Raquel Medeiros Vinci

Frédéric Mestdagh

Christof Van Poucke

Carlos Van Peteghem

Bruno De Meulenaer
email: bruno.demeulenaer@ugent.be

Quenten Denon

A two year investigation towards an effective quality control of incoming potatoes as an acrylamide mitigation strategy in French fries

Keywords: Potatoes, Processed foods Acrylamide, Incoming potatoes, Quality control, Colour, Reducing sugars

A two year investigation towards an effective quality control of incoming potatoes as an acrylamide mitigation

The current entrance quality control used in the French fries industry is done based on colour evaluation with USDA/Munsell colour chart (after a short frying test, typically 180°C for 3 min). Based on a study carried out during 2 consecutive potato storage seasons the possibility of a more effective entrance control of the raw potato tubers in order to identify batches of potatoes prone to acrylamide formation was evaluated. The current entrance control was compared to two other colour evaluation methods (CIE L*a*b* colour parameters and a process-specific Agtron analyzer) and to reducing sugar contents determination. Seasonal variability did not affect the slopes of the linear correlation models, for most of the parameters studied. The determination of colour formation measured by the Agtron methodology and reducing sugars content allowed a better identification of batches of potatoes prone to acrylamide formation when compared to the current entrance control. Different scenarios represented by decision trees for quality control measures for incoming potatoes were evaluated while considering the investigation value of 600 Acrylamide is formed in starchy foods processed at high temperatures (> 120°C) commonly 2 applied in frying, baking and roasting [START_REF] Tareke | Analysis of 341 acrylamide, a carcinogen formed in heated foodstuffs[END_REF]. This contaminant is mainly formed 3 through Maillard reactions between reducing sugars and specific amino acids such as asparagine 4 [START_REF] Stadler | Acrylamide from Maillard reaction products[END_REF][START_REF] Mottram | Acrylamide is formed in the Maillard 332 reaction[END_REF][START_REF] Becalski | Acrylamide in foods: occurrence, 298 sources, and modeling[END_REF]. Potato products are strongly 5 susceptible to acrylamide formation since this food commodity contains asparagine and reducing 6 sugars. Because, the amount of free asparagine in potato tubers generally largely exceeds the 7 amount of reducing sugars; it is the latter which determines the degree of acrylamide formation 8 and Maillard browning upon frying [START_REF] Amrein | Potential of acrylamide 293 formation, sugars, and free asparagine in potatoes: A comparison of cultivars and farming 294 systems[END_REF][START_REF] De Wilde | 310 Influence of storage practices on acrylamide formation during potato frying[END_REF]. Since the in potato products [START_REF] Amrein | Potential of acrylamide 293 formation, sugars, and free asparagine in potatoes: A comparison of cultivars and farming 294 systems[END_REF][START_REF] Becalski | Acrylamide in French fries: influence of free amino acids and sugars[END_REF][START_REF] Elmore | Changes in free amino acids and sugars in potatoes due to sulfate fertilization and the 321 effcet on acrylamide formation[END_REF][START_REF] Rommens | Low-acrylamide French 334 fries and potato chips[END_REF] [START_REF] De Wilde | 310 Influence of storage practices on acrylamide formation during potato frying[END_REF]. 1), the correlation models obtained for both 152 cultivar seasons needed to be compared in order to evaluate the feasibility of using an entrance 153 control parameter as an acrylamide mitigation strategy. Accordingly, slopes of linear models 154 obtained for both seasons were compared statistically for significant differences by means of a t-155 test with 95% confidence level (Table 2). The t values were calculated based on the difference 156 between the two slopes divided by the standard error of the difference between slopes. Slopes of 157 the correlation models obtained between entrance parameters and acrylamide in the final product 158 were not significantly different between both seasons with an exception of the models obtained 159 with the a* parameter (Table 2). Models obtained between colour parameters after two stage 160 frying and acrylamide in the final product were not significantly different either. On the other 161 hand, significant differences were observed between slopes of the models regarding parameters 162 evaluated in the final product (reducing sugars after blanching and colour determination after 2) and THVs were determined for each relevant parameter on basis of 182 these combined datasets. In addition, the application of these THVs to the combined data-set 183 resulted in a preliminary analysis of sample categorization based on their predicted and actual 184 acrylamide levels in the final product (Table 3). For interpretation of the results, batches of 185 potatoes with acrylamide contents < 600 µg kg -1 are considered as "acceptable", while for 186 acrylamide contents > 600 µg kg -1 potato batches are considered as "unacceptable". In practice, 3). Worth noting is the fact that 56% of those rejected samples correspond to 198 samples "unnecessarily rejected", since the actual acrylamide content of the final product is lower 199 then the EU indicative level. Measuring colour with Agtron at entrance reduces the number of 200 rejected samples to 16% and in addition the number of "unnecessarily rejected" samples is 201 reduced to 40%. The determination of reducing sugars at entrance allows even a further reduction 202 of rejected samples to 12%, out of which, 37% are rejected "unnecessarily". This implies that 4, 6 203 and 11 % of the samples would be unnecessarily rejected using the THV's for reducing sugars,

204

Agtron and USDA, respectively. Therefore the application of reducing sugars and Agtron could 205 represent a significant decrease in unnecessary costs that industry would have treating these 206 "unacceptable" potato lots, compared to a THV system based on the currently used USDA 207 measurement.

209

Another important aspect to consider when selecting a quality control parameter regarding 210 incoming potatoes, and when evaluating the THVs application is the percentage of non-211 conformities, such as the percentage of accepted samples with acrylamide >600 µg kg -1 . . These 212 percentages ranged from 1 to 3% for the entrance parameters (fried at 180°C, 3 min) (Table 3).

213

The lowest percentage (1%) was obtained with the application of the THV for Agtron. The combination Agtron/RS as an entrance control shows no improvements towards acrylamide 243 prediction when compared to the application of RS by itself. The application of the USDA/RS 244 combination as entrance control further decreased the percentage of rejected samples to 11%, 245 although no improvements were found for the total "unnecessarily rejected" samples (4%).

246

Furthermore, the application of the latest combination as entrance control resulted in an increase 247 of accepted samples with acrylamide >600 µg kg -1 (4%). The combination USDA/Agtron/RS was 248 also tested and same results as USDA/RS were obtained.

250

It should be kept into account that the THV for quality control tools mentioned in this study are 251 directly related to the hypothetical acrylamide level of 600 µg kg -1 and thus are restricted to this 252 acrylamide limit, blanching and frying conditions used in this study. Moreover, dextrose and 253 sodium acid pyrophosphate which are typically added in industrial practices were not taken into 254 account in this correlation study. However, obtained results demonstrate that even though 255 seasonal variability was observed for the potato varieties used for this study, similar models were 256 obtained for both potato storage seasons. Consequently the determination of THVs as an entrance 257 control tool seems feasible, for standardized frying conditions. Moreover, from a food safety 258 point of view, a more optimal quality control could be adopted for incoming potatoes in order to

9

  discovery of acrylamide in foods, stakeholders have promptly supported the scientific community 10 in finding the mechanisms of acrylamide formation and possible mitigation strategies from farm-11 to-fork. Food Drink Europe (CIAA/FDE) established a Technical Acrylamide Expert Group since 12 2003 and created the "Acrylamide Toolbox" which is available online 13 (www.fooddrinkeurope.eu). This toolbox presents a regularly updated description of the possible 14 mitigation strategies that could be employed to reduce acrylamide levels in foods contributing the 15 most for acrylamide intake (CIAA/FDE 2011). Nonetheless, acrylamide is still present in our 16 diets and moreover from a risk characterization perspective presents a very low MOE (margin of 17 exposure) (200-50) (FAO/WHO 2005). Consequently, regulatory authorities urge for the 18 reduction of acrylamide levels in food. Potato products, French fries in particular, are one of the 19 greatest contributors to dietary acrylamide intake (EFSA 2011). To date no legal limits regarding 20 acrylamide contents in foods have yet been established. Just recently, the European Commission 21 published a Recommendation on the website from the European Commission's Directorate 22 General for Health and Consumer Policy (DG SANCO) on investigations into the levels of 23 acrylamide in food (EC 2011). Member States are recommended to investigate cases where 24 foodstuffs after final preparation, exceed prescribed acrylamide indicative 25 values. For French fries and potato crisps, these indicative values are 600 and 1000 µg/kg, 26 respectively. 27 Current industrial practices in French fries production include an entrance quality control based 29 on colour evaluation with USDA/Munsell colour chart. Colour is evaluated after a short frying 30 test, typically 180°C for 3 min. This quality control relates raw material to colour specifications of final product (customers demand) and accordingly, either raw material is rejected for 32 processing or appropriate process adjustments are taken (e.g. blanching conditions). The ability 33 of the current entrance control to predict acrylamide formation in the final product was 34 previously compared to two other colour evaluation methods (CIE L*a*b* colour parameters and 35 a process-specific Agtron analyzer) and to reducing sugar contents determination (Medeiros 36 Vinci et al. 2010). Since multiple pre and post-harvest factors may impact acrylamide formation

  37

  in homogenized potato sample as described previously(Mestdagh et 79 al. 2005). After aqueous extraction, using [2,3,3-D 3 ]acrylamide as internal standard, the 80 acrylamide extract was further cleaned up by solid-phase extraction. The extract was analyzed 81 using LC-MS/MS with positive electrospray ionization. disaccharides in homogenized potato sample were assessed as described earlier (De 84 Wilde et al. 2005). Briefly, after aqueous extraction, addition of an internal standard (phenyl-β-85 D-glucopyranoside), clean-up and drying under nitrogen, the filtrate was derivatized and injected 86 in a GC equipped with a flame-ionization detector.87Analysis of dry matter 88The dry matter content was determined, based on an official AOAC method (1990). Briefly, 5 g 89 of homogenized potatoes were mixed with calcinated sea sand and placed in the oven at 105°C 90 until a constant weight was obtained. French fries was determined using a Konica Minolta spectrophotometer CM-2500d 94 (Konica Minolta, Osaka, Japan), operating in the CIE L*a*b* colour space, an Agtron process E15-FP, Nevada, USA) and using an USDA/Munsell colour chart (VIVA, Den 96 Haag, The Netherlands), as described in Medeiros Vinci et al. (2010).

  was performed using SPSS 17.0 (SPSS inc., Chicago, IL) to 100 determine linear relationships between different parameters studied and acrylamide. Significant 101 differences between slopes of linear models was accessed by Student t-test. potato tubers from Western Europe, representing the most frequently used in 105 potato processing industry, were used for this study. Five sampling points were taken from 106 January to June in 2008 and repeated in 2009. Table 1 presents reducing sugar contents of the 10 107 potato varieties considered in this study during both sampling seasons and Figure 1 shows the 108 acrylamide contents of French fries prepared from the different potato varieties after blanching 109 and a two stage frying. Frying conditions were standard (par-fried at 180°C for 90 s and final 110 frying performed at 175°C for 150 s) throughout all sampling points since e.g. moisture content 111 may greatly affect acrylamide and colour formation. As shown in

163

  blanching and two stage frying). The models obtained with the a* parameter were not 164 significantly different and therefore were an exception. The comparison of the correlation study 165 performed in 2008 and 2009 suggests that even though seasonal differences were observed 166 regarding the raw material used during both seasons, similar linear models were obtained for 167 parameters evaluated. This seems to be significantly relevant taking into account that 168 10 different potato varieties were used and five sampling points were taken over a complete 169 storage season. In the previous study(Medeiros Vinci et al. 2010) the linear relationships 170 between the acrylamide content of the final product and the colour parameters considered or the 171 reducing sugar content were used, in combination with a hypothetical acrylamide limit, to define threshold values (THVs) for the latter product parameters. This would enable the potato 173 processing industry to identify in a better way lots of potatoes which are prone to acrylamide 174 formation. This concept was again evaluated for the present dataset.

  linear models and evaluation of threshold values (THVs) 177In contrast to our previous report(Medeiros Vinci et al. 2010) in which a hypothetical acrylamide 178 limit of 500 µg kg -1 was used, new threshold values were calculated based on the new EU 179 indicative level of 600 µg kg -1 . Only the data of the parameters which presented linear models not 180 significantly different for both seasons were considered and thus they were combined in one data 181 set (2008+2009) (Table

  < 600 µg kg -1 would have been in conformity (in respect to the 192 acrylamide limit) but would represent unnecessary "extra-precautions for industry" (e.g., extra 193 blanching) which translates in unnecessary costs. 194 195 The comparison of the different entrance parameters demonstrated that the current method 196 applied as entrance control (USDA) would result in the highest percentage of rejected samples 197 (19%) (Table

  214

  the total "unnecessarily rejected" samples (from 6 to 5%). However accepted 239 samples with acrylamide values > 600 µg kg -1 increased from 1% to 3% when applying Agtron 240 measurement individually versus the combined USDA/Agtron. 241 242

259Figure 1 .Figure 2 .Figure 3 .

 123 Figure captions

  Relation between reducing sugar content in raw potatoes (A) and colour Agtron after one stage frying (B) with acrylamide in French fries after blanching and two stage frying. Grey triangles and broken line represent 2008 data and black squares and solid line 2009 represent the data. 362x121mm (96 x 96 DPI) Possible combinations of different entrance parameters for acrylamide prediction in the final product. RS expressed as g 100 g-1 FW. 239x236mm (120 x 120 DPI) F o r P e e r R e v i e w O n l y

39 further elaborated in the following potato storage season. The goals were to evaluate if the 40 models observed in 2008 were still valid for another growth season and thus if a robust tool could

  al.

	73	step (as described for the unblanched). Frying and par-frying were conducted in a semi-
	74	professional thermostated deep-fryer (Fritel 2505, Belgium). The repeatability of acrylamide
	75	formation in French fries prepared according to these frying conditions presents a RSD of 15%
		F o F o
		r r
		P P
		e e r e e r
	38	2008; Shepherd et al. 2010), the study performed by Medeiros Vinci et al. (2010) was therefore R e v i e w R e v i e w
	41 42	be identified which allows the industry to identify incoming potato lots which are more prone for O n O n
		acrylamide formation during the final frying. This would allow French fries producers to take l y l y
	44	appropriate measures in order to lower acrylamide contents in French fries as reasonably
	45	possible. Different scenarios considering the possible quality control measures for incoming
		potatoes were evaluated, categorizing samples based on predictions of threshold values and
		6

Table 1 ,

 1 reducing sugar 112 contents of the 10 varieties sampled, varied throughout both seasons. Some varieties (e.g. Bintje and Fontana) tended to reach a higher reducing sugar content by the end of the implementation, using a colour parameter as an acrylamide predictor in raw 144 material is more practical than analyzing sugar contents. Agtron was given as the example since 145 this was the colour parameter which generally presented higher correlation coefficients within all 146 different frying procedures (Table2). This equipment is already typically used by French fries 147 industry for final product quality control. Thus, the current quality control of incoming potatoes,

	114
	115	storage season, probably due to senescent sweetening. Remarkably, this was not observed for
	116	Russet Barbank, a variety know to be particularly vulnerable for this phenomenon. For other
		varieties however, no clear trend of the reducing sugar content could be observed, also probably
	119

113 Asterix, 117 due to within batch variations. The variability of the raw material has previously been reported in 118 another study (De Meulenaer et al. 2008) and represents one of the challenges that potato growers 150 151 Due the variability observed in raw material (Table

Table captions 361

 captions Table1. Dry matter (DM) of raw potato tubers and reducing sugar contents (RS) before (UB) and

	362
	363	after standardized blanching treatment (B) for different potato varieties throughout 2008 and
	364	2009 potato storage seasons.
	365
	366	Table 2. Pearson correlation coefficients (R) and slopes of linear models obtained in correlation
	367	studies of 2008, 2009 and combined data 2008+2009. Parameters: entrance (reducing sugars-raw
	368	material; colour-after one stage frying); non blanched-two stage fried (2 Sf) (colour); blanched-
	369	two stage fried (Bl + 2 Sf) (reducing sugars-blanched potato strips; colour-after blanching and
	370	two stage frying).
	371

Table 3 .

 3 Application of THVs of data 2008+2009 and an acrylamide limit of 600 µg/kg to

	372
	373	samples of the study (2008+2009) regarding acrylamide formation in French fries. RS expressed
	374	as g/100 g FW. Parameters: entrance (reducing sugars-raw material; colour-after one stage
		frying); non blanched-two stage fried (2 Sf) (colour); blanched-two stage fried (Bl + 2 Sf)
	376
	377
	378

375

(reducing sugars-blanched potato strips; colour-after blanching and two stage frying). 259x157mm (96 x 96 DPI)

Table 3 .

 3 Mean ± SD of dry matter contents of raw potato tubers throughout 2008 and 2009. ND -not determined. Slopes significantly different, (p<0.05) for models obtained in 2008 and 2009. N/A -not applicable. Percentage calculated based on total number of samples (2008+2009), N= 93; b Percentage calculated based on accepted samples when applying investigation value for acrylamide and THV for the respective parameter; c Percentage calculated based on rejected samples when applying investigation value for acrylamide and THV for the respective parameter.

	Table 1
			Maris Piper	Asterix	Bintje	Russet Burbank	Innovator	Agria	Victoria	Fontana	Lady Olympia	Markies
	2008	DM (%) a	20.7±3.0 19.4±1.3 20.9±2.2 20.5±1.7 21.6±1.1 19.3±1.0 16.6±0.9 19.7±2.6 19.4±1.9 20.2±2.1
	Jan	RS (g 100 g -1 FW) UB	0.35	0.37	ND	0.21	0.133	0.12	ND	ND	ND	ND
		B	0.19	0.17	0.07	0.15	0.07	0.03	ND	ND	ND	ND
	Feb	UB	0.28	0.27	ND	0.19	0.17	0.14	0.05	0.04	0.05	0.03
		B	0.24	0.12	0.04	0.07	0.05	0.11	0.02	0.04	0.02	0.02
	Mar	UB	0.33	0.32	0.11	0.19	0.12	0.17	0.04	0.03	0.05	0.01
		B	0.17	0.14	0.06	0.05	0.04	0.06	0.02	0.01	0.03	0.01
	May	UB	0.25	0.25	0.16	0.06	0.19	0.08	0.03	0.03	0.02	0.02
		B	0.11	0.07	0.07	0.04	0.08	0.05	0.02	0.01	0.01	0.01
	Jun	UB	0.50	0.50	0.36	ND	ND	0.10	0.10	0.14	ND	0.03
		B	0.21	0.22	0.25	ND	ND	0.04	0.07	0.08	ND	0.02
	2009	DM (%) a	18.8±1.9 18.2±2.3 20.6±3.0 20.8±3.0 18.7±2.7 19.7±2.7 17.7±3.5 20.6±1.1 20.2±2.2 19.0±2.1
	Jan	RS (g 100 g -1 FW) UB	ND	0.13	0.11	0.23	0.04	0.02	0.10	0.07	0.05	0.01
		B	ND	0.07	0.08	0.12	0.04	0.02	0.07	0.06	0.04	0.01
	Feb	UB	0.47	0.14	0.14	0.18	0.09	0.03	0.08	0.04	0.16	0.02
		B	0.21	0.05	0.09	0.15	0.04	0.02	0.04	0.03	0.03	0.01
	Mar	UB	0.56	0.10	0.12	0.09	0.08	0.02	0.05	0.07	0.03	0.02
		B	0.34	0.08	0.09	0.08	0.05	0.01	0.03	0.05	0.02	0.01
	May	UB	0.43	0.25	0.25	0.17	0.15	0.03	0.05	0.09	0.04	0.06
		B	0.26	0.06	0.1	0.12	0.06	0.01	0.03	0.05	0.02	0.01
	Jun	UB	0.49	0.50	0.40	0.15	0.23	0.03	0.03	0.17	0.04	0.29
		B	0.31	0.16	0.28	0.15	0.17	0.03	0.02	0.09	0.04	0.17

a a a

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgments

This research was financed by Flanders' FOOD, EPPA and Belgapom. Frédéric Mestdagh is a 283 post-doctoral researcher supported by the research Foundation-Flanders (FWO). The support 284 provided by PinguinLutosa Foods (Belgium) for the necessary practical arrangements was greatly 285 appreciated. The authors thank the steering committee of EUPPA for fruitful discussions during

Determining reducing sugars at entrance or colour measurement with USDA would at least


Christine H Müller

Richard Huggins

Wen-Han Hwang

Consistent estimation of species abundance from a presence-absence map

Keywords: Presence Absence Map, Consistency, Gamma-Poisson Model

The estimation of the abundance of a species using the presence or absence of the species over a grid of cells simplifies data collection but the resulting statistical analysis is challenging. Several estimators have been proposed but their properties are unknown. Here we consider a generalized gamma-Poisson model which allows dependencies across the grid and develop a new estimator for this model. It is shown that this estimator is consistent, allowing us to conclude that it is indeed possible to estimate abundance from presence-absence maps.

Introduction

In ecological studies where the interest is in estimating the number of individuals of a certain species in a given area it is often too laborious to count all individuals in an area. A more labor effective procedure is to divide the area in M cells and to determine for each cell whether the species is present or not. The data may be thought of as a presence-absence map that identifies the occupied cells and it is often represented by a I × J-matrix (y ij ) i=1,...,I, j=1,...,J with M = I • J cells, where y ij = 1 if the i, j cell is unoccupied cell and 0 otherwise. Then the question is whether it is possible to estimate the total number N of the species in the area from these data in general. The analysis of these data presents an interesting statistical challenge [START_REF] Kunin | Extrapolating species abundance across spatial scales[END_REF] The resolution of the problem requires distributional assumptions on the unknown numbers X ij of individuals in each cell. Suppose that there are N individuals so that N = I i=1 J j=1 X ij . The simplest approach is to assume that each individual is independent of other individuals and the probability of its occurrence in a given cell is 1/M. Then (X ij ) i=1,...,I, j=1,...,J has a multinomial distribution where each cell has the same probability and the abundance N can be estimated using maximum likelihood. Alternatively, one may suppose that X ij are independently and identically distributed with a common Poisson distribution with mean λ. In this case, the probability a cell is empty is e -λ . Let m 0 = I i=1 J j=1 y ij be the number of empty (unoccupied) cells. Then by the method of moments, we may estimate λ by log(M/m 0 ) so that an estimate of N is M log(M/m 0 ). However, since the assumption of independently and identically distributed random variables does not reflect the spatial aggregation (clustering) typically observed in nature, this method often underestimates the true abundance [START_REF] He | Estimating species abundance from occurrence[END_REF] and hence is not recommended in literature.

In order to reflect the spatial aggregation, He and Gaston (2000) used the negative binomial distribution, i.e., they assumed that the X ij follow the negative binomial distribution. Although it is well known that the negative binomial distribution is more appropriate than the Poisson distribution for modeling aggregated organisms and populations (Pielou 1977), estimation is more difficult as there are two parameters in the model and they are not identifiable from a presence-absence map. To circumvent this difficulty, [START_REF] He | Estimating species abundance from occurrence[END_REF] generate a coarser map, where adjacent pairs of cells are merged from the base map and impose a constraint on the clumping parameter of the coarser map. The resulting estimator for the species abundance was shown to outperform other methods in several empirical studies (He andGaston 2000, 2007). Nevertheless, He andGaston (2000, 2007) do not provide theoretical justification for the constraint and this weakness is criticized by Conlisk et al. (2007). By noting that the negative binomial distribution arises from a gamma-Poisson mixture model, Hwang and He (2011) proposed a generalized gamma-Poisson mixture model. In this model, a gamma random variable is first associated with each cell, as in the negative binomial model, but correlations between the cells are modelled by allowing the conditional mean of the Poisson counts to be a weighted sum of all the gamma random variables. [START_REF] Hwang | Estimating abundance from presence-absence maps[END_REF] used some gamma approximations to find an estimator. When applied to forest census data [START_REF] Condit | Changes in a tropical forest with a shifting climate: results from a 50 ha permanent census plot in Panama[END_REF], they showed that the estimator based on the generalized gamma Poisson model could improve estimation of abundance from occurrence data. To date, however, no statistical properties of the estimators of [START_REF] He | Estimating species abundance from occurrence[END_REF] and [START_REF] Hwang | Estimating abundance from presence-absence maps[END_REF] are known. Even consistency has not been proven.

In this study, we develop a new consistent estimator under the generalised gamma-Poisson model. In Section 2 we reformulate the model of [START_REF] He | Estimating species abundance from occurrence[END_REF] using the gamma-Poisson framework and then describe the generalised gamma-Poisson model. In Section 3 we derive the proposed estimator for the generalised gamma-Poisson model whose consistency is shown in Section 4. This consistency proof is not straightforward and needs some tricky arguments which are given in the Appendix. The estimator is applied to real data in Section 5. The results are discussed in Section 6.

Model

For simplicity, assume that I and J are even. Let λ ij be the mean abundance at the cell (i, j). This mean may depend on environmental characteristics, such as the slope of the land, the soil content and the humidity gradient. Given the means λ ij , we assume that the abundances X ij follow Poisson distributions and they are independent of each other. If the λ ij are independent and identically distributed gamma distributed random variables with parameters k and β so that E(λ ij ) = kβ and var(λ ij ) = kβ 2 , then X ij are independent and each X ij has a negative binomial distribution. The parameter k is called the clumping parameter (Pielou, 1977) as it corresponds to the degree of aggregation of individuals in a cell. It is easy to see the absence probability P (X ij = 0) is 1/(1 + β) k . As a consequence, the likelihood function based on the observed y ij is

1 (1 + β) k m0 1 - 1 (1 + β) k M -m0 .
The likelihood is not identifiable as there are two unknown parameters but only a single statistic m 0 . In order to estimate the parameters, He and Gaston (2000) create a coarser map where adjacent cells in the original base map are merged, collapsing a 2 × 2 cell to a single cell. That is, let

X q (2i-1)(2j-1) = X (2i-1)(2j-1) + X (2i-1)(2j) + X (2i)(2j-1) + X (2i)(2j) , for i = 1, • • • , I/2 and j = 1, • • • , J/2,
which is the number of individuals in the merged quadruple cells. The coarser quad map then indicates whether the X q (2i-1)(2j-1) is larger than zero or not. Denote by m q 0 the number of empty cells in the coarser map. Noting N = Mkβ, He and Gaston (2000) solved the equations m 0 /M = P (X ij = 0) and 4m q 0 /M = P (X q ij = 0) to estimate N and k. Simple algebra shows that X q (2i-1)(2j-1) has a negative binomial distribution with the clumping parameter 4k. Nevertheless, [START_REF] He | Estimating species abundance from occurrence[END_REF] suppose that X q (2i-1)(2j-1) has the same distribution as X ij , which has clumping parameter k and mean kβ. This presumption is mainly based on empirical evidence and obviously contradicts the theoretical calculations. In addition, [START_REF] Conlisk | The impossibility of estimating a negative binomial clustering parameter from presence-absence data: A comment on He and Gaston[END_REF] showed that there is no solution from the negative binomial model under the correct distribution. To overcome the weaknesses of this approach, Hwang and He (2011) proposed an extension in the gamma-Poisson model framework. This extension yields a generalized gamma-Poisson mixtured model that has meaning in the ecological context.

To develop this model, recall that the mean level λ ij represents the environmental characteristics of the immediate area of study, hence the variables λ ij would tend to vary smoothly over nearby cells. Hence, it is unrealistic to assume independence of the λ ij . It is more appropriate to consider the spatial correlation of the λ ij over the cells so that the λ ij in neighboring cells are more similar than between those that are further apart. This observation motivated [START_REF] Hwang | Estimating abundance from presence-absence maps[END_REF] to model the mean levels as a gamma random field. Formally, let V st , s, t ∈ Z Z = {0, -1, 1, -2, 2, . . .}, be independent and identically gamma distributed random variables with parameters a and b (so that E(V st ) = ab and var(V st ) = ab 2 ). Given 0 ≤ ρ < 1, we define the mean levels

λ ij = ∞ s=-∞ ∞ t=-∞ ρ |i-s|+|j-t| V st ,
which is a spatial moving average of the V st . The parameter ρ reflects the strength of the spatial correlation of the mean levels λ ij . It is then assumed that the conditional distributions of the X ij given V st = v st for all s, t ∈ Z Z are Poisson with mean λ ij and that the X ij and X kl are conditionally independent given V st = v st , s, t ∈ Z Z for (i, j) = (k, l). As the joint distribution of X ij reduces to the usual independent negative binomial model when ρ = 0, it is called the generalized gamma-Poisson mixture model. Hwang and He (2011) used gamma approximations to find an estimator, however, the consistency of this estimator remains an open question. Here we propose a new moment estimator whose consistency can be established.

Estimation

Since the conditional distribution of

X ij given V st = v st , s, t ∈ Z Z, is Poisson with parameter λ ij = ∞ s=-∞ ∞ t=-∞ ρ |i-s|+|j-t| v st , we see that E(X ij ) = ∞ s=-∞ ∞ t=-∞ ρ |i-s|+|j-t| E(V st ) = ∞ s=-∞ ρ |i-s| ∞ t=-∞ ρ |j-t| ab = 1 + ρ 1 -ρ 2 ab,
and thus

E (N) = I i=1 J j=1 E(X ij ) = I J 1 + ρ 1 -ρ 2 ab. (1)
Once we have estimators a, b, ρ of a, b, and ρ this yields the estimator

N = I J 1 + ρ 1 -ρ 2 a b, (2)
and our main task is to estimate these parameters using presence-absence data.

Let Y ij = 1 {0} (X ij ) be an indicator of whether the cell (i, j) is empty or not. The variables

Y d i(2j-1) = Y i(2j-1) Y i(2j) and Y q (2i-1)(2j-1) = Y (2i-1)(2j-1) Y (2i-1)(2j) Y (2i)(2j-1) Y (2i)(2j)
indicate whether the double and quadruple cells are empty. Our approach is based on the method of moments and to this end we need:

Proposition 1 a) E(Y ij ) = ∞ s=-∞ ∞ t=-∞ 1 + b ρ |i-s|+|j-t| -a , b) E(Y d ij ) = ∞ s=-∞ ∞ t=-∞ 1 + b ρ |i-s| (ρ |j-t| + ρ |j+1-t| ) -a , c) E(Y q ij ) = ∞ s=-∞ ∞ t=-∞ 1 + b (ρ |i-s| + ρ |i+1-s| )(ρ |j-t| + ρ |j+1-t| ) -a . Define M 0 = I i=1 J j=1 Y ij , M d 0 = I i=1 J/2 j=1 Y d i(2j-1)
and

M q 0 = I/2 i=1 J/2 j=1 Y q (2i-1)(2j-1)
with realizations m 0 , m d 0 , and m q 0 . Then, a, b, and ρ can be estimated by solving

1 I J m 0 = E (a,b,ρ) 1 I J M 0 = ∞ s=-∞ ∞ t=-∞ 1 + b ρ |s|+|t| -a , (3)
2 I J m d 0 = E (a,b,ρ) 2 I J M d 0 = ∞ s=-∞ ∞ t=-∞ 1 + b ρ |s| (ρ |t| + ρ |t+1| ) -a , 4 I J m q 0 = E (a,b,ρ) 4 I J M q 0 = ∞ s=-∞ ∞ t=-∞ 1 + b (ρ |s| + ρ |s+1| )(ρ |t| + ρ |t+1| ) -a .

Consistency

To establish consistency we let IJ → ∞ so that the number of cells observed increases. As we suppose the occupancy probabilities do not depend on I or J this implies that N → ∞. This corresponds to increasing the size of the map.

Theorem 1 If (a, b, ρ) ∈ (0, ∞) × (0, ∞) × [0, 1), then 1 I J M 0 , 2 I J M d 0 , 4 I J M q 0 ⊤ is a weakly consistent estimator of E (a,b,ρ) 1 I J M 0 , E (a,b,ρ) 2 I J M d 0 , E (a,b,ρ) 4 I J M q 0 ⊤ as I J → ∞.
The consistency of the estimators ( a, b, ρ) and thus of N follows from the following general proposition, which is similar to those of Amemiya (1973, Lemma 3), White (1980, Lemma 2.2) and White (1981, Theorem 2.1).

Proposition 2 Let be Θ ⊂ IR L compact, g : Θ → IR L continuous satisfying g(θ) = g(θ ′ ) for all θ, θ ′ ∈ Θ with θ = θ ′ , and X N an L dimensional random vector. If X N is a weakly consistent estimator for g(θ), then

θ N ∈ arg min θ∈Θ X N -g(θ)
is a weakly consistent estimator of θ.

Since E (N) defined in (1) is a continuous function of (a, b, ρ), Theorem 1 implies at once:

Theorem 2
The abundance estimator N given by [START_REF] Condit | Changes in a tropical forest with a shifting climate: results from a 50 ha permanent census plot in Panama[END_REF], where (a, b, ρ) ∈ Θ for a compact parameter space Θ ⊂ (0, ∞) × (0, ∞) × (0, 1) are solutions of (3), satisfies N -E(N) = o P (IJ).

Hence we need only to prove Theorem 1. This proof is not straightforward because of the dependence structure and therefore given together with the proof of Proposition 1 in the Appendix.

Application to real data

We consider data from a complete census of a tropical rain forest in Barro Colorado Island (BCI), Panama [START_REF] Condit | Changes in a tropical forest with a shifting climate: results from a 50 ha permanent census plot in Panama[END_REF]). The study area was a 50-hectare (500×1000 m) rectangle plot referred to as the BCI plot. The tree plot was established in 1980 and has been surveyed six times so far. The data from the 1985 census are used in this study. The data records the exact location of species for each free-standing tree with diameter at breast height (dbh) at least 1 cm in the plot. In the 1985 census, there are 238,018 trees representing 299 species. We illustrate the computation of the estimator on the species Faramea occidentalis. Here N = 25094, m 0 = 7824, m d 0 = 1994 and m q 0 = 357. There were 20000 -7824 = 12176 occupied cells, giving a lower bound on N. We solved the estimating equations (3) with the products from negative to positive infinity replaced by products from -T to T by minimizing an objective function consisting of the sums of squares of the differences. We considered ρ = 0.05, 0.06, . . . , 0.99 and for each value of ρ we found values of a and b to minimize the objective function. In Figure 1 we plot the value of the minimum and the estimated N as functions of ρ for T = 60. The minimum was quite flat as a function of ρ near the minimum, however, a plot of the logarithm of the minimum, not given here, did show a distinct minimum. At this minimum, â = 0.00066, b = 3.27, and ρ = 0.92 and the estimated abundance was N = 24774. In the second part of the figure we see that the estimated value of N is also quite flat and is close to the true value for a range of values of ρ. The minimum value of the sums of squared differences of (3) and N as functions of ρ ∈ (0, 1) for Faramea occidentalis. In the second plot, the horizontal line gives the known value of N (solid line) and the lower bound on N (dashed line).

Discussion

We have demonstrated analytically that the new estimator is consistent and conclude that for the mixed gamma-Poisson model it is possible to estimate abundance from presence-absence maps in a quite general setting.

The assumption of a compact parameter space is required to prove Proposition 2 and hence Theorem 2, but since the compact parameter space can be arbitrarily large, this is not a strong restriction. We also require g(θ) = g(θ ′ ) for all θ, θ ′ ∈ Θ with θ = θ ′ . This is an identifiability condition which is usually difficult to verify in nonlinear settings with more than two parameters. Here this means that the solutions a, b, ρ of the equations given by ( 3) are unique, i.e. (a, b, ρ) is identified by 1 I J m 0 , 2 I J m d 0 , 4 I J m q 0 ⊤ . This is not the case for ρ = 0 or a = 0 or b = 0, but we conjecture that identifiability holds for ρ > 0, a > 0, b > 0. Since ρ = 0 does not satisfy the conditions of the Theorem, we have shown consistency only in the case where the cell counts are dependent. Moreover, consistency is obtained as IJ → ∞ with the cell occupancy probabilities remaining the same so that N → ∞. An alternate approach not considered here would be to allow the grid to become finer so that N is the number of individuals in a given area. In this case, the large sample properties are problematic as for a fine enough grid, N will be the total number of occupied cells.

We were not concerned with the practical application of the new estimator but have focused on consistency. In particular we have not addressed the goodness of fit of the model. The precision of the estimator may be addressed using bootstrap methods according to Hall 1988.

[9] Kunin, W.E., Hartley, S., and Lennon, J.J. (2000). Scaling down: On the challenge of estimating abundance from occurrence patterns. The American Naturalist 156, 560-566.

[10] Pielou, E.C. (1977). Mathematical Ecology. John Wiley, New York.

[11] White, H. (1980). Nonlinear regression on cross-section data. Econometrica 48, 721-746.

[12] White, H. (1981). Consequences and detection of misspecified nonlinear regression models. J. Amer. Statist. Assoc. 76, 419-433.

A Proofs

To prove Theorem 1, some propositions must be proved. As Proposition 1 and Proposition 3 below are related we combine their proofs.

Proposition 3 a) E(Y ij • Y kl ) = ∞ s=-∞ ∞ t=-∞ 1 + b [ρ |i-s|+|j-t| + ρ |k-s|+|l-t| ] -a , b) E(Y d ij • Y d kl ) = ∞ s=-∞ ∞ t=-∞ 1 + b [ρ |i-s| (ρ |j-t| + ρ |j+1-t| ) + ρ |k-s| (ρ |l-t| + ρ |l+1-t| )] -a , c) E(Y q ij • Y q kl ) = ∞ s=-∞ ∞ t=-∞ 1 + b [(ρ |i-s| + ρ |i+1-s| )(ρ |j-t| + ρ |j+1-t| ) + (ρ |k-s| + ρ |k+1-s| )(ρ |l-t| + ρ |l+1-t| )] -a .
Proof of Propositions 1 and 3. Since a Poisson random variable X with parameter λ satisfies P (X = 0) = exp{-λ}, using the moment generating function of the gamma distribution and the independence of the V st , we have

E(Y ij ) = P (X ij = 0) = ∞ s=-∞ ∞ t=-∞ E exp{-ρ |i-s|+|j-t| V st } = ∞ s=-∞ ∞ t=-∞ (1 + b ρ |i-s|+|j-t| ) -a
which is Proposition 1 a). Since X ij and X kl are conditionally independent given V st with s, t ∈ Z Z, Proposition 1 b) and Proposition 3 a) follow from

E(Y ij • Y kl ) = P (X ij = 0, X kl = 0) = E exp{- ∞ s=-∞ ∞ t=-∞ (ρ |i-s|+|j-t| + ρ |k-s|+|l-t| ) V st } .
Proposition 1 c) is a special case of Proposition 3 b). Since b) and c) of Proposition 3 can be proved similarly, only the proof of c) is shown here: ). Then there exists a constant c only depending on a, b, ρ such that

E(Y q ij • Y q kl ) = P (X ij = 0, X i(j+1) = 0, X (i+1)j = 0, X (i+1)(j+1) = 0, X kl = 0, X k(l+1) = 0, X (k+1)l = 0, X (k+1)(l+1) = 0) = E exp - ∞ s=-∞ ∞ t=-∞ ρ |i-s|+|j-t| + ρ |i-s|+|j+1-t| + ρ |i+1-s|+|j-t| + ρ |i+1-s|+|j+1-t| + ρ |k-s|+|l-t| + ρ |k-s|+|l+1-t| + ρ |k+1-s|+|l-t| + ρ |k+1-s|+|l+1-t| V st . Proposition 4 Let be (a, b, ρ) ∈ (0, ∞)×(0, ∞)×[0, 1
I i=1 I i=1 J j=1 J j=1 ln E(Y ij Y kl ) E(Y ij ) E(Y kl ) ≤ c I J, I i=1 I i=1 J/2 j=1 J/2 j=1 ln E(Y d i(2j-1) Y d k(2l-1) ) E(Y d i(2j-1) ) E(Y d k(2l-1) ) ≤ c I J, I/2 i=1 I/2 i=1 J/2 j=1 J/2 j=1 ln E(Y q (2i-1)(2j-1) Y q (2k-1)(2l-1) ) E(Y q (2i-1)(2j-1) ) E(Y q (2k-1)(2l-1) ) ≤ c I J.
To prove Proposition 4, we need Proof.

Lemma 1 Let α i (s) = ρ |i-s| + ρ |i+1-s| for i = 1, . . . , I. Then I i=1 I k=1 ∞ s=-∞ α i (s)α k (s) ≤ 12 I, for ρ = 0, 1 (1-
If ρ ∈ (0, 1), then α i (s) = ρ i-s + ρ i+1-s = ρ i-s (1 + ρ) ≤ 2 ρ i-s for s ≤ i and

α i (s) = ρ s-i + ρ s-(i+1) = ρ s-i-1 (ρ + 1) ≤ 2 ρ s-i-1 for s ≥ i + 1. Analogous bounds hold for α k (s). Let i ≤ k. Then ∞ s=-∞ α i (s)α k (s) ≤ i s=-∞ (2 ρ i-s ) (2 ρ k-s ) + k s=i+1 (2 ρ s-i-1 ) (2 ρ k-s ) + ∞ s=k+1 (2 ρ s-i-1 ) (2 ρ s-k-1 ) = 4 ρ k-i ∞ s=0 (ρ 2 ) s + ρ k-i-1 (k -i) + ρ k-i ∞ s=0 (ρ 2 ) s = 4 ρ k-i 1 1 -ρ 2 + ρ k-i-1 (k -i) + ρ k-i 1 1 -ρ 2 ≤ 4 ρ |k-i|-1 1 -ρ 2 (2ρ + |k -i|) . 9
Similarly, the same upper bound holds for i ≥ k. As

I i=1 I k=1 ρ |k-i| |k -i| = 2 (1 -ρ) 4 (I -1)ρ -2Iρ 2 + (I + 1)ρ 3 + (I + 1)ρ I+1 -2Iρ I+2 + (I -1)ρ I+3 ≤ 2 (1 -ρ) 4 10I,
we see that

I i=1 I k=1 ∞ s=-∞ α i (s)α k (s) ≤ I i=1 I k=1 4 ρ |k-i|-1 1 -ρ 2 (2ρ + |k -i|) ≤ 4 1 -ρ 2 2 I i=1 I k=1 ρ |k-i| |k -i| + 2 I + 1 ρ I i=1 I k=1 ρ |k-i| |k -i| ≤ 4 1 -ρ 2 2 2 (1 -ρ) 4 10 I + 2 I + 1 ρ 2 (1 -ρ) 4 10 I ≤ 1 (1 -ρ 2 )(1 -ρ) 4 4 • 42 I + 4 ρ 20 I .
If ρ = 0, we obtain using

0 0 = 1 ∞ s=-∞ α i (s)α k (s) ≤ 4, for |k -i| ≤ 1, 0, for |k -i| > 1,
as required.

Proof of Proposition 4.

To complete the proof of Proposition 4, let

α 0 i (s) = ρ |i-s| , α i (s) = ρ |i-s| + ρ |i+1-s| , β 0 j (t) = ρ |j-t| , β j (t) = ρ |j-t| + ρ |j+1-t| .
Setting

α 1 i (s) = α 0 i (s), β 1 j (t) = β 0 j (t), α 2 i (s) = α 0 i (s), β 2 j (t) = β j (t), α 3 i (s) = α i (s), β 3 j (t) = β j (t),
and

Y 1 ij = Y ij , Y 2 ij = Y d ij , Y 3 ij = Y q ij , Propositions 1 and 3 provide for m = 1, 2, 3 E(Y m ij ) = ∞ s=-∞ ∞ t=-∞ 1 + b α m i (s)β m j (t) -a , E(Y m ij Y m kl ) = ∞ s=-∞ ∞ t=-∞ 1 + b [α m i (s)β m j (t) + α m k (s)β m l (t)] -a .
The inequality ln(1 + x) ≤ x for all x ≥ 0 yields

ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) = ln ∞ s=-∞ ∞ t=-∞ 1 + b [α m i (s)β m j (t) + α m k (s)β m l (t)] -a 1 + b α m i (s)β m j (t) -a (1 + b α m k (s)β m l (t)) -a = a ∞ s=-∞ ∞ t=-∞ ln 1 + b α m i (s)β m j (t) (1 + b α m k (s)β m l (t)) 1 + b [α m i (s)β m j (t) + α m k (s)β m l (t)] = a ∞ s=-∞ ∞ t=-∞ ln 1 + b 2 α m i (s)β m j (t) α m k (s)β m l (t) 1 + b [α m i (s)β m j (t) + α m k (s)β m l (t)] a>0,b>0,ρ≥0 ≤ a ∞ s=-∞ ∞ t=-∞ ln 1 + b 2 α m i (s)β m j (t) α m k (s)β m l (t) ≤ a ∞ s=-∞ ∞ t=-∞ b 2 α m i (s) β m j (t) α m k (s) β m l (t).
Let I 1 = {i; i = 1, . . . , I}, J 1 = {j; j = 1, . . . , J}, I 2 = {i; i = 1, . . . , I}, J 2 = {j; j = 2n -1 with n = 1, . . . , J/2}, I 3 = {i; i = 2n -1 with n = 1, . . . , I/2}, J 3 = {j; j = 2n -1 with n = 1, . . . , J/2}.

Then for m = 1, 2, 3 using Lemma 1, and noting α m i (s) ≤ α i (s) for i = 1, . . . , I and β m j (t) ≤ β j (t) for j = 1, . . . , J we see that

i∈I m j∈J m k∈I m l∈J m ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) ≤ a i∈I m j∈J m k∈I m l∈J m ∞ s=-∞ ∞ t=-∞ b 2 α m i (s) β m j (t) α m k (s) β m l (t) ≤ a I i=1 J j=1 I k=1 J l=1 ∞ s=-∞ ∞ t=-∞ b 2 α i (s) β j (t) α k (s) β l (t) = a b 2 I i=1 I k=1 ∞ s=-∞ α i (s) α k (s) J j=1 J l=1 ∞ t=-∞ β j (t) β l (t) ≤ a b 2 12 2 I J, for ρ = 0 a b 2 1 (1-ρ 2 )(1-ρ) 4 4 • 42 + 4 ρ • 20 2 I J, for ρ ∈ (0, 1) = c I J,
where the constant c depends only on a, b, ρ.

Proposition 5 For (a, b, ρ) ∈ (0, ∞) × (0, ∞) × [0, 1) a) var(M 0 ) ≤ c I J, b) var(M d 0 ) ≤ c I J, c) var(M q 0 ) ≤ c I J,
where c is a constant only depending on a, b, and ρ.

Proof of Proposition 5.

Using the same notation as in the proof of Proposition 4, we have

M 0 = i∈I 1 j∈J 1 Y 1 ij , M d 0 = i∈I 2 j∈J 2 Y 2 ij , M q 0 = i∈I 3 j∈J 3 Y 3 ij .
Hence for m = 1, 2, 3 we have only to prove

var i∈I m j∈J m Y m ij ≤ c I J. First note that E(Y m ij ) = E(Y m kl ) and cov(Y m ij , Y m kl ) ≤ var(Y m ij ) = var(Y m kl )
for all i, k = 1, . . . , I, j, l = 1, . . . , J, and m = 1, 2, 3. This implies

var(Y m 11 ) E(Y m 11 ) 2 = var(Y m ij ) E(Y m ij ) 2 ≥ cov(Y m ij , Y m kl ) E(Y m ij ) 2 = E(Y m ij Y m kl ) -E(Y m ij ) E(Y m kl ) E(Y m ij ) E(Y m kl ) = E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -1,
so that

E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) ≤ var(Y m 11 ) E(Y m 11 ) 2 + 1 =: c 1 (4)
for all i, k = 1, . . . , I, j, l = 1, . . . , J, and m = 1, 2, 3. The mean value theorem provides

E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -1 = exp ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -exp(0) = exp(θ ijkl ) ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -0 with θ ijkl ∈ 0, ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) if ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) ≥ 0
and

θ ijkl ∈ ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) , 0 if ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) < 0.
In particular, θ ijkl satisfies because of (4)

θ ijkl ≤ ln var(Y m 11 ) E(Y m 11 ) 2 + 1 = ln(c 1 )
so that

E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -1 ≤ exp(ln(c 1 )) ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) = c 1 ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl )
.

Then Proposition 4 provides

var i∈I m j∈J m Y m ij = i∈I m j∈J m k∈I m l∈J m cov(Y m ij , Y m kl ) = i∈I m j∈J m k∈I m l∈J m E(Y m ij Y m kl ) -E(Y m ij ) E(Y m kl ) E(Y m ij ) E(Y m kl ) E(Y m ij ) E(Y m kl ) = i∈I m j∈J m k∈I m l∈J m E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) -1 E(Y m 11 ) 2 ≤ i∈I m j∈J m k∈I m l∈J m c 1 ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) E(Y m 11 ) 2 = c 1 E(Y m 11 ) 2

  , He and Gaston 2000, Kunin et al. 2000, Conlinsk et al. 2007, He and Gaston 2007).

Figure 1 :

 1 Figure1: The minimum value of the sums of squared differences of (3) and N as functions of ρ ∈ (0, 1) for Faramea occidentalis. In the second plot, the horizontal line gives the known value of N (solid line) and the lower bound on N (dashed line).

ρ 2 )

 2 (1-ρ) 4 4 • 42 I + 4 ρ 20 I , for ρ ∈ (0, 1), for all I ∈ IN.

  i∈I m j∈J m k∈I m l∈J m ln E(Y m ij Y m kl ) E(Y m ij ) E(Y m kl ) ≤ c 1 E(Y m 11 ) 2 c 2 I J = c I J,where as c 1 , E(Y m 11 ) 2 , and c 2 only depend on a, b, and ρ so does c. Proof of Theorem 1. Theorem 1 now follows from Proposition 5 a), b) and c) and Chebyshev's inequality.


Myriam Rakho
email: rakho@univ-mlv.fr

Matthieu Constant
email: mconstan@univ-mlv.fr

Evaluating the impact of some linguistic information on the performances of a similarity-based and translation-oriented Word-Sense disambiguation method

In this article, we present an experiment of linguistic parameter tuning in the representation of the semantic space of polysemous words. We evaluate quantitatively the influence of some basic linguistic knowledge (lemmas, multi-word expressions, grammatical tags and syntactic relations) on the performances of a similarity-based Word-Sense disambiguation method. The question we try to answer, by this experiment, is which kinds of linguistic knowledge are most useful for the semantic disambiguation of polysemous words, in a multilingual framework. The experiment is about 20 French polysemous words (16 nouns and 4 verbs) and we make use of the French-English part of the sentence-aligned EuroParl Corpus for training and testing. Our results show a strong correlation between the system accuracy and the degree of precision of the linguistic features used, particularly the syntactic dependency relations. Furthermore, the lemma-based approach absolutely outperforms the word form-based approach. The best accuracy achieved by our system amounts to 90%.

Introduction

In word sense disambiguation (WSD) task, multiple experiments of parameter tuning in the representation of the semantic spaces have been carried out (Pancardo-Rodguez & al., 2005;Crestan & al., 2003). In this paper, we present the results of a similar experiment, looking, among all the cooccurrents of a polysemous word, for the best candidates to be employed as dimensions of its semantic space : those which are discriminative enough to give a WSD method the ability for distinguishing its different senses. For this purpose, we varied the combinations of some basic linguistic knowledge in the semantic spaces (lemmas, multi-word expressions, grammatical tags and syntactic relations). The semantic spaces, built from pre-classified instances of the ambiguous words in their occurring contexts, are used in examplar-based classification methods like k Nearest Neighbors algorithm (kNN) [START_REF] Veenstra | Memory-based Word Sense disambiguation[END_REF], Support Vector Machines (SVM, Keok Lee & al., 2004), Semantic Classification Trees (SCT, Loupy & al., 2000) and other methods that use context-similarity measures [START_REF] Apidianaki | Data-driven semantic analysis for multilingual WSD and lexical selection in translation[END_REF]. The classes associated with the training instances in the semantic space of a word can be its senses as they are defined in traditional lexical resources (dictionaries, thesauri) [START_REF] Gale | A method for disambiguating word senses in a large corpus[END_REF]. In a multilingual framework, the classes can also be its translation equivalents in one [START_REF] Kaji | Unsupervised word sense disambiguation using bilingual corpora[END_REF] or more [START_REF] Crego | Plusieurs langues (bien choisies) valent mieux qu"une : traduction statistique multi-source par renforcement lexical[END_REF] other languages. For our experiment, we chose the machine translation (MT) oriented and similarity-based method described in [START_REF] Apidianaki | Data-driven semantic analysis for multilingual WSD and lexical selection in translation[END_REF]. Our corpus for training and testing was the French (SL) and English (TL) aligned version of the sentence-aligned EuroParl corpus (Koehn, 2003), and we evaluated the performances of our WSD method for the disambiguation of 20 polysemous words (16 nouns and 4 verbs). In the first section of this article, we give a description of the WSD method we used for this experiment. The second section is our definition of the semantic spaces. And in the third section, we propose an analysis for the results of our experiment.

Description of our WSD method

We used for this experiment the MT oriented method described in [START_REF] Apidianaki | Data-driven semantic analysis for multilingual WSD and lexical selection in translation[END_REF].

The training instances, for a given word, are the vectors of cooccurrents representing the SL segments (SL_segment) in the aligned corpus in which this word occurs. The classes are its translation equivalents (EQV) in the corresponding TL parts. [START_REF] Apidianaki | Data-driven semantic analysis for multilingual WSD and lexical selection in translation[END_REF] takes things in two steps. First, the EQVs of the word are grouped into clusters representing its various senses, and the new instance is assigned the most suitable cluster. And secondly, the most probable EQV is chosen among those in this cluster. We evaluated the first step of this method. Clustering the EQVS. First, every EQV is associated with a unique vector (EQV i _segment) which contains the union of the components of all the SL_segments with which it is associated. Every cooccurrent (j) is assigned as many relative weights (rw ij ) as classes (i) with which it is associated. The value of rw ij is the discriminating potential of j between EQV i and the other EQVs of the word (see [START_REF] Apidianaki | Data-driven semantic analysis for multilingual WSD and lexical selection in translation[END_REF]. We then build a similarity matrix of EQVs using the weighted Jaccard coefficient (WJ, Grefenstette, 1994). While all the cooccurrents are taken into account when computing the relative weights, the computation of EQVs similarity can be made either taking into account all the cooccurrents or only the syntactic cooccurrents and the neighbors (parameter sim, described in section 3). Clusters of semantically similar EQVs are built, in which the similarity between all EQVs is equal or higher than the average of all the similarities in the matrix. Every cluster is represented by a vector containing all the cooccurrents that appear in all the EQV i _segment of its components at once.

The decision function used for determining the cluster of the new instance (w new ) is defined as follows :  The similarity between the context vector of w new and the vector of every cluster is calculated on the basis of their intersection : it is the ratio between, on one hand, the sum of the relative weights of the cooccurrents that appear in both vectors, and, on the other hand, the product of the sizes of the two vectors.  And the most similar cluster is assigned to w new .

Our definition of the semantic space(s)

Representing the dimensions of the semantic spaces have been done using various kinds of linguistic knowledge. Table 1 describes the corresponding parameters and their modalities. The linguistic preprocessing of every vector corresponding to a SL_segment was done in three steps :  step 1 : type and comp parameters are applied to the whole vector.  step 2 : ctxt_type parameter is applied. The output of this step is a vector in which the components that belong to the same kind of context (thematic, neighbours or syntactic cooccurrents) are assigned the same absolute weight (aw). Three kinds of vectors are obtained, corresponding to the three combinations of contexts we have tested :

-a neighbours and thematic vector is a vector in which two kinds of contexts are represented : the neighbours of the ambiguous word and the other (thematic) cooccurrents of the word. The first ones are distinguished by a strong absolute weight (aw=2) while aw is 1 for the other (thematic) cooccurrents ; -a syntactic and thematic vector is a vector in which the direct syntactic cooccurrents are distinguished from the other cooccurrents by the same weighting procedure as in the preceding kind of vector ;

-a thematic vector is a vector in which all the (thematic) components of the original vector are assigned the same absolute weight (aw=1).  step 3 : ctxt_component is applied differently to every kind of context represented in the vector. For a given cooccurrent j, all its absolute weights in the context vectors that are associated with a given EQV i are summed. Its weight relatively to this EQV (rw ij , section 2) is then multiplied by this sum. Thus, the neighbours and the syntactic cooccurrents are favoured when computing EQVs similarity. We give in table 2 below an example, using, as a context of the word "article", the SL sentence : "Selon l'article 22 du règlement, vous voulez que les membres dressent un compte-rendu détaillé de leurs activités". This example illustrates the "neighbour (in bold) and thematic" context. In the "vectorial representation" row, we put in brackets the value of aw for each cooccurrent.

NLP tools.

The NLP tools we used for preprocessing the corpus are Unitex [START_REF] Paumier | Unitex 2.0 User Manual[END_REF], for multi-word expressions extraction, TreeTagger (Schmidt, 1995), for lemmatization and grammatical tagging, and the Xip parser online demo (Xerox Incremental Parser, Aït-Mokhtar & al., 2002), for the detection of syntactic relations.

Evaluation

We evaluate the parameter combinations defined in section 2 above, in order to find the more relevant one for our WSD method, in terms of representativeness for the different senses of a word. We use for that the French-English part of the sentence-aligned EuroParl corpus, which consists of about one million sentences, that is 30 million words for each language version. We evaluate the disambiguation of 20

Parameters combination :

Step 1 : type=form#tag ; comp=yes

Step 2 : ctxt_type = neighbours and thematic

Step 3 : ctxt=1+3 (1 for neighbours ; 3 for thematic cooccurrents)

Training and testing data

We first manually built a bilingual lexicon in which each SL word is associated with its various TL translations (EQV) in the aligned corpus. Then, for every word, we extracted a corpus, consisting of sub-corpora for its EQVs : one sub-corpus consists of the SL part (SL_segment) of all the aligned segments in the corpus in which the word is translated by the EQV concerned. Every sub-corpus contains around one thousand SL_segments, 80% of which are used for training and 20% for testing.

Table 3 describes the words of our task : it summarizes the mean polysemy from monolingual and multilingual points of view (the number of usages of the words according to the French (Larousse, 2009) dictionary and the number of TL EQVs in our corpus, respectively), the size of the training and testing sets for each word and the size of the sub-corpora for each EQV (minimal and maximal size).

In the bilingual lexicon, each word is associated with all the TL EQVs that are used to translate it the aligned corpus. Table 4 gives the lexicon entries for the word compte and their part-of-speech (POS). Figure 1 is an illustration of the extraction of the semantic space for the word article.

Evaluation metrics

The metrics used for the evaluation are :  recall : the ratio between the number of correct predictions and the number of reference instances  enriched precision : the ratio between the number of correct predictions and the number of predictions  f-score : (2* (recall * precision)) / (recall * precision) A prediction is considered as correct if the selected cluster for w new contains the EQV used as its reference translation in the aligned segment of the SL part which contains w new .

Best scores obtained

The best score obtained for our WSD method amounts to 90.5%. This score is equally obtained with the "neighbours and thematic" context or with the "syntactic and thematic" context, both defined with the following parameters combination : type is form#tag, comp is yes, ctxt is 3 for the thematic cooccurrents and any value for the neighbours or the syntactic cooccurrents, and sim is strong.

Parameters evaluation

4.4.1.

Quantitative evaluation Due to lack of space, we cannot show in this extended abstract all the results we have obtained. The diagrams in figure 1 represent the evolution of the f-score depending on ctxt parameter, when type is form#tag and comp is yes. For parameters type and comp, we give the most significant scores only.

4.4.2.

Global tendencies In this sub-section, we propose several conclusions concerning the linguistic parameters we have drawn from the results of the experiment. Table 5 illustrates quantitatively the interaction between the linguistic informations. Each cell of the table contains the highest score obtained with all the parameter combinations in which both the row entry and the column entry modalities of the two parameters concerned are activated. Concerning ctxt_comp parameter, only its application to the thematic context is represented, since we found that it have no influence for the neighbors and the syntactic coocurrents. We can draw, from this table, the global tendency for every parameter. For example, in the row and column that represent type parameter, the values in the form#tag (in dark gray in the table) part are always higher than the ones in the form part (in light gray), whatever the parameter with which type is combined.

4.4.3.

Our findings In this sub-section, we propose several conclusions concerning the linguistic parameters we have drawn from the results of the experiment.

Parameter sim.

The best score (90.5%) falls to 73.9% when sim is all : the similarity between two EQVs is computed using all their cooccurrents. Representing the different usages of a polysemous word is then more precise when using "syntactic patterns". However, the thematic context cannot be ignored, since the best score fell to 74% when only the syntactic cooccurrents and the neighbors were considered in the computation of the relative weights of the cooccurrents (rw ij ).

Parameter ctxt_type.

We observe that the f-scores are higher when neighbors and syntactic cooccurrents are used in the semantic spaces.

Parameter type.

The representation of the semantic spaces was more precise with form#tag value for this parameter. With the optimal combination, we observed a strong decrease when type is form (81.4%). Then, we can say that the morpho-syntactic component plays a significant role in the representation of the linguistic context of the words. Parameter comp. The f-scores were better when multi-word expressions were considered as single units : the best-score falls to 81.8% when comp is no. This is explained by the fact that the sense, for this kind of expressions, cannot be induced by a semantically compositional process. So, inserting the sense of the lexical units composing them in the semantic space of a word is literally incorrect.

Linguistic parameters and

Parameter ctxt_comp. This parameter was very influential for the performances of our WSD method. The grammatical filtering was absolutely bad. And the influence of the lemmatization varies depending on the kind of context. It was good for the thematic context, but had no influence for the neighbors and for the syntactic cooccurrents.

Conclusion.

Making use of distributional hypothesis in order to describe the semantic space of the words cannot be done by considering all the co-textual elements in a homogenous way. Then, each word can be caracterized by a multifaceted representation of its local and global contexts of usage in which each kind of co-textual element (neighbors as left and right lexical and grammatical context, syntactic cooccurrents, first and second order thematic cooccurrents, predicates and arguments, semantic roles, and so on) has to be taken into account. Moreover, each kind of co-textual element has then to be favoured depending on the goal of the task. For example, [START_REF] Baroni | Using cooccurrence statistics and the Web to discover synonyms in technical language[END_REF]) used narrow windows of size 2 and 5 (immediate lexical neighbors) to discover synonymy relations.

4.4.4.

Analysis of the evolution of the scores Figure 1 is a diagram representation of the evolution of the scores following ctxt_comp parameter. The evolution of the labels of the Y-axis shows us consistent relation between the scores and ctxt_comp parameter. In fact, in the two first diagrams, the scores are formed of four groups, corresponding to the fours modalities of ctxt_comp applied to the thematic context (b part in the a+b labels). This four groups are in the following ascending order : [2 1 4 3], which provides two rankings in the scores :

-a first order in terms of words and lemmas : (2,1) < (4,3), so words < lemmas ; -and a second order in terms of grammatical filtering :

2<1 and 4<3, so filtering < no filtering. In the first diagram, within this first ranking, a second ranking is observed that follows the modality of ctxt_comp applied to the syntactic cooccurrents (a part in the a+b labels). This second ranking is a corroboration of the order we observed in the first ranking, relating to the grammatical filtering : in the four groups, we effectively observe that (4,2) < (3,1). And in the second diagram, we observe an identical second ranking relating to the lemmatization and that corroborate the fact that lemmas are better than words : 3<1 in the four groups. The third diagram represents the scores obtained when no distinction is made between the neighbors and syntactic cooccurrents on one hand, and the other (thematic) cooccurrents. Once again, we observe the same ranking concerning the grammatical filtering : (2,4)<(1,3). But the lemmas are better than words (80 vs. 77.7%, respectively) when this filtering is applied, while the words are (slightly) better than lemmas (81.6 vs. 81.3%) when no grammatical filtering is applied.

Conclusion and future work

We have evaluated the impact of some linguistic knowledge on WSD performances using a classification method based on the kNN algorithm. The best scores were obtained with five different parameter combinations, what corroborate [START_REF] Habert | Les linguisitques de corpus[END_REF])"s conclusion according to which the best linguistic model does not exist, theoretically. Every kind of linguistic knowledge has fluctuating effects depending particularly on the other kind of linguistic knowledge it is combined with and on the NLP application for which the WSD method concerned will be a sub-task. Various improving factors should be considered, like combining both neighbors and syntactic cooccurrents, or using neighbors from windows of size higher than 1 and second order cooccurrences from every kind of context. Besides, we could define the semantic spaces differently according to the grammatical tag of the ambiguous word as suggested by [START_REF] Habert | Les linguisitques de corpus[END_REF] (adjectival and adverbial cooccurrents are certainly more semantically informative for nouns than for verbs, for example). We could also extend the training corpora and define more fine-grained entries in the bilingual lexicon by using more than one TL. "thematic" context Finally, we should do the same experiment using WSD methods based on other learning techniques, like SVM and SCT, and complete our evaluation with a comparative one, both in monolingual and multilingual frames.

  Vectorial representation : article#nom(4) -l#det:art(2) -22#card(2) -selon#prp(1)du#prp(1)règlement#nom(1)vous#pro :per(1)vouloir#ver :pres(1)que#kon(1)le#det :art(1)membre#nom(1)dresser#ver :pres(1)un#det :art(1)compte-rendu#nom(1)détailler#ver :pper(1)de#prp(1)leur#det :pos(1)activité#nom(1)

Figure 1 :

 1 Figure 1: Building the semantic space of the word article

  Selon l'article 22 du règlement, vous voulez que les members dressent un compte rendu détaillé de leurs activités Under rule 22 of the rules of procedure, you want us, as members, to give you an exact account of what we do, at what time.

	TL_segment :
								French polysemous
				words,	16	of	which	are	nouns	(article,
	Linguistic	Correponding	Parameter
	knowledge	parameter	modalities			Signification
	Nature of a	ctxt_type	neighbour a unit appearing in a window of size 1 before and after the word
	cooccurrent		syntactic a unit linked to the word by a direct syntactic relation (Subject, Object, …)
			thematic a unit that is neither a neighbor nor a syntactic cooccurrent
	Type of a	type	form	a form only (word or lemma, depending on the modality of ctxt_comp
	cooccurrent			parameter described below)
			form#tag a form and its grammatical tag (word#tag or lemma#tag)
	Lexical form	comp	no	the context vectors are composed of simple word-token units
			yes	multi-word expressions are considered as single units
	Grammatical	ctxt_comp	1	the cooccurrents of the	All the components of the vector
	form and grammatical category		2 3	ambiguous word cooccurrents lemmas of the	Only the nouns, verbs and adjectives are used All the components of the vector

4

Only the nouns, verbs and adjectives are used EQVs similarity sim all WJ coefficient using all the components of the two vectors strong WJ coefficient using only the strong components in the two vectors Table 1: Parameters for the representation of the semantic spaces of the words The ambiguous word : article SL_segment :

Table 2 :

 2 An illustration for the neighbours and thematic context.

barrage, cadre, compte, conclusion, culture, matière, passage, produit, raison, rapport, reserve, société, traitement and vol)

while 4 are verbs (lever, monter, porter and saisir).

Table 3 :

 3 Description of the words and their sub-corpora

	SL linguistic form.POS : {TL EQVs}
	compte.N : {account}
	en fin de compte.Adv. Loc. : {in the end, ultimately}
	se rendre compte de.V : {realise}
	rendre des comptes.V : {be accountable}
	tenir compte de.V : {take into consideration, take into
	account, take account of, give consideration to, be aware
	of, consider}
	compte tenu de.Prep. Loc. : {because (of), considering}
	Cour des Comptes.N : {Court of Auditors}

Table 4 :

 4 Lexicon entries for the word compte

Table 5 :

 5 Global tendencies of the linguistic parameters and their interaction with each other Figure 1: Impact of lemmatisation and grammatical filtering : f-score depending on ctxt_comp parameter (comp is yes and type is form#tag) (reminder : 1:words ; 2:filtered words ; 3:lemmas ; 4:filtered lemmas)

	their modalities	all	sim	strong	form	type form#tag	no	comp	yes	ctxt_comp (1 vs 3) words lemmas
	type	form form#tag	81.6 81.6		81.6 90.5
	comp	no yes	81.3 81.3		81.8 90.5	81.6 81.4	81.8 90.5
	ctxt_comp	words	81.6		82.7	80.8	82.7	80.9		82.7
	(1 vs 3)	lemmas	81.3		90.5	81.7	90.5	81.1		90.5
	ctxt_comp	filtered	80		84.6	75.9	84.9	76		84.5	75.8	84.5
	(2 vs 4)	all	81.6		90.5	81.6	90.5	85.7		90.5	82.7	90.5


Thibaut Le Naour

Nicolas Courty

Sylvie Gibet

Fast Motion retrieval with the distance input space

Accessing and querying large motion databases can be a time consuming operation since motions are by essence big multivariate time series. However, with the help of good dimensionality reduction methods, this problem can be alleviated. Yet, it appears that the intitial motion representation space if of great importance with regards to the final performances. This paper advocates the idea that the distance representation space (i.e. ensemble of joint-to-joint distances of the skeleton), though bigger, can yield a powerful representation of the motion. In order to validate this idea, comparisons with rotation and position based representations are experimented in a motion retrieval task performed with the HDM05 motion database. The final resulting retrieval algorithm is very fast with satisfying performances.

Introduction and related work

There is a large number of tehniques to represent a motion. Associated to these representations, motion retrieval algorithms needs efficient similarity measures. Tradionally, these similarity measures can be separated in two categories: numerical and logical. Among numerical approaches, dimensionality reduction has the advantage to offer a representation smaller than original motion without losing major information. Alexa and Muller[?] are the first ones to introduce PCA technique, which consists in to project a maximum of information on the axis which describe the most significant of the motion. In 2007, Xiang and Zhu[?] presents a new non-linar method: isometric feature mapping based on geodesic distances. Kovar and al [?] proposes a new method based on the distances between point clouds of motion postures. He uses dynamic time warping algorithm (DTW) in particulary to bring out spatial and temporal relations between movements. DTW is an efficient algorithm but it is expensive in time calculation. Other approaches like geometric representation associated to logical laws provide an abstract characterization of the motion. Müller et al.[?] pre-process the motion database by dividing motions according to qualitative geometric features. Later motion template defined by matrices (row represent spatial information and column temporal data) were introduced in [?]. More recently Liang et al.[?] presented an original method based on kinematic and dynamic constraints to retrieve a motion.

In this paper, we highlight the importance of using relevant representations of movements to achieve very fast motion retrieval. We show that a distancebased motion representation, coupled to the inner product in the latent space leads to good recognition rates with a very low computational cost. Our results are tested for movements contained in the database HDM05.

Motion representation

We propose to use the spatial relationships between joint positions in order to characterize a motion. More specifically, a motion is expressed as the evolution of Euclidean distances between skeleton joints over time. This space will be referred to as the distance space. For each posture, we extract a graph G = (V , E, d) where V is the set of vertices and E is the set of edges which are changing over time. The segments composing the skeleton being fixed during the animation, we do not retain them to express the movement. d is therefore the set of distances of V × V → R + . We express in particular the Euclidean distance between the point x i and the point x j , for every (x i , x j ) ∈ E. The spatial structure of the graph can be expressed by the set of equations:

∀(x i , x j ) ∈ E, x i -x j = d ij , (1)
where . is the Euclidian norm. Finally, a motion is represented by a n × m matrix M where n is number of distance and m number of frame composing motion. Figure 1 illustrates this graph for one posture. Contrary to positional or rotational representations, it is interesting to observe that an important amount of data is thus necessary to represent a posture. For example, in a case of a skeleton with 31 joints, 465 distances values are necessary to define the posture, where only 31 * 3 = 93 values are needed in the position space. Information characterizing motion being most of the time redundant (joint orientations, positions, or distances), statistical methods have been introduced to reduce the representation space. In particular, classical methods use Principal Component Analysis (PCA) to decompose motion into an orthogonal subspace, in which each principal components drive a decreasing amount of variance. This approach allows to compress motion information, and to give a simplified and significant representation of motion through the first eigenaxis. We postulate here that the high-dimensional space induced by distances will adapt particularly well to such linear dimensionality reduction techniques, and bring eigenaxis that will capture well the essence of motions. This assumption is inspired by the classical kernel methods theory that first embed data into high-dimensional before conducting linear statistical techniques.

Methodology and results

In order to retrieve motion which are similar to an example motion in the database, we use a similarity measure which is based on the first axis of the PCA method applied on the matrix of distances characterizing each movement. More precisely, for two motions represented by the matrices M i and M j , the similarity measure is defined by the inner product between the principal vectors V i and V j , with V i is the first eigen vector extracted through a singular value decomposition (SVD) applied to M i . Let us note here that this measure allows to work independently of the temporal length of the considered sequences.

Our method has been tested on the database HDM05[?], which contains a collection of 497 motions distributed in 78 different classes. After removing the example-motion from the database, we define for each movement a criteria by searching the nearest neighbors of a movement and testing its class. Hence, one entry c i,j of the confusion matrix will be the number of times a motion of class i has been associated to a motion of class j normalized by the number of motions in class i. Of course, the best retrieval algorithm would give high values along the diagonal. The test is completed for all the movements, without any prior treatment. The results are illustrated in figure 2. It shows the confusion matrices when searching a motion represented by the first eigen vector resulting from the PCA, applied on distances (a), positions (b), and orientations (c). Furthermore, the searching is performed for the first two nearest neighbors. We observe a strong coherency between searched movements and the results in the case of movements represented by distances, and inversely there is no correlation for movements represented by positions or rotations. This results show the pertinence to use distances to represent a motion than positions and rotations. The efficiency rate of our algorihm so the number of success motion retrieve by 2-nearest neighbors algorithm for all motion, is 68.8% and of success without any prior processing of the database (13.7% for positions and 13.9% for rotations).

The comparison time is negligible since the comparison of a motion with the rest of the database corresponds to the inner product of two 465-values vectors.

Conclusion

Our objective was to provide a fast motion retrieval method, capable of finding quickly motions similar to the example motion in a large motion database. Our method is based on a motion representation in the distance space, a motion being characterized by the evolution over time of distances between the skeleton joints. The motion is then reduced by using a principal component analysis. With such an approach, searching a motion within the database is performed simply and quickly through the inner product of the first PCA-axis for each pair of movements. We showed that it is interesting to represent a motion in distance space, in particulary because our technique need more information than position and rotation space to describe a motion. Moreover our technique which can be carried out in real time on non-pretreated motions from the database HDM05, is promising, with a success rate close to 70%. Future works will consider thorough comparisons with state-of-the-art algorithms in motion retrieval.

Fig. 1 .

 1 Fig. 1. Creation of the graph associated to a posture: in yellow the skeleton and in green dotted line the edges of variable lengths.

Fig. 2 .

 2 Fig. 2. Confusion matrices obtained using distance (a), positional (b) and rotational (c) representation space of the motions