domain: spi chim phys

text:
Romain Rouxel

Michele Diego

Fabio Medeghini

Paolo Maioli

Francesco Rossella

Fabrice Vallée

Francesco Banfi

Aurelien Crut

Natalia Del Fatti

Aurélien Crut

Natalia Del Fatti

Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object

Keywords: nanoscale heat transfer, single-particle spectroscopy, metal nano-objects, Kapitza resistance, thermoplasmonics

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

INTRODUCTION

A detailed understanding of nanoscale heat transfer is required in many technologies. It is for instance essential to manage the crucial problem of heat dissipation from nanoscale components in nanoelectronics, to create nanostructured materials with good thermoelectric properties and to maximize the efficiency of nanoparticle-based photothermal therapies. [START_REF] Cahill | Nanoscale Thermal Transport[END_REF][START_REF] Cahill | Nanoscale Thermal Transport. II. 2003-2012[END_REF][START_REF] Pop | Heat Generation and Transport in Nanometer-Scale Transistors[END_REF][START_REF] Snyder | Complex Thermoelectric Materials[END_REF] Nanoscale heat transfer fundamentally differs from that at macroscopic scale for two major reasons. First, the thermal resistances limiting heat transfer at interfaces ("Kapitza resistances" [START_REF] Swartz | Thermal Boundary Resistance[END_REF][START_REF] Stoner | Conductance and Heat Flow between Solids at Temperatures from 50 to 300 K[END_REF] ) play a much greater role in the former case, and rule the cooling rate of small nanostructures. [START_REF] Wilson | Colloidal Metal Particles as Probes of Nanoscale Thermal Transport in Fluids[END_REF][START_REF] Juvé | Cooling Dynamics and Thermal Interface Resistance of Glass-Embedded Metal Nanoparticles[END_REF][START_REF] Giannetti | Thermomechanical Behavior of Surface Acoustic Waves in Ordered Arrays of Nanodisks Studied by Near-Infrared Pump-Probe Diffraction Experiments[END_REF][START_REF] Giannetti | Ultrafast Laser Pulses to Detect and Generate Fast Thermomechanical Transients in Matter[END_REF] Second, Fourier law of heat conduction, which assumes a diffusive motion of heat carriers, ceases to be applicable to describe heat propagation over distances smaller than their characteristic mean free paths (ballistic regime) [START_REF] Chen | Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles[END_REF][START_REF] Siemens | Quasi-Ballistic Thermal Transport from Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-Ray Beams[END_REF] and in situations where the causality relation between the temperature gradient and heat flux becomes manifest (hydrodynamic regime). [START_REF] Gandolfi | International Journal of Heat and Mass Transfer Accessing Temperature Waves : A Dispersion Relation Perspective[END_REF][START_REF] Simoncelli | Generalization of Fourier's Law into Viscous Heat Equations[END_REF] A lot of the current understanding of nanoscale heat transfer has been obtained using noncontact optical pump-probe techniques. The main idea of this approach consists in investigating the dissipation of the energy injected by a pump light beam in a sample by monitoring the induced modifications of the sample optical properties, e.g. by measuring the transmission or reflection of a probe beam. This general strategy can be implemented in various ways, differing by the types of light sources, spectroscopy methods and samples used. For instance, pump-probe techniques can work either in the frequency domain [START_REF] Schmidt | A Frequency-Domain Thermoreflectance Method for the Characterization of Thermal Properties[END_REF][START_REF] Regner | Broadband Phonon Mean Free Path Contributions to Thermal Conductivity Measured Using Frequency Domain Thermoreflectance[END_REF] (modulating pump power, the use of continuous beams being possible in this case) or time domain [START_REF] Wilson | Anisotropic Failure of Fourier Theory in Time-Domain Thermoreflectance Experiments[END_REF] (using time-delayed pump and probe pulses, which can be spatially overlapped or separated), and can be applied to both bulk and nanostructured materials, provided that they are partly absorbing. [START_REF] Minnich | Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths[END_REF][START_REF] Johnson | Direct Measurement of Room-Temperature Nondiffusive Thermal Transport Over Micron Distances in a Silicon Membrane[END_REF] Time-resolved experiments involving absorbing nanostructures (playing the role of both heaters and thermometers) dispersed in a dielectric environment are particularly interesting to study nanoscale heat transfer. Indeed, the sizes of the heated volumes correspond in this case to those of the absorbing nano-objects (which can be of the order of a few or a few tens of nm), rather than that of the pump beam, whose lateral size is limited by diffraction to about half of its wavelength, i.e. a few hundreds of nm (and which constitutes the relevant length scale for heating of bulk materials and thin films). This approach has for instance been used to measure the thermal conductances at the interface between metal nanoparticles and their environment, and to investigate the dependence of these conductances on composition of nanoparticles, environment and interfaces (e.g., nature and length of surfactant molecules). [START_REF] Wilson | Colloidal Metal Particles as Probes of Nanoscale Thermal Transport in Fluids[END_REF][START_REF] Hu | Heat Dissipation for Au Particles in Aqueous Solution: Relaxation Time versus Size[END_REF][START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF][START_REF] Park | Plasmonic Sensing of Heat Transport at Solid-Liquid Interfaces[END_REF][START_REF] Banfi | Temperature Dependence of the Thermal Boundary Resistivity of Glass-Embedded Metal Nanoparticles[END_REF] The relation between the measured time-resolved optical signals and the thermal kinetics in the nanoparticles and their local environment has also been recently modeled, [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF][START_REF] Gandolfi | Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles[END_REF][START_REF] Ferrera | Thermometric Calibration of the Ultrafast Relaxation Dynamics in Plasmonic Au Nanoparticles[END_REF] allowing more precise estimations of interface conductances in cases where environment heating significantly influences time-resolved signals. Additionally, some time-resolved experiments performed on arrays of metal nanowires patterned on crystalline substrates with high phonon mean free paths (e.g. sapphire and silicon) have demonstrated the failure of Fourier law at the nanoscale. [START_REF] Siemens | Quasi-Ballistic Thermal Transport from Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-Ray Beams[END_REF][START_REF] Hoogeboom-Pot | A New Regime of Nanoscale Thermal Transport: Collective Diffusion Increases Dissipation Efficiency[END_REF][START_REF] Frazer | Engineering Nanoscale Thermal Transport: Size-and Spacing-Dependent Cooling of Nanostructures[END_REF][START_REF] Hu | Spectral Mapping of Thermal Conductivity through Nanoscale Ballistic Transport[END_REF][START_REF] Zhang | Length Dependent Thermal Conductivity Measurements Yield Phonon Mean Free Path Spectra in Nanostructures[END_REF] All the studies mentioned above were performed on ensembles of nanostructures. The signals measured in these investigations thus represent an average over nanostructures affected by an unavoidable dispersion of their morphology and thermal contact with the environment. This approach makes analysis of the amplitude of time-resolved signals extremely challenging, as it would require to precisely know the density of the nanoparticles and their size dispersion, and to take into account the different efficiencies with which their heating is generated and probed (due to their different locations in the non-uniform pump and probe beams, and to the dispersion of their optical responses resulting from their morphological differences). Single-particle methods, which have already been applied to study the linear optical response, [START_REF] Crut | Optical Absorption and Scattering Spectroscopies of Single Nano-Objects[END_REF][START_REF] Hu | Dark-Field Microscopy Studies of Single Metal Nanoparticles: Understanding the Factors That Influence the Linewidth of the Localized Surface Plasmon Resonance[END_REF] internal thermalization [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] and acoustic vibrations [START_REF] Ruijgrok | Damping of Acoustic Vibrations of Single Gold Nanoparticles Optically Trapped in Water[END_REF][START_REF] Staleva | Vibrational Dynamics of Silver Nanocubes and Nanowires Studied by Single-Particle Transient Absorption Spectroscopy[END_REF][START_REF] Crut | Acoustic Vibrations of Metal Nano-Objects: Time-Domain Investigations[END_REF][START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF] of metal nano-objects, constitute an ideal way to overcome these limitations. In this work, we apply singleparticle time-resolved spectroscopy to the study of the cooling dynamics of individual metal nanoparticles: gold nanodisks (NDs) supported on sapphire. We report quantitative investigations of the sensitivity of the measured signals to ND heating, by measuring the proportionality factor between optical cross-section changes and nanoparticle temperature rise. This sensitivity is shown to strongly depend on the probe wavelength used, in agreement with the results of previous thermo-optical models. [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF][START_REF] Gandolfi | Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles[END_REF] By performing experiments on a large number of individual NDs with different diameters and thicknesses, we also characterize the size-dependence of ND cooling dynamics. Finite-element modeling (FEM) was used to demonstrate that ND cooling dynamics are essentially determined by the thermal resistance at the gold-sapphire interface, whose value can be determined by comparison between the measured and simulated signals.

EXPERIMENTAL AND THEORETICAL METHODS

Nanodisk fabrication and morphological characterization. Gold NDs with a diameter spanning the D = 60-190 nm range were nanopatterned by a combination of electron beam lithography, Au thermal evaporation and lift-off techniques on the optically polished surface of a 480 μm thick (0001) α-Al2O3 single crystal sapphire substrate. [START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF][START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] A large (10 μm) separation between adjacent NDs was chosen so as to allow optical investigation of a single ND and to avoid optical and thermal inter-ND coupling effects. Atomic force and scanning electron microscopy characterizations of the NDs were performed to estimate the thicknesses and diameters of the produced NDs, respectively. Spatial modulation microscopy/spectroscopy. Localization and linear optical characterization of individual NDs were performed by spatial modulation spectroscopy (SMS). [START_REF] Arbouet | Direct Measurement of the Single-Metal-Cluster Optical Absorption[END_REF] This microscopy/spectroscopy far-field optical technique is based on the periodic displacement of a single nano-object in the focal spot of a tightly focused light beam, which induces a modulation of the transmitted light power (Fig. 1a). The measured signal being related to the extinction cross-section σext of the investigated nano-object via accessible experimental parameters (size of the illuminating spot and amplitude of the spatial modulation), SMS enables σext to be quantitatively determined as a function of the illumination wavelength λ and light polarization angle θ. SMS experiments were performed using a tunable Ti:sapphire oscillator combined with a visible optical parametric oscillator and a frequency-doubling BBO crystal as a light source, allowing optical studies in the 375-1040 nm wavelength range. The light beam delivered by this source was focused on the sample by a 100X microscope objective, the direction of its linear polarization being controlled by a wire grid polarizer.

The size of the laser spot at the sample surface, close to the diffraction limit (about 0.7 λ full-width at half-maximum), was precisely determined by two-dimensional fitting of the SMS images of small NDs obtained using a translation stage. Spatial modulation of the sample was performed at f = 1.5 kHz frequency with 300 nm modulation amplitude and lock-in detection at 2f. Quasi-circular NDs were identified from the weak dependence of their extinction on θ. [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] Time-resolved experiments. The ultrafast dynamics of individual NDs were monitored using single-particle femtosecond pump-probe spectroscopy, [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF][START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF] an approach based on the illumination of the investigated nanoparticle with two spatially overlapping, tightly focused and time-delayed femtosecond light pulses, referred to as pump and probe (Fig. 1c). Time-resolved measurements on individual NDs were performed by combining the SMS microscope with a two-color pump-probe setup based on the femtosecond laser source described in the previous paragraph, delivering ∼150 fs pulses.

The oscillator pulse train was split in two parts to generate the pump and probe beams. The pump beam was produced using second harmonic generation (yielding λpp in the 375-510 nm range), while the probe beam was either the signal from the Ti:sapphire oscillator (with λpr=690-1040 nm in this case) or the output of the optical parametric oscillator (λpr=500-730 nm). A mechanical chopper operating at 30 kHz was used to modulate the pump power. The pump-induced relative changes of probe beam transmission, ΔTr/Tr, were measured using synchronous detection as a function of the time interval separating pump and probe pulses, controlled by a mechanical delay line. This raw measured signal is directly related to the transient changes of the extinction cross-section σext(λpr) of the investigated nano-object at probe wavelength λpr via ∆Tr/Tr= -∆σext (λpr)/Spr, with Spr=πdFWHM 2 /(4 ln2) the area of the probe spot and dFWHM its full width at half maximum. [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] As Spr is an experimentally accessible parameter, these experiments provide access to the ultrafast variations of ∆σext as a function of the pump-probe delay. Frequent optical realignments were performed so as to maintain the pump and probe beams centered on the investigated ND.

Optical simulations. Modeling of the ND plasmonic response was performed by FEM, using the RF module of the COMSOL commercial software. Gold NDs were described as circular cylinders characterized by their height h and diameter D. Their inhomogeneous environment resulting from the fact that they are supported on a sapphire substrate was explicitly included in the model, refractive indexes of 1.76 and 1 being used for the sapphire substrate and the above medium, respectively. A perfectly matched layer was used around the simulation domain to avoid spurious finite-size effects.

Gold dielectric function ε was modeled based on Johnson and Christy tables of bulk gold dielectric function ε JC 40 (the most common choice in the literature although other tables are available [START_REF] Aspnes | Optical Properties of Au: Sample Effects[END_REF][START_REF] Olmon | Optical Dielectric Function of Gold[END_REF] ). ε was written as ε(ω)=ε JC (ω)+ωp 2 /(ω(ω+iγ0))-ωp 2 /(ω(ω+iγ)), which corresponds to describing the dielectric response of gold conduction electrons by a Drude term [START_REF] Crut | Optical Absorption and Scattering Spectroscopies of Single Nano-Objects[END_REF] involving the bulk gold plasma frequency ωP (ℏωP = 9.01 eV) and an optical scattering rate γ of the conduction electrons larger than that of bulk gold (ℏγ0 ≈ 50 meV). [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] ND illumination by a linearly polarized plane wave was considered. Absorption and scattering cross-sections were deduced from the computed electric field and summed to yield the extinction one. Simulations were performed for different D and h values until a good agreement with the investigated experimental spectrum was reached, as shown in Fig. 1a. The ∂σext/∂ε1 and ∂σext/∂ε2 partial derivatives (Fig. 1b) were deduced from the calculations leading to an optimal reproduction of experimental spectra by first reproducing them with slightly modified ε1 or ε2 values and then performing discrete derivation.

Cooling dynamics simulations. The cooling dynamics of gold NDs were modeled using the Heat Transfer module of COMSOL. Gold NDs were described as perfect cylinders. The sapphire substrate was explicitly included in the simulations as a half-sphere of 1 µm radius (large enough to ensure that the choice of boundary conditions at its external border does not influence the computed dynamics). Gold and sapphire were described using their bulk thermal parameters, i.e. cAu= 2.5 J.cm -3 .K -1 and cAl2O3=3.1 J.cm -3 .K -1 volumetric heat capacities, and ΛAu=317 W.m -1 .K -1 and ΛAl2O3=25 W.m -1 .K -1 thermal conductivities (note that the small temperature excursions in our experiments make unnecessary to consider their temperature-dependence). Perfect contact and finite thermal conductance G were assumed at their interface. Simulations were performed in the time domain with a 10 ps time step, the axial symmetry of the system being exploited to reduce computational costs. A ∆T0 initial temperature rise was defined in the ND. ND temperature was seen to remain quasi-uniform throughout the cooling process, as expected from a Bi = hG/ΛAu<<1 Biot number.

RESULTS AND DISCUSSION

Two samples of sapphire-supported Gold NDs with different ND thicknesses (h≈18 and 40 nm, with ND diameters ranging from 60 to 190 nm in both cases) were made. The extinction spectra of the investigated single NDs were determined using SMS (Fig. 1a). [START_REF] Crut | Optical Absorption and Scattering Spectroscopies of Single Nano-Objects[END_REF][START_REF] Arbouet | Direct Measurement of the Single-Metal-Cluster Optical Absorption[END_REF] As discussed in detail in a recent paper, [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] combination of SMS measurements with numerical modeling of ND optical response constitutes a powerful tool for the morphological characterization of single NDs, especially since their electron microscopy imaging is challenging (large substrate thickness preventing the use of transmission electron microscopy and substrate charging degrading the resolution of scanning electron microscopy images). On the one hand, polarization-dependent extinction measurements allow the detection of minute deviations of actual ND shapes from the targeted cylindrical ones, as they induce a polarization-dependent extinction cross-sections and lift the degeneracy of ND in-plane dipolar localized surface plasmon resonances (SPRs). [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] On the other hand, comparison of the measured extinction spectra with those generated using optical numerical simulations allows one to estimate the dimensions of the investigated NDs, which may differ from those targeted in the nanofabrication process. [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] The extinction spectrum of one of the investigated NDs (referred to as ND1 in the following) is shown in Fig. 1a as an example. It displays a marked resonance near 700 nm wavelength, associated to the in-plane SPR of ND1. This spectrum was found to be quasi-independent of polarization, thus suggesting a quasi-circular ND section. ND1 extinction spectrum could be well reproduced by FEM simulations accounting for the sapphire substrate supporting the NDs, using D=101 nm and h=21 nm for the ND diameter and thickness (note that using an optical scattering rate γ of conduction electrons larger than that of bulk gold γ0, with ℏγ0 ≈ 50 meV, is necessary to reproduce the measured SPR width, with ℏγ= ℏγ0+100 meV for the ND shown in Fig. 1a, as in previous analyses on NDs of similar sizes [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] ; the large SPR width probably results from the high polycrystallinity of nanostructures produced using electron beam lithography). These deduced D and h values, are in reasonable agreement (about 10% larger) with those measured with atomic force and scanning electron microscopies, respectively. FEM simulations also enable to determine the sensitivity of the extinction spectra of NDs to changes in the gold dielectric function ε=ε1+i ε2, which is an essential ingredient to analyze the effect of ND heating on its optical response, as discussed in detail below. This sensitivity is quantified by ∂σext/∂ε1 and ∂σext/∂ε2 partial derivatives (Fig. 1b). These derivatives both present enhanced amplitudes in the SPR domain but display different spectral shapes, reflecting the fact that ε1 and ε2 mostly affect SPR position and width, respectively.

The ultrafast dynamics of individual NDs were monitored using single-particle femtosecond pump-probe spectroscopy, [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF][START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF] an approach based on the illumination of the investigated nanoparticle with two spatially overlapping, tightly focused and time-delayed femtosecond light pulses, referred to as pump and probe (Fig. 1c, also see the Methods part). The transient ∆σext variations measured using λpp=410 nm and λpr=690 nm pump and probe wavelengths for ND1 are presented as an example in Fig. 1c. The measured signal contains signatures of the sample excitation by pump pulse absorption and the induced relaxation mechanisms (processes 1-3 in Fig. 1c). [START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF] The sharp peak observed at short timescales reflects the gold ND ultrafast excitation and internal thermalization on a ≈1 ps timescale by electron-electron and electron-phonon scattering mechanisms (process 1 in Fig. 1c). This impulsively launches, by a displacive excitation mechanism induced by thermal dilation, ND acoustic vibrations, which appear on nanosecond timescales as damped oscillations in time-resolved signals (process 2 in Fig. 1c) and have been the scope of recent studies. [START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF][START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] This feature partly overlaps with a monotonic decay of the signal associated to ND cooling (i.e., dissipation of the thermal energy injected by the pump pulse, also occurring on a nanosecond timescale, process 3 in Fig. 1c). The contribution of the cooling process to the time-resolved signals can be obtained upon subtraction of ND internal thermalization and vibrational response contributions, respectively fitted by an exponential decay and a sum of damped sinusoids, each corresponding to a detected vibrational mode. The deduced thermal component, shown in the inset of Fig. 1c, presents a quasi mono-exponential decay over the whole investigated range of pump-probe delays (up to 2.5 ns). More generally, mono-exponential decays were observed at long time scales (>500 ps) for all signals, but deviations from a mono-exponential behavior were observed at short time scales for some of the investigated NDs (in particular in the case of excitation with high pump fluence, as discussed below). All thermal components could however be accurately reproduced using a biexponential decay function A1 exp(-t/τ1)+ A2 exp(-t/τ2), with A1>A2 amplitudes and characteristic times of the order of τ1 ≈ 2 ns and τ2 ≈ 200 ps in the case of h=18 nm NDs.

Time-resolved signals being averaged over multiple acquisitions, care must be exerted to ensure that no modification of the ND thermal dynamics occurs during the course of the experiments.

Controlling the pump-induced initial ND heating is critical in this context, as shown in Fig. 2 for experiments on ND2, a ND fabricated with the same parameters as ND1. For a pump beam fluence at the particle position of Fpp=1 J.m -2 (deduced from measurements of pump beam power and spot size), successive acquisitions were reproducible, corresponding to a quasi-exponential response with A1/A2=4.4 (black line in Fig. 2a). However, increasing pump fluence to Fpp=2.7 J.m -2 led to marked modifications of the cooling dynamics (red line in Fig. 2a), manifesting in biexponential fits with A1/A2 ratio reduced to 1.8. These changes were preserved after return to the initial lower pump fluence (blue line in Fig. 2a). Additionally, they were correlated with a permanent modification of the extinction spectrum of the ND (Fig. 2b), whose SPR became red shifted and with larger maximal amplitude after ND2 exposure to the 2.7 J.m -2 fluence. In the following, thermal investigations were performed in a low excitation regime to avoid such perturbations.

The initial temperature rise of an internally thermalized metal nanoparticle corresponding to a given Fpp value was estimated as ∆T0= uabs /cAu, with cAu the volumetric heat capacity of gold (cAu = 2.5 J.cm -3 .K -1 ) and uabs=σabs(λpp)Fpp/V the absorbed energy density, determined by pump fluence Fpp, ND volume V and absorption cross-section at pump wavelength σabs(λpp). V and σabs(λpp) are obtained from optical FEM calculations. Specifically, V is computed using the sizes used in the simulation to achieve an optimal reproduction of the experimental extinction spectrum (Fig. 1a). FEM simulations also allow to separate the contributions of absorption and scattering to extinction, and thus to estimate σabs(λpp).

This analysis led to ∆T0≈15 and 40 K for the two fluence values used in Fig. 2. For the highest fluence, the ultrafast initial electronic excitation and subsequent electron-lattice thermalization at T0+∆T0 appear sufficient to induce permanent modifications in the ND or its local environment. Understanding the precise mechanisms underlying this effect would require further investigations, but is out of the scope of this paper, where only ∆T0 values below 15 K were considered after these preliminary studies, so as to avoid modification of ND linear and ultrafast optical responses during the course of the experiments.

The effect of probe wavelength on the time-resolved signals measured on ND1 is illustrated in Fig. 3. In our experimental setup based on a Ti:sapphire oscillator, either frequency-doubled or driving an optical parametric oscillator (see the Methods part), the wavelengths of the pump and probe beams cannot be chosen fully independently; therefore their parameters could not be kept constant throughout this series of experiments. In order to isolate the effect of probe wavelength from that of pump excitation, the measured transient extinction cross-section changes ∆σext were divided by the estimated initial temperature rise ∆T0, with ∆T0≈10 K for the Fpp≈0.7 J/m² typical pump beam fluence used in these experiments. Fig. 3a shows that the probe wavelength considerably affects the amplitude of time-resolved signals, but weakly their temporal dynamics. This is best appreciated by separately comparing the temporal dynamics (inset of Fig. 3a) and amplitudes (Fig. 3a-b) of the thermal components of the measured time-resolved signals, obtained by subtraction of the internal thermalization and vibrational contributions. Indeed, the inset of Fig. 3a shows that all thermal components overlap once normalized by their initial value, while Fig. 3b allows one to appreciate their variations of sign and amplitude in the spectral range of the ND SPR (Fig. 1a).

These observations can be understood considering the simplified situation of uniform temperatures in both the nanoparticle and its local surrounding medium (assumed homogeneous here), noted Tp and Tm [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF][START_REF] Gandolfi | Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles[END_REF] (environment heterogeneity and Tm spatial variations being conversely included in the more complete FEM simulations of ND thermal dynamics presented below). In this simplified case, time-and probe wavelength-dependent transient extinction changes are related, to first order, to the temporal evolutions of Tp and Tm via

( ) ( ) ( ) ( ) ( ) t T A t T A t m pr m p pr p pr ext ∆ + ∆ = ∆ λ λ λ σ , (1) with ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ε λ ε σ λ λ ε λ ε σ λ ε λ ε σ λ m m m ext m p ext p ext p dT d A dT d dT d A ∂ ∂ = ∂ ∂ + ∂ ∂ = 2 2 1 1 (2)
where ε=ε1+iε2 is the complex dielectric function of the metal nanoparticle, and εm that of its environment, assumed real. Eq. 1 shows that transient extinction changes generally reflect a combination of the thermal dynamics in the nanoparticle and its environment, with λpr-dependent weights. This may lead to a complex dependence on probe wavelength of time-resolved signals, as was experimentally demonstrated in the case of gold nanoparticles in ethanol. [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF] This effect originates from the different dynamics of the temperatures in the nanoparticle and its local environment in the context of actual time-resolved experiments. Indeed, the former is suddenly increased by pump pulse absorption and monotonously decays towards its equilibrium value, while the latter reaches its maximum with a position-dependent delay, due to the time needed for the heat generated in the nanoparticle to be transferred and to propagate into its environment. [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF][START_REF] Gandolfi | Ultrafast Thermo-Optical Dynamics of Plasmonic Nanoparticles[END_REF] In the present study, the similarities of the dynamics measured using a large number of probe wavelengths and the monotonous decay of |∆σext| as a function of time demonstrated by the inset of Fig. 3a imply that sapphire substrate heating has a negligible influence on the measured signals. This can be ascribed to the weak sensitivity of sapphire dielectric function to temperature changes, with dεm/dT ≈ 4.10 -5 K -1 in the visible range, [START_REF] Thomas | Frequency and Temperature Dependence of the Refractive Index of Sapphire[END_REF] to be compared to the 10 -3 K -1 typical values of |dε1/dT| and |dε2/dT| for gold in the same spectral range. [START_REF] Wilson | Thermoreflectance of Metal Transducers for Optical Pump-Probe Studies of Thermal Properties[END_REF][START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] Additionally, the high thermal conductivity of sapphire (25 W.m -1 .K -1 ) favors heat dissipation in the substrate away from the ND vicinity, and thus limits the temperature rise in this region.

The negligible contribution of environment heating to time-resolved signals means that the heating-induced changes of the ND extinction cross-section are proportional to the ND temperature rise at all times (this temperature being well-defined for pump-probe delays exceeding the few ps duration required for complete ND thermalization by electron-electron and electronphonon interaction processes). In particular, the Ap coefficient entering eq. 1 can be computed as the ratio between the initial amplitude of the thermal signal and the estimated initial ND temperature rise ∆T0. The Ap spectrum of ND1 retrieved from time-resolved experimental signals is shown in Fig. 3b. Its spectral shape resembles that of the ∂σext/∂ε2 derivative shown in Fig. 1b: it displays large and negative values near the central position of the SPR (dashed blue line in Fig. 3b-c) and smaller, positive values on the SPR sides.

Ap can also be theoretically estimated from eq. 2, which requires knowledge of ∂σext/∂ε1,2 and dε1,2/dT derivatives for the investigated gold ND. The determination of ∂σext/∂ε1,2 (Fig. 1b) from optical FEM computations has been presented above. The temperature dependence of gold dielectric function has been the object of multiple recent measurements and models. [START_REF] Wilson | Thermoreflectance of Metal Transducers for Optical Pump-Probe Studies of Thermal Properties[END_REF][START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF][START_REF] Chen | Dielectric Function Dependence on Temperature for Au and Ag[END_REF][START_REF] Shen | Temperature-and Roughness-Dependent Permittivity of Annealed/Unannealed Gold Films[END_REF][START_REF] Reddy | Temperature-Dependent Optical Properties of Gold Thin Films[END_REF][START_REF] Zhang | Dielectric Function of Polycrystalline Gold Films: Effects of Grain Boundary and Temperature[END_REF][START_REF] Magnozzi | Plasmonics of Au Nanoparticles in a Hot Thermodynamic Bath[END_REF] The thermal derivatives dε1,2/dT extracted from these measurements strongly depend on the type of gold sample used (continuous or granular, annealed or not, deposited thickness) and on temperature. As a result, the values of the dε1,2/dT coefficients of gold reported in the literature are dispersed, in particular for the dε1/dT derivative, as ε1 presents large values but weak relative changes with temperature in the visible range.

To evaluate the impact of these uncertainties on Ap spectra, two different dε1,2/dT tables were used for computing Ap (Fig. 3c): one experimentally determined by Wilson and Cahill, [START_REF] Wilson | Thermoreflectance of Metal Transducers for Optical Pump-Probe Studies of Thermal Properties[END_REF] and one theoretically estimated from the expected temperature dependences of Drude and interband components of gold dielectric function, as discussed in ref. [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] (see figure 6 of this reference). The dε1,2/dT values present significant differences in the red/near infrared range relevant to the present experiments. For instance, at 800 nm wavelength, the former table yield dε1/dT≈3.10 -5 K -1 and dε2/dT≈2.10 -3 K -1 , [START_REF] Wilson | Thermoreflectance of Metal Transducers for Optical Pump-Probe Studies of Thermal Properties[END_REF] while the latter one corresponds to dε1/dT≈1.10 -3 K -1 and dε2/dT≈3.10 -3 K -1 . [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] As a result, the computed Ap spectra differ depending on the chosen table (Fig. 3c). However, they both present spectral variations in good agreement with the experimental results, close to those of the ∂σext/∂ε2 derivative (Fig. 1b) which result from the fact that |dε2/dT|>|dε1/dT|in the investigated spectral range for both considered dε1,2/dT tables (see eq. 2). The measured and computed Ap amplitudes, which have maximal values of a few tens of nm 2 .K -1 also show a good quantitative agreement considering the high number of parameters involved in the analysis and the large uncertainties affecting dε1,2/dT coefficients. The proportionality between the measured time-resolved signals and ND transient temperature rise was exploited for systematic investigations of the size-dependence of ND cooling dynamics. Time-resolved signals were collected on 60 different NDs of different sizes. In all the experiments, sample illumination with a λpp=410 nm, low fluence (<1 J.m -2 ) pump beam was used in order to achieve moderate (≈10 K) initial temperature rises and avoid ND modification throughout the measurements (Fig. 2). Since probe wavelength choice is not critical to achieve ∆σext ∝∆Tp (as shown by Fig. 3b), a λpr=510 nm wavelength away from the ND SPR domain (Fig. 1a) was used as it reduces the relative contribution of ND acoustic vibrations to the measured time-resolved signals. All bare time-resolved signals were processed by subtraction of internal thermalization and vibrational signatures, as in Figs. 1a and3a. Fig. 4a presents representative normalized thermal signals obtained for NDs with different thickness (h=18 and h=40 nm, average thicknesses of the two samples used determined by AFM) and diameter (from 60 to 180 nm), showing that their cooling kinetics is strongly dependent on h and weakly dependent on D in this size range. All measured thermal components present an exponential decay at large time scales (>500 ps delays), with an associated decay time τ1 of about 0.7 and 1.7 ns for h=18 and h=40 nm NDs, respectively (as discussed above, some NDs present a biexponential dynamics, whose description requires a second characteristic time τ2<<τ1). Such exponential behaviour suggests an interface-limited cooling dynamics. [START_REF] Wilson | Colloidal Metal Particles as Probes of Nanoscale Thermal Transport in Fluids[END_REF][START_REF] Juvé | Cooling Dynamics and Thermal Interface Resistance of Glass-Embedded Metal Nanoparticles[END_REF] Indeed, in the case of a Biot number Bi = hG/ΛAu <<1 (with ΛAu the gold ND thermal conductivity), which guarantees that the ND remains isothermal throughout the thermal relaxation process, and of a substrate heat conductivity high enough to ensure negligible modification of substrate temperature near the ND being cooled (two relevant assumptions in the case of Au NDs on sapphire), integrating the interfacial heat current density Jint=G∆Tp (Fig. 4b) over the ND surface S in contact with the substrate leads to V•cAu•d∆Tp/dt=G•S•∆Tp. This results in an exponential decay of ∆Tp with a characteristic time τ=cAuh/G, cAu being the volumetric heat capacity of gold and G the Kapitza thermal conductance of the nanostructure/substrate interface. [START_REF] Caddeo | Thermal Boundary Resistance from Transient Nanocalorimetry: A Multiscale Modeling Approach[END_REF] Such an exponential decay, with a characteristic time linearly scaling with ND height, has been experimentally reported for permalloy NDs nanopatterned on a Si substrate [START_REF] Giannetti | Ultrafast Laser Pulses to Detect and Generate Fast Thermomechanical Transients in Matter[END_REF] and the transient thermal dynamics theoretically investigated for the case of Cu NDs deposited on Si [START_REF] Banfi | Ab Initio Thermodynamics Calculation of All-Optical Time-Resolved Calorimetry of Nanosize Systems: Evidence of Nanosecond Decoupling of Electron and Phonon Temperatures[END_REF] . Within the context of the present study, the τ1∝h scaling is in approximate agreement with the 2.4x smaller τ1 values determined for h=18 nm NDs as compared to h=40 nm ones, and their weak dependence on ND diameter. Furthermore, preliminary G estimation via G= cAuh/τ leads to G values in the order of 60 MW.m -2 .K -1 for the gold-sapphire interface, which is consistent with the 20-70 MW.m -2 .K -1 range reported in the literature for thin gold films. [START_REF] Stoner | Conductance and Heat Flow between Solids at Temperatures from 50 to 300 K[END_REF][START_REF] Xu | Measurement of Interfacial Thermal Resistance by Periodic Heating and a Thermo-Reflectance Technique[END_REF][START_REF] Freedman | Thermal Interface Conductance across Metal Alloy-Dielectric Interfaces[END_REF][START_REF] Blank | Influence of the Thickness of a Nanometric Copper Interlayer on Au/Dielectric Thermal Boundary Conductance[END_REF][START_REF] Blank | Towards a Coherent Database of Thermal Boundary Conductance at Metal/Dielectric Interfaces[END_REF] A more thorough modeling [START_REF] Banfi | Ab Initio Thermodynamics Calculation of All-Optical Time-Resolved Calorimetry of Nanosize Systems: Evidence of Nanosecond Decoupling of Electron and Phonon Temperatures[END_REF] of ND cooling dynamics requires to include the effect of the finite heat conductivity Λm of the sapphire substrate (associated to a Jm=-Λm grad(Tm) heat current density according to Fourier law, Fig. 4b), which was achieved using FEM numerical simulations (described in the Methods part). The dynamics computed using the complete FEM model for NDs with 18 or 40 nm thicknesses and 80 or 160 nm diameters (solid lines in Fig. 4c) are slightly slower than the purely exponential ones with τ1=cAuh/G decay time obtained by assuming infinitely fast heat diffusion in the substrate for the same G=70 MW.m -2 .K -1 considered value of interface conductance (dashed lines), the difference increasing with ND diameter. They also deviate from a purely exponential decay, as also observed in a previous theoretical work addressing the cooling of a metal ND in contact with a dielectric substrate, in the case where the latter does not remain isothermal during the cooling process. [START_REF] Banfi | Ab Initio Thermodynamics Calculation of All-Optical Time-Resolved Calorimetry of Nanosize Systems: Evidence of Nanosecond Decoupling of Electron and Phonon Temperatures[END_REF] However, their non-exponential character is noticeable only at time scales exceeding those considered in the experiments. The fact that ND cooling dynamics are primarily limited by heat transfer at the ND-substrate interface is confirmed by comparing the cooling dynamics computed by assuming a purely interface-limited process (corresponding to infinite ΛAl2O3 sapphire thermal conductivity, dashed lines in Fig. 4c) and a purely diffusion-limited process (corresponding to infinite G, dotted lines in Fig. 4c). As expected, the latter dynamics are strongly non-exponential, dependent on ND diameter and considerably faster than the former ones for the whole range of ND sizes explored in the experiments. It is however interesting to note that the large difference between the time scales associated to interfacial heat transfer and heat diffusion in the sapphire substrate demonstrated here for gold NDs would be strongly reduced for some other NDs compositions allowing better interface conductance with the substrate, or for nanoparticles with larger lateral sizes (or nanostripes or thin films). Regarding ND composition, interfacial heat transfer would be for instance about 5 times faster for a nickel ND than for a gold one, as G≈300 MW.m -2 .K -1 for a nickel-sapphire interface. [START_REF] Blank | Towards a Coherent Database of Thermal Boundary Conductance at Metal/Dielectric Interfaces[END_REF] Regarding nanoparticle size, Fig. 4c shows that increasing ND diameter increases the thermal cooling time scales.

The less efficient cooling predicted for a heat source with large lateral size (the limiting case corresponding to a continuous film, i.e. D/h >>1) as compared to the smallest ones, can be understood from the fact that in the latter case of a small nano-object heat can spread inside the substrate in three dimensions (hot spot behavior), while in the former case it mainly propagates in the direction perpendicular to the interface (the less efficient dissipation of the heat generated close to the interface in the former case leading to higher temperature rises in this region).

The difference between the dynamics computed using the complete FEM model and in the simple interface-limited case (in which substrate temperature remains uniform and unmodified throughout the cooling process) implies that extracting a precise G value from time-resolved experimental data requires their comparison with results of the complete FEM model, especially for the NDs with the largest diameter values, i.e. approaching the continuum film case (for which the isothermal substrate hypothesis is less satisfied). To do so, the measured cooling times τ1 were compared to those extracted from exponential fits of the FEM-simulated cooling dynamics using different G values (Fig. 4d). The cooling times determined for h=40 nm NDs are well reproduced by calculations assuming G=(60±10) MW.m -2 .K -1 . Those of h=18 nm NDs correspond to a slightly larger G=(75±10) MW.m -2 .K -1 average value. The small difference between the G values extracted from the two ND thicknesses, corresponding to two distinct samples fabricated during two different processing sessions, could be due to either small differences in the gold-sapphire interface in these two samples or to uncertainties associated to the determination of their average thicknesses by AFM.

CONCLUSIONS

In conclusion, the cooling dynamics of individual supported nanoparticles (gold NDs with 60-190 nm diameters and 18-40 nm thicknesses) were quantitatively investigated using an optical pumpprobe spectroscopy approach combined with a spatial modulation microscope. The use of moderate pump fluences, limiting lattice heating to <15 K values, was essential to avoid permanent modification of the measured signals during the experiments. The absolute amplitudes of the transient extinction cross-section changes, induced by ND heating, were seen to strongly depend on probe wavelength, in good agreement with the results of a thermo-optical model. All experimental signals measured on NDs with diameters down to 60 nm could be theoretically reproduced using Fourier law with bulk sapphire thermal properties, and a gold-sapphire thermal interface conductance of about 70 MW.m -2 .K -1 showing weak size dependence in the investigated size range and consistent with previous measurements on thin films. As demonstrated by FEM investigations, the cooling dynamics measured here are mostly limited by the gold-sapphire interface, the characteristic times associated with interfacial heat transfer from the ND to the substrate being considerably longer than those associated with substrate heat diffusion. These experiments may be extended to other compositions and geometries of the nano-objects and their supporting substrate, opening the way to direct investigations of the different nanoscale thermal transport regimes at the single-particle level.

Figure 1 .Figure 2 .

 12 Figure 1. Linear and time-resolved optical spectroscopies of an individual gold ND supported on a sapphire substrate. a) Absolute extinction spectrum of a single ND (referred to as ND1, black dots) measured by spatial modulation spectroscopy, whose principle is shown on the right (see also the main text). ND1 extinction spectrum is well reproduced by FEM calculations performed with D=101 nm and h=21 nm ND diameter and thickness (red line). b) ∂σext/∂ε1 (blue) and ∂σext/∂ε2 (green) derivatives deduced from FEM calculations. c) Ultrafast extinction changes of ND1 following its impulsive excitation, measured using time-resolved pump-probe spectroscopy (principle shown on the right) with λpp = 410 nm pump wavelength, λpr = 690 nm probe wavelength and 0.7 J.m -2 pump fluence. The signal contains signatures of three physical processes: ND excitation and internal thermalization (1), damped acoustic vibrations (2) and cooling (3). The normalized thermal part of the signal, obtained by subtraction of the contributions of the two former processes, is shown in the inset on a semilogarithmic scale, highlighting its quasi-exponential character.

1 )Figure 3 .

 13 Figure 3. Probe wavelength effect on the measured time-resolved signals. a) Time-resolved signals measured on ND1 with probe wavelengths λpr= 520 nm (black), 660 nm (red), 675 nm (green), 718 nm (blue), 730 nm (orange) (with λpp = 410 nm for all these signals) and 790 nm (purple, with λpp = 395 nm), divided by the estimated initial ND temperature rise ∆T0. The normalized thermal components of these signals are shown in the inset in lin-log scale (same color code). b) Ap coefficients deduced from the amplitude of the thermal components of time-resolved signals measured on ND1 (eq. 1). c) Ap coefficients computed based on eq. 2 using the ∂σext/∂ε1 and ∂σext/∂ε2 spectra deduced from FEM calculations (Fig.1b) and dε1,2/dT values taken from refs.[START_REF] Wilson | Thermoreflectance of Metal Transducers for Optical Pump-Probe Studies of Thermal Properties[END_REF] (solid black line) and 45 (dashed red line). The dashed blue line indicates the central position of the SPR.

Figure 4 .

 4 Figure 4. Morphological dependence of the cooling dynamics of individual gold NDs. a) Normalized thermal components of experimental time-resolved signals measured on individual gold NDs with h=18 nm thickness (estimated by AFM) and diameters of about 60 (black), 120 (blue) and 180 nm (cyan), and h=40 nm thickness and diameters of about 70 (red), 120 (purple) and 180 nm (yellow) (estimated

ACKNOWLEDGMENTS

This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The authors thank Antoine Gallet and Marco Gandolfi for their help at initial stages of the project and Fabien Vialla for useful discussions. Francesco Banfi acknowledges financial support from Université de Lyon in the frame of the IDEXLYON Project -Programme Investissements d' Avenir (ANR-16-IDEX-0005) and from Université Claude Bernard Lyon 1 through the BQR Accueil EC 2019 grant.

Notes

The authors declare no competing financial interest.


Tom Obry
email: tom.obry@laas.fr

Louise Travé-Massuyès
email: louise@laas.fr

Audine Subias
email: subias@laas.fr

DyClee-C: a clustering algorithm for qualitative data based diagnosis

Sensors are multiplying on machines, networks and living things. Reasoning and extracting knowledge from this huge amount of data is among nowadays challenges. In databased diagnostic applications, large amounts of data are often available but a key issue is that data remain unlabelled because labelling would require too much time and imply prohibitive costs. The different situations, e.g. normal or faulty, must hence be learned from the data. Clustering methods, also qualified as unsupervised classification methods, can then be used to create groups of samples according to some similarity criterion. The different groups can supposedly be associated to different situations.

Numerous algorithms have been developed in recent years for clustering numeric data but these methods are not applicable to qualitative (categorical) data. However, in many application domains, qualitative features are key to properly describe the different situations. This paper presents DyClee-C, an extension of the numeric feature based DyClee algorithm to qualitative data. DyClee-C is applied to two data sets: a soybean data set to diagnose the disease soybean plants and a breast cancer data set to assess the current diagnosis in terms of recurrence events and prognose possible relapse.

INTRODUCTION

In the digital age, the amount of data that is recorded by organizations and companies is enormous. If this data is to have added value, it must be possible to extract relevant information automatically. This is why data mining methods appear to be crucial. Among them, clustering methods have an essential role to play. Indeed, data often remains unlabelled because labelling would require too much time and imply prohibitive costs. In diagnostic applications, the different situations, e.g. normal or faulty, must hence be learned from the data. Clustering methods, also qualified as unsupervised classification methods, can then be used to create groups of samples according to some similarity criterion. The different groups can supposedly be associated to different situations.

In the field of clustering, many unsupervised learning algorithms exist. Among the most well-known, we find K-Means [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF], [START_REF] John | Algorithm as 136: A k-means clustering algorithm[END_REF], DBSCAN [START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF] and hierarchical ascending classification (HAC) [START_REF] Gabor | Hierarchical clustering via joint between-within distances: Extending ward's minimum variance method[END_REF]. These algorithms use a numeric distance criterion such as the classical Euclidean distance or the Manhattan distance for high dimensional data sets to determine sample sim-ilarity and closest clusters. For this reason, most algorithms are not applicable to qualitative data, that is nominal or ordinal. However, qualitative features are key to properly describe the different situations in many application domains.

As a matter of fact, the metrics quoted above do not make possible to calculate a distance between two samples described by qualitative features. Some algorithms have been developed to overcome this problem, such as K-modes [START_REF] Huang | Extensions to the k-means algorithm for clustering large data sets with categorical values[END_REF], ROCK [START_REF] Guha | Rock: A robust clustering algorithm for categorical attributes[END_REF] or SQUEEZER [START_REF] He | Squeezer: an efficient algorithm for clustering categorical data[END_REF]. All these algorithms use their own notion of similarity to create clusters.

In this paper, an extension of the DyClee numeric clustering algorithm is proposed [START_REF] Barbosa Roa | Dyclee: Dynamic clustering for tracking evolving environments[END_REF], [START_REF] Andrea | A data-based approach for dynamic classification of functional scenarios oriented to industrial process plants[END_REF], [START_REF] Barbosa | A data-based dynamic classification technique: A two-stage density approach[END_REF]. The method, named DyClee-C, applies to qualitative data. It must be considered as a building block of the mixed numeric/qualitative DyClee version that is under construction. After the presentation of DyClee-C, DyClee-C is applied to two data sets: a soybean data set to diagnose the disease soybean plants and a breast cancer data set to assess the current diagnosis in terms of recurrence events and prognose possible relapse.

The paper is organized as follows. Section 2 presents the basic principles and the different steps of the DyClee algorithm. The qualitative extension DyClee-C is introduced in section 3. The tests on the soybean and breast cancer data sets are presented in section 4. Finally, section 5 provides conclusions and perspectives of this work2 .

THE DYCLEE ALGORITHM

The Dynamic Clustering algorithm for tracking Evolving Environments (DyClee) is an unsupervised learning method. DyClee implements a distance and density based algorithm that features several properties like handling non convex, multi-density clustering with outlier rejection, and it achieves full dynamicity. All these properties are not generally found together in the same algorithm and DyClee hence pushes forward the state of the art in this respect. The first step qualified as distance-based step operates at the rate of the data stream and creates micro-clusters putting together data samples that are close in the sense of the L1-norm. Micro-clusters are stored in the form of summarized representations including statistical and temporal information. The second step qualified as density-based step uses a density-based approach applied to the micro-clusters to create the final clusters. A cluster is defined as a set of connected micro-clusters where every in-side micro-cluster presents high density and every boundary micro-cluster exhibits either medium or high density. Figure 1 illustrates how the DyClee algorithm works. DyClee only handles numeric data sets making use of the KD-Tree algorithm for grouping the micro-cluster at the begining of the density-based step. However, the KD-tree does not process qualitative data. A KD-Tree is a spacepartitioning data structure for organizing points in a kdimensional space [START_REF] Maneewongvatana | It's okay to be skinny, if your friends are fat[END_REF]. KD-Trees are a useful data structure for several applications, such as nearest neighbor search. The work presented in this paper aims to propose a dynamic clustering approach able to manage qualitative data and capture large data dimensions. For this purpose, an alternative to the KD-Tree is proposed and integrated with the DyClee method, thus leading to DyClee-C.

QUALITATIVE EXTENSION DYCLEE-C

The extension to qualitative data relies mainly on the removal of micro-clusters and on an alternative solution to KD-Tree for finding neighbors. Indeed, in DyClee, the hypercubes (i.e the micro-clusters) make it possible to represent a similar population. Centers of micro-clusters (µC) can be calculated by averaging, for each attribute, all the samples present in the µC. In the case of qualitative data, we decided to experiment an algorithm that does not make use of the micro-cluster concept. In consequence, the distance-based step of DyClee, during which the samples are assigned to the µCs, is removed and the density-based step directly deals with the samples. In DyClee, clusters are formed by searching in each group found by the KD-Tree, sub-regions of dense and semi-dense microclusters. In the DyClee-C extension, they are formed from the results of the Locality Sensitive Hashing (LSH) [START_REF] Indyk | Approximate nearest neighbors: towards removing the curse of dimensionality[END_REF] algorithm, [START_REF] Leskovec | Mining of massive datasets[END_REF], [START_REF] Gionis | Similarity search in high dimensions via hashing[END_REF]. This method is used in several applications such as clustering, searching for nearest neighbors in large dimensions and the detection of similar images.

The following sections detail the principles of the LSH algorithm, how the clusters are formed and they present several parameters of the method allowing to refine the clustering results.

Locality Sensitive Hashing algorithm

The LSH algorithm is an appropriate alternative to the KD-Tree since, on the one hand, this algorithm is able to process numerical and qualitative data, on the other hand this algorithm is a solution to the problem of the curse of dimensionality [START_REF] Jerome H Friedman | On bias, variance, 0/1-loss, and the curse-of-dimensionality[END_REF], [START_REF] Rust | Using randomization to break the curse of dimensionality[END_REF], [START_REF] Verleysen | The curse of dimensionality in data mining and time series prediction[END_REF]. LSH makes it possible to reduce the dimensionality of large data sets. LSH refers to a family of hash functions that associate samples in buckets: similar samples (in the sense of a matching measure of similarity) are assigned to the same bucket while dissimilar samples are assigned to different bucket. Figure 2 illustrates this method: classic hash functions assign each sample to a different bucket while the hash functions belonging to the family of hash functions of the LSH assign close samples to the same bucket. The intuition behind this family of hash functions is that the probability that two samples share the same bucket is related to the distance that separates these two samples. The greater the distance, the lower the probability that the two samples share the same bucket [START_REF] Indyk | Approximate nearest neighbors: towards removing the curse of dimensionality[END_REF]. Definition 3.1 gives the conditions for two samples to share the same bucket. Definition 3.1. A family of hash functions is called (d1, d2, p1, p2) -sensitive for all x and y ∈ S, where S is a set of samples if the two following conditions are fulfilled:

• d(x,y) ≤ d1 ⇒ Pr h∈H [h(x) = h(y)] ≥ p1 • d(x,y) ≥ d2 ⇒ Pr h∈H [h(x) = h(y)] ≤ p2
where d(., .) is the distance between two samples with values between 0 and 1, h(.) is the result of the hash function h applied to a sample. p1 and p2 are probability thresholds for two samples to share the same bucket, with values between 0 and 1. To find sample's neighbors, all samples that share the same basket as some sample at least once on the t hash tables are considered neighbors. This technique is used to find groups in DyClee-C. Figure 3 illustrates the search for nearest neighbors for a sample. The neighbors of the sample q (noted q for query point) are all samples that share the same bucket as the q point at least once. Here, the neighbors of q are the red and yellow dots.

Identification of clusters with DyClee-C

In DyClee, clusters are created from groups of micro-clusters found by the KD-Tree. Let M = {µC1, ..., µCz, ..., µC k }, the set of micro-clusters found with µCz the z-th micro-cluster and k the number of micro-clusters. D = {D1, ..., Dz, ..., D k } is the set of densities corresponding to micro-clusters with Dz the density of the z-th micro-cluster. G = {G1, ..., Gz, ..., G l } is the set of micro-cluster groups detected by the KD-Tree, Gz is the z-th micro-cluster group and l is the number of micro-cluster groups. Note that the number of microclusters k is generally much bigger than the number of groups of micro-clusters l. For each group, the densest micro-cluster subregions are searched. A cluster is created if a micro-cluster is dense and if its neighbors are dense or semi-dense inside a group. If a micro-cluster is outlier, all the samples in this micro-cluster are considered noise.

DyClee offers two approaches to find clusters: the global approach and the local approach.

In the global approach, a micro-cluster µCz is said to be dense if Dz is greater than or equal to the two global density thresholds which are the median and average density of all micro-clusters D. A micro-cluster µCz is semi-dense if Dz is greater than or equal to one of the two global density thresholds. Finally, a micro-cluster µCz is said to be outlier if Dz is less than or equal to the two global density thresholds. In the case of DyClee-C, as explained previously, the concept of micro-cluster no longer exists. The notion of density of a micro-cluster Dz is replaced by the number of neighbors of each categorial sample. The neighbor of a sample is spotted by a connection. Let O = {O1, ..., Oz, ..., On }, the set of samples with Oz, the z-th sample of O and n the number of samples. C = {CO1, ..., COz, ..., COn } is the set of connections of all samples with COz, the number of connections of Oz. CG = {CG1, ..., CGz, ..., C Gl } is the set of samples connections for all groups of samples found by the LSH algorithm, with CGz the number of connections of samples in the group z and l the number of groups of qualitative samples found.

As part of the qualitative global approach, a sample Oz is dense if COz is greater than or equal to the global density thresholds which are then the median and average of the number of connections of each sample. Oz is semi-dense if COz is greater than or equal to the global density thresholds. Finally, If the sample is semi-dense, the sample is added to the cluster but his neighbors are not sought. This approach allows for a cluster with a dense center and semidense cluster edges. A cluster is created when no new neighbor is detected. The operation continues with a new sample. Algorithm stops when all the samples have been analyzed (lines 4 to 24). Clusters.add(cluster)

In DyClee-C, the principle of creating clusters by local approach is unchanged but instead of comparing a sample's connections to the median and the average of the connections of all samples O, COz is compared to the connection thresholds calculated from the median and the average of the connections of samples in the CGz group. The conditions for a sample to be dense, semi-dense and outlier are given by inequalities (4). The pseudo code associated with the local approach is given by Algorithm 2.

To illustrate this approach, the example of Figure 6 is used. The density thresholds change in relation to the overall approach. Each group has its own density thresholds. Group 1 composed of G1 = {1, 2, 3, 4, 5, 5, 6, 7, 8} has for median(DG1) = 2 and average(DG1) = 2,25 and group 2 composed of G2 = {9, 10, 11, 12, 12, 13, 14} has for median(DG2) = 2.5 and average(DG2) = 2.75. The Cl1 cluster remains unchanged from the overall approach. Concerning the Cl2 cluster, samples 9 and 11 are added to sample 14 and are also considered outliers because CO9 and CO11 are less than median(DG2) and average(DG2). The Cl2 cluster is therefore composed of Cl2 = {10, 12, 13}. Figure 6 illustrates the clusters found by applying the local approach.

The methods for generating clusters in DyClee-C have been detailed in the previous section. Several parameters allow the user to improve clustering results by adding some knowledge to the data. These parameters are detailed below.

Settings of Dyclee-C

The final clusters are found using one of the two approaches described in the previous section. These results can be refined O ← samples that are not in Clusters 25: end while Figure 6. Clusters found by applying the local approach with optional parameters. In this section, three parameters are described. The first is called U nclassed accepted, the second is called minimum samples and the last parameter presented is called n clusters. These parameters are tuned in function of the problem encountered and evaluated with cluster validity methods [START_REF] Liu | Understanding of internal clustering validation measures[END_REF] like those they are presented in section 4. In Dy-Clee, clusters are composed of dense (center) and semi-dense (edge of the cluster) micro-clusters. Outliers are considered unrepresentative and their samples are rejected (considered noise). Depending on the context, it may be interesting not to consider outliers but, on the contrary, to assign all the data to a cluster. When the parameter U nclassed accepted is activated, all the samples must be assigned to a cluster, i.e. there is no outlier rejection.

The second parameter is called minimum samples and allows you to set the minimum number of samples that a cluster must contain to be considered as a final cluster. Indeed, depending on the application, small clusters may not be representative. In the case where a nominal situation has to be analysed, small clusters can represent abnormal situations. These clusters are no longer considered as final clusters and the samples assigned to them are marked as noise. The equation (5) allows you to set the size that clusters must have to be considered as final clusters.

|Clz| ∨ l i=0 |Cli| |Cl| (5)
with |Clz| the size of the cluster z to be evaluated and |Cl| is the number of cluster groups.

The use of the parameter is illustrated in Figure 7 with three clusters Cl = {Cl1, Cl2, Cl3}. Cluster 1 is given by Cl1 = {1, 2, 3, 4, 5, 6, 7, 8}, cluster 2 by Cl2 = {9, 10, 11, 12, 13, 14} and the third by Cl3 = {15, 16, 17}. To be considered as final clusters, all Cl clusters must have a size greater than the threshold defined in the equation ( 6):

T hreshold = |Cl1| + |Cl2| + |Cl3| |Cl| = 8 + 6 + 3 3 = 5, 67 (6)
The Cl1 and Cl2 clusters are larger than 5.67 (respectively 8 and 6) and are therefore considered final clusters. On the other hand, since the Cl3 cluster has a size equal to 3, the samples that compose it (15, 16 and 17) are considered as noise (in the Figure 7 in grey). The last parameter is called n clusters and allows you to consider the most important n clusters as final clusters. Samples belonging to the remaining clusters are assigned to the final n clusters. Let be A = {A1, ..., Ai, ..., Am}, all the qualitative attributes of the set of samples O, with m the number of attributes. c = {c1, ..., cz, ..., c l } is the set of cluster centers Cl with cz the cluster center Clz. A cz center is defined by cz = {cz1, ..., czi, ..., czm} with czi, the i-th component of the z center. The terms of an attribute Ai are noted M od(Ai) = {mi1, ..., mij, ..., mip} with mij, the j-th term of the attribute Ai and p, the number of terms of the attribute Ai. The frequency of a modality mij is noted f r(mij). The i component of the cz center is defined as the most frequent modality of the Ai attribute (see equation [START_REF] Gionis | Similarity search in high dimensions via hashing[END_REF]. czi = max(f r(mi1, ..., f r(mij), ..., f r(mip)).

(

When the n clusters parameter is active, the distance between samples not in the final n clusters is calculated and samples are assigned to the nearest center. Figure 8 The section 4 presents the different tests performed on the DyClee-C.

EVALUATION

DyClee-C was tested on well-known UCI Machine Learning data sets in the clustering domain like Zoo, Congressional Voting Records, Soybean and Breast Cancer. The two latter data sets have been selected to be reported in this paper because they correspond to a diagnosis problem. As they have a large number of dimensions, it allows to test how DyClee-C handles data sets with many qualitative attributes. DyClee-C was compared to the K-modes clustering algorithm [START_REF] Huang | Extensions to the k-means algorithm for clustering large data sets with categorical values[END_REF]. Note that an initialization method for the centers of clusters that is not present in the Huang paper is also tested. Indeed, this new method was introduced by [3] 11 years after the original paper. Training phases have been realized on a part of the data sets to find correct combinations of parameters. To evaluate clusters, three validity measures are used: the purity, recall and the precision. Purity is the ratio between the sum of the number of elements correctly assigned in each class i (noted T Pi) and the number of samples in the data set. This measure is described in the equation [START_REF] Guha | Rock: A robust clustering algorithm for categorical attributes[END_REF] where T P i is the real positive rate of the i th class, k is the number of classes and N is the number of samples in the data set.

P urity = k i=0 T Pi N (8)
The recall corresponds to the ratio between the number of elements correctly assigned to the i th class and the number of elements belonging to the i th class (noted T Pi + F Ni). This measure is described in the equation [START_REF] John | Algorithm as 136: A k-means clustering algorithm[END_REF] with Ri, the recall of the i th class, T Pi, the real positive rate in the i th class and F Ni class, the false negative rate in the i th class. A false negative is a result where the model incorrectly predicts the negative class.

Ri = T Pi T Pi + F Ni (9)
The last measure is the precision which corresponds to the ratio between the number of elements correctly assigned to the i th class and the number of elements assigned to the i th class (noted T Pi + F Pi). This measure is described in the equation [START_REF] He | Squeezer: an efficient algorithm for clustering categorical data[END_REF], with Pi, the accuracy of the i th , T Pi class, the true positive rate for the i th and F Pi class, the false positive rate for the i th class. A false positive is a result where the model incorrectly predicts the positive class.

Pi = T Pi T Pi + F Pi (10)

Soybean data set

The first data set is the "Soybean" [START_REF] Ryszard | UCI machine learning repository[END_REF]. This data set consists of 47 soybean plants with 35 qualitative attributes and 4 classes. Attributes correspond to the characteristics of the plant (size of the seed, leaves,...). The classes correspond to diseases specific to soybean plants. Two tests with two different settings were performed on this data set. The first one was done with the parameter U nclassed accepted enabled. The parameter of K-Modes corresponding to the number of clusters is k clusters = 4. Original classes of the "Soybean" data set, results of DyClee-C and K-Modes algorithms with both initializations [START_REF] Huang | Extensions to the k-means algorithm for clustering large data sets with categorical values[END_REF], [START_REF] Cao | A new initialization method for categorical data clustering[END_REF]) are shown in Table 1. Results in Table 1 show that classes C2 and C3 were perfectly detected by both algorithms tested. However, the C1 and C4 clusters were poorly formed by DyClee-C while K-Modes found the right classes. The reason behind the DyClee-C misclassification is that DyClee-C do not compute the distance between every samples. Indeed, the Locality Sensitive Hashing consider as nearest neighbors samples that share the same bucket at least once (section 3.1). As K-Modes method measures the dissimilarity between the whole data set, the Huang's algorithm is more precise. Moreover, K-Modes method needs the number of clusters as a parameter. As an unsupervised clustering algorithm, this kind of information is normally not known in advance. Purity, recall and precision measures for DyClee-C and K-Modes algorithms are shown in Table 2.

C
As K-Modes method found correct classes, purity, recall and precision measures corresponding are equal to 100%. The second test highlights outliers and their influence on DyClee-C's results. Parameters enabled for this test Table 3 represents the classes of the "Soybean" data set and the clusters found by DyClee-C when the outliers detected by the parameter minimum samples are not taken into account. The results show that in this configuration, all classes were found by DyClee-C. The three validity measures associated to this result are equal to 100%.

Breast Cancer data set

The second data set is the "Breast Cancer" [START_REF] Tan | UCI machine learning reposi-tory[END_REF]. This data set consists of 286 instances and 9 qualitative attributes and 2 classes. An instance corresponds to a patient and each attribute is an information about the patient and his pathology (age, which breast has the tumor, the degree of the malignancy,...). Some attributes have missing values and are represented by a "?" in the data set. These special values are considered as a modality for the test. Classes are instances which have a no recurrence events and recurrence events. Parameters enabled of DyClee-C for this test are U nclassed accepted and n clusters = 2. The parameter of K-Modes corresponding to the number of clusters is k clusters = 2. Original classes of the "Breast Cancer" data set, results of DyClee-C and K-Modes (with both initializations) are shown in Table 4 While purity score is slightly higher for DyClee-C than K-Modes, the method presented in this paper is clearly better for recall and precision score than the Huang's algorithm.

CONCLUSIONS AND PERSPECTIVES

In this article, an extension of DyClee, a dynamic clustering algorithm for numeric data, is presented. This extension, called DyClee-C, allows one to apply the basic concepts of Dy-Clee to qualitative data. The KD-Tree has been replaced by the LSH algorithm, which makes it possible to form groups of qualitative samples with large dimensions. The two approaches, global and local density, to generate clusters have been modified to capture qualitative data. The concept of micro-cluster density is replaced by the number of neighbours of a sample, i.e. its connections. Thus, a sample with a certain number of connections is more dense than a sample with few connections. Three parameters have been adapted in order to refine the clusters obtained. The first one allows one to identify and reject outliers, the second one only keeps clusters with a size larger than the average cluster size and the last one allows one to consider the most important n clusters as final clusters.

For the tests, DyClee-C has been compared to the K-Modes algorithm on two well-known data sets with several qualitative features. The Breast Cancer data set shows that DyClee-C is able to detect classes even if the samples have missing modalities. The Soybean data sets illustrates the capacity of DyClee-C to detect outliers. The obtained results are promising and show that clustering can be used for diagnosis purposes even for data bases with qualitative features. In the tests, evaluation is achieved by comparing the clustering results to the ground truth clusters, which is obviously improper. Indeed, unsupervised classification algorithms are used when labels are not known. Subsequently, further evaluation is planned to be carried out with the agnostic evaluation criterion proposed in [START_REF] Ducharlet | A multi-phase iterative approach for anomaly detection and its agnostic evaluation[END_REF]. Also, a mixed version of DyClee to manage both numerical and qualitative data is under development.

Figure 1 .

 1 Figure 1. Overview of DyClee

Figure 2 .

 2 Figure 2. Assignment of samples with hash functions with the LSH

Figure 3 .

 3 Figure 3. Searching neighbors with the LSH algorithm

  These conditions are represented by the inequalities (1), where median(D1, ..., D k ) and average(D1, ..., D k ) correspond respectively to the median and average density of micro-clusters in M . µCzdense ⇔ Dz ≥ median(D1, ..., D k ) ∧Dz ≥ average(D1, ..., D k ) µCzsemi -dense ⇔ Dz ≥ median(D1, ..., D k ) ∨Dz ≥ average(D1, ..., D k ) µCzoutlier ⇔ Dz < median(D1, ..., D k ) ∧Dz < average(D1, ..., D k ) (1)

  Oz is outlier if COz is less than the global density thresholds. These conditions are represented by the inequalities[START_REF] Andrea | A data-based approach for dynamic classification of functional scenarios oriented to industrial process plants[END_REF], where median(CO1, ..., C Ok ) and average(CO1, ..., C Ok ) correspond respectively to the median and mean of the set of connections of C. Ozdense ⇔ COz ≥ median(CO1, ..., C Ok ) ∧COz ≥ average(CO1, ..., C Ok ) Ozsemi -dense ⇔ COz ≥ median(CO1, ..., C Ok ) ∨COz ≥ average(CO1, ..., C Ok ) Ozoutlier ⇔ COz < median(CO1, ..., C Ok ) ∧COz < average(CO1, ..., C Ok ) (2) Algorithm 1 implements the creation of clusters with the global approach of DyClee-C. From line 1 to 3, the density thresholds median(CO1, ..., C Ok ) and average(CO1, ..., C Ok ) are calculated from the set of connections of all the samples of C. When a dense sample is detected, his neighbors are searched for and analyzed. If these are dense neighbors of neighbors are sought.

O

  ← samples that are not in Clusters 25: end while To illustrate the global approach, two groups found by the LSH, are shown in Figure 4. The first qualitative group is made up of eight samples G1 = {1, 2, 3, 4, 5, 6, 7, 8} and the second is composed of six samples G2 = {9, 10, 11, 12, 13, 14}. The set C = {5, 3, 1, 3, 2, 1, 2, 1, 2, 5, 2, 3, 3, 1} corresponds to the connections of all samples. The overall density thresholds, i.e. the median and the mean of the connections present in C, are respectively equal to median(CO1, ..., C Ok ) = 2 and average(CO1, ..., C Ok ) = 2.43. A cluster is composed of dense samples in the center and semi-dense samples at the cluster boundaries.

Figure 4 .

 4 Figure 4. Example of groups of qualitative samples detected by the LSH algorithm

Figure 5 .

 5 Figure 5. Clusters found by applying the global approach

  Ozdense ⇔ COz ≥ median(CGz) ∧COz ≥ average(CGz) Ozsemi -dense ⇔ COz ≥ median(CGz) ∨COz ≥ average(CGz) Ozoutlier ⇔ COz < median(CGz) ∧COz < average(CGz)

Figure 7 .

 7 Figure 7. Left: parameter minimum samples disabled. Right: parameter minimum samples enabled

  takes the example of Figure 7. In this example, the two most important clusters are considered as final clusters (Cl1 and Cl2). Samples 15, 16 and 17 of Cl3 are reassigned to the nearest clusters. Samples 15 and 17 are therefore assigned to the Cl1 cluster and sample 16 to the Cl2 cluster.

Figure 8 .

 8 Figure 8. Left: parameter n clusters disabled. Right: parameter n clusters enabled with n = 2

  Algorithm 1 DyClee-C global approach Require: Set of samples O, set of connexions of samples C

	5:	toAnalyze = []
	6:	cluster = []
	7:	toAnalyze ← O.pop()
	8:	while toAnalyze not empty do
	9:	for i ∈ toAnalyze do
	10:	if i → Dense then
	11:	N eighbors ← LSH.query(i)
	12:	for neighbor ∈ N eighbors do
	13:	if (neighbor is Dense ∧ neighbor / ∈ cluster)
		then
	14:	cluster.add(neighbor)
	15:	toAnalyze.add(neighbor)
	16:	else if (neighbor is Semi -dense ∧ neighbor
		/ ∈ cluster) then
	17:	cluster.add(neighbor)
	18:	end if
	19:	end for
	20:	end if
	21:	end for
	22:	end while
	23:

1: Oz → Dense such that COz ≥ median(CO1, ..., C Ok ) ∧ COz ≥ average(CO1, ..., C Ok ) 2: Oz → Semi -dense such that COz ≥ median(CO1, ..., C Ok ) ∨ COz ≥ average(CO1, ..., C Ok ) 3: Clusters = [] 4: while O not empty do

  Algorithm 2 DyClee-C local approach Require: Set of samples O, set of connexions of samples C, Set of groups of samples CG 1: Oz → Dense such that COz ≥ median(CGz) ∧ COz ≥ average(CGz) 2: Oz → Semi -dense such that COz ≥ median(CGz) ∨

		COz ≥ average(CGz)
	3: Clusters = []
	4: while O not empty do
	5:	toAnalyze = []
	6:	cluster = []
	7:	toAnalyze ← O.pop()
	8:	while toAnalyze not empty do
	9:	for i ∈ toAnalyze do
	10:	if i → Dense then
	11:	N eighbors ← LSH.query(i)
	12:	for neighbor ∈ N eighbors do
	13:	if (neighbor is Dense ∧ neighbor / ∈ cluster)
		then
	14:	cluster.add(neighbor)
	15:	toAnalyze.add(neighbor)
	16:	else if (neighbor is Semi -dense ∧ neighbor
		/ ∈ cluster) then
	17:	cluster.add(neighbor)
	18:	end if
	19:	end for
	20:	end if
	21:	end for
	22:	end while
	23:

Clusters.add(cluster)

24:

Table 1 .

 1 Clusters found by DyClee-C and K-Modes for the Soybean data set

		1	C 2	C 3	C 4
	Soybean	17	10	10	10
	DyClee-C	26	10	10	1
	K-Modes (Huang)	16	14	10	7
	K-Modes (Cao)	17	10	10	10

Table 2 .

 2 table Purity, recall and precision scores of DyClee-C and K-Modes algorithm for the Soybean data set are U nclassed accepted and minimum samples. The latter makes it possible to eliminate samples belonging to clusters with a size smaller than the average cluster size.

		Purity	Recall	Precision
	DyClee-C	70%	77%	85%
	K-Modes (Huang)	78%	55%	52%
	K-Modes (Cao)	100%	100%	100%
		C 1	C 2	C 3	C 4
	Soybean	10	9	9	6
	Result	10	9	9	6

Table 3 .

 3 table Clusters found by DyClee-C for the Soybean data set with outliers removed

Table 4 .

 4 . table Clusters found by DyClee-C and K-Modes for the Breast Cancer data setDyClee-C have detected better groups than the K-Modes algorithm. Few samples have been misclassified by DyClee-C while they are more samples in the wrong class with the K-Modes algorithm. Purity, recall and precision measures for DyClee-C and K-Modes algorithms are shown in Table5.

		C 1	C 2
	Breast Cancer	201	85
	DyClee-C	207	79
	K-Modes (Huang)	149	137
	K-Modes (Cao)	183	103

Table 5 .

 5 table Purity, recall and precision scores of DyClee-C and K-Modes algorithm for the Breast Cancer data set

This work is part of the CIFRE PhD project n o 2017/0047 in collaboration with the company ACTIA. It is also related to the French Occitanie Region project Evolearn.