Irene Mustieles Marin

Juan M Asensio

Bruno Chaudret

Bimetallic Nanoparticles Associating Noble Metals and First-Row Transition Metals in Catalysis

Bimetallic nanoparticles (NPs) are complex systems with properties that far exceed those of the individual constituents. In particular, association of a noble metal and a first-row transition metal are attracting increasing interest for applications in catalysis, electrocatalysis, and magnetism, among others. Such objects display a rich structural chemistry thanks to their ability to form intermetallic phases, random alloys, or core-shell species. However, under reaction conditions, the surface of these nanostructures may be modified due to migration, segregation, or isolation of single atoms, leading to the formation of original structures with enhanced catalytic activity. In this respect, Zakhtser et al. report in this issue of ACS Nano the synthesis and study of the chemical evolution of the surface of a series of PtZn nanostructured alloys. In this Perspective, we report some selected examples of bimetallic nanocatalysts and their increased activity compared to the corresponding pure noble metal, with a special focus on Pt-based systems. We also discuss the mobility of the species present on the catalyst surface and the electronic influence of one metal to the other. Bimetallic nanocrystals, which comprise a noble metal (in particular, a platinum group metal) and a non-noble metal (in general, a first-row transition metal), are attracting a lot of interest because of

the important modifications to their properties (e.g., chemical, mechanical, electrical, magnetic, etc.) that are induced by the presence of the two metals and their interactions with one another. [START_REF] Zhou | Noble-Metal Based Random Alloy and Intermetallic Nanocrystals: Syntheses and Applications[END_REF] Alloys have been used since the beginning of metallurgy, when tin was employed to harden copper in the Bronze Age. One particularly interesting example is brass (alloy of copper and zinc) for which the preferential dealloying of zinc has been studied since the 19th century. Although the properties of bulk alloys have been known for a long time, the situation becomes more complex when reaching the nanoscale. Thus, for nanocrystals, the stoichiometry may be more difficult to control and the surface chemistry may be responsible for changes in stoichiometry, the most common effect being surface oxidation. These systems can give rise to disordered (or random) alloys or to ordered alloys, also known as intermetallics. Intermetallic compounds display an atomically ordered structure where the atoms occupy the lattice following a defined stoichiometry. [START_REF] Zhou | Noble-Metal Based Random Alloy and Intermetallic Nanocrystals: Syntheses and Applications[END_REF] Pt-based catalysts are active for many reactions, including alkane dehydrogenation and CO oxidation, but are also electrocatalysts for the oxygen reduction reaction (ORR) and the methanol oxidation reaction (MOR). Furthermore, it has been observed in many cases that, when combined with another metal, their catalytic properties are enhanced. This enhancement enables spectacular transformations in catalysis, including a recent demonstration of the possibility to oxidize methane selectively into methanol. [START_REF] Bai | Activating and Converting CH4 to CH3OH via the CuPdO2/CuO Nanointerface[END_REF] However, precise knowledge of the composition or surface structure of the system is highly challenging because of the dynamics present in these nano-objects, which depend on external conditions. For instance, in this issue of ACS Nano, Zakhtser et al. report the synthesis and study of the chemical evolution of the surface of a series of PtZn nanostructured alloys. [START_REF] Zakhtser | Chemical Evolution of Pt-Zn Nanoalloys Dressed in Oleylamine[END_REF] This research highlights the mobility of the surface atoms when exposed to different conditions. The authors found that zinc was preferentially oxidized when the particle was in contact with air, and that the Pt-Zn phase was not stable and showed a tendency to lose Zn. Depending on the zinc concentration, either ultrathin islands of ZnO or a thin shell of ZnO at the surface of the Pt particles resulted (Figure 1). Upon exposure to CO, the size of ZnO patches reduced due to a preferential binding of the CO molecule to the Pt surface. It turns out that ZnO islands play an important role in the catalytic oxidation of CO. This inverse oxide/metal configuration takes advantage of the promotion of the catalytic activity by incorporating nanostructures of base metal oxides at the surface of a catalyst. Furthermore, Zakhtser et al. also describe the leaching of zinc when PtZn nanoparticles (NPs) were left for several days at room temperature in non-degassed toluene, a phenomenon which leads to the formation of a Pt-enriched surface. [START_REF] Zakhtser | Chemical Evolution of Pt-Zn Nanoalloys Dressed in Oleylamine[END_REF] These observations highlight the dynamic behavior of bimetallic nano-alloys that can eventually be segregated or can give rise to the isolation of one atom of an element at the surface of another, leading to a large complexity of situations, some aspects of which we will discuss here. Using selected recent examples, we will underscore the importance of such bimetallic compounds in catalysis and electrochemistry and give evidence for their atomic mobility and for the electronic influence of one metal on the other.

Structural and Electronic Considerations of Bimetallic Systems

Bimetallic NPs have been studied ever since nano-objects first aroused interest. [START_REF] Toshima | Bimetallic nanoparticles-novel materials for chemical and physical applications[END_REF] The expected gains concern both physical and chemical properties and, hence, may include catalytic, magnetic, photophysical, and electronic properties. Thus, alloying between two metals leads to site isolation of specific atoms but also induces a change in the electronic structures of the resulting alloy due to the presence of neighboring atoms. [START_REF] Föttinger | In Situ Spectroscopy of Complex Surface Reactions on Supported Pd-Zn, Pd-Ga, and Pd(Pt)-Cu Nanoparticles[END_REF] Bimetallic PdCu NPs were reported in 1993 by J. S. Bradley in collaboration with our group [START_REF] Bradley | Synthesis of monodispersed bimetallic palladium-copper nanoscale colloids[END_REF] and by Toshima, who has been a pioneer in the field. [START_REF] Toshima | Bimetallic nanoparticles-novel materials for chemical and physical applications[END_REF] The NPs were prepared by an organometallic approach, and noted two interesting features. First, prior reduction of Pd induced the reduction of the Cu precursor, which was not otherwise reduced in the reaction conditions; 6 this finding appears to be a general trend for the preparation of bimetallic NPs. Second, the atomic mobility within the NPs induced by surface chemistry. Under a CO atmosphere, the surface of the particles was enriched with Pd up to the point that only Pd-CO stretches were visible by infrared (IR) spectroscopy. However, when the system was then placed under vacuum and CO was rapidly reintroduced, the main feature observed was the presence of a Cu-CO stretch. Reversible atom migration was then highlighted, with Pd and Cu being attracted in turn at the surface of the NPs. This point is important regarding the adaptation and migration of atoms within NPs. [START_REF] Bradley | Surface Chemistry on Colloidal Metals. Reversible Adsorbate-Induced Surface Composition Changes in Colloidal Palladium-Copper Alloys[END_REF] However, in most cases, migrations are induced by surface oxidation and thus are irreversible, as in the case of the PtZn alloy reported by Zakhtser et al. [START_REF] Zakhtser | Chemical Evolution of Pt-Zn Nanoalloys Dressed in Oleylamine[END_REF] The field has grown tremendously since that time and the understanding of the chemistry of these species has made considerable progress thanks to theoretical approaches. For instance, Tsai et al. calculated that PdZn alloys exhibit valence electron densities of states similar to that of Cu and significant chemical shifts of Pd 3d states with respect to pure Pd. [START_REF] Tsai | Intermetallic: A Pseudoelement for Catalysis[END_REF] It is therefore reasonable to observe that the activity of PdZn in steam reforming of methanol is similar to that of Cu. This finding led the authors to the concept of pseudo-elements, which can be of interest for designing new intermetallic systems. [START_REF] Tsai | Intermetallic: A Pseudoelement for Catalysis[END_REF] As far as chemical properties are concerned, the first advantage of an association between a Pt metal and a main group element or a catalytically inactive species may be to induce site selectivity when the NPs can offer several tracks of reactivity. This selectivity was noted in early studies on MSn bimetallic nanocatalysts (M = Ru, Rh, Pt) prepared by surface organometallic chemistry for which researchers used techniques including NMR to elucidate the bonding and coordination environment on the surface. [START_REF] Didillon | Surface organometallic chemistry on metals. Evidence for a new surface organometallic material, Rh[Sn(n-C4H9)x]y/SiO2, obtained by controlled hydrogenolysis of tetra-n-butylstannane on a rhodium/silica catalyst[END_REF] These systems were first used for the steric bulk induced by the presence of Sn which, when present on the surface of Ru NPs prevents the hydrogenation of phenyl rings during the reduction of acetophenone or styrene. [START_REF] Bonnefille | Tin-decorated ruthenium nanoparticles: a way to tune selectivity in hydrogenation reaction[END_REF] Similar selectivity was also found for unsaturated aldehydes and ketones, the carbonyl group of which can be selectively hydrogenated on RhSn NPs. [START_REF] Didillon | The impact of surface organometallic chemistry in heterogeneous catalysis: A new class of highly chemoselective hydrogenation catalysts, RhsSn(n-C4H9)2/SiO2[END_REF] Other systems associating a catalytic metal to an inert one are expected not only to lead to a partial deactivation of the surface but also to increase its selectivity. In this respect, FeRu NPs were shown to reduce the C=O bond of acetophenone selectively whereas Ru NPs of the same size would first reduce the phenyl ring. [START_REF] Kelsen | How to Modulate Catalytic Properties in Nanosystems: The Case of Iron-Ruthenium Nanoparticles[END_REF] This research was extended to benzylideneacetone and furfuralacetone, a platform molecule derived from biomass. In both cases, C=O and C=C bonds were selectively reduced and no reduction of the phenyl or furan rings was observed, in contrast to pure Ru NPs. Although this reactivity was expected, it was surprising to observe an enhancement of the catalytic activity of the bimetallic NPs compared to the pure Ru NPs. The explanation is not straightforward but has to do with the polarization of Ru induced by Fe or to the better coordination of oxygen species to partially oxidized surface Fe atoms. [START_REF] Luska | Enhancing the Catalytic Properties of Ruthenium Nanoparticle-SILP Catalysts by Dilution with Iron[END_REF] A somewhat similar effect can be found in the decarbonylation versus hydrodeoxygenation (HDO) of hydroxymethyl furfural. Pino et al. observed the PdFe nanocatalysts were selective for HDO whereas Pd NPs of the same size and on the same support predominantly led to decarbonylation and the formation of furan (Figure 2). The reason given for this shift in selectivity is that the use of an alloying metal with a strong affinity for oxygen, such as Fe, enhances the selectivity toward C-O hydrogenolysis and disfavors other reaction pathways such as C-C cleavage. Copyright 2017 Elsevier.

Bimetallic NPs are complex systems with properties that far exceed those of the individual constituents, [START_REF] Föttinger | In Situ Spectroscopy of Complex Surface Reactions on Supported Pd-Zn, Pd-Ga, and Pd(Pt)-Cu Nanoparticles[END_REF] as illustrated by case studies of ordered intermetallics (PdZn, Pd 2 Ga) and substitutional alloys (PdCu, PtCu). For both PdZn and PdGa, the active phase is created under reaction conditions by deposition of Pd and further reduction of the metal oxide support, which has become a typical route to access this family of alloys. This method not only leads to isolation of Pd sites but also induces electronic modification of Pd by the Zn or Ga neighbors. The stability of the active phase is an issue because partial decomposition of PdZn or Pd 2 Ga produces Pd islands that reduce selectivity.

Selective Hydrogenation

A first application of these nanocatalysts is the use of their encumbered surface for selective hydrogenation reactions, which we will limit to two representative examples. In an early example, Bronstein et al. showed that the formation of a PdZn alloy enhances catalytic activity in the hydrogenation of dehydrolinalool. Introducing Zn in the Pd lattice induced a change in its electronic configuration and overall geometric structure. The authors ascribed the nature of the active centers on the particle surface as the main effect responsible for the increase in the catalytic activity. In other terms, the site isolation could enhance the catalytic activity. [START_REF] Bronstein | Structure and Properties of Bimetallic Colloids Formed in Polystyrene-block-Poly-4-vinylpyridine Micelles: Catalytic Behavior in Selective Hydrogenation of Dehydrolinalool[END_REF] Zhou et al. reported a PdZn intermetallic catalyst presenting both high selectivity and activity in the semi-hydrogenation of acetylene. The Pd-Zn-Pd surface pattern displays a weak π-bonding pattern for ethylene adsorption, which prevents further hydrogenation and facilitates its desorption. This pattern is responsible for the high chemoselectivity for acetylene semi-hydrogenation, whereas the moderate σ-bonding mode for acetylene on the Pd sites was responsible for the high hydrogenation activity (Figure 3). Copyright 2016 American Chemical Society.

Alkane Dehydrogenation

A more challenging application resulting from the same electronic effect is alkane dehydrogenation.

Researchers have used modification of the Pt catalyst by main group elements to limit the isomerization and hydrogenolysis activities of the catalyst. In addition, the presence of these elements is reported to limit sintering and to facilitate coke elimination. [START_REF] Sattler | Catalytic Dehydrogenation of Light Alkanes on Metals and Metal Oxides[END_REF] Tin is the most-often employed promoter and it gives rise to the formation of a Pt-Sn alloy during catalysis. Platinum assists the reduction of Sn; the origin of the effect is both a limitation of the extension of the Pt surface and an electronic effect. Thus, alloyed metallic Sn or Sn 2+ species in close contact with Pt are able to transfer electrons to the 5d band of platinum atoms, which will modify the adsorptive and catalytic properties of the metal. [START_REF] Yang | First-Principles Calculations of Propane Dehydrogenation over PtSn Catalysts[END_REF] Other metals have been studied as promoters of Pt activity in the dehydrogenation of alkanes as well. Most notably, Zn prevents undesired side reactions such as coke formation and isomerization in a way similar to that of Sn. [START_REF] Silvestre-Albero | Modification of the catalytic behaviour of platinum by zinc in crotonaldehyde hydrogenation and iso-butane dehydrogenation[END_REF] Thus, Zn-promoted Pt NPs have been shown to be efficient and selective catalysts for alkane dehydrogenation. Similar to the studies regarding alkyne hydrogenation but instead at high temperature, it has been proposed that Pt sites in PtZn alloys are more electron-rich than pure Pt, which would weaken the Pt-olefin bonds and promote olefin desorption, hence increasing the reaction rate. [START_REF] Camacho-Bunquin | Atomically Precise Strategy to a PtZn Alloy Nanocluster Catalyst for the Deep Dehydrogenation of n-Butane to 1,3-Butadiene[END_REF] Theoretical calculations of the partial d-orbital density of states (DOS) suggested that the d band center of the Pt DOS on PtZn(111) is shifted toward lower energies respect to that of Pt(111).

In another work, Cybulskis et al. prepared Pt 1 Zn 1 intermetallic NPs (iNPs) supported on SiO 2 with isolated Pt surface sites, leading to near 100% C 2 H 4 selectivity in ethane dehydrogenation (EDH). In addition, PtZn showed a 6-fold higher turnover rate (TOR) per Pt surface at 600 °C with respect to monometallic Pt/SiO 2 . As discussed above, density functional theory (DFT) studies reveal that the addition of Zn modifies the energy of Pt 5d electrons, promoting TOR, while incorporation of Zn into the catalyst surface explains the enhanced product selectivity (Figure 4).

CO Oxidation

Naitabdi et al. reported that pure Pt and bimetallic PtZn NPs showed similar behaviors during catalytic CO oxidation reactions. Both types of NPs are prone to being encapsulated by the support whereas pure Zn NPs are not encapsulated. [START_REF] Naitabdi | CO oxidation activity of Pt, Zn and ZnPt nanocatalysts: a comparative study by in situ near-ambient pressure X-ray photoelectron spectroscopy[END_REF] For other oxides such as iron oxides, an interesting phenomenon could be evidenced: Pt NPs partially covered with FeOx nanopatches displayed higher activities in CO oxidation than neat Pt NPs, whereas those with a full Fe oxide shell at the surface showed much lower activity. Characterizations by X-ray diffraction, transmission electron microscopy (TEM), and X-ray photoelectron spectroscopy (XPS) suggested that Pt promoted the reduction of interfacial Fe oxide. [START_REF] Xu | Interface-Confined FeOx Adlayers Induced by Metal Support Interaction in Pt/FeOx Catalysts[END_REF]

Carbon Oxides Hydrogenation

Hydrogenation mostly concerns PdGa alloys, which, like PdZn, can form intermetallic phases. García-Trenco et al. used these catalysts for the hydrogenation of CO 2 into methanol in a single step. [START_REF] García-Trenco | Pd2Ga-Based Colloids as Highly Active Catalysts for the Hydrogenation of CO2 to Methanol[END_REF] Interestingly, colloidal Pd 2 Ga-based catalysts were shown to catalyze the hydrogenation of CO 2 into methanol efficiently in solution. The catalysts were synthesized by thermal decomposition of Pd(II) acetate and Ga(III) stearate, which generated Pd 0 nanoparticles of ca. showed some morphological changes of the NPs during the first hour of reaction, which the authors ascribed to the presence of some dealloying. [START_REF] Gentzen | An intermetallic Pd2Ga nanoparticle catalyst for the single-step conversion of CO-rich synthesis gas to dimethyl ether[END_REF]

Electrochemistry

The preparation of bimetallic alloys can lead to significant enhancement of the efficiency of Pt electrodes through different effects as a result of the mobility and/or oxidation of one component, for example zinc. A first effect can be the induction of nanostructuration of a Pt electrode through the dealloying and oxidation of a PtZn alloy. For example, Huang et al. obtained a PtZn alloy by electrodeposition of a Zn salt on a Pt electrode. Immersion in acidic ZnCl 2 -EMIC ionic liquid can provide a simple yet effective way to fabricate nanostructured Pt electrodes with high surface area. [START_REF] Huang | Formation of Nanoporous Platinum by Selective Anodic Dissolution of PtZn Surface Alloy in a Lewis Acidic Zinc Chloride-1-Ethyl-3-methylimidazolium Chloride Ionic Liquid[END_REF] Kang et al. showed that PtZn and Pt 3 Zn NPs are active toward methanol oxidation. In this system, the spherical NPs exhibited higher activity than the cubic ones and the intermetallic Pt 3 Zn showed better performance than the alloy phase PtZn. Although the PtZn NPs showed lower activity toward methanol oxidation than other comparable phases (PtMn, PtCo or PtCu), the PtZn NPs were advantageous in terms of poisoning tolerance and the low cost of Zn. [START_REF] Kang | Synthesis, Shape Control, and Methanol Electro-oxidation Properties of Pt-Zn Alloy and Pt3Zn Intermetallic Nanocrystals[END_REF] PtZn iNPs supported on multiwalled carbon nanotubes (MWNTs) are active catalysts toward formic acid and methanol electrooxidation (Figure 5). [START_REF] Miura | Synthesis of Intermetallic PtZn Nanoparticles by Reaction of Pt Nanoparticles with Zn Vapor and Their Application as Fuel Cell Catalysts[END_REF] Sub-4-nm PtZn iNPs, synthesized using a sacrificial mesoporous silica shell, exhibited ca. 10 times higher mass activity toward MOR in both acidic and basic solutions compared to larger PtZn iNPs synthesized on MWNT. Density functional theory calculations demonstrate that MOR on smaller PtZn iNPs is energetically more favorable than on larger iNPs, due to their high density of corner sites and the lower-lying energetic pathway. The calculations also predicted that PtZn systems follow a "non-CO" pathway for MOR because of the stabilization of the OH* intermediate by Zn atoms, whereas pure Pt systems form highly stable COH* and CO* intermediates, leading to catalyst deactivation. In addition to electronic modifications, the influence of the compressive strain induced by the formation of bimetallic NPs has been studied by Jia et al. [START_REF] Jia | Activity Descriptor Identification for Oxygen Reduction on Platinum-Based Bimetallic Nanoparticles: In Situ Observation of the Linear Composition-Strain-Activity Relationship[END_REF] and Strasser et al. [START_REF] Strasser | Lattice-strain control of the activity in dealloyed core-shell fuel cell catalysts[END_REF] for PtCo and PtCu alloys, respectively, as electrocatalysts in ORR. The geometric effect in this case is generated by a mismatch between the Pt-enriched shell and the base-metal-enriched core that it coats. In situ measurements demonstrated that this configuration generates a compressive strain that modifies the Pt-Pt distance, depending on the alloy composition. Through calculations and experimental measurements, the authors demonstrated that this compressive strain induces a downshift of the d band center, which is translated in a weakening of the chemisorption of the oxygenated species leading to an increase in the catalytic activity.

Magnetic Properties

Bimetallic NPs comprising a platinum group metal and a first-row ferromagnetic metal may also display interesting physical properties. Such bimetallic compositions can give rise to new and interesting properties when preparing hard (FeRh, CoRh, CoPd, FePt, CoPt) or soft (CoRu) magnets.

These magnetic materials are also dynamic and possess interesting applications in catalysis. For example, Wu et al. recently described monitoring the reactivity of CoPd alloy NPs under catalytic CO oxidation conditions using ambient pressure X-ray spectroscopy and TEM. As for the PtZn system, the catalysis process induces a reconstruction of the catalysts, leaving CoOx on the NP surface. The synergy between Pd and CoOx coexisting on the surface promotes the catalytic activity of the bimetallic catalysts. [START_REF] Wu | Bimetallic synergy in cobalt-palladium nanocatalysts for CO oxidation[END_REF] In the case of magnetic properties, the chemical order within the particle is of utmost importance, perhaps even more so than in catalysis. Many studies have been devoted to FePt NPs and, specifically, to the formation of the fully ordered, face-centered tetragonal phase (fct). This phase exhibits a large coercivity and can therefore store magnetic information. However, the ordering occurs at high temperatures, which limits their applicability. [START_REF] Sun | Recent Advances in Chemical Synthesis, Self-Assembly, and Applications of FePt Nanoparticles[END_REF] In the case of CoPt, the hardening of the magnetic properties depends on the concentration of Pt. Hence, for Co x Pt y NPs of 1.0-1.5 nm diameter, coercive fields of 0.52, 1.76, and 2.06 T were found for Co 3 Pt, CoPt, and CoPt 3 compositions, respectively. [START_REF] Ely | Nanoscale Bimetallic CoxPt1-x Particles Dispersed in Poly(vinylpyrrolidone): Synthesis from Organometallic Precursors and Characterization[END_REF] In the case of CoRh NPs, a strong increase of magnetization of both cobalt and rhodium results from the alloying process for a series of 2 nm NPs of variable compositions. [START_REF] Zitoun | Magnetic Enhancement in Nanoscale CoRh Particles[END_REF] Ordering has nevertheless been observed in solution in the case of FeCo NPs where we could evidence the formation of the ordered B2 phase under relatively mild conditions (Figure 6). 36 Another aspect that we have not explored in this Perspective but that we expect will develop in the coming years is the possibility of combining physical and chemical properties in a single entity to address the onset of catalytic reactions by an external optical or magnetic stimulus. We are presently engineering FeC x @Ru, [START_REF] Asensio | Hydrodeoxygenation Using Magnetic Induction: High-Temperature Heterogeneous Catalysis in Solution[END_REF] FeCo@C, [START_REF] Martínez-Prieto | Ultrastable Magnetic Nanoparticles Encapsulated in Carbon for Magnetically Induced Catalysis[END_REF] and FeNi 3 @Ni NPs for CO 2 methanation, 39 methane dry reforming, or hydrodeoxygenation of biomass platform molecules in solution in mild conditions. [START_REF] Mustieles Marin | Hydrodeoxygenation and hydrogenolysis of biomass-based materials using FeNi catalysts and magnetic induction[END_REF] Here, all the phenomena of alloying, dealloying, and oxidation that we have highlighted in this Perspective can be present and can participate in the catalysis under the magnetic stimulus. [START_REF] Asensio | Hydrodeoxygenation Using Magnetic Induction: High-Temperature Heterogeneous Catalysis in Solution[END_REF][START_REF] Martínez-Prieto | Ultrastable Magnetic Nanoparticles Encapsulated in Carbon for Magnetically Induced Catalysis[END_REF][START_REF] De Masi | Engineering Iron-Nickel Nanoparticles for Magnetically Induced CO2 Methanation in Continuous Flow[END_REF][START_REF] Marbaix | Tuning the Composition of FeCo Nanoparticle Heating Agents for Magnetically Induced Catalysis[END_REF][START_REF] Asensio | To heat or not to heat: a study of the performances of iron carbide nanoparticles in magnetic heating[END_REF][START_REF] Mustieles Marin | Hydrodeoxygenation and hydrogenolysis of biomass-based materials using FeNi catalysts and magnetic induction[END_REF]

Conclusions and Perspectives

The development of reproducible and scalable synthetic methods for the preparation of controlled bimetallic NPs still remains a challenge. The synthesis of such objects can involve complex methodologies if one wants to control the composition and the surface of the NPs precisely. In addition, if we consider the few selected examples described here, the vision of a stable alloy is vanishing as the surfaces appear as dynamic entities subjected to the reaction conditions. Thus, operando techniques or in situ characterization under catalytically relevant conditions may arise as the most fruitful tools to understand this dynamic behavior. Dynamics can also lead to the formation of catalytically active single atoms such as isolated Pt, recently characterized at the surface of Cu NPs, which display a particularly high activity for the hydrogenolysis of glycerol. [START_REF] Zhang | Platinum-copper single atom alloy catalysts with high performance towards glycerol hydrogenolysis[END_REF] Regarding this alloy formation as a consequence of the dynamic behavior, it has been evidenced that in some cases reduction of the support led to catalyst destructuration and formation of oxide patches on the surface of the NPs, which have been shown to enhance the catalytic activity. Such phenomena have been known for a long time in heterogeneous catalysis but more examples are being reported that

Figure 1 .

 1 Figure 1. Exposition of Pt-Zn nanoparticles (NPs) under air leads to formation of ZnO sheets that leach from the NP surface. The PtZn NPs displaying a Pr-rich outer layer with ZnO nano-islands showed enhanced performance in catalysis. Reprinted from ref 3. Copyright 2020 American Chemical Society.

14

 14

Figure 2 .

 2 Figure 2. Pd-Fe nanoparticles (NPs) bind furan rings more weakly than do pure Pd NPs. Thus, ring hydrogenation and decarbonylation pathways are disfavored, enhancing the selectivity in the hydrodeoxygenation of biomass-derived platform molecules. Reprinted with permission from ref 14.

16

 16

Figure 3 .

 3 Figure 3. PdZn alloys selectively catalyze the semihydrogenation of acetylene into ethylene thanks to the easy desorption of the alkenes from the surface of the bimetallic catalyst. Reprinted from ref 16.

21

 21

Figure 4 .

 4 Figure 4. Intermetallic PtZn nanoparticles supported on SiO 2 display higher activities in ethane hydrogenation thanks to the modification of the d-band center. TOR, turnover rate. Reprinted from ref 21. Copyright 2017 American Chemical Society.

24 Scheme 1 .

 241 Scheme 1. Synthesis of Pd 2 Ga nanoparticles (NPs) by reduction of Ga(III) stearate promoted by Pd NPs, which are obtained in situ by reduction of Pd(II) acetate at high temperatures. Reprinted from ref 24. Copyright 2017 American Chemical Society.

29

 29

Figure 5 .

 5 Figure 5. (a) Schematic representation of the synthesis of PdZn nanoparticles (NPs) supported on multiwalled carbon nanotubes. First, the Pt precursor is loaded and reduced to yield Pt NPs coated in a SiO 2 shell. Then, addition of the Zn precursor and removal of the SiO 2 shell led to the intermetallic system. MWNT, multiwalled carbon nanotubes. (b-f) Transmission electron microscopy micrographs for each step of the synthetic pathway. Reprinted from ref 29. Copyright 2009 American Chemical Society.

Figure 6 .

 6 Figure 6. Formation of the ordered B2 phase in intermetallic FeCo nanoparticles (NPs) led to interesting magnetic properties that make these NPs suitable for applications in microelectronics (induction-based systems). Reprinted from ref 36. Copyright 2019 American Chemical Society.

profit from these mobilities to enhance activity and/or selectivity of the catalysts. Such is the case for Pt-the prototypical catalyst-which is as important for electrochemistry (electrolysis, fuel cells) as for oil refining (in particular, cracking and, now, biomass conversion). Modulation of its reactivity may require an additional metal to prevent drawbacks (e.g., bimetallics for fuel cells: PtRu, PtZn, etc.) such as the poisoning of the catalyst by CO. Modification of the electronic properties of the nanocatalysts, which are now well-understood by modeling, in particular by DFT calculations, opens the door to a fine tuning of the electron density on the NP surface for specific reactions (e.g., the alkane dehydrogenation or the selective hydrogenation reactions described hereabove). In addition, we have seen that the addition of a second metal not only prevents reaction drawbacks but in many cases enhances the catalytic activity because of the modification of the d-band center of the Pt DOS.

Further complexifications will likely lead to novel polymetallic NPs that will be able to perform more complex reactions. Finally, the combination of the intrinsic physical properties of the nano-objects, whether optical or magnetic, and their chemical properties open new avenues to a rich catalysis based on complex but well-mastered nanostructures.

ACKNOWLEDGMENT

The authors thank funding from the European Research Council ERC-Advanced Grant MONACAT 2015-694159.

The authors declare no conflict of interest.


Jean-Patrick Baudry

Adrian E Raftery

Gilles Celeux

Kenneth Lo

Raphael Gottardo

Combining Mixture Components for Clustering

Keywords: BIC, entropy, flow cytometry, mixture model, model-based clustering, multivariate normal distribution BIC, entropie, cytologie, modèles classification non supervisée, distribution normale multivariée

Model-based clustering consists of fitting a mixture model to data and identifying each cluster with one of its components. Multivariate normal distributions are typically used. The number of clusters is usually determined from the data, often using BIC. In practice, however, individual clusters can be poorly fitted by Gaussian distributions, and in that case model-based clustering tends to represent one non-Gaussian cluster by a mixture of two or more Gaussian distributions. If the number of mixture components is interpreted as the number of clusters, this can lead to overestimation of the number of clusters. This is because BIC selects the number of mixture components needed to provide a good approximation to the density, rather than the number of clusters as such. We propose first selecting the total number of Gaussian mixture components, K, using BIC and then combining them hierarchically according to an entropy criterion. This yields a unique soft clustering for each number of clusters less than or equal to K; these clusterings can be compared on substantive grounds. We illustrate the method with simulated data and a flow cytometry dataset.

Agréger des composantes d'un mélange pour la classification non supervisée

Résumé : La classification non supervisée par les modèles de mélange consiste à ajuster un modèle de mélange aux données, puis à identifier une classe à chacune des composantes obtenues. Le nombre de classes est habituellement choisi d'après les données, typiquement par le critère BIC. Cependant, en pratique, certains groupes peuvent être mal approchés par une distribution gaussienne et cette démarche mène souvent à représenter un tel groupe par un mélange de plusieurs composantes gaussiennes. Lorsque le nombre de composantes est effectivement interprété comme le nombre de classes, ce dernier risque d'être surestimé. La cause en est que BIC estime le nombre de composantes gaussiennes nécessaire pour que le mélange obtenu approche correctement la densité des données, et pas le nombre de classes en tant que tel. Nous proposons de sélectionner dans un premier temps le nombre total de composantes gaussiennes du mélange à ajuster, K, par BIC, puis de combiner hiérarchiquement ces composantes d'après un critère d'entropie. Nous obtenons ainsi une classification pour chaque nombre de classes inférieur ou égal à K; la nature des données et les connaissances a priori les concernant peuvent alors permettre de comparer ces différentes classifications. Nous illustrons cette méthode sur des données simulées et par une application à des données de cytologie.

INTRODUCTION

Model-based clustering is based on a finite mixture of distributions, in which each mixture component is taken to correspond to a different group, cluster or subpopulation. For continuous data, the most common component distribution is a multivariate Gaussian (or normal) distribution. A standard methodology for model-based clustering consists of using the EM algorithm to estimate the finite mixture models corresponding to each number of clusters considered and using BIC to select the number of mixture components, taken to be equal to the number of clusters [START_REF] Fraley | How many clusters? Answers via model-based cluster analysis[END_REF]. The clustering is then done by assigning each observation to the cluster to which it is most likely to belong a posteriori, conditionally on the selected model and its estimated parameters. For reviews of model-based clustering, see [START_REF] Mclachlan | Finite Mixture Models[END_REF] and [START_REF] Fraley | Model-based clustering, discriminant analysis and density estimation[END_REF]. [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF] argued that the goal of clustering is not the same as that of estimating the best approximating mixture model, and so BIC may not be the best way of determining the number of clusters, even though it does perform well in selecting the number of components in a mixture model. Instead they proposed the ICL criterion, whose purpose is to assess the number of mixture components that leads to the best clustering. This turns out to be equivalent to BIC penalized by the entropy of the corresponding clustering.

We argue here that the goal of selecting the number of mixture components for estimating the underlying probability density is well met by BIC, but that the goal of selecting the number of clusters may not be. Even when a multivariate Gaussian mixture model is used for clustering, the number of mixture components is not necessarily the same as the number of clusters. This is because a cluster may be better represented by a mixture of normals than by a single normal distribution.

We propose a method for combining the points of view underlying BIC and ICL to achieve the best of both worlds. BIC is used to select the number of components in the mixture model. We then propose a sequence of possible solutions by hierarchical combination of the components identified by BIC. The decision about which components to combine is based on the same entropy criterion that ICL implicitly uses. In this way, we propose a way of interpreting the mixture model in clustering terms by identifying a set of mixture components with each cluster. Finally, ICL could be helpful for identifying the number of clusters among the solutions provided by the designed hierarchy. This number of clusters can be different from the number of components chosen with BIC.

Often the number of clusters identified by ICL is smaller than the number of components selected by BIC, raising the question of whether BIC tends to overestimate the number of groups. On the other hand, in almost all simulations based on assumed true mixture models, the number of components selected by BIC does not overestimate the true number of components [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF][START_REF] Mclachlan | Finite Mixture Models[END_REF][START_REF] Steele | Importance sampling methods for inference in mixture models and missing data[END_REF]. Our approach resolves this apparent paradox.

In Section 2 we provide background on model-based clustering, BIC and ICL, and in Section 3 we describe our proposed methodology. In Section 4 we give results for simulated data, and in Section 5 we give results from the analysis of a flow cytometry dataset. There, one of the sequence of solutions from our method is clearly indicated substantively, and seems better than either the original BIC or ICL solutions. In Section 6 we discuss issues relevant to our method and other methods that have been proposed.

MODEL SELECTION IN MODEL-BASED CLUSTER-ING

Model-based clustering assumes that observations x = (x 1 , . . . , x n ) in R nd are a sample from a finite mixture density

p(x i | K, θ K ) = K k=1 p k φ(x i | a k ), (1)
where the p k 's are the mixing proportions (0 < p k < 1 for all k = 1, . . . , K and k p k = 1), φ(. | a k ) denotes a parameterized density, and θ K = (p 1 , . . . , p K-1 , a 1 , . . . , a K ). When the data are multivariate continuous observations, the component density is usually the d-dimensional Gaussian density with parameter a k = (µ k , Σ k ), µ k being the mean and Σ k the variance matrix of component k.

For estimation purposes, the mixture model is often expressed in terms of complete data, including the groups to which the data points belong. The complete data are

y = (y 1 , . . . , y n ) = ((x 1 , z 1 ), . . . , (x n , z n )),
where the missing data are z = (z 1 , . . . , z n ), with z i = (z i1 , . . . , z iK ) being binary vectors such that z ik = 1 if x i arises from group k. The z i 's define a partition P = (P 1 , . . . , P K ) of the observed data x with P k = {x i such that z ik = 1}.

From a Bayesian perspective, the selection of a mixture model can be based on the integrated likelihood of the mixture model with K components [START_REF] Kass | Bayes factors[END_REF], namely

p(x|K) = p(x|K, θ K )π(θ K )dθ K , (2)
where π(θ K ) is the prior distribution of the parameter θ K . Here we use the BIC approximation of [START_REF] Schwarz | Estimating the dimension of a model[END_REF] to the log integrated likelihood, namely

BIC(K) = log p(x|K, θK ) - ν K 2 log(n), ( 3
)
where θK is the maximum likelihood estimate of θ K and ν K is the number of free parameters of the model with K components. [START_REF] Keribin | Consistent estimation of the order of mixture models[END_REF] has shown that under certain regularity conditions the BIC consistently estimates the number of mixture components, and numerical experiments show that the BIC works well at a practical level [START_REF] Fraley | How many clusters? Answers via model-based cluster analysis[END_REF][START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF][START_REF] Steele | Importance sampling methods for inference in mixture models and missing data[END_REF]).

There is one problem with using this solution directly for clustering. Doing so is reasonable if each mixture component corresponds to a separate cluster, but this may not be the case. In particular, a cluster may be both cohesive and well separated from the other data (the usual intuitive notion of a cluster), without its distribution being Gaussian. This cluster may be represented by two or more mixture components, if its distribution is better approximated by a mixture than by a single Gaussian component. Thus the number of clusters in the data may be different from the number of components in the best approximating Gaussian mixture model.

To overcome this problem, [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF] proposed estimating the number of clusters (as distinct from the number of mixture components) in model-based clustering using the integrated complete likelihood (ICL), defined as the integrated likelihood of the complete data (x, z). ICL is defined as

p(x, z | K) = Θ K p(x, z | K, θ)π(θ | K)dθ, (4)
INRIA where p(x, z | K, θ) = n i=1 p(x i , z i | K, θ) with p(x i , z i | K, θ) = K k=1 p z ik k [φ(x i | a k )] z ik .
To approximate this integrated complete likelihood, [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF] proposed using a BIC-like approximation, leading to the criterion

ICL(K) = log p(x, ẑ | K, θK ) - ν K 2 log n, (5)
where the missing data have been replaced by their most probable values, given the parameter estimate θK .

Roughly speaking, ICL is equal to BIC penalized by the mean entropy

Ent(K) = - K k=1 n i=1 t ik ( θK ) log t ik ( θK ) ≥ 0, ( 6
)
where t ik denotes the conditional probability that x i arises from the kth mixture component (1

≤ i ≤ n and 1 ≤ k ≤ K), namely t ik ( θK ) = pk φ(x i |â k ) K j=1 pj φ(x i |â j )
.

Thus the number of clusters, K , favored by ICL tends to be smaller than the number K favored by BIC because of the additional entropy term. ICL aims to find the number of clusters rather than the number of mixture components. However, if it is used to estimate the number of mixture components it can underestimate it, particularly in data arising from mixtures with poorly separated components. In that case, the fit is worsened.

Thus the user of model-based clustering faces a dilemma: do the mixture components really all represent clusters, or do some subsets of them represent clusters with non-Gaussian distributions? In the next section, we propose a methodology to help resolve this dilemma.

METHODOLOGY

The idea is to build a sequence of clusterings, starting from a mixture model that fits the data well. Its number of components is chosen using BIC. We design a sequence of candidate soft clusterings with KBIC , KBIC -1,. . . ,1 clusters by successively merging the components in the BIC solution.

At each stage, we choose the two mixture components to be merged so as to minimize the entropy of the resulting clustering. Let us denote by t K i,1 , . . . , t K i,K the conditional posterior probabilities that x i arises from cluster 1, . . . , K with respect to the K-cluster solution. If clusters k and k from the K-cluster solution are combined, the t i,j 's remain the same for every j except for k and k . The new cluster k ∪ k then has the following conditional probability:

t K i,k∪k = t K i,k + t K i,k .
Then the resulting entropy is:

-

n i=1   j =k,k t K ij log t K ij + (t K ik + t K ik ) log (t K ik + t K ik )   . (7)
Thus, the two clusters k and k to be combined are those maximizing the criterion:

-

n i=1 {t K ik log(t K ik ) + t K ik log(t K ik )} + n i=1 t K i k∪k log t K i k∪k among all possible pairs of clusters (k, k ). Then t K-1 i,k , i = 1, . . . , n, k = 1, . . . , K -1 can be updated.
Any combined solution fits the data as well as the BIC solution, since it is based on the same Gaussian mixture; the likelihood does not change. Only the number and definition of clusters are different. Our method yields just one suggested set of clusters for each K, and the user can choose between them on substantive grounds. our flow cytometry data example in Section 5 provides one instance of this.

If a more automated procedure is desired for choosing a single solution, one possibility is to select, among the possible solutions, the solution providing the same number of clusters that ICL. An alternative is to use an elbow rule on the graphic displaying the entropy variation against the number of clusters. Both these strategies are illustrated in our examples.

The algorithm implementing the suggested procedure is given in the Appendix.

SIMULATED EXAMPLES

We first present some simulations to highlight the posssibilities of our methodology. They have been chosen to illustrate cases where BIC and ICL do not select the same number of components.

Simulated Example with Overlapping Components

The data, shown in Figure 1(a), were simulated from a two-dimensional Gaussian mixture.

There are six components, four of which are axis-aligned with diagonal variance matrices (the four components of the two "crosses"), and two of which are not axis-aligned, and so do not have diagonal variance matrices. There were 600 points, with mixing proportions 1/5 for each non axis-aligned component, 1/5 for each of the upper left cross components, and 1/10 for each of the lower right cross components. We fitted Gaussian mixture models to this simulated dataset. BIC selected a 6-component mixture model, which was the correct model; this is shown in Figure 1(b). ICL selected a 4-cluster model, as shown in Figure 1(c). The four clusters found by ICL are well separated.

Starting from the BIC 6-component solution, we combined two components to get the 5cluster solution shown in Figure 1(d). To decide which two components to merge, each pair of components was considered, and the entropy after combining these components into one cluster was computed. The two components for which the resulting entropy was the smallest were combined.

The same thing was done again to find a 4-cluster solution, shown in Figure 1(e). This is the number of clusters identified by ICL. The entropies of the combined solutions are shown in Figure 2, together with the differences between successive entropy values. There seems to be an elbow in the plot at K = 4, and so we focus on this solution. Note that there is no conventional formal statistical inferential basis for choosing between different numbers of clusters, as the likelihood and the distribution of the observations are the same for all the numbers of clusters considered.

In the four-cluster solution, the clusters are no longer all Gaussian; now two of them are modeled as mixtures of two Gaussians each. Note that this four-cluster solution is not the same as the four-cluster solution identified by ICL: ICL identifies a mixture of four Gaussians, while our method identifies four clusters of which two are not Gaussian. Figure 1(f) shows the true classification. Only 3 of the 600 points were misclassified.

Simulated Example with Overlapping Components and Restrictive Models

We now consider the same data again, but this time with more restrictive models. Only Gaussian mixture models with diagonal variance matrices are considered. This illustrates what happens when the mixture model generating the data is not in the set of models considered. BIC selects more components than before, namely ten (Figure 3a). This is because the generating model is not considered, and so more components are needed to approximate the true distribution. For example, the top right non-axis-aligned component cannot be represented correctly by a single Gaussian with a diagonal variance matrix, and BIC selects three diagonal Gaussians to represent it. ICL still selects four clusters (Figure 3b).

In the hierarchical merging process, the two components of one of the "crosses" were combined first (Figure 3c), followed by the components of the other cross (Figure 3d). The nondiagonal cluster on the lower left was optimally represented by three diagonal mixture components in the BIC solution. In the next step, two of these three components were combined (Figure 3e). Next, two of the three mixture components representing the upper right cluster were combined (Figure 3f). After the next step there were five clusters, and all three mixture components representing the lower left cluster had been combined (Figure 3g).

The next step got us to four clusters, the number identified by ICL (Figure 3h). After this last combination, all three mixture components representing the upper right cluster had been combined. Note that this four-cluster solution is not the same as the four-cluster solution got by optimizing ICL directly. Strikingly, this solution is almost identical to that obtained with the less restrictive set of models considered in Section 4.1.

The plot of the combined solution entropies against the number of components in Figure 4 suggests an elbow at K = 8, with a possible second, less apparent one at K = 4. In the K = 8 solution the two crosses have been merged, and in the K = 4 solution all four visually apparent clusters have been merged. Recall that the choice of the number of clusters is not based on formal statistical inference, unlike the choice of the number of mixture components. Our method generates a small set of possible solutions that can be compared on substantive grounds. The entropy plot is an exploratory device that can help to assess separation between clusters, rather than a formal inference tool.

Circle/Square Example

This example was presented by [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF]. The data, shown in Figure 5(a), were simulated from a mixture of a uniform distribution on a square and a spherical Gaussian distribution. Here, for illustrative purposes, we restricted the models considered to Gaussian mixtures with spherical variance matrices with the same determinant. Note that the true generating model does not belong to this model class.

In the simulation results of [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF], BIC chose two components in only 60% of the simulated cases. Here we show one simulated dataset in which BIC approximated the underlying non-Gaussian density using a mixture of five normals (Figure 5b). ICL always selected two clusters (Figure 5c). The progress of the combining algorithm is shown in Figure 5(d-f). The final two-cluster solution, obtained by hierarchical merging starting from the BIC solution, is slightly different from the clustering obtained by optimizing ICL directly. It also seems slightly better: ICL classifies seven observations into the uniform cluster that clearly do not belong to it, while the solution shown misclassifies only three observations in the same way. The true labels are shown in Figure 5(g). The entropy plot in Figure 6 does not have a clear elbow.

FLOW CYTOMETRY EXAMPLE

We now apply our method to the GvHD data of [START_REF] Brinkman | High-content flow cytometry and temporal data analysis for defining a cellular signature of graft-versus-host disease[END_REF]. Two samples of this flow cytometry data have been used, one from a patient with the graft-versus-host disease (GvHD), and the other from a control patient. GvHD occurs in allogeneic hematopoietic stem cell transplant recipients when donor-immune cells in the graft attack the skin, gut, liver, and other tissues of the recipient. GvHD is one of the most significant clinical problems in the field of allogeneic blood and marrow transplantation.

The GvHD positive and control samples consist of 9,083 and 6,809 observations respectively. Both samples include four biomarker variables, namely, CD4, CD8β, CD3 and CD8. The objective of the analysis is to identify CD3 + CD4 + CD8β + cell sub-populations present in the GvHD positive sample. In order to identify all cell sub-populations in the data, we use a Gaussian mixture model with unrestricted covariance matrix. Adopting a similar strategy to that described by [START_REF] Lo | Automated gating of flow cytometry data via robust model-based clustering[END_REF], for a given number of components, we locate the CD3 + sub-populations by labeling components with means in the CD3 dimension above 270 CD3 + . This threshold was based on a comparison with a negative control sample as explained by [START_REF] Brinkman | High-content flow cytometry and temporal data analysis for defining a cellular signature of graft-versus-host disease[END_REF].

We analyze the positive sample first. A previous manual analysis of the positive sample suggested that the CD3 + cells could be divided into approximately five to six CD3 + cell subpopulations [START_REF] Brinkman | High-content flow cytometry and temporal data analysis for defining a cellular signature of graft-versus-host disease[END_REF]. ICL selected nine clusters, five of which correspond to the

INRIA -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 Simulated observed data. n=200 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X 1 X 2 BIC solution. K=5. ENT=36 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X 1 X 2 ICL solution. K=2. ENT=10 (a) (b) (c) -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X 1 X 2 Combined solution. K=4. ENT=22 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X 1 X 2 Combined solution. K=3. ENT=13 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X 1 X 2 Combined solution. K=2. ENT=5 (d) (e) (f) -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 X1 X2
The True Complete Data with 2 clusters. CD3 + population (Figure 7(b)). Compared with the result shown in [START_REF] Lo | Automated gating of flow cytometry data via robust model-based clustering[END_REF], the CD4 + CD8β -region located at the bottom right of the graph is not represented. BIC selected 12 components to provide a good fit to the positive sample, seven of which are labeled CD3 + (Figure 7(a)). The CD4 + CD8β + region seems to be encapsulated by the cyan, blue, yellow and red components. Starting from this BIC solution, we repeatedly combined two components causing maximal reduction in the entropy. The first two combinations all occurred within those components originally labeled CD3 -, and the CD4 vs CD8β projection of the CD3 + sub-populations remains unchanged.

However, when the number of clusters was reduced to nine, the purple cluster representing the CD3 + CD4 + CD8β -population was combined with the big CD3 -cluster, resulting in an incomplete representation of the CD3 + population (Figure 7 (c)). Hence, the combined solution with ten clusters, in which seven are labeled CD3 + , seems to provide the most parsimonious view of the positive sample whilst retaining the seven important CD3 + cell sub-populations. Note that the entropy of the combined solution with nine clusters (1409) was smaller than that of the ICL solution (3236). The entropy plot (Figure 8) suggests an elbow at K = 10 clusters, agreeing with our more substantively-based conclusion.

Next we analyze the control sample. A satisfactory analysis would show an absence of the CD3 + CD4 + CD8β + cell sub-populations. ICL chose five clusters, two of which correspond to the CD3 + population (Figure 9 (b)). The black cluster at the left of the graph represents the CD4 -region. The red cluster at the rigth of the graph represents the CD8 -region. It seems that it misses a part of this cluster near the black cluster. This suggests that the ICL solution could be improved.

BIC selected 11 components, four of which are labeled CD3 + (Figure 9(a)). The black component on the left side does not extend to the CD4 + region. However, contrary to previous findings in which CD4 + CD8β + cell sub-populations were found only in positive samples and not in control samples, a green component is used to represent the observations scattered within the CD4 + CD8β + region.

Similar to the result for the positive sample, when we combined the components in the BIC solution, the first few combinations took place within those components initially labeled CD3 -. When only six clusters remained, the blue and red components in Figure 9(a) combined, leaving the CD3 + sub-populations to be represented by three clusters (Figure 9(c)).

After two more combinations (K=4), the green component merged with a big CD3 -cluster. Finally we had a "clean" representation of the CD3 + population with no observations from the CD3 + CD4 + CD8β + region. One last combination (K=3) merged the two remaining CD3 - clusters, resulting in the most parsimonious view of the control sample with only three clusters but showing all the relevant features (Figure 9(d)). Once again, the entropy of the combined solution (106) was much smaller than that of the ICL solution (496). Note that in this case we ended up with a combined solution that has fewer clusters than the ICL solution. The plot of the entropy of the combined solution against the number of clusters (Figure 10) suggests an elbow at K = 7, but substantive considerations suggest that we can continue merging past this number.

INRIA

DISCUSSION

We have proposed a way of addressing the dilemma of model-based clustering based on Gaussian mixture models, namely that the number of mixture components selected is not necessarily equal to the number of clusters. This arises when one or more of the clusters has a non-Gaussian distribution, which is approximated by a mixture of several Gaussians. Our strategy is as follows. We first fit a Gaussian mixture model to the data by maximum likelihood estimation, using BIC to select the number of Gaussian components. Then we successively combine mixture components using the entropy of the conditional membership distribution to decide which components to merge at each stage. This yields a sequence of possible solutions, one for each number of clusters, and in general we expect that users would consider these solutions from a substantive point of view.

The underlying statistical model is the same for each member of this sequence of solutions, in the sense that the likelihood and the modeled probability distribution of the data remain unchanged. What changes is the interpretation of this model. Thus standard statistical testing or model selection methods cannot be used to choose the preferred solution.

If a data-driven choice is required, however, we suggest inspecting the entropy plot and looking for an elbow, or choosing the number of clusters selected by ICL. An inferential choice could be made, for example using the gap statistic [START_REF] Tibshirani | Estimating the number of clusters in a data set via the gap statistic[END_REF]. However, the null distribution underlying the resulting test does not belong to the class of models being tested, so that it does not have a conventional statistical interpretation in the present context. It could still possibly be used in a less formal sense to help guide the choice of number of clusters.

Our method preserves the advantages of Gaussian model-based clustering, notably a good fit to the data, but it allows us to avoid the overestimation of the number of clusters that can occur when some clusters are non-Gaussian. The mixture distribution selected by BIC allows us to start the hierarchical procedure from a good summary of the data set. The resulting hierarchy is easily interpreted in relation to the mixture components. We stress that the whole hierarchy from K to 1 clusters might be informative. Note that our method can also be used when the number of clusters K * is known, provided that the number of mixture components in the BIC solution is at least as large as K * .

One attractive feature of our method is that it is computationally efficient, as it uses only the conditional membership probabilities. Thus it could be applied to any mixture model, and not just to a Gaussian mixture model, effectively without modification. This includes latent class analysis [START_REF] Lazarsfeld | The logical and mathematical foundations of latent structure analysis[END_REF][START_REF] Hagenaars | Applied Latent Class Analysis[END_REF], which is essentially model-based clustering for discrete data.

Several other methods for joining Gaussian mixture components to form clusters have been proposed. [START_REF] Walther | Detecting the presence of mixing with multiscale maximum likelihood[END_REF] considered the problem of deciding whether a univariate distribution is better modeled by a mixture of normals or by a single, possibly non-Gaussian and asymmetric distribution. To our knowledge, this idea has not yet been extended to more than one dimension, and it seems difficult to do so. Our method seems to provide a simple alternative approach to the problem addressed by [START_REF] Walther | Detecting the presence of mixing with multiscale maximum likelihood[END_REF], in arbitrary dimensions.

Wang and Raftery (2002, Section 4.5) considered the estimation of elongated features in a spatial point pattern with noise, motivated by a minefield detection problem. They suggested first clustering the points using Gaussian model-based clustering with equal spherical covariance matrices for the components. This leads to the feature being covered by a set of "balls" (spherical components), and these are then merged if their centers are close enough that the components are likely to overlap. This works well for joining spherical components, but may not work well if the components are not spherical, as it takes account of the component means but not their shapes. [START_REF] Tantrum | Assessment and pruning of hierarchical model based clustering[END_REF] proposed a different method based on the hierarchical model-based clustering method of [START_REF] Banfield | Model-based Gaussian and non-Gaussian clustering[END_REF]. Hierarchical model-based clustering is a "hard" clustering method, in which each data point is assigned to one group. At each stage, two clusters are merged, with the likelihood used as the criterion for deciding which clusters to merge. [START_REF] Tantrum | Assessment and pruning of hierarchical model based clustering[END_REF] proposed using the dip test of [START_REF] Hartigan | The dip test of unimodality[END_REF] to decide on the number of clusters. This method differs from ours in two main ways. Ours is a probabilistic ("soft") clustering method that merges mixture components (distributions), and while that of [START_REF] Tantrum | Assessment and pruning of hierarchical model based clustering[END_REF] is a hard clustering method that merges groups of data points. Secondly, the merging criterion is different. [START_REF] Li | Clustering based on a multilayer mixture model[END_REF] assumed that the number of clusters K is known in advance, used BIC to estimate the number of mixture components, and joined them using k-means clustering applied to their means. This works well if the clusters are spherical, but may not work as well if they are elongated, as the method is based on the means of the clusters but does not take account of their shape. The underlying assumption that the number of clusters is known may also be questionable in some applications. [START_REF] Jörnsten | Mixture models with multiple levels, with application to the analysis of multifactor gene expression data[END_REF] extended Li's method so as to apply it to multifactor gene expression data, allowing clusters to share mixture components, and relating the levels of the mixture to the experimental factors.

A Algorithm

Choose a family of mixture models: {M Kmin , . . . , M K max }. Complete Gaussian mixture models are suggested: M K contains any mixture with K Gaussian components. Here is the algorithm we work with: For K = KBIC , . . . , (K min + 1):

Choose the clusters l and l to be combined at step K → K -1 :

(l, l ) = argmax (k,k )∈{1,...,K} 2 , k =k

- n i=1 t K ik log(t K ik ) + t K ik log(t K ik ) + n i=1 (t K ik + t K i k ) log(t K ik + t K i k ) , where t K ik = f K k (xi) P K j=1 f K j (xi)
is the conditional probablity of component k given the Kcluster combined solution.

INRIA

Define the densities of the combined clusters for the (K-1) clusters solution by combining l and l : for k = 1, . . . , (l ∧ l -1), (l ∧ l + 1), . . . , (l ∨ l -1) {f

K-1 k = f K k } f K-1 l∧l = f K l + f K l for k = l ∨ l , . . . , (K -1) {f K-1 k = f K k+1 } 4.

Figure 1 :

 1 Figure 1: Simulated Example 1. (a) Simulated data from a 6-component 2-dimensional Gaussian mixture. (b) BIC solution with 6 components. (c) ICL solution with 4 clusters. (d) Combined solution with 5 clusters. (e) Combined solution with 4 clusters. (f) The true labels for a 4-cluster solution. In (b) and (c) the entropy, ENT, is defined by equation (6) with respect to the the maximum likelihood solution, and in (d) and (e) ENT is defined by equation (7).

  Figure 2: (a) Entropy values for the K-cluster Combined Solution, as defined by equation (7), for Simulated Example 1. (b) Differences between successive entropy values.

Figure 3 :Figure 4 :

 34 Figure 3: Simulated Example 2. The data are the same as in Simulated Example 1, but the model space is more restrictive, as only Gaussian mixture models with diagonal covariance matrices are considered. See Fig.1 legends for explanations about ENT. (a) BIC solution with 10 mixture components. (b) ICL solution with 4 clusters. (c) Combined solution with 9 clusters. (d) Combined solution with 8 clusters. (e) Combined solution with 7 clusters. (f) Combined solution with 6 clusters. (g) Combined solution with 5 clusters. (h) Combined solution with 4 clusters. (i) True labels with 4 clusters.

Figure 5 :

 5 Figure 5: Circle-Square Example. See Fig.1 legends for explanations about ENT. (a) Observed data simulated from a mixture of a uniform distribution on a square and a spherical Gaussian distribution. (b) The BIC solution, with 5 components. (c) The ICL solution with 2 clusters. (d) The combined solution with 4 clusters. (e) The combined solution with 3 clusters. (f) The final combined solution, with 2 clusters. (g) The true labels.

Figure 6 :Figure 7 :

 67 Figure 6: (a) Entropy values for the K-cluster Combined Solution, as defined by equation (7), for the Circle-Square Example. (b) Differences between successive entropy values.

Figure 8 :

 8 Figure 8: (a) Entropy values for the GvHD positive sample. (b) Differences between successive entropy values.

Figure 9 :

 9 Figure 9: GvHD control sample. Only components labeled CD3 + are shown. (a) BIC Solution (K=11). (b) ICL Solution (K=5). (c) Combined Solution (K=6). (d) Combined Solution (K=3).

Figure 10 :

 10 Figure 10: (a) Entropy values for the K-cluster Combined Solution for the GvHD Control Sample. (b) Differences between successive entropy values.

  1. Compute MLE(K) for each model using the EM algorithm:∀K ∈ {K min , . . . , K max }, θK = arg max θ K ∈Θ K log p(x | K, θ K ) the density f K k of each combined cluster k for each K from KBIC to K min : ∀k ∈ {1, . . . , KBIC }, f

  To select the number of clusters through ICL: ) . is the conditional probability of component k given the MLE for the model with K Gaussian components.

	KICL =	argmin K∈{Kmin,...,Kmax}	-log p(x | K, θK ) -	n i=1	K k=1	t ik ( θK ) log t ik ( θK ) +	ν K 2	log n ,
	where t ik ( θK ) =	pK k φ(xi|â K k ) P K j=1 pK j φ(xi|â K

j

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