M Baloul

E Cherrier
email: estelle.cherrier@ensicaen.fr

C Rosenberger
email: christophe.rosenberger@ensicaen.fr

Challenge-based Speaker Recognition For Mobile Authentication

User authentication is a major trend to guarantee the security of electronic transactions when using mobile devices such as tablets or mobile phones. Biometrics is for us the only real user authentication method. In this article, we propose to realize a speaker recognition approach to achieve this goal. We use a challenge-based method to avoid the replay attack (especially if the impostor has recorded the user's voice). In this case, free text recognition is realized. Experimental results on the CMU database show very good results, while providing low computation times.

I. . Introduction

The wide and recent development of smartphones and the correlated growing request to access online services (home banking, e-government, e-commerce. . . ) has involved a need for mobile secure authentication. Among the existing solutions (static passwords, one time passwords, X509 certificates, coding tables. . . ), challenge-based biometric authentication represents a promising proposal. Like any biometric system, a challenge-based biometric solution must meet essential requirements to address security and respect for privacy such as: confidentiality, unlinkability, resistance to replay attacks, revocability.

Within the biometric research field, challenge-based approaches are related to dynamic authentication that can be solved using a behavioral modality, such as mouse dynamics, keystroke dynamics, speaker recognition, etc. . . Behavioral biometrics has the advantage of being non-intrusive, in the sense that speaking, typing on a keyboard. . . is natural and simple for the user, therefore such modalities are globally well accepted.

Similarly to all biometric systems, challenge-based ones consist of two steps. The first step concerns the user enrolment: enrolment means first the capture of the biometric raw data, the features extraction to define a model (which is stored as a reference) of each genuine user and its storage (if the template meets some quality requirements). The second step called verification, used either for authentication or identification purposes, considering a challenge, must predict if the user has the expected behavior face to the challenge: as for example, type an unknown sentence on a keyboard or tell an unknown sentence. . . Since behavioral biometrics is involved, it must be difficult for an intruder to imitate the correct behavior.

Concerning mobile phones, some biometric sensors are already present in the object itself, providing them with inherent biometric abilities: we can mention the microphone, the webcam, the touch pad (and for some of them a fingerprint reader). Therefore, a challenge based on the way the mobile's owner speaks seems rather obvious and natural.

Challenge-based speaker recognition on a mobile phone belongs to the wide research field of textindependent speaker recognition. Indeed, to be authenticated, the mobile's owner will have to utter an unknown sentence or an unknown word, which is precisely text-independent speaker verification. Among the intense literature on this topic, we just refer the reader to the thorough survey paper [START_REF] Kinnunen | An overview of text-independent speaker recognition: From features to supervectors[END_REF] and the associated references. Using classical speaker recognition techniques to design an authentication system based on a biometric challenge on a mobile phone is not straightforward. Indeed, some constraints, inherent to the use of a mobile device, must be taken into account from the design step: the quality of the sound acquisition depends on the characteristics of the embedded microphone and the environment, the complexity of the embedded algorithms must be adapted to the capacity of the smartphone in terms of memory and processing power. The aim of this paper is twofold. First, how to find a simple solution that could be further embedded in a mobile, among the existing speaker recognition techniques? Second, what are the performances of the selected method applied to a suited database, in terms of EER, recognition rate and verification time?

The outline of the paper is the following: in Section II, we detail the different steps of a challenge based biometric speaker recognition for mobile devices. Both stages of enrollment and verification will be considered within the constraints inherent to the mobile context. Some methods of the literature will be presented and the most adapted one will be described. Section III presents the proposed method based on MFCC (Melfrequency cepstral coefficients) characterization of the voice signal. Section IV is dedicated to the experimental protocol description and the obtained experimental results. At the end of the paper, the conclusion of our study and some perspectives will be given. We conclude and give some perspectives of this work.

II. . Text-independent speaker recognition

The human voice is a complex information-bearing signal, depending on physical and behavioral characteristics. The raw speech signal, uttered by any person, is extremely rich in the sense that it involves high dimensional features. To perform efficient speaker recognition, one must reduce this complexity, while keeping sufficient information in the extracted feature vector. Some methods for speaker recognition have become popular, since few decades, which are gathered in the survey paper [START_REF] Kinnunen | An overview of text-independent speaker recognition: From features to supervectors[END_REF]. Here, we briefly recall the textindependent speaker recognition process, where five steps are considered.

• Signal acquisition

Microphones and analog-digital converter are used to record and digitize the user's voice. At the end of this step, a numerical vector representing the uttered speech is available. The duration of speech recording depends on the desired accuracy.

• Speech signal preprocessing The speech signal is not a stationary signal since the vocal tract is continuously deformed and the model parameters are time-varying. But, it is generally admitted that these parameters are constant over sufficiently small time intervals. Classically, the signal is divided into frames of 25 milliseconds. This division into frames leads to discontinuities in the temporal domain, and inevitably to oscillations in the frequential domain. Among the possible solutions to avoid this phenomenon (see [START_REF] Harris | On the use of windows for harmonic analysis with the discrete fourier transform[END_REF] for example), Hamming windows are applied. Besides, within the uttered text, silence zones can lead to performance degradation, so they must be removed. The reference [START_REF] Hautam | Improving speaker verification by periodicity based voice activity detection[END_REF] presents a voice activity detection (VAD) method based on realtime periodicity analysis, which enables silence removal. This method is also applied in [START_REF] Fränti | Developing speaker recognition system: from prototype to practical application[END_REF]. In case of noisy signal, it can be filtered to reduce the noise level.

• Feature extraction Based on the speech signal registration and preprocessing, features are extracted to define a model corresponding to the user. Ideally, these features must be robust to intrinsic variability of the user's voice (due to stress, to disease), to noise and distorsion, to impersonation. The most widely employed methods involve short-term spectral features. We just cite two of them: MFCC (Melfrequency cepstral coefficients) introduced by [START_REF] Davis | Comparison of parametric representations for mmonosyllabic word recognition in continuously spoken sentences[END_REF], and LPCC (linear predictive cepstral coefficients) proposed by [START_REF] Huang | Spoken language processing: a guide to theory, algorithm, and system development[END_REF], a detailed overview can be found in [START_REF] Kinnunen | An overview of text-independent speaker recognition: From features to supervectors[END_REF]. According to numerous studies, MFCC reveals to be more robust and efficient in practice.

• Speaker modeling Once these features have been extracted on each frame, the corresponding model or template design requires a training phase. We mention here the most popular techniques. GMM (Gaussian mixture model) [START_REF] Reynolds | Robust text-independent speaker identification using Gaussian mixture speaker models[END_REF] is a method based on a modeling of the statistical distribution of the extracted features. This method exhibits excellent performances, but is not suited to a challenge-based biometric system, owing to its computational cost.

The VQ (vector quantization) method [START_REF] Soong | A vector quantization approach to speaker recognition[END_REF] is based on LBG algorithm [START_REF] Linde | An algorithm for vector quantizer design[END_REF], [START_REF] Burton | Text-dependent speaker verification using vector quantization source coding[END_REF]. This process permits, after clustering, to describe a voice sample by a model vector having a predefined fixed size, whatever the initial length of the signal. Besides, the most recent method SVM (Support vector machine) [START_REF] Vapnik | Statistical learning theory[END_REF] consists of binary classifiers, developed to allow the separation of complex data in large spaces. One SVM must be trained for each genuine user.

• Speaker recognition These four previous steps correspond to the user enrolment phase. In the last recognition step, two problems can be considered: user authentication (the system must verify a claimed identity, through one vs. one comparison) or user identification (the system must check if the user is a genuine user, through one vs. multiple comparisons depending of the number of genuine users in the database). For GMM based modeling, the recognition relies on a likelihood estimation and the output is a probability. For VQ based modeling, the recognition test is classically performed through Euclidean distance computation. Whereas for SVM based modeling, the test phase uses the same process as the training phase. The reference [START_REF] Matrouf | Lia gmm-svm system description : Nist sre08[END_REF] shows that the performances are at least as good as that of GMM based recognition. Notice that the acquisition conditions may be worse in this step than in the enrolment step, where the stored model must be of high quality.

For GMM or VQ based modeling, the recognition test is classically performed through the Euclidean distance computation, with less parameters for the VQ. Whereas for SVM based modeling, the test phase uses the same process as the training phase. The reference [START_REF] Matrouf | Lia gmm-svm system description : Nist sre08[END_REF] shows that the performances are at least as good as that of GMM based recognition. Notice that the acquisition conditions may be worse in this step than in the enrolment step, where the stored model must be of high quality.

Many papers propose to use MFCC combined with SVM to perform speaker recognition, we mention just a few: in [START_REF] Kamruzzaman | Speaker Identification using MFCC-Domain Support Vector Machine[END_REF] for text-dependent speaker identification with neural networks, in [START_REF] Chen | Speaker verification using mfcc and support vector machine[END_REF] for text-dependent speaker verification, in the project [START_REF] Fränti | Developing speaker recognition system: from prototype to practical application[END_REF] for a thorough implementation.

The main contribution of this paper is to analyze the performances of the previous algorithms, chosen among the most efficient of the literature, depending on different sets of parameters. We do not propose any new method, we combine existing methods to design a text-independent authentication process, applied to a realistic (concerning the acquisition conditions) database. The proposed analysis concerns a trade-off between the performance, and the computational cost. It is the first step in the design of an implementation on a mobile device. Now, we detail how the aforementioned speaker recognition techniques can be adapted to a mobile context.

III. . Challenge-based speaker recognition

In a challenge-based biometric system, the enrolment phase is not different from that of any biometric system. Concerning speaker recognition, the user is asked to speak during a predefined time. Then, the preprocessing, the feature extraction and user modeling are performed to generate a template stored in a database. In the test mode and after extracting features, a distance between these parameters and the claimed model is calculated and compared to a given threshold, if it is a verification; and between these parameters and all models that exist in the database, if it is an identification.

In this paper, as in most of papers dealing with text-independent speaker recognition, we consider a MFCC based method owing to its robustness and better performances as a feature extraction method [START_REF] Kinnunen | An overview of text-independent speaker recognition: From features to supervectors[END_REF], [START_REF] Fränti | Developing speaker recognition system: from prototype to practical application[END_REF] and VQ for modeling [START_REF] Kabir | Vector quantization in text dependent automatic speaker recognition using mel-frequency cepstrum coefficient[END_REF]. In figure III, we present the general diagram of a speaker recognition system in training and test mode.

Fig. 1. General architecture of speaker recognition system

As mentioned in section II, the first step in speaker recognition is silence removal. In this paper, we resort to the simple method proposed in [START_REF] Giannakopoulos | A method for silence removal and segmentation of speech signals, implemented in matlab[END_REF] to remove silence. The computation time of this approach is low, which is a very important property for mobile authentication. This method is based on the extraction of two particular audio features, namely signal energy and spectral centroid, defined below. Let x(n), n ≥ 0 stand for one sentence of the database, and n the current discrete time. This signal x(n) is divided into N frames of 50 milliseconds, denoted x i (n), for i = 1, N . For each frame x i (n), one defines:

-Signal Energy: the energy of the i th frame is computed as follows

E(i) = 1 N N n=1 |x i (n)| 2 (1)
-Spectral Centroid: we can compute the spectral centroid C i of the i th frame by the following formula

C i = N k=1 (k + 1)X i (k) N n=1 X i (k) (2)
where X i (k), k = 1..., N stands for the i th discrete Fourier transform coefficient of the i th frame.

After computing these two feature sequences (Energy and Centroid), they will be compared to two thresholds T E and T C based on the energy sequences and spectral centroid sequences respectively. We describe the process to determine T E the same method is applied to determine T C .

The histogram of the energy sequence is computed and then a smoothing filter is applied (a median filter). The threshold T E is estimated as follows from the local maxima of the histogram as follows:

Let M 1 and M 2 denote the positions of the first and second local maxima respectively. Then, compute:

T E = wM 1 + M 2 w + 1 (3)
where w is a user defined parameter.

The voiced frames are determined as the frames whose both feature values (i.e. Energy and Centroid) are larger then the two thresholds T E and T C respectively. After removing silence from the voice signal, the features extraction and modeling steps can be applied. In this paper, we use the MFCC and VQ algorithms detailed in the reference [START_REF] Velisavljevic | Mini-project: : An automatic speaker recognition system[END_REF].

The figure 2 illustrates the computation of the MFCC coefficients, which is briefly detailed below.

Consider again a particular sentence x(n) as before. The voice signal is divided into small frames x i (n) of

y i (n) = x i (n) * w(n) ( 4
)
where y i (n) is the transformed signal, x i (n) is the considered frame and W (n) is the Hamming window defined by:

W (n) = 0.54 -0.46cos( 2πn 256 -1 ) (5)
for

0 ≤ n ≤ N -1.
The Fourier transform of each frame is computed, the next step is performed in the frequency domain. The human voice spectrum is not linearly distributed, therefore, we use a Mel scale filter bank to represent the wide spectrum. A given frequency f in Hz can be converted into the Mel scale [START_REF] Muda | Voice recognition algorithms using mel frequency cepstral coefficient and dynamic time warping techniques[END_REF]:

M EL(f ) = 2595 * log 10 (1 + f 700 ) (6)
In general, 20 Mel filters are required for high accuracy. We apply after a logarithmic compression and a discrete cosine transform. Finally, the discrete amplitudes of the resulting cepstrum are called the MFCCs coefficients [START_REF] Davis | Comparison of parametric representations for mmonosyllabic word recognition in continuously spoken sentences[END_REF].

The resulting MFFC coefficients of each sentence x(n) are 20 dimensional vectors, each vector will be represented by a given number of centroids (between 8 and 256), resulting in a vector template of fixed size modeling each user. This step is called the Vector Quantization (VQ), it is done by the LBG algorithm [START_REF] Kabir | Vector quantization in text dependent automatic speaker recognition using mel-frequency cepstrum coefficient[END_REF]. One advantage of using VQ is to reduce the computational cost. The obtained centroids are used to model the user. At the enrolment step, for each user, a specific model is calculated and stored in the database.

At the verification step, after the extraction of the query MFCC coefficients, we compute the Euclidean distance between these parameters and the model of the claimed reference; the obtained distance is compared to a given threshold.

Let M F CC(n, p) be the MFCC coefficients of a given user and V Q(n, q) the query reference model, p >> q:

M F CC =        M F CC11 M F CC12 ... M F CC1p M F CC21 M F CC22 ... M F CC2p . . . M F CCn1 M F CCn2 ... M F CCnp        (7) V Q =        V Q11 V Q12 ... V Q1q V Q21 V Q22 ... V Q2q . . . V Qn1 V Qn2 ... V Qnq        (8)
To compute the Euclidean distance between the two matrices M F CC and V Q, we proceed as follows.

For each column M F CC j , j = 1, p, we calculate the Euclidean distance between this column and the nearest column

V Q k , k = 1, q.
The considered distance is the sum of the p resulting distances, following:

ED (M F CC,V Q) = p j=1 min 1≤k≤q {dist(M F CCj, V Q k )} (9)
where :

dist(M F CCj, V Q k ) = n l=1 (M F CC l,j -V Q l,k ) 2 (10)
where M F CC l,j (respectively V Q l,k ) is the coefficient of the M F CC matrix (resp. the V Q matrix) at row l and column j (resp. row l and column k).

The final decision is the result of all this process, it depends on an operational threshold: if the distance ED (M F CC,V Q) is lower than this threshold, the user is authenticated by the system, otherwise it is rejected.

In the next section, we characterize the proposed method in terms of performance and computation time.

We also analyze the impact of parameters on efficiency such as the number of centroids to consider for the quantization step or the number of samples to generate the model of the user.

IV. Experimental results

In this paper, we consider the PDA database (PDAm data set) [START_REF] Obuchi | PDA speech database, carnegie mellon university[END_REF] proposed by CMU (Carnegie Mellon University). It consists of voice signals collected by a PDA device. 50 sentences of about 4 to 8 seconds are uttered by 16 users. The users work at CMU, they are native speakers of American English. The voice samples are recorded at 44.1kHz sampling rate. The original data was then downsampled to both 16kHz and 11.025kHz, see [START_REF] Obuchi | PDA speech database, carnegie mellon university[END_REF] for more details.

In this paper, we quantify the performance of the proposed method as follows:

• The number of sentences used in the training step are varied (1, 10, 20, 30 and 40 sentences among 50). For each value, the number of centroids used in the VQ method are also varied.

• We compute the ROC (Receiver Operating Curve) curve that gives the performance behavior of the biometric system for any value of the decision threshold (for verification purpose).

• The performance is also evaluated through the computation of the EER (Equal Error Rate).

• The second performance criterion is the recognition accuracy.

• We intend to determine the best tradeoff between a low EER, a small number of centroids which influence the memory space and a small number of sentences for the enrolment step which has an impact on the execution time.

• We propose to evaluate the time necessary for the enrolment step, depending on the previous chosen parameters, on a PC, since this step could be performed on a server side. In table I, we present the EER value of the biometric system by varying the number of sentences and the number of centroids used in the VQ modeling. We can see that the more centroids we use, the better is the performance. As the number of centroids used for the enrolment step has an impact of the computation time, we try to find a tradeoff between efficiency and computation time.

We obtain the best EER = 0.83 with 30 sentences (which is equivalent to about 3 minutes of recorded voice) and 64 centroids for the VQ modeling. This performance is interesting for a low cost biometric solution. In order to avoid the replay attack, the voice can be analysed in order to match the challenge.

Figure IV shows the corresponding ROC curve. We can see that for the FRR value F AR = 10 -4 equals 20% which is not bad for a low cost solution. In table II, we present the recognition rate of the system by varying the number of sentences and the number of centroids used for the VQ modeling. These results are

TABLE II. Authentication rate for different numbers of sentences and centroids

Now, bearing in mind the initial purpose of mobile implementation, we intend to estimate the computation time of both steps: enrolment and verification. The enrolment step will probably be done on a PC used as server, so this estimation has been performed with c Matlab, with the selected values for the parameters (number of sentences and number of centroids). We proceed as follows: to estimate the enrolment time, consider 30 seconds of the recorded voice and design a model of this voice signal with 64 centroids. It takes 35.6 seconds, so for 3 minutes of speech signal it will take about 3 minutes and 34 seconds in c Matlab environment. With a C implementation, we can expect to decrease by ten the computation time.

For the verification time, a sample of 5 seconds of voice signal is selected for the verification process. It takes 2.53 seconds of time processing in c Matlab. Even if it is difficult to compare the computation time between c Matlab on laptop and a mobile phone, we think that this computation time is a good estimate on what we could achieve on a mobile device.

V. . Conclusion and perspectives

We proposed in this article a free-text speaker recognition method. The features we used are Mel-frequency cepstral coefficients. The vector quantization allows to handle fixed-size feature vectors. We optimized the processing chain in order to have a good tradeoff between efficiency and computation time. Recognition results on the CMU database (that represents operational conditions) are satisfying with a EER value equal to 0.83.

Perspectives of this work are to implement on a mobile phone the proposed method to realize an offline user authentication.

Fig. 2 .

 2 Fig. 2. Calculation process of MFCC coefficients

TABLE I . EER for different numbers of sentences and centroids satisfying

 I as the False Acceptance Rate is in general low and it is very easy and fast to ask the user to make another capture for the verification step.

	Fig. 3

. ROC curve for 30 sentences and 64 centroids

			Number of sentences
		1	10	20	30	40
	VQ 8	91.58	94.06	93.75	94.69	96.88
	VQ 16	92.47	92.81	93.54	94.38	95.62
	VQ 32	92.86	93.28	93.54	94.38	95.62
	VQ 64	92.98	93.75	93.54	94.38	95.62
	VQ 128	92.6	93.59	93.54	94.38	95.62
	VQ 256	92.86	93.75	93.33	94.38	95.62


est destinée au dépôt

  An integrated framework combining Bio-Hashed minutiae template and PKCS15 compliant card for a better secure management of fingerprint cancelable templates Rima Belguechi, Estelle Cherrier, Christophe Rosenberger, Samy Ait-Aoudia To cite this version: Rima Belguechi, Estelle Cherrier, Christophe Rosenberger, Samy Ait-Aoudia. An integrated framework combining Bio-Hashed minutiae template and PKCS15 compliant card for a better secure management of fingerprint cancelable templates. Elsevier Journal on Computers & Security, 2013, 15 p. hal-00999083


Syed Zulkarnain
email: syed-zulkarnain.syed-idrus@ensicaen.fr

Syed Idrus

Estelle Cherrier
email: estelle.cherrier@ensicaen.fr

Christophe Rosenberger
email: christophe.rosenberger@ensicaen.fr

Patrick Bours
email: patrick.bours@hig.no

Soft Biometrics For Keystroke Dynamics

Keywords: Biometrics, keystroke dynamics, soft biometrics, pattern recognition, computer security

Keystroke dynamics is a viable and practical way as an addition to security for identity verification. It can be combined with passphrases authentication resulting in a more secure verification system. This paper presents a new soft biometric approach for keystroke dynamics. Soft biometrics traits are physical, behavioral or adhered human characteristics, which have been derived from the way human beings normally distinguish their peers (e.g. height, gender, hair color etc.). Those attributes have a low discriminating power, thus not capable of identification performance. Additionally, they are fully available to everyone which makes them privacy-safe. Thus, in this study, it consists of extracting information from the keystroke dynamics templates with the ability to recognise the hand(s) used (i.e. one/two hand(s)); the gender; the age category; and the handedness of a user when he/she types a given password or passphrase on a keyboard. Experiments were conducted on a keystroke dynamics database of 110 users and our experimental results show that the proposed methods are promising.

Introduction

Keystroke dynamics is an interesting and a low cost biometric modality as it enables the biometric system to authenticate or identify an individual based on a person's way of typing a password or a passphrase on a keyboard [START_REF] Giot | Greyc keystroke: a benchmark for keystroke dynamics biometric systems[END_REF][START_REF] Giot | Keystroke dynamics overview[END_REF]. It belongs to the class of behavioral biometrics, in the sense that the template of a user reflects an aspect of his/her behavior. Among the behavioral biometric modalities, we can mention signature analysis, gait recognition, voice recognition, or keystroke dynamics. Generally speaking, the global performances of keystroke dynamics based authentication systems are lower than the popular morphologic modalities based authentication systems (such as fingerprints, iris, etc. . . ) [START_REF] Maltoni | Handbook of fingerprint recognition[END_REF]. Besides, the main advantage of resorting to keystroke dynamics [START_REF] Bours | Continuous keystroke dynamics: A different perspective towards biometric evaluation[END_REF][START_REF] Giot | Keystroke dynamics overview[END_REF] to authenticate a user relies in its low cost. Indeed, for this modality, no extra sensor is required. The fact that the performances of keystroke dynamics are lower than other standard biometric modalities can be explained by the variability of the users behavior. One solution to cope with this variability is to study soft biometrics, first introduced by Jain et al. in [10]. In this paper "soft biometric traits" are defined as "characteristics that provide some information about the individual, but lack the distinctiveness and permanence to sufficiently differentiate any two individuals". For example Jain et al. consider gender, ethnicity, and height as complementary data for a usual fingerprint based biometric system.

Soft biometrics allows a refinement of the search of the genuine user in the database, resulting in a computing time reduction. For example, if the capture corresponds to a male according to a soft biometric module, then, the standard biometric authentication system can restrict its research area to male users, without considering female ones.

Concerning keystroke dynamics, an original approach is presented in the work of Epp et al. [START_REF] Epp | Identifying emotional states using keystroke dynamics[END_REF], strongly linked with the behavioral feature of keystroke dynamics. The authors show that it is possible to detect the emotional state of an individual through a person's way of typing. In this case, detecting anger and excitation is possible in 84% of the cases. Gender recognition is dealt in the work of Giot et al. in [START_REF] Giot | A new soft biometric approach for keystroke dynamics based on gender recognition[END_REF]: they show that it is possible to detect the gender of an individual through the typing of a fixed text. The gender recognition rate is more than 90% and the use of this information in association to the keystroke dynamics authentication, reduces the Equal Error Rate (EER) by 20%. The work of Syed-Idrus et al. [START_REF] Syed-Idrus | 9th Summer School for Advanced Studies on Biometrics for Secure Authentication: Understanding Man Machine Interactions in Forensics and Security Applications[END_REF] show that it is possible to detect users' way of typing by using one finger (i.e. one hand) and more than one fingers (i.e. two hands) with 80% correct recognition accuracy performed on a dataset with three passwords.

The objective of this paper is twofold. First, we present a new data collection of 110 users, both from France and Norway. This new benchmark will be released to the scientific community. Then, we propose an extended study of soft biometrics for keystroke dynamics on this new database. We are interested in the criteria that can influence the way of typing of the users. We test if it is possible to predict if the user:

1. types with one or two hands 2. is a male or a female 3. belongs to a particular age category 4. is right-or left-handed Indeed, predicting these soft features may help an authentication system (which is not considered here) to reduce the computing burden while in search of the genuine user in the database. This paper is organised as follows. Section 2 is devoted to the description of the proposed methodology: the characteristics of the database are described, together with the data collection process, and the tools that are used for analysis purposes. In Section 3, we present the obtained results while Section 4 presents the conclusions and the future works to be addressed.

Biometric Database

For the purpose of this study, we have created a new biometric database. We have performed data collection and also an experiment in two locations: France and Norway, but in fact the subjects originate from 24 different countries who are either studying or residing in one of the countries concerned. The selection of subjects from a myriad of nationalities would eliminate the possibility of bias sampling and thereby enhance the credibility of our experiment. Thus, a total of 110 people had volunteered to participate in this experiment where 70 of them were located in France and 40 in Norway. Table 1 shows the statistics repartition of handedness among men/women and the number of each category with respect to the other categories. For the creation of the biometric database, some experimentation tools are required such as a laptop; two external keyboards (French keyboard for users in France and Norwegian keyboard for users in Norway) i.e. AZERTY and QW-ERTY, respectively; and an application to collect the keystroke dynamics data. The location and position of the hardware are in a fixed position and immoveable throughout the session for the authenticity of the outcomes. According to experts, the best password is a sentence [3]. Hence, for the purpose of this study for keystroke dynamics, we present 5 passphrases as shown in Table 2, which are between 17 and 24 characters (including spaces) long, chosen from some of the well-known or popular names or artists (known both in France and Norway), denoted P 1 to P 5 . We asked all of the participants to type these 5 different passphrases 20 times. We use the GREYC Keystroke software [START_REF] Giot | Greyc keystroke: a benchmark for keystroke dynamics biometric systems[END_REF] to capture biometric data. A screenshot of this software is shown in Figure 1. We define two classes of the way of typing; gender category; age category; and handedness category denoted C 1 and C 2 , respectively as follows:

• Way of typing: C 1 = One Hand: only one hand is used (right/left depends if the user is right/left-handed person); C 2 = Two Hands: both hands are used.

• Gender: C 1 = Male; C 2 = Female.

• Age: C 1 = < 30 years old; C 2 = ≥ 30 years old.

• Handedness: The data that we had obtained from the 5 passphrases as listed in Table 2 are keystroke dynamics data, as mentioned earlier in the article. Keystroke dynamics data consist of information containing a field of four timing values namely: the timing pressure of when the two buttons are pressed (ppTime); the timing release of when the two buttons are released (rrTime); the timing of when one button is released and the other is pressed (rpTime) that is the latencies between keystrokes; and the timing of when one button is pressed and the other is released (prTime) that is the time durations of keystrokes [START_REF] Giot | Greyc keystroke: a benchmark for keystroke dynamics biometric systems[END_REF][START_REF] Giot | Keystroke dynamics overview[END_REF]. These data however, do not match exactly to the durations and latencies of keys because their ordering is based on time and not key code [START_REF] Giot | Greyc keystroke: a benchmark for keystroke dynamics biometric systems[END_REF]. We use the keystroke template vector, which is the concatenation of the four mentioned timing values to perform our data analysis by classifying two classes for each category.

C 1 = Right-handed; C 2 = Left-handed.

Data Collection Process

Each user has to type each passphrase P j , j = 1..5 for each hand class C i , i = 1, 2, 10 times without errors. If there are typing errors, the current entry has to be cancelled and the user will have to resume until 10 successful entries for both classes of hand have been recorded into the system. If the user is a right-handed person, he/she will only need to use the right hand to key-in the passphrases in a normal typing pace, and similarly for the left-handed people. At the end of the data collection, we have a total of 11000 data samples (= 5 passphrases x 2 classes of hand x 110 users x 10 entries) in the proposed biometric database. For each user, 7 out of 10 samples are used for both training and test data. The first three entries for each user will not be taken into account because a leeway was given to the users to allow them to train themselves for each of the given passphrase.

Data Analysis

In this subsection, we present our methodology. We used a Support Vector Machine (SVM) [START_REF] Hearst | Support vector machines. Intelligent Systems and their Applications[END_REF][START_REF] Steinwart | Support vector machines[END_REF][START_REF] Vapnik | Statistical learning theory[END_REF] to perform the classification. This classifier is aimed at maximising the margins between the considered classes. From the data obtained, it takes a set of input data and predicts, for each given input, which possible class it belongs to. It consists of an evaluation of the class (C i ) recognition rate in function of the ratio of data kept for the learning stage and hence we use a Library for Support Vector Machines (LIBSVM) developed by [START_REF] Chang | Libsvm: A library for support vector machines 3. Durgahee, A.: The best password is a sentence: says expert[END_REF] with default values. Two steps are involved: the first is devoted to a learning process, while the second is a test of the resulting SVM. The input data for the SVM that will be used to predict the 'test' data are a fixed percentage of the whole data. The remaining data are then used for test purpose, leading to a recognition rate. For data analysis, we recall that we are interested in soft biometrics criterion that can be applied to our biometric database: one or two hand(s); male or female, age < 30 or ≥ 30 years old, right/left-handed. In the sequel, we focus on the first experiment, which consists in recognising if the user types with one or two hands. The other categories are dealt with in the same manner.

The computation of SVM process is done for 100 iterations for each percentage of the learning ratio (we selected between 1% and 90% of total data to define the training set of data) and calculating the average to produce the recognition rate, which is supposed to be more consistent. Among the existing kernels for the SVM, we classically use the Radial Basis Function (RBF) kernel function. This kernel nonlinearly maps samples into a higher dimensional space so it can handle nonlinear relation between class labels and attributes. We have to search which is the best couple (C, γ) for each type of pattern. For best accuracy performance, we have to set two parameters (C = 128.0 is the penalisation coefficient of the SVM, γ = 0.125 is the parameter of the kernel), introduced by [START_REF] Hsu | Soft biometric traits for personal recognition systems[END_REF]. Here, each data file is normalised in order to have input values in the range ({-1; 1}).

To validate our performance metrics, we perform the computation of Confidence Interval (CI). A CI is necessary when it is associated with the recognition rate of the soft biometric trait. Therefore, a CI computation may reinforce the confidence in the obtained results. It is based on a re-sampling, which consists of a random draw with a replacement of new values of example from the test base. For each draw, the data will be randomly selected i.e. different data obtain in each selection. This will be done i times in order to calculate the CI (ideally, i=1000 draws). The CI can be determined based on the percentiles of the normal distribution. It represents a measure of confidence on the estimated error rate i.e. the smaller the interval is, the more reliable the calculated error rate is. Here, the CI at 95% is defined by Equation ( 1), where the recognition rate is estimated from the initial sample, i the number of draws, and σ the variance of the i recognition rates calculated for the different draws.

CI = 1.96 × Σ(rate) ± σ(rate) √ i (1)

Experimental Results

We had performed several simulations with SVM for computations on three different aspects of the data. The first results deal with the averaged (over 100 iterations) recognition rates for the four soft categories for different percentage of training data, from 1% to 90%. Then these results are completed by confidence intervals computation, based on a re-sampling and shuffling of the data.

• Hand Recognition Figure 2(a) illustrates the results of the recognition rates on different learning ratios with one hand (C 1 ) and two hands (C 2 ) for five different passphrases P 1 to P 5 . In this experiment, the results are promising, since from the ratio of 50% of total data used for training the SVM, the recognition rate is over 90%.

• Gender Recognition Figure 2(b) illustrates the results of the recognition rates on different learning ratios with males (C 1 ) and females (C 2 ) for passphrases P 1 to P 5 . The recognition rate, depending on the considered passphrase, is between 65% and 90% for a ratio over or equal to 50%.

• Age Category Recognition Figure 2(c) illustrates the results of the recognition rates on different learning ratios with < 30 years old (C 1 ) and ≥ 30 years old (C 2 ) for passphrases P 1 to P 5 . The recognition rate for a ratio over 50% is slightly less than that of other soft criteria, namely between 65% and 82%.

• Handedness Recognition Figure 2(d) illustrates the results of the recognition rates on different learning ratios with right-handed (C 1 ) and left-handed (C 2 ) for passphrases P 1 to P 5 . The obtained recognition rate tends to vary more than for other soft categories, but stays between 70% and 90%, which are nevertheless quite good results.

• Confidence Intervals and Discussion

Table 3 shows the CI computed for the different categories (i.e. hand(s), gender, age, handedness): the previous results are coherent with the obtained CI. Notice that we add an extra Gender category: the first line corresponds to the previous simulation, with 78 males and 32 females. The second line in this category corresponds to an equilibrated case, where we have considered only 32 males to have the same number of males and females. Not surprisingly, the results are significantly better with the equilibrated classes, with an increase of 5 to 10%. We also add an extra Age category: < 32 and ≥ 32 years old. The results are slightly the same as those of initial age category. 96% ± 0.1% 96% ± 0.1% 95% ± 0.1% 94% ± 0.1% 94% ± 0.1% Gender (78m/32f ) 64% ± 0.3% 64% ± 0.3% 63% ± 0.3% 71% ± 0.3% 68% ± 0.3% Gender (32m/32f ) 74% ± 0.3% 69% ± 0.3% 70% ± 0.2% 78% ± 0.2% 76% ± 0.2% Age (< 30 ≥) 64% ± 0.2% 64% ± 0.2% 63% ± 0.2% 69% ± 0.2% 69% ± 0.2% Age (< 32 ≥) 63% ± 0.2% 63% ± 0.2% 64% ± 0.2% 67% ± 0.2% 69% ± 0.2% Handedness (12rh/12lh) 72% ± 1.2% 73% ± 1.2% 72% ± 1.2% 72% ± 1.3% 73% ± 1.2%

From the previous results, we are able to see that the performances differ from one soft category to another because of the different criteria involved in the analysis mentioned earlier in the article.

In this paper, we propose a new soft biometric approach for keystroke dynamics. It consists of predicting the user's way of typing by defining the number of hands used to type (one or two), the gender, the age category, and handedness, whereby the results were promising. Another part of this work is the creation of a substantial database, with 110 users, from France and Norway, with 100 samples per user. The obtained results could be used as a reference model to assist the biometric system to better recognise a user by a way he/she types on a keyboard. Hence, it would strengthen the authentication process by hindering an impostor trying to enter into the system. Having made a video recording during the data collection session, we also plan to exploit the video capture to further enhance the performances by using a fusion method as our future work. Another work in progress consists in studying the fusion of several soft categories, to enhance the recognition.

Fig. 2 .

 2 Fig. 2. Average recognition rates

Table 1 .

 1 Repartition of samples

	User

70 (France); 40 (Norway) Gender 78 males (47 from France, 31 from Norway); 32 females (23 from France, 9 from Norway) Age Category < 30 years old (37 men, 14 women); (between 15 and 65 years old) ≥ 30 years old (41 men, 18 women) Handedness 98 right-handed (70 men, 28 women); 12 left-handed (8 men, 4 women)

Table 2 .

 2 Passphrases

	Password	Description	Size
	P1	leonardo dicaprio	17-char
	P2	the rolling stones	18-char
	P3	michael schumacher 18-char
	P4	red hot chilli peppers 22-char
	P5	united states of america 24-char

Fig. 1. GREYC keystroke software

Table 3 .

 3 Confidence interval computation at 50% learning ratio for 5 passphrases

	P1	P2	P3	P4	P5
	Hand(s)


A. Privacy protection in the e-health context

Security and privacy of data medical records is investigated since the mid 90s by Anderson [START_REF] Anderson | A security policy model for clinical information systems[END_REF], where several principles on e-health data management were proposed. Specific regulations to the e-health context have been developped, such as the HIPAA for United States (2006, [US 06]), or the recommendation for European Union (1997, [Cou97]). This last recommendation has allowed to define the personal data. These data cover any information related to an identifiable individual. Moreover, the medical data refer to all personal data concerning the health of an individual. Consequently, medical data belong to the patient, who can require his/her own sovereignty on them.

Quantin et al. in [QCF + 09] suggest the de-centralized approach for e-health information systems. This solution, combined with the unlinkability principle, gives rise to an identity management based on a local identifier for each patient in each hospital. The aggregation of personal health information is thus prevented. A first solution is the management of these local identifiers by a central trusted authority, with a global table of local identifiers. This authority transfers the required identity from a service to another. However, as the centralized approach, this approach presents problems. Indeed, the identifier table is vulnerable to data removal or disclosure.

These vulnerabilities can be avoided. The identity management shall not store the relationship between different used local identifiers on the database. This solution is proposed by Deng et al. in the context of e-health providers [START_REF] Deng | Towards a cross-context identity management framework ehealth[END_REF], [DSDC + 09]. Nevertheless, access rights are not really taken into account according to the role of the medical employee and no authentication between different hospitals is realized. Moreover, the unlinkability principle involves another problem. Indeed, in their scheme, a doctor requests the local identifier to the identity provider with a patient description. Consequently, these data are linked in a table. The patient description could be an alternative to a global identifier and provides a possibility of aggregation.

Ateniese and Medeiros [START_REF] Ateniese | Anonymous eprescriptions[END_REF] propose others alternatives in the context of privacy in healthcare in suggesting a group signature scheme. Similarly, De Decker et al [START_REF] Decker | A privacy-preserving ehealth protocol compliant with the belgian healthcare system[END_REF] use privacy preserving credentials. Nevertheless, both schemes require major changes in existing e-health infrastructures.

B. Privacy requirements in e-health information systems

The privacy in medical records is realized according several criteria. Firstly, the use and the collection of data are safeguarded. Then, the anonymity and the unlinkability principle must be ensured. Moreover, the patient has control over their data. Finally, the patient's registers must be decentralized.

Security requirements: Each medical record requires an access control, with the name of the people or the group of people who may read or modify data. All access must be marked on the record with the name, the date and the time. The access control ensures data confidentiality and data integrity, verifying that only authorized people could access medical records. Moreover, a direct application of the security principle of computarized data files requires encrypted databases. Availability of medical data must be ensured by hospitals and only to authorized medical personal, even if medical data are shared between several healthcare providers.

Medical data minimization: This principle equally provides the anonymity of medical data, including a possible reversibility in emergency cases. Unlinkability property involves effective measures preventing the aggregation of personal health information, inside the hospital and between different e-health providers. Moreover, an accurate access to right policy is used for data minimization. For instance, a doctor can fully access medical information of his/her patients, contrary to a nurse who will have a limited access. In both cases, they do not need to know patient's administrative information, according to the data minimization principle.

Sovereignty of medical records:

The doctor creating the medical data or the hospital storing them are not the owners of this information. Indeed, all medical information belongs to the patient. Consequently, patients have control over their data and the access to this information. Moreover, the transfer between two institutions must be realized with his/her consent. Obviously, the emergency scenario, where the life of the patient depends on external information, must take into account. A typical characteristic of an e-health context is the trusted relationship between a doctor and his/her patient. Implicitly, this latter gives the doctor his/her consent to access his/her medical record. Nevertheless, this consent does not concern all the medical records of all institutions. This issue is confirmed by the numerous illegal information flows which have been discovered in the British NHS. Besides, Anderson [START_REF] Anderson | Under threat: patient confidentiality and nhs computing[END_REF], [START_REF] Anderson | Patient confidentiality and central databases[END_REF] recalls this conflict between the consent of patient and the need to know of doctors. De-centralized system: In many countries, as for the British NHS system, a national database of medical records has been considered in order to improve the availability of medical records. However, strong criticisms of practitioners combined with negative reactions of public opinion have emerged for several reasons. For example, in case of system problem, a complete failure of medical information could be caused by the centralized data. Moreover, a major disclosure of these data would also have a strong national consequence for the privacy of citizens [START_REF] Anderson | Under threat: patient confidentiality and nhs computing[END_REF], [START_REF] Anderson | Patient confidentiality and central databases[END_REF]. Thus, a direct consequence of the data sensitivity principle is the use of de-centralized systems.

III. GLOBAL APPROACH OF THE E-HEALTH ARCHITECTURE

A. Identity management

In the proposed scheme, a Public Key Infrastructure is used. The PKI is a hierarchical infrastructure with one certification authority, depending on the current country. This authority allows the generation and the storage of public/private keys pair. The private key allows to sign the certificates of different hospitals, clinics and other e-health providers H 1 ,..., H n . Thus, a trusted communication is created between them. Then, these institutions generates and stores a pair of public/private keys. These latters are used to sign the certificate of each medical employee M i . Moreover, each medical personnel has a numerical identity (login/password) provided by his e-health provider 1 .

In this architecture, the identity derivation service IDS is considered as the trust part, by creating and managing local pseudonyms for the patient from his/her global identifier Gid. This number identifies the patient in all health structure and is used during a transfer of medical records. Gid is a very sensitive data. Consequently, the storage of a table of all patients' global identifiers in a given institution would be very dangerous for the patients' privacy. Thus, instead of store the global identifier, the identity derivation service IDS computes two local identifiers from the patient's global identifier. Two identity controllers are then used to check identities and access rights for data inside the e-health provider and in other institutions. The access controller specific medical information is called M AC (Medical Access Controller), the corresponding administrative data is the AAC (Administrative ccess Controller).

Consequently, in our scheme (see Fig. 1), two databases are considered: one for administrative records (first name, last name, civil status, address, phone number,...) and another for medical records. These bases correspond to two local identifiers Lid 1 and Lid 2 . At admission to the hospital, the patient may be given a wristband (or contactless smartcard) equipped with a RFID tag. This wristband contains his/her name, first name, his/her Lid 2 , used as a pseudonym for the patient in his/her medical records and also one secret point explained in the Section IV.

During the registration phase of a patient P , described in the frame III-A.1, the identity derivation service IDS uses two secret keys K 1 and K 2 . Then, it applies the encryption scheme AES to the patient global identifier Gid. The AES is used with K 1 and K 2 in order to respectively compute Lid 1 and Lid 2 . The global identifier is deleted to avoid accidental disclosure of this very sensitive information. Moreover, 1 A secure smart card for medical personnel would increase the security of this authentication. The card could contain the digital certificate of the holder, with the signature of the certification authority. During the consultation, the patient presents his/her wristband to the doctor as a proof of his/her consent for the access and the update of his/her medical record by the doctor. The communication between the RFID tag and the doctor's NFC tablet allows the transfer of Lid 2 . As a result, the doctor has to maintain a list of his/her patients along with the list of local identifiers in his/her personal tablet. In each consulted service, the patient presents his/her medical wristband. Thus, after check of access controllers, the service could access his/her medical record. The requestor's identity is recorded by the access controllers together with a date and time stamp.

Thus, the doctor's authentication authenticates the doctor to M AC as a doctor affiliated with the hospital. The knowledge of Lid 2 allows M AC to extract the information of the affected patient. The access procedure to medical record is similar for each employee (see Frame III-A.2). However, as detailled in the Section IV, this knowledge is not enough to be considered as the affected patient's physician, and consequently to have access rights.

III.A.2. Access medical data with the local identifier

1) The doctor provides his/her login/password and Lid2 of his/her patient to M AC. 2) M AC controls the doctor's identity, personal certificate and access rights. 3) M AC accepts or refuses authorization to medical data and replies to the request using Lid2. The requester's identity, the date and the time are registered.

In emergency case, if the patient is unconscious, an authorized doctor or service needs to access the patient's medical records, the concerned medical personnel gives the patient's identity instead of Lid 1 or Lid 2 (see Frame III-A.3). If the patient is already registered, AAC can retrieve Lid 1 from the patient's identity. Then, IDS retrieves Gid from K 1 and Lid 1 , and then Lid 2 from Gid and K 2 . Finally, M AC registers the requester's identity, with the date, the time and the type of requested medical information. After the operation, the patient is informed of this process.

III.A.3. Access medical data without the local identifier

1) The doctor provides his/her login/password and an patient's identity, such as the name, to AAC. 2) AAC controls the doctor's identity, the personal certificate and the access rights. 3) AAC retrieves Lid1 with the patient identity. 4) AAC gives Lid1 to IDS. 5) IDS computes the patient's Gid with Lid1 and K1. 6) IDS computes Lid2 with Gid and K2. 7) IDS deletes Gid and gives Lid2 to M AC. 8) M AC accepts or refuses authorization to the medical record and replies to the corresponding request with Lid2. The identity of the requester, the date and the time are registered.

B. Data transfer between hospitals

The collaboration between two hospitals or clinics for cross communications of medical records is sometimes required. This collaboration ensures the continuity of the patient's medical treatment and the availability of medical data. The communication must be realized on a secure channel with the consent of the patient (except, of course, in some exceptional cases such as an emergency).

This architecture easily supports this requirement, through the PKI infrastructure and the patient's local identifier. More precisely, an authorized medical personnel, authenticated by his own medical access controller, can request external information. In this case, the identity derivation service recovers the global identifier (without storage) and transfers it to the second hospital, which can retrieves the requested information.

IV. ENCRYPTION OF DATABASES

In order to access patient's medical records, the employee provides information on his/her rights and on his/her patient. If an employee is authorized, the access controller M AC transfers the requested medical records. However, employees have not the same rights in a hospital. For instance, we can considered the following simplified scheme, described in the table 2: the doctor and the patient can access and complete medical records, whereas the nurse only need access prescriptions and therefore has no right diagnosis. In order to manage the employees' access rights in medical institutions, it is necessary to be able to add actors to the systems. Moreover, for practical reasons, the actors must have one unique private key. The principle of Shamir's secret sharing perfectly applies [START_REF] Shamir | How to share a secret[END_REF]. The idea is to divide the secret into several parts distributed to participants and is based on the fact that n points are enough to define a (n -1)-degree polynomial. Thus, in order to find the secret, a certain number of (n -1) points must unite.

During the patient's registration, a (n -1)-degree polynomial is attributed to the patient. Thus, in addition to the local identifiers, his/her wristband with RFID tag contained one secret point of this polynomial. Concerning the employees, during the consultation, the patient presents his/her wristband to the employee. Then, a secret point of polynomial is created by the server for this employee. The server must also store (n -1) points of this polynomial. Indeed, if the patient or the employee wants access these data, the combination of his/her secret point with the (n-1) points of server allows to retrieve the initial polynomial. Thus, the intercept represents the encryption key of medical patient data. Similarly, the decryption of administrative data appeals to the secretary's secret point and the (n -1) server points. The Fig. 3 gives an overview of the solution. The cryptographic calculations can be executed inside an HSM or in implementing crypto-processors as explained by Sklavos in [START_REF] Sklavos | On the hardware implementation cost of crypto-processors architectures[END_REF].

Consequently, the access right checks by AAC and M AC are realized through this protocol. Thus, if a rogue doctor obtains Lid 2 of a patient who he/she does not follow, he/she will not have a point of polynomial. So, he/she does not access the data. Moreover, when he/she accesses data, the history is updated. As indication, in the administrative file, the name, first name and age must not be encrypted to ensure potential admission to the emergency.

V. SECURITY AND PRIVACY ANALYSIS

A. Security requirements

In order to access patient's data, the medical employee must be authenticate thanks to a classical system of login and password and their certificates. Therefore, only relevant employees can access or update information inside the institution. Moreover, the files of databases, the contactless transactions are encrypted and all exchanges are authenticated. Consequently, the confidentiality and integrity of personal data are ensured by the access controllers. Finally, the medical data can be read in the exceptional emergency case. Thus, the availability of personal data is also respected.

B. Medical data minimization

Firstly, the use of different local identifiers for one patient inside one hospital allows to ensure the anonymity of medical data. Thus, if the contents of the medical records and of the administrative records are stolen, it is not possible to aggregate these information without the knowledge of the secret keys. In the same way, the possession of two medical data files of two different institutions does not allow to bind these information.

Moreover, only the identity derivation service knows the relation between Lid 1 , Lid 2 and Gid, and is able to generate them. Reciprocally, the medical access controller accesses to the medical databases, but M AC does not know the link between the local identifiers. Moreover, M AC is not able to access administrative database. The same constraint is observed for administrative access controller. Thus, the distinction between identity manager, controllers and databases allows to ensure the unlinkability principle.

Finally, the use of Shamir's secret sharing manages the access rights and ensures that only relevant information are provided to medical employee. Indeed, each employee has one secret point which allows to find the (n -1)-degree polynomial associated with (n -1) points of trust server. Moreover, if the server is damaged, its (n -1) points do not allow to find the sensitive (n -1)-degree polynomial. Consequently, the minimization principle is also ensured. Moreover, the proposed access control policy takes into account the modifications of employee's access rights. For instance, if a nurse has not access to patient data, his/her point is deleted. In addition, the points of other members with access rights are recomputed.

C. Sovereignty of medical records

The exchange between the patient and the employee through the patient's wristband and the employee's NFC tablet allows to provide the Lid 2 and to generate a secret point for the employee. The consent of the patient for the access to his medical record is thus realized. Moreover, when an employee accesses to the patient's data, the identity of the employee, the date and the time of the request are registered. Consequently, the data sovereignty principle is ensured and strengthened.

VI. CONCLUSION

The protection of sensitive medical information in an ehealth system is a complex challenge for security and privacy protecting technologies. The proposed solution preserves the privacy of the patient, under strong constraints of the medical system. The access system of encrypted data, based on the Shamir's secret sharing, presents an efficient solution, that could be integrated in existing health infrastructure.

Figure 1 .

 1 Figure 1. Organization inside an hospital

Figure 2 .

 2 Figure 2. Access rights function of actors

Figure 3 .

 3 Figure 3. Shamir's secret sharing in a e-health system

ACKNOWLEDGMENT

The authors would like to thank Coline Migonney for her participation during the section encryption of database, as well as the BULL company and the ANRT association for their financial support.


Damien Passemier
email: damien.passemier@gmail.com

Jianfeng Yao

Estimation of the Number of Spikes, Possibly Equal, in the High-Dimensional Case

Keywords: Spiked population model, high-dimensional covariance matrix, random matrix theory, Tracy-Widom law

Estimation of the Number of Spikes, Possibly Equal, in the High-Dimensional Case

Damien Passemier, Jian-Feng

Introduction

The spiked population model has been introduced in [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF], and appears in many scientific fields. In wireless communications, a signal emitted by a source is modulated and received by an array of antennas, and the reconstruction of the original signal is directly linked to the inference of "spikes". In psychology literature, the strict factor model is equivalent to the spiked population model, and the number of factors has a primary importance [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF]. Similar models can be found in physics of mixture [START_REF] Kritchman | Determining the number of components in a factor model from limited noisy data[END_REF], [Nad] or population genetics. More precisely, we consider the following spiked population model for the observed signals x(t):

x(t) = q 0 k=1 f k (t)a k + σn(t) = Af (t) + σn(t), (1)
1 where f (t) = (f 1 (t), . . . , f q 0 (t)) * ∈ R q 0 are q 0 independent random signals (q 0 < p) with mean zero and unit variance; A = (a 1 , . . . , a q 0 ) is a p × q 0 full rank matrix (mixing weights); and σ ∈ R is the unknown noise level, n(t) ∼ N (0, I p ) is a p-dimensional Gaussian white noise independent of f (t).

The population covariance matrix Σ of x(t) equals AA * + σ 2 I p and has the spectral decomposition W * ΣW = σ 2 I p + diag(α 1 , . . . , α q 0 , 0, . . . , 0) where W is an unknown basis of R p and α 1 ≥ α 2 ≥ • • • ≥ α q 0 > 0. As in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF], we rewrite the spectral decomposition of Σ as W * ΣW = σ 2 diag(α ′ 1 , . . . , α ′ q 0 , 1, . . . , 1), with α ′ i = α i /σ 2 + 1.

Notice that Model (1) is called a strict factor model in [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF]. It should not be confused with those factor model widely used in econometric and finance which have generally a more complex structure. For instance, the term strict factor model is used in these fields to refer to a noise with a general diagonal covariance, while in the so-called approximate factor model or the dynamic factor model (see e.g. [START_REF] Chamberlain | Arbitrage, factor structure, and meanvariance analysis on large asset markets[END_REF] and [START_REF] Forni | The generalized dynamic factor model: identification and estimation[END_REF]), the noise is crosssectionally correlated unlike the white noise structure assumed in (1), and the factors f (t) could be a time-series unlike the i.i.d. structure assumed in (1). In sum, these factor models have a much more complex structure than the spiked population model (1) considered in this paper.

A fundamental inference problem in Model (1) is the determination of the number of spikes q 0 . Many methods have been developed, mostly based on information theoretic criteria, such as the minimum description length (MDL) estimator, Bayesian model selection or Bayesian Information Criteria (BIC) estimators, see [START_REF] Wax | Detection of signals by information theoretic criteria[END_REF] for a review. Nevertheless, these methods are based on asymptotic expansions for large sample size and may not perform well when the dimension of the data p is large compared to the sample size n. To our knowledge, this problem in the context of high-dimension appears for the first time in [START_REF] Combettes | Signal detection via spectral theory of large dimensional random matrices[END_REF]. Recent advances have been made using the random matrix theory by [Har] or Onatski [START_REF] Onatski | Testing hypotheses about the numbers of factors in large factor models[END_REF] in economics, and Kritchman & Nadler [START_REF] Kritchman | Determining the number of components in a factor model from limited noisy data[END_REF] in chemometric literature.

Several studies have also appeared in the area of signal processing from high-dimensional data. Everson & Roberts [ER00] proposed a method using both random matrix theory (RMT) and Bayesian inference, while Ulfarsson & Solo combined RMT and Stein's unbiased risk estimator (SURE) in [START_REF] Ulfarsson | Dimension estimation in noisy PCA with SURE and random matrix theory[END_REF]. In [START_REF] Rao | Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples[END_REF] and [START_REF] Nadler | Nonparametric detection of signals by information theoretic criteria: performance analysis and an improved estimator[END_REF], the authors proposed some estimators using information theoretic criteria. Finally in [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF], Kritchman & Nadler constructed an estimator based on the distribution of the largest eigenvalue (hereafter refereed as the KN estimator). In [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF], we have also introduced a new method based on recent results of [START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF] and [START_REF] Paul | Asymptotics of sample eigenstructure for a large dimensional spiked covariance model[END_REF] in random matrix theory. It is worth mentioning that for high-dimensional time series, an empirical method for the estimation of the spike number has been recently proposed in [START_REF] Lam | Estimation of latent factors for high-dimensional time series[END_REF] and [START_REF] Lam | Factor modeling for high-dimensional time series: inference for the number of factors[END_REF].

In all the cited references above, spikes are assumed to be distinct. However, we observe that when some of these spikes are close each other, the estimation problem becomes more difficult and these algorithms need to be modified. We refer this new situation as the case with possibly equal spikes and its precise formulation will be given in Section 2.2. The aim of this work is to extend our method [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF] to this new situation and to compare it with the KN estimator, that is known in the literature as one of the best estimation methods.

The rest of the paper is organized as follows. In Section 2, the estimation problem of the number of possibly equal spikes is introduced, and our estimator is then proposed with a proof for its asymptotic consistency. Section 3 provides simulation experiments to assess the finite-sample quality of our estimator. In Section 4, we analyze the influence of a tuning parameter C used in our procedure and propose an automatic calibration method of the parameter. Next, we carry out simulation experiments in Section 5 to compare our method to the benchmark KN estimator. Conclusions then follow and the appendix collects all the proofs.

Main results

The sample covariance matrix of the n p-dimensional i.i.d. vectors considered at each time t, (x i = x(t i )) 1≤i≤n is

S n = 1 n n i=1 x i x * i .
Denote by λ n,1 ≥ λ n,2 ≥ • • • ≥ λ n,p its eigenvalues. Our aim is to estimate q 0 on the basis of S n . We first recall our previous result of [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF] in the case of different spikes.

Next, we propose an extension of the algorithm to the case with possibly equal spikes. The consistency of the extended algorithm is established.

2.1. Previous work: estimation with different spikes. We consider the case where the (α i ) 1≤i≤q 0 are all different, so there are q 0 distinct spikes. It is assumed in the sequel that p and n are related so that when n → +∞, p/n → c > 0. Therefore, p can be large compared to the sample size n (high-dimensional case).

Moreover, we assumed that

α ′ 1 > • • • > α ′ q 0 > 1 +
√ c for all i ∈ {1, . . . , q 0 }; i.e all the spikes α i 's are greater than σ 2 √ c. For α = 1, we define the function

φ(α) = α + cα α -1 .
Baik and Silverstein [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF] proved that, under a moment condition on x, for each k ∈ {1, . . . , q 0 } and almost surely, λ n,k -→ σ 2 φ(α ′ k ). They also proved that for all 1 ≤ i ≤ L with a prefixed range L and almost surely,

λ n,q 0 +i → b = σ 2 (1 + √ c) 2 .
The estimation method of q 0 in [PY12] is based on a close inspection of differences between consecutive eigenvalues δ n,j = λ n,jλ n,j+1 , j ≥ 1.

Indeed, the results quoted above imply that a.s. δ n,j → 0, for j ≥ q 0 whereas for j < q 0 , δ n,j tends to a positive limit. Thus it becomes possible to estimate q 0 from index-numbers j where δ n,j become small. More precisely, the estimator is qn = min{j ∈ {1, . . . , s} :

δ n,j+1 < d n }, (2)
where s > q 0 is a fixed number big enough, and d n is a threshold to be defined. In practice, the integer s should be thought as a preliminary bound on the number of possible spikes.

In [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF], we proved the consistency of qn providing that the threshold satisfies d n → 0, n 2/3 d n → +∞ and under the following assumption on the entries of x.

Assumption 1. The entries x i of the random vector x have a symmetric law and a subexponential decay, that means there exists positive constants D, D ′ such that, for all

t ≥ D ′ , P(|x i | ≥ t D ) ≤ e -t .
2.2. Estimation with possibly equal spikes. As said in the Introduction, when some spikes are close each other, estimation algorithms need to be modified. More precisely, we adopt the following theoretic model with K different spike strengths α 1 , . . . , α K , each of them can appear n k times (equal spikes), respectively. In other words,

spec(Σ) = (α 1 , . . . , α 1 n 1 , . . . , α K , . . . , α K n K , 0, . . . , 0 p-q 0 ) + σ 2 (1, . . . , 1 p ) = σ 2 (α ′ 1 , . . . , α ′ 1 n 1 , . . . , α ′ K , . . . , α ′ K n K , 1, • • • , 1 p-q 0
).

with

n 1 + • • • + n K = q 0 .
When all the spikes are unequal, differences between sample spike eigenvalues tend to a positive constant, whereas with two equal spikes, such difference will tend to zero. This fact creates an ambiguity with those differences corresponding to the noise eigenvalues which also tend to zero. However, the convergence of the δ n,i 's, for i > q 0 (noise) is faster (in O P (n -2/3 )) than that of the δ n,i from equal spikes (in O P (n -1/2 )) as a consequence of Theorem 3.1 of Bai & Yao [BY08]. This is the key feature we use to adapt the estimator (2) to the current situation with a new threshold d n . The precise asymptotic consistency is as follows.

Theorem 1. Let (x i ) (1≤i≤n) be n copies i.i.d. of x which follows the model (1) and satisfies Assumption 1. Suppose that the population covariance matrix Σ has K non null and non unit eigenvalues

α 1 > • • • > α K > σ 2 √ c with respective multiplicity (n k ) 1≤k≤K (n 1 + • • • + n K = q 0 )
, and pq 0 unit eigenvalues. Assume that p n → c > 0 when n → +∞. Let (d n ) n≥0 be a real sequence such that d n = o(n -1/2 ) and n 2/3 d n → +∞. Then the estimator qn is consistent; i.e q n → q 0 in probability when n → +∞.

Compared to the previous situation, the only modification of our estimator is a new condition

d n = o(n -1/2 ) on the convergence rate of d n . The proof of Theorem 1 is postponed to Appendix.
There is a variation of the estimator defined as follows. Instead of making a decision once one difference δ k is below the threshold d n (see (2)), one may decide once two consecutive differences δ k and δ k+1 are both below d n , i.e. define the estimator to be q * n = min{j ∈ {1, . . . , s} : δ n,j+1 < d n and δ n,j+2 < d n }.

(3)

It can be easily checked that the proof for the consistency of qn applies equally to q * n under the same conditions as in Theorem 1. This version of the estimator will be used in all the simulation experiments below. Intuitively, q * n should be more robust than qn . We notice that eventually more than two consecutive differences could be used in (3). However, our simulation experiments have shown that using more consecutive differences does not improve significantly. So we limit ourselves the simulation study to two consecutive differences.

Implementation issues and overview of simulation experiments

The practical implementation of the estimator q * n depend on two unknown parameters, namely the noise variance σ 2 and the threshold sequence d n . As for an estimate of σ 2 , we used in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF] an algorithm based on the maximum likelihood estimate

σ 2 = 1 p -q 0 p i=q 0 +1 λ n,i .
As explained in [START_REF] Kritchman | Determining the number of components in a factor model from limited noisy data[END_REF] and [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF], this estimator has a negative bias. Hence the authors developed an improved estimator with a smaller bias. We will use this improved estimator of the noise level when it is unknown.

It remains to choose a threshold sequence d n . As argued in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF], we use a sequence d n of the form Cn -2/3 √ 2 log log n, where C is a "tuning" parameter to be adjusted. In our Monte-Carlo experiments, we shall consider two choices of C: the first one is manually tuned and used to assess some theoretical properties of the estimator q * n ; and the second one is a data-driven and automatically chosen value that is used in real-life applications. This automatic choice is introduced in Section 4.

In the remaining of the paper, we conduct extensive simulation experiments to assess the quality of the estimator q * n including a detailed comparison with a benchmark detector from the literature.

In all experiments, data are generated with the assigned noise level σ 2 = 1 and empirical values are calculated using 500 independent replications. Table 1 gives a summary of the parameters in the experiments. One should notice that both the given value of σ 2 = 1 and the estimated one, as well as the manually tuned and the automatic chosen values of C are used in different scenarios. There are in total three sets of experiments. The first set (Figures 12and Models A, B), given in this section, illustrates the convergence of our estimator in the situation of equal spikes. The second set of experiments (Figures 34and Models D-K) addresses the performance of the automatic tuned C and they are reported in Section 4. The last set of experiments (Figures 5-6-7), reported in Section 5, are designed for a comparison with the benchmark detector KN. 3.1. Comparison between the case of equal spikes and different spikes. In Figure 1, we consider the case of a single spike α and analyze the probability of misestimation as a function of the value of α, for (p, n) = (200, 800), c = 0.25 and (p, n) = (2000, 500), c = 4. We set C = 5.5 for the first case and C = 9 for the second case (all manually tuned). The noise level σ 2 = 1 is given. The estimator performs well: we recover the threshold from which the behavior of the spike eigenvalues differ from the noise ones ( √ c = 0.5 for the first case, and 2 for the second). Next we keep the same parameters while adding some equal spikes. In Figure 2, we consider Model A: (α 1 , α 2 , α 3 ) = (α, α, 5), 0 ≤ α ≤ 2.5 and Model B: (α 1 , α 2 , α 3 ) = (α, α, 15), 0 ≤ α ≤ 8. The dimensions are (p, n) = (200, 800) and C = 6 for Model A, and (p, n) = (2000, 500) and C = 9.9 for Model B.

There is no particular difference with the previous situation: when spikes are close or even equal, or near to the critical value, the estimator remains consistent although the convergence rate becomes slower. Overall, these experiments demonstrate that the proposed estimator is able to find the number of spikes.

On the choice of C: an automatic calibration procedure

In the previous experiments, the tuning parameter C has been selected manually on a case by case basis. This is however untenable in a real-life situation. We now provide an automatic calibration of this parameter. The idea is to use the difference of the two largest eigenvalues of a Wishart matrix (which correspond to the case of no spike): indeed, our algorithm should stop once two consecutive eigenvalues are below the threshold d n corresponding to a noise eigenvalue. As we do not know precisely the distribution of the difference between eigenvalues of a Wishart matrix, we approximate the distribution of the difference between the two largest eigenvalues λ1 -λ2 by simulation under 500 independent replications. We then take the mean s of the 10th and the 11th largest spacings, so s has the empirical probability P( λ1 -λ2 ≤ s) = 0.98: this value will give reasonable results. We calculate a C by multiplying this threshold by n 2/3 / 2 × log log(n). The result for various (p, n), with c = 1 and c = 10 are displayed in Table 2. The values of C are quite close to the values we would have chosen in similar settings (For instance, C = 5 for c = 1 and C = 9.9 or 11 for c = 10), although they are slightly higher. Therefore, this automatic calibration of C can be used in practice for any data and sample dimensions p and n.

To assess the quality of the automatic calibration procedure, we run some simulation experiments using both C and the manually tuned C. We consider in Figure 3 the case c = 10. On the left panel we consider Model D (α = (6, 5)) and Model E (α = (6, 5, 5)) (upper curve). On the right panel we have Model F (α = (10, 5)) and Model G (α = (10, 5, 5)) (upper curve). The dotted lines are the results with C manually tuned. Using the automatic value C causes only a slight deterioration of the estimation performance. We observe significantly higher error rates in the case of equal spikes for moderate sample sizes.

The case c = 1 is considered in Figure 4 with Models H (α = 1.5) and I (α = (1.5, 1.5)) (upper curve) on the left and Model J (α = (2.5, 1.5)) and K (α = (2.5, 1.5, 1.5)) (upper curve) on the right.

Compared to the previous situation of c = 10, using the automatic value C affects a bit more our estimator (up to 20% of degradation). Nevertheless, the estimator remains consistent. Furthermore, we have to keep in mind that our simulation experiments have considered critical cases where spikes eigenvalues are close: in many of practical applications, theses spikes are more separated so that the influence of C will be less important. Kritchman & Nadler. In their papers [START_REF] Kritchman | Determining the number of components in a factor model from limited noisy data[END_REF] and [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF], Kritchman & Nadler develop a different method to estimate the number of spikes. In this section we compare by simulation our estimator (PY) with this method (KN). Notice that these authors have compared their estimator KN with existing estimators in the signal processing literature, based on the minimum description length (MDL), Bayesian information criterion (BIC) and Akaike information criterion (AIC) [START_REF] Wax | Detection of signals by information theoretic criteria[END_REF]. In most of the studied cases, the estimator KN performs better. Furthermore in [START_REF] Nadler | Nonparametric detection of signals by information theoretic criteria: performance analysis and an improved estimator[END_REF], this estimator is compared to an improved AIC estimator and it still has a better performance. Thus we decide to consider only this estimator KN for the comparison here.

Method of Kritchman & Nadler and comparison

Algorithm of

In the absence of spikes, nS n follows a Wishart distribution with parameters n, p. In this case, Johnstone [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] has provided the asymptotic distribution of the largest eigenvalue of S n .

Proposition 1. Let S n be the sample covariance matrix of n vectors distributed as N (0, σ 2 I p ), and λ n,1 ≥ λ n,2 ≥ • • • ≥ λ n,p be its eigenvalues. Then, when n → +∞, such that p n → c > 0

P λ n,1 σ 2 < β n,p n 2/3 s + b → F 1 (s), s > 0 where b = (1+ √ c) 2 , β n,p = 1 + p n 1 + n p 1 3
and F 1 is the Tracy-Widom distribution of order 1.

Assume the variance σ 2 is known. To distinguish a spike eigenvalue λ from a noise one at an asymptotic significance level γ, their idea is to check whether

λ n,k > σ 2 β n,p-k n 2/3 s(γ) + b (4)
where s(γ) verifies F 1 (s(γ)) = 1-γ and can be found by inverting the Tracy-Widom distribution. This distribution has no explicit expression, but can be computed from a solution of a second order Painlevé ordinary differential equation. The estimator KN is based on a sequence of nested hypothesis tests of the following form: for k = 1, 2, . . . , min(p, n) -1,

H (k) 0 : q 0 ≤ k -1 vs. H (k) 1 : q 0 ≥ k . For each value of k, if (4) is satisfied, H (k) 0
is rejected and k is increased by one. The procedure stops once an instance of H (k) 0 is accepted and the number of spikes is then estimated to be qn = k -1. Formally, their estimator is defined by

qn = argmin k λ n,k < σ 2 β n,p-k n 2/3 s(γ) + b -1.
Here σ is some estimator of the noise level (as discussed in Section 3). The authors proved the strong consistency of their estimator as n → +∞ with fixed p, by replacing the fixed confidence level γ with a sample-size dependent one γ n , where γ n → 0 sufficiently slow as n → +∞. They also proved that lim p,n→+∞ P (q n ≥ q 0 ) = 1.

It is important to notice here that the construction of the KN estimator differs form ours, essentially because of the fixed alarm rate γ. We will discuss the issue of the false alarm rate in the last section. 5.2. Comparison with our method. In order to follow a real-life situation, we assume that σ 2 = 1 is unknown and we estimate it for both methods. Furthermore, we use the automatic calibration procedure to choose the constant C in our method. We give a value of γ = 0.5% to the false alarm rate of the estimator KN, as suggested in [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF] and use their algorithm available at the author's homepage. We consider in Figure 5 Model D: (α 1 , α 2 ) = (6, 5) and and Model J: (α 1 , α 2 ) = (2.5, 1.5).

For both Model D and J, the performances of the two estimator are close. However the estimator PY is slightly better for moderate values of n (n ≤ 400) while the estimator KN has a slightly better performance for larger n. The difference between the two estimators are more important for Model J (up to 5% improvement).

Next we consider in Figure 6 Model E: (α 1 , α 2 , α 2 ) = (6, 5, 5) and Model K: (α 1 , α 2 , α 2 ) = (2.5, 1.5, 1.5), two models analogous to Model D and J but with two equal spikes.

For Model E, the estimator PY shows superior performance for n ≤ 500 (up to 20% less error): adding an equal spike affects more the performance of the estimator KN. The difference between the two algorithms for Model K is higher than in the previous cases; the estimator PY performs better in all cases, up to 10%. In Figure 7 we examine a special case with no spike at all (Model L). The estimation rates become the so-called false-alarm rate, a concept widely used in signal processing literature. The cases of c = 1 and c = 10 with σ 2 = 1 given are considered. In both situations, the false-alarm rates of two estimators are quite low (less than 4%), and the KN one has a better performance. In summary, in most of situations reported here, our algorithm compares favorably to an existing benchmark method (the KN estimator). It is also important to notice a fundamental difference between these two estimators: the KN estimator is designed to keep the false alarm rate at a very low level while our estimator attempts to minimize an overall misestimation rate. We develop more in details these issues in next section. 5.3. Influence of C on the misestimation and false alarm rate. In the previous simulation experiments, we have chosen to report the misestimation rates. However, to have a fair comparison to either the KN estimator or any other method of the number of spikes, the different methods should have comparable false alarm probabilities. This section is devoted to an analysis of possible relationship between the constant C and the implied false alarm rate. Following [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF], the false alarm rate γ of such an algorithm can be viewed as the type I error of the following test H 0 : q 0 = 0 vs. H 1 : q 0 > 0, that is the probability of overestimation in the white noise case. Recall the step k of the algorithm KN tests H

(k) 0 : q 0 ≤ k -1 vs. H (k)
1 : q 0 ≥ k. In [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF], the authors argue that their threshold is determined such that

P(reject H (k) 0 |H (k) 0 ) ≈ γ.
of overestimation is kept to a very low level). Otherwise, if the goal is also to minimize the overall misestimation rates i.e. including underestimation errors, our algorithm can be a good substitute to the KN estimator. One could think of choosing C in each case to have a probability of overestimation kept fixed at a low level, but in this case the probability of underestimation would be high and the performance of the estimation would be poor: our estimator is constructed to minimize the overall misestimation rate.

Concluding remarks

In this paper we have considered the problem of the estimation of the number of spikes in high-dimensional data. When some spikes have close or even equal values, the estimation becomes harder and existing algorithm need to be re-examined or corrected. In this spirit, we have proposed a new version of our previous algorithm. Its asymptotic consistency is established. We compare our algorithm to an existing competitor proposed by Kritchman & Nadler (KN, [START_REF] Kritchman | Determining the number of components in a factor model from limited noisy data[END_REF], [START_REF] Kritchman | Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory[END_REF]). From our extensive simulation experiments in various scenarios, overall our estimator can have smaller misestimation rates, especially in cases with close and relatively low spike values (Figure 6) or more generally for almost all the cases provided that the sample size n is moderately large (n ≤ 400 or 500). Nevertheless, if the primary aim is to fix the false alarm rate and the overestimation rates at a very low level, the KN estimator is preferable.

Our algorithm depends on a tuning parameter C. By comparison, the KN estimator is remarkably robust and a single value of γ = 0.5% was used in all the experiments. In Section 5 we have provided a first approach to an automatic calibration of C which is quite satisfactory. More investigation is needed in the future for this calibration in order to further improve the performance of our estimator.

Proposition 2. Assume that the entries x i of x satisfy E( x i 4 ) < +∞, α ′ j > 1 + √ c for all 1 ≤ j ≤ K and have multiplicity n 1 , . . . , n K respectively. Then as p, n → +∞ so that p n → c, the n k -dimensional real vector √ n{λ n,jφ(α ′ k ), j ∈ J k } converges weakly to the distribution of the n k eigenvalues of a Gaussian random matrix whose covariance depends on α ′ k and c.

Proposition 3 is a direct consequence of Proposition 5.8 from Benaych-Georges, Guionnet and Maida [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF]: Proposition 3. Assume that the entries x i of x have a symmetric law and a sub-exponential decay, that means there exists positive constants D, D' such that, for all t ≥ D', we have P(|x i | ≥ t D ) ≤ e -t . Then, for all 1 ≤ i ≤ L with a prefixed range L, n 2 3 (λ n,q 0 +ib) = O P (1).

We also need the following lemma: Lemma 1. Let (X n ) n≥0 be a sequence of positive random variables which weakly converges to a probability distribution with a continuous cumulative distribution function. Then for all real sequence (u n ) n≥0 which converges to 0, P(X n ≤ u n ) → 0.

Proof. As (X n ) n≥0 converges weakly, there exists a function G such that, for all v ≥ 0, P(X n ≤ v) → G(v). Furthermore, as u n → 0, there exists N ∈ N such that for all n ≥ N , u n ≤ v. So P(X n ≤ u n ) ≤ P(X n ≤ v), and lim

n→+∞ P(X n ≤ u n ) ≤ lim n→+∞ P(X n ≤ v) = G(v).
Now we can take v → 0: as (X n ) n≥0 is positive, G(v) → 0. Consequently, P(X n ≤ u n ) → 0.

Proof. of Theorem 1. The proof is essentially the same as Theorem 3.1 in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF], except when the spikes are equal. We have {q n = q 0 } = {q 0 = min{j : δ n,j+1 < d n }} = {∀j ∈ {1, . . . , q 0 }, δ n,j ≥ d n } ∩ {δ n,q 0 +1 < d n }.

Figure 1 .

 1 Figure 1. Misestimation rates as a function of spike strength for (p, n) = (200, 800) and (p, n) = (2000, 500).

Figure 2 .

 2 Figure 2. Misestimation rates as a function of spike strength for (p, n) = (200, 800), Model A and (p, n) = (2000, 500), Model B.

Figure 3 .

 3 Figure 3. Misestimation rates as a function of n for Models D, E (left) and Models F, G (right).

Figure 4 .

 4 Figure 4. Misestimation rates as a function of n for Models H, I (left) and Models J, K (right).

Figure 5 .

 5 Figure 5. Misestimation rates as a function of n for Model D (left) and Model J (right).

Figure 6 .

 6 Figure 6. Misestimation rates as a function of n for Model E (left) and Model K (right).

Figure 7 .

 7 Figure 7. False-alarm rates as a function of n for c = 1 (left) and c = 10 (right).

Table 1 .

 1 Summary of parameters used in the simulation experiments. (L: left, R: right)

	Fig. Mod.	spike		Fixed parameters	Var.
	No. No.	values	p, n	c	σ 2	C	par.
	1		(α)	(200, 800) 1/4 Given (2000, 500) 4	5.5 9	α
	2	A B	(α, α, 5) (α, α, 15) (2000, 500) 4 (200, 800) 1/4 Given	6 9.9	α
	3L	D E	(6, 5) (6, 5, 5)		10	Given 11 and auto n
	3R	F G	(10, 5) (10, 5, 5)		1	Given	5 and auto n
	4L	H I	(1.5) (1.5, 1.5)		1	Given	5 and auto n
	4R	J K (2.5, 1.5, 1.5) (2.5, 1.5)		1	Given	5 and auto n
	5L D	(6, 5)		10 Estimated	Auto	n
	5R	J	(2.5, 1.5)		1 Estimated	Auto	n
	6L	E	(6, 5, 5)		10 Estimated	Auto	n
	6R K (2.5, 1.5, 1.5)		1 Estimated	Auto	n
	7	L	No spike		1 Estimated 10	Auto	n

Table 2 .

 2 Approximation

	(p,n)	(200,200)(400,400)(600,600) (2000,200)(4000,400)(7000,700)
	Value of s	0.340	0.223	0.170	0.593	0.415	0.306
	C	6.367	6.398	6.277	11.106	11.906	12.44

of the threshold s such that P( λ1 -λ2 ≤ s) = 0.98.

Acknowledgment The authors are grateful to two Referees and the Associate Editor for their helpful comments that have led to many improvements of the paper.

More precisely, they give an asymptotic bound of the overestimation probability: they show that for n = 500 and p > 10, this probability is close to γ.

Since for our method, we do not know explicitly the corresponding false alarm rate, we recall in Table 3 the results from Figure 7, which were for the cases c = 1 and c = 10, with the automatic value C, under 500 independent replications. As seen previously, the false alarm rates of our algorithm are much higher than the false alarm rate γ = 0.5% of the KN estimator. Nevertheless, and contrary to the KN estimator, the overestimation rate of our estimator will be different from the false alarm rate, and will depend on the number of spikes and their values. Indeed, we use the gaps between two eigenvalues, instead of each eigenvalue separately. Consequently, there is no justification to claim that the probability P(q n > q|q = q 0 ), for q 0 > 1 will be close to P(q n > 0|q = 0). To illustrate this phenomenon, we use the settings of Model E and K (q 0 = 3) and we evaluate the overestimation rate using 500 independent replications (note that the corresponding false alarm rates are those in Table 3). The results are displayed in Table 4. We observe that these overestimation rates are lower than the false alarm rates given in Table 2: this suggests that no obvious relationship exists between the false alarm rate γ and the overestimation rates for our algorithm.

Furthermore, the overestimation rates of the KN estimator are 0 in all cases: it means that misestimation is mostly underestimated.

In summary, if the goal is to keep overestimation rates at a constant and low level, one should employ the KN estimator without hesitation (since by construction, the probability Appendix A. Proof of Theorem 1 Without loss of generality we will assume that σ 2 = 1 (if it is not the case, we consider λ n,j σ 2 ). For the proof, we need the following Propositions 2 and 3 from the literature. Proposition 2 is a result of Bai and Yao [START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF] which derives from a CLT for the n kpacked eigenvalues

Case of j = q 0 + 1. In this case, δ n,q 0 +1 = λ n,q 0 +1λ n,q 0 +2 (noise eigenvalues). As d n → 0 such that, n 2/3 d n → +∞, and by using Proposition 3 in the same manner as in the proof of Theorem 3.1 in [PY12], we have

Case of 1 ≤ j ≤ q 0 . These indexes correspond to the spike eigenvalues.

• Let I 1 = {1 ≤ l ≤ q 0 |card(J l ) = 1} (simple spike) and I 2 = {l -1|l ∈ I 1 and l -1 > 1}. For all j ∈ I 1 ∪ I 2 , δ n,j corresponds to a consecutive difference of λ n,j issued from two different spikes, so we can still use Proposition 3 and the proof of Theorem 3.1 in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF] to show that P(δ n,j < d n ) → 0, ∀j ∈ I 1 .

For all j ∈ I 3 , it exists k ∈ {1, . . . , K} such that j ∈ J k .

• If j + 1 ∈ J k then, by Proposition 2, X n = √ nδ n,j converges weakly to a limit which has a density function on R + . So by using Lemma 1 and that d n = o(n -1/2 ), we have

Consequently, as previously, δ n,j corresponds to a consecutive difference of λ n,j issued from two different spikes, so we can still use Proposition 2 and the proof of Theorem 3.1 in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF] to show that P(δ n,j < d n ) → 0.

• The case of j = q 0 is considered as in [START_REF] Passemier | On determining the number of spikes in a high-dimensional spiked population model[END_REF].

Conclusion. P(δ n,q 0 +1 ≥ d n ) → 0 and q 0 j=1 P(δ n,j < d n ) → 0, therefore P(q n = q 0 ) -→ n→+∞ 1.