dataesr/scientific-paragraphs-categorization huggingface dataset:
Note that more general concepts of efficiency can be used as in  [11] :  (4) , which depends on θ(0) and more generally any function h(θ(0), x 0 (T ), E(γ)). Eff ∼ x 0 (T ) 2 M 11 (γ(0))/E(γ), with M = (M ij ) ij defined in The main contribution of this article is that in the optimal control theory frame, one can construct a one parameter family of geodesics strokes for the metric defined by g, providing a foliation of the interior of the triangle T whose boundary corresponds to the abnormal geodesic stroke, see Fig. 2. A unique stroke of this family is the solution which maximizes the efficiency. A whole section is devoted to the analysis of the optimal status of the abnormal triangle stroke, completing the explicit evaluation of the efficiency, outlined in  [6]. This construction is based on the application of the Maximum Principle  [18, 23]  supplemented by second order optimality conditions  [7]  and such curves are numerically evaluated using continuation and shooting methods implemented in the HamPath software  [12].
Numerical simulations. Numerical simulations using the HamPath software are presented on Fig. 4  based on the geometric analysis and to achieve the computations of strokes corresponding respectively to simple loops, limaçons and eight shape curves. Note that they are performed using the Euclidean cost. • Simple loops. They are obtained by continuation of small circles. • Limaçons. They are generated by perturbations of simple loops, by period doubling. • Eight-shapes. They appear only on the sides of the triangle.
Applications: numerical simulations We present below a sequence of simulations using the HamPath software, considering the Euclidean cost 2π 0 (u 2 1 + u 2 2 ) dt, and the mechanical energy cost (3). Also we consider a non symmetric case where we minimize the cost 2π 0 (u 2 1 -cos(θ 1 )/2 u 1 u 2 +1/3u 2 2 ) dt and the simple loop strokes are not anymore symmetric with respect to the line Σ (see Fig. 8 ). Remark 3. 3. The existence of minimizers is guaranteed to the fact that efficiency of small strokes tends to zero when the amplitude goes to zero, thanks to Green formulae. In particular we shall consider: • Different kinds of normal strokes for the mechanical cost: simple loop, limaçon, eight and computations of conjugate points (see Fig. 5 ). Only simple loops are candidates to be optimal strokes.
A method of clustering a collection of datasets measured on the same individuals, called CLUSTATIS, was introduced and applied to the segmentation of the subjects participating in a projective mapping experiment or a free sorting task. A refinement of this method of clustering is proposed. It consists in segmenting the subjects while discarding those subjects who can be considered as atypical because they do not fit to the pattern of any cluster. This strategy of analysis requires the determination of a threshold parameter that delineates the boundary between the main clusters and the noise cluster that contains the atypical subjects. An appropriate selection of this parameter is proposed. The general strategy of analysis is illustrated on the basis of a simulation study and data from projective mapping and free sorting experiments.
Introduction CLUSTATIS is a method of cluster analysis of datasets that pertain to the same individuals. In all the applications considered hereinafter, the individuals refer to products and we shall henceforth designate them as such. CLUSTATIS can be applied to situations where the datasets do not necessary refer to the same variables  (Llobell, Cariou, Vigneau, Labenne, Qannari, 2018). It was successfully applied to the case of projective mapping, also called Napping, and free sorting data. In these experiments, the data from each subject are recorded in a specific dataset. The aim is to find homogeneous clusters of datasets, which for the two tasks considered herein (i. e. , Napping and free sorting) will result in a segmentation of the subjects participating to these tasks. However, it very often occurs that there are atypical datasets which do not fit to any particular pattern.
Notwithstanding, these datasets are assigned to one or the other of the clusters and, as a result, the homogeneity of the clusters can be affected and the performance of the clustering algorithm can be deteriorated  (Dave, 1991). A strategy for identifying atypical subjects and discarding them is discussed and applied to three case studies pertaining to projective mapping/Napping and free sorting experiments. As a matter of fact, we follow a strategy of analysis similar to that described by  Vigneau, Qannari, Navez and Cottet (2016)  for the cluster analysis of variables. We also refer to this paper for a general discussion and a brief review of several strategies to deal with atypical data in cluster analysis. In sensory analysis, the atypical subjects that we seek to identify may correspond to outlying subjects, subjects with no clear assessment, undecided subjects, etc.
This kind of behaviour was discussed in the context of preference studies by  Westad, Hersleth and Lea (2004). The strategy for identifying atypical datasets is to include in the optimization criterion associated with CLUSTATIS an additional cluster " + 1" besides the main clusters of datasets. All the datasets that do not seem to fit to the pattern of any of the main clusters will be assigned to the " + 1" cluster. Dave (1991)  who was the first to formalize the concept of the " + 1" strategy, refers to this additional cluster as a 'noise cluster'. A noteworthy feature of this strategy of analysis is that it is simple and intuitively appealing. In the framework of projective mapping/Napping task, three segmentation methods were discussed by  Berget, Varela & Naes (2019)  within the framework of projective mapping.
We outline a comparison of CLUSTATIS with two of these methods on the basis of a case study. The paper is organised as follows. In the "Material and methods" section, we start by sketching a brief reminder of CLUSTATIS and discuss new properties of this method of analysis. In particular, we outline a new formalization of the optimisation criterion which will turn out to be central to the identification of the atypical datasets. We briefly recall how CLUSTATIS can be applied to data from projective mapping and free sorting experiments. Thereafter, we introduce the " + 1" strategy for identifying atypical subjects for the CLUSTATIS method. We put a particular emphasis on how to select the threshold parameter that delineates the boundary between the main clusters and the noise cluster. We finish this section by a comparison of methods.
In section 3, we outline a simulation study to demonstrate the appropriateness of the selected threshold parameter for the determination of the noise cluster. Thereafter, we illustrate the general strategy of analysis on the basis of data from a projective mapping/Napping (subsection 3. 2). A second case study is particularly devoted to the comparison of CLUSTATIS with alternative methods (subsection 3. 3). A third case study is dedicated to the application of CLUSTATIS to free sorting data (subsection 3. 4). We end the paper by a general discussion and concluding remarks (section 4).
CLUSTATIS Let us consider datasets denoted by ,… , , which are assumed to be column centred. These datasets are measured on the same products but the variables may differ from one dataset to another. CLUSTATIS revolves around the STATIS method  (Lavit, Escoufier, Sabatier, Traissac, 1994; Schlich, 1996). The cornerstone of this strategy of analysis is the scalar product matrices associated with the datasets at hand: = ,…,.
=. These are × symmetric matrices which reflect the spatial configuration of the products since the entry corresponding to the row and column gives the scalar product between the and products. Central to the STATIS method is the RV coefficient which reflects the similarity between two configurations  (Robert & Escoufier, 1976; Schlich, 1996; El Ghaziri & Qannari, 2015) : ( , ) = , ‖ ‖‖ ‖ = ( ) ( ) ( ). We will also assume that ‖ ‖ = 1 (! = 1, … , ) which means that these matrices have been standardized by dividing each original by its norm, ‖ ‖ = #$%&'( ). This standardization is performed to account for differences among the datasets such as different scaling, different number of columns, etc. CLUSTATIS seeks homogeneous clusters of datasets in such a way that the datasets in each cluster are as close as possible to a group average dataset.
Formally, CLUSTATIS aims at minimizing the following criterion: ) = ∑ ∑ || - -∈/ 0 1 234 5 - (6) || 7 (1) Clearly, this criterion pinpoints the fact that we are seeking clusters of datasets in such a way that, in each cluster, 8 2 , the matrices are clustered around an average matrix, (6). The weights 5 -in each cluster are given by the STATIS method. Thus, those datasets in a given cluster which have less agreement with the other datasets in this cluster will have small weights. Contrariwise, those datasets which are more in line with the general point of view of the cluster under consideration will be allotted relatively larger weights. For solving this optimization criterion, hierarchical and partitioning algorithms were proposed  (Llobell et al. , 2018). Although it is advised to perform both algorithms in conjunction, each one of them can stand by itself to provide a solution to the clustering problem.
We shall focus herein on the partitioning algorithm and discuss how it can be extended to identify atypical datasets (i. e. , subjects). This algorithm is akin to the 9-means algorithm and requires the choice of the number of clusters and a starting partition of the datasets into clusters. The starting partition can be derived from the hierarchical cluster analysis  (Llobell et al. , 2018)  or can be chosen by the practitioner. Alternatively, one may run a multi-start procedure consisting in performing the partitioning algorithm several times (e. g. , 50 times) with, at each time, a staring partition chosen at random. Eventually, the partition that corresponds to the smallest value of criterion ) is chosen. An interesting ANOVA-like decomposition is the following:  (6) ||² that we seek to minimize.
We can also show that the percentage of explained variation given by = = ∑ ; (6) ; 7 1 234 ∑ || : -34 || 7 > is equal to ∑ ? @ (0) A 0B@ : , where C 4 (2) is the largest eigenvalue of the matrix of pairwise coefficients between the datasets in the 9 cluster. This index reflects the overall within-clusters homogeneity. The homogeneity in the 9 cluster can be assessed by ? @ (0) : (0) where (2) is the number of datasets in the 9 cluster. We can also show that at the convergence of the clustering algorithm, we have ∑ ∑ 7 ( -∈/ 0 1 234 , (6) ) = ∑ C 4 (2) 1 234 and ) = -∑ C 4 (2) 1 234. This means that CLUSTATIS can also be defined as a method that seeks to maximize the quantity ∑ ∑ 7 ( -∈/ 0 1 234 , (6) ), which reflects the within-cluster homogeneity. The application of CLUSTATIS to projective mapping data is straightforward.
To each subject ! (! = 1, … , ), we associate a two dimensional dataset which provides the coordinates of the products under study on the sheet of paper given by this subject. Although all the datasets have two columns, these columns may convey different meanings from one respondent to another. Thus, CLUSTATIS is indicated to analyse this kind of data and segment the respondents. More details are given in  Llobell et al. (2018). An illustration of this strategy of analysis together with the demonstration of the interest of detecting atypical subjects are outlined in section 3. The application of CLUSTATIS to the case of free sorting task was also discussed in  (Llobell et al. , 2018). The data from the subject ! (! = 1, … , ) who has sorted the products at hand into D -clusters are recorded into a matrix E ( xD -) consisting of dummy variables.
Discarding atypical datasets with the "G+1" strategy The guiding principle of this strategy of analysis is to set aside those datasets that are weakly linked to all the group average configurations associated with the various clusters. it was formalized by  Dave (1991)  who introduced the concept of "noise cluster" or " + 1" cluster, which collects the objects (i. e. , datasets in our context) that have been discarded. In a first step, the strategy of analysis requires to select a threshold value, H, between 0 and 1 below which the link between two datasets as measured by the RV coefficient is considered as very weak. Formally, we seek to maximize the following criterion: I J = ∑ ∑ KL -2 7 K , (6) M + L -(1N4) H 7 M : -34 1 234 (3) Where L -2 is the Kronecker symbol which is equal to 1 if the dataset ! belongs to cluster 9 and 0 otherwise, with the constraint ∑ L -2 = 1 1N4 234.
This constraint stipulates that a dataset belongs to one and only one cluster, including the noise cluster. The rationale behind this optimization is very simple. Basically, for each (! = 1, … , ), we seek the cluster 9 for which K , (6) M is the largest. Then we assess whether K , (6) M > H. If so, then is assigned to cluster 9, otherwise, it is assigned to the cluster "K+1". This means that we filter out those datasets whose degrees of association, as measured by the coefficient , with the main groups are below a threshold value, H. All in all, the partitioning algorithm runs as follows. Step 0 (initial partition). Choose a partition into clusters of datasets (i. e. , subjects) as a starting point. This can be done by a random assignment of the datasets at hand to clusters or by cutting the hierarchical tree at the level corresponding to clusters.
Step 1 (group average datasets). In each cluster 8 2 , the group average (6) is determined by means of the STATIS method. Step 2 (changing clusters). New clusters of subjects are formed by moving each dataset, , to the cluster 8 2 for which the quantity ( , (6) ) is the largest providing that this quantity is larger than H otherwise, is assigned to the cluster "K+1". Steps 1 and 2 are iterated until the datasets stop changing clusters.
Choice of the threshold value P The problem that yet needs to be solved is how to select an appropriate parameter H. Since this parameter is akin to a similarity measure between two datasets as assessed by the similarity measure , it is comprised between 0 and 1. For H = 0, the "K+1" cluster is empty, and as H increases, more datasets are likely to join this noise cluster. For H = 1, all the datasets are set in the noise cluster. Dave (1991)  suggested a strategy for choosing an appropriate value for ρ in a clustering problem based on distances between objects. This consists, in a first step, of computing the average of the squared distances between each object and each center of the various clusters. Thereafter, this average should be multiplied by a constant. However, few details are given regarding this constant except that it should be small.
Since the parameter ρ is akin to an RV coefficient, this tremendously restricts the field of prospection in comparison to a cluster analysis on the basis of distances. As a result, we are in a similar situation to the cluster analysis of variables by the CLV method. In this context,  Vigneau et al. (2016)  proposed several strategies for the selection of the parameter associated with the noise cluster. Among these strategies, we single out the one consisting in exploring the range between 0 and 1 by considering several values of ρ and examining the evolution of the within-cluster homogeneity, on the one hand, and, on the other hand, the evolution of the number of objects (i. e. , variables in the context of CLV and datasets in the context of CLUSTATIS) that are assigned to the noise cluster.
This strategy can be easily adapted to the CLUSTATIS context as will be illustrated on the basis of a case study. More importantly, we propose to directly define a value for the parameter ρ consisting in calculating the average of the RV coefficients of each dataset with the center of its own group and the center of the nearest group. Formally, the selected value H that we propose is the following: H = ∑ RST( , ( 6 ) )N ST( , K6 UM V W XB@ 7: (4) where (6 ) is the average scalar products matrix obtained by means of the STATIS method of the cluster to which the ! dataset belongs, and, likewise, (6 Y) is the average scalar products matrix of the nearest cluster to the ! dataset. In order to pinpoint the idea behind this choice, we refer to Figure  1  where, for the sake of illustration, the fictitious datasets to be clustered are represented as points in a twodimensional space.
In reality, they are embedded in a space of much higher dimension. We aim to circumscribe the points of each cluster in a circle so that the points that are close to their own center are inscribed in this circle whereas, the points that are straddling two or more clusters are assigned to the cluster "K+1". The first circles, with the smallest radius, encompass the datasets whose RV coefficients with their own center is larger than the average of the RV coefficients of all the datasets with their own center. These circles leave many points outside. The largest circles encompass the datasets whose RV coefficients with their own center is larger than the average of the RV coefficients of the datasets with all the centers of all the groups, which would be in line with Dave's idea  (Dave, 1991).
These circles are far too large and, as a result, very few points are left outside. The intermediate circles correspond to the solution that we propose, namely in each circle, we find those datasets whose RV coefficients with their own center is larger than the average of the RV coefficients of all the dataset with their own center as well as with the center of the nearest cluster. This choice seems to be more appropriate. It is worth emphasizing that herein the criterion of appropriateness means that we aim at defining clear clusters with improved within homogeneity. The datasets that are left aside are not necessarily outliers but are merely deemed not fitting the pattern of any cluster.
Projective mapping: a simulation study The aim of the simulation study is to confirm the efficiency of CLUSTATIS and demonstrate the appropriateness of the proposed parameter H for correctly setting aside the atypical datasets. We strictly followed a simulation framework proposed by  Berget et al. (2019). More precisely, projective mapping data were simulated by considering the case of 100 subjects and 9 products. The subjects were assumed to belong to three clusters A, B and C. More specifically, the data were simulated in such a way that the clusters A and B have different perception of the products and, the cluster C contains subjects who should fall in the noise cluster since they "organise the samples more or less randomly and, hence represent noise"  (Berget et al. , 2019). The typical configurations in clusters A and B are given in Figure  2.
Projective mapping: the smoothies data. The data which are used to illustrate CLUSTATIS together with the strategy of identification of atypical datasets pertain to a Projective Mapping/Napping experiment and can be found in the R package SensoMineR  (Lê and Husson, 2008). They concern 8 smoothies which were evaluated by 24 consumers. Therefore, we have 24 datasets; each dataset being associated with a consumer and consists of the x-y coordinates of the eight smoothies on the sheet of paper. The dendrogram associated with the hierarchical cluster analysis and the variation, delta, of the criterion ) of the CLUSTATIS analysis are shown in Figure  3. It turns out that, when moving from a solution with three clusters to a solution with two clusters, the variation of the criterion is relatively large.
This should arouse the suspicion that, at this step, the merging of two clusters is yielding a heterogeneous cluster. Therefore, we should retain the solution with three clusters. Besides providing a clustering of the datasets at hand, CLUSTATIS simultaneously performs STATIS within each cluster, yielding weights associated with the various subjects which reflect the agreement with the general point of view of the cluster. These weights are also used in CLUSTATIS to compute a group average configuration associated with each cluster. The weights associated with the various subjects arranged in a decreasing order are given in Table  1. Clearly, it appears that subject S2 stands out from the other subjects in cluster 1. Similarly, subjects S8 and S7 stand out from the other subjects in cluster 3.
Projective mapping: the yoghurt data The aim of this section it to compare CLUSTATIS with the Proclustrees and SCR strategies which were used to analyse projective mapping data  (Berget et al. , 2019). For this purpose, we use the same yoghurt data that were used by these authors and which were kindly made available to us. These data concern a projective mapping experiment involving 100 subjects who were asked to assess 12 commercial blueberry yoghurt packages marketed in Norway. For more details regarding these data, we refer to three articles (Naes,  Berget, Liland, Ares & Varela, 2017; Varela et al. , 2017; Berget et al. , 2019). The dendrogram obtained by means of CLUSTATIS (Figure  6 ) indicates to choose two clusters of subjects. This finding agrees with that derived from the dendrogram obtained by means of Proclustrees  (Berget et al.
Free sorting: the aromas data The data which are used to illustrate CLUSTATIS with the "noise cluster" option can be found in the R package FreeSortR  (Courcoux, 2016). They concern 16 aromas which were evaluated by 31 subjects who are students in food science but who are not experts in oenology and have not received any training in wine tasting. The sixteen aromas, which are known to be perceptible in white or red wines, were selected from "Le nez du vin" box set  (Edition Jean Lenoir, 2006). Examples of such aromas are: Lemon, Grapefruit, Honey, Butter, Grilled bread, Grilled hazelnut, etc. The aromas coded by a three-digit random number were presented simultaneously to the subjects in glass vials in a random order. Each subject formed groups of the 16 aromas according to their perceived similarities and differences.
The data from each subject were encoded into dummy variables and standardized as indicated in section 2. 1. The datasets thus obtained were submitted to CLUSTATIS with the noise cluster option. Five clusters can be considered in this case (dendrogram not shown for the sake of saving space). The selected noise parameter based on equation (  4 ) is H = 0. 66. This entails that four subjects were set in the noise cluster. Table  3  gives the within-cluster homogeneity before and after the removal of the atypical subjects. In particular, we can see that the removal of the four subjects resulted in an increase of nearly 4% of the overall homogeneity index. The Figure  8  shows the configuration of the aromas associated with the first cluster of subjects with and without the noise cluster option.
IV. Conclusion We believe that the case studies together with the simulation study discussed in this paper have confirmed the interest and the efficiency of the CLUSTATIS strategy of segmentation of a panel of subjects in a projective mapping and a free sorting experiments. The range of application of this strategy of analysis is larger than these two experiments and include experiments such as sensory profiling with fixed or free choice vocabulary, preference data which include several liking attributes, etc. This method of analysis is flexible enough to offer the option of discarding respondents who do not fit to any cluster. This yields a better, more stable and easier to interpret segmentation of the subjects. An advantage of the CLUSTATIS approach over competing strategies of analysis is its consistency since the whole strategy is contained in the optimization criterion that seeks to define homogeneous clusters.