Nachwa Abou Bakr
email: nachwa.aboubakr@inria.fr2james.crowley@inria.fr

James Crowley

Histogram of Oriented Depth Gradients for Action Recognition

Keywords: Vision, Profondeur, Reconnaissance d'Actions, Codage MPEG Vision, Depth, Action Recognition, MPEG encoding

Cet article présente des expériences menées sur la perception de la profondeur de mouvements dans le cadre de la reconnaissance visuelle d'actions à partir de mesures locales extraites de séquences vidéo RGBD encodées selon la norme MPEG. Nous démontrons que ces mesures peuvent être combinées avec des descripteurs RGB spatio-temporels locaux pour l'élaboration d'une méthode de calcul efficace pour la reconnaissance d'actions. Des vecteurs de Fisher sont utilisés pour encoder et concaténer un descripteur de profondeur avec les descripteurs existants RGB locaux. Ces vecteurs sont ensuite utilisés pour la reconnaissance d'actions manuelles à l'aide d'un classifieur linéaire SVM. Nous comparons l'efficacité de ces mesures à l'état de l'art sur deux jeux de données récents pour la reconnaissance d'actions dans des environnements culinaires.

Introduction

Visual action recognition is the process of labelling image sequences with action labels. In this work we investigate the use of dense local descriptors for action recognition. Dense sampling has been shown to improve results when compared to sparse interest points for action recognition [START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF]. Requirements for reduced computational time have lead some authors to investigate methods that extract motion vectors directly from compressed video signals. For example, methods have been proposed to extract flow directly from MPEG encoded video [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF]. The resulting method has been shown to provide useful description of Dense Trajectories (DT) [START_REF] Wang | Action recognition by dense trajectories[END_REF], at the cost of some loss of recognition accuracy. This has led us to ask if this approach can be extended to provide fast computation of dense local descriptors from the depth channel of MPEG encoded RGBD image sequences. In the following, we report the results on experiments in recognition of action labels using features extracted from depth channel of Kinect-like sensors using MPEG encoded motion flow. We describe a new descriptor based on oriented gradients of depth extracted from the motion channel of RGBD image sequences. This descriptor is used in addition to the MPEG Flow (MF) descriptors based on the RGB channels from the same sequence. Local video descriptors using Dense Trajectories [START_REF] Wang | Action recognition by dense trajectories[END_REF] and MPEG Flow [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF] are employed as baseline descriptors for comparison. DT, as a video descriptor for action recognition, starts by sampling interest points densely at regular fixed grids. Detected interest points are then tracked over time and points for which no motion is detected are eliminated. This set of trajectories are then used as space-time interest points and descriptors are computed around those points. Four descriptors have been used in DT : Histogram of Oriented Gaussian (HOG), Histogram of Optical Flow (HOF), Motion Boundary Histogram in x,y directions (MBH x , MBH y ). An analysis on computational requirements of DT shows that 61% of the running time is spent on the computation of optical flows. Reducing this computational burden is an interesting challenge. The MF method exploits the estimated motion vectors used for video compression and uses them as interest points. This change provides a significant improvement in the time required for extraction of features at the cost of a small decrease in recognition accuracy [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF]. Both DT and MF are video descriptors computed around motion trajectories. The primary differences between these two methods are that the interest points to be described in MF are at sparse positions of motion fields. In DT, points are sampled at the positions of a regular fixed grid, and they are tracked over time to build trajectories. Motion is encoded in DT using Gunnar-Farnebäck optical flows while MF uses estimated motion fields during video compression. Our goal is to determine if it is possible to improve MF results by exploiting data from depth channel of Kinect-like sensors.

Histogram of Oriented Depth Gradients

Depth can capture the structural features of objects. In addition to encoding the position of objects in space. Thus it is reasonable to ask if we can improve the results of MF video descriptor by including the depth channel to better describe the scene. To do so, we compute gradients of depth which encodes the structure of the depth image. As in MF, interest points are a sparse set of points extracted while compressing video. This ensures describing area around motion points. Practically, a volume of 32 × 32 × 15 is computed around interest points. This volume is divided by a grid of 2 which produces a grid cell of 16 × 5. For each grid cell, we extract the gradients by applying the Sobel filter [1, 0, -1] in a 16 × 16 pixels. Then, an 8-orientation bins discretized histogram is built to encode the gradient orientations. Gradient magnitudes are used to weight the gradient orientations. For each temporal slice the gradient is normalized using l2-normalization.

After that descriptors are encoded by transforming the collection of local video descriptors {x i , . . . , x N } into a fixedsize vector representation. Following [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF], we chose to use Fisher Vectors (FV) [START_REF] Sánchez | Image classification with the fisher vector : Theory and practice[END_REF] for descriptor encoding. particular, only first-order differences of FV are used. Action categories are then classified by a linear one-vs-rest SVM classifier.

Experiments

Experiments are performed on two publicly available datasets : 50 salads [START_REF] Stein | Combining embedded accelerometers with computer vision for recognizing food preparation activities[END_REF] and Actions of Cooking Eggs (ACE) [START_REF] Shimada | Kitchen scene context based gesture recognition : A contest in ICPR2012[END_REF]. For 50 salads dataset ; we used provided annotations of core actions only as verbs without taking into account their corresponding objects ; in total, we used 6 different action labels. ACE also provides 6 different annotated actions but only one level of granularity which includes pre-and post movements of performed actions. For the evaluation, we measured the mean of Average Precision (mAP). We consider two image sequences ; one provides an RGB view of the scene and the other encodes scene depth. To study the impact of motion depth image sequences in describing actions, we extracted a Histogram of Oriented Depth Gradients (HODG) from depth image sequences. We extracted as well RGB descriptors of DT 1 and MF 2 . Combining both descriptors improves the results of MF method on two datasets as shown in Table 1 where RGB+HODG is a concatenation of encoded descriptors by FV. We compare the results to DT since MF follows the same structure and set of descriptors.

DT approach achieved the best results in this experiment at the cost of the computation time. In the other hand, adding HODG to MF descriptors achieves better results than [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF] while preserving the computational cost 3 . Table 2 shows the number of processed frames per second while computing both RGB-based and HODG using the implementation of DT and MF. From the tables, we can also see that HODG alone performs relatively good in action recognition.

Conclusion

In this work, we reported an improvement of the results of MF method by using the benefit of depth channel in Kinectlike sensors. Since depth channels provide separate images that can be treated in parallel with no additional cost or sequentially while preserving a close to real-time capability.

The idea is to combine the gradients of depth image sequences (HODG) extracted at sparse motion fields which are estimated in video compression to the set of RGB descriptors of MF [START_REF] Kantorov | Efficient feature extraction, encoding and classification for action recognition[END_REF]. Both RGB and Depth descriptors are encoded using FV. We found that trajectories of depth image sequences, described by HODG, can describe manipulation actions with an acceptable recognition precision. To this end, we evaluated our proposed method on two recent datasets ; 50 salads, and ACE. We recorded the computational time during extraction of descriptors using both methods. Results show a significant improvement in recognition precision with almost no significant additional cost.

TABLE 1 -

 1 The results of our experiments using mAP.

	DT (fps) MF (fps)

TABLE 2 -

 2 The number of processed frames per second while descriptors extraction for 50 salads dataset.


Bessem Bouraoui
email: bouraoui@lsiit.u-strasbg.fr

Christian Ronse
email: ronse@lsiit.u-strasbg.fr

Joseph Baruthio
email: baruthio@ipb.u-strasbg.fr

Nicolas Passat
email: passat@lsiit.u-strasbg.fr

Philippe Germain
email: phil.germain@free.fr

Gray-level hit-or-miss transform based region-growing for automatic segmentation of 3D coronary arteries

Introduction

In this paper we propose a fully automatic algorithm for coronary artery extraction from X-ray data (3D-CT scan, 64 detectors). Growing and thresholding methods, in their most general form, are not sufficient to extract only the whole coronary arteries, because of the properties of these images [START_REF] Lorenz | Simultaneous segmentation and tree reconstruction of the coronary arteries in MSCT images[END_REF]. Finding appropriate methods is known to be a challenging problem because of the data imperfections such as noise, heterogeneous intensity and contrasts of similar tissues. We deal with these challenges by employing discrete geometric tools to fit on the arteries form independently from any perturbation of the data.

Background

Region-growing

Basically, region-growing methods are based on two main concepts: seed-point(s) and expansion criterion. Starting from the seed-point(s), and exploring the currently segmented object neighborhood, if the expansion criterion is satisfied for a given point, this point is added to the object. With the data we are interested in, it can be assumed that vessels are brighter than the surrounding tissues, but a simple thresholding is not sufficient to define an expansion criterion. To reach our objective two requirements are then essential [START_REF] Passat | Region-growing segmentation of brain vessels: An atlas-based automatic approach[END_REF]:

• no user seed-point selection but an automatically detected one; • a expansion criterion independent of the variation of image properties.

Gray level hit-or-miss transform

The hit-or-miss (HMT) in binary images is a classic mathematical morphology operator [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF], that uses two structuring elements A and B. The operation is made by testing two conditions: A has to belong to the object, while B has to belong to the background. This binary operator is not easily extended to gray level images. We use a definition of HMT operator [START_REF] Naegel | Grey-level hit-or-miss transforms -Part II: Application to angiographic image processing[END_REF] that assigns to A and B gray-levels a and b, respectively. Following this definition, the graylevel (GL) HMT compares the minimum intensity a min in A to the maximum intensity b max in B: if

a min > (b max + a -b)
, then the point is selected by the GL HMT.

Images and coronary artery anatomy

The considered images do not cover only the heart, but a large part of the trunk, including the lungs, the spinal column, the liver and a part of the stomach (Fig. 1(a)). With 512 3 as dimension, a reduction of the search space becomes necessary. There are two main coronary arteries (the left one and the right one): both of them originate from the beginning of the aorta, immediately above the aortic valve. The aorta is the largest single blood vessel in the body, corresponding to a large tubular structure whose axial section is circular at almost 3 cm from the aortic valve and presenting a wall sufficiently thick to be visualized in the data.

Segmentation method 4.1 Heart-zone automatic detection

Using a GL HMT, the detection of the heart zone is realized with two ad hoc structuring elements: a sphere for A, and for B one of two spheres whose centers are horizontally symmetrical w.r.t. the center of A (Fig. 1(b)). The lungs correspond to the background which has to be fit by B, while the heart corresponds to the object which has to be fit by A.

As result, only the heart zone is kept, realizing an important reduction of the search area (Fig. 1(c)).

Aorta automatic detection

About 2 to 3 cm above the aortic valve, an axial section of the aorta presents a circular shape, with a radius varying from 2 cm to 4 cm, this value depending on the patient and the conditions of the image acquisition. Our goal is to localize this circular shape using a GL HMT with multi-sized structuring elements. A horizontal disk (A) with a variable radius models the object. It is surrounded by one background point on every quarter (B) (Fig. 2(a)). The center of the detected circular shape will be used as seed for a first region-growing, the expansion criterion of which will involve two parameters: intensity and texture [START_REF] Flureau | Multi-object segmentation of cardiac MSCT data using SVM classifiers and connectivity algorithm[END_REF]. This region-growing will be stopped by the aorta wall on the sides and by the aortic valve down (Fig. 2(a)).

Coronary arteries seed point detection

The coronary arteries are the only tubular structures that generate from the aorta. The idea is to search this tubular form in the wall of the aorta. Following a strategy close to the one proposed in [START_REF] Naegel | Segmentation automatique de l'entrée de la veine porte dans le foie par application de la transformée en tout-ou-rien en niveaux de gris[END_REF], a GL HMT is applied, with a sphere as object and 4 points surrounding this sphere as background. These points belong to the plane normal to the artery axis. The choice of these points defines the axis direction of the searched coronary artery. 13 structuring elements, corresponding to the 13 discrete principal directions are then used (Fig. 2(b)).

Coronary arteries detection

Starting from the two seed-points previously detected on the aorta wall, a region-growing is performed, applying the GL HMT on every neighbor as criterion. The involved structuring elements are the same as those described in the previous subsection. The obtained result is illustrated in Fig. 2(b).

Discussion and conclusion

We have proposed a fully automatic segmentation method for the coronary arteries. The method was tested on 4 coro-scan images, visually providing satisfactory results. Validation work on 40 other patients is currently in progress. Some improvements are in progress. As mentioned above, images have a rather considerable size, which makes the cost in computing time prohibitive, particularly during the heart zone detection and the segmentation of the aorta (Table 1). A multi-resolution approach has then been integrated. On the first two steps, an under-sampling to 64 3 size image is carried out.

Once the aorta is detected, an over-sampling enables to get back the original size. This approach enables the application of the algorithm with a lower complexity.

Table 1. Effects of the sampling on different detection.

Object / Volume 512 3 64 3 Heart zone 45 mn 5 mn Aorta 10 mn 1 mn Coronary arteries 1 mn -

Final algorithm

Input: Original image 512 3 (Fig. 1(a)).

-Under-sampling to get a 64 3 image.

-Heart zone detection by GL HMT application (Fig. 1(c)).

-Aorta segmentation.

-Circular section aorta detection: center used as seed point.

-Region-growing with intensity and texture as criterion (Fig. 2(a)).

-Over-sampling of the aorta to a 512 3 image.

-Search of the two coronary arteries seed points on the aorta wall, using GL HMT.

-Coronary arteries detection with a region-growing using GL HMT as criterion.

Figure 1 .Figure 2 .

 12 Figure 1. Heart zone detection


Fellow, IEEE Bernard Picinbono
email: bernard.picinbono@lss.supelec.fr

Time Intervals and Counting in Point Processes

Keywords: Counting, point processes, time measurements. I

Time point processes can be analyzed in two different ways: by the number of points in arbitrary time intervals or by distance between points. This corresponds to two distinct physical devices: counting or timeinterval measurements. We present an explicit calculation,v alid for arbitrary regular processes,of the statistical properties of time intervals such as residual or life time in terms of counting probabilities.

For this calculation, we show that these intervals must be considered as random variables defined by conditional distributions.

at a very lowlevel of intensity requires the use of statistical properties of photons or photoelectrons [START_REF] Pierce | The capacity of photon counting channel[END_REF], [START_REF] Saleh | Fundamentals of Photonics[END_REF]. On the other hand, at a macroscopic level many areas such as traffic problems or computer communications require the use of point process statistics [START_REF] Gelenbe | Introduction to Queuing Networks[END_REF].

There are two approaches to describe point processes theoretically or to study them experimentally.

The first one makes use of counting procedures in one or several nonoverlapping time intervals. The appropriate physical devices for this approach are counters. A limit aspect of counting appears in coincidence experiments in which the time intervals of counting are so small that they can only contain one or zero point [START_REF] Macchi | The coincidence approach to stochastic point proccesses[END_REF].

On the otherhand, it is possible to analyze point processes by using time intervals between points measurements. This introduces the concept of residual time, orsur vival time, orw aiting time of order n, which is the time distance between an arbitrary time instant and the nth point of the processes following this instant. It is also possible to study the life time which is the time distance between successive ornonsuccessive points of the processes.

In the stationary case, the calculation of the probability distributions of residual or life times in terms of counting probabilities is known [START_REF] Mcfadden | The axis crossing intervals of random functions[END_REF], [START_REF] Mcfadden | On the lengths of intervals in a stationary point process[END_REF]. However, in many practical situations, the stationarity assumption cannot be introduced and it appears that the direct transposition of the results obtained in the stationary case is not possible. The main reason is that time intervals must be considered as random variables (RV) defined by conditonal distributions.We shall see that this remark is of no importance in the stationary case but it must be taken into account for nonstationary processes. The omission of this fact has resulted in many incorrect expressions appearing in classical books on point processes. This is one of the reasons for analyzing the problem again and more carefully.

Before going further, let us introduce some general concepts and notation that will be used throughout the correspondence. As indicated in the title, we are interested in time point processes, which means that the points are time instants.

We assume that the point processes studied are defined only in a time interval (T i ; T ), where T i and T are the beginning and the end of the processes, respectively. For the sake of simplicity we take T i as the origin of time, or T i = 0.

We denote by N [t 1 , t 2 ) the number of points in the interval [t 1 , t 2 ). It is a discrete-valued RV and the point process is entirely defined if for any set of nonoverlapping intervals[t i , t i +∆t i ) the joint probability distribution of the RVs {N [t i , T i +∆t i ]} is known. These probabilities are denoted counting probabilities, and we shall use the notations

p i (t, τ ) = P {N (t, τ ) = i}. (1)

II. RESIDUAL TIME OF ORDER n

A. General Results

Let t be an arbitrary time instant satisfying 0 = T i ≤ t ≤ T . The residual time of order n is the RV R n (t) equal to the distance between the origin T i and the nth point of the process posterior to t.

It is fundamental to note that this RV does not exist if there are less than n points posterior to t, or if the event N [t; T ) < n is realized. Consequently, the distribution function (DF) of R n (t) defined by

F h (t, τ ) = P [R n (t) ≤ τ ], t ≤ τ ≤ T is the conditional probability F n (t, τ ) = P ({N [t, τ ) ≥ n}|{N [t, T ) ≥ n}) = P ({N [t, τ ) ≥ n}.{N [t, T ) ≥ n}) P (N [t, T ) ≥ n) (2)
As τ < T , the numerator is equal to

P (N [t, τ ) ≥ n) = 1 - n-1 i=0 p i (t, τ ) (3)
The denominator has the same structure but p n (t, τ ) is replaced by π i = p i (t, T ). This yields

F n (t, τ ) = 1 1 -n-1 i=0 π i 1 - n-1 i=0 p i (t, T ) . (4)
It results directly from (2) that F n (t, τ ) is effectively a DF, which means a nondecreasing function varying from 0 to 1 when τ varies from t to T .

The probability density function (pdf) f n (t, τ ) is the derivative of this expression with respect to τ , when it does exist, or

f n (t, τ ) = -1 1 -n-1 i=0 π i n-1 i=0 ∂p i (t, τ ) ∂τ . (5)
It is clear that (4) or ( 5) establish a relation between counting probabilities and statistics of the residual time, which is the objective of this correspondence.

B. Stationary Case

In this case, p i (t, τ ) = p i (t -τ )) and T tends to infinity. As a result, p i = 0 and (4) becomes

F n (t, τ ) = 1 - n-1 i=0 p i (τ -t) ,
where p i (τ -t) is the probability of counting i points between t and τ . This is the classical expression for stationary point processes.

This shows clearly the difference between the stationary and nonstationary cases. When the point process is stationary, it is not necessary to consider a conditional distribution because the event introducing the condition is realized with probability 1. Indeed, except when the process has a zero density, there is always an infinite number of points posterior to any time instant t.

d n (m) = n-1 i=0 p i = exp(-m) 1 + m + ... + m n-1 (n -1)! (6)
This is obviously the probability that a Poisson RV of mean m takes a value smaller than n. With this notation (4) can be written as

F n (t, τ ) = 1 -d n (m) 1 -d n (M )
where

m = m(t, τ ) = τ t λ(θ)dθ, M = M (t, τ ) = T t λ(θ)dθ (7)
In order to calculate the pdf, we note that the derivative of d n (m) defined by ( 6) with respect to m is -m n-1 exp(-m)/(n -1)!. Consequently, the pdf of the residual time is

f n (t, τ ) = 1 1 -d n (M ) e m(t,τ ) m n-1 (t, τ ) (n -1)! = λ(τ ), 0 ≤ τ ≤ T. (8)
This expression can be obtained directly from the properties of a Poisson process. Note that the factor

[1-d n (M )] -1 in (8) ensures that f n (t, τ
) is indeed a pdf. This factoris usually forgotten, which introduces a function that is not a pdf. This, for example, is the case for [7, eq. (3.

3)], [8, eq. (2.20)], [9, eq. (2.3.6)],

or [10, eq. (8.8)]. The reason for these expressions is that the calculations do not take into account the fact that the residual time is an RV defined by a conditional DF.

In order to visualize the effect of the nonstationarity, consider the example of a Poisson process with a density equal to λ in the interval [0, T ) and zero outside. The value of the pdf of the residual time

R n (0) = R n is f n (τ ) = 1 1 -d n (M ) (λτ ) n-1 (n -1)! λe -λτ , 0 ≤ τ ≤ T (9)
and zero outside this interval. It is easy to calculate the mean value of this RV which is

E(R n ) = n λ 1 -d n+1 (M ) 1 -d n (M ) = n λ c n (M ), (10)
where d n and M are defined just above. When T → ∞, c n (M ) → 1 and E(R n ) tends to the value n/λ. This is in agreement with the fact that in a stationary Poisson process of density λ the distances between successive points are independent and identically distributed (i.i.d.) random variables of mean value 1/λ.

Thus, the term c n (M ) can be considered as a correction factor due to the nonstationary character of the process. This correction factor is a function of M and it is clear that the normalization factor 1 -

d n (M )
in the pdf is fundamental to obtain the correct mean value.

DRAFT January 28, 2018

The difference between stationary and nonstationary Poisson processes with constant density is especially important when the density becomes very small. In the stationary case, the mean value 1/λ tends to infinity when λ → 0. This is of course impossible in our particular case of Poisson process with a density equal to zero outside the interval [0, T ). For λ → 0, the mean value E(R n ) defined by ( 10) tends to T [n/n + 1]. For example, for n = 1 this gives T = 2, and this can also be obtained from the pdf f 1 (τ ) which tends to 1/T and shows that the RV R 1 is uniformly distributed in the interval |0, T ).

D. Compound Poisson Processes

Compound Poisson processes, sometimes called doubly stochastic processes [START_REF] Snyderand | Random Point Processes in Time and Space[END_REF], are Poisson point processes in which the density λ(t) is a random function [START_REF] Picinbono | Random Signals and Systems[END_REF]. They play an important role in many areas of physics orinfor mation sciences and especially in optical communications. Indeed, it can be shown by various arguments that they describe the point process of the detection of photons and that the random density is proportional to the random intensity of the optical field [START_REF] Sudarshan | Fundamentals od Quantum Optics[END_REF].

Forthese processes, the calculation of p i and π i require an ensemble average over λ(t). Then all the previous calculations can be used again with the only difference that d n (m) and d n (M ) are replaced by their expectation values with respect to λ(t). The pdf of the residual time thus becomes

f n (t, τ ) = 1 1 -E[d n (M )] ×E [ τ t λ(θ)dθ] n-1 (n -1)! λ(τ ) exp - τ t λ(θ)dθ , (11)
In the case where λ(t) is stationary, E[d n (M )] = 0, and we find once again a known classical expression (see [10, p. 348]).

III. MULTIPLE TIME DISTANCES

Instead of studying a single time distance it is possible to jointly analyze two or several such distances.

In order to simplify the presentation, we will restrict ourselves to the analysis of the case of two distances of order one, the extensions to other cases introducing only notational complexity but no conceptual difficulties.

Considerthe two RVs R 1 (t) and R 2 ((t) as defined at the beginning of Section II. By construction, they satisfy the condition R 1 (t) < R 2 ((t). Let f (t, τ 1 , τ 2 ) be theirjoint pdf. A calculation transposing to this case the method used above yields

f (t, τ 1 , τ 2 ) = -1 1 -π 0 -π1 ∂ 2 p 10 (t, τ 1 , τ 2 ) ∂τ 1 ∂τ 2 (12)
January 28, 2018 DRAFT for t ≤ τ 1 ≤ τ 2 ≤ T and zero otherwise. In this expression, p 10 (t, τ 1 , τ 2 ) is the probability that the intervals [t, τ 1 ) and [τ 1 , τ 2 = contain 1 and 0 points, respectively.

In the case of a nonstationary Poisson process, this pdf is

f (t, τ 1 , τ 2 ) = 1 1 -d 2 (M ) λ(τ 1 )λ(τ 2 ) exp[-m(t, τ 2 )], (13)
where d 2 (M ) is defined by ( 6) and [START_REF] Cox | Point Processes[END_REF]. It is easy to verify that this pdf is normalized in its domain of definition t ≤ τ 1 ≤ τ 2 ≤ T .

IV. CONCLUSION

The purpose of this correspondence was to calculate the statistical properties of the distances between points of a point process in terms of the statistics of counting in some time intervals. This problem was solved a long time ago for stationary processes, but the extension of the results for nonstationary processes has so far been presented in an incorrect way.

We have shown that this extension requires the consideration of time distances between points as conditional random variables and the errors appearing in literature stemmed from overlooking this point.

The consequences of this were analyzed in this correspondence for the residual time and some numerical examples on pure and compound Poisson processes illustrate the calculations.

The analysis was extended forthe life time of any regularpoint process. It was shown that the pdf of the life time can be deduced in terms of counting probabilities in one single interval.

Finally, the procedure was extended for multiple time distances and the expression of the statistics of the distance was given in terms counting probabilities in adjacent intervals.

  Consider a nonstationary Poisson process defined by a density λ(t) equal to zero if t is not in the interval [T i , T ). For the following calculations, it is worth introducing the quantity d n (m) associated to any Poisson distribution of mean value m and defined by

	C. Poisson Processes
	January 28, 2018	DRAFT


Jorge Martinez-Gil
email: jorge.martinez-gil@scch.at

CoTO: A Novel Approach for Fuzzy Aggregation of Semantic Similarity Measures

Keywords: Knowledge-based analysis, Text mining, Semantic similarity measurement, Fuzzy logic

Semantic similarity measurement aims to determine the likeness between two text expressions that use different lexicographies for representing the same real object or idea. There are a lot of semantic similarity measures for addressing this problem. However, the best results have been achieved when aggregating a number of simple similarity measures. This means that after the various similarity values have been calculated, the overall similarity for a pair of text expressions is computed using an aggregation function of these individual semantic similarity values. This aggregation is often computed by means of statistical functions. In this work, we present CoTO (Consensus or Trade-Off) a solution based on fuzzy logic that is able to outperform these traditional approaches.

Introduction

Textual semantic similarity measurement is a field of research whereby two terms or text expressions are assigned a score based on the likeness of their meaning [START_REF] Pirro | A semantic similarity metric combining features and intrinsic information content[END_REF]. Being able to accurately measure semantic similarity is considered of great relevance in many computer related fields since this notion fits well enough in a number of particular scenarios. The reason is that textual semantic similarity measures can be used for understanding beyond the literal lexical representation of words and phrases. For example, it is possible to automatically identify that specific terms (e.g., Finance) yields matches on similar terms (e.g., Economics, Economic Affairs, Financial Affairs, etc.) or an expert on the treatment of cancer could also be considered as an expert on oncology or tumor treatment.

The detection of different formulations of the same concept or text expression is a key method in a lot of computer-related disciplines. To name only a few, we can refer to a) data clustering where semantic similarity measures are necessary to detect and group the most similar subjects [START_REF] Batet | Ontology-based semantic clustering[END_REF], b) data matching which consists of finding some data that refer to the same concept across different data sources [START_REF] Martinez-Gil | Reverse ontology matching[END_REF], c) data mining where using appropriate semantic similarity measures can help to facilitate both the processes of text classification and pattern discovery in large texts [START_REF] Couto | Semantic similarity over the gene ontology: family correlation and selecting disjunctive ancestors[END_REF], or d) automatic machine translation where the detection of terms pairs expressed in different languages but referring to the same idea is of vital importance [START_REF] Chen | Bilingual Sense Similarity for Statistical Machine Translation[END_REF].

Traditionally, this problem has been addressed from two different points of view: semantic similarity and relational similarity. However, there is a common agreement about the scope of each of them [START_REF] Batet | An ontology-based measure to compute semantic similarity in biomedicine[END_REF].

Semantic similarity states the taxonomic proximity between terms or text expressions [START_REF] Pirro | A semantic similarity metric combining features and intrinsic information content[END_REF]. For example, automobile and car are similar because they represent the same notion concerning means of transport.

On the other hand, the more general notion of relational similarity considers relations between terms [START_REF] Punuru | Learning non-taxonomical semantic relations from domain texts[END_REF]. For example, nurse and hospital are related (since they belong to the healthcare domain) but they are far from represent the same real idea or concept. Due to its importance in many computer-related fields, we are going to focus on semantic similarity for the rest of this paper.

There are a lot of semantic similarity measures for identifying semantic similarity. However, the best results have been achieved when aggregating a number of simple similarity measures [START_REF] Do | COMA -A System for Flexible Combination of Schema Matching Approaches[END_REF]. This means that after the various similarity values have been calculated, the overall similarity for a pair of text expressions is computed using an aggregation function of the individual semantic similarity values. This aggregation is often computed by means of statistical functions (arithmetic mean, quadratic mean, median, maximum, minimum, and so on) [START_REF] Martinez-Gil | Smart Combination of Web Measures for Solving Semantic Similarity Problems[END_REF]. Our hypothesis is that these methods are not optimal, and therefore, can be improved. The reason is that these methods are following a kind of compensative approach, and therefore they are not able to deal with the non-stochastic uncertainty induced from subjectivity, vagueness and imprecision from the human language. However, dealing with subjectivity, vagueness and imprecision is exactly one of the major purposes of fuzzy logic. In this way, using techniques of this kind should help to outperform current results in the field of semantic similarity measurement. Therefore, the major contributions of this work can be summarized as follows:

• We propose CoTO (Consensus or Trade-Off), a novel technique for the aggregation of semantic similarity values that appropriately handles the non-stochastic uncertainty of human language by means of fuzzy logic.

• We evaluate the performance of this strategy using a number of general purpose and domain specific benchmark data sets, and show how this new approach outperforms the results from existing techniques.

The rest of this paper is organized as follows: Section 2 describes the state-of-the-art concerning semantic similarity measurement. Section 3 describes the novel approach for the fuzzy aggregation of simple semantic similarity measures. Section 4 describes our evaluations and the results that have been achieved. Finally, we draw conclusions and put forward future lines of research.

Related Work

The notion of textual semantic similarity represents a widely intuitive concept. Miller and Charles wrote:

...subjects accept instructions to judge similarity of meaning as if they understood immediately what is being requested, then make their judgments rapidly with no apparent difficulty [START_REF] Miller | Contextual correlates of semantic similarity[END_REF]. This viewpoint has been reinforced by other researchers in the field who observed that semantic similarity is treated as a property characterized by human perception and intuition [START_REF] Resnik | Semantic Similarity in a Taxonomy: An Information-Based Measure and its Application to Problems of Ambiguity in Natural Language[END_REF]. In general, it is assumed that not only are the participants comfortable in their understanding of the concept, but also when they perform a judgment task they do it using the same procedure or at least have a common understanding of the attribute they are measuring [START_REF] O'shea | Benchmarking short text semantic similarity[END_REF].

In the past, there have been great efforts in finding new semantic similarity measures mainly due it is of fundamental importance in many application-oriented fields of the modern computer science. The reason is that these techniques can be used for going beyond the literal lexical match of words and text expressions. Past works in this field include the automatic processing of text and email messages [START_REF] Lamontagne | Textual Reuse for Email Response[END_REF], healthcare dialogue systems [START_REF] Bickmore | Health dialog systems for patients and consumers[END_REF], natural language querying of databases [START_REF] Erozel | Natural language querying for video databases[END_REF], question answering [START_REF] Moschitti | Kernels on Linguistic Structures for Answer Extraction[END_REF], and sentence fusion [START_REF] Barzilay | Sentence Fusion for Multidocument News Summarization[END_REF].

On the other hand, according to Sanchez el al. [START_REF] Sanchez | Ontology-based information content computation[END_REF]; most of these existing semantic similarity measures can be classified into one of these four main categories.

1. Edge-counting measures which are based on the computation of the number of taxonomical links separating two concepts represented in a given dictionary [START_REF] Leacock | Combining Local Context and WordNet Similarity for Word Sense Identification, WordNet: An Electronic Lexical Database[END_REF].

2. Feature-based measures which try to estimate the amount of common and non-common taxonomical information retrieved from dictionaries [START_REF] Petrakis | X-similarity: computing semantic similarity between concepts from different ontologies[END_REF].

Information theoretic measures which try to determine similarity between concepts as a func-

tion of what both concepts have in common in a given ontology. These measures are typically computed from concept distribution in text corpora [START_REF] Jiang | Semantic similarity based on corpus statistics and lexical taxonomy[END_REF]. [START_REF] Batet | Ontology-based semantic clustering[END_REF]. Distributional measures which use text corpora as source. They look for word co-occurrences in the Web or large document collections using search engines [START_REF] Bollegala | A Web Search Engine-Based Approach to Measure Semantic Similarity between Words[END_REF].

It is not possible to categorize our work into any of these categories. The reason is that we are not proposing a new semantic similarity measure, but a novel method to aggregate them so that individual measures can be outperformed. In this way, semantic similarity measures are like black boxes for us.

However, there are several related works in the field of semantic similarity aggregation. For instance COMA, where a library of semantic similarity measures and friendly user interface to aggregate them are provided [START_REF] Do | COMA -A System for Flexible Combination of Schema Matching Approaches[END_REF], or MaF, a matching framework that allow users to combine simple similarity measures to create more complex ones [START_REF] Martinez-Gil | An Ontology Matching Framework[END_REF].

These approaches can be even improved by using weighted means where the weights are automatically computed by means of heuristic and meta-heuristic algorithms. In that case, most promising measures receive better weights. This means that all the efforts are focused on getting more complex weighted means that after some training are able to recognize the most important atomic measures for solving a given problem [START_REF] Martinez-Gil | Evaluation of two heuristic approaches to solve the ontology meta-matching problem[END_REF]. There are two major problems that make these approaches not very appropriate in real environments: First problem is that these techniques require a lot of training efforts.

Secondly, these weights are obtained for a specific problem and it is not easy to find a way to transfer them to other problems. As we are going to see in the next section; CoTO, the novel strategy for fuzzy aggregation of atomic measures that we present here, represents an improvement over traditional statistical approaches, and do not incur in the drawbacks from the heuristic and meta-heuristic ones, since it does not require any kind of training or knowledge transfer.

Fuzzy aggregation of semantic similarity measures

Currently, the baseline approach for computing the degree of semantic similarity between a pair of text expressions is based on an aggregation function of the individual semantic similarity values. This approach has proven to achieve very good results in practice. The idea is simple: to use quasi-linear means (like the median, the arithmetic mean, the geometric mean, the root-power mean, the harmonic mean, etc.) for getting the overall similarity score. In this way, we do not rely in an sole measure for taking important decisions. If there are some individual measures that do not perform very well for a given case, their effects are blurred by other measures that perform well. However, all these approaches present a major drawback: none of the operators is able to model in some understandable way an interaction between the different semantic similarity measures.

To overcome this limitation, first we develop a fuzzy membership function to capture the importance of different semantic similarity measures, and then use an operator for aggregation of multiple similarity measures corresponding to different features of semantic similarity. Experimental evaluations included in the next section will confirm the suitability of the proposed method.

Fuzzy modeling of semantic similarity

During a long time, similarity in general and semantic similarity in particular have been unknown and intangible attributes for the research community. According to O'Shea et al. the question that had to be faced was: Is similarity just some vague qualitative concept with no real scientific significance? [START_REF] O'shea | Benchmarking short text semantic similarity[END_REF]. To answer the question a broad survey of the literature, taking in as many fields as possible, was conducted. This revealed a generalized abstract theory of similarity [START_REF] Tversky | Features of Similarity[END_REF], tying in with well-respected principles of measurement theory, many uses as both a dependent and independent variable in the fields of Cognitive a specific value is one, then the linguistic statement corresponding to the linguistic term applies in all respects (e.g. old for an age of 80 years). If, in contrast, it is zero, then there is absolutely no agreement (e.g. "very young" for an age of 80 years).

Since most fuzzy sets have a universe of discourse consisting of the real line R, it would be impractical to list all the pair defining a membership function. A more convenient and concise way to define a membership function is to express it as a mathematical formula. This can be expressed by means of the following equation. The parameters a, b, c, d (with a < b <= c < d) determine the x coordinates of the four boundaries of the underlying membership function.

m(x; a, b, c, d) = max min x -a b -a , 1, d -x d -c , 0
In our case, we have three linguistic terms for assessing the degree of semantic similarity between two terms or text expressions: bad, fair and good1 . Our membership function states the membership of each of these linguistic terms in the form of a trapezoid bounded between zero and one. Figure 1 shows us this more clearly: each linguistic value can belong to one of the three linguistic terms. Sometimes, a given linguistic value can belong (to some extent) to two or more different linguistic terms. For example, the semantic similarity for the word pair vehicle-motorbike can be assessed as 0.4 fair and 0.6 good (maybe 4 experts said fair and 6 experts said good). This fact allows us to model semantic similarity in a non-compensative way, thus, a much more flexible way that traditional approaches. As a result, more sophisticated aggregation schemes can be proposed.

Fuzzy aggregation of atomic measures

In the field of semantic similarity measurement, aggregation functions are generally defined and used to combine several numerical values (from the different semantic similarity measures to be aggregated) into a single one, so that the final result of the aggregation takes into account all the individual values in a given manner. The fundamental similarity measures which cover many specific characteristics from text strings are the most widely used measures in state of the art. However, the real issue arises when these similarity measures give different results for the same scenario. Different techniques have been used to aggregate the results of different similarity measures. Most of them have reached a high level of success [START_REF] Martinez-Gil | Smart Combination of Web Measures for Solving Semantic Similarity Problems[END_REF].

In fuzzy logic, things are a little bit different. Values can belong to either single numerical or non numerical scale, but the existence of a weak order relation on the set of all possible values is the minimal requirement which has to be satisfied in order to perform aggregation. Nevertheless, the values to be aggregated belong to numerical scales, which can be of ordinal or cardinal type. Once values are defined, it is possible to aggregate them and obtain new value defined on the same scale, but this can be done in many different ways according to what is expected from the aggregation operation, what is the nature of the values to be aggregated, and what kind of scale has been used [START_REF] Grabisch | Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes[END_REF].

It is necessary to remark that aggregation is a very extensive research field in which numerous types of aggregation functions or operators exist. They are all characterized by certain mathematical properties and aggregate in a different manner. But in general, aggregation operators can be divided into three categories [START_REF] Grabisch | Fuzzy integral for classification and feature extraction[END_REF]: conjunctive, disjunctive and compensative operators.

• Conjunctive operators combine values as if they were related by a logical AND operator. That is, the result of combination can be high only if all the values are high.

• Disjunctive operators combine values as an OR operator, so that the result of combination is high if at least one value is high.

• Compensative operators which are located between min and max (bounds of the t-norm and tconorm families). In this kind of operators, a bad (good) score on one criterion can be compensated by a good (bad) one on another criterion, so that the result will be medium. To be more formal, our CoTo aggregation operator on a fuzzy set (2 ≥ n) is defined by a function

h : [0, 1] n → [0, 1]
which follows these axioms:

• Boundary condition: h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = 1

for all i ∈ N n , if a i ≤ b i for all i ∈ N n , then h(a 1 , a 2 , ..., a n ) ≤ h(b 1 , b 2 , ..., b n ); that is, h is
monotonic increasing in all its arguments.

• Continuity: h is a continuous function.

We need a fuzzy associative matrix for implementing our strategy. A fuzzy associative matrix expresses fuzzy logic rules in tabular form. These rules take n variables as input, mapping cleanly to a vector. Linguistic terms are bad (the two text entities to be compared are not similar at all), fair (the two text entities to be compared are moderately similar) and excellent (the two text entities to be compared are very similar). A linguistic term reaches a consensus when it receives the highest number of votes, in that case its associated fuzzy set will be the result of the aggregation process. In case, two or more linguistic terms may receive the same major amount of votes2 , two or more fuzzy sets will be combined in a desirable way to produce a single fuzzy set. This is exactly the purpose of our CoTo aggregation operation. Our final overall score will be computed by means of the trade-off of their respective associated fuzzy sets. This trade-off can be achieved by any of the traditional processes of producing a quantifiable result by means of defuzzification.

Even once the fuzzy model has been defined, it is necessary to configure some parameters concerning the fuzzy terms from the model. This means that it is necessary to perform a parametric study about the degree of overlapping between trapezoids, number of linguistic terms, defuzzification method, etc.

for deciding when a pair of text expressions is going to be considered or not semantically equivalent.

This can be achieved by means of parameter tuning and refinement. Parameter tuning consists of optimizing the internal running parameters in order to maximize the fulfillment of our goal (replicate the human behavior when assessing the semantic similarity of each of the expression pairs contained in the benchmark data sets we are going to solve). In this case, we refer to the the maximization of efficiency (or error minimization) so that the benchmark data set can be solved with a minimum number of errors.

This fact is of vital importance since the smaller number of errors, the greater the quality of the results obtained by our strategy.

For the defuzzification process, we have chosen the method Center of Gravity (CoG) (a.k.a. fuzzy centroid method) to find the final non-fuzzy value associated with the semantic similarity between the text expressions to be compared. This classical method consists of computing the center of gravity for the area under the curve determined by the rules triggered, and we have chosen it because this method represents a trade-off between the rules triggered (what it is exactly the Trade-Off we have mentioned along the manuscript). This method can be computed as it is expressed in the following formula:

CoG = b x=a µ A (χ)x b x=a µ A (χ)
This method is similar to the formula for calculating the center of gravity in physics. The weighted average of the membership function or the center of the gravity of the area bounded by the membership function curve is computed to be the most crisp value of the fuzzy quantity.

Figure 2 shows us a summary of the whole process for clarification purposes. This process starts by encoding a numerical value into a linguistic term by matching the given value within the limits of the existing fuzzy sets these linguistic terms represent. Then, each linguistic term serve as an input for the rule engine which implements the aggregation operator (CoTO). One of the advantages of fuzzy logics is that the design of complex rules engines become an intuitive task (mainly due to the proximity of the linguistic terms to natural language). In a further step, the rule engine triggers the rules that configure the resulting fuzzy set. Finally, the final aggregated score is retrieved by computing the CoG of the resulting fuzzy set.

Evaluation

The comparison between the aggregated value for semantic similarity measures and human similarity judgments is going to be calculated in terms of correlation between the two sets of ratings, thereby giving us a qualitative assessment of the correlation with human similarity judgments. This in turn is an indication of the usefulness in, for example, an information retrieval task. In this section we summarize the main experiments and the results obtained in our study. We have used three different benchmark data sets. Firstly, we aim to measure semantic similarity for general terms, to do that we are going to use the Miller-Charles benchmark data set [START_REF] Miller | Contextual correlates of semantic similarity[END_REF] which is intended for measuring the quality of artificial techniques when assessing the semantic similarity of general words. Secondly, we are going to test our approach using two domain specific benchmark data sets from the biomedical field. First of them is called the Biomedical Medical Subject Headings (MeSH) [START_REF] Pedersen | Measures of semantic similarity and relatedness in the biomedical domain[END_REF] and it is intended for measuring the quality of artificial techniques when assessing the semantic similarity of very specific words belonging to the field of the biomedicine. The second one is a benchmark data set concerning medical disorders that was created by Pedersen et al. in collaboration with Mayo Clinic experts [START_REF] Pedersen | Measures of semantic similarity and relatedness in the biomedical domain[END_REF]. Finally, we discuss the result from our experiments.

It is important to remark, that our technique is going to be compared to baseline aggregation methods.

This baseline strategy consists of using the following family of means:

x(m) = 1 n • n i=1 x m i 1 m
By choosing different values for the parameter m, the following types of means are obtained: m → ∞ maximum, m = 2 quadratic mean, m = 1 arithmetic mean, m → 0 geometric mean, m = -1 harmonic mean, m → -∞ minimum. It is also necessary to explain that all tables, except those for the Miller & Charles ratings, are normalized into values in [0, 1] range for ease of comparison. This means that we cannot include geometric and harmonic means since we allow the value 0 when assessing semantic similarity and this may involve a error concerning division by zero.

In summary, from a strictly mathematical point of view, solving this problem consists on obtaining the maximum value for the Pearson Correlation Coefficient [START_REF] Ahlgren | Requirements for a cocitation similarity measure, with special reference to Pearson's correlation coefficient[END_REF] of two numeric vectors, one generated by human experts and other generated by a computational algorithm. The final result can vary between -1 (results from humans and the proposed algorithm are exactly the opposite) to 1 (results from humans and the proposed algorithm are exactly the same). Obviously, our challenge is to obtain a score of 1 what may mean that our solution is able to perfectly replicate human behavior. It is also important to remark that we compute the p-value for each result. The p-value is the value representing the probability to find the given result if the correlation coefficient were in fact zero (null hypothesis). If this probability is lower than the conventional 5.0 • 10 -2 , then the correlation coefficient can be considered as statistically significant.

General purpose data set

First experiment is performed by using the Miller-Charles benchmark data set [START_REF] Miller | Contextual correlates of semantic similarity[END_REF] which is a widely used reference data set for evaluating the quality of new semantic similarity measures for word pairs.

The rationale behind this way to evaluate quality is that each result obtained by means of artificial techniques may be compared to human judgments. Therefore, the ultimate goal is to replicate human behavior when solving tasks related to semantic similarity without any kind of supervision. Table 1 shows us the 30 word pairs of the data set. The columns called WordA and WordB represent the word pairs belonging to the Miller-Charles benchmark data set. This collection of word pairs ranges from words which are not similar (for instance, rooster-voyage) to word pairs that are synonyms according to human judgment (for instance, automobile-car). Column called Human represent the opinion provided by people. This opinion was originally given in numeric score in the range [0, 4] where 0 stands for no similarity between the two words from the word pair and 4 stands for complete similarity. There is no problem when assessing semantic similarity using values belonging to the interval [0, 1] since the Pearson correlation coefficient is invariant against a linear transformation.

For the first experiment, we aim to smartly aggregate semantic similarity measures between terms which consists of using dictionaries. These measures are: a) Hirst, b) Jiang, c) Resnik, d) Leacock and e) Lin. A detailed description of these measures is out the scope of this work, but some explanatory insights are described in [START_REF] Budanitsky | Evaluating WordNet-based Measures of Lexical Semantic Relatedness[END_REF]. For us, it is enough to know these single measures are the state-of-the-art in the field of semantic similarity measurement [START_REF] Chaves-Gonzalez | Evolutionary algorithm based on different semantic similarity functions for synonym recognition in the biomedical domain[END_REF]. Table 2: Results for the aggregation of the different semantic similarity measures based on measures taking advantage of dictionaries. Some baseline strategies are able to reduce the risk of using only one measure for taking decisions. However, CoTO beats all simple similarity measures and compensative operators. Moreover, the results are statistically significant.

Table 2 shows the results for the aggregation of the different semantic similarity measures based on dictionary measures. Some traditional aggregation strategies (baseline approach) are in line with the best single measures, but the best score is achieved by using CoTO. It is also important to remark that the results we have achieved are statistically significant.

In our second experiment, we aim to determine the degree of semantic similarity between terms which consists of using the knowledge inherent in the historical search logs from the Google search engine. We have decided to perform our first experiment using four semantic similarity measures exploiting historical search patterns on the Web [START_REF] Martinez-Gil | Semantic similarity measurement using historical google search patterns[END_REF]. These semantic similarity measures are: a) frequent Table 3: Results for the aggregation of the different semantic similarity measures based on measures taking advantage of Google historical data. Some baseline aggregation strategies outperform single measures. However, CoTO beats all simple similarity measures and compensative operators. Moreover, the results are statistically significant.

measures is a distributional measure. The reason is these measures try to determine the likeness between terms by means of a smart analysis of their occurrences in the historical web search logs from Google.

Table 3 shows the results for the aggregation of the different semantic similarity measures based on measures taking advantage of Google historical data. Some traditional aggregation strategies (Quadratic mean, Arithmetic mean and Median) outperform single measures, but the best score is achieved, once again, by using CoTO.

Now we propose a new experiment using another kind of semantic similarity measure: the Normalized Google Distance [START_REF] Cilibrasi | The Google Similarity Distance[END_REF]. This semantic similarity measure consists of computing the the number of hits returned by the Google search engine for a given set of keywords. The rationale behind this is that terms with similar meanings in a natural language sense tend to be close in units of Normalized Google Distance, while words with dissimilar meanings tend to be farther apart. This approach only uses the But CoTO beats all simple measures and compensative operators.

probabilities of search terms extracted from the web corpus in question. Assuming that x and y are the terms to be compared, the formula is:

N GD(x, y) = max{log f (x), log f (y)} -log f (x, y) log M -min{log f (x), log f (y)}
Additionally, to perform this experiment we are using also other web search engines Ask, Bing and Yahoo!. This idea was introduced in [START_REF] Martinez-Gil | Smart Combination of Web Measures for Solving Semantic Similarity Problems[END_REF]. Then, we are going to aggregate the values using our family of means and to compare the results with our CoTO strategy.

Table 4 shows the results for the aggregation of the different s measures based on Google Distance over popular web search engines (Ask, Yahoo!, Bing and Google). Once again, some baseline aggregation strategies (Quadratic mean, Arithmetic mean and Median) outperform single measures. But once again, CoTO beats all simple semantic similarity measures and compensative operators.

Domain specific data sets

MeSH, the first of the biomedical benchmark data sets is composed by a set of 36 word pairs extracted from the MeSH data set [START_REF] Pedersen | Measures of semantic similarity and relatedness in the biomedical domain[END_REF]. Table 5 shows us a part of this data set. The columns called ExpressionA and ExpressionB represent the text expressions belonging to this benchmark data set. Column called Human represent the opinion provided by 8 medical experts. The similarity between text expressions have been also assessed between 0 and 1. Therefore, this data set ranges from biomedical expressions which are not similar (for instance, Anemia-Appendicitis) to expression pairs that are synonyms according to expert judgment (for instance, Antibiotics-Antibacterial Agents).

Table 6 shows the results for the aggregation of the different semantic similarity measures based on cutting-edge similarity measures from the biomedical domain. Explaining each of them is out of the scope of this work, but a detailed description can be found in [START_REF] Chaves-Gonzalez | Evolutionary algorithm based on different semantic similarity functions for synonym recognition in the biomedical domain[END_REF]. Once again, the strategy CoTo (Consensus or Trade-Off) is able to beat all the single measures as well as all the compensative operators by a wide margin.

Concerning the second benchmark data set from the biomedical domain. It was created by Pedersen et al. in collaboration with experts from the Mayo Clinic [START_REF] Pedersen | Measures of semantic similarity and relatedness in the biomedical domain[END_REF]. Table 7 shows us this data set. We have to say that the columns called ExpressionA and ExpressionB represent the expressions pairs belonging to the Pedersen-Mayo Clinic benchmark data set. This collection of 30 text expressions ranges from cases which are not similar (for instance, Hyperlidpidemia-Metastasis) to other cases that are synonyms according to the expert judgment (for instance, Renal failure-Kidney failure). Column called Human represents the rating provided by experts from the biomedical domain.

Table 8 shows the results for the aggregation of the different semantic similarity measures based on cutting-edge similarity measures. Detailed description for each of these algorithms can be found in [START_REF] Chaves-Gonzalez | Evolutionary algorithm based on different semantic similarity functions for synonym recognition in the biomedical domain[END_REF].

Once again, we have that the strategy CoTO (Consensus or Trade-Off) is able to beat, once again, all the single measures as well as all the compensative operators by a wide margin.

Discussion

Results show us that our CoTO strategy is able to consistently beat existing approaches based on compensative operators when solving both general purpose and domain specific data sets. In fact, CoTO has outperformed all existing semantic similarity measures and aggregation methods in all experiments performed in this study. Moreover, the results obtained were statistically significant. The reason is that unlike baseline aggregation techniques based on compensative operators, this aggregation strategy requires a consensus or at least a trade-off between between majority opinions. This means that dissident votes are not taken into account to compute the overall semantic similarity score. Therefore, our initial hypothesis seems to be true: dissident values may decrease the final quality of the overall semantic similarity score, so this fact can help to outperform current aggregation techniques based on compensative operators.

The major reason for getting these good results is that fuzzy logic and the classic compensative approach try to address different forms of uncertainty. Whereas both fuzzy logic and the classic compensative approach can represent degrees of certain kinds of subjective judgment, CoTO uses the concept of fuzzy set membership, i.e., how much a variable is in a set (there is not necessarily any uncertainty about this degree), and the classic compensative approach uses the concept of subjective probability, i.e., how probable is it that a variable is in a set. The technical consequence of this distinction is that CoTO relaxes the axioms of the classical compensative approach, which are derived from adding uncertainty, but not degree, to the crisp values of subjective judgments.

Conclusions & Future Work

In this work, we have presented a novel approach for the fuzzy aggregation of semantic similarity measures. This novel approach can be summarized using the motto Consensus or Trade-off what means that atomic semantic similarity measures have to be aggregated without reflecting dissident recommendations in case of a consensus have been reached or using a high degree of trade-off in case a recommendation consensus from atomic measures does not exist. Results show us that this novel approach is able to consistently beat existing approaches based on compensative operators when solving both general purpose and domain specific data sets.

In future, demanding applications where high accuracy of understanding of the user intent is needed, the stakes are high and the users may present adversarial or disruptive characteristics in interacting with systems will require the use of very precise semantic similarity measures. We want to investigate what happens when the amount of linguistic terms for assessing semantic similarity measurement is increased.

Additionally, it could be interesting to explore the horizontal aggregation of semantic similarity measures, i.e. the aggregation of single measures of different nature. Positive results in this context could lead to computers to be able to recognize and predict the semantic similarity between text expressions without requiring any kind of human intervention.

Figure 1 :

 1 Figure1: Fuzzy degrees of semantic similarity using three linguistic terms. Please note that, in this case, each linguistic value can belong (to some extent) to two different linguistic terms

Figure 2 :

 2 Figure 2: Overall summary of the fuzzy aggregation process: a) Values from n semantic similarity measures (ssm) are fuzzificated into linguistic terms, b.1) a rule engine determines if there is a consensus between linguistic terms and triggers n rules (rt) accordingly, b.2) if there is a consensus, only a resulting fuzzy set will be generated, if not, two o more fuzzy sets representing the (two or more) most voted choices will be generated, c) the CoG of the resulting set(s) is computed

  co-occurrence of terms in search patterns, b) computation of the relationship between search patterns, c) outlier coincidence in search patterns, and d) forecasting comparisons. Each of these semantic similarity

•

  Monotonicity: For any pair a 1 , a 2 , ..., a n and b 1 , b 2 , ..., b n of n-tuples such that a i , b i ∈ [0, 1]

Table 1 :

 1 Miller-Charles benchmark data set. Human ratings are between 0 (not similar at all) and 4

	WordA	WordB	Human
	rooster	voyage	0.08
	noon	string	0.08
	glass	magician	0.11
	cord	smile	0.13
	coast	forest	0.42
	lad	wizard	0.42
	monk	slave	0.55
	forest	graveyard	0.84
	coast	hill	0.87
	food	rooster	0.89
	monk	oracle	1.10
	car	journey	1.16
	brother	lad	1.66
	crane	implement	1.68
	brother	monk	2.82
	implement tool	2.95
	bird	crane	2.97
	bird	cock	3.05
	food	fruit	3.08
	furnace	stove	3.11
	midday	noon	3.42
	magician	wizard	3.50
	asylum	madhouse	3.61
	coast	shore	3.70
	boy	lad	3.76
	journey	voyage	3.84
	gem	jewel	3.84
	automobile car	3.92
	(totally similar)

Table 4 :

 4 Results for the aggregation of the different semantic similarity measures based on GoogleDistance over web search engines. Some baseline aggregation strategies outperform single measures.

Table 6 :

 6 Results for the aggregation of the different semantic similarity measures based on cutting-edge similarity measures. The strategy CoTO (Consensus or Trade-Off) is able to beat all the single measures as well as all the compensative operators by a wide margin. The results are statistically significant.Moreover, the results are statistically significant.

	ExpressionA	ExpressionB	Human
	Renal failure	Kidney failure	1.000
	Heart	Myocardium	0.750
	Stroke	Infarct	0.700
	Abortion	Miscarriage	0.825
	Delusion	Schizophrenia	0.550
	Congestive heart failure Pulmonary edema	0.350
	Metastasis	Adenocarcinoma	0.450
	Calcification	Stenosis	0.500
	Diarrhea	Stomach cramps	0.325
	Mitral stenosis	Atrial fibrillation	0.325
	C. pulmonary disease	Lung infiltrates	0.475
	Rheumatoid arthritis	Lupus	0.275
	Brain tumor	Intracranial hemorrhage	0.325
	Carpel tunnel syndrome Osteoarthritis	0.275
	Diabetes mellitus	Hypertension	0.250
	Acne	Syringe	0.250
	Antibiotic	Allergy	0.300
	Cortisone	Total knee replacement	0.250
	Pulmonary embolus	Myocardial infarction	0.300
	Pulmonary fibrosis	Lung cancer	0.350
	Cholangiocarcinoma	Colonoscopy	0.250
	Lymphoid hyperplasia	Laryngeal cancer	0.250
	Multiple sclerosis	Psychosis	0.250
	Appendicitis	Osteoporosis	0.250
	Rectal polyp	Aorta	0.250
	Xerostomia	Alcoholic cirrhosis	0.250
	Peptic ulcer disease	Myopia	0.250
	Depression	Cellulites	0.250
	Varicose vein	Entire knee meniscus	0.250
	Hyperlidpidemia	Metastasis	0.250

Table 7 :

 7 Mayo Clinic biomedical benchmark. Human ratings are between 0 (not similar at all) and 1

(totally similar)

We will investigate approaches using a larger amount of linguistic terms in the future

For example, a scheme with 5 semantic similarity measures, where bad receives 1 vote, fair receives 2 votes and excellent receives 2 votes

Acknowledgments

We would like to thank the reviewers for their time and consideration. This work has been funded by Vertical Model Integration within Regionale Wettbewerbsfaehigkeit OOE 2007-2014 by the European Fund for Regional Development and the State of Upper Austria.

Method