F B Oliveira

C C S Couto-Paz

H S Martins
email: efmartins@unb.br

C J Tierra-Criollo

is with DEMAR C A Coste

Evidence of Current Perception Threshold (CPT) to assess neural fiber function in healthy subjects suggests greater discrimination for stimuli at 1, 250 and 3000 Hz than at 5, 250 and 2000 Hz. Similar data are not yet described for subjects affected by diabetes mellitus. This study proposes to provide a comparative database of parameters obtained with sinusoidal electrical stimulation applied at 1, 5, 250, 2000 and 3000 Hz in subjects affected and non-affected by the diabetes. Ninety subjects were recruited to compose the control (n=45) and diabetic (n=45) groups. The CPT intensities and the reaction times obtained for left and right feet stimulation show responses characterized by weaker intensities (533 to 1671 µA) and longer delays (1.24 to 1.42 s) at low frequencies (1 and 5 Hz) than the intensities (3965 to 5685 µA) and delays (0.96 to 1.12 s) obtained at high frequencies (2000 and 3000 Hz).

Moreover, the low frequency stimulation trials evoked up to 73% of the self-reports corresponding to C-fiber sensations while high frequencies evoked up to 60% of the self-reports related to Aβ-fiber sensations. Moreover, the subjects affected by diabetes needed a stronger intensity of stimulation current in order to perceive consistently the sensations evoked by Aβfibers (hypoesthesia). In addition, the findings reinforce the suggestion that the discrimination between sensations related to different neural fibers is increased for stimuli at 1, 250 and 3000 Hz for both groups.

I. INTRODUCTION

Neural fibers can be classified by their size into large or small fibers according to different functions. Large fibers (6 to 12 µm) mediate myotatic reflexes, touch, pressure, vibration and proprioception. Small fibers (0.2 to 5 µm) mediate sensation of temperature and pain, as well autonomic function [START_REF] Papanas | New diagnostic tests for diabetic distal symmetric polyneuropathy[END_REF][START_REF] Said | Diabetic neuropathy[END_REF]. Nerve conduction analyses are breaking new ground by diagnosing and assessing neuropathies and, in particular, the Current Perception Threshold (CPT) -the weakest current intensity capable of evoking a perception -has been reported and suggested as a screening tool to evaluate the damage of the nerve fibers in the peripheral nervous system [START_REF] Nather | Assessment of sensory neuropathy in patients with diabetic foot problems[END_REF][START_REF] Zheng | Correlation of current perception threshold and somatosensory evoked potential in diabetes[END_REF]. In this context, sinusoidal electrical currents can be used at different frequencies to test different fibers and sensations [START_REF] Nather | Assessment of sensory neuropathy in patients with diabetic foot problems[END_REF][START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF]. Martins et al. [START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF] presented findings obtained from healthy individuals that show better discrimination for stimuli at 1, 250 and 3000 Hz than normally applied at 5, 250 and 2000 Hz. Given that the same findings were not present from subjects affected by diabetes mellitus, the primary aim of this study was to provide a comparative database of parameters obtained from sinusoidal electrical stimulation at different frequencies for subjects affected or non-affected (controls) by diabetes mellitus type 2, allowing a choice of frequencies that would promote better clinical discrimination.

II. METHODS

A. Subjects

Forty-five subjects affected by diabetes mellitus type 2, ages between 56 and 85 years, were recruited to form the diabetic group (table 1). A convenience sampling procedure was performed using subjects who attended a public recruitment forum. Inclusion criteria included: (1) the diagnosis of diabetes mellitus type 2 and (2) a minimal cognition of 19 points evaluated by a modified version of the Mini-mental state examination [START_REF] Folstein | Mini-mental and son[END_REF]. All participants presenting other types of disabilities were excluded. Another 45 subjects non-affected by diabetes mellitus were recruited to compose a control group (table 1). All participants (n=90) signed a consent form approved by the Research Ethics Committee issued by the Foundation of Education and Research in the Health Sciences, Distrito Federal, protocol number 160.752.

B. Procedures

The Michigan Neuropathy Screening Instrument (MNSI) [START_REF] Moghtaderi | Validation of Michigan neuropathy screening instrument for diabetic peripheral neuropathy[END_REF] was applied to identify diabetic peripheral neuropathy. In this clinical scoring system, the subject has to reach a score equal or superior to 8 to be diagnosed as diabetic peripheral neuropathy. A NeuroStim System [START_REF] Martins | Sistema para estudo do limiar de percepção de corrente elétrica com forma de onda arbitrária[END_REF] was used to deliver electrical stimulation following the procedures developed by Martins et al. [START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF]. A sinusoidal current stimulation was applied at frequencies of [START_REF] Papanas | New diagnostic tests for diabetic distal symmetric polyneuropathy[END_REF][START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF]250,2000 skin over the hallux of the left and right feet for control and diabetic groups. The subjects pressed a button in their dominant hand whenever they felt any somatosensory perception. Such information was used to determine the Current Perception Threshold (CPT) and the Reaction Times (RT).

Just after stimulation sequences, the subjects were asked to describe their perceived sensations in their own words (self-report). Those sensations were classified as small Cfiber sensations (reported by terms like prick, pang, twinge, burn or other similar terms); large Aβ-fiber sensations (reported by terms like squeeze, pressure, movement, vibration or other similar terms) or combined sensations. Few subjects had no words to describe their perceptions and were classified as "Not reported" (table 2). In theory, the self-report perceived after a low frequency stimulation (1 and 5 Hz) should be associated to C-fiber sensations, whereas that one perceived after a high frequency stimulation (2000 and 3000 Hz) should be associated to Aβfiber sensations. Given that, each self-report presenting the expected association was defined as "consistent perception". The perceptions evoked after 250 Hz stimulations are mainly related to Aδ-fibers and their self-reported terms can be confused with those describing C-fiber and Aβ-fiber sensations. For this reason, Aδ-fiber sensations were not considered in this analysis.

The statistical inferences and the graphs were generated by GraphPad Prism 5 and differences were considered when the statistical test resulted in p<0.05. D'Agostino-Pearson test expressed a non-Gaussian distribution for the majority of the variables used in this work. Consequently, we applied a non-parametric test for all analyse.

III. RESULTS

A majority, 97.77% (n=44), in the control group and 95.55% (n=43) in the diabetic group were composed of women. Both groups included overweight senior participants (25.00 < BMI < 29.99) with preserved cognition (MMS > 19 points) as observed in table 1. The diabetic group did not presented any subject diagnosed as diabetic peripheral neuropathy, including MNSI scores ranging from 0 to 6 points (MNSI < 8).

TABLE I. CHARACTERISTICS OF THE NON-AFFECTED (CONTROL) AND AFFECTED (DIABETIC) GROUPS

Values are presented by mean ± the Standard Deviation (SD). Wilcoxon Matched-Pairs Signed Rank Test did not detect significant differences (p>0.05) between controls and diabetic groups. The comparisons were made between the control and diabetic groups for all frequencies (1, 5, 250, 2000 and 3000 Hz) and sides (left and right): BMI -Body Mass Index; MMS -Mini-Mental Score; MNSI -Michigan Neuropathy Screening Instrument; CPT -Current Perception Threshold and RT -Reaction Time.

A. Perceptions self-reported

In table 2, we observe a larger contingent of consistent perceptions for 1 than 5 Hz stimulations in the left feet from the control group. For the diabetic group, the contingent of consistent perceptions obtained from 1 Hz was 9% less than that obtained from 5 Hz stimulation just in the left feet. Comparing the contingent of consistent perceptions observed in high frequency stimulations, we found a higher contingent of consistent perceptions from 3000 than 2000 Hz in the right feet for both groups. Values are presented for each group by percentages of the sample separated in columns by frequencies [START_REF] Papanas | New diagnostic tests for diabetic distal symmetric polyneuropathy[END_REF][START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF]250,2000 and 3000 Hz). The images of the feet differentiate the control (drawn in dark lines) from the diabetic (drawn in light lines) group. Kinds of sensations were divided by their relationship with C or Aβ fibers from the left and right foot (lines). The significant increases (↑) or decreases (↓) of the self-reported contingents indicating a consistent perception were indicated by arrows. The significant discrepancies (p<0.05) were detected between the proportions observed from stimulations at 1 and 2000 Hz when, respectively, compared to the proportions expected from stimulations at 5 and 3000 Hz by the Chi-Square Test.

B. Sinusoidal electrical stimulation at different frequencies

As observed in figure 1, the CPT intensities at low frequency stimulations (1 and 5 Hz) generally demanded weaker intensities than those at higher frequencies (2000 Hz and 3000 Hz). The intensities of the stimulations applied at 1 Hz were significantly different from the intensities observed at 250, 2000 and 3000 Hz in all experimental conditions. Furthermore, from the right foot of the diabetic group, the intensity of the CPT at 1Hz was also different from the intensity observed at 5 Hz (figure 1).

The same did not occur for intensities observed from the 5 Hz stimulations. As observed in figure 1, there were significant differences from intensities at 2000 and 3000 Hz, but they were not different from those observed at 250 Hz. In addition, for all experimental conditions, we did not find differences among the pairs of intensities obtained between 2000 Hz and 3000 Hz.

Despite the fact that the statistical test for matched-pairs did not find significant differences between control and diabetic groups (table 1) for CPT intensities, a significant variance of means was detected between the control and diabetic groups for CPT applied at 2000 and 3000 Hz and the multiple comparison post-test identified significant increase between the left foot of the control group and the right foot of the neuropathic group, as indicated by the arrows in figure 1. As observed in figure 2, in general, the RT at low frequency stimulations (1 and 5 Hz) were longer than those at higher frequencies (2000 and 3000 Hz). The delay to perceive the stimulations applied at 1 Hz were significantly different from those observed at 2000 and 3000 Hz in all experimental conditions. Furthermore, from both feet of the control group (right and left sides), the delay of the RT at 1Hz was also different from the delay observed at 250 Hz. However, none difference of delay was observed between 1 and 5 Hz (figure 2).

The same occurs to the delays observed from 5 Hz stimulations, including a significant difference observed between 5 and 250 Hz for the right foot from the diabetic group. The delays at 2000 and 3000 Hz were significantly different from those observed at 1 and 5 Hz, but were not different from those observed at 250 Hz in all experimental conditions (figure 2).

In addition, for all experimental conditions (figure 2), we did not find differences among the pairs of delays obtained between 2000 Hz and 3000 Hz. As previously observed by Martins et al. [START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF][START_REF] Martins | Sistema para estudo do limiar de percepção de corrente elétrica com forma de onda arbitrária[END_REF], this study showed the same pattern of stimulation characterized by weaker CPT intensities and longer RT delays at low frequencies (1 and 5 Hz) than the intensities and delays at high frequencies (2000 and 3000 Hz). Moreover, the stimulation at 250 Hz presented intermediate intensities and delays. Koester and Sielgelbaum [START_REF] Koester | Local signaling passive electrical properties of the neuron[END_REF] explain that the small fibers have a more increased capacitance than the large fibers, then the low frequency stimulations promote depolarization of the small fibers whereas the high frequency stimulation do the same for large fibers. The middle frequency stimulations should have an intermediate behavior.

For the control group, more numerous contingents of consistent perceptions were found for 1 and 3000 Hz than 5 and 2000 Hz stimulations, reinforcing that a better discrimination is found at these frequencies as previously suggested [START_REF] Martins | Current perception threshold and reaction time in the asssessment of sensory peripheral nerve fibers through sinusoidal electrical stimulation at different frequencies[END_REF][START_REF] Martins | Sistema para estudo do limiar de percepção de corrente elétrica com forma de onda arbitrária[END_REF]. As the non-affected group, the group affected by diabetes presented almost the same proportions of self-reports, apart from a consistent perception reduced to 9% in the left foot during 1 Hz stimulations (table 2). In addition, a kind of left-right-footedness was observed in the self-reports.

As concluded by Tan [START_REF] Tan | Cerebral somatosensory potentials evoked by posterior tibial nerve stimulation: lateralization and relation to handedness in lefthanded normal subjects[END_REF], an asymmetric organization interhemispheric of the somatosensory potentials evoked by stimulations of the left and right sides could contribute to a consistent perception more lateralized due to stronger sensory feedback correlated with a hand preference. However, we did not conducted tests to identify hand and foot preferences for further discussion, a limitation of this study.

Although the subjects affected by the diabetes had similar age, gender proportion, body composition, stimulation parameters, and delay of perception than the non-affected subjects (table 1), an increased intensity of current was applied to evoke sensations at 2000 and 3000 Hz (figure 1) for the diabetic group, identifying an hypoesthesia [START_REF] Hohne | Reduced plantar cutaneous sensation modifies gait dynamics, lower-limb kinematics and muscle activity during walking[END_REF] in the Aβ-fibers which was not clinically detected by the MNSI [START_REF] Moghtaderi | Validation of Michigan neuropathy screening instrument for diabetic peripheral neuropathy[END_REF] (4.27±4.04). This result supports a possible advantage to early diagnosis of the diabetic peripheral neuropathy commonly developed in this population.

V. CONCLUSION

The comparative data in this study showed that the subjects affected by the diabetes mellitus needed a higher intensity of stimulation to perceive the sensations evoked by Aβ-fibers (hypoesthesia). In addition, the findings suggest greater dissimilarities obtained for stimuli at 1, 250 and 3000 Hz than 5, 250 and 2000 Hz, reinforcing their use for both groups.

Figure 1 .

 1 Figure 1. Connecting-lines graph indicating median and interquartile range for the intensities of Current Perception Threshold (CPT) analyzed at different frequencies (1, 5, 250, 2000 and 3000 Hz). The CPT intensities were grouped from non-affected (control ○ -thick lines) and affected (diabetic • -fine lines) subjects considering right (continuous lines) and left (discontinuous lines) body sides. The significant differences -detected from Kruskal-Wallis Test followed for Dunn's Multiple Comparison Post-Test and considering p<0.05 -are presented together with the legend (upper graph area) by indicative lines linking pairs of frequencies. The significant increases observed for the diabetic groups -detected by the same statistical test -were indicated by arrows (↑).

Figure 2 .

 2 Figure 2. Connecting-lines graph indicating median and interquartile range for the reaction time (RT) delays analyzed in different frequencies (1, 5, 250, 2000 and 3000 Hz). The RT delays were grouped from non-affected (control ○ -thick lines) and affected (diabetic • -fine lines) subjects considering right (continuous lines) and left (discontinuous lines) body sides. The significant differences -detected from Kruskal-Wallis Test followed for Dunn's Multiple Comparison Post-Test and considering p<0.05 -were also presented together with the legend (upper graph area) by indicative lines linking pairs of delays.

and 3000 Hz to the Current perception threshold through sinusoidal electrical stimulation at different frequencies in a comparative assessment for subjects affected and non-affected by diabetes mellitus

	Franassis B. Oliveira, Emerson Fachin-Martins, Clarissa C.S. Couto-Paz, Henrique R. Martins,
	Member, IEEE, Carlos Julio Tierra-Criollo and Christine Azevedo Coste, Member, IEEE

TABLE II .

 II

PROPORTIONS OF SELF-REPORTED PERCEPTIONS BY GROUPS

ACKNOWLEDGMENT

We thank A. R. Bezerra, D. I. Oliveira-Azevedo and K.K.P. Botelho -the undergraduates in physical therapy of the University of Brasilia and State University of Goias -for their support during the experimental procedures to collect the data. We also thank A. P. Murray for the English review.

*Research supported by FAPEG-CAPES and MCTI. E. Fachin-Martins is with the NTAAI -Núcleo de Tecnologia Assistiva, Acessibilidade e Inovação,


Yosra Ben Slimen
email: yosra.benslimen@orange.com

Sylvain Allio
email: sylvain.allio@orange.com

Julien Jacques
email: julien.jacques@univ-lyon2.fr

Model-Based Co-clustering for Functional Data

Keywords: coclustering, données fonctionnelles, algorithme SEM-Gibbs model-based coclustering, functional data, SEM-Gibbs algorithm

Nous présentons dans ce travail un algorithme de coclustering pour données fonctionnelles. Cet algorithme repose sur le modèle des blocs latents utilisant une modélisation gaussienne des composantes principales fonctionnelles et un algorithme SEM-Gibbs pour l'inférence.

Introduction

With the introduction of new technologies and services in mobile networks, the complexity of these latter have increasingly grown creating an heterogeneous environment where different architectures (micro-, macro-, pico-, femto-cells) and different radio access technologies (GSM, UMTS, LTE, . . .) coexist. In this context, mobile operators need to deal with the new challenges in order to provide a top quality of services without increasing costs.

The quality of services is measured by data captured from the network. These data are generated from different sources such as probes, robots and key performance indicators (KPI). They are used for different needs such as network maintenance, optimization, troubleshooting and management. They are also used for self-organizing networks e.g for self-healing and self-optimization tasks.

KPIs are measurements used to monitor quality of service perceived by the user and the network performance. The KPIs concern different network elements such as transceivers, cells, sites, . . . and they are defined by mathematical formulas derived from different counters, computed periodically from the network. Therefore, KPIs are continuous data that are computed with different temporal granularities (hourly or less, daily, weekly). Figure 1 provides an illustration of 20 daily evolutions of 30 KPIs, with a temporal granularity of 15 minutes. Some KPIs are common across the different radio access technologies while others are specific to each one of them. Even under same technology, the counters/KPIs may differ from one constructor to another. Therefore, as the number of technologies, services, cell types, and constructors grows, the number of KPIs observed by the support team becomes enormous and network planning and operating becomes more complex. However, nowadays, the number of procedures, e.g related to optimization and fault management, that are manually carried out is still considerable. This has triggered a significant research effort, gathered under the term "automatic networking".

In this context, with this work, we are interested in KPIs and the study of their behaviors. We define a clustering of not only the different KPIs, but also of observations. The proposed model can help the technical support team of mobile operators and it can afford additional information to self-organizing networks by discovering new relationships between the different KPIs and new similarities between their behaviors in different days. In other terms, we define a model for co-clustering of both KPIs and days.

The rest of this paper is organized as follows. Section 2 presents the data representation and its pre-processing. Section 3 presents the latent block model for functional data that we propose and Section 4 details its inference. Finally, Section 5 describes briefly the ongoing work.

The data

The data under study are a sample of n observations (days), each observation being described by a set of p curves (functional features, KPIs). The statistical model underlying data represented by (multivariate) curves is a stochastic process with continuous time:

X = {X(t)} t∈[0,T ] with X(t) = (X 1 (t), . . . , X p (t)) ∈ R p , p ≥ 2.
A sample path of X is represented by a set of p curves.

Transformation of the observed discretized curves

In practice, data are generally observed at discrete time points and with some noise. In order to reflect the functional nature of data, smoothing methods consider that the true curve belongs to a finite dimensional space spanned by some basis of functions. Let us assume that each observed curve x ij (1 ≤ i ≤ n, 1 ≤ j ≤ p) can be expressed as a linear combination of basis functions {φ j } =1,...,m j :

x ij (t) = m j =1 a ij φ j (t), t ∈ [0, T ]. (1)
The basis expansion coefficients a ij = {a ij } =1,...,m j can be estimated by least square smoothing (see [START_REF] Ramsay | Functional data analysis[END_REF] for instance). In this work, the same basis {φ } =1,...,m is used for all the functional features (KPIs).

Principal components analysis for functional data

Principal components analysis is often used to give a simple representation of the functional data. Principal components analysis for functional data (FPCA, [START_REF] Ramsay | Functional data analysis[END_REF]) consists in computing the principal components C h and principal factors f h of the Karhunen-Loeve expansion:

X(t) = µ(t) + h≥1 C h f h (t), t ∈ [0, T ]. (2)
When curves are assumed to be decomposed into a finite basis of function [START_REF] Govaert | Co-Clustering[END_REF], FPCA consists in a classical PCA of the basis expansion coefficients using a metric defined by the inner product between the basis functions. FPCA has been extended to multivariate functional data in [START_REF] Jacques | Model-based clustering of multivariate functional data[END_REF].

In theory, the number of principal components are infinite. However, in practice, due to the fact that the curves are observed at discrete time points and then approximated on a finite basis of functions, the maximum number of components one can compute is equal to the number m of basis functions used for approximation.

Latent block model for functional data

In a co-clustering study, the goal is to gather observations as well as (functional) features into clusters. For this, we use the latent block model which assumes that data into a block (defined by a cluster of observations and a cluster of features) are independent and identically distributed. Let K r be the number of clusters in row and let K c be the number of clusters in column.

Let x = (x ij ) 1≤i≤n,1≤j≤p be the matrix of curves (x ij : x ij (t), t ∈ [0, T ]) whose rows are observations (days) and whose columns are the functional features. Let c = (c h ij ) 1≤i≤n,1≤j≤p,1≤h≤m be the principal components resulting from a (univariate) FPCA of all the curves x, without distinction between curves from different observations or different features. In the following, the straightforward ranges for i, j, h, k r and k c will be omitted for simplicity of notations.

The latent block model for functional data we propose is defined by: p(x; θ) = v∈V w∈W p(v; θ)p(w; θ)f (c|v, w; θ)

where,

• V is the set of all possible partitions of the rows into K groups, W is the set of all possible partitions of the columns into M groups,

• p(v; θ) = ikr α v ikr
kr , α kr being the row-mixing proportion and v = (v ikr ) ikr with v ikr = 1 if observation i belongs to the row cluster k r , 0 otherwise, • p(w; θ) = jkc β w jkc kc , β kc being the column-mixing proportion, and w = (w jkc ) jkc with w jkc = 1 if the functional covariate j belongs to the column cluster k c , 0 otherwise,

• f (c|v, w; θ) = ijkrkc p(c ij ; µ krkc , Σ krkc ) v ikr w jkc with, -c ij = (c h ij ) 1≤h≤m are the functional principal components of x ij (t),
p(•; µ krkc , Σ krkc ) is the m-variate Gaussian density with mean µ krkc = (µ hkrkc ) 1≤h≤m and diagonal variance matrix Σ krkc with diagonal (σ 2 hkrkc ) 1≤h≤m , • θ = (α kr , β kc , µ krkc , Σ krkc ) 1≤kc≤Kc,1≤kr≤Kr .

Inference via a SEM-Gibbs algorithm

Let us assume that a FPCA has been carried out on the whole data set of curves, and then each curve x ij is represented by its principal components c h ij for h = 1, . . . , m. Inference of the latent block model is computationally infeasible with an EM algorithm [START_REF] Govaert | Co-Clustering[END_REF] and we choose to use its stochastic version SEM coupled with a Gibbs sampling. Starting from an initial value of the parameter θ (0) and of the missing data w (0) , the q th iteration of the partial SEM-Gibbs alternates the following SE and M steps.

SE step Execute a small number (at least 1) of successive iterations of the two following steps:

1. generate the row partition v

(q+1) ikr |c, w (q) for all 1 ≤ i ≤ n, 1 ≤ k r ≤ K r : p(v ikr = 1|c, w (q) ; θ (q) ) = α (q) kr f kr (c i |w (q) ; θ (q) ) k r α (q) k r f k r (c i |w (q) ; θ (q) )
where c i = (c h ij ) j,h and f kr (c i |w (q) ; θ (q) ) = jkc p(c ij ; µ

(q) krkc , Σ (q)
krkc ) w (q) jkc 2. generate the column partition w

(q+1) jkc |c, v (q+1) for all 1 ≤ j ≤ p, 1 ≤ k c ≤ K c : p(w jkc = 1|c, v (q+1) ; θ (q) ) = β (q) kc f kc (c j |v (q+1) ; θ (q) ) k c β (q) k c f k c (c j |v (q+1) ; θ (q) )
where c j = (c h ij ) i,h and f kc (c j |v (q+1) ; θ (q) ) = ikr p(c ij ; µ

(q) krkc , Σ (q) krkc ) v (q+1) ikr M step Estimate θ (q+1) conditionally on v (q+1) , w (q+1) : α (q+1) kr = 1 n i v (q+1) ikr β (q+1) kc = 1 p j w (q+1) jkc µ (q+1) krkc = 1 n (q+1) krkc i j c v (q+1) ikr w (q+1) jkc ij Σ krkc (q+1) = 1 n (q+1) krkc -1 i j (c ij -µ (q+1) krkc ) t (c ij -µ (q+1) krkc ) v (q+1) ikr w (q+1) jkc
where n (q+1)

krkc = i j v (q+1) ikr w (q+1) jkc .
Choosing the parameter estimation and the final partition After a brun in period, the final estimation θ of the parameter θ is defined by the mean of sample distribution. The final partition is estimated by maximum a posteriori according to θ.

Ongoing work

In this paper, we defined a co-clustering model for functional data. Our ongoing work includes the establishment of an experimental study in order to test our approach. The experimentation is composed of tests with simulated data as well as tests with real data. The results will be presented during the conference. Our model is characterized to be generic so it can be applied to any application that uses multivariate functional data and that aims to discover the relationship between them and the different observations. In particular, in the mobile network domain, our model can extract new knowledge about KPIs which will help the technical support team of network operators in optimization and troubleshooting tasks. Moreover, it can be applied as a pre-processing step of self-organizing networks procedures in order to explore the enormous number of the used KPIs.

The main advantage of this model is that it offers a way to get labeled data, which is very hard to obtain in the area of mobile network. Actually, in some use cases such as self-healing, where there is no dataset available (neither issued from live networks nor artificial), designing and evaluating data mining approaches is still a challenge that should be tackled. Therefore, starting from unlabeled data, we proposed a technique of clustering which result i.e the different clusters can be addressed to mobile network experts who can identify them. The resulted labeled data will allow the use of supervised techniques and prediction of new observations.

Figure 1 :

 1 Figure 1: Sample of daily evolutions of 30 KPIs.