S Gota
email: gota@belem.saclay.cea.fr.

M Gautier

L Douillard

N Thromat

P Duraud

Le F~vre

surface science

Influence of the substrate oxidation state in the growth of copper clusters on A1_20_3(0001) surface: a XANES and EXAFS study

Introduction

The study of the electronic and structural properties of metal clusters supported on metal oxides has attn~cted interest in recent years [START_REF] Henrich | The Surface Science of Metal Gxydes[END_REF]. These systems present interesting properties from both fundamental and "~:echnological points of view. On the one hand, the study of small clusters provides information about the t~ansition from isolated atoms to bulk matter, a subject of key importance in solid state physics. On the other hand, their large surface to bulk ratio induces a high chemical reactivity in oxidation and in catalysis processes, which supplies interesting applications for industry. In addition, the knowledge of the metal-oxide interface properties is important in the preparation of MOS (metal-oxide-semiconductor) devices and in the provision of oxide supported metal conductor pathways in integrated circuits.

In view of the importance of surface effects in thin metal films supported on oxides, one can reasonably expect that the state of the oxide substrate (oxygen-rich or reduced, reconstructed or not) determines the nature of the bonds at the interface and, in consequence, likewise the morphology of the supported metal clusters.

In order to investigate this phenomenon, we have chosen to study the first stages of the copper growth onto alumina as a model system. The Cu/AI203 interface is widely used in heterogeneous catalysis and microelectronics, where metallisation of aluminium oxide is an important technological concern.

The experimental work reported in the literature ~ncerning the Cu/AI203 system was performed using as substrates either oxidised polycrystalline aluminium [2][3][4][5][6][7], or monocrystalline a-alumina [8][9][10][11][12]. In both cases, controversy remains concerning the growth mode of the vapour deposited Cu thin films. Two growth modes, Stransld-Krastanov [4,9,12] (layer growth up to one or few monolayers followed by three-dimensional crystallite growth, SK) and Volmer-Weber [2,3,10,11] (formation of threedimensional crystallites without a preceding adsorbed layer, VW) are proposed.

In our opinion, the lack of agreement is caused by the use of experimental methods sensitive in an indirect way to the structural parameters of the interface. A commonly used method consists in measuring the AES (Auger electron spectroscopy) intensities from both the adsorbate and substrate atoms as a function of the adsorbate average thickness (or a related parameter). These plots present characteristic behaviours associated with the different growth modes [13]. The completion of the first monolayer is manifested by a linear increase (decrease) of the adsorbate (substrate) AES intensity from the lowest coverage, followed by a change of slope in both substrat¢ and adsorbate curves. In the case of a layer-by-layer growth (Frank van der Merwe growth mode, FM), the plot presents successive changes of slope corresponding to the formation of each following layer. In the SK growth mode, the first break is followed by a smooth increase (decrease) of the intensity of the adsorbate (substrate). In contrast, the VW growth mode presents a smooth variation of the AES intensity from the beginning of the adsorbate deposition. The limitation of this method is the difficulty to discriminate between V-W and S-K modes at the very first stages. Taking into account the experimental error of these measurements, it is quite hard to clearly distinguish between .~ straight line and a smooth behaviour of the intensity at very low coverages (< monolayer). As discussed by Diebold et al. [14], these AES plots can not be considered as conclusive determination of the growth mechanism.

This handicap is the reason why different groups give quite opl~site interpretations to AES plots from the Cu/AI203 interface showing similar trends. Chen et al. [4] and Moiler and Guo [9] associate a slight change of the slope with the completion of a "smooth" first layer, followed by the formation of three-dimensional crystallites (SK growth mode). In contrast, di Castro and Polzonetti [2,3] suggest that the growth mechanism take place in two successive steps: in a first step, the Cu atoms diffuse on the alumina surface or form small dispersed clusters. The break of the curve corresponds to the formation of big clusters with similar properties to those of the bulk metal (VW growth mode). This second hypothesis is supported by the weak interaction founded between Cu and Al203 by UPS (ultraviolet photoemission spectroscopy) [2,5]. The formation of a complete first Cu overlayer suggested by Chen et al. implies a rather strong Cu-Al203 interaction in contrast with their own HP~ELS (high resolution electron energy loss spectroscopy) measurements which show that the adsorption of copper does not significantly affect the (AI-O) surface phonon mode [4].

Very recently, Varma et al. [12] have studied the growth mode of Cu on (0001) a-alumina surface using ARXPS (angle resolved X-ray photoemission spectroscopy). For a thin film, the effective thickness d(0) as a function of the polar emission angle 0 can be obtained by a mathematical expression [15] involving the measured photoelectron core level intensities from the adsorbate and substrate atoms. The calculated values of d(0) should be independent of 0 for the case of a uniform thickness adsorbate film. However, the presence of clusters will provide angle dependent values of d(O). They found an angle dependent d(0) curve for coverages up to 3 ,A, while for lower coverages, d(0) remains angle independent. In view of these results, they infer a SK growth mode. We consider that this technique is not well adapted for the determination of the Cu/Al203 interface growth mode, because there exist troublesome difficulties for the determination of the Cu 3p and Al 2p photoelectron core levels intensities used in the d(0) calculation formula. Firstly, the Cu 3p and Al 2p photoelectron peaks are separated by only 2 eV in energy. Because of the experimental energy resolution, a deconvolution procedure into the Cu3P3/2, Cu3pz/2 and Al2p is required. This procedure adds significant inaccuracies. Secondly, the Cu3p intensity for coverages lower than 3 ~ is extremely weak. These two facts lead us to estimate that these ARXPS results do not provide any conclusive proof for the Cu/Al203 interface growth mode.

In contrast to the techniques discussed above, surface extended X-ray absorption fine structure spectroscopy [START_REF] Fur A Review | see for example, X-ray Absorption: Principes, Applications[END_REF][START_REF] Citrin | [END_REF][18] (SEXAFS) is a well adapted and powerful tool to investigate directly the structure of the metal/oxide interfaces. Indeed, the description of the XANES (X-ray absorption near edge structure) region of the spectra between 0 and about 50 eV above the threshold is mainly dependent on the geometrical structure of a finite cluster of atoms around the absorbing atom [19]. For extracting structural information from a XANES experimental spectrum, a multiple scattering (MS) analysis is needed because of the important photoelectron mean free path and the strong scattering power of the. atoms in matter for low kinetic energies. At higher energies, the photoelectron mean free path and the atomic scattering power become gradually smaller and a single scattering (SS) regime is reached [20].

X-ray absorption spectroscopy has been already successfully used by Montane et al. [21] to investigate the structure of free copper microclusters embedded in a solid argon matrix. They have estimated an average particle size in good agreement with mass-spectroscopy measurements. Also, Apai et al.

[22] have studied by SEXAFS the Cu and Ni clusters formed by vapour deposition onto amorphous carbon substrates.

We have studied, by means of X-ray absorption spectroscopy (XANES, EXAFS), the influence of the a-alumina oxidation state in the formation of the Cu/AI203 interface, following the procedure outhned below:

(1) At first, we have prepared in situ both oxidised and reduced (0001) a-alumina surfaces and have evaporated equal amounts of copper (~ 80% monolayer) on each. (2) Next, in order to point out differences in the Cu thin films morphology when deposited either on a reduced or on a oxidised a-alumina substrate, we have performed X-ray absorption experiments at the CuK edge on each Cu/Al203 interface. Under our experimental conditicns, the analysis of the obtained spectra allows us to infer the follow-ing results concerning the Cu/A1203 interface growth: (a) Whatever the alumina (b) first stages of the oxidation state, the Cu/AI203 interface grows via the Volmer-Weber mode rather than the Stranski-Krastanov mode, which has been suggested by some authors in the literature. The morphology of supported Cu clusters strongly depends on the oxidation state of the alumina surface. For equal amounts of copper, much larger clusters are found on reduced a-AI203(0001) than on oxidised substrates.

Experimental

The SEXAFS measurements were carried out on the wiggler DW21 beamline of the storage ring DCI at LURE (Orsay, France). The energy of the positrons in the machine is 1.85 GeV and the current is about 300 mA. The beam is focused in the vertical direction by a cylindrical mirror. We have used a Si(311) double crystal monochromator, the third diffraction order being cut by the mirror. The base pressure in the WXAS experimental station during the data acquisition was lower than 5 × 10-1o Torr. This UHV chamber is equipped with a LEED (low energy electron diffraction) device and an electron gun (5 keV) coaxial with a CMA (cylindrical mirror analyzer) permitting AES measurements. The Cu K edge (8976 eV) spectra were performed at normal incidence, and recorded in the fluorescence mode using a Si-Li detector. The count rates were around 3500 counts/s, which is much smaller than the saturation rates of the counting system. For studying the XANES region, the absorption spectra were recorded between 8950 and 9050 eV, using energy steps of 0.5 eV. For studying the EXAFS oscillations, the energy range was 8950-9650 eV and the energy step 2 eV.

The samples were thin slabs of monocrystalline a-Al20 3 with one face polished up to optical use specifications and oriented perpendicular to the (0001) surface. The samples were previously heated up to 900°C in air for two hours to restore surface stoichiometry. We have prepared two quite different substrates: (a) A reduced a-A1203(0001) surface. Oxygen va-cancies were created by Ar + bombardment (PA,+=8X 10 -5 Torr, 600 eV energy, current of 5.4 mA) at room temperature. This surface does not exhibit a (1 x 1) LEED pattern characteristic of the bulk hexagonal symmetry [23].

(b) An oxidised A]203(0001) surface. The oxygenrich surface was prepared by an Ar + bombardment similar to that described above, followed by a heating at 800°C under 0 2 pressure (Po2 = 5 X 10 -7 Torr) during 30 min. The temperature was measured simultaneously by a W-Re thermocouple and an optical pyrometer. Befoie Cu deposition, the surface cleanliness was checked by AES. The two Cu/Al20 3 interfaces were prepared by evaporation of pure copper heated by a tungsten wire• The evaporation rate (1 A/8 s) was carefully calibrated with the help of a quartz crystal microbalance before each deposition. The equivalent deposited thickness was 3 A on both reduced and oxidised substrates. We estimate that it corresponds to an "equivalent coverage" of 80% ML [10] referring to ideal layer by layer deposition, whereas we observe cluster growth. The evaporation was carried out just before the X-ray absorption data acquisition. We have additionally checked that an equal quantity of Cu atoms was condensed on both the reduced and the oxidised substrates by measuring the absorption jump edge, which was found identical in both cases, sponding to 80% ML of copper deposited onto aalumina, resemble more that of metallic copper l(a). The features labelled as A, B and C in l(a), which are distinctive of a metallic fcc copper XANES spectra [25], clearly exist, however smoothed, in the l(b) and l(c) spectra. In contrast, the typical strong peaks present in the l(d) Cu20 XANES spectra do not arise either in the l(b) or in the l(c) data.

3, Results

XANES results

EXAFS results

Fig. 2 shows the normalised raw X-ray absorption spectra of: (a) 80% ML of copper deposited on a reduced A1203(0001) surface. (b) 80% ML of copper deposited on an oxidised A1203(0001) surface. The continuous line is a cubic spline curve fit of the raw data.

In this work, data processing for extracting x(k) from the raw data has been done following standard procedures described elsewhere [20,26]. De-gli~ching has been performed by linear interpolation between two physical points. A linear background contribution from previous edges has been subtracted from the experimental signal, giving the It(E) spectrum. For determining /.~o, the low frequency atomic-like absorption coefficient, the experimental signal It(E)

was fitted with polynomial splines using a leastsquares procedure. E o has been taken at the maximum of the derivative of the jump edge. (b) 80% ML of copper deposited on a reduced a-A1203(0001) surface. (c) 80% ML of copper deposited ori an oxidised a-A1203(0001) surface.

We can observe a strong foremost peak, corresponding to the first coordination shell, which is located at the same R position in the three RDF. For the case of metallic copper, this peak is originated by 12 Cu atoms placed at r = 2.54 A from the absorber one [21,27]. However, the intensity of this peak is weaker in the 3(b) and 3(c) RDF compared with that in the reference. The second and third peaks in the metallic Cu RDF correspond respectively to a second shell of 6 atoms located at r = 3.61 A and a third shell of 24 atoms located at r = 4.42 A [21,27]. The co~xesponding peaks in the 3(b) and 3(c) RDF pre- sent relative intensities which are quite distorted compared to those in the reference. In addition, these peaks are slightly shifted in the 3(c) RDF related to those in the metallic Cu RDF.

0,04

Io ML C~Os

The EXAFS modulations at high energies (~ 50 eV above the absorption edge) are well described by a short range single scattering X-ray absorption theory [28]. We have focused our data analysis in the structure determination of the first and second shells of neighbours around the emitter. For isolating their contribution in the RDF curves, we have used smooth windows of 1.6-3.0 ,~ and 3.0-3.74 ,~, respectively.

For assuring the correctness of our results, the obtained Fourier back transformed spectra have been fitted following two independent procedures: for one, we have used the amplitude and the phase shifts computed by McKale et al. [29]. For another, we have used those extracted from the reference copper sample. Remarkable good fits have been obtained by only using one type of atom (Cu) for both the first and the second shell. Structural parameters giving the best least-squares fits are summarised in Table 1, where dc~_c~ represents the mean distance, N the average number of atoms and Atr 2 the mean square relative displacement of each shell. The error bars in the detehnination of each parameter are also given. We have performed series of fits weighting the EXAFS data in several powers of k ~ (i ---0,1,2,3). In Table 1, the mean values of these series of fits are shown. Only slight deviations of about 1% in the distances and about 10% in the coordination numbers have been noticed, which illustrate the accuracy of our EXAFS structural determination. Fig. 4 shows the best least-squares fits obtained for the structural parameters given in Table 1. Open circles represent experimental signal and full line the simulation using the amplitude and phase shifts tabulated by McKale et al. The left of the figure shows the fits corresponding to 80% ML of copper deposited on a reduced Al203(0001) surface and the right of the figure those corresponding to 80% ML of copper deposited on an oxidised Al203(0001) surface.

For the case of the reduced substrate, the photoabsorber has a first shell environment similar to that of bulk copper atoms. For the second shell, the bond distance coincides with that of bulk copper, whereas the mean coordination number is significantly weaker (4 atoms instead of 6).

For the case of the oxidised substrate, the structural parameters are quite different, although the same equivalent quantity of Cu was deposited on both reduced and oxidised sub,~trates. The first and second shell bond distances are slightly distorted compared with the metal copper reference. Nevertheless, this small distortion might be assumed to be inside the error bars. However, the number of neighbours in the first and second shells has dramatically decreased (8 instead of 12 and 3 instead of 6 in bulk copper, rc~spectively).

Discassion

Experimental evidence of Volmer-Weber growth mode for the Cu /Ale0 3 interface

Because of the strong interaction between the photoelectron and the atoms neighbouring the emitter, XANES spectroscopy provides a unique probe for the local structure determination. All the features specific of metallic copper are present (however smoothed) in the l(b) and l(c) spectra. This fact implies that the deposited copper atoms reach the same local environment as atoms into metallic bulk copper. This result suggests the existence of small copper clusters, i.e., a VW growth mode for the Cu/Al203 interface, as proposed by di Castro et al. [2,31. In order to argue our conclusion, let us assume the hypothesis of a SK growth mode, i.e. Cu atoms form a complete monolayer before the formation of clusters. At this point, we must remember that: (a) Copper atoms bound preferentially to oxygen rather than to aluminium atoms [5,7]. In consequence, for an equivalent deposit of 80% ML, a large number of Cu-O bonds should appear at the Cu/alumina interface. (b) The value of the Auger parameter (Acu, sensitive to the different oxidation states of copper) was measured by Gautier et al. for the system Cu/Al203. For very weak coverages ( << 1 ML) it corresponds to Cu(I) [10,11]. In this framework, the presence of a large number of Cu-O bonds should induce the presence of the Cu20 l(d) typical features in the l(b) and l(c) spectra. In other words, copper atoms forming a monolayer over the alumina substrate should not have at all the environment of Cu atoms in an fcc structure, and consequently, there would not be any reason for their XANES spectra to resemble that of metallic copper. It is clear that the l(b) and l(c) spectra do not present any feature existing in the Cu20 spectra but on the contrary, a marked metallic character.

Additional evidence of a VW growth mode can be obtained by the EY~FS data analysis. The fits of the backscattered Fourier spectra have been satisfactorily performed considering only Cu neighbours. The best least-squares fits have been reached for an fcc environment in both the first and the second coordination shells. Our EXAFS analysis does not bring into light any signature of possible Cu-O bonds.

In a previous work, Gautier et al. have measured the ~ES spectra of 30% and 60% ML of Cu deposited onto a stoichiometric a-AI,O~(0001) surface exhibiting a (1 x 1) LEED pattern [ 10,11 ]. As in the present work, both spectra closely resemble that of metallic copper. Summarising our results and those from Refs. [10,11], we can conclude that copper atoms cluster in an fcc structure when deposited onto alumina, independently of the oxidation state and crystallinity of the a-Al20~(0001) substrate. This conclusion is supported by previous UPS [2] and HREELS [4] measurements, which clearly reveal a weak interaction between copper and AI:O~. Under our experimental conditions, we believe that the $g growth mode proposed by means of AES [4,9] and ARXPS [12] can be ruled out. AES plots cannot be considered as a convincing determination of the growth mechanism because of the difficulty in distinguishing the curves behaviour for very low coverages.

Copper cluster size estimations by means of shell by shell multiple scattering calculations

Frahm, Barbee and Wa~burtop [27] have followed the formation of Cu clusters on a W substrate by in situ quick-scanning EXAFS, for depositions from I to 72 equivalent ML thickness. In both XANES and EXAFS regions of spectra, the increase of the Cu thickness results in an evolution towards the typical Cu fcc structure, A similar trend is found in shell by shell multiple scattering calculations [25,30].

In our case, we observe that the A and B features are more pronounced in the l(b) spectra corresponding to a reduced alumina substrate than in the l(c) spectra corresponding to an oxidised substrate. As the structure of the spectra develops with the number of shells, this fact may indicate that, for an equal quantity of deposited Cu, the size of the deposited clusters is bigger on the reduced than on the oxygen-rich substrate. In this view, we have performed a sequence of MS calculations for spherical fcc Cu clusters, starting with one shell and running up to four shells. The XANES calculations have been performed using the program MXAS [31], developed by the theoretical group of the LNF-INFN (Laboratori Nazionali di Frascati-lstituto Nazionali di Fisica Nucleare) at Frascati, Italy. The theoretical approach is based on the real-space cluster method, and includes all the MS contributions. The phase shifts were derived from a one-electron potential built by superposition of neutral atomic charge densities [32], assuming the muffin-tin approximation. An energy-dependent exchange and correlation complex Hedin-Lundqvist potential [START_REF] Hedin | Solid State Physics[END_REF] The complex Hedin-Lundqvist potential was used. radius r = 5.105 ,~) is needed for predicting the existence of the (A) and (B) features present in the experiment. These structures are originated from single and multiple scattering events involving the fourth and higher shells of atoms.

A comparison between our experimental data and MS calculations tends to associate a size close to 4 shells (r = 5 A) for the clusters deposited on the oxidised alumina substrate and quite higher than 4 shells (r >> 5 A) for those deposited on the reduced one. We can exclude smaller cluster sizes because their corresponding calculated spectra do not reproduce the position and shape of the A and B features, which only reach the bulk position (not the intensity) when at least the fourth shell is taken into account.

However, structural disorder reasonably exists in clusters compared to bulk and contributes as well to broaden the sharp features characteristic of a wellordered fcc Cu spectra. Indeed, missing atoms at the clusters surface, slight deviations from the spherical symmetry and a non-homogeneous cluster size and shape distribution cannot be excluded. In the study of Frahm et al. [27], even for a coverage of 72 equivalent ML, all the modulations are weaker than those of a typical fcc Cu spectra. They connect this observation with the existence of disorder. So, keep- ing in mind that we compare our experiments with calculations simulating perfect spherical clusters, we propose that this rough estimation of the cluster size should be considered with some caution.

Copper cluster size estimations by means of EXAFS results

The peak positions in the RDF corresponding to both the reduced and the oxidised substrates remain nearly unchanged with respect to the reference. This fact matches with the first-and second-neighbour bond distances obtained by EXAFS analysis, which are very close to that of bulk copper (see Table 1). This result evidences that the Cu clusters supported onto alumina stabilise ia the metallic fcc structure from coverages of about one monolayer. A similar result was found by Montane et al. for Cu clusters embedded in a solid argon matrix. In contrast, in the case of Cu clusters supported on a graphite surface, De Crescenzi et al. [START_REF] Crescenzi | [END_REF] and Apai et al. [22] found that the fcc Cu distances are only reached for a O cluster radius larger than ~ 25 A.

We notice a reduction of the overall intensity in the 3(b) and 3(c) RDF compared with the copper 3(a) one. In the same way, the numbers of Cu-Cu bonds obtained by EXAFS analysis for the thin films are always smaller than the Cu fcc predicted ones. Rather, the average coordination numbers are much lower for an oxidised substrate than for a reduced substrat¢ (see Table 1). This fact originates mainly from cluster size effects, Indeed, the decrease of the average, coordination number is caused by the increase of the fraction of atoms at the cluster surface, when the cluster size decreases [21]. What is more, we can extract more precise information about the clusters size by comparison of the experimental mean coordination numbers with those calculated for a given cluster size [21], Fig. 6 shows the calculated coordination numbers of (a) the first shell and (b) the second shell for a spherical cluster radius range from 3.6 to 18 ~. The coordination numbers are increasing functions of the cluster radius, upward tending to the bulk values. Even for big clusters of about 25 shells of atoms (r ~ 18 A J, the bulk values are not yet reached. Note that, when a new shell containing an important number of atoms is added, the mean coordination number decrease compared to the im-mediate precedent shell. Our experimental values are also plotted together with their error bars.

For the oxidised substrate, the experimental mean coordination numbers of 8.1 for the first shell and 2.85 for the second shell match with a cluster radius averaging between 4.5 and 5.5 A (about 55 atoms, 4 shells). This result implies firstly, that the clusters shape is very close to the ideal fcc spherical one, and secondly, the cluster size distribution is rather nniform. For the reduced substrate, the first shell coordination number is that of metallic copper (12). This value is strictly reached in the case of the infinite solid. In contrast, the second shell coordination number is smaller than the corresponding bulk one (4.3 instead of 6J and corresponds (see Fig. 6b) to fcc clusters on average r = 8.5 A (about 250 atoms, 12 shell~). This discrepancy in the cluster size estimation has several origins: on the one hand, the presence of missing atoms at the cluster surface. Indeed, the second shell coordination number is more sensitive to vacancies at the cluster surface than the first one. On the other hand, the presence of particles with different sizes which contribute to the mean EXAFS experimental values. Distortions from fcc spherical symn~etry are less probable, because they imply also a change in the interatomic distances. Anyway, we must keep in mind that the coordination numbers obtained from EXAFS analysis are highly correlated to the mean free path and the Debye-Waller term in the EXAFS equation, which includes dynamic and static disorder.

Our EXAFS analysis allows us to assess that the ~.opper cluster radius on t.b~. oxidised alumina is very close to 5 ~, whereas it is at least 85 ~ for the case of the reduced substrate. Therefore, taking into account that the same amount nf Cu atoms was deposited on both surfaces, the di~.erence in the cluster nucleation is obviously related with the a-Al203 surface degree of oxidation. Indeed, Kasowsky and co-workers have found, by means of UPS, that Cu atoms bond preferentially to surface oxygen atoms instead of AI atoms [5]. So, having less oxygen atoms available for chemical bonds on the reduced substrate, copper atoms begin to interact between each other earlier than on an oxidised substrate, giving rise to bigger clusters.

In previous work, Gautier et al. [10,11] have studied the evolution of the Cu Auger parameter for

Fig. 1 Fig. 1 .

 11 Fig. 1 shows a comparison between the normalised Cu K edge absorption spectra ~(E) (XAHES region) of: (a) A clean metallic copper foil (total yield detection). (b) 80% ML of copper deposited on a reduced a-AI203(0001) surface. (c) 80% ML of copper deposited on an oxidised a-A1203(0001) surface. (d) A Cu20 powder (transmission mode detection).This spectra has been kindly provided by H. Tolentiao from Ref.[24], Both the l(b) and l(c) XANES spectra, cone-

Fig. 3 Fig. 2 .Fig. 3 .

 323 photon energy (eV) Fig.2. Normalised raw )(.-ray absorption spectra for (a) 80% ML of copper deposited on a reduced A1203(0001) surface and (b) 80% ML of copper deposited on an oxidised A1203(0001) surface. Both spectra have been measured in fluorescence mode at normal incidence. The energy range is 8950-9650 eV and the energy step 2 eV. The continuous line is a cubic spline curve fit of the raw data. A linear background contribution from previous edges has been subtracted.

Fig, 4 ,

 4 Fig,4, Best least-squares fits obtained for the structural parameters given in TableI. Open circles represent experimental signal and full line the simulation using the amplitude and phase shifts tabulated by McKa~e et al. For isolating the contribution of the first and second shells in the RDF, back Fourier transformation was applied between 1.6-3.0 ~ and 3.0-3.74 ~k, respectively, in the RDF of Fig.3. The left of the figure shows the fits corresponding to 80% ML of copper deposited on a reduced A1203(0001) surface and on the right of the figure those corresponding to 80% ML of copper deposited on an oxidised A1203(000!) surface.

Fig. 5 .

 5 Fig.5. Calculated XANES spectra for Cu fcc clusters with different sizes: I shell; 2 shells; 3 shells; 4 shells. The number of atoms and the radius of each cluster are indicated above the corresponding curve. Calculations are performed applying the real-space cluster method and include all the MS contributions. The complex Hedin-Lundqvist potential was used.

Fig. 6 .

 6 Fig. 6. Calculated coordination numbers of (a) the first shell and (b) the second shell for cluster radius from 3.6 to 18 ~,. Our experimental values are plotted together with their bar errors.

Table 1

 1 Structural parameters giving the best least-squares fits for 80% ML of copper deposited on (a) a reduced and (b) an oxidised (0001) A]203 substrate, where d o_ce represents the mean distance, N the average number of atoms and Atr 2 the mean square relative displacement of each shell

			dcu_c u					N (± 10%)				AO.2 (~2)
			(4-0.02 ,A)I s* (+0.05 ,g,)2 "d								(4-0.5 X 10 -3)
	(a) 80% ML of Cu on reduced A1203(0001)
	McKale phase shifts
	P' shell		2.54						11.6					0.012
	2 nd shell		3.61						4.6					0.~6
	Cu fcc reference phase shifts
	1 't shell		2.54						12.~					0.012
	2 nd shell		3.604					4.0					0.005
	(b) 80% ML of Cu on oxidised A1203(0001)
	McKale phase shifts
	1 u shell		2.525					8.0					0.~7
	2 "d shell		3.64						2.7					0.~6
	Cu fcc reference phase shifts
	1" shell		2.525					8.20					0.006
	2 ad shell		3.64						3.0					0.~6
						m m.c~c~..s.,,.a '	.%				SO ML Cu/Al:O s oskl~4d
									•					I It e, oord. ~ll
														t
									•						o
	0,~
	....	~	....	i,,i,,-.,	-z	....	J	....	....	i	....	I	....	I..~lil..*l	....
	j 0,~ J

Acknowledgements

We are grateful to D. Chandesris, D. Ragonnet and H. Magnan (LURE, Orsay) for their help and assistance during the X-ray absorption experiments. We acknowledge H. Tolentino and A. Fontaine for providing us the Cu20 X-ray absorption spectra. We thank C.R. Natoli from INFN (Istituto Nazionali di Fisica Nucleare, Frascati, Italy) for providing the full multiple scattering MXAS code. We also thank F.M.F. de Groot for useful discussions and advice.

Reterences

two quite different ot-gl203 surfaces: a stoichiometric (1 × 1) one and a reconstructed 3¢r~ -× 3~R ± 9 ° one, this later being in a highly reduced state [35]. For the (1 × 1) surface, the A cu parameter reaches the metal value for an equivalent deposited coverage of 2 ML, whereas for the ¢3i-× V'31 R + 9 ° reduced surface, the Cu(O) oxidation state is ~'eached for a much lower coverage of 0.5 ML. This result is interpreted by a difference in the cluster size in both surfaces, with a larger size on the reduced surface.

Conclusions

Summarising the XANES results on the cu/gl203 interface, we can infer that it grows predominantly via the Volmer-Wc, ber mode. The formation of metalli-" particles from the very early stages of the deposition is caused by the weak interaction between Cu and the AI203 surface. The Stranski-Krastanov growth mode propo~,ed by other authors in the literature can be ruled out under our experimental conditions. We clearly show that the cluster size is larger on a reduced A1203(0001) surface than on an oxidised one by u~ing a self-consistent data analysis. This is caused by the predilection of Cu atoms for bonding to O atoms rather to Al atoms.


Anna Aniszewska-Stępień

Romain Hérault

Guillaume Hacques

Ludovic Seifert

Gilles Gasso

Learning from partially labeled sequences for behavioral signal annotation

Keywords: partially labeled data, hidden Markov model, constrained Viterbi algorithm, behavioral signal labeling, climbing pattern discovery

Herewith, we present a learning procedure that allows to deal with a partially labeled sequence dataset, i.e. when each sequence in the train dataset may contain labeled as well as unlabeled chunks. In our application case, this occurs when motor activity has been manually annotated (due to the recognition based on the video recording) and independently registered by the measuring system of high precision (touch sensors): human annotation misses some events that have been captured by the sensors. In the general setting, we aim at predicting the labels for a new fully unlabeled movement sequence, while the training has been performed on the partially labeled dataset. For this purpose we propose to use classical sequence model (hidden Markov model) that is furnished with a constrained Viterbi algorithm, which gives us a quick access to the hard approximation of the correct labeling sequences. We demonstrate, that this simple modification that constrained Viterbi provide, allows the HMM model to be trained on sparse data, and overall results in surprisingly high log-likelihood and accuracy level in annotating the partially labeled behavioral sequences in climbing. The same time we show the way to access correct labeling of the unannotated signal that can be helpful in various sport science studies for movement pattern sequential prediction.

Introduction

In the real word, we very rarely posses the full information about the environment. For a machine learning prediction task, in the ideal setting, we shall dispose reliable and fully described data to train the model for pattern prediction in the newly observed data set; whereas this condition is hardly ever met in practice. Even for the experimental training data set, there can be missing information due to the errors or gaps in annotation, especially if the large information set is labeled by human and/or the labels are not trivially accessed. Depending on the type of data (discrete or continuous) and the suspected distributions of labels, there can be different approaches to solve this problem. In this type of tasks (e.g. behavioral signal annotation, entity recognition, protein structure recognition), we want to label the observation sequences with a respective sequences that consist of the finite set of labels. Furthermore, the provided training sequences have some labels missing, while the missing ratio vary from sample to sample. Thus, all our training sequences may be partially labeled, but we still need to build a model that fully annotates the validation/test unlabeled sequences.

Proposed approach In the following article we will study a generative model that has been usually used in the context of sequential learning [START_REF] Dietterich | Machine learning for sequential data: A review[END_REF] adjusted to learn on sparsely labeled sequences. Namely, we will construct the procedure for the hidden Markov model to be learned from the sequences with voids in annotation, thanks to the Viterbi algorithm, that is modified in a way to find the optimal path passing by few constrained, known states. The proposed algorithm learns the distribution of the states with hidden Markov model and uses constrained Viterbi algorithm to predict the missing labels in the sequences; iteration over these two steps until convergence, emerges the stable sequence labeling close to the true one, which has been evaluated herewith in terms of log-likelihood and similarity (accuracy) ratio. We will show, how this relatively simple approach, resolves complex task, that is frequently encountered when we dispose only very little knowledge about the observation states. Our analysis aims at proving the efficiency of the proposed approach within the task of labeling the observation behavioral signal for which we never access the full information about the proper labels (not even for the training data sample). This situation is potentially frequently present in sport, when the access to the whole range of annotation is difficult and expensive (in terms of time and expert knowledge), while we need to draw generalized knowledge about the patterns present in large data sets, which as in our case, are registered time series. The motor task that produced the signal of our interest is the relative climber's position on the wall in the moment of touching the hold. The state (label) that we would like to find, is the kind of limb used for the support. The so obtained designation is crucial to find climbers profiles in the skill acquisition patterns discovery [START_REF] Boulanger | Automatic sensor-based detection and classification of climbing activities[END_REF]. We suppose however, that the proposed approach in the same way may be applied to design the patterns of subjects' strategy during any other motor task.

Article structure The paper is organized as follows. After introducing he state-ofthe-art (Section 2), we illustrate the context in which the model has been tested and we provide the characteristics of the model (Section 3). Then, in Section 4, we detail the model and the adjustments for partial learning. The experiments and evaluation methods for the results are defined in Section 5. Next, we specify the type of evaluation used and finally we interpret the results and conclude (Section 6). example is either fully labeled or fully unlabeled [START_REF] Chapelle | Semi-supervised Learning[END_REF]. The techniques that are used for semi-supervised sequential learning (within the broader area of structural learning) are multiple and deeply studied (mostly in the framework of NLP) over a number of recent years (e.g. deep learning models with pre-training word vectors [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] or self-training [START_REF] Clark | Semi-supervised sequence modeling with cross-view training[END_REF], [START_REF] Dai | Semi-supervised sequence learning[END_REF], co-training [START_REF] Blum | Combining labeled andunlabeled data with co-training[END_REF], conditional random fields [START_REF] Veeramachaneni | A simple semi-supervised algorithm for named entity recognition[END_REF] and many other). We would like to however insist in the distinction of the above mentioned situation from the one we study in the present article: we here address observable sequences which are incompletely annotated, in which the presence of gaps and voids within each sequence reduce the labelling ratio (that is 1 for fully labelled sequence). Notwithstanding thus obtained partially labeled data sets are omnipresent, the situation of scarce access to the experimental data for sequential learning, has not yet been extensively explored in the literature.

One known discriminative approach to address this challenge has been proposed by Fernandes and Brefeld [START_REF] Fernandes | Learning from Partially Annotated Sequences[END_REF]. They used structured perceptron furnished with Viterbi algorithm to cope with the non-labeled parts of the sequences. This allowed them to train the perceptron on only the labelled parts of the sequences. Thus obtained model (simple transductive loss-augmented perceptron, STLAP), in spite of its simplicity, results in proper annotation for the initial labeling ratio larger than 0.5, which is comparable with standard supervised or semi-supervised methods. Also, in [START_REF] Fernandes | Using Wikipedia for crosslanguage named entity recognition[END_REF] the hidden Markov model with Baum-Welch algorithm has been compared by the same authors to the above mentioned STLAP in the partial training framework (see also [START_REF] Scheffer | Active hidden Markov models for information extraction[END_REF]). The distinction with our approach however is that in our contribution we apply constrained Viterbi to even simpler model (hidden Markov model), for which we claim very good label annotation level. A separate branch of research employs conditional random fields, for example in Li et al. [START_REF] Li | Large scale sequential learning from partially labeled data[END_REF] different learning models are trained and subsequently combined for large scale sequential learning by adapting the idea of ensemble training. The result obtained on some types of partially labeled data were comparable to basic conditional random field approach, however the concept of many learning models may not be applicable to some type of data, especially when the transition between the tokens in the sequence is crucial as in our case. Conditional random fields combined with deep learning has been also recently studied in [START_REF] Zhou | Learning with annotation of various degrees[END_REF] in order to learn with annotation of various degrees (unsupervised, fully supervised and partially labeled sequences) in a unified way. Surprisingly, almost all the above mentioned papers claimed that the outcomes received for the partially labeled training data result in comparable performance or even outperform the ones received for the fully labeled data set (as they may avoid overfitting). Even though we do not study this property in the present article, some detailed studies of the phenomenon have been presented in [START_REF] Ning | Partial or complete, that's the question[END_REF] and [START_REF] Zhi | Partially-typed NER datasets integration: Connecting practice to theory[END_REF]. Due to this fact and facing costly annotations of partially labeled data, there is a growing number of publications in the field, as for example the recent experiments in combining contradictory partial annotations from different datasets conduced by Huang et al. [START_REF] Huang | Learning a unified named entity tagger from multiple partially annotated corpora for efficient adaptation[END_REF]. Additionally, and given that the above models have been tested the data only empirically, the required statistical consistency justification for the published discriminative models used for structural learning has been provided by Antoniuk et al. in [START_REF] Antoniuk | Consistency of structured output learning with missing labels[END_REF].

Application context

The partially labeled learning has been studied mostly for the linguistic tasks (entity recognition, natural language processing) or biological sequence analysis (protein structure, DNA sequences). Our application in sport science requires to work on the behavioral signal that is difficult to be systematically processed due to its innate complexity and its spatio-temporal continuous nature. For a sport science application later purpose (climbers' patterns discovery, as in [START_REF] Boulanger | Automatic sensor-based detection and classification of climbing activities[END_REF]), we study the following case. The climbing data (mass center position combined with handhold touch time of the climber while moving up the artificial wall, Fig. 1(a)) is recorded by a camera (for the mass center) and, separately, by the wall sensors (for hold touch time). Hold touches that are visible on the video recording have been manually annotated by which kind of foot or hand (left hand, right hand, left foot, right foot) performs the action. Naturally, so obtained labeling contains a lot of voids as part of the events recorded by the hold sensors are not present or hidden in the video used by human for annotating. Meanwhile, the manual annotation has been rendered only for a subset of the recordings. Since human annotation is very costly (it requires expert knowledge and is time consuming) and at the same time -as being based on the visual attribution -it is not precise, hence we face two challenges simultaneously: 1) the labeling does not cover the whole set of experimentally collected recordings and 2) annotations do carry the mistakes or gaps in the describing sequences. Thus, dealing with these both difficulties (as well as with possible noise) is not a trivial task. One another difficulty we encounter when annotating only subset of the recordings, is the supposed difference in the distributions between the annotated and the unannotated set. Due to the climbing learning protocol evolution, the sequences were not stationary, Fig. 2. The protocol (as explained in [START_REF] Hacques | Climbers' learning dynamics : an exploratory study[END_REF]) consisted in thirteen climbing sessions: three test sessions and ten training sessions. The test sessions aimed at scanning the behavioral repertoire of the participants. More specifically, to assess to what extent the participants could perform hand alternations (i.e., they used the two different hands on two subsequent movements) and hand repetitions (i.e., they used the same hand on two subsequent movements). In this purpose, they had to climb three different routes that either encouraged to perform alternations, repetitions or both coordination patterns (i.e., the Neutral route). Additionally on the test sessions, participants climbed the three routes in three different instruction conditions: a free condition (to observe the spontaneous behavior), an alternation condition (i.e., they were invited to perform as much alternations as they could on the routes) and a repetition condition (i.e., they were invited to perform as much repetitions as they could). Then, on the training sessions, participants climbed on more sophisticated routes where they were instructed to use all the handholds in a bottom-up order and to climb as fluently as possible, that is, avoiding saccades and stops during the ascents. That way, they had to search for the most adapted chain of movements for the route. According to their training group, participants climbed one to ten different routes during the training sessions. All groups had one route in common that they climbed at least three times per session (i.e., the Control route). Finally, the training protocol started (i.e., first ascent of the first training session) and ended (i.e., last ascent of the tenth training session) with one trial on a Transfer route for which they had no additional practice during the training sessions. This route was designed to assess whether the participants were able to find adapted chain of movements on new routes with training.

In many cases we dispose large registered data sets that can be labelled manually based on observations. Therefore in the whole process human mistakes in the labeling and the presence of artifacts in the recordings are inevitableboth error sources can prompt missing values in the labeling sequences. We would like to use the subset of partially labeled observations (recording of handhold touch time combined with the mass centre position, see Fig. 1 (b)) to predict the labeling (i.e left or right, hand or foot) of the whole dataset. As the data set consists of the pair sequences (observations and incomplete labels), thus we will apply the sequential learning with hidden Markov model. The challenging part is to manage the missing parts of the sequences and learn from the annotated parts to fill up the gaps with constrained Viterbi algorithm. Our task is similar to entity recognition with the average labeling ratio about one third of the sequence.

Methods

HMM Viterbi training on fully labeled sequences

The sequential learning task [START_REF] Dietterich | Machine learning for sequential data: A review[END_REF] is to find the relation between the observation sequence o = (o 1 , o 2 , ..., o T ) and its pair label sequence l = (l 1 , l 2 , ..., l T ), where each l t ∈ {L 1 , L 2 , ..., L n } = L (assuming the sequences o are some continuous trains of measurements of length T and labels are the limbs: LH, RH, LF, RF). For HMM supervised learning framework [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications on speech recognition[END_REF], with fully labeled pairs (o, l), this relationship is described with a generative model m. It is subsequently applied, with the use of Viterbi algorithm [START_REF] Viterbi | Error bounds for convolutional codes and an asymptotically optimum decoding algorithm[END_REF], on new sequences of observations in order to find most likely corresponding label sequences (Fig. 3(a)). We notify, that by applying the Viterbi algorithm we obtain hard label assignation, whereas by using Baum-Welsh algorithm we would get probabilistic one. The former hard assignation is necessary for the next step of the processing in sport science: climb profiling by clustering (not presented here). Seemingly, there is a relation within each pair (o t , l t ) all along the sequences, which we describe with probability of observation emission e s (o t ) = P (o t |l t = s). In case of a Markov process, there is no dependence between non-adjacent labels, whereas, there is one between the adjacent labels in the sequence l, that is described with the probability of transition f rs = P (l t+1 = r|l t = s). Both mentioned types of conditional probabilities, that in case of sequences apply to every element of sequences o and l, in form of matrices E = (e s (o t )) s∈L,1≤t≤T and F = (f rs ) r,s∈L , as well as initial probabilities p s = P (l 1 = s), adjusted to data pairs (o, l) are the parameters of our model m = (p, E, F ). We find the optimal parameters by maximizing the joint probability P (o, l). Once the model parameters are found, we predict new labels for unlabeled observations o with Viterbi algorithm. The algorithm, starting from p and propagating through the observation sequence o, stores the probabilities of most likely path of labels l that generated o and same time, the most likely label sequence. The resulting optimal l is the sequence of argmax, once we have found all the probabilities. At first, in the framework of hidden Markov model, we calculate the joint probability for the sequence of observations o and the sequence of labels l as

P (o, l) = p s T t=1 P (o t |l t ) T -1 t=1 P (l t+1 |l t ) (1
Subsequently, with the Viterbi algorithm, the parameters, in a recursive manner and through T steps, serve to compute the maximum-likelihood label sequence l given the observation sequence o .

The recurrence is used to track the v and w intermittent variables:

v 0,s = p s (3) v t,s = max r v t-1,r f rs e s (o t ) (4)
w t,s = arg max r v t-1,r f rs (5)
and that for the last estimate label element

lT = arg max r v T,r (6)
by backtracking we find the whole estimate sequence lt = w t+1, lt+1 .

Thus, the ultimate sequence l is the optimal sequence of labels. The optimality of the Viterbi algorithm can be shown as in [START_REF] Omura | On the Viterbi decoding algorithm[END_REF].

Our contribution: HMM Viterbi training on partially labeled sequences

In case of partially labeled sequences in training set, we must adapt the basic supervised HMM procedure in order to train only from the labeled chunks of the sequences. For this purpose we will use the constrained Viterbi algorithm [START_REF] Cao | A novel product coding and recurrent alternate decoding scheme for image transmission over noisy channels[END_REF], [START_REF] Fernandes | Learning from Partially Annotated Sequences[END_REF]. Firstly, we can notify that for unsupervised learning (when all the observations are unlabeled), we can randomly initialize the model parameters m and use the Expectation-Maximization algorithm to find both the m and the sequence of labels in fully unsupervised framework. The herewith technique combines the supervised and unsupervised learning so that initially and in each EM iteration, we take advantage of the labeled chunks of the sequences and constrain the unsupervised scenario. Namely, we initialize the model with random parameters or with pre-training held on only labeled chunks of the sequences (l t , o t ), for which we get interim parameters m. In the next step, we find the interim labels l for the partially labeled sequence (Fig. 4) with the use of constrained Viterbi, that is enforced to move through the already known label chunks, by maximizing the probability of the most likely label path. Indeed, in the constrained scheme, the traditional Viterbi algorithm operates on the unlabeled segments of sequences, but with the first and the last states labeled (which are the fixed adjacent tokens), treating each chunk sequence independently (Fig. 3(b) ). Subsequently, we use the fully labeled sequences l to generate new model parameters m and further adjust the labeling on the initially unlabeled parts of the sequences. While EM finds the local optimum, we iterate until convergence to find the best representation in terms of log-likelihood of the label sequence joint probability (Algorithm 1).

Algorithm 1 HMM with constrained Viterbi

1: Initialization: train model m with only fully labeled chunks of sequences (o, l) 2: repeat 3:

Step 1 (E): complete the gaps in the sequences of labels with constrained Viterbi algorithm (parameters calculated with model m), to get fully labeled estimates (o, l) 4:

Step 2 (M): update model m due to new labels (o, l) 5: until End condition: iteration number 5 Results

Experimental setting

We test the model on the following data:

1. synthetic data set from known model parameters where some annotations are randomly discarded, 2. experimental climbing data -chunks of full sequences with some annotations randomly discarded, 3. experimental climbing data -original partially labelled sequences.

For the synthetic data generation we used the Gaussian bi-variate distribution for the four label states. All the four states have the same covariance S, they differ on their mean m. For each of the two dimension of the mean, we used either -µ or µ, leading to 4 possibles states as shown in Equation 8:

m 1 = µ µ , m 2 = µ -µ , m 3 = -µ µ , m 4 = -µ -µ , S = σI. (8)
So defined synthetic data distribution was to approximate the statistics of the climbing data. To apply random label discarding in fully labeled sequences, the level of gap ratio has been fixed (0.25). For the experimental sport data set (points 2 and 3), we dispose the 497 partially annotated sequences of observations (simple sessions), which are divided into train and test sets. The labeling with four types of limbs, reaches the annotation level of about 0.3 for the sequences of the average length of 20 tokens. Finally, the established model will be used for the annotation of overall number of about 3700 sequences recorded with touch sensors. In order to evaluate model performance in case of the first two data sets (where ground true labeling has been initially known), we calculated accuracy measure which is similarity ratio (averaged sum over whole sequence, scoring 0 for mismatch and 1 for match on each token in the sequence) employed on the fully labeled initial sequence l (before label removal) vs the sequence estimated by the model l. Hence, the evaluation score for a whole sequence l and its prediction l reads

d(l, l) = 1 T T t=1 d t (l t , lt ) (9)
with the loss function d t defined over two tokens l t and lt as

d t (l t , lt ) = 1 for l t = lt , 0 otherwise. ( 10
)
To test the hidden Markov model with constrained Viterbi we used synthetically generated data with known parameters. The missing labels was to best resemble the true annotation present in real data (the gap ratio of about 0.3). We compared the predicted model parameters with the original counterpart as well as the scoring in two sequences: label sequence generated with original model and label sequence estimated with predicted model parameters. The same way of evaluation has been employed for experimental data with full annotations. In these cases we are able to compare the resulting fully labelled sequences with the initial true labels, before label removal. For all the dataset however, the likelihood of the observation sequences knowing the model parameters is always accessible whether the true labels are fully known, partially known or totally unknown. To sum-up, in case of fully labelled sequences altered artificially to partially labelled ones (dataset 1 and 2), we can compute a similarity ratio comparing the estimated labels to the true ones. Otherwise (dataset 3), only the likelihood of the HMM is accessible as an evaluation measure.

Evaluation

Synthetic data set As explained in point 1, firstly we have tested the model on artificially generated data (with either random initialization or pre-trained on labeled chunks of sequences). The number of sequences in trained and test sets was 1000, the average length of sequences was about 20 tokens, with label gap ratio equal to 0.25. We analyse the log-likelihood over the joint probability of observation and labeling sequences that evolves across the iterations for label distribution with mean 1.5 and covariance 0.2 (Fig. 5). This value determines how well the label sequence describes their pair observation sequence, based on the label distribution. As expected, we observe monotonic increase of log-likelihood over the iterations, as well as for the similarity score computed for unannotated chunks. The stability is reached after 2nd iteration. The log-likelihood evolution proves that the model fits well with the experimental data and at the same time leads to the generation of optimal labels. Likewise, score analysis confirm increasing recognition of labels if compared to the ground true annotations in the subsequent steps. Climbing data set We first explore the real experimental dataset with artificial deletion of the annotations (with an alteration rate of 0.3). On the artificially deleted slots, we do have knowledge of the true labeling, thus we can compute the similarity ratio. After alternation, first, we pre-trained the HMM on the fully labeled chunks of sequences, bypassing the voids. Then, the HMM was trained using the constrained Viterbi in the manner exposed in the Methods 4, with all the partially labelled sequences. In the Figure 6(a), we observe learning through iterations (growth of the log-likelihood), and at the same time -the annotations of the labels are mostly correct (the maximum similarity ratio is over 0.9). Based on this fact, we studied the case of partially labeled original sequences registered by touch sensors (Point 3). Similarly, for experimental sport data we observe monotonic growth of the log-probability function, which is -as in synthetic data case and the artificially discarded labelings in sport data -depicting learning across the iterations. While the ratio of unannotated labels in all three experimental data cases was about 0.3, we may expect that the results are comparable. Like in the previous data sets, in the original partially labeled sport data, the model log-likelihood converges also very fast, within two iterations (Fig. 6(b)). In this case, unlike the previous two cases, since the true labels remain unknown, the similarity ratio is not accessible. The end-values of log-likelihood function however, are lower than in the artificially label discarded sport data case. We may suppose that this fact could have an impact on the quality of the label prediction. Although we do not dispose any other objective evaluation method, the convergence of the model may be a proof of leading towards the correct solution. We assume that the mislabelling, if applicable, might have occurred in case the artifacts appear in the collected data (e.g. when the sensor registered a touch made not by the hand or foot).

To complete the analysis, we pursued the machine learning evaluation method in which for synthetic data train set to test set ratio was 1000:1000 and for the sport data annotated sequences we split the data set with the ratio 400:97. Namely, we verified the accuracy of the model trained on one subset of partially labeled data (train set), when applied on the previously unused subset of data with no labeling (test set). For this setting, the similarity score evaluated on the new test sequences were not substantially different from the previously observed training evaluation (Tab.1), and confirmed correct label attribution.

Data set training testing synthetic 0.99 0.99 climbing 0.91 0.90 Table 1. Similarity score for synthetic data set vs climbing data set.

Conclusion and perspective

In the present article, we have been studying how the generative model (HMM) with a simple adjustment (constrained Viterbi) deals with the learning the partial labelling in the training sequences. It is a predictive model, that allows to access the distribution of the limbs used while climbing the artificial wall. We tested the approach on the real climbing data as well as on the synthetic data, that was alike. We claim that, based on the log-likelihood value, which specifies how good is the description that we find for the measured data, due to our experiments we can judge on model performance. We must admit however, that the additional challenge we face in our application is the discrete set of states (labels). This situation renders difficult to measure the effective labelling ratio based only on log-likelihood value. We have compared the log-likelihood ratio with similarity score computed for unannotated chunks in cases, when the full labelling has been known in order to assert patterns of likelihood function. We observed that it proves apt attribution of labels in case of the data set, when the prior full labelling is not accessible, as in the sport data set. We have demonstrated, that the log-likelihood matches scoring and can be used for model evaluation in case of training on only partially labeled data sets.

Perspective We can show experimentally that the log-likelihood convergence with a general tendency of monotonic increase (and not decrease) approximates correct label attribution. In a broader setting though, HMM with constrained Viterbi may not be a perfect label predictor for the continuous behavioral signal, but is a simple way to approximate labelling of unknown sequence, with scarce prior knowledge about training sample. Therefore, to outperform our algorithm, some more sophisticated method, as for instance neural networks would have to be applied. Likewise, the convergence of optimality function for the leading model (not only for the Viterbi constrained, but its implementation into the framework of EM-like full Algorithm 1), requires more systematic research and theoretically correct proof. Although the model is simplistic and its output may not be faultless, it is still sufficient for the first approximation of the behavioral signal labelling in sport application and it can be the baseline for the further research on the subject, that could be tested in the contexts of different paradigms in human movement science.

Fig. 1 .

 1 Fig. 1. (a) Experimental setting for climbing of the artificial wall (the mass-center trajectory of the climber is marked with the yellow line). The signal is composed by the time train of the distance between the climber's hip and limb position, when touching the hold. Different types of climbing routes and training paradigms have been used all along the data collection protocol. (b) Position of climber's body center collapsed to 1D by summation (x + y) -evolution in time. (c) Observation signal as the relative position of the body centre vs the limb that is touching the hold (collapsed to 1D by summation).

Fig. 2 .

 2 Fig. 2. The protocol of data collection with session profile (number of ascends in each session is provided in the box). The PREtest, POSTtest and RETention sessions (yellow) stand for the partially labeled sequences in our training dataset. They consist of simple routes (Neutral, Alternation and Repetition type); Transfer route opens and closes the Training Session. Each of the 10 Training Sessions (the unannotated data set, for which we predict the labels with our model) starts with the Control Route.

  ) and we estimate the parameters of model m by maximizing log-likelihood m = arg max m n i=1 log P (o, l|m).

Fig. 3 .Fig. 4 .

 34 Fig. 3. (a) Standard Viterbi algorithm.The magenta path is the optimal one, ending with the maximum v value (magenta circle). All the possible paths (in cyan) are explored. At each tracking/forward step the most likely path to each state is recorded (dark magenta). At the end, only the path that leads to the most likely ending state is retained (magenta). (b) Constrained Viterbi algorithm. Here, only the paths that contain the first and last pre-fixed states are explored during the forward stage.

Fig. 5 .

 5 Fig. 5. Synthetic data set (generated with mean parameter µ = 1.5 and covariance parameter σ = 0.2): log-likelihood value evolution (left) and similarity ratio (right). The labels have been discarded with ratio 0.25. The applied initialization was either random (blue) or pre-trained (orange) on the labeled chunks.

Fig. 6 .

 6 Fig. 6. Sport data set. (a) Sequences originally fully labeled but with artificial label deletion: log-likelihood evolution (left) and similarity ratio (right). In order to apply the model, the labels have been synthetically discarded with ratio of about 0.25. (b) original data, partially labeled with gap ratio about 0.3: log-likelihood evolution.

Acknowledgement

This work was supported by a grant from the French National Agency of Research (reference: ANR-17-CE38-0006 DynACEV).