domain: math shs

text:
Theognosia Michailidou
email: edtgm@leeds.ac.uk

Non-mathematics majors doing statistics: Factors behind performance

Keywords: Statistics education, cognitive and non-cognitive factors, achievement

Main objectives and theoretical foundation of the research study

Nowadays, a large proportion of undergraduate university students are required to undertake at least one statistics or statistics-related course. The main consideration guiding my research study is the exploration of several cognitive and non-cognitive factors, the "other" outcomes of statistics education [START_REF] Schau | Students' attitudes: The "other" important outcome in statistics education[END_REF], and their relationship with the students' academic performance in introductory statistics courses offered at tertiary institutions. Cognitive, affective and motivatio na l factors under investigation include: students' feelings, attitudes and beliefs about statistics; students' anxiety feelings and perceived self-efficacy regarding statistics; motivational orientatio ns, achievement goals, students' interest in and engagement with statistics (such as effort, persistence, learning strategies); resilient behaviour characteristics when learning and studying statistics; and prior mathematics or/and statistics background and performance. A proposed theoretical causal model, including direct and indirect effect of these variables on achievement in statistics, will be tested. Moreover, the study focuses on exploring and documenting non-mathematicians' perceptions, behaviors, challenges and experiences when completing a statistics course.

My theoretical foundation is supported by some key theories: Social-cognitive learning theory [START_REF] Bandura | Motivational processes affecting learning[END_REF] and Self-efficacy theory [START_REF] Bandura | Self-efficacy: Toward a unifying theory of behavioral change[END_REF], Expectancy-Value theory [START_REF] Wigfield | Expectancy-value theory of achievement motivatio n[END_REF] and Achievement Goal theory (Dweck, 1986). Some ideas behind these theories have been adapted to statistics learning experiences and they have been used as a guide for both the quantitative and qualitative strands of my research work.

Research design and methodology, and data collection and analysis procedures

In order to accomplish the research aims and goals, a mixed-methods research design (that is a combination of quantitative and qualitative data collection methods) has been employed. A selfreported questionnaire, which was designed and developed specifically for the purposes of the doctoral study, has been administered to a larger sample of students in university classroom settings.

It comprised of open-and closed-ended questions, and Likert-type questions on a 5-point scale.

Individual (and pair) face-to-face semi-structured interviews were conducted with a sample of participants who had completed the questionnaire and consented to be interviewed.

The data collection procedure can be summarized in three many stages (not in chronological order): quantitative data-gathering (executed in two phases -at the beginning and at the end of the period of instruction of various statistics courses); qualitative data-gathering; and participants' final grades obtained at the end of a statistics course from each of the instructors. The data collection was carried out over a period spanning two academic semesters, the fall semester 2015 and the spring semester 2016. I collected data from six universitiesall the recognized universities (both public and private) operate in Cyprus. I observed and gathered data from 35 statistics classes (23 courses) taught by 15 different instructors. I collected over 1000 questionnaires and I executed 60 face-to-face intervie ws. The participants, with a variety of mathematics background and experience, came from diverse academic departments and degree programmes.

The quantitative data will be analysed using a variety of methods: basic statistical techniq ues (descriptive statistics, reliability estimates, correlation coefficients, χ 2 tests, t-tests, analysis of variance and regression analysis); advanced statistical methods (multilevel analysis, factor analys is and structural equation modelling techniques). Τhe qualitative data will be coded using thematic analysis approach.

Potential significance and contribution

This study may act as a springboard for further research in Cyprus and cross-cultural comparisons between Cyprus and other countries. The major findings and recommendations of the research work may constitute a helpful tool for the statistics instructors and stakeholders of statistics education in implementing interventions, innovations and instructional strategies/ practices to develop and improve the quality, the efficacy and the relevance of statistics courses that provided for nonmathematics majors.

The poster for the CERME-10 conference includes a brief introductory statement of the background, context and objectives of this doctoral study as well as information about the research design and methodology and data collection and analysis processes. Also, some preliminary findings (along with visual and graphical representations) are presented.

Proceedings of CERME10


Wei Tan

Guy A Boy

Keywords: Commercial Aircraft, Onboard Information System, Human-Centered Design, Tangible Interactive System, Avionics, Context

Pilots currently use paper-based documentation and electronic systems to help them perform procedures to ensure safety, efficiency and comfort on commercial aircrafts. Management of interconnections among paper-based operational documents can be a challenge for pilots, especially when time pressure is high in normal, abnormal, and emergency situations. This dissertation is a contribution to the design of an Onboard Context-Sensitive Information System (OCSIS), which was developed on a tablet. The claim is that the use of contextual information facilitates access to appropriate operational content at the right time either automatically or on demand. OCSIS was tested using humanin-the-loop simulations that involved professional pilots in the Airbus 320 cockpit simulator. First results are encouraging that show OCSIS can be usable and useful for operational information access. More specifically, contextsensitivity contributes to simplify this access (i.e., appropriate operational information is provided at the right time in the right format. In addition, OCSIS provides other features that paper-based documents do not have, such as procedure execution status after an interruption. Also, the fact that several calculations are automatically done by OCSIS tends to decrease the pilot's task demand.

Introduction

An airplane consists of a number of mechanical and computerized systems. An airplane cannot stop or brake in the air, and fuel is consumed during the entire flight. Consequently, flight time is limited. Flight crewmembers have all the capacities and limitations of any human being; they can be qualified as human operators. They typically collaborate, communicate, and cooperate with each other to execute flight tasks. Actions performed by flight crewmembers in the cockpit must adhere to procedures in context. Onboard paper-based operational documents barely provide context-sensitive information. Therefore, context has to be handled by pilots. Procedures execution is a major safety factor. Until now, specific pilot roles have been supported by paper-based documentation in both operations and training. It is a pilot's job to make decisions, act, communicate, cooperate, and coordinate operationally, with procedures established in operational documentation developed through airline policy and governmental regulation. All the procedures and information can be found from documents. The onboard documents can be categorized into four kinds of documents: flying documents, which are related to all flight operations; systems documents, which include systems' theory, principles, and controls; navigation documents, which are the charts that pilots use on the flight deck; and performance documents, which provide operational data for all flight phases such as takeoff, landing, and go-around [START_REF] Tan | From commercial aircraft operational procedures to an onboard context-sensitive information system[END_REF].

However, onboard paper-based documents are not the only resources that pilots have in the cockpit. Several other onboard systems can enhance the pilot's awareness of aircraft status (i.e., aircraft states). They can provide very comprehensive information on the state of the aircraft in an integrated way. Taking Airbus Electronic Centralized Aircraft Monitor (ECAM) as an example, it provides actions together with corresponding flight parameters to pilots who have to deal with related malfunctions. It provides steps to handle failures for a large number of situations [2].

2

State of art

Tablet-based systems onboard

Pilots are familiar with paper-based manuals, which are easy to use, tag, mark, and retain, even though they are heavy and difficult to carry. Nobody can permanently remember all procedures and technical knowledge, particularly, under time pressure. Now many applications on iPad that contain paper charts information are available. Moreover, Boeing introduced a tablet-based version of the paper Quick Reference Handbook (QRH) used by flight crews in 2013 [3]. The Interactive QRH offers advanced navigation and search capabilities to enable the pilot to easily find the proper checklist (see Fig. 1). The Interactive QRH also simplifies non-normal checklist use, especially for those checklists in which the correct condition must be selected from two or more choices. In line with technological innovation, Airbus has developed iPad EFB solutions to provide airlines with an alternative to PC operating system EFB devices (see Fig. 1). With -FlySmart with Airbus‖ applications on iPad, pilots will be able to compute performance calculations and also consult Airbus Flight Operations Manuals from a light hand-held device [START_REF]Combining Airbus' EFB content with the world's most versatile mobile device[END_REF][START_REF][END_REF].

Even though Electronic Flight Bag (EFB) and Onboard Information Systems (OIS) are advanced tools to assist pilots' work, operational documents are still in their original format and arrangement. Not all abnormal procedures are available on the ECAM, nor do all types of aircrafts have ECAM or some other electronic systems to process and display procedures. Unlike paper, computer support enables easy contextualiza-tion of provided information. The Onboard Context-Sensitive Information System (OCSIS) is introduced into commercial aircraft cockpit to make the flight safe, efficient, and comfortable by providing assistance in normal and abnormal situations and enhanced capabilities of interaction with other onboard systems.

Human-centered design approach

-A human-in-the-loop (HITL) simulation is a modeling framework that requires human interaction. This approach is typically called participatory design. The emergence of HITLS technologies, therefore, enables researchers and practitioners to investigate the complexity of human-involved interactions from a holistic, systems perspective [6]‖.

Fig. 2. Human-Centered Design approach [7]

As shown in Fig. 2, the model typically represents reality in a simplified way. It proposes important elements and their relevant interconnections in an appropriate, orchestrated manner. The simulation represents the interaction that brings the model to life, which can be used to improve understanding of interactions among different elements that the model implements. It is also used to improve the model itself and eventually modify it [START_REF] Boy | From automation to tangible interactive objects[END_REF]. HITLS is used early on during the design phase. Consequently, Human-Centered Design (HCD) has been used to incrementally improve OCSIS toward an acceptable mature version (i.e., incremental prototype development, test, and modification) [START_REF] Boulnois | The Onboard Context-Sensitive Information System for Commercial Aircraft[END_REF]. Modeling OCSIS requires pilots' involvement, and interaction with other onboard systems. The process can be run on a flight simulator, which in turn produces experimental data that could be used to improve OCSIS.

2.3

Context-sensitive procedures -Context is any information that can be used to characterize the situation of an entity. An entity is a person, place, or object that is considered relevant to the interaction between a user and an application, including the user and applications themselves [START_REF] Dey | Understanding and using context[END_REF]‖. Pilots must accomplish flight tasks (e.g., cockpit preparation, takeoff, approach, and landing procedures) within the appropriate context depend on flight phases. We use Interaction Blocks to represent, implement, and handle context-sensitive procedures (see Fig. 3). Fig. 4 provides an example of iBlock for flaps setting. An interaction block is defined by: a set of actions; and a situation pattern that includes triggering preconditions and a context pattern; and post-conditions that include a goal and abnormal conditions [START_REF] Boy | Cognitive function analysis[END_REF]. Let's take the example of the -Flaps Set‖ procedure in the approach scenario (see Fig. 4). Its triggering preconditions consist of -Flaps position‖ (i.e., visible on the E/WD screen) and -Flaps handle position‖ (i.e., visible on the Flaps lever). When the -Flaps position‖ on E/WD equals to the value of the -Flaps handle position,‖ then the goal is reached and the pilot can continue to next procedure; otherwise, a pop-up window is displayed on OCSIS to inform about an abnormal situation, and the -Flaps Locked‖ procedure needs to be executed. This procedural knowledge representation was developed during the 1990s to represent operational procedures in aircraft cockpits and led to deeper investigations on context representation also. It is therefore very appropriate for context-sensitive procedural information representation.

Design of OCSIS prototype

OCSIS is currently programed using Objective-C, an object-oriented language available on Apple's hardware, on Xcode, the Integrated Development Environment for Objective-C. Human-Centered Design institute (HCDi) is equipped with two commercial aircraft simulators, A320 and B737, that were developed using Prepare3d, a Lockheed Martin software, which allows vehicle control (e.g., marine, terrestrial, air, or spatial) in a totally virtual world that is simulated to be close to reality. The first prototype of OCSIS is applying A320 procedures and references. Once the application starts on the iPad, a Welcome page is displayed. The default/initial page displays procedures and actions that crews need to perform or have performed. A menu is provided to select other tabs at the top of this page (see Fig. 5).

Fig. 5. OCSIS's Menu

At the end of the 1990s, EURISCO and Airbus carried out a study with 60 commercial pilots on how electronic documentation could be structured [START_REF] Blomberg | Information Needs for Flight Operations: Human-Centered Structuring of Flight Operations Knowledge[END_REF]. Results showed that it could be best structured into three information levels. We adopted this re-organized structure for OCSIS (see Fig. 6): Fig. 6. Level 1 and Level 2 information of APU Bleed set Level 1: Need to know or safety-critical information that the pilot needs to have immediately.

Level 2: Nice to know or short explanations of Level 1. Level 3: Understand technical knowledge on systems' principles and troubleshooting.

In order to keep consistency of Airbus's philosophy and make all the actions are easy to be understand, Dynamic Color System (DCS) is designed to enhance the pilot's perception and comprehension of the current situation., as shown in Table 1. Different colors stand for different meanings that provide the pilot with a direct and swift status of procedures. The items marked with -*‖ are updated after a series of formative evaluations. Once the action is completed and OCSIS can access the related parameters' status, it automatically becomes -green.‖ In the current version of OCSIS, this kind of automation is done for only the a few parameters, which can be detected (i.e., colors change automatically). For all other parameters, the pilot marks them -DONE‖ manually (i.e., the action line becomes green). The pilot can also postpone an action by selecting the -WAIT‖ button, and then the action line becomes amber (see Fig. 7). Fig. 7. Cyan, green, amber, and white for action status -Scenarios of human-computer interaction help us to understand and to create computer systems and applications as artifacts of human activityas things to learn from, as tools to use in one's work, as media for interacting with other people [START_REF] Carroll | Five reasons for scenario-based design[END_REF]‖. The current version of OCSIS includes several normal procedures and two abnormal scenarios. -Initial Approach‖ and -Final approach‖ are the scenarios that we choose for normal procedures. -Fuel Leak‖ and -Flaps Locked‖ are the scenarios that we choose for abnormal procedures. Context patterns trigger procedures in real-time both in normal and abnormal situations. In an abnormal situation such as -Flaps Locked,‖ OCSIS will immediately inform the pilot about this malfunction by displaying a popup information window (see Fig. 8). Pilots can become aware of the problem through the pop-up window and start following actions. If they choose to do it later, a reminder line (see Fig. 9) will be displayed at the bottom of the interface, which directs to additional -Flaps Locked‖ procedures. The first testing was carried out at HCDi's simulator lab. Four pilots with flight experience were chosen as participants. They were involved in two sessions. The first session consisted of performing all required procedures both in normal and abnormal scenarios using paper-based manuals. The second session consisted of performing the same procedures using OCSIS. There was an interval of a few days between the two sessions for each person.

Pilot participants provided excellent feedback on actions to use OCSIS, look and feel, and other usability criteria. The results showed that every pilot understood how to use OCSIS. They all reported that OCSIS was easy to hold and use (see Fig. 10). Pilots provided feedback on information icons, color, and size, which was used to improve user interaction with OCSIS. Results showed that all pilots, except one (who said that size of the items was not big enough), felt comfortable with OCSIS information display and interaction (see Fig. 11). Based on pilots' feedbacks we made improvements on part of the icons' color and buttons' function to OCSIS, also changed flight phase titles' size, position and function to remove the ambiguity. A grey color code was added to the system representing the -Not Applicable‖ case to assist pilots' decision-making. There were procedures that are embedded into other procedures that cause recursivity issues, e.g., the Engine 1(2) Relight procedure is embedded into the -Fuel Leak‖ procedure. During this phase of testing, we simply added the content of the Engine 1(2) Relight procedure into the -Fuel Leak‖ part, but we subsequently developed a generic hyperlinked iBlock system within OCSIS that enables to automatically put into an operational sequence.

An integration survey was administered to the pilots. Results showed that all pilots prefer that the iPad be fixed in the cockpit rather than available as a handheld device. Four options were suggested (see Fig. 12): (1) Next to side-stick for each pilot; (2) On the windshield with a flexible arm; (3) In the pedestal or on the instrument panel as a unit; (4) Inside a box at the side of the pedestal.

Fig. 12. Options of iPad's location

The second testing was carried out at a flight training center in China. Twenty-two A320 pilots, including eleven captains and eleven first officers, participated in the testing, performing as aircrew in A320 simulators. We used the same route and protocols as the first testing. During the testing, we observed in the -Fuel Leak‖ scenario pilots easily established the failure and excluded the irrelevant procedures (e.g., -Fuel imbalance‖ procedure). In the -Flaps Locked‖ scenario, OCSIS reduced the chance of wrong calculation of landing distance and approach speed. Regarding OCSIS look and feel, pilots were asked to evaluate OCSIS usability in terms of colors, interactive buttons, and other devices available on the OCSIS iPad. Results showed that three pilots had difficulty during training to select buttons (see Fig. 13).

Fig. 13. Look and feel feedback

Two pilots thought OCSIS was not reactive enough; this was due to the fact that data-link was not available on the simulator at that time. We corrected that. Pilot-OCSIS interaction was evaluated using pilots' feedback on information icons, color, and size (see Fig. 14). The results showed that pilot-OCSIS interaction is satisfactory in general.

Fig. 14. Interaction feedback

Based on pilots' feedback, we made improvements to OCSIS after second testing: -Engine Shutdown‖ and -Engine Relight‖ procedures were included in the -Fuel Leak‖ scenario. Pilots may shut down engine to check if fuel leak is from engine or somewhere else. When pilots move the thrust lever to idle position, the low-speed rotor (N1) of engine is going to be 0, which triggers the -Engine Failure‖ procedure both on ECAM and OCSIS that increases redundancy. Pilots should perform ECAM actions first and then come back to OCSIS to complete additional actions of -Engine Shutdown‖ procedure. This being done, pilots can continue executing the -Fuel Leak‖ procedure, and pilots had to remember to go back procedure. Pilots may decide to relight the engine if they check and discover that the fuel leak is not coming from the engine but from elsewhere. The -Engine Relight‖ procedure is provided on the same page (see Fig. 15). Fourteen pilots preferred the iPad located close to each side-stick (Position 1 in Fig. 12). Eight pilots preferred the iPad installed in the middle of instrument panel (Position 2 in Fig. 12). We chose Position 1 as the test location of OCSIS in the third testing. A flexible arm was set up to hold the iPad near the side-stick on the right side of the simulator.

The third testing was held with six pilots participated, and three additional pilots took the Nielsen's ten Usability Heuristics survey [START_REF] Nielsen | Usability Engineering[END_REF], using the same protocols and timelines as first and second testing. The results of user-system interaction questionnaires showed that every pilot understood and was satisfied with using OCSIS and with OCSIS's user interaction, as well as the location of OCSIS in the cockpit. Pilots were required to assess OCSIS look and feel by evaluating display usability in terms of color, buttons, and other OCSIS devices. The results show that all pilots had no trouble to understand the system (see Fig. 16). Pilot-OCSIS interaction was assessed on the basis of pilots' feedback on information icons, color, and size (see Fig. Based on Nielsen's ten Usability Heuristics survey [START_REF] Nielsen | Usability Engineering[END_REF] which helped us prioritize issues with respect to those that users found critical to those that may not be critical, we made the following improvements to OCSIS after third testing mainly on design decision phase: 1. Quick maneuverability to specific procedures/menus instead of scrolling (see Fig. 18 and Fig. 19). It is possible to fit each flight phase on a single page, and pilots can move left and right to review procedures for other flight phases. In the meantime, a menu to select a particular flight phase can be set at the top of the screen.

Pilots should be free to move through menus; information flow progress is saved on each page.

Discussion

OCSIS was first designed to be used as a tangible interactive system (TIS) onboard a commercial aircraft [START_REF] Tan | Iterative Designs of Onboard Context-Sensitive Information System (OCSIS)[END_REF]. Onboard paper-based documentation has been used from the beginning of aviation history and is tangible for pilots to use. Tablets and some applications (e.g., Jeppesen Mobile FliteDeck) have been authorized to be used on the flight deck by the FAA. Consequently, we considered that tablets are tangible objects that can support OCSIS software. This is physical tangibility, but figurative tangibility should be tested [START_REF] Boy | Tangible Interactive Systems -grasping the world with computers[END_REF]. Figurative tangibility in this case means for pilots to keep correct cognition using OCSIS. The testing studies were conducted to provide a first set of methods and tools with this figurative tangibility assessment. And it is possible for us to extend this assessment. The testing studies that were performed showed that this hypothesis was confirmed on a fully equipped cockpit simulator in realistic flight operations scenarios with professional pilots. It is obvious that more iteration is needed and will be implemented in the near future to get a mature version of OCSIS. This paper provides a first iteration of participatory design of OCSIS. More generally, it shows the shift from the traditional automation approach, where additional software was added to the cockpit and induced some kinds of rigidity that sometimes resulted in unexpected situations, to TIS design, where tangibility has to be tested using situation awareness models and criteria [START_REF] Tan | Contribution to the Onboard Context-Sensitive Information System (OCSIS) of Commercial Aircraft[END_REF]. Of course, the concept of tangibility is more complex and will require more investigation.

Conclusion

This research and HCD effort are based on both participatory design and agile development (i.e., at the end of each phase, the system is testable in an HITLS environment). This is now typical for the design and development of tangible interactive objects [START_REF] Boy | From automation to tangible interactive objects[END_REF], and more generally tangible interactive systems (TISs) [START_REF] Boy | Tangible Interactive Systems -grasping the world with computers[END_REF], where the problem is no longer automation but the search for tangibility. Modeling and simulation are very useful to explore possibilities and drawbacks of these TISs. The quality of both simulation capabilities and pilot participants is crucial [START_REF] Boulnois | The Onboard Context-Sensitive Information System for Commercial Aircraft[END_REF]. If the issues traditionally raised by human factors and ergonomics specialists when engineering work is done are now posed at the beginning of the design phase in a virtual world (i.e., virtual engineering is part of HCD), new kinds of questions would emerge from this practice, that is, tangibility [START_REF] Tan | Iterative Designs of Onboard Context-Sensitive Information System (OCSIS) for Commercial Aircrafts[END_REF].

OCSIS is a comprehensive system that aims to make flight of commercial aircraft safe, efficient, and comfortable. The multiple usability evaluations and user-centered assessments performed on the system can discover the maximum number of issues. First designs of OCSIS were based on our creativity process, in the sense of synthesis and integration, on previous expertise and experience in the commercial aviation domain, more specifically, work done by Blomberg, Speyer and Boy [START_REF] Blomberg | Information Needs for Flight Operations: Human-Centered Structuring of Flight Operations Knowledge[END_REF] on the three layers of electronic operations documentation and Ramu's [START_REF] Ramu | Contextual operational documentation effectiveness on airline pilots' performance[END_REF] dissertation work on onboard context-sensitive information systems. This work is typically based on human-computer interaction (HCI), hypertext, and context-sensitive information systems work. Usability testing brought us a series of usability issues that helped us to concretely improve OCSIS. Although not all the solutions could be addressed due to time, these can be addressed in further work.

Fig. 1 .

 1 Fig. 1. Interactive QRH in B737NG [3] on the left and Airbus FlySmart [5] on the right

Fig. 3 .Fig. 4 .

 34 Fig. 3. Interaction block representation[START_REF] Boy | Cognitive function analysis[END_REF]

Fig. 8 .Fig. 9 .

 89 Fig. 8. -Flaps Locked‖ triggering

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Look and feel feedback

Fig. 15 .

 15 Fig. 15. Engine Relight procedure.

  [START_REF] Tan | Iterative Designs of Onboard Context-Sensitive Information System (OCSIS)[END_REF]

  ).

Fig. 16 .

 16 Fig. 16. Look and feel feedback

Fig. 17 .

 17 Fig. 17. Interaction feedback

Fig. 18 .Fig. 19 .

 1819 Fig. 18. Procedure's headings before improvement

Fig. 20 .Fig. 21 .

 2021 Fig. 20. Checklist's icon before improvement

Fig. 22 .Fig. 23 .

 2223 Fig. 22. The title of abnormal procedure before improvement

Table 1 .

 1 Color codes

	Color	Representation
	Cyan	Actions to be performed
	Green	Actions performed
		Marked as performed
	Amber	Postponed actions or checks
		*The title of abnormal procedures
		Cautions
	Red	*The title of emergency procedures
		Warnings
	White	Notes
		More information for actions
		*The title of flight phase
		*Normal Checklists
	Magenta	Restrictions or constraints
	Grey	*Not applicable for current context

-Ready to do‖ actions are in cyan.

This work was done while Dr. Boy was Dean and University Professor at Florida Institute of Technology. Ref.: Tan, W. & Boy, G.A. (2018). Tablet-based Information System for Commercial Air-craft: Onboard Context-Sensitive Information System (OCSIS). HCI International Conference, Orlando, Florida.

Acknowledgments

We thank Sebastien Boulnois and Anouk Mirale, who developed a fair amount of OCSIS software, and Varun Korgaonkar who designed the usability questionnaires for the third testing. Thanks to our HCDi colleagues, Jarrett Clark, provided software development support for this project. Dr. Alexandre Lucas Stephane, helped me with the design of testing and the analysis of experimental results, and Dr. Ondrej Doule, who helped me with the design theory and methods. Thanks to all who participated in the various tests and provided me with their time and great feedback to improve OCSIS. Thanks to Dr. Barbara K. Burian, Dr. Divya Chandra, Dr. Christophe Kolski, and Dr. Scott Winter, who provided me with feedback, suggestions, and comments on this project.

Tablet-based Information System for Commercial Aircraft: Onboard Context-Sensitive Information System

(OCSIS)


Ye Jiang

Aik Kong

Zhenmin Lee

Bin Tang

Anthony Ma

Haizhou Larcher

Li

Larcher

Plda

Kong Aik Lee
email: kalee@i2r.a-star.edu.sg

Zhenmin Tang

Bin Ma

Anthony Larcher

Haizhou Li

PLDA Modeling in I-Vector and Supervector Space for Speaker Verification

Keywords: speaker verification, i-vector, probabilistic LDA

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

Introduction

Over recent years, many approaches based on the use of Gaussian mixture models (GMM) in a GMM-UBM framework [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture model[END_REF] have been proposed to improve the performance of speaker verification system. Inspired by the joint factor analysis (JFA) approach [START_REF] Kenny | Speaker and session variability in GMM-Based speaker verification[END_REF][START_REF] Kenny | A study of inter-speaker variability in speaker verification[END_REF], it was shown in [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF] that the problem of session variability can be coped with by confining the variability within a low-dimensional subspace, referred to as the total variability space, in the parameter space of GMM.

The idea of defining a total variability space is to extract low dimensional identity vectors (the so-called i-vectors), by which speech segments of variable-length can be represented as fixedlength vectors [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF]. Such a representation greatly simplifies the modeling and scoring processes in speaker verification. For instance, we could assume that the i-vectors are generated from a Gaussian density instead of the mixture of Gaussian densities usually considered for the case of acoustic features [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture model[END_REF]. In this regard, linear discriminant analysis (LDA) [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF], probabilistic LDA (PLDA) [START_REF] Prince | Probabilistic linear discriminant analysis for inferences about identity[END_REF], and the heavy-tailed PLDA [START_REF] Kenny | Bayesian speaker verification with heavy-tailed priors[END_REF] have shown to be effective for such fixed-length data. In this paper, we shall focus on PLDA with Gaussian prior instead of heavy-tailed prior as it was recently shown in [START_REF] Garcia-Romero | Analysis of i-vector length normalization in speaker recognition systems[END_REF] that the advantage of the heavytailed assumption diminishes with a simple length-normalization on the i-vector preceding PLDA modeling.

Since the total variability space is always spanned by a lowrank rectangular matrix, a dimension reduction process is also imposed by the i-vector extractor. In this paper, we advocate the use of uncompressed form of i-vector. Similar to that in [START_REF] Dehak | Front-end factor analysis for speaker verification[END_REF], our extractor converts speech sequence into fixed-length vector, but retains its dimensionality in the full supervector space. Modeling of speaker and session variability is then carried out with PLDA, which has shown to be effective in handling high-dimensional data [START_REF] Prince | Probabilistic linear discriminant analysis for inferences about identity[END_REF]. By doing so, we avoid reducing the dimensionality of the i-vector twice -first in the extraction process and second in the PLDA model. Any dimension reduction procedure will unavoidably discard information. Our intention is therefore to keep the full dimensionality till the scoring stage with PLDA and to investigate the performance of PLDA in the supervector space.

The downside of using uncompressed form of i-vector (we call this i-supervector hereafter to avoid confusion) with PLDA is that we have to deal with large matrices. The size of the matrices becomes enormous when more sessions are available for each speaker in the development data 1 . One option is to estimate the subspaces in a decoupled manner, which might lead to suboptimal solution [START_REF] Kenny | Speaker and session variability in GMM-Based speaker verification[END_REF][START_REF] Kenny | A study of inter-speaker variability in speaker verification[END_REF]. In this paper, we show how the subspaces can be jointly estimated by partitioning large matrices into sub-matrices, thereby making the matrix inversion and the joint estimation feasible. A significant advantage of PLDA approach for speaker verification is the use of Bayes factor [START_REF] Brümmer | A farewell to SVM: Bayes factor speaker detection in supervector space[END_REF] in computing the verification score. In this regard, we show how to manipulate large matrices efficiently in computing the PLDA score. In addition, we also look into various normalization methods and introduce the use of Gaussianized rank-norm for PLDA.

The paper is organized as follows. In Section 2, we look at the scenario where dimension reduction is performed twice when PLDA is used on i-vector. Section 3 shows that inversion of large matrices encountered in PLDA can be solved by exploiting some inherent structure of the matrices. Section 4 deals with PLDA scoring and introduces the Gaussianized rank norm. We present some experimental results in Section 5 and conclude the paper in Section 6.

PLDA for i-vector and i-supervector

From i-vector to i-supervector

An i-vector represents a variable-length speech utterance as a low-dimensional vector (low as compared to the dimensionality of the mean supervector). The generative equation is given by

m Tx , (1)
where m and are the mean supervectors of the speaker (and session) dependent GMM and the UBM, respectively. The subspace spanned by the columns of the matrix T captures the speaker and session variability (hence the name total variability).

The latent variable x is taken to be a low-dimensional random vector with a standard normal distribution. For each observation sequence , the i-vector is given by the posterior mean of the latent variable x , i.e., | E x . Since T is always a lowrank rectangular matrix, the dimensionality D of the i-vector is much smaller compared to that of the supervector, i.e., D C F . Here, F is the dimensionality of the acoustic feature and C is the number of mixture in the GMM.

Consider the case where we allow the latent variable to grow into the full supervector space, in which D C F . The generative equation is now given by m Dz .

(

) 2
Similar to that in (1), the i-supervector is taken as the posterior mean of the latent variable:

| E z
. The difference here is that D is a CF-by-CF diagonal matrix so that the i-supervector has the same dimensionality as the mean supervector m. The isupervector extractor can easily be implemented by adopting the diagonal model in JFA [START_REF] Kenny | Speaker and session variability in GMM-Based speaker verification[END_REF][START_REF] Kenny | A study of inter-speaker variability in speaker verification[END_REF] with a slight modification. The matrix D is trained per utterance instead of per speaker basis in order to capture both speaker and session variability.

PLDA

The advantage of i-vector and i-supervector representations is that they represent a speech segment as a fixed-length vector instead of a variable-length sequence of vectors. Let ij be a fixed-length vector representing the j-th session of the i-th speaker with the assumption that each speaker has multiple sessions in the development set. Taking ij as input, PLDA assumes that it is generated from a Gaussian density, as follows

T T | , ij ij p μ F F G G Σ , (3)
where μ denotes the mean vector and

T T Γ F F G G Σ is the covariance matrix.
The modeling capability of PLDA is determined by the structured modeling of the covariance Γ . To understand this, we rewrite (3) in the form of marginal density, as follows | ,

ij ij i ij i ij i ij p p p p d d h w h w h w , (4)
where the conditional and prior densities are given by

| , | , ij i ij i ij p h w μ F h G w Σ , (5) 0
, i p h I , (6) 0
, ij p w I . (7)
In the above equations, i h is the speaker-specific latent variable pertaining to the i-th speaker, while ij w is the session-specific latent variable corresponds to the j-th session of the i-th speaker. The low-rank matrices F and G are used to model the subspaces pertaining to speaker and session variability, while the diagonal matrix Σ mops up the remaining variability. Looking at [START_REF] Prince | Probabilistic linear discriminant analysis for inferences about identity[END_REF], the mean of the conditional distribution is given by

T T T | , ij i ij i ij E h w μ F G h w . ( 8
)
Comparing ( 1) and ( 8), we see that both i-vector extractor and PLDA model involve dimension reduction via similar form of subspace modeling. This observation motivates us to explore the use of PLDA on i-supervector in the original supervector space. The extraction process serves as the front-end which converts a variable-length sequence to a fixed-length vector without reducing the dimension.

Joint-estimation of posterior means

We estimate the parameters , , , μ F G Σ of the PLDA model using the expectation maximization (EM) algorithm. To this end, we assume that our development set consists of speech samples from N speakers each having J number of sessions. Notice that the number of sessions J could be different for each speaker. In the E-step, the inference of the posterior means involves the inversion of the precision matrix L, as shown below:

1 T 1 T 1 T 1 T 1 T 1 T 1 1 T 1 T 1 T 1 T 1 0 0 . 0 0 0 0 J F Σ F I F Σ G F Σ G FΣ G G Σ F G Σ G I L G Σ F GΣ G I G Σ F G Σ G I (9)
The matrix is large as we consider the joint inference of latent variables representing the speaker, i h , and for all sessions

,1 i w , ,2 i w , , , i J w
from the same speaker. The size of the matrix increases with the number of sessions J, though more sessions is always desirable for more robust estimation of parameter.

The matrix L possesses a unique structure since all sessions from the same speakers are tied to one speaker-specific latent variable. As shown in [START_REF] Stolcke | Nonparametric feature normalization for SVM-based speaker verification[END_REF], the matrix L can be partitioned into four smaller blocks. Let A, B, C, and Φ denotes the four sub- matrices, the inverse is given by

1 1 1 1 1 1 1 A B M M B Φ L C Φ Φ CM Φ Φ CMBΦ . ( 10
)
Inversion of Φ , in the left hand side of (10), is simple as it is block diagonal. Let

1 T 1 Q G Σ G I and T 1 Λ QG Σ F , the sub-matrix M is given by 1 T 1 J M F Σ F GΛ I . ( 11
)
Using above results, it can be shown that the posterior means of the speaker and session specific latent variables are given by

T 1 T T 1 1 1 J J i i j i j j j E h M F Σ MΛ G Σ , (12)
T 1 ij ij i E E w Q G Σ Λ h , (13)
where ij ij μ denotes the i-vectors or i-supervectors centralized with the global mean ,

1 ij i j N J μ .
Similarly, the M-step can also be formulated in terms submatrices. Let T ij i ij w h w by appending i h to each session ij w belonging to the same speaker. We update the PLDA model, as follows:

1 T , , ij ij ij ij i j i j E E F G w w w , ( 14
) T T , 1 ij ij ij ij i j diag E NJ Σ F G w . ( 15
)
The second-order moment in ( 14) is obtained for each individual session of all speakers, as follows

T ij ij ij ij E E E M MΛ w w w w Λ M Q Λ M Λ . ( 16
)
Notice that in (14), the subspaces F and G are estimated jointly, whereby the correlation between speaker and session variability are considered.

PLDA verification score

Given two i-supervectors (or i-vector) 1 2 , that correspond to the train and test segments, respectively, the verification score is calculated as the log-likelihood ratio between two hypotheses {H 0 , H 1 }:

1 2 0 1 2 1 log , | log , | s p H p H . (17)
Here, 0 H and 1 H correspond to the models as shown in Fig. 1. The model 0 H hypothesizes that 1 2 , belong to the same speaker and hence share the same speaker-specific latent variable 1,2 h . On the other hand, the model 1 H hypothesizes that they belong to different speakers and hence have separate latent variables 1 h and 2 h . The solution for (17) can be given by

2 T 2 2 1 T 1 1 1 1 1 1 2 2 2 l l l l l l l s I I K K K ( 18
)
where K is a constant consisting of the determinant of matrices, which diminishes when score normalization is applied. The other two variables are defined as follows:

1 T K F GG Σ F , ( 19
) 1 T l l F GG Σ μ . ( 20
)
The matrix inversion 1 T

GG Σ

can be solved using the matrix inversion lemma, as follows

1 1 T 1 1 T 1 T 1 GG Σ Σ Σ G G Σ G I G Σ . ( 21
)
Notice that in (20), the centralized vector is projected onto the subspace F where speaker information co-vary the most (i.e., dimension reduction), while de-emphasizing the subspace pertaining to channel variability. Notice that the transformation matrix 1 T F GG Σ , in (19) into (20), should be computed by multiplying F to the right-hand-side of (21).

Another prerequisite for good performance with PLDA is that the i-supervectors have to follow a normal distribution, as in (3). It has been shown in [START_REF] Garcia-Romero | Analysis of i-vector length normalization in speaker recognition systems[END_REF], for the case of i-vector, that whitening followed by length normalization help toward this goal. However, whitening is not feasible for i-supervector due to data scarcity. To this end, we propose in this paper a Gaussianized version of rank norm [START_REF] Stolcke | Nonparametric feature normalization for SVM-based speaker verification[END_REF]. The i-supervector is processed element-wise with warping functions mapping each dimension to a standard Gaussian distribution (instead of uniform distribution as in rank norm). To put it mathematically, let

m m l m m b B b r B , ( 22
)
where denotes the cardinality of a set. The Gaussianized value is then obtained by using the inverse CDF of a standard Gaussian distribution (i.e., the probit function), as follows

1 2 erf 2 1 m m l r , ( 23
)
where 1 erf denotes the inverse error function. This can then be followed by length normalization prior to PLDA modeling.

Experiment

Experiments were carried out on the core task (short2-short3) of NIST SRE08. We use two well-known metrics in evaluating the performance, namely, equal error rate (EER) and minimum detection cost (MinDCF). Two gender-dependent UBMs consisting of 512 Gaussians were trained using data drawn from the SRE04. Speech parameters were represented by a 54dimensional vector of mel frequency cepstral coefficients (MFCC) with first and second derivatives appended.

The total variability matrix T in (1) and D in (2) were both trained with similar set of data drawn from Switchboard, SRE04 and SRE05. We use 500 factors for T, while D is a diagonal matrix by definition. The dimensionality of i-vector is therefore 500, while i-supervector is of dimensionality CF = 27648. The rank of the matrices F and G in the PLDA model is set to 300 and 200, respectively, for the case of i-supervector. For i-vector, best result is found with the rank of F set to 300 and using a full matrix for Σ , in which G is no longer required. This observation is consistent with that reported in [START_REF] Kenny | Bayesian speaker verification with heavy-tailed priors[END_REF].

Feature and score normalization

The first set of experiments is to investigate the effectiveness of different normalization methods on i-vector and i-supervector prior to PLDA modeling (i.e., length normalization, whitening and Gaussianized rank norm) and on the score (we used s-norm as reported in [START_REF] Kenny | Bayesian speaker verification with heavy-tailed priors[END_REF]). For simplicity, we used only telephone data and report the results on det6 (i.e., tel-tel subtask) in TABLE I. Length normalization (len) always outperforms raw for both i-vector and i-supervector. Whitening followed by length normalization (white+len) further improves the performance for i-vector. Similarly in the case of i-supervector, we used Gaussianized rank-norm followed by length normalization (grank+len) to cope with the high dimensionality. Finally, we also notice that s-norm gives consistent improvement for both i-vector and i-supervector.

Without any normalization (raw) i-supervector performs better than i-vector. One possible reason is that the Gaussian assumption in (3) can be better fulfilled in the supervector space with higher dimensionality compared to that of the i-vector. However, after applying a full normalization (white +len+snorm, grank+len+snorm), i-vector outperforms isupervector. This is consistent for both MALE and FEMALE sets in terms of EER and MinDCF. Notice that i-vector gain huge improvement from length normalization. For the MALE case, we observed 20.0% and 6.5% of relative improvement in EER when length normalization was applied on i-vector and isupervector, respectively. One avenue to explore for isupervector is a better normalization method beside length normalization and Gaussianized rank-norm. , are from the same speakers. The alternative hypothesis states that they are from different speakers.

Performance comparison

We compared the performance of i-supervector and i-vector using JFA as baseline and under different train-test channel conditions, namely, det1 (int-int), det4 (int-tel), det5 (tel-mic) and det6 (tel-tel) as defined in NIST SRE08 short2-short3 core task. The PLDA models used for i-vector and i-supervector were the same as described in Section 5.1. In addition, we included microphone data (drawn from SRE05 and SRE06) for the whitening transform, Gaussianized rank-norm and s-norm to handle the interview (int) and microphone (mic) channel conditions. The JFA was trained as follows. The eigenvoice loading matrix (with 300 factors) was train on telephone data drawn from Switchboard. The eigenchannel loading matrix (with 150 factors) was trained using both telephone and microphone data drawn from SRE04, SRE05 and SRE06. Finally, the diagonal model was trained using telephone data drawn from SRE04. TABLE II shows the results when full normalization (i.e., white+len+snorm for i-vector, grank+len+snorm for isupervector, and zt-norm for JFA) was applied. Here, we consider the EER and MinDCF by pooling together the male and female scores. Similar to the observation in Section 5.1, i-vector gives better performance than i-supervetor for the case with full normalization, except in det5 where i-supervector gives a much lower EER though the MinDCF is slightly worse. This again shows that current normalization strategy (Gaussianized ranknorm followed by length normalization) thought effective, has to be further improved. Compared to the JFA baseline, PLDA methods (both i-vector and i-supervector) give competitive performance with slightly lower EER and MinDCF in det4, det5 and det6.

Conclusions

We have introduced the use of uncompressed form of i-vector (i.e., the i-supervector) for PLDA-based speaker verification. Similar to i-vector, an i-supervector represents a variable-length speech utterance as a fixed-length vector. But different from ivector, we keep the total variability space having the same dimensionality as the original supervector space. To this end, we showed how manipulation of large matrices can be done efficiently in training and scoring with the PLDA model. We also introduced the use of Gaussianized rank-norm for feature normalization prior to PLDA modeling.

Compared to i-vector, we found that i-supervector performs better when no normalization (on both feature and score) was applied. This suggests that Gaussian assumption imposed by PLDA becomes less stringent and easier to fulfill in the higher dimensional i-supervector space. However, the performance improvement given by the high dimensionality diminishes when full normalization is applied. As such, current normalization strategy, though effective, has to be improved for better performance. This is a point for future work. Finally, it is also evident that PLDA methods based on i-vector or i-supervector give competitive performance compared to the state-of-the-art JFA.

  denotes the elements of the isupervector l . We first get the normalized rank of m l with respect to a background set m B , as follows :

Figure 1 :

 1 Figure 1: PLDA for verification task. The null hypothesis H 0 states that the observations 1 2, are from the same speakers. The alternative hypothesis states that they are from different speakers.

TABLE I :

 I Performance comparison of various normalization methods on i-vector and i-supervector evaluated on NIST SRE08, det6 subtask of short2-short3.

	I-VECTOR		MALE EER MinDCF	FEMALE EER MinDCF
	raw	6.1785	3.1206	8.1486 3.7028
	len	4.9411	2.6286	6.4409 3.0581
	white+len	4.5458	2.4546	6.3193 3.0065
	white+len+snorm	4.3478	2.2155	6.1530 3.0034
	I-SUPERVECTOR		MALE EER MinDCF	FEMALE EER MinDCF
	raw	5.2632	2.6605	6.7976 3.3368
	len	4.9199	2.6271	6.3667 3.3624
	grank+len	4.8982	2.6676	6.0976 3.2588
	grank+len+snorm	4.5888	2.3737	6.2639 3.1132
	TABLE II: Performance comparison under various train-test channel
	conditions: det1, det4, det5 and det6 subtasks of short2-short3,
	NIST SRE08.
		det1 (int-int)	det4 (int-tel)
		EER		MinDCF	EER	MinDCF
	i-vector	7.2964		3.5189	5.7919	2.7576
	i-supervector	7.8769		3.6724	5.9421	3.0541
	JFA	7.1404		3.7991	7.6815	3.0350
		det5 (tel-mic)	det6 (tel-tel)
		EER	MinDCF	EER	MinDCF
	i-vector	6.0462		2.0975	5.5602	2.7556
	i-supervector	4.7554		2.2475	5.7740	2.8949
	JFA	7.2690		2.6126	6.8126	3.0960

The number of sessions is usually limited in face recognition for which PLDA was originally proposed in[START_REF] Prince | Probabilistic linear discriminant analysis for inferences about identity[END_REF].

Acknowledgements

The work of Ye Jiang is partially supported by Jiangsu Provincial Natural Science Foundation of China under Grant No. CX2211_0261.


Pierre-Michel Bousquet
email: pierre-michel.bousquet@univ-avignon.fr

Anthony Larcher
email: alarcher@i2r.a-star.edu.sg

Driss Matrouf
email: driss.matrouf@univ-avignon.fr

Jean-Franc ¸ois Bonastre
email: jean-francois.bonastre@univ-avignon.fr

Oldřich Plchot
email: iplchot@fit.vutbr.cz

Variance-Spectra based Normalization for I-vector Standard and Probabilistic Linear Discriminant Analysis

Keywords: i-vectors, probabilistic linear discriminant analysis, speaker recognition

I-vector extraction and Probabilistic Linear Discriminant Analysis (PLDA) has become the state-of-the-art configuration for speaker verification. Recently, Gaussian-PLDA has been improved by a preliminary length normalization of i-vectors. This normalization, known to increase the Gaussianity of the i-vector distribution, also improves performance of systems based on standard Linear Discriminant Analysis (LDA) and "two-covariance model" scoring. But this technique follows a standardization of the i-vectors (centering and whitening ivectors based on the first and second order moments of the development data). We propose in this paper two techniques of normalization based on total, between-and within-speaker variance spectra 1 . These "spectral" techniques both normalize the i-vectors length for Gaussianity, but the first adapts the ivectors representation to a speaker recognition system based on LDA and two-covariance scoring when the second adapts it to a Gaussian-PLDA model. Significant performance improvements are demonstrated on the male and female telephone portion of NIST SRE 2010.

Introduction

Based on Factor Analysis (FA), the Total Variability space framework has become a new standard for speaker verification systems. Total Variability space provides a compact representation of speech sessions, so called i-vectors, that carries out the classification task into a low dimensional factor space rather than in the GMM-super-vector space. By nature, the Total Variability space contains the overall between-utterance variability and it is ultimately the role of intersession compensation methods to define how speakers are discriminated from one another in this subspace.

The initial i-vector speaker recognition system developed by Dehak et al. [START_REF] Dehak | Front-End Factor Analysis for Speaker Verification[END_REF][START_REF] Dehak | Support Vector Machines versus Fast Scoring in the Low-Dimensional Total Variability Space for Speaker Verification[END_REF] enhanced discrimination in Total Variability space by reducing the dimension of i-vectors using standard Linear Discriminant Analysis (LDA) and Within-Class Covariance Normalization (WCCN). Classification was performed by using a cosine kernel. Later, normalized and weighted versions of LDA have been proposed [START_REF] Mclaren | Sourcenormalised and weighted LDA for robust speaker recognition using I-Vectors[END_REF] to address the issue of unbalanced development data.

Since generative models have been introduced, i-vectors are seen as an observation from a probabilistic model, ignoring their extraction mechanism. The two-covariance model introduced in the speaker recognition field by N. Brummer et al. [START_REF] Brummer | The speaker partitioning problem[END_REF] proposes an efficient alternative to existing scoring. This generative model is a particular case of Probabilistic Linear Discriminant Analysis (PLDA), itself a special case of Factor Analysis (FA) [START_REF] Simon | Probabilistic linear discriminant analysis for inferences about identity[END_REF] as it considers a single component. Gaussian-PLDA (G-PLDA) assumes Gaussian priors of both channel and speaker factors. Introduced in [START_REF] Kenny | Bayesian speaker verification with heavytailed priors[END_REF], Heavy-tailed version of PLDA (HT-PLDA) replaces Gaussian distributions with Student-t distributions and was shown to substantially improve the performance.

Recently it has been shown [START_REF] Garcia | Analysis of i-vector length normalization in speaker recognition systems[END_REF][START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF] that preliminary standardization and length normalization of i-vectors (normalizing centered and scaled i-vectors by their magnitude) significantly improves performance of Gaussian-based systems like two-covariance scoring [START_REF] Brummer | Fully Bayesian likelihood ratio vs i-vector length nor-malization in speaker recognition systems[END_REF], Mahalanobis Scoring [START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF] or Gaussian PLDA [START_REF] Garcia | Analysis of i-vector length normalization in speaker recognition systems[END_REF]. In particular, Gaussian-PLDA preceded by a standardization and length normalization compares very favorably with the Heavy Tailed version of PLDA [START_REF] Garcia | Analysis of i-vector length normalization in speaker recognition systems[END_REF]. This work deals with the length normalization step.

Once norms of all the i-vectors are equal, the representation space becomes a spherical surface of finite volume. In this context, we propose two normalization techniques based on total, between-and within-speaker distributions of development data. The first one is adapted to a speaker recognition system based on LDA and two-covariance scoring. This normalization aims at enhancing the optimization criterion of LDA. The second one is adapted to a Gaussian-PLDA model. It moves data towards an appropriate starting point in the optimization landscape of Gaussian-PLDA.

Next two sections give first a description of the i-vector paradigm then describe LDA, PLDA and two-covariance models. Section 4 describes the two "variance-spectra based normalization" techniques after presenting a simple visualization tool dedicated to the analysis of development dataset before and after transform. Section 5 analyses the behaviour of these techniques on the telephone portion of NIST SRE 2010.

I-vector paradigm

The i-vector approach has become state of the art in the field of speaker verification [START_REF] Dehak | Front-End Factor Analysis for Speaker Verification[END_REF][START_REF] Dehak | Support Vector Machines versus Fast Scoring in the Low-Dimensional Total Variability Space for Speaker Verification[END_REF]. In this approach, an i-vector extractor converts a sequence of acoustic vectors into a single lowdimensional vector representing the whole speech utterance. The speaker-and session-dependent super-vector s of concatenated Gaussian Mixture Model (GMM) means is assumed to obey a linear model (Factor Analysis) of the form:

s = m + Tw ( 1
)
where m is the mean super-vector of the Universal Background Model (UBM), T is the low-rank variability matrix obtained from a large dataset by MAP estimation [START_REF] Kenny | Joint factor analysis of speaker and session variability: Theory and algorithms[END_REF] and the standardnormally distributed latent variable w is the resulting i-vector.

I-vectors recognition systems

To improve comparison of i-vectors in a speaker verification trial, dimensionality reduction techniques, like Linear Discriminant Analysis (LDA) [START_REF] Dehak | Front-End Factor Analysis for Speaker Verification[END_REF][START_REF] Dehak | Support Vector Machines versus Fast Scoring in the Low-Dimensional Total Variability Space for Speaker Verification[END_REF][START_REF] Brummer | Fully Bayesian likelihood ratio vs i-vector length nor-malization in speaker recognition systems[END_REF], can be applied. I-vectors can also be seen as observations from a probabilistic generative model (two-covariance model [START_REF] Brummer | The speaker partitioning problem[END_REF], Probabilistic Linear Discriminant Analysis [START_REF] Simon | Probabilistic linear discriminant analysis for inferences about identity[END_REF][START_REF] Kenny | Bayesian speaker verification with heavytailed priors[END_REF]). In the following, these dimension reduction techniques and generative models used in our experiments are described.

Linear Discriminant Analysis

Standard Linear Discriminant Analysis (LDA) is a technique for dimensionality reduction that projects the data onto a subspace which satisfies the requirement of maximizing betweenclass variance and minimizing within class variance. The LDA optimization problem (finding a basis of this subspace) can be defined according to the following ratio:

J (v) = v t Bv v t Wv ( 2
)
where B is the between-speaker covariance matrix and W the within-speaker covariance matrix defined by:

B = S s=1 ns n (ys -µ) (ys -µ) t (3)
W = 1 n S s=1 ns i=1 (w s i -ys) (w s i -ys) t ( 4
)
where ns is the number of utterances for speaker s, n is the total number of utterances, w s i are the i-vectors of sessions of speaker s, ys is the mean of all the i-vectors of speaker s and µ represents the overall mean of the training dataset.

This ratio of Equation 2 is often referred to as the Rayleigh coefficient for space direction v. The optimal subspace is comprised by the first eigenvectors (those with highest eigenvalues) of W -1 B.

In the speaker recognition domain, it turned out that better performance were achieved [START_REF] Dehak | Front-End Factor Analysis for Speaker Verification[END_REF][START_REF] Dehak | Support Vector Machines versus Fast Scoring in the Low-Dimensional Total Variability Space for Speaker Verification[END_REF] by replacing B and W with the scatter-matrices:

S B = S s=1 (ys -µ) (ys -µ) t (5)
S W = S s=1 1 ns ns i=1 (w s i -ys) (w s i -ys) t (6)
Unlike B and W, these matrices do not take into account prior of each speaker, i.e. the amount of utterances per speaker training sample. In the following, we refer to LDA B,W (resp. LDA S B ,S W ) for linear discriminant analysis based on the Rayleigh coefficient computed with B and W (resp. S B and S W ).

Probabilistic Linear Discriminant Analysis

PLDA Model

Introduced in [START_REF] Simon | Probabilistic linear discriminant analysis for inferences about identity[END_REF], the Gaussian Probabilistic Linear Discriminant Analysis (PLDA) is a generative i-vector model which assumes that each p-dimensional i-vector w of a speaker s can be decomposed as:

w = µ + Φys + Γz + ε (7)
This model comprises two parts: (i) the component µ + Φys which depends only on the speaker, (ii) the noise component Γz + ε which is different for every session and represents within-speaker noise. Matrix Φ is rectangular, with rvoices columns (rvoices < p) providing a basis for a speaker subspace, usually called "eigenvoices". Likewise, Γ is rectangular, with r channels columns providing a basis for a channel subspace, usually called "eigenchannels". Standard normal priors are assumed for ys and z. Lastly the residual term ε is assumed to be Gaussian with zero mean and diagonal covariance Σ.

The particular case of r channels = p (full-dimensional matrix Γ) is equivalent to the version of the PLDA proposed in [START_REF] Kenny | Bayesian speaker verification with heavytailed priors[END_REF]: the eigenchannels are removed from Equation 7 and the residual noise is assumed to have a full covariance matrix. The PLDA model becomes:

w = µ + Φys + ε ( 8
)
where Φ is a p × r matrix (r < p) and ε a p-dimensional vector with a full covariance matrix.

PLDA scoring

After estimation of the PLDA meta-parameters, the speaker verification score given two i-vectors w1 and w2 is the likelihoodratio described by Equation 9, where the hypothesis θtar states that inputs w1 and w2 are from the same speaker and the hypothesis θnon states they are from different speakers. score = log P (w1, w2|θtar) P (w1, w2|θnon)

As proposed by S.Prince [START_REF] Simon | Probabilistic linear discriminant analysis for inferences about identity[END_REF][START_REF] Simon | Computer Vision: Models Learning and Inference[END_REF], the generative Equation 7 under assumption θtar looks like:

w1 w2 = µ µ + Φ Γ 0 Φ 0 Γ   y z1 z2   + ε1 ε2
(10) and under assumption θnon:

w1 w2 = µ µ + Φ 0 Γ 0 0 Φ 0 Γ    y1 y2 z1 z2   + ε1 ε2
(11) For the Gaussian-PLDA case, all the marginal likelihoods are Gaussian and the score 9 can be evaluated analytically [START_REF] Simon | Computer Vision: Models Learning and Inference[END_REF]. The final expressions do not include the hidden variables. Numerator of Equation 9 is equal to:

N w1 w2 ; µ µ , ΦΦ t +ΓΓ t +Σ ΦΦ t ΦΦ t
ΦΦ t +ΓΓ t +Σ (12) and denominator to:

N w1 w2 ; µ µ , ΦΦ t +ΓΓ t +Σ 0 0 ΦΦ t +ΓΓ t +Σ (13)

The two-covariance model

Described in [START_REF] Brummer | The speaker partitioning problem[END_REF][START_REF] Bishop | Pattern recognition and machine learning[END_REF], the two-covariance model can be seen as a particular case of the Probabilistic Linear Discriminant Analysis [START_REF] Burget | Discriminatively trained probabilistic linear discriminant analysis for speaker verification[END_REF]. It consists of a simple linear-Gaussian generative model in which an i-vector w of a speaker s can be decomposed in:

w = ys + ε ( 14
)
where the speaker model ys is a vector of the same dimensionality as an i-vector, ε is Gaussian noise and

P (ys) = N (µ, B) , (15)
P (w|ys) = N (ys, W) . (16)
N denotes the normal distribution, µ represents the overall mean of the training dataset, B and W are the between-and within-speaker covariance matrices defined in Equations 3, 4.

Under assumptions (15)( 16), the score from Equation 9 can be expressed as:

s = N (w1|y, W) N (w2|y, W) N (y|µ, B) dy i=1,2 N (wi|y, W) N (y|µ, B) dy (17)
The explicit solution of ( 17) is given in [START_REF] Brummer | The speaker partitioning problem[END_REF].

I-vector normalizations

In the following, we call "normalization" any transformation that projects i-vectors onto the surface area of the p-dimensional sphere of radius 1, and "length normalization" the straight division of i-vectors by their Euclidean norm. In the last paragraphs of this section, we propose two variance-spectra based normalization techniques, one for LDA system and the other for PLDA system. First, we introduce a visualization tool intended to better analyze the spectral distributions of a dataset.

Spectral graph

Given an i-vector dataset expressed in a given basis, we call "spectral graph" a visualization of the total and related speaker, session variabilities across the space dimensions.

To proceed, matrices B and W of Equation 3 and 4 are computed. The total covariance matrix is equal to B + W. The spectral graph displays the three series diag(B+W), diag(B) and diag(W), which contain the proportion of total, speaker and session variance per dimension.

Figure 1 shows the spectral graph of female development dataset described in section 5, in the eigenvector basis of total covariance matrix 2 . Dimension of the i-vector space is p = 600. Note that the "total" curve is just the eigenvalue • the total covariance matrix is far from the identity matrix: its eigenvalue spectrum is not flat, though Factor Analysis-Total Variability assumes that i-vectors are standard-normally distributed;

• the speaker spectrum is highly correlated with the total spectrum. As FA-Total Variability ignores the speaker information, the latent variable seems to induce a bias into the training sample;

• the session spectrum is flatter, thus less correlated with other spectra.

Length normalization

Proposed in [START_REF] Garcia | Analysis of i-vector length normalization in speaker recognition systems[END_REF][START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF], the length normalization (scaling the ivectors by their magnitude) is intended to make them more Gaussian. It can be shown 3 that, in a p high-dimensional space, the spherical surface of radius 1 is the maximal density shell of a normal distribution of covariance matrix p -1 I. As a result, length normalization combats bad modelling assumptions, but also dataset shift between development and trial i-vectors. However, two points caught our attention:

• Length normalization is preceded by a standardization (centering and whitening) based on he first and second order moments of the development data, It could be of interest to explore other normalizations based on total, speaker or noise covariance of the training dataset. We call "variance-spectra based normalization" such normalizations.

• In the current framework, normalization is applied previously to a discriminative technique, here LDA or PLDA.

We propose to apply normalizations to transform the development and trial i-vector representations in such a way they better fit assumptions of the following discriminant techniques.

Figure 2 displays the spectral graph of female development dataset, in the eigenvector basis of total covariance matrix, after length normalization. We observe that this technique does not modify spectral distributions. We present below two variancespectra based normalizations suitable for LDA and PLDA respectively. LDA assumes that it exists a subspace of significantly high speaker variability and low noise variability. Thus projecting the data onto this subspace would improve speaker discrimination.

There is usually no optimal solution to the LDA problem, i.e. finding axes simultaneously maximizing the speaker variance and minimizing the session variance. The aim of a variance-spectra based normalization before LDA is to increase the Rayleigh quotient on the surface of the sphere in such a way that the dataset moves towards the optimal solution. We presented in Interspeech 2011 [START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF] an algorithm intended to prepare i-vectors for scoring. Given a training dataset T , the algorithm is:

Algorithm for training dataset for i = 1 to nb iterations
Compute mean µi and total covariance matrix Σi of T

For each w of T :

w ← Σ -1 2 i (w -µ i ) Σ -1 2 i (w -µ i ) Σ -1 2 i
can be computed using a singular value decomposition. For each test vector of an evaluation set the same algorithm is applied, but with the successive parameters µ i and Σi computed on the training dataset.

Algorithm for test vectors

Given a test vector w,

for i = 1 to nb iterations: w ← Σ -1 2 i (w -µ i ) Σ -1 2 i (w -µ i )
In each iteration, i-vectors of the training dataset are standardized then length normalized. With only one iteration, the algorithm is just a standardization followed by normalization, already used in current i-vector based systems [START_REF] Garcia | Analysis of i-vector length normalization in speaker recognition systems[END_REF][START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF]. We showed empirically in [START_REF] Bousquet | Intersession compensation and scoring methods in the i-vectors space for speaker recognition[END_REF] that this algorithm makes the global mean µ i tending to 0 and the covariance matrix Σi to p -1 I (the identity matrix divided by the dimension p of the Total Variability space). Note that this algorithm is stationary after several iterations.

Wv = p -1 -λ v ( 18
)
Thus v is an eigenvector of W for the eigenvalue p -1 -λ . Accordingly, sum of B and W eigenvalues for each axis is equal to p -1 .

Therefore the first eigenvectors of B are the solution of the LDA B,W optimization problem of Equation 2. As a same orthogonal basis contains speaker and session principal subspaces (often called eigenvoices and eigenchannels) and as i-vectors lie on a sphere, we call EigenFactors Radial (EFR) this variancespectra based normalization technique. It is worth noting that after EFR, retaining the r first dimensions of the eigenvector basis of B becomes equivalent to removing the (p -r) first dimensions of W. This involves that LDA after normalization is equivalent to Nuisance Attribute Projection 4 .

Variance-spectra based normalization for PLDA

Method

Considering a normalization of i-vectors that would be adapted to PLDA modeling, we raise the following issue: once all ivectors lie on a spherical surface, it is difficult to estimate the within-class covariance matrix. To depict this fact, Figure 4 shows a two-dimensional example: i-vectors of three speakers lie on a circle, with a good level of isolation between classes. The arrows indicate the first principal axis (1D) of session variability for each speaker and the central red arrows indicate the first principal axis of W. Estimating the principal axis of the 1.0 q q q q q q q spk 1 spk 2 spk 3

session variability over all speakers by the mean of the three axes gives a poorly accurate result. More generally, it can be questioned to base a generative model on an overall linear estimation of session covariances meanwhile data lie on a spherical surface. Facing this fact, the most appropriate alternative seems to apply a transformation which keeps data on the nonlinear surface for Gaussianity, but without any principal directions of session variability nor without dependences between directions. Thus, to have a spherical within-class covariance matrix (a spherical matrix is a matrix σ 2 I).

Finding a way to conciliate length normalization and spherical noise matrix is greatly facilitated by the previous variance-spectra based normalization. Based on the fact that the EFR normalization makes Σ tending to p -1 I, which is spherical, we propose to carry out a similar algorithm, by replacing Σ with W. The algorithm becomes: 4 Details on Nuisance Attribute Projection (NAP) can be found in [START_REF] William | SVM based speaker verification using a GMM supervector kernel and NAP variability compensation[END_REF].

Algorithm for training dataset for i = 1 to nb iterations

Compute mean µ i and within-class cov. matrix Wi of T For each w of T :

w ← W -1 2 i (w -µ i ) W -1 2 i (w -µ i )
For each test vector of an evaluation, the same algorithm is applied, with the successive parameters µ i and Wi computed on the training dataset.

Algorithm for test vectors

Given a test vector w,

for i = 1 to nb iterations: w ← W -1 2 i (w -µ i ) W -1 2 i (w -µ i )
We call Spherical Nuisance normalization this technique, because the combined session and residual nuisances have a spherical covariance matrix 5 . 5, W is also close to be diagonal. Note that since W tends to a matrix σ 2 I (σ scalar) this algorithm is stationary after some iterations. The main point here is that the energy of the B-spectrum has been maintained. Less than 200 axes contain a major proportion of the speaker variability. The part of speaker variance in the total variance even increased (from 41% to 50% for the male dataset and 40% to 47% for the female dataset).

PLDA initialization

PLDA meta-parameters are estimated through an iterative Expectation Maximization (EM) algorithm. In the absence of any other information, speaker and channel matrices Φ and Σ of Equation 7 are randomly initialized. After the above algorithm, it can be taken advantage of the new representation to initialize the PLDA matrices. To proceed, development and evaluation data are expressed in the eigenvector basis of B (each vector is multiplied by the B eigenvector matrix sorted by decreasing order of eigenvalues). In this basis, which is the one of Figure 5, the r first dimensions contain the principal speaker variability. Hence, the speaker matrix is initialized with:

Φ = Ir,r 0p-r,r (19)
where Ir,r is the r × r identity matrix and 0p-r,r is the (pr) × r null matrix. Γ matrix is initialized by the Cholesky decomposition of W, so that W = ΓΓ t . By this way, the estimation algorithm is initialized at an meaningful starting point in the optimization landscape of the PLDA EM-algorithm. PLDA will refine this model by distributing speaker variabilities in the subspace and normalizing anisotropic nuisances.

Experiments and results

Experimental setup

All our experiments were carried out on the extended-core telephone-telephone condition (i.e., condition 5) from the NIST SRE 2010 evaluation. For all our experiments, we have used the i-vectors provided by Brno University of Technology (BUT) [START_REF] Matejka | Full-covariance UBM and heavy-tailed PLDA in I-Vector speaker verification[END_REF].

Voice Activity Detection

Speech/silence segmentation is performed by BUT Hungarian phoneme recognizer [START_REF] Schwarz | Hierarchical structures of neural networks for phoneme recognition[END_REF], where all phoneme classes are linked to the speech class. More details on VAD are provided in [START_REF] Brummer | ABC system description for NIST SRE 2010[END_REF].

Feature Extraction

We use MFCC 19 + energy augmented with their delta and double delta coefficients, making 60 dimensional feature vectors.

The analysis window has 20 ms with shift of 10 ms. First we remove silence frames according to VAD and after that we apply short-time cepstral mean and variance normalization which uses a window of 300 frames.

GMM UBM Training

One gender-independent UBM was represented as a full covariance 2048-component GMM. It was trained on the 7823 female and 7779 male segments balanced over the telephone/telmic/interview channels from NIST SRE 2004[START_REF] Kenny | Joint factor analysis of speaker and session variability: Theory and algorithms[END_REF][START_REF] Bishop | Pattern recognition and machine learning[END_REF] and 2008 data. The variance flooring was used in each iteration of EM algorithm during the UBM training [START_REF] Matejka | Full-covariance UBM and heavy-tailed PLDA in I-Vector speaker verification[END_REF].

I-vector Extractor Training

Gender-dependent i-vector extractor was trained on the following telephone data: NIST SRE 2004[START_REF] Kenny | Joint factor analysis of speaker and session variability: Theory and algorithms[END_REF][START_REF] Bishop | Pattern recognition and machine learning[END_REF], Switchboard II Phases 2 and 3, Switchboard Cellular Parts 1 and 2, Fisher English Parts 1 and 2 giving 9627 female speakers in 45195 segments of speech, and 6989 male speakers in 33645 segments of speech (both after VAD). The results are reported with 600 dimensional i-vectors.

LDA and PLDA Training

The gender-dependent training lists for two-covariance, LDA and PLDA are from sessions of NIST SRE 2004[START_REF] Kenny | Joint factor analysis of speaker and session variability: Theory and algorithms[END_REF][START_REF] Bishop | Pattern recognition and machine learning[END_REF], Switchboard II Phases 2 and 3 and Switchboard Cellular Parts 1 and 2, telephone only, nominal length higher than 180 seconds. No minimum amount of utterances per speaker has been stated. This gives 21475 sessions of 1575 speakers for male, 27155 sessions of 2012 speakers for female.

Results

Tables 1, 2, 3 give comparison result for the core extended condition 5 (telephone-telephone) from the NIST SRE 2010 evaluation, between systems based on three techniques: twocovariance scoring, LDA followed by two-covariance scoring and PLDA, in terms of Equal Error Rate (EER) and normalized minimum Decision Cost Functions (DCF) for the two operating points as defined by NIST for the SRE 2010 evaluations.

Two-covariance scoring

Table 1 compares performance of two-covariance scoring (Equation 17) applied without any transformation of the initial data and after a simple length normalization (without any standardization or other spectral technique) of both development and evaluation data. Length normalization involves a great enhancement of performance for both genders. Data become more suitable for a scoring based on the two-covariance model. 2 shows the performance of LDA systems, all followed by two-covariance scoring. LDA based on S b and Sw (LDA S B ,S W ) or based on B and W (LDA B,W ), are applied after either length normalization or n iterations of variance-spectra based normalization (EFRnorm n iter). For all systems, the best LDA dimension reduction is dim = 80.

Compared to Table 1, both LDA techniques improve performance in terms of EER and minDCF. Standardization before normalization (EFRnorm 1 iter.) enhances performance for female evaluation but not for male evaluation. Two iterations of standardization and normalization (EFRnorm 2 iter.) achieve the best performance for male and female evaluation, but only if LDA based on B and W is carried out. Beyond two iterations, performance are stationary. These experiments show that standardization and i-vector representation after two iterations, which is adapted to LDA based on B and W, is the most robust of these LDA-based systems. 3 presents the results of the baseline Gaussian-PLDA model (according to Equation 7) following a length normalization. The rank of the speaker-matrix Φ is 80 and the rank of the channel-matrix Γ is 600 (full-dimension).

For the male evaluation, 100 iterations of the EM algorithm have been required to achieve optimum performance. For the female evaluation, this value increased to 300 iterations. As the speaker and channel matrices of the PLDA are randomly initialized, performance of the system are subject to a variability. Therefore, the displayed values are the mean performance yielded by repeating 10 times the same experiment. For male evaluation, EERs vary in the range of 1.15 to 1.33 and minD-CFs from 0.31 to 0.34. For female evaluation, EERs vary in the range of 1.78 to 1.86 and minDCFs from 0.34 to 0.35. 3 gives the mean result of the best Gaussian PLDA system following standardization and lengthnormalization (EFR): two iterations of these normalization have been required and the optimal ranks of Φ and Γ are 80 and 600. Compared to first row (a single length-normalization), no improvement of performance is observed. Third row of Table 3 shows results of the same Gaussian PLDA system following two iterations of Spherical Nuisance normalization for both development and evaluation data. As before, the rank of speaker-matrix Φ is 80 and the rank of channelmatrix Γ is 600. For both genders, 100 iterations of the PLDA algorithm have been required to achieve optimum performance. Speaker and channel matrices are randomly initialized and the displayed values are the mean performance yielded by repeating 10 times the same experiment. For male evaluation, EERs vary in the range of 1.04 to 1.13 and minDCFs from 0.29 to 0.32. For female evaluation, EERs vary in the range of 1.73 to 1.84 and minDCFs from 0.33 to 0.34. The preliminary Spherical Nuisance normalization improves mean performance for both genders, in terms of EER and minDCF.

Fourth row of Table 3 shows results of the same Gaussian PLDA system following two iterations of Spherical Nuisance normalization. Here, speaker and channel matrices have been initialized with the procedure described in paragraph 4.4.2. As before, the rank of speaker-matrix Φ is 80 and the rank of channel-matrix Γ is 600. For the male evaluation, only 10 iterations of the EM algorithm have been required to achieve optimum performance, and only 2 iterations for the female evaluation. Beyond, performances are almost stationary for both genders. However, when performing only one iteration of EM, the system yields slightly worse EERs: 1.15% for male evaluation and 1.75% for female evaluation. This result shows that the PLDA modelling remains necessary to optimize i-vector based speaker recognition engines.

The Spherical Nuisance normalization preliminary applied makes the PLDA algorithm quickly converge to an optimization point which yields significant gain of performance, compared to baseline. In rows two and three of Table 3, two iterations of Spherical Nuisance normalization are performed. When performing only one iteration, performance are slightly worse (EER of 1.10% for male evaluation, 1.81% for female evaluation) but increasing the number of iterations doesn't improve much as it seems that this algorithm converges after 2 or 3 iterations. It achieves a relative improvement of the PLDA system of 14.8%-point on the EER for the male evaluation, and of 4.76%-point for the female evaluation. Note that with Lnorm+G-PLDA system (first row of Table 3), an equivalent initialization of speaker and channel matrices does not improve the performance of this system.

Conclusion

This paper presented two i-vector normalization techniques based on variance-spectra of training dataset. Those normalizations aim at adapting the i-vector representation to a speaker discriminative system.

In the case of Linear Discriminant Analysis followed by two-covariance scoring, we show that the preliminary algorithm of standardization and length-normalization can be iterated to enhance the optimization criterion of LDA and makes the ivector distribution more Gaussian. We also suggest the best within-and between-covariance matrices to use. In the case of Probabilistic Linear Discriminant Analysis, the lower-bound maximization EM algorithm is dependent of the initial representation of data, i.e. algorithm converges to a local maximum-likelihood point in the optimization landscape of the PLDA. In this context, Spherical Nuisance normalization, also an iterative algorithm, moves data from a local optimum to a better one in a different part of the parameter space.

We show that after performing two iterations of Spherical Nuisance normalization and initializing the PLDA matrices according to the new i-vector representation, our system achieves optimum performance after only a reduced number of EM iterations (between 2 and 10). Nevertheless, EM estimation of the PLDA parameters remains necessary as the system yields slightly worse performance when running only one iteration of EM (1.15% EER for male evaluation and 1.75% EER for female evaluation).

While conducting this work, we observed that, after applying two iterations of Spherical Nuisance normalization, removing the channel matrix from the PLDA model (channel rank of Γ = 0) and keeping only a diagonal noise covariance matrix does not degrade the best performance reported in this paper. After Spherical Nuisance normalization, the simplest speakerfactor analysis model (low rank speaker subspace and diagonal noise covariance matrix) is able to optimize the performance of our i-vector-based speaker recognition engine. This needs to be analyzed and clarified by future investigations.

The spectral graph of initial development data is a simple and useful visualization tool for speaker recognition. We observed that the spectral distributions in regard to the latent speaker variable may differ from an i-vector extraction configuration to another so as the performance gap between speaker recognition systems (LDA, PLDA, ...). Analysing this graph can help to assess the adequacy of an i-vector extraction configuration to these speaker recognition systems.

Figure 1 :

 1 Figure1: Spectral graph of female development data (before any transformation). For the 600 dimensions (x axis), the y axis shows the total, speaker and session parts of variance.

Figure 2 : 4 . 3 .

 243 Figure 2: Spectral graph of female development data after application of length normalization.

Figure 3 :Figure 3

 33 Figure 3: Spectral graph of female development data, in the eigenvector basis of B, after 3 iterations of variance-spectra based normalization (Eigen Factor Radial).

Figure 4 :

 4 Figure 4: Two-dimensional example of session variability onto a spherical surface. I-vectors of three speakers are shown. All i-vectors are of norm 1. Arrows indicate the first principal session axis of each speaker. Central red arrows indicate the first principal axis of W.

Figure 5 :Figure 5

 55 Figure 5: Spectral graph of female development data, in the eigenvector basis of B, after application of 3 iterations of Spherical Nuisance normalization.

Table 1 :

 1 Comparison of two-covariance scoring without and with length normalization of i-vectors for the core extended condition 5 (tel-tel) from the NIST SRE 2010 evaluation, in terms of EER and minDCF.

		male	female
		EER	DCF	EER	DCF
	2cov	4.47 0.44 4.75 0.48
	length-norm + 2cov 1.38 0.34 2.34 0.44
	5.7.2. LDA and two-covariance scoring
	Table

Table 2 :

 2 Comparison of five systems based on LDA for the core extended condition 5 (tel-tel) from the NIST SRE 2010 evaluation, in terms of EER and minDCF. All systems use twocovariance scoring.

			male	female
			EER	DCF	EER	DCF
	length-norm	LDA S B ,S W	1.27 0.31 2.27 0.38
	EFRnorm 1 iter. LDA S B ,S W 1.36 0.33 2.29 0.39
	EFRnorm 1 iter. LDA B,W	1.36 0.30 1.89 0.35
	EFRnorm 2 iter. LDA S B ,S W 1.30 0.32 2.30 0.39
	EFRnorm 2 iter. LDA B,W	1.27 0.31 1.89 0.35
	5.7.3. PLDA
	First row of Table

Table 3 :

 3 Comparison of three systems based on PLDA for the core extended condition 5 (tel-tel) from the NIST SRE 2010 evaluation, in terms of EER and minDCF.

		male	female
		EER	DCF	EER	DCF
	length-norm + G-PLDA	1.22 0.32 1.81 0.34
	EFRnorm 2 iter. + G-PLDA	1.27 0.35 1.94 0.35
	SphNnorm + G-PLDA	1.08 0.31 1.77 0.34
	SphNnorm + G-PLDA init. 1.04 0.29 1.73 0.33
	Second row of Table

We speak of "variance-spectra" by analogy with the spectrum of a matrix, i.e. the set of its eigenvalues.

Here and in the following, only female dataset spectral graphs are shown, since it turned out that all the male dataset graphs are almost identical.

http://ontopo.wordpress.com/2009/03/10/reasoning-in-higherdimensions-measure/

The case of isotropic noise ε ∼ N 0, σ 2 I is referred to as "homoscedastic residuals model"[START_REF] Tipping | Mixtures of probabilistic principal component analyzers[END_REF].

Acknowledgments

We thank Pavel Matějka for providing the i-vectors of BUT. Special thanks to Niko Brümmer for the matlab PLDA system and for helpful discussions throughout the NIST-SRE2011 Workshop in Atlanta.