the top of the form. As a word "Developmental psychology" is written at the center, write down words associated from the word in order you associated in a manner to connect them. In order to make the order of associated words clear, draw lines from the origin word for association to those associated. Also note numbers to indicate the order of association. You don't need to write link-label. You are allowed to use 30 minutes for writing. Then, let's get started!" Then, instructing, "Stop!" after 30 minutes elapsed, the forms were collected. A multi-regression analysis was performed based on stepwise way using relation explanation score as a dependent variable and number of bifurcated concepts/number of direct connections, number of correct responses, number of cross-links, and previous knowledge survey score as independent variables. Mean value and standard deviation of each variable is shown in Table 1. Correlation coefficients between each variable are shown in Table 2. In addition, Table 3 shows bifurcation number, standard partial regression coefficient β of previous knowledge, as well as significance probability p of t and VIF.

3.Results

*

4.Discussion

Positive reaction counts refer to the appropriate number of vocabulary associated with conceptional relationship. It is a high number in considering the limited time of 30 minutes which indicates that students are keeping the memory of the study contents for a long time. On the other hand, Table 1 indicates that students do not have a deep understanding of intentional and extensional concepts as well as conceptional relationship despite of their low relative explanation score. In comparison with a concept divergent and a regular concept mapping, the introductory concept mapping (ICM) enables us to theoretically draw unlimited lines that are connected directly to the initial stimulus thereby it is reasonable to say why the number of lines become large. (See Fig. 3 and Fig. 4). Fig. 3 Prior knowledge score is consisted of basic knowledge questions and it doesn't provide a high performance. On the other hand, Table 2 shows a significantly high relative explanation score for the prior knowledge, the positive reaction counts as well as the concept divergent just like the previous studies while its correlation for cross-link is low. It is assumed that cross-link was not sufficiently generated as they didn't have enough study contents and time for the course unit in order to discover a new relevance in conceptional relationships. This is why the cross-link was not extracted as a valid variable by multiple regression analysis as shown in Table 3.

Table 2 shows a significantly high relative explanation score for the prior knowledge and the concept divergent just like the previous studies. Because a number of memory concepts divided by a number indicates a high correlation for cross-link, it is assumed that correlation was enhanced due to the high average concepts associated with direct reaction words towards stimulus words. In [START_REF] Minagawa | The Relation between Elements of Introductory Concept Map and the Score of Multiple-Choice Test[END_REF][START_REF] Minagawa | The effect of the Introductory Concept Map creation in shorttime[END_REF], cross-link was largely found among students with high academic achievement which is believed due to the correlation of their high academic ability and creativity. However, it is hard to say cross-link happens naturally but requires for a professor to provide sufficient advice or training to students. It is also preferable to have a high motivation for study.

According to Novak et al., cross-link is believed to be an expression of extremely highly abstract thought. Therefore, it can be judged that participants in the present experiment were a bit lack of that aspect, but it may be true in any way that cross-link is the most difficult one to be created. Among variables for which higher correlation between intra-concept relation is expected, number of bifurcated concepts and previous knowledge remained after a multi-regression analysis for the results of the experiment this time. Quantity and quality of previous knowledge is quite important for connecting intraconcept relation directly. In addition, as it is required for concept bifurcation a good deal of knowledge on what kinds of meanings the given concept can have, this result is believed to be rational. As it is believed to lead to content understanding support by taking a strategy to instruct about cross-link and the like in advance in addition to recruiting more motivating experiment participants, we intend to study focusing on such aspect in the future. In this study, sophomore in nursing department were targeted, however, we have done similar experiments with college students in early childhood education department as well. Both majors are directly related to the practice, however, it is believed that proper study for developmental psychology and its theories are required.

Fig. 2

 2 Fig. 1 An example of the Concept Map, "Rock"

Fig. 4

 4 Fig.4 Student B's Introductory Concept Map, "Developmental Psycholpogy" in Japanese

  Student A's Concept Map, "apparent movement" in Japanese

	Table 3 Standard regression coefficient β of the number of branch and the preceding knowledge, score
	number of	β	t	p	VIF
	branches	.44	2.69	.013	1.571
	prior knowledge	.43	2.63	.015	1.571


Sanggyu Shin
email: shin@aiit.ac.jp

Yoichi Seto
email: seto.yoichi@aiit.ac.jp

Study of Cancelable Biometrics in Security Improvement of Biometric Authentication System

Keywords: cancelable biometrics, biometric, authentication system

Recently, there is a widespread use of biometric authentication systems. This is because biometric systems have become open and large scale and enrolment and authentication systems are separate. Many methods have been proposed for cancelable biometrics technology in biometric systems. However, the security criterion in such is indefinite in cancelable biometrics technology. Moreover, there is still no work on the systematic study of the safety of biometric authentication systems. In this paper, we consider the cancelable biometric techniques from the perspective of the safety of the system. In addition, we also verify the effect on the security precaution of the liveness detection techniques using Fault Tree Analysis, a risk evaluation method about data protection and spoofing prevention techniques.

Introduction

In biometric authentication systems, the biometric data that becomes the reference for comparison is called template data. A template is biometric information that is used at authentication. The need for template sharing has been recognized because using the same template between applications makes inheriting trust among organizations (e.g., passports and back cards) possible. Furthermore, it enhances stability and reduces cost of system development and operation. Template protection technologies aim at preventing the prediction of raw biometrics data from templates and at preventing leaked templates from being reused by unauthorized users (cancelling leaked templates and reissuing new valid templates).

When the template data leaks, problems such as the spoofing of other users and leakage of private data occur. These problems are usually addressed by the cryptography technology and the system technology of tamper-proof devices such as smart cards.

Various methods have been published for security techniques now known as cancelable biometrics [START_REF] Clancy | Secure smartcard based fingerprint authentication[END_REF]- [START_REF] Soutar | Biometric encryption[END_REF]. However, there has been no systematic study on the criteria for security in biometric authentication systems.

Paul proposed to tackle the problem and present a novel solution for cancelable biometrics in multimodal system. They developed a new cancelable biometric template generation algorithm using random projection and transformationbased feature extraction and selection. The performance of their proposed algorithm was validated on multi-modal face and ear database [START_REF] Paul | Multimodal Cancelable Biometrics[END_REF]. They also present a novel architecture for template generation in the context of situation awareness system in real and virtual applications [START_REF] Paul | Situation awareness of cancelable biometric system[END_REF]. In addition, Rathgeb presented comprehensive survey of biometric cryptosystems and cancelable biometrics [START_REF] Rathgeb | A survey on biometric cryptosystems and cancelable biometrics[END_REF].

The purpose of this paper is to discuss and evaluate the cancelable biometrics techniques from the viewpoint of ensuring safety of the systems.

First, the location and the type where the threat is generated on the biometric authentication systems model are explored, and the meaning of cancelable biometrics as the countermeasure technology to those threats is described.

Next, we propose a scheme for evaluating the effectiveness of cancelable biometrics. After that, the effectiveness of the security precaution with the liveness detection techniques is verified by using FTA (Fault Tree Analysis), which is a quantitative risk analysis and is the evaluation approach about the data protection and the proofing prevention technology then the superiority of the liveness detection technique is presented.

Finally, to know the open problems with the current template protection technologies, we summarize our evaluation result, especially from technical and security points of view.

Threat medel for a biometric authentication system

Many biometric data is inherently exposed (e.g., faces and fingerprints). It has little value without linking information to person and it is a countermeasure preventing a leak. Liveness detection should be done at sensors and since it has no alternative technologies to it.

Fig. 1 shows the processing diagram of a biometric authentication system. At first, raw biometric data of an individual is captured from the user. For every input, biometric data changes due to the body and sensor device condition in the environment.

In the enrollment process, the correction processing is included in the feature extraction from raw biometric data of an individual, and the extracted characteristics of individual is stored in the database as template data.

In the authentication process, the identification data that specifies the user is inputted from a sensor device and the corresponding template data is selected from the database. Two sets of features are matched, then the degree of similarity is obtained by a matching process. It is assumed that the authentication succeeds if the degree of similarity exceeds a threshold. As a result, the user can access the application.

The number in Fig. 1 shows the location under the threats of attack in a biometric authentication system. The threats at each location are explained as follows. It is very difficult to execute this attack because the feature extraction processing and the matching processing are often done in the same system. However, when an extracted feature data is transmitted to the matching processing by the Internet, this attack becomes possible by substituting the packet data. (5) Tampering with stored template data: An illegal user makes the falsification of a template data stored in the database, such that an unfair user obtains an illegal attestation and gains access or a fair user obtains an illegal attestation and is denied access. (6) Re-input of stored biometric data: Biometrics that remains on the sensor devices is automatically inputted again without user input. (7) Attack on the transfer from template data storage to matching processing:

When the template stored in the database is transferred to the matching processing through the communication channel, the template data is illegally changed. (8) Replacement of threshold value: The threshold value is set to the given value by rewriting the judgment policy in order to get the intended result. (9) Attack on matching process: The matching process is attacked, and the matching result is replaced with an arbitrary score. (10) Substituting of final decision data: The judgment result of the authentication is substituted.

The countermeasures shown in Table 1 are effective ways to prevent the above-mentioned attacks. The threats in categories (4), ( 5) and ( 7) relate to the theft of template data and those in categories ( 1) and ( 6) relate to counterfeit use (spoofing) when biometric data is captured and attested. For example, the development of countermeasure techniques for the following attacks will become important in the future.

-Study of fabricating a counterfeit fingerprint with cheap material such as the gelatin, which can be made for a short time. -A method of the transformation of biometric data of protection and each application of biometric data by the encryption for the template protection and storing. -A peculiar attack that counters dictionary attack to the biometric authentication.

Another problem includes the copying process when the data used for the biometric authentication loses reliability.

When reliability is lost, the authentic method such as using the key, the token, and the password, etc. can nullify these attestation devices as many times as you want. But, there is a limit in the number of times of nullification for biometrics.

In the security requirement for a biometric authentication system, the cancelable biometric techniques are not exclusive but are one of the measures technologies.

The problem of the template data leakage is divided into the problem of spoofing due to reuse and privacy concerns. Biometrics was originally exposed, and at 1:1 matching, the individual can be specified if there is a link to other information. Therefore, there is an opinion that biometrics is not privacy.

As for the problem of reuse, the measures technique when reusing such as the liveness detection technique is more effective than the encryption of data and nullification of data if biometrics can specify the individual by the link information.

It is necessary to examine effectiveness compared with the competing measures techniques, for instance, cryptography. T he analysis of the effectiveness of the measures technique is described in Chapter 4.

Systematization of the cancelable biometrics technique

Cancelable biometrics techniques

Cancelable biometrics is a collective term of template protection techniques that nullifies it when an original biometric data is made invisible from the template not to be restorable in the biometric authentication systems.

There are two kinds templates. One is for image (signal) data input from the sensor and the other is for the feature data that used for processing. There are two kinds of nullification methods: based on encryption techniques and on image processing.

Encryption-based approach stores encrypted templates and decrypts templates at authentication time. Image processing-based approach matches encrypted templates without decrypting the data and provides the abilities of invisibility of original biometric data and canceling lost templates. It is also a mixed technology of biometrics and encryption as various mechanisms are used to implement these functionalities. Generally, security strength is unclear.

Fig. 2 shows typical processing flow of the cancelable biometrics. The biometric data is distorted by the conversion processing using a one way transformation function in an enrollment or an authentication processing. The template generated by a nonreciprocal method is stored in the smart card and the database.

The biometric data causes a different swerve at each registration. Therefore, when the reliability of biometric information is lost, the misinterpretation method used at that time is changed. Furthermore, it only has to register again based on a new conversion coefficient.

The technical details of a cancelable biometrics based on image processing is discussed thoroughly in [START_REF] Shimizu | An evaluation of biometric template protection methods[END_REF].

The groupings by PET (Privacy Enhancing Technologies) [START_REF] Shimizu | A study on the effectiveness of cancelable biometric technology in biometric authentication systems[END_REF]- [START_REF] Seto | Proposal to develop and operate for useful biometric application systems[END_REF] are classified techniques of privacy protection that can be applied to the systematization of cancelable biometrics. Techniques of cancelable biometrics and the dynamic key generation algorithm developed now can be classified into four categories (A to D) in reference to the classified techniques of PET, as shown in Table 2. Category A is a method by noninvertible conversion. Conversion can be applied to both image (signal) and feature region. An example of conversion in the image area includes morphing and the block substitution. For example, the block structure is allocated in the block substitution according to a feature point in former image area, it arranges at every the block, and the scramble is done. However, not noninvertible conversion in this case but former image can often be computed reversibly.

Therefore, security strength is low, and the conversion in the image space is classified into category B.

Category C is the method to use information on fake data such as Fuzzy Vault.

Category D is the method of the data protection using the public key cryptography and zero knowledge proof technology. Furthermore, category D also proposes data protection techniques that use a general cryptographic protocol in ISO TC68 that is the technical committee intended for financial services.

Currently, data protection technologies in biometric data are perceived to be immature. Therefore, the empty column in Table 2 represents template protection technologies that could be proposed in the future.

Evaluation of validity of the cancelable biometrics

There are two technologies for the protection of template data: image processing based methods and cryptographic based methods.

Cryptographic template protection technology is open to the public, which allows evaluation of security strength by third parties. For example, FIPS140-2 is maintained as a safety standard at the implementation level.

Application Existence of an application where image processing-based template protection methods have the precedence over cryptographic-based.

On the other hand, from a technical point of view, image processing-based template protection methods have the following problems..

-Proper evaluation of the template protection technique is insufficient.

-The third party evaluation cannot be done because the algorithm and the interface are unpublished. -The security strength evaluation scheme is not established, and an objective evaluation concerning a one way transformation, accuracy preservation and the processing performance is not done (Refer to Table 3). -There is a possibility that it is technically immature as described in paragraph 3.1, and a more effective method will be developed in the future. -There is no appropriately applied actual case.

-The individual data not in the database but in the smart card model is general from a viewpoint of a restriction of law and safety. -Cancelable biometrics is a technology that assumes storage in a database, and a powerful application that can demonstrate superiority over template protection technology of cryptography based methods is not known. -The priority is low in the viewpoint of the security risk. Details are presented in Chapter Chapter 4. Therefore, it is enough not to ensure that the template data is kept secret and discuss the safety of the system.

As a solution to these problems, security evaluation techniques will be established and an appropriate application will be developed.

Evaluation of countermeasure techniques for securing the system

Two points should be considered: technical validity and standardization. If biometrics have been exposed. the possibility that it is acquired by a malicious person without the consent of the user is high. Therefore, for the safety of a biometric authentication system, measures against reuse are more important than those against theft.

In reusing the template data, there are two kinds of abuse such as the capturing of the biometrics of the counterfeit and spoofing due to hacking to the system. The liveness detection function is necessary for the former and it is necessary for data transfer to be secure for the latter.

Applying the standardized cryptography technology for the data protection in the channel is an advantage in terms of cost and safety. Moreover, liveness detection measures that should be carried out when body information is acquired from the sensor, and image processing techniques become indispensable, as no alternative technology exists.

Therefore, from a practical viewpoint, it is thought that the development of liveness detection technology is a higher priority than the data protection technique of the image processing based cancelable biometrics.

In this chapter, the liveness detection technique is discussed as a spoofing prevention in the sensor aiming to prove the above-mentioned hypothesis quantitatively. Then, encryption and cancelable biometrics are taken up as data protection techniques in the channel and the database. The effectiveness of each technique is evaluated with respect to safety.

In this study, FTA (Fault Tree Analysis) is adopted as a quantitative risk evaluation method. FTA is a technique for making a logic diagram with a tree structure known as the Fault Tree that shows the causal relation of the generation process of the threat, the calculation of the probability of occurrence of the threat based on this logic diagram, and evaluation of the risk.

Applied to the security measures in the information system, FT is composed of making the threat a root [START_REF] Shimizu | A study on the effectiveness of cancelable biometric technology in biometric authentication systems[END_REF], and uniting events derived according to the causal relation hierarchically by using the logic gate of the logical add and the logical product. The probability of occurrence of the root threat can be obtained by giving the probability of occurrence for each lower event after the FT is made.

Fig. 3 shows FT made for the spoofing attack. The probability of occurrence of a basic event of the liveness detection, the encryption, and the cancelable biometrics was given to FT under assumption that each technology was applied, and the probability of occurrence of the threat was calculated. Table 4 shows the probability of occurrence of the basic event used by this analysis. It was roughly distinguished that the probability of occurrence obtained an actual figure in three stages because it was difficult. The living body detection technology was assumed to be the one that it was possible to detect it surely when body information not to be alive was presented.

Table 5 shows the numerical results. The risk type is usually defined by the product of the probability of occurrence and the size of the loss. Here it was assumed that the loss that occurred as a result of the spoofing was the same for each threat, and so the risk is evaluated only by the probability of occurrence.

It is understood to have lowered the liveness detection in addition while the cryptography and the cancelable biometrics lower the probability of occurrence (0.15 and 0.13, respectively) under the situation in which measures are not done compared with the probability 0.33 that the threat is generated from the results in Table 5. The following conclusions can be derived from above.

-In these technologies, the liveness detection is the most effective.

-From viewpoint of prevention of reuse to effectiveness of this level cryptography and cancelable biometrics that is data protection technology. -The cryptography technology is standardized, and it is an advantage if an objective evaluation approach for safety has been established.

The template protection technology currently proposed is still immature but it is possible that it would evolve sufficiently in the future. Technical merits and demerits can't be judged objectively because the precision and the security strengths aren't estimated enough.

Conclusions

Living body measure is most effective in these technologies. However, encryption technology is standardized and there is a merit by which objective evaluation technique is established about safety.

The viewpoint of an effective methodology that secured the safety of biometric authentication systems was considered. The cancelable biometrics technique was systematized based on Privacy Enhancing Technology, and the possibility that a new technology will be developed in the future was shown in Chapter 4.

It proposed the evaluation items such as one way transformation, accuracy preservation and processing performance that clarified the effectiveness of the cancelable biometrics. The effectiveness of the technologies were quantitatively compared by using FTA method for the reuse problem in a biometric authentication system, when biometrics data had been leaked and exposed. According to the FTA analysis, when compared the cryptography and liveness detection, the effect of cancelable biometrics on the safety of biometric authentication systems was small.

Fig. 1 .

 1 Fig. 1. Possible attack points in a biometric authentication system.

( 1 )

 1 Attack on sensor input of fake biometrics: The photograph of face, counterfeit fingerprint or signature are put on the sensor. (2) Attack on the transfer data from sensor to feature extraction processing: Biometric data captured from the sensor is replaced with information attacking the network or the bus. (3) Making replacement of extracted feature data: The feature extraction processing is attacked with the Trojan horse etc., and an arbitrary feature is set instead of an actual feature. (4) Illegal conversion of body data: Body data is replaced with counterfeit data.

Fig. 2 .

 2 Fig. 2. The processing flow of cancelable biometrics.

Fig. 3 .

 3 Fig. 3. Fault tree for an impersonation attack.

Table 1 .

 1 Threats and countermeasures.

	No Threats	Countermeasures
	(1) Attack on sensor input of fake biometrics	Liveness detection
	(2) Attack to transfer data from sensor to feature ex-	Encryption
	traction processing
	(3) Making replacement of extracted feature data	Digital signature
	(4) Illegal conversion of body data	Encryption
		Cancelable biometrics
	(5) Tamper with stored template data	Physical security
		Cancelable biometrics
	(6) Re-input of stored biometric information	Liveness detection
		Challenge & Response
	(7) Attack to transfer from template data storage to Encryption
	matching processing	Cancelable biometrics
	(8) Tampering authentication parameters
	(9) Replacement of threshold	Digital signature
	(10) Data

Table 2 .

 2 Grouping by PET of template protection techniques.

	Principle	Methods	Ref.
		A1. Non-reversible transformation
		of images
	One-way function	A2. Non-reversible transformation	[6]
		of templates
		B. Image morphing
	Common key cryptography NA
	Public key cryptography	D1.	[5] [7]
	Secret sharing function	NA
	Blind signature	NA
	Zero knowledge protocol	D2	[4]
	Proxy network	NA
	Fake information	C1. Fuzzy vault	[1] [3]
		C2. Convolution random pattern	[2]
	Privacy language	NA

Table 3 .

 3 Evaluation axis of the cancelable biometrics.

	Items	Description
	Hardness of reconstructing	Proof of being essentially unable to reconstruct
	original data	original biometric data from converted data by a
		one-way function etc.
	Preservation of authentica-	Proof that authentication precision over converted
	tion precision	data is no lower than that over pre-converted
		(original) data
	Performance	Demonstration of the performance of a conversion
		algorithm for practical use
	Application	Existence of an application where image
		processing-based template protection methods
		have the precedence over cryptographic-based.

Table 4 .

 4 Occurrence probabilities of basic events.

		Event probability (When a measure is applied)
	Events	Nothing	Cryptography Cancelable	Liveness
				biometrics	detection
	Obtaining physical infor-	0.1	0.1	0.1	0.1
	mation from sensors
	Obtaining exposed phys-	0.3	0.3	0.3	0.3
	ical information outside
	the system
	Reconstruction of physi-	0.3	0.3	0.3	0.3
	cal information from ex-
	posed biological informa-
	tion
	Derivation of transfor-	-	-	0.1	-
	mation parameters of
	cancelable biometrics
	Reconstruction of physi-	0.1	0.1	0.1	0.1
	cal information from fea-
	tures
	Reconstruction of physi-	0.3	0.3	0.3	0.3
	cal information from im-
	ages
	Analysis of encryption	-	0.01	-	-
	keys
	Data Encryption	1.0	0	1.0	1.0
	Interception of commu-	0.1	0.1	0.1	0.1
	nication
	Database Access	0.2	0.2	0.2	0.2
	Unauthorized reading of	0.1	0.1	0.1	0.1
	IC card
	Presentation of not phys-	1.0	1.0	1.0	1.0
	ical information that is
	not live

Table 5 .

 5 Probability of occurrence of threats.

	Applied technology	Probability of occurrence of	Range of Reduction Proba-
		threats	bility
	Nothing	0.33	-
	Cryptography	0.18	0.15
	Cancelable biometrics	0.20	0.13
	Liveness detection	0	0.33


Daishi Watabe

Takanari Minamidani

Hideyasu Sai

Takaharu Taiyo Maeda

Takaharu Yamazaki

Jianting Cao

Estimating a Shooting Angle in Ear Recognition

Keywords: ear recognition, normalizing shooting angle, Gabor feature

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Background

Ear shape is unique to every individual and has been used in forensic science over the past 40 years [START_REF] Iannarelli | Iannarelli system of ear identification[END_REF]. In particular, ear prints left on walls have been used in identification of criminals, most notably in the Netherlands [START_REF] Meijerman | Earprints in forensic investigations[END_REF]. A detailed survey on using ear shapes for forensic purposes is available in [START_REF] Meijerman | Earprints in forensic investigations[END_REF], where historical studies and present issues are detailed. Furthermore, detailed surveys of automatic ear recognition systems are available in [START_REF] Abaza | A Survey on Ear Biometrics[END_REF][4] [START_REF] Yuan | A Review of Recent Advances in Ear Recognition[END_REF], where databases, algorithms, experimental conditions, and accuracies are presented. Whereas masks and sunglasses often purposely obscure facial features, ear shapes can be all that is required to identify subjects. However, the shooting angle of an ear from a surveillance camera is usually not the same as that for a facial image. Hence, accounting for such differences is necessary [START_REF] Moriyoshi | Morphological Classification of the Ear Components and the Correlation between Morphological Characteristics in Each Component[END_REF] [START_REF] Moriyoshi | Personal Identification from Ear Images in the Different Orientations[END_REF].

Related studies

Moriyoshi [START_REF] Moriyoshi | Morphological Classification of the Ear Components and the Correlation between Morphological Characteristics in Each Component[END_REF][7] thoroughly investigated the effect of differences in the shooting angle in the context of forensic science. As far as image processing and computer vision are concerned, a few studies have variations in the angle of the shot [START_REF] Watabe | ICA , LDA , and Gabor Jets for Robust Ear Recognition , and Jet Space Similarity for Ear Detection[END_REF] [START_REF] Watabe | Ear biometrics using jet space similarity[END_REF]. However, these studies were limited to in-plane rotations. In single-view-based ear recognition studies, the authors improved the robustness of the method for off-angle rotation [START_REF] Watabe | Empirical Evaluations of a Single-view-based Ear Recognition when Rotated in Depth[END_REF] [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF], in which feature vectors of various poses are estimated from a single-view image and recognition processes are performed without using prior knowledge of shooting angles of the input images. This is done by taking a correlation against the averaged estimated feature vectors over various poses. Although this averaging process may contaminate final accuracy, we did not use information on the shooting angle of input ear images because the estimation method of the shooting angle of an ear image is not established. One may point out the usability of the shooting angle of face images, for which various estimation methods are well-established. However, because individual variations in the ear overhang angle are considerable, the use of the estimated face angles as an ear angle is not feasible. We do not know, however, how promising it is to pursue the direction of seeking a method of estimating the shooting angle of an ear. If such an estimation method is established, it will be possible to improve the accuracy of the single-view-based ear recognition system, by taking the correlation against estimated feature vectors of a specified angle, but not against contaminated averaged feature vectors of unspecified shooting angles.

Aim of this study

An initial attempt to estimate the shooting angle of an ear image is presented. Using these estimated shooting angles, a few estimation methods for the feature vector of other shooting angles are compared experimentally. We examine the possibility of improving the robustness of ear recognition by estimating the shooting angle of an ear image.

2

Proposed method

Outline

In Subsections 2.2 and 2.3, our method for estimating the shooting angle of an input image is explained. For completeness, the method we used for ear recognition is summarized briefly in Subsections 2.4, 2.5, and 2.6.

Gabor features of ear minutiae

To fix a baseline, we used the Gabor features for the various methods described below: Let be a point in a plane. A 2D plane wave defined by wave vector and modified by a Gaussian function is called a Gabor function (Eq. ( 1)):

  2 2 2 2 2 2 ( ) exp exp exp 2 2 i                                  k k x x k x . (1)
Here denotes the width of this function determined by the Gaussian function. The factor is a compensation term that eliminates averages. This condition is required from wavelet theory, but if is large enough, the factor can be ignored. Gabor functions are characterized as localized wavy shapes in various directions determined by plane waves. Gabor filters, i.e., convolutions with these Gabor functions, extract the direction and wavelength of these localized wavy shapes of an image near the point under consideration. Wavy shapes in various directions also characterize the outer ear. Thus, endpoints, junctions, and protuberances of the ridges of the outer ear are selected as feature points. Wavy shapes near these feature points are measured and

Gabor configurations

Five wavelengths, 4,4 2,8,8 2,16 , were adopted as Gabor filters to cover the various widths of ridges along the ear that appear in the experimental data. Furthermore, to cover all directions evenly, eight directions corresponding to π/8 rotations are employed. To realize these settings, we set σ = 2π and

𝐤 = (𝑘 𝑥 , 𝑘 𝑦 ) = 2𝜋 𝜆 (cos 𝜋 8 𝜇 , sin 𝜋 8 𝜇 ), (2)
for λ = 4, 4√2, 8, 8√2, 16 and 𝜇 = 0, … ,7 in Eq. ( 1).

We implemented these Gabor filters using a mask of 101×101 pixels for the convolution window, and this convolution was performed using the fast Fourier transform. Using this bank of Gabor filters, Gabor feature vectors were sampled at the feature points as indicated in Fig. 1 and then stacked into one vector with maximum dimensions of 560 (=80×7). Furthermore, when phases were ignored, that is only taking absolute values, these vectors became 280 (=40×7) dimensional.

Estimating shooting angles of an input image

Given an input ear image of unknown shooting angle, we can compute the similarity scores between this input ear image and the ear images of a known shooting angle from a database. Where this angle is close to the unknown shooting angle of an input ear image, it is anticipated that the number of ear images with higher scores for their similarities may be large. Based on this concept, we examined the following algorithm:

1. First, similarity scores between an input ear image and images of known shooting angles in a database are computed. 2. The summation scores for these similarity scores above a given threshold within a shooting angle are obtained. If there is no sample with a similarity score higher than the threshold, this algorithm returns a failure for the shooting angle estimation. 3. The above process 1-2 is repeated through various shooting angles in the database. 4. Finally, the shooting angle with the maximum summation is returned as an estimated shooting angle of an input image.

Similarity scores are given by normalized-cross-correlations to the phase-ignored Gabor feature vectors. A threshold is employed in order not to contaminate the estimation accuracy of shooting angle through using the lower scores of the similarities of non-similar ears. This threshold is obtained by maximizing the estimation accuracies of shooting angles through a survey of threshold-values using leave-one-out cross validation strategy.

Estimation of Gabor features after off-angle rotation for a single registration image using a linear jet transformation

For completeness, the method used in [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF] is outlined. Locally, near the feature points, the subject is approximated by a tangent plane. The tangent plane does not have depth. Hence, the image of this plane rotated in depth can be estimated. This estimated image reflects local features under pose variations near the feature points. Similar to the tangent plane, Gabor jets only represent local features. Motivated by this, we explore the benefits of Gabor jets for subjects rotated in depth. The following outlines the reproduction method using Gabor jet estimates of subjects with different poses [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF] [START_REF] Watabe | Empirical Evaluations of a Single-view-based Ear Recognition when Rotated in Depth[END_REF]. Let the x-y coordinates be set on the camera plane and the z-axis set perpendicular to this plane. Suppose that a subject plane, initially placed parallel to the camera plane, is rotated by ϕ around its x-axis and then θ around its y-axis. By observing the transformations of unit vectors, a point on the subject plane initially at is transformed to x given by: .

If this plane is initially placed at 11 ( , )

 and not parallel to the camera plane, the above transformation is:

. ( 4
)
Under this transformation, the transformation of the Gabor jets corresponding to the pose change can then be estimated. In what follows, is denoted as for simplicity. Components of the transformed Gabor jets are obtained by convoluting the Gabor function with the transformed image . Using , this is .

(

) 5
Assuming the following approximation: ,

the Gabor jet transformation is simply written as:

. ( 7
) ( , ) xy  u cos sin sin , 0 cos             x Au A 1 2 2 1 1 ( , ) ( , )       x A A u 1 2 2 1 1 ( , ) ( , )      AA A () I 1 Ax , ' '  x Au x Au ' ( ) ( ') ( ') ' ( ') ( ') | | j I d I d          1 k k k x A x x x x u u Au A u '' ' ( ') | | ( ) ( ') c     k kk k k Au A A u '' ' ' ( ) ( ) ( ) j j   k kk k k x c A u
Once is obtained, the transformation of the Gabor jets can be estimated using:

.

Matrix is obtained by multiplying both sides of Eq. ( 6) by and integrating both sides. Two of the variables are difficult to determine. One is 22 ( , )

 , which depends on the poses of input images. This is an unknown in real scenarios. In [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF], we solved this issue by producing the Gabor feature of many other poses in advance. The other unknown variable is 11 ( , )  , which represents the normal vector of the tangent plane at each feature point. Because this variable is difficult to determine from a single-view-image, some type of statistical modeling is necessary. In [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF], this model was produced using an exhaustive search of smaller equal error rates in the variable  and  using a five-fold cross validation strategy.

Estimation of Gabor features after off-angle rotation for a single registration image using principal component analysis (PCA)

To estimate the Gabor feature vectors for other poses, the feature vectors taken at the registration and input angles are stacked into one feature vector for the same person in a training set. Because phases are ignored and absolute values taken, a 560 (=40×7×2) dimensional vector is obtained. Such stacked feature vectors are created for all training datasets and subjected to PCA. For testing the sample, the Gabor feature vector at the registration angle and the null data are stacked into one vector. Using the principal component subspace, the Gabor feature vector at the input angles are estimated as a sub-vector of the back-projected stacked feature vector. Similar to Subsection 2.5, a five-fold cross validation is used to create training and test sets. The principal component subspaces serves 3D statistical modeling for estimating feature vectors of other poses.

Estimation of Gabor features after off-angle rotation for a single registration image using multiple regression analysis (MRA)

To estimate the Gabor feature vectors taken from different shooting angles, the normal equation is solved to obtain the regression coefficients that describe each component of the Gabor feature as a linear combination of the components of Gabor features of the registration angle. With phases ignored, a set of 280(=40×7) normal equations is solved for training sets and used to estimate the Gabor features of the input angles for test sets. Similar to Subsection 2.5, a five-fold cross validation is used to create training and test sets. The regression coefficients obtained serve 3D statistical modeling that can be used to estimate feature vectors for other poses.

)) c  A kk CA () '( ) ( )  A j x C j u C '' ( ')  k u 2.8 (

Creating the linear discriminant analysis matrix using the estimated features for input images with unknown poses

To fix a baseline for comparison, similar to the method described in [START_REF] Yamamoto | Human and Object Interaction Processing (HOIP) project[END_REF], all the estimated feature vectors, as illustrated in Subsections 2.5, 2.6 and 2.7, are subject to multiple linear discriminant analysis (LDA), thereby creating the LDA matrix of discriminant vectors. After applying this LDA matrix to both the input and registration feature vectors, a normalized correlation is computed to obtain the similarity between the input and registration feature vectors taken from different shooting angles.

Experiment

Database of feature vectors for the experiment

Experiments were performed using the database of Gabor feature vectors at ear feature points of images from the human and object interaction processing (HOIP) database [START_REF] Yamamoto | Human and Object Interaction Processing (HOIP) project[END_REF] obtained in our previous studies [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF]. The HOIP database consisted of facial images of 300 subjects photographed from 504 (72 yaw angles every 5° and 7 roll angles every 15°) directions, where the size of the ear fitted approximately within a 70×90 pixel window. By mirror reflecting images, left profile images of 600 people were subjected to Gabor feature computations. Thus a database of Gabor feature vectors for 600 people was obtained.

3.2

Shooting angles and number of visible feature points in the experiments.

To examine robustness against yaw-angle pose variations, verification experiments were performed using an ear image of the true left profile of a registration image taken from 85°. Input data were taken from yaw angles varying from 40° to 80°, every 10°. One hundred and sixty-two subjects corresponding to images with seven visible feature points at all angles were selected for input, registration, and training data. In these datasets, there was a single biometric sample for each identity in each angle.

3.3

Experiment for estimating the shooting angle.

For the algorithm presented in Subsection 2.4, a threshold was determined by maximizing the estimation accuracy of the shooting angle using a leave-one-out cross validation strategy as follows:

1. First, the set of images at a 40° angle were selected as input images. An image from this image set was selected and treated as an input image of an unknown shooting angle. All other images with the same identity as this input image were removed from the image sets of yaw angles varying from 40° to 80°, every 10°. 2. Second, the shooting angle for this input image was estimated using the algorithm presented in Section 2.4. 3. Repeating this process over all the images in the image sets at 40° angles provided an estimation accuracy at a 40° angle. 4. Performing the above steps 1-3 similarly on sets of images at 50° to 80° angles provided estimation accuracies for each angle. The accuracies from 40° to 80° angles were averaged. 5. A survey search of threshold-values that maximized the averaged accuracy was performed based on the coarse-to-fine approach.

Experiment for examining robustness

The effect of our proposed method using the estimated shooting angle, robustness was examined as follows:

1. From the feature vectors of registration data taken from 85°, feature vectors for the yaw angles 40°, 50°, 60°, 70° and 80° were estimated using algorithms LJT, PCA and MRA as demonstrated in 2.5-2.7. 2. As in Subsection 2.8, the LDA matrix was created using these estimated and registration datasets. Using this matrix, the registration, input and estimated datasets were all transformed into a coordinated dataset where discrimination was easier. 3. Similarity scores were obtained using normalized cross correlations taken against the 85° angle registration feature vector, and the estimated feature vector of the estimated shooting angle. Equal error rates were obtained from ROC using the algorithm in Section 3.5.

In summary, the following six cases were compared;

 LJT, 85° angle registration of the feature vector  LJT, estimated feature vector of the estimated shooting angle.  PCA, 85° angle registration of the feature vector  PCA, estimated feature vector of the estimated shooting angle  MRA, 85° angle registration of the feature vector  MRA, estimated feature vector of the estimated shooting angle Similarity scores obtained using the 85° registration feature vector corresponded to our previous method in [START_REF] Watabe | Improving the Robustness of Single-View-Based Ear Recognition When Rotated in Depth[END_REF]. Where the shooting angle estimation failed, the registration angle for 85° was used instead for correlation computation. A small number of such cases depended on the shooting angle (~2%).

The validity metrics for the experiments

As validity metrics for the verification experiments, the ROC, and the equal error rate EER are commonly used (10.6.3 of [START_REF]ISO/IEC: Information technology -Biometric performance testing and reporting -Part 1: Principles and framework[END_REF]). For computing these metrics, we used the algorithm recommended in Annex. F.1-2 of [START_REF]ISO/IEC: Information technology -Biometric performance testing and reporting -Part 1: Principles and framework[END_REF].

3.6

Results of the experiment for estimating shooting angle.

The accuracy at each threshold is demonstrated in Fig. 3. When the threshold value is 0.88, the maximum averaged accuracy is 46.6%. This is a somewhat encouraging result as an initial attempt, because this accuracy far exceeds that of a random answer (20%) to the question of five selective answers (40°, 50°, 60°, 70°, 80°). However, there seems to be considerable room to improve the estimation accuracy of the shooting angle.

Result of the experiment to examine robustness

Using the estimated shooting angle as determined in Subsection 3.6, equal error rates at various input yaw angles were obtained, as in Fig. 4. . The results using PCA and MRA were not particularly good. Equal error rates for the estimated data of the estimated shooting angles were worse than the equal error rates using registration data without shooting angle normalization. Similar to our previous report [START_REF] Watabe | Comparison of Ear Recognition Robustness of Single-View-Based Images Rotated in Depth[END_REF], the number of subjects may not be sufficient for accurately determining the principal component subspace and regression matrix. However, using LJT, the estimated data for the estimated shooting angle perform as accurately as our previous method using averaged estimated feature vectors of various poses.

Discussion and Conclusions

An initial attempt to estimate the shooting angle of an ear image is presented. Although the estimation accuracy was 48.8% and far exceeds the accuracy of a random answer (20%) to a question of five selective answers, there seems to be considerable room to improve it because the presented estimation algorithm for the shooting angle is not sophisticated. Using this estimated shooting angle, the estimation method of feature vector of other poses-PCA MRA and LJT-are examined. Although none perform beyond the accuracy of our previous method using averaged estimated feature vectors of various poses, LJT performs as accurately as our previous result. Hence, using LJT and refining the accuracy of the estimated shooting angle by improving the algorithm, there may be a chance to improve the robustness of singleview-based ear recognition.

Fig. 1 .

 1 Fig. 1. Feature points of an ear.

Fig. 2 .

 2 Fig. 2. Example of input images. (Yaw angles 0°, 90°, and 180° corresponding to frontal face, true left profile, and back)

Fig. 3 .

 3 Fig. 3. Accuracy at various thresholds.

Fig. 4 .

 4 Fig. 4. Equal error rate at various yaw angles.

Acknowledgments

The facial database was used with the permission of Softopia Japan Foundation. This work was supported by KAKENHI 22700219 and 80337609.


Jacek Grekow
email: j.grekow@pb.edu.pl

Audio Features Dedicated to the Detection of Four Basic Emotions

Keywords: Music emotion recognition, Audio feature extraction, Music information retrieval

In this paper, we decided to study the effect of extracted audio features, using the analysis tool Essentia, on the quality of constructed music emotion detection classifiers. The research process included constructing training data, feature extraction, feature selection, and building classifiers. We selected features and found sets of features that were the most useful for detecting individual emotions. We examined the effect of low-level, rhythm and tonal features on the accuracy of the constructed classifiers. We built classifiers for different combinations of feature sets, which enabled distinguishing the most useful feature sets for individual emotions.

Introduction

One of the most important elements when listening to music is the expressed emotions. The emotions contained in music can alter or deepen the emotional state of the listener. For example, the Funeral March listened to during a funeral deepens the emotional state of the departed's loved ones; while light and relaxing music listened to at home after a hard day's work can restore the listener's good mood. The elements of music that affect the emotions are timbre, dynamics, rhythm, and harmony. Changes in the types of instruments used, the dynamics, rhythm, and harmony change the emotions found in the music.

In the era of the Internet, searching music databases for emotions has become increasingly important. Automatic emotion detection enables indexing files in terms of emotions [START_REF] Grekow | Emotion Based MIDI Files Retrieval System[END_REF]. Automatic emotion detection also enables creating visual emotion maps of musical compositions [START_REF] Grekow | Mood tracking of musical compositions[END_REF].

In this paper, we decided to study the effect of extracted audio features, using the analysis tool Essentia [START_REF] Bogdanov | ESSENTIA: an Audio Analysis Library for Music Information Retrieval[END_REF], on the quality of constructed music emotion detection classifiers. We selected features and found sets of features that were the most useful for detecting individual emotions. We examined the effect of lowlevel, rhythm and tonal features on the accuracy of the constructed classifiers.

Studies on emotion detection in music are mainly based on two popular approaches: categorical or dimensional. The categorical approach [START_REF] Lu | Automatic mood detection and tracking of music audio signals[END_REF][5][6] [START_REF] Xu | Source Separation Improves Music Emotion Recognition[END_REF] describes emotions with a discrete number of classes -affective adjectives. In the dimensional approach [8][9][10], emotions are described as numerical values of valence and arousal. In this way, the emotion of a song is represented as a point on an emotion space. In this work, we used the categorical approach.

An important phase in emotion detection is feature extraction. There are several other studies on the issue of emotion detection using different audio tools for musical feature extraction. Studies [6][11][12] used a collection of tools that use the Matlab environment called MIR toolbox [START_REF] Lartillot | MIR in Matlab (II): A Toolbox for Musical Feature Extraction from Audio[END_REF]. Feature extraction library jAudio [START_REF] Mckay | jAudio: a feature extraction library[END_REF] was used in studies [START_REF] Xu | Source Separation Improves Music Emotion Recognition[END_REF] [START_REF] Lin | Exploration of music emotion recognition based on midi[END_REF]. Feature sets extracted from PsySound [START_REF] Cabrera | PSYSOUND: A computer program for psychoacoustical analysis[END_REF] were used in paper [START_REF] Aljanaki | Computational modeling of induced emotion using GEMS[END_REF], while studies [START_REF] Yang | A regression approach to music emotion recognition[END_REF][16] used the Marsyas framework [START_REF] Tzanetakis | Marsyas: A framework for audio analysis[END_REF]. The Essentia [START_REF] Bogdanov | ESSENTIA: an Audio Analysis Library for Music Information Retrieval[END_REF] library for audio analysis was used in studies [START_REF] Laurier | Automatic Classification of Musical Mood by Content-Based Analysis[END_REF] [START_REF] Sarasua | Support Vector Machine Active Learning for Music Mood Tagging[END_REF].

There are also papers devoted to the evaluation of audio features for emotion detection within one program. Song et al. [START_REF] Song | Evaluation of Musical Features for Emotion Classification[END_REF] explored the relationship between musical features extracted by MIR toolbox and emotions. They compared the emotion prediction results for four sets of features: dynamic, rhythm, harmony, and spectral features.

An important paper in the area of music emotion recognition was written by Yang et al. [START_REF] Yang | Machine Recognition of Music Emotion: A Review[END_REF], who did a comprehensive review of the methods that have been proposed for music emotion recognition. Kim et al. [START_REF] Kim | State of the Art Report: Music Emotion Recognition: A State of the Art Review[END_REF] presented another paper surveying the state of the art in automatic emotion recognition.

Music Data

In this research, we use four emotion classes: energetic-positive, energetic-negative, calm-negative, calm-positive. They are presented with their abbreviations in Table 1 and cover the four quadrants of the two-dimensional Thayer model of emotion [START_REF] Thayer | The Biopsychology of Mood and Arousal[END_REF]. They correspond to four basic emotion classes: happy, angry, sad, and relaxed. To conduct the study of emotion detection, we prepared two sets of data. One set was used for building one common classifier for detecting the four emotions, and the other data set for building four binary classifiers of emotion in music. Both data sets consisted of six-second fragments of different genres of music: classical, jazz, blues, country, disco, hip-hop, metal, pop, reggae, and rock. The tracks were all 22050Hz Mono 16-bit audio files in .wav format.

The author of this paper, a music expert with a university musical education, labeled the music samples. The music expert listened to six-second music samples and then labeled them with one of the emotions (e1, e2, e3, e4). In the case when the music expert was not certain which emotion to assign, such a sample was rejected. In this way, each file was associated with only one emotion/label.

The first training data set for emotion detection consisted of 324 files, 81 files labeled as e1, 81 files labeled as e2, 81 files labeled as e3, and 81 files labeled as e4.

We obtained the second training data from the first set. It consisted of four sets of binary data. For example, data set for binary classifier e1 consisted of 81 files labeled e1 and 81 files labeled not e1 (27 files each from e2, e3, e4). In this way, we obtained four binary data sets (consisting of examples of "e" and "not e") for four binary classifiers e1, e2, e3, e4.

Feature Extraction

For feature extraction, we used Essentia [START_REF] Bogdanov | ESSENTIA: an Audio Analysis Library for Music Information Retrieval[END_REF], a tool for audio analysis and audiobased music information retrieval. Essentia is an open-source C++ library, which was created at Music Technology Group, Universitat Pompeu Fabra, Barcelona.

We used Essentia version 2.0.1 (published in 02/2014), which contains a number of executable extractors computing music descriptors for an audio track: spectral, time-domain, rhythmic, tonal descriptors, and returning the results in YAML and JSON data formats.

The use of Essentia software entailed getting through installation procedures and the documentation. Launching the program required compiling the source code (C ++) and installing additional libraries.

Extracted features by Essentia are divided into three groups: low-level, rhythm and tonal features (Table 2).

Essentia also calculates many statistic features: the mean, geometric mean, power mean, median of an array, and all its moments up to the 5th-order, its energy, and the root mean square (RMS). To characterize the spectrum, flatness, crest and decrease of an array are calculated. Variance, skewness, kurtosis of probability distribution, and a single Gaussian estimate were calculated for the given list of arrays.

The previously prepared, labeled by emotion, music data sets served as input data for the Essentia tool used for feature extraction. For each 6-second file from the music data set, we obtained a representative single feature vector. The obtained lengths of feature vectors had 471 features.

Results

The Construction of One Classifier Recognizing Four Emotions

We built classifiers for emotion detection using the WEKA package [START_REF] Witten | Data Mining: Practical machine learning tools and techniques[END_REF]. During the construction of the classifier, we tested the following algorithms: J48, Ran-domForest, BayesNet, IBk (K-nn), SMO (SVM). The classification results were calculated using a cross validation evaluation CV-10. The first important result was that during the construction of the classifier we obtained the highest accuracy among all tested algorithms for SMO algorithm. SMO was trained using polynominal kernel. The results obtained for SMO algorithm are presented in Table 3. The result (classifier accuracy) improved to 64.50% after applying attribute selection (attribute evaluator: WrapperSubsetEval [START_REF] Kohavi | Wrappers for feature subset selection[END_REF], search method BestFirst).

The confusion matrix (Table 4), obtained during classifier evaluation, shows that the most recognized emotion was e2 (F-measure 0.727), and the next emotions were e1 and e3 (F-measure 0.653 and 0.65). The hardest emotion to recognize was e4 (F-measure 0.544).

From the confusion matrix, we can conclude that usually fewer mistakes are made between the top (e1, e2) and bottom (e3, e4) quadrants of the Thayer model. At the same time, recognition of emotions on the valence axis (positivenegative) is more difficult. The most important features (with group abbreviation) after applying attribute selection were:

-Energy of the Erbbands (L), -MFCC (L), -Onset Rate (R), -Beats Loudness Band Ratio (R), -Key Strength (T), -Chords Histogram (T).

In the selected features, we have a representative of low-level (L), rhythm (R) and tonal (T) features. This means that features of each of the three groups are important/useful during emotion detection.

The results were not satisfactory; classifier accuracy was too low (64.50%). It is difficult to build a good classifier that differentiates four emotions equally well. Some emotions have better recognition (e2) and others worse (e1, e3, e4), which lowers total classifier accuracy.

The Construction of Binary Classifiers

To improve emotion detection accuracy, we decided to build specialized binary classifiers for each emotion. A binary classifier algorithm can better analyze data sets for the presence of a given emotion.

During the construction of the binary classifiers, we tested the following algorithms: J48, RandomForest, BayesNet, IBk (K-nn), and SMO (SVM) on the prepared binary data. We calculated the classification results using a cross validation evaluation CV-10. Once again, we obtained the best results for SMO algorithm. The results are presented in Table 5. Accuracy improved (3-14 percentage points) for all four classifiers after applying attribute selection (attribute evaluator: WrapperSub-setEval, search method BestFirst).

The best results were obtained for emotion e2 (90.74%) and the worst results for emotion e4 (77.16%). We can conclude that in our case, emotions with a negative valence (e2, e3) are recognized better by approx. 10 percentage points than emotions with a positive valence (e1, e4). The obtained binary classifier accuracy results were higher (12-26 percentage points) than the accuracy of one classifier recognizing four emotions.

Table 6 presents the most important features obtained after feature selection (attribute evaluator: WrapperSubsetEval, search method BestFirst) for each emotion. In each feature set, we had a representative of low-level, rhythm features, even though we had different sets for each emotion. Only in the case of classifier e4, tonal features were not used. The energy of the bands was important for e1, e2, and e4 classifiers, but they differed as to which bands they pertain: e1 -Barkbands, e2 -Erbbands, and Melbands, e4 -Barkbands and Erbbands. High Frequency Content, which is characterized by the amount of high-frequency content in the signal is important for e3 and e4 classifiers. Beats Loudness Band Ratio (the beat's energy ratio on each band) was very important for emotion The of individual feature sets L, R or T did not have better results than their combinations. Combining feature sets R+T (rhythm and tonal features) improved classifier results in the case of classifiers e1 and e2. Combining feature sets L+T (low-level and tonal features) improved classifier results in the case of classifiers e2, e3 and e4.

Conclusions

In this paper, we studied the effect of extracted audio features, using the analysis tool Essentia, on the quality of constructed music emotion detection classifiers. The research process included constructing training data, feature extraction, feature selection, and building classifiers.

We built a classifier recognizing four basic emotions, but its accuracy was not satisfactory (64.50%). We then built binary classifiers dedicated to each emotion with accuracy from 77% to 90%. We obtained information on which features are useful in the detection of particular emotions.

We examined the effect of low-level, rhythm and tonal feature sets on the accuracy of the constructed binary classifiers. We built classifiers for different combinations of feature sets, which enabled distinguishing the most useful feature sets for individual emotions. The obtained results present a new and interesting view of the usefulness of different feature sets for emotion detection.

Classifier accuracy could be better. The process of searching for and assessing new features describing audio files continues.

Table 4 .

 4 Confusion matrix for the best result classified as -a b c d a

Table 1 .

 1 Description of mood labels

	Abbreviation	Description
	e1	energetic-positive
	e2	energetic-negative
	e3	calm-negative
	e4	calm-positive

Table 2 .

 2 The feature set obtained from Essentia

	Group	Group	Features
		Abbreviation
	Low-level features	L	Average Loudness
			Energy of the Barkbands
			Energy of the Erbbands
			Energy of the Melbands
			Dissonance
			Dynamic Complexity
			HFC (High Frequency Content)
			Pitch Salience
			Silence Rate
			Spectral Centroid
			Spectral Complexity
			Spectral Energy
			Spectral Energy Band High
			Spectral Energy Band Low
			Spectral Energy Band Middle High
			Spectral Energy Band Middle Low
			Zero Crossing Rate
			GFCC (Gammatone Feature Cepstral Coefficients)
			MFCC (Mel-Frequency Cepstral Coefficients)
	Rhythm features	R	Beats Loudness
			Beats Loudness Band Ratio
			BPM (The mean of the most salient tempo)
			BPM Histogram
			Danceability
			Onset Rate
	Tonal features	T	Chords Changes Rate
			Chords Number Rate
			Chords Strength
			Key Strength
			Chords Histogram
			HPCP (Harmonic Pitch Class Profile)

Table 3 .

 3 Accuracy obtained for SMO algorithm

	Accuracy
	Before attribute 59.26%
	selection
	After attribute 64.50%
	selection

Table 5 .

 5 Classifier accuracy for emotions e1, e2, e3, and e4 obtained for SMO

	Classifiers Classifiers Classifiers Classifiers
	for e1	for e2	for e3	for e4
	Before attribute 66.05%	87.04%	77.16%	65.43%
	selection
	After attribute 80.86% 90.74% 87.03%	77.16%
	selection

Table 7 .

 7 Classifier accuracy for emotions e1, e2, e3, and e4 obtained for combinations of feature sets

	Features set Classifiers Classifiers Classifiers Classifiers
		for e1	for e2	for e3	for e4
	L	72.22%	88.27%	79.62%	77.77%
	R	73.45%	82.09%	81.48%	72.83%
	T	72.22%	81.48%	76.54%	69.75%
	L+R	77.16%	88.88%	77.77%	77.16%
	L+T	79.01% 90.74% 86.41% 80.24%
	R+T	80.86% 90.12% 79.01%	73.45%
	All (L+R+T) 80.86% 90.12% 87.03% 77.16%

Acknowledgments. This paper is supported by the S/WI/3/2013.

detection because it was used in all sets. Another important feature was the tonal feature: Chords Histogram, which was used by e2 and e3 classifiers. Feature sets seem to logically describe the nature of each emotion. More energetic emotions are described by features pertaining to energy and rhythm, and more calm emotions by parameters such as rhythm and the amount of high frequency.

Evaluation of Different Combinations of Feature Sets

During this experiment, we evaluated the effect of various combinations of feature sets -low-level (L), rhythm (R), tonal (T) -on classifier accuracy obtained for SMO algorithm. We calculated the classification results using a cross validation evaluation CV-10. To improve the classification results, we used attribute selection (attribute evaluator: WrapperSubsetEval, search method BestFirst). The obtained results are presented in Table 7.

The obtained results indicate that the use of all groups (low-level, rhythm, tonal) of features resulted in the best accuracy in most cases (e1, e2, e3). The only exception was classifier e4, where using the set L+T (low-level, tonal) had better results (80.24%) than using all features -accuracy 77.16%.


Radko Mesiar
email: radko.mesiar@stuba.sk

Integrals based on monotone measure: optimization tools and special functionals

Keywords: Capacity, Choquet integral, Copula, Copula-based integral, Decomposition integral, Sugeno integral, Universal integral

Integrals on finite spaces (e.g., sets of criteria in multicriteria decision support) based on capacities are discussed, axiomatized and examplified. We introduce first the universal integrals, covering the Choquet, Shilkret and Sugeno integrals. Based on optimization approach, we discuss decomposition and superdecomposition integrals. We introduce integrals which are universal and decomposition (superdecomposition) ones and integrals constructed by means of copulas. Several distinguished integrals are represented as particular functionals. Finally, we recall also OWA operators and some their generalizations.

Introduction

Though integrals are, in general, introduced on general measurable spaces, in many computer-supported applications the finite spaces are considered, such as the sets of criteria, sets of rules, sets of players, etc. Therefore, in this paper we will consider universes X n = {1, ..., n} , n ∈ N, and σ-algebras 2 X n . More, we restrict our considerations into functions f : X n → [0, 1], i.e., we will consider n-dimensional vectors x = (x 1 , ..., x n ) = ( f (1), ..., f (n)) ∈ [0, 1] n as our integrands. Note that these integrands can be seen as membership functions of fuzzy subsets of X n , too [START_REF] Zadeh | Fuzzy Sets[END_REF]. The information about the weights of subsets of X n (e.g., groups of players, groups of criteria, etc.) is condensed into a capacity (fuzzy measure) m : 2 X n → [0, 1], which is supported to be monotone and m( / 0) = 0, m(X n ) = 1. Our overview of integrals on X n , i.e., of discrete integrals is organized as follows. In Section 2, we bring the concept of universal integrals originally introduced in [START_REF] Klement | A universal integral as common frame for Choquet and Sugeno integral[END_REF]. Section 3 is devoted to the decomposition integrals of Even and Lehrer [START_REF] Even | Decomposition-integral: unifying Choquet and concave integrals[END_REF] and the superdecomposition integrals of Mesiar et al [START_REF] Mesiar | Superdecomposition integrals[END_REF]. Here we describe also all integrals which are simultaneously universal and decomposition (superdecomposition). In Section 4, we introduce several copula-based universal integrals. Section 5 brings the characterization of some distinguished integrals as particular functionals. Finally, some concluding remarks are added, recalling OWA operators and some of their generalizations.

Though universal integrals were introduced in [START_REF] Klement | A universal integral as common frame for Choquet and Sugeno integral[END_REF] for any measurable space (X, A), any monotone measure m and all A-measurable functions f : X → [0, ∞], for our purposes we constrain them to act on finite spaces, and we will consider capacities and membership functions of fuzzy sets. For n ∈ N, let M n denote the set of all capacities on

X n . Definition 1. A mapping I : ∞ n=1 M n × [0, 1] n → [0, 1] is called a (discrete) universal integral whenever (i) there is a semicopula ⊗ : [0, 1] 2 → [0, 1] (i.e.,

an increasing binary function on [0, 1]

with neutral element e = 1) such that for any n ∈ N, m ∈ M n and

x = c • 1 A ∈ [0, 1] n , c ∈ [0, 1], it holds that I(m, x) = c ⊗ m(A), (ii) for any (m i , x i ) ∈ M n i × [0, 1] n i , i = 1, 2, such that for any t ∈ [0, 1], m 1 {i ∈ X n 1 | x (1) i ≥ t} ≤ m 2 { j ∈ X n 2 | x (2) j ≥ t} it holds that I(m 1 , x 1 ) ≤ I(m 2 , x 2 ).
Note that I is then increasing in both coordinates and -For an arbitrary semicopula ⊗ : [0, 1] 2 → [0, 1], the smallest universal integral that is linked to ⊗, is given, for any (m, x) ∈ M n × [0, 1] n , by

I(m 1 , x 1 ) = I(m 2 , x 2 ) whenever m 1 {i ∈ X n 1 | x (1) i ≥ t} = m 2 { j ∈ X n 2 | x
I ⊗ (m, x) = sup x i ⊗ m j ∈ X n | x j ≥ x i | i ∈ X n = sup x σ(i) ⊗ m ({σ(i), . . . , σ(n)}) , (1)
where σ : X n → X n is an arbitrary permutation such that x σ(1) ≤ . . . ≤ x σ(n) . In particular, if ⊗ = ∧ (minimum) then I ∧ = Su is the Sugeno integral [START_REF] Sugeno | Theory of Fuzzy Integrals and its Applications[END_REF], if ⊗ = • (product) then I • = Sh is the Shilkret integral [START_REF] Shilkret | Maxitive measure and integration[END_REF], and if ⊗ = T is a strict norm, then I T is the Weber integral [START_REF] Weber | Two integrals and some modified versions -critical remarks[END_REF], compare also N-integral of Zhao [START_REF] Zhao | N) fuzzy integral[END_REF]. -The Choquet integral [START_REF] Choquet | Theory of capacities[END_REF] that is given by

Ch(m, x) = n ∑ i=1 x σ(i) -x σ(i-1) • m ({σ(i), . . . , σ(n)}) (2)
(x σ(0) = 0 by convention), is a universal integral linked to the product. -The arithmetic mean of integrals Ch and Su, I = 1 2 (Ch + Su), is also a universal integral, and it is linked to the semicopula ⊗ :

[0, 1] 2 → [0, 1], a ⊗ b = 1 2 (ab + min{a, b}).
Note that the class of all discrete universal integrals is a convex bounded partially ordered set. Its bottom I T D is linked to the drastic product t-norm T D (the smallest semicopula), and, for all (m, x) ∈ M n × [0, 1] n , is given by

I T D (m, x) = max max x i | m j ∈ X n | x j ≥ x i = 1 , m ({i ∈ X n | x i = 1}) . Note that if m(A) < 1 whenever A = X n , then I T D (m, x) = max {min {x i | i ∈ X n } , m ({i ∈ X n | x i = 1})} .
On the other side, the top universal integral I ∧ is linked to the minimum ∧ (which is the greatest semicopula) and given by

I ∧ (m, x) = essup m (x) ∧ m(Supp(x)) = min max x i | m j ∈ X n | x j ≥ x i > 0 , m ({i ∈ X n | x i > 0}) .
Several other kinds of universal integrals will be discussed in Sections 3 and 4.

Decomposition and superdecomposition integrals

For X n = {1, . . . , n}, any non-empty subset B ⊆ 2 X n is called a collection, and any nonempty set H ⊆ 2 2 Xn \{ / 0} of collections is called a decomposition system. Denote by X n the set of all decomposition systems. We recall several examples of decomposition systems:

-H i = {(A 1 , . . . , A i ) | (A 1 , . . . , A i ) is a chain}, i = 1, . . . , n; -H i = {(A 1 , . . . , A i ) | (A 1 , . . . , A i ) is a dis joint system o f subsets}, i = 1, . . . , n; -H * = 2 X n \ { / 0}; -H A = {B ⊆ X n | A ⊆ B}.
Recently, Even and Lehrer [START_REF] Even | Decomposition-integral: unifying Choquet and concave integrals[END_REF] introduced decomposition integrals based on the idea of decomposition systems. These integrals can be viewed as a modification of the idea of the lower integral (inner measure). Similarly, the upper integral (outer measure) inspired the concept of superdecomposition integrals [START_REF] Mesiar | Superdecomposition integrals[END_REF].

Definition 2. For a given capacity m : 2 X n → [0, 1] and a fixed decomposition system

H ∈ X n , the corresponding decomposition integral I H ,m : [0, 1] n → [0, ∞[ is given by I H ,m (x) = sup ∑ j∈J a j m(A j ) | (A j ) j∈J ∈ H , a j ≥ 0, j ∈ J, ∑ j∈J a j 1 A j ≤ x , (3) and the corresponding superdecomposition integral I H ,m : [0, 1] n → [0, ∞[ is given by I H ,m (x) = inf ∑ j∈J a j m(A j ) | (A j ) j∈J ∈ H , a j ≥ 0, j ∈ J, ∑ j∈J a j 1 A j ≥ x . (4)
Note that:

- [START_REF] Yang | The pan-integral on fuzzy measure space (in Chinese)[END_REF]; -I H * = CV E is the concave integral of Lehrer [START_REF] Lehrer | A new integral for capacities[END_REF]; -I H * = CEX is the convex integral introduced in [START_REF] Mesiar | Superdecomposition integrals[END_REF]; -I H Xn = Min;

I H 1 = I H 1 = Sh is the Shilkret integral [18]; -I H n = I H n = Ch is the Choquet integral [1]; -I H n = PAN is the PAN integral of Yang
-I H Xn = Max.
The relationships between these integrals are visualized in Figure 1. -

I H 1 ,m (x) = I H 1 ,m (x) = 0.4 (Shilkret integral); -I H 2 ,m (x) = 0.49; -I H 3 ,m (x) = I H 3 ,m (x) = 0.53 (Choquet integral); -I H 2 ,m (x) = 0.59; -I H 3 ,m (x) = 0.65 (PAN integral); -I H * ,m (x) = 0.65 (concave integral); -I H * ,m (x) = 0.49 (convex integral); -I H X 3 ,m (x) = 0.4; -I H X 3 ,m (x) = 0.7.
Decomposition and superdecomposition integrals are positively homogeneous. Hence, if a universal integral I is also a decomposition (superdecomposition) integral on each X n , n ∈ N, then it is necessarily linked to the product

•, I(m, c • 1 A ) = c • m(A).
The integrals, which are both universal and decomposition integrals or universal and superdecomposition integrals, were characterized by Mesiar and Stupňanová in [START_REF] Mesiar | Decomposition integrals[END_REF]. Note that we will use the same notation H i , i ∈ N, for decomposition systems related to the chains of length at most i, independently of the underlying space X n . Obviously, for a fixed n ∈ N, then H n = H n+k for each k ∈ N.

Theorem 1. Let I be a universal integral, I = Ch, which is also a decomposition (superdecomposition) integral on each X n , n ∈ N. Then I = I H k for some k ∈ N (I = I H k for some k ∈ N).

Recall that I H 1 = Sh is the Shilkret integral. Moreover, for a fixed n ∈ N, I H n = Ch is the Choquet integral on X n , and then

I H n = I H n+1 = I H k for each k ≥ n.
Similarly,

I H n = Ch = I H k for each k ≥ n.
The family I H n n∈N I H n n∈N forms a chain of universal integrals

I H 1 ≤ I H 2 ≤ . . . ≤ I H k ≤ . . . ≤ Ch ≤ . . . ≤ I H k ≤ . . . ≤ I H 2 ≤ I H 1 , Ch = sup k∈N I H k = inf k∈N I H k .
Note also that the only integral on X n , which is both decomposition and superdecomposition integral, is the Choquet integral.

Copula-based universal integrals

Copulas (of dimension 2) are in fact joint distribution functions (restricted to [0, 1] 2 ) of random vectors (X,Y ) such that both X and Y are uniformly distributed over [0, 1]. Two extremal copulas are the minimum M, M(x, y) = min{x, y} and the Fréchet-Hoeffding lower bound W given by W (x, y) = max{0, x + y -1}. The third distinguished copula is the independence copula Π, Π(x, y) = x • y. For more details see [START_REF] Nelsen | An Introduction to Copulas[END_REF]. Note that any copula C : [0, 1] 2 → [0, 1] is in a one-to-one correspondence with a probability measure

P C : B([0,1] 2 ) → [0, 1]
with uniformly distributed margins. In particular, for a Borel subset E ⊆ [0, 1] 2 we have:

-P M (E) = µ 1 ({x ∈ [0, 1] | (x, x) ∈ E}), -P W (E) = µ 1 ({x ∈ [0, 1] | (x, 1 -x) ∈ E}), -P Π (E) = µ 2 (E),
where µ i is the standard Lebesgue measure on Borel subsets of R i , i = 1, 2. Inspired by Imaoka [START_REF] Imaoka | On a subjective evaluation model by a generalized fuzzy integral[END_REF], Klement et al. [START_REF] Klement | Measure-based aggregation operators[END_REF] introduced copula-based integrals

I C : n∈N M n × [0, 1] n → [0, 1]
given by

I C (m, x) = P C (u, v) ∈ [0, 1] 2 | v ≤ m({i ∈ X n | x i ≥ u}) = n ∑ i=1 C x σ(i) , m ({σ(i), . . . , σ(n)}) -C x σ(i-1) , m ({σ(i), . . . , σ(n)}) , (5)
σ being a permutation ensuring x σ(1) ≤ . . . ≤ x σ(n) .

Note that I Π = Ch is the Choquet integral and I M = Su is the Sugeno integral, and that I C is a universal integral linked to ⊗ = C. Recently, we introduced [START_REF] Klement | Universal integrals based on copulas[END_REF] hierarchical families of copula-based universal integrals, which extend the results from Section 4.

M n × [0, 1] n → [0, 1], given by I C,k (m, x) = sup k ∑ j=1 C x σ(i j ) , m A i j ) -C x σ(i j-1 ) , m A i j ,
where A i j = σ(i j ), . . . , σ(n) and 0 = i 0 < i 1 < . . . < i k ≤ n, and

I C,k (m, x) = inf k ∑ j=1 C x σ(i j ) , m Ãi j-1 -C x σ(i j-1 ) , m Ãi j-1 ,
where Ãi j-1 = σ(i j-1 ) + 1, . . . , σ(n) and

0 = i 0 < i 1 < . . . < i k ≤ n, with convention σ(i 0 ) = 0, are universal integrals linked to ⊗ = C. Moreover, I C,1 ≤ I C,2 ≤ . . . ≤ I C ≤ . . . ≤ I C,2 ≤ I C,1 ,
and, for a fixed n ∈ N,

I C,n = I C = I C,n . Obviously, if C = Π, then I Π,k = I H k and I Π,k = I H k , k ∈ N.
Recall that the discrete Lebesgue integral in our framework is just the weighted arithmetic mean, L(x) = ∑ n i=1 w i x i , and it is related to the probability measure P on X n , where P({i}) = w i . L can be seen as an additive aggregation function. Note that A : [0, 1] n → [0, 1] is an aggregation function [START_REF] Grabisch | Aggregation Functions[END_REF] if and only if A is nondecreasing in each coordinate and A(0, ..., 0) = 0, A(1, ..., 1) = 1. Hence an aggregation function is the Lebesgue integral once A(x + y) = A(x) + A(y) for all x, y, x + y ∈ [0, 1] n , and the corresponding capacity (probability) P is given by P(E) = A(1 E ). For several other distinguished integrals we have their characterization as special functionals:

-Choquet integral is a comonotone additive aggregation function [START_REF] Schmeidler | Integral representation without additivity[END_REF]; -Sugeno integral is a comonotone maxitive and min-homogeneous aggregation function [START_REF] Marichal | An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework[END_REF]; -Shilkret integral is a comonotone maxitive and positively homogeneous aggregation function [START_REF] Grabisch | Aggregation Functions[END_REF]; concave integral is the smallest concave functional on [0, 1] n satisfying I(1 E ) ≥ m(E) [START_REF] Even | Decomposition-integral: unifying Choquet and concave integrals[END_REF]; convex integral is the greatest convex functional on [0, 1] n such that I(1 E ) ≤ m(E) [START_REF] Mesiar | Superdecomposition integrals[END_REF].

Recall that two vectors x, y ∈ [0, 1] n are comonotone if x i > x j excludes y i < y j , i.e., (x ix j ) (y iy j ) ≥ 0 for each i, j ∈ X n .

Concluding remarks

We have introduced and discussed several kinds of discrete integrals with respect to capacities. As a particular example, often exploited in numerous applications, we recall OWA operators [START_REF] Yager | On ordered weighted averaging aggregation operators in multi-criteria decision making[END_REF] as the mappings OWA w : [0, 1] n → [0, 1], which are defined by

OWA w (x) = n ∑ i=1 w i x σ(i) ,
where w = (w 1 , . . . , w n ) ∈ [0, 1] n is a normed weighting vector, i.e.,

n ∑ i=1 w i = 1.
Considering a capacity m : 2

X n → [0, 1] given by m(A) = card(A) ∑ i=1
w n-i+1 , Grabisch [START_REF] Grabisch | Fuzzy integral in multicriteria decision making[END_REF] showed that OWA w = Ch m is the Choquet integral with respect to the capacity m. A capacity, which only depends on the cardinality of measured sets, is called a symmetric capacity. It can be viewed as a basis for generalizations of OWA operators. The first generalization yields OMA (Ordered Modular Average) operators, which were introduced by Mesiar and Zemánková in [START_REF] Mesiar | The Ordered Modular Averages[END_REF]:

Definition 3. Let m : 2 X n → [0, 1
] be a symmetric capacity generated by a normed weighting vector w = (w 1 , . . . , w n ), and let C :

[0, 1] 2 → [0, 1] be a fixed copula. Then the integral I C (m, •) : [0, 1] n → [0, 1] is called an OMA operator.
Note that OMA operators were characterized axiomatically as comonotone modular symmetric idempotent aggregation functions, i.e., an OMA : [0, 1] n → [0, 1] satisfies the following properties:

(i) OMA(x∨y)+OMA(x∧y) = OMA(x)+OMA(y) for any comonotone couple x, y ∈ [0, 1] n (i.e., for x and y with the property (x ix j )(y iy j ) ≥ 0 for any i, j ∈ X n ); (ii) OMA(x) = OMA(x σ(1) , . . . , x σ(n) ) for any permutation σ : X n → X n ; (iii) OMA(c, . . . , c) = c for any c ∈ [0, 1]. By [START_REF] Grabisch | Aggregation Functions[END_REF], OMA C,m is given by

OMA C,m (x) = n ∑ i=1 f i (x σ(i) ), where f i : [0, 1] → [0, 1] is given by f i (t) = C(t, w i + . . . + w n ) -C(t, w i+1 + . . . + w n )).
Thus, f i is an increasing 1-Lipschitz function that satisfies f i (0) = 0, f i (1) = w i , and If C = Π then f i (t) = w i t, and in that case OMA operators are simply OWA operators (and they are characterized by comonotone additivity, which is a genuine property of the Choquet integrals [START_REF] Schmeidler | Integral representation without additivity[END_REF]).

For C = M, f i (t) = max{0, min{t -(w i+1 + . . . + w n ), w i }}, and hence

OMA M,m (x) = n i=1
x σ(i) ∧ (w i + . . . + w n ),

i.e., OMA M,m is the ordered weighted maximum [START_REF] Dubois | A review of fuzzy aggregation connectives[END_REF]. Other kinds of OWA generalizations are based on Theorem 2, and again, they can be introduced for an arbitrary copula C. We restrict our considerations to the independence copula Π only. Then, for a fixed n ∈ N and a symmetric capacity m on X n generated by a normed weighting vector w = (w 1 , . . . , w n ), all integrals I Π,k (m, •) and I Π,k (m, •), k ∈ {1, . . . , n}, can be viewed as generalizations of OWA operators and

I Π,n (m, •) = I Π,n (m, •) = OWA w .
In general, for a fixed k ∈ {1, . . . , n}, it holds

I Π,k (m, x) = max k ∑ j=1
x σ(i j )x σ(i j-1 ) • (w i j + . . . + w n ) , where 0 = i 0 < i 1 < . . . < i k ≤ n, and

I Π,k (m, x) = min k ∑ j=1
x σ(i j )x σ(i j-1 ) • (w i j-1 +1 + . . . + w n ) , with 0 = i 0 < i 1 < . . . < i k ≤ n.

( 2 )

 2 j ≥ t} for each t ∈ [0, 1] (compare the equality of the expected values of random variables with the same distribution function). Moreover, I(m, 1 A ) = m(A) and I(m, c • 1 X n ) = c for all m ∈ M n , A ⊆ X n and c ∈ [0, 1]. We recall several distinguished examples:

Fig. 1 . 1 .

 11 Fig. 1. Hasse diagram of some decomposition and superdecomposition integrals

Theorem 2 .

 2 Let C : [0, 1] 2 → [0, 1] be a copula and fix k ∈ N. Then the mappings I C,k and I C,k : n∈N

  n

  t) = t for any t ∈ [0, 1].

Acknowledgment. The support of grant APVV-0013-14 is kindly acknowledged.