Su Ruan
email: su.ruan@greyc.ismra.fr

Jinhao Xue

Bruno Moretti

Marinette Revenu

Daniel Bloyet

Fuzzy Knowledge-based Recognition of Internal Structures of the Head

Nous proposons une méthode basée sur la connaissance a priori pour la segmentation et la reconnaissance des formes des structures internes du cerveau en IRM. Les connaissances sur les formes des structures et les distances entre elles, provenant de l'atlas de Talairach, sont modélisées par un champ flou en utilisant une analogie avec la distribution du potentiel d'électrostatique. Une sursegmentation est d'abord effectuée sur le cerveau pour obtenir des régions homogènes. La reconnaissance des structures est ensuite obtenue par la classification des régions utilisant un algorithme génétique, suivie par un affinement au niveau du pixel. Les connaissances floues modélisées sont utilisées dans ces deux étapes. La performance de la méthode proposée est validée par référence aux résultats manuels en utilisant 4 indices de quantification.

Abstract -We propose a knowledge-based method for segmenting and recognizing internal brain structures viewed by MRI (Magnetic

Resonance Imaging). The knowledge about shapes of the structures and relative distances between them, derived from Talairach stereotaxic atlas, is fuzzified by analogy with the electrostatic potential distribution. The brain is firstly over-segmented. Then the recognition of the cerebral structures is achieved by the region-wise labeling using GAs (Genetic Algorithms), followed by a voxel-wise amendment using parallel region growing. The fuzzy knowledge is used both to design the fitness function of GAs, and to conduct the region growing. The performance of our proposed method has been quantitatively validated by 4 indexes with respect to manually labeled images.

Introduction

Automated labeling of structures is complicated, facing difficulties due to overlapping intensities, anatomical variability in shape, size, and orientation, partial volume effects, as well as noise perturbations, intensity inhomogeneities, and low contrast in images. Therefore, it is needed to supplement domain knowledge, to achieve labeling like what radiologists do. In recent years, many reports have been published in this direction in terms of atlas-based ( model-based, or knowledge-based ) segmentation of neuroanatomical structures .

One intuitive strategy to use knowledge for labeling, named as registration-segmentation paradigm by Collins et al. [START_REF] Collins | Automatic 3D model-based neuroanatomical segmentation[END_REF] , is to register and transfer labels of a pre-labeled atlas onto the MRI images to be segmented. The performance of this strategy over-relies on the accuracy of the registration employed, which suffers from the limited degrees of freedom of the transformations, and from anatomical inter-individual variability (in orientation, shape, size, and position). Another important strategy is to integrate statistical knowledge of intensity and position into a shape model, and to locate the structures which match the model. Staib et al. [START_REF] Staib | An integrated approach for locating neuroanatomical structure from MRI[END_REF] used a gradient-based parametric deformable shape model, integrating region information and prior probability knowledge about mean shape and variation of the structures. Gonzalez Ballester et al. [START_REF] Ballester | Combined statistical and geometrical 3D segmentation and measurement of brain structures[END_REF] guided the segmentation by statistical shape knowledge built from data sets of pre-labeled structures. Several researchers used ASM (Active Shape Models) to label brain structures [START_REF] Cootes | The use of active shape models for locating structures in medical images[END_REF] [START_REF] Duta | Segmentation and interpretation of MR brain images: an improved active shape model[END_REF]. ASM are parametric deformable models of shape and appearance of flexible objects, which restrict the possible deformation using shape template and intensity model, both generated through statistics of training sets.

Besides the two strategies aforementioned, the use of the fuzzy sets theory to represent structural information has been introduced by I.Bloch [START_REF] Bloch | On fuzzy distances and their use in image processing under imprecision[END_REF] [START_REF] Bloch | Fuzzy relative position between objets in image processing: a morphological approach[END_REF]. From these fuzzy sets, the atlasbased recognition using fuzzy fusion is proposed [START_REF] Géraud | Atlas-guided recognition of cerebral structures in MRI using fusion of fuzzy structural information[END_REF]. The recognition is carried out sequentially on one structure at a time. The information got from the atlas is updated after the recognition of each structure in order to guide the next one. Inversely to this method, our method use a parallel process.

The contribution of this paper is twofold. First, the structural knowledge from the Talairach stereotaxic atlas is modeled with fuzzy sets. The fuzzification is carried out by analogy with the electrostatic potential distribution in the vicinity of hollow structures with uniform surface charge density. Second, the fuzzy sets obtained for each structure are used to search the optimal labels from over-segmented regions using GAs (Genetic Algorithms).

Fuzzification of the knowledge

The Talairach atlas is well-accepted in medical image processing, owing to its contribution to the delineation and labeling of numerous brain neuroanatomical structures (one sample slice of this atlas is shown in We define the degree of fuzziness to represent the degree of belonging to one structure. Since our recognition process searches for the structures in parallel, the interactions between different structures have to be taken into account. We propose a model which is constructed by analogy with the electrostatic potential distribution in the vicinity of conducting objects. The potential of any point relative to one structure is in fact influenced by all structures. More precisely, any structure s (

[ ] N s , 1 ∈
) is considered as an isolated conductor in the electrical equilibrium, i.e. all the charges are distributed on its 3-D outer surface S s . The electrostatic potential at voxel x located outside any structure can be expressed as: [START_REF] Collins | Automatic 3D model-based neuroanatomical segmentation[END_REF] where ε 0 is a constant of vacuum permitivity, ζ(x') denotes the charge density at point x' on S s . If ζ(x') is considered as an uniform distribution, the equation ( 1) can be simplified as:

(

where n s denotes the number of voxels v i s (i=1, ...,n s ) on the surface of the structure s in the atlas T V , and d(x,v i s ) is a distance from x to v i s . The potential distribution of the structure s is firstly normalized by its maximal value to guaranty p s (x) ∈ [0,1]. The fuzzy membership value related to the structure s is than defined by:

(

where . . The structure j has the maximal potential value.

Recognition method

We propose a coarse-to-fine strategy to achieve the brain structure recognition, based on both the fuzzy structural knowledge from the Talairach stereotaxic atlas, and on the information from the MRI images.

Two datasets were made available in our method. One is the Talairach stereotaxic atlas TO ; another one is a volume of MRI images V T , which have been registered into the stereotaxic coordinates system using MPItool 1 software package. The volume V T is used as a registration template. To obtain the coarse location of brain structures in a new volume MRI VO , the AIR software package developed by Woods et al. [START_REF] Woods | MRI-PET registration with automated algorithm[END_REF] is used to register the prepared registration template V T onto VO. A matrix of mapping parameters is thus obtained. TO is resampled using this matrix, and the resampled result T V can be superimposed automatically onto VO. This resampled Talairach atlas is then fuzzified by the way described in the previous section.

Over-segmentation with fuzzy MRF

We expect the over-segmentation to fulfill the following requirements: 1) not to result in tremendous redundant regions like those generated by the watershed algorithm, thus to be able to reduce the computation complexity of the following GAs in searching optimal labeling of regions; 2) to segment brain tissues like GM (Gray Matter ) and CSF (Cerebro-Spinal Fluid), which contain most of the important neuroanatomical structures; 3) to describe partial volume effects quantitatively; and 4) to decrease the information loss before labeling. In this context, we applied a scheme using the combination of partial volume modeling [START_REF] Ruan | Brain tissue classification of magnetic resonance Images using partial volume modeling[END_REF] and fuzzy MRF (Markov Random Fields) [START_REF] Salzenstein | Parameter estimation in hidden fuzzy Markov random fields and image segmentation[END_REF], which can generate regions of pure tissues: GM, WM (White Matter),and CSF, and also mixtures of these pure tissues. The fuzzy MRF models simultaneously "pure" voxels using Dirac functions, and "mixed" voxels using Lebesgue measure. Moreover, fuzzy MRF can preserve information through fuzzy membership of voxels belonging to a given class, as well as statistical context information through MRF. It is justified for our requirements.

Estimation of statistical moments

The statistical mean M s and variance σ s 2 of each structure s are chosen to describe the intensity of the structure, and estimated from a set Ω s which includes regions reliably belonging to s. Using a fuzzy MRF segmentation, oversegmented regions are obtained along with their fuzzy membership values to different tissues (CSF, GM and WM). The regions classified into Ω s should have large intersection areas with s in the registered Talairach atlas T V , and high values of fuzzy membership to brain tissues, e.g. high membership to GM while regarding s as caudate, and putamen, and membership to CSF while regarding s as ventricle. With the help of the fuzzy model of ROI and the set Ω s , we can estimate the mean M s and variance σ s 2 with a higher precision. The way of estimation can be written as follows:

(4)

∫ - = ' ) ' ( 4
1 ) ( 0 ζ πε ∑ = s n s i s v x d x p 1 ) , ( 1 ) ( ( )
     ≠ = = s j if ) ( / ) ( ) ( s if ) ( ) ( max max x p x p x p j x p x j s s j s µ )) ( ( sup ) ( ] , 1 [ max x p x p i N i j ∈ = 2 4 3 1 ∑ Ω ∈ = s x s s x f x Z x M ) ( ) ( 1 ) ( µ (5)
where , and f(x) is the intensity of the voxel x.

Region-wise recognition using Gas

GAs are stochastic search methods which use analogy with some mechanisms of evolution in nature. They are used widely in optimization, artificial intelligence, involving brain structures labeling, conducted by an objective function which is referred as fitness function in GAs related references [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF] [START_REF] Sonka | Knowledge -based interpretation of MR brain images[END_REF]. The most application-dependent component of GAs is the fitness function, which affects significantly the performance and computation complexity of GAs.

Our fitness function has been designed as the multiplication of two items, based on the fuzzy model of ROI and the statistical mean estimated above. The first item relates to intersected shapes with the same label between the atlas TV and images V o , as :

(6)
where r s µ is the membership value corresponding to the structure s in the region r.

is a normalization coefficient, where n r represents the amount of voxels in the region r .

The second item measures the correlation of intensity between regions and the corresponding structures, as: [START_REF] Bloch | Fuzzy relative position between objets in image processing: a morphological approach[END_REF] where m r is the mean of region r , and M s r denotes the mean of the structure s in the region r, which are estimated in equations [START_REF] Duta | Segmentation and interpretation of MR brain images: an improved active shape model[END_REF]. I max ( I min ) is the maximum (minimum) intensity of the images.

Voxel-wise amendment conducted by knowledge

The only use of region-wise labeling is no doubt coarse and insufficient to achieve an accurate labeling of neuroanatomical structures, because the over-segmentation suffers significantly from the overlap of intensity ranges among different structures like caudate, thalamus and putamen.

However, after the region-wise labeling, the majority of voxels has been correctly labeled, and can be considered as seeds for refinement. In this context, we choose a parallel region growing algorithm to achieve voxel-wise amendment, conducted by the structural and statistical knowledge formerly obtained. In this procedure, we make the different structures grow simultaneously, to avoid one structure to grow into another one located in its proximity and whose intensity is overlapping.

Some of the knowledge applied are derived from the atlas T v , such as shape, spatial position, and distance information represented by the fuzzy sets. Some others are derived from the MRI images V O and its over-segmented version, such as the statistical moments

Results and quantitative validation

In this study, the subjects were scanned with a GE Signa 1.5 Tesla scanner, employing a T1-weighted SPGR pulse sequence. The parameters of the SPGR sequence were TR=30ms, TE=7ms, flip angle=40 °, image size= 256x256x124 voxels, and voxel size =0.94x0.94x1.2 mm3.

The performance of the method is demonstrated by labeling four important neuroanatomical structures: caudate, thalamus, ventricle and putamen. These The quantitative validation is carried out by comparing the results obtained by our method denoted as L a , with the "ground truth" obtained manually denoted as L g . Seven images were chosen randomly for the comparison. Two indices of quantitative measures were calculated from the statistical moments of distance histogram [START_REF] Moretti | Phantom-based performance evaluation: application to the brain segmentation from magnetic resonance images[END_REF]. In fact, we calculated the distance of any contour voxel x s of the structure s in L a to the corresponding contour of s in L g : the distance histogram, revealing the spatial distribution of recognition errors, is built from the superimposition of L a onto the distance map of L g , from which some characteristics are extracted, such as mean, standard deviation (SD). The

1 1 ) ( 1 µ ∑ = = t n r r n N 1 1 2 2 ] ) ( )[ ( 1 ) ( s s x s s M x f x Z x - = ∑ Ω ∈ µ σ ∑ Ω ∈ = s x s x Z ) ( µ
two other indices measured are the false positive ratio representing the error due to the misrecognition of the structure s ( γ fp s ) , and false negative representing the error due to the loss of desired voxels of s (γ fn s ). The results obtained by these four indices are shown in Table 1. At first sight, it appears that the mean discrepancy between L g and L a is always less than 1 voxel, which is quite satisfactory. The low SD indicates that the labeling errors are peakwise, which entails a concentration of labeling errors around 1 voxel. We can also deduce that almost all the false ratios are less than 10% (except γ fp s for caudate due to its smaller size).

Conclusion

An automatic, knowledge-based method to segment and recognize brain neuroanatomical structures (ventricle, caudate, thalamus and putamen) from MRI images has been developed. The structural knowledge derived from Talairach stereotaxic atlas is fuzzified to represent the impression and uncertain information for guiding the recognition process. Quantitative validation has also been performed to demonstrate the performance of the proposed method, with manually labeled images considered as "ground truth" using 4 quantitative indices. The results show that this method is promising for quantitative analysis of brain neuroanatomical structures. The direction of knowledge integration is certainly worthy of further investigation in image segmentation and recognition.

  FIG 1.a). Although it indicates approximate shape, position and relationship of the brain structures, it is far from the real brain (FIG 1.b). Therefore, we use fuzzy sets to model the imprecision and the uncertainty of the structure information.

  a) one sample slice of Talairach atlas: putamen (1), ventricle (2), thalamus (3) and caudate (4), (b) one slice of MRI brain volume.

2

 2 four structures are clearly visible close to the center of the axial Talairach atlas (see FIG 1.a). The fuzzification of the Talairach atlas is carried out by the method described in section 2. The slice of index 67 is shown in FIG 2 to illustrate the four obtained fuzzy sets corresponding to the four main structures. The over-segmentation applied to the MRI dataset generates 2878 regions in the slice shown in FIG 1 (b). The result of this sample image is shown in FIG 3 (a). The volume rendering result is shown in FIG 3 (b). The results are visually satisfied. Fuzzy sets derived from the Talaraich atlas (corresponding to FIG 1(a)): (a) ventricle; (b) putamen; (c) caudate; (d) thalamus.

1

 1

  results. (a) The four obtained structures superimposed to the original MRI image (FIG 1(a)). (b) Volume rendering of the 4 recognized structures .

  TAB. 1: Quantitative validation results with statistical moments of distance histogram and with false positive and false negative ratios.

	Ventricule	Means -0.890	SD 0.482	γ fp 0.07. s	s 0.018 γ fn
	Thalamus	-0.801	0.908	0.090	0.063
	Putamen	-0.819	0.651	0.083	0.049
	Caudate	-0.598	1.422	0.112	0.086


Kevin B Li

Semi-Independent Sliced Inverse Regression

Keywords:

published or not. The documents may come

Introduction

Forecasting using many predictors has received a good deal of attention in recent years. The curse of dimensionality has been turned into a blessing with the abundant information in large datasets. Various methods have been originated to extract efficient predictors, for example, non-stationary factor model (NFM), Bayesian model averaging, Lasso, boosting, etc. Among them, non-stationary factor model is conceptually appealing in economics because it is structurally consistent with log-linearlized models such as nonstationary stochastic general equilibrium models. The recent development in statistics provides a new method of dimension reduction in regression for large-dimensioned data. The literature stems from [START_REF] Duan | Slicing regression: a link-free regression method[END_REF], and [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF], which proposed a new way of thinking in the regression analysis, called sliced inverse regression (SIR). SIR reverses the role of response y and predictors x. Classical regression methods mainly deal with the conditional density f (y|x). SIR collects the information of the variation of predictors x along with the change of the response y, by exploring the conditional density h(x|y). Usually the 1 dimension of the response is far more less than the dimension of the predictors, hence, it is a way to avoid the "curse of dimensionality". The traditional SIR does not work well for highly correlated data, because the algorithm requires the inverse of the covariance matrix. This is not feasible when the number of variables N is greater than the number of observations T , which happens a lot in economics studies. In addition, the economic variables are often highly correlated, due to the derivation formula or same category. This makes the covariance matrix ill-conditioned, causes the inverse matrix lack of precision and too sensitive to the variation of matrix entries, and leads to a false or unstable result. There are some extensions of SIR for the highly collinearity data and "T < N " problems, for example, regularized sliced inverse regression ( [START_REF] Zhong | RSIR: Regularized Sliced Inverse Regression for motif discovery[END_REF], [START_REF] Li | Sliced inverse regression with regularizations[END_REF]) and partial inverse regression ( [START_REF] Li | Partial inverse regression[END_REF]). In this paper, we propose a new method of dimension reduction, called the semiindependent sliced inverse regression (SISIR) method, for many predictors in a data rich environment. We evaluate its property theoretically and use it for forecasting economic series. Comparison in terms of pseudo out-of-sample forecasting simulation shows the advantage of our method. The remaining of the paper is organized as follows. Section 2 introduces semi-independent SIR method with its statistical property. An application on an economic dataset is given in Section 3. Conclusions with some discussions are provided in Section 4.

Modeling and methods

The regression model in [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] takes the form of

y = g(β ′ 1 x, β ′ 2 x, . . . , β ′ K x, ), (2.1)
where the response y is univariate, x is an N -dimensional vector, and the random error is independent of x. Many methods can be used to find the e.d.r.-directions, for example, principal component analysis might be the most commonly used one in economics. But unlike these methods, SIR not only reduces dimensions in regression but also integrates the information from both predictors and response. Moreover, different from the classical regression methods, SIR intends to collect information on how x changes along with y. That is to say, instead of estimating the forward regression function η(x) = E(y|x), inverse regression considers ξ(y) = E(x|y). Compared with η(x), the inverse regression function ξ(y) depends on one-dimensioned y, which makes the operation much easier. [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] showed that using SIR method, the e.d.r.-directions can be estimated by solving Cov E(x|y)

β j = ν j Cov(x)β j , (2.2)
where ν j is the jth eigenvalue and β j is the corresponding eigenvector of Cov E(x|y) with respect to Cov(x). During the forecasting procedure, the covariance matrices can be replaced by their usual moment estimates.

Semi-independent sliced inverse regression

In this section, we introduce grouping methodology with the sliced inverse regression to improve the performance of SIR on collinear data. Assume that the variables of interest can be grouped into several blocks, so that two variables within the same block are correlated to each other, and any two variables belonging to different blocks are independent. In practice, an orthogonalization procedure can be applied to reduce the correlations between blocks in order to fit our assumption. Thus, we can group the variables according to their correlations in order to find the e.d.r-directions, because there is no shared information between groups. The grouping method we use is hierarchical grouping with complete linkage. The dissimilarity is defined as 1 -|Correlation|. The algorithm for the semi-independent SIR method can be described as following.

1. Standardize each explanatory variable to zero mean and unit variance.

2. Cluster x (N ×1) into x 1 ′ • • • x c ′ ′ based on the correlation matrix of x, where x i is N i × 1, c
i=1 N i = N , and c is the number of groups, which will be determined by cross-validation.

3. Restricted to each group, perform SIR method and pick up k i SIR directions based on the sequential chi-square test [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF], say θ

(i) j , j = 1, . . . , k i , i = 1, . . . , c.
4. Collect all the SIR variates obtained from the groups, say {θ

(i)′ j x i , i = 1, 2, . . . , c, j = 1, 2, . . . , k i }. 5. Let λ l = 0 1 ′ θ (i)′ j 0 2 ′ ′ , l = 1, . . . , m, m = c i=1 k i ,
where 0 1 and 0 2 are zero column vectors with dimension i-1 k=1 N k and c k=i+1 N k , respectively. Denote Λ = (λ 1 , λ 2 , . . . , λ m ). The variates {θ

(i)′ j x i } can be written in a vector form as (λ ′ 1 x, . . . , λ ′ m x) ′ = Λ ′ x.
6. Then, perform SIR method one more time to the pooled variates Λ ′ x to reduce dimensions further, and get the e.d.r.-directions (γ 1 , γ 2 , . . . , γ v ), where v is also determined by the sequential chi-square test. Denote Γ = (γ 1 , γ 2 , . . . , γ v ), the final SISIR variates we chose are Γ ′ Λ ′ x.

7. Estimate the values of forecasting series using the SISIR variates Γ ′ Λ ′ x.

Linear models are used in this paper.

Note that the matrices

Γ is m × v, Λ is N × m, so Γ ′ Λ ′ x is v × 1.
Therefore, we only use v factors to build the final model for forecasting y, instead of using N variables based on the original dataset.

Statistical property of semi-independent SIR

[37] established the unbiasedness for the e.d.r.-directions found by SIR, assuming the following linearity condition.

Linearity Condition 1. For any b ∈ R N , the conditional expectation E(b ′ x|β ′ 1 x, . . . , β ′ K x) is linear in β ′ 1 x, . . . , β ′ K x.
The linearity condition is not easy to verify, however, [START_REF] Eaton | A characterization of spherical distributions[END_REF] showed when x is elliptically symmetrically distributed, for example, multivariate normally distributed, the linearity condition holds. Furthermore, [START_REF] Hall | On almost linearity of low dimensional projections from high dimensional data[END_REF] showed that elliptical symmetric distribution is not a restrictive assumption, because the linearity condition holds approximately when N is large even if the dataset is not elliptically symmetric. Without loss of generality, we assume each variable in x has been standardized to zero mean and unit variance for our discussion. [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] proved the following theorem, Theorem 1. Assume Linearity Condition 1, the centered inverse regression curve E(x|y) is contained in the space spanned by Σ x β j , j = 1, . . . , K, where Σ x is the covariance matrix of x.

Similar unbiasedness property can be proved for our semi-independent SIR. Theorem 2. Under certain linearity conditions, E(x|y) is contained in the space spanned by Σ x ΛΓ.

Theorem 2 describes the desirable property that there is no estimation bias. The e.d.r.-space estimated by our SISIR method contains the true inverse regression curve.

Orthogonalization

For a given dataset X with dimension N × T , and groups X 1 , . . . , X c , the correlations between these groups need to be reduced to zero, to achieve group-wise independence. QR decomposition along with projection operators is used to perform the orthogonalization. To begin with, use QR decomposition to find the orthogonal bases of the first group X 1 , named as Q 1 . Next, project the second group X 2 onto the space of span{Q 1 } ⊥ , which is the orthogonal complement of the space spanned by X 1 , named as X 2 * ,

X 2 * = (I -Q 1 Q 1 ′ )X 2 . (2.3)
Then use QR decomposition again to find the orthogonal bases of X 2 * , named as Q 2 , and project X 3 onto the space of span{Q 1 , Q 2 } ⊥ , named as X 3 * . Keep doing such process till the last group X c , we will get a new sequence of groups X 1 , X 2 * , . . . , X c * , in which every two groups are orthogonal, and the new sequence contains all the information of the original dataset X.

Regularization

Due to the high correlations between the series within each group, the covariance matrices of each group Σ x i are ill-conditioned, which make them hard to be inversed. We suggest a regularized version of the covariance matrix to overcome this issue ( [START_REF] Friedman | Regularized discriminant analysis[END_REF]).

Σ x i (τ ) = (1 -τ )Σ x i + τ trΣ x i N i I N i , (2.4)
where τ ∈ [0, 1] is the shrinkage parameter. This is similar to the ridge version proposed by [START_REF] Zhong | RSIR: Regularized Sliced Inverse Regression for motif discovery[END_REF], which replaces Σ x i with Σ x i + τ I N i . The shrinkage parameter τ can be chosen by cross-validation. Note when τ = 1, the regularized covariance matrix will degenerate to a diagonal matrix whose diagonal elements are the means of the eigenvalues of Σ x i . In such case, the chosen e.d.r.-direction is one of the input series, and the other series, which may also contain information for the predictors, are discarded.

Comparison between SISIR and SIR

Before applying the proposed SISIR method to real data, consider the following simulated example first to compare the performance of SISIR and SIR methods. Let x 1 , x 2 , • • • , x 10 be independent and identically distributed (i.i.d.) with multivariate normal distribution N (0, Σ), where Σ is a 10 × 10 covariance matrix with 1 at diagonal and 0.9 at off-diagonal. The random error e is independent to x i 's, and follows normal distribution N (0, 0.1). The response y is simulated using the following formula,

y = 10 j=1 j × x j + e
Root mean square error (RMSE) is considered as a criterion to evaluate the prediction performance.

RMSE = T i=1 ŷi -y i 2 T , (2.5)
where ŷi is the ith predicted value of response, y i is the ith observed value, and T is the number of observations. We simulate 300 observations at each run under above conditions. In SISIR, the parameters c and τ are chosen as c = 10, τ = 0.5 to minimize RMSE. Table 1 presents the means and standard deviations for the RMSE of SISIR and SIR across 100 runs. From Table it is clear that SISIR has much smaller RMSE than SIR. In fact, our other simulations, which are not presented here, show that SISIR performs even better when the sample size T decreases to N .

Application

The dataset contains 143 quarterly economic variables from 13 economic categories. We use 109 of them as explanatory variables, since the other 34 are just high-level aggregates of the 109. All 143 variables are used for forecasting purpose. The correlation plot of the 109 predictor series after logarithm and/or differencing transformation is showed in Figure 1, which demonstrates that there do exist some highly correlated blocks. Therefore, our semi-independent method is necessary for this dataset. For the purpose of comparison, similar rolling pseudo out-of-sample forecasting simulation is used. The main steps can be described as follows,

1. From 1985 to 2008, at each date t, use cross-validation, which is described below, to the most recent 100 observations to choose the parameter c and τ in SISIR based on mean square error.

2. Use the chosen ĉ and τ , apply SISIR one more time to predict ŷt+h , where h is the forecasting period.

3. Calculate the RMSE for the forecasting procedure,

RMSE = T t=1 y t+h -ŷt+h 2 /T .
The steps for cross-validation are described as follows,

1. Regress y t+h and x t on the autoregressive terms 1, y t , y t-1 , y t-2 , y t-3 , in order to eliminate the autoregressive effect. Denote the residuals as ỹt+h and xt . We compare our method with the non-stationary factor model using the first five principle components (NFM-5), which was claimed to be no worse than any other shrinkage methods according to Stock and Watson (2011). Autoregressive model of order 4 (AR(4)) is used as a benchmark, all RMSEs are recorded as the ratio relative to AR(4). Smaller relative RMSE indicates better forecasting performance. Table 2 presents the number of series with smaller RMSE than AR(4) model for SISIR and NFM-5. We can see that for forecasting period h = 1, if SISIR is used, there are 97 series out of 143 have smaller RMSE than the benchmark AR(4) model. If NFM-5 is used, only 85 series out of 143 have smaller RMSE than AR(4) model. The differences become even larger for big forecasting period, when h = 4 the number of series of SISIR increases to 115 while the number of NFM-5 decreases to 53. Table 3 presents the distributions of the RMSEs for AR(4), NFM-5, and SISIR methods. When h = 1, the first quartile of the relative RMSE of SISIR is just 0.768, which is much smaller than the relative RMSE of NFM-5 (0.961), and the median relative RMSE of SISIR is 0.907, while NFM-5 has 0.993. When h = 2 and h = 4, SISIR improves the forecasting results of AR(4) for more than 3/4 of the series. The relative RMSE of SISIR at first, second, and third quartile are all smaller than those of NFM-5. From Table 2 and3, one can tell that SISIR improves the forecasting results significantly compared to the NFM-5 method, especially for longer forecasting period. Table 4 presents the median RMSE relative to AR(4) model by category via cross-validation. Column "S&W" reports the smallest relative RMSE Stock and Watson got using NFM-5 and other shrinkage methods in their 2011 paper. Comparing all these results, SISIR method has smaller median relative RMSEs for more than 70% of these categories among three forecasting period, which demonstrates its superiority again. Rate, Inventories, Interest Rates, etc. But it does not work well for some others, such as Housing, Money, Exchange Rates. Figure 2 plots six series from both SISIR favored and no-favored categories. Three of them in Figure 2a are from SISIR favored categories and three of them in Figure 2b are from SISIR no-favored categories. From these plots, one can see that the responses of the SISIR no-favored series are quite disordered. They are more like white noises, the variations are big but the changes of x means are not distinct. The inverse regression method is aimed to detect the variation of E(x|y). If the conditional expectations of x do not have much difference for different values of y, the estimation for the e.d.r.-directions will be inaccurate, and will lead to the poor performance on forecasting.

Let

ℑ(t) = {1, • • • , t -2h -3, t + 2h + 3, • • • , 100}, at each date t = 1, • • • , 100 -h,

Conclusion

Sliced inverse regression now becomes a popular dimension reduction method in computer science, engineering and biology. In this article, we bring it to economic forecasting model when there is a large number of predictors and high collinearity. Compared to the classical non-stationary factor model, SIR retrieves information not only from the predictors but also from the response. Moreover, our semi-independent regularized SIR has the ability to handle highly collinearity or "T < N " data. The simulation confirms that it offers a lot of improvements over NFM-5 model on the economic dataset.

After finding the SISIR variates, we use linear models for forecasting the responses y, because scatter plots for SISIR variates and y values show strong linear relationships, and the results are desirable. But one may use polynomials, splines, Lasso, or some other more advanced regression techniques for different situations. Based on its basic idea, there are more than one generalizations of SIR using higher order inverse moments. For instance, SAVE, SIR-II ( [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF]), DR ( [START_REF] Li | On directional regression for dimension reduction[END_REF]), and SIMR ( [START_REF] Ye | Sliced inverse moment regression using weighted chi-squared tests for dimension reduction[END_REF]). Our semi-independent algorithm can also be applied to these methods for highly collinearity data, and good performance is expected. Above all, we can conclude that the semi-independent regularized sliced inverse regression is a powerful tool in forecasting using many predictors. It may not be limited in economic forecasting, and can also be applied to dimension reduction or variable selection problems in social science, microarray analysis, or clinical trails when the dataset is large and highly correlated.

  find the e.d.r-directions and linear regression model using SISIR and observations ỹi and xi , i ∈ ℑ(t).

3 . 4 .

 34 Use the e.d.r-directions and linear regression model from the above step at date t to predict ỹt+h . For fixed h, parameters (c, τ ) are chosen by minimizing the sum of squared forecasting error,

Figure 1 :

 1 Figure 1: Plot of Correlations of the 109 Predictor Series

Figure 2 :

 2 Figure 2: Plots of the Forecasting Values (△) vs. Real Observations (•) from 1985 to 2008.

Table 2 :

 2 Number of Series with Smaller RMSE than AR(4) Model

		NFM-5 SISIR
	h = 1	85	97
	h = 2	59	109
	h = 4	53	115

Table 4

 4 also indicates the performance of SISIR varied across categories. It has outstanding performance for some categories, such as Industrial Production, Unemployment

Table 3 :

 3 Distributions of Relative RMSEs by Pseudo Out-of-Sample Forecasting

	Method	Percentiles
		0.050 0.250 0.500 0.750 0.950
	AR(4)	1.000 1.000 1.000 1.000 1.000
	NFM-5	0.874 0.961 0.993 1.022 1.089
	SISIR	0.621 0.768 0.907 1.048 1.372
		(a) h = 1

Table 4 :

 4 Median Relative RMSE for Forecasting by Category of Series

			h = 1			h = 2			h = 4
	Category	NFM-5 S&W SISIR NFM-5 S&W SISIR NFM-5 S&W SISIR
	1. GDP Components	0.905	0.905	1.079	0.907	0.870	0.807	0.906	0.906	0.839
	2. Industrial Production	0.882	0.882	0.669	0.861	0.852	0.694	0.827	0.827	0.745
	3. Employment	0.861	0.861	0.849	0.861	0.859	0.803	0.844	0.842	0.823
	4. Unempl. Rate	0.800	0.799	0.771	0.750	0.723	0.723	0.762	0.743	0.647
	5. Housing	0.936	0.897	1.220	0.940	0.902	1.081	0.926	0.882	0.807
	6. Inventories	0.900	0.886	0.856	0.867	0.867	0.764	0.856	0.856	0.784
	7. Prices	0.980	0.970	0.865	0.977	0.961	0.892	0.963	0.948	0.797
	8. Wages	0.993	0.938	0.967	0.999	0.919	0.960	1.019	0.931	1.031
	9. Interest Rates	0.980	0.946	0.849	0.952	0.928	0.892	0.956	0.949	0.822
	10. Money	0.953	0.926	1.000	0.933	0.921	0.950	0.909	0.909	0.927
	11. Exchange Rates	1.015	0.981	0.974	1.015	0.980	1.108	1.036	0.965	1.150
	12. Stock Prices	0.983	0.983	0.840	0.977	0.955	0.893	0.974	0.961	1.039
	13. Cons. Exp.	0.977	0.977	0.765	0.963	0.960	1.082	0.966	0.955	0.963
				10

(a) From SISIR Favored Categories (b) From SISIR No-Favored Categories


Ivana Roche
email: ivana.roche@inist.fr

Dominique Besagni
email: dominique.besagni@inist.fr

Claire François
email: claire.francois@inist.fr

Marianne Hörlesberger
email: marianne.hoerlesberger@ait.ac.at

Edgar Schiebel
email: edgar.schiebel@ait.ac.at

A bottom-up approach to assess the interdisciplinarity of journals from a multidisciplinary corpus of bibliographical records 1

This work is investigating the possibility of assessing the interdisciplinarity of scientific and technological journals without using any taxonomy or classification scheme as those usually adopted in bibliographical databases. For that, we start from a large corpus of bibliographic records from which we extract terms, either keywords already present or obtained by text mining techniques. With the help of a clustering method, that corpus is split into clusters defining a given number of scientific fields. Those fields and the keywords indexing each document are the basis of the calculation of an interdisciplinarity score applying the diffusion model approach. Then, we calculate an interdisciplinarity indicator for each journal by combining the scores obtained by its articles.

Introduction

Interdisciplinarity and its corollary, specificity, are useful characteristics to locate metaphorically a journal in the scientific and technological (S&T) literature landscape. Previous works were done on the issue of subject classification and the creation of coherent journal sets. Usually these approaches are based on data available from Thomson Reuters' Journal Citation Reports (JCR) where the citations got by each publication are aggregated at the journal level, i.e. the work by Leydersdorff and Rafols (2011) that makes use of measures of interdisciplinarity like network indicators or unevenness indicators. In a recent study [START_REF] Thijs | Bibiographic coupling and hierarchical clustering for the validation and improvement of subject-classification schemes[END_REF] introduced a very interesting approach building a network among journals based on bibliographic coupling. Hybrid approaches based on citation-based and lexical similarities are also known [START_REF] Janssens | A hybrid mappping of information science[END_REF] as well as approaches combining several indicators of journal specificity based on textual coherence and research communities [START_REF] Boyack | Multiple dimensions of journal specificity: Why journals can't be assigned to disciplines[END_REF]. In our work, an exclusively content-based approach is developed to determine from a large multidisciplinary corpus of bibliographical records an indicator measuring the interdisciplinarity of each record and, from that, the interdisciplinarity of the journals where these documents were published. Our approach is directly inspired on the methodology developed in the context of the DBF project [START_REF] Hörlesberger | A concept for inferring 'frontier research' in grant proposals[END_REF], the goal of which was to infer attributes of 'frontier research' in peer-reviewed research proposals under the scheme of the European Research Council (ERC). To this end, indicators across scientific disciplines and in accord with the strategic definition of frontier research by the ERC are elaborated, exploiting textual proposal information and other scientometric data of grant applicants. In particular, an indicator was devised to characterized any project that "… pursues questions irrespective of established disciplinary boundaries, involves multi-, inter-or trans-disciplinary research that brings together researchers from different disciplinary backgrounds, with different theoretical and conceptual approaches, techniques, methodologies and instrumentation, perhaps even different goals and motivations" (EC, 2005) and that we defined as interdisciplinarity. That indicator is built upon the basic assumption and previously successfully tested concept [START_REF] Schiebel | An advanced diffusion model to identify emergent research issues: the case of optoelectronic devices[END_REF] that the frequency of occurrence and distribution of discipline specific keywords in scientific documents can be used to classify and characterize disciplines. The concept is consistent with the practice of bibliometric clustering, where the contents of each cluster (e.g., words and articles, or cited references and articles) are ranked by some index (e.g. TF-IDF) of specificity to the cluster. In the next section the developed methodology is presented followed by some preliminary results.

Methodology

Our methodology is based on a diffusion model approach [START_REF] Schiebel | An advanced diffusion model to identify emergent research issues: the case of optoelectronic devices[END_REF][START_REF] Roche | Identification and characterisation of technological topics in the field of Molecular Biology[END_REF], developed in the context of a previous project aiming at detecting emerging technologies, that evaluates the status of each term in a considered discipline by measuring its so-called degree of diffusion. The diffusion model is founded on the assumption that new findings in a research field are published in journals, conference proceedings, books etc. That S&T literature is collected in bibliographical databases where the content of each document is represented with a set of keywords. Keywords that describe the innovative results occur in the first stage in an unusual manner. In the second stage the research intensifies and established keywords are used. In later stages, the results cross the disciplinary barrier by diffusing to other research fields where they follow a similar evolution cycle. Consequently, the diffusion status is obtained by the calculation for each keyword of a diffusion degree that can be either "unusual", "established" or "cross-section". Two pragmatic approaches are successively employed to realise this categorisation. Firstly, the so-called Home Technology terms (H-T terms) are defined. We assumed keywords which are specific for a field occurred with a higher probability in that field rather than in others. The probability is defined by the frequency of one term in a field divided by the number of articles in this field, namely the relative term frequency (rtf Field ). For a term, we calculate its rtf Field in each field and the field with the highest probability is declared to be its Home Technology. So after this assignment we obtain for each field the list of its H-T terms. Therefore the complete terminology associated to a field consists of the union of its H-T term list and the set of terms imported from the other fields. Secondly, we use the Gini index (or also Gini coefficient), a measure of statistical dispersion developed by the Italian statistician Corrado Gini [START_REF] Gini | Measurement of inequality of incomes[END_REF] at the beginning of the 20 th century. The Gini index (GINI) is a measure of the inequality of a distribution and it varies from 0 to 1, a value of 0 expressing total equality and a value of 1 maximal inequality. It is commonly used as a measure of inequality of the income or wealth of the countries. It is, in this study, employed as a measure of the dispersion of a term in a scientific domain. A Gini index equal to 0 means a completely uniform distribution and indicates that the term occurs in all the considered fields of the domain. Conversely, a Gini index of 1 tells us that the term is very specifically limited to the only field where it appears. If we consider a set of n Home Technologies, the Gini index of a term can be calculated by the Brown formula:

   1 1 1 0 1 n k k k k k GINI X X Y Y         
where X is the cumulative share of Home Technologies, and Y the cumulative share of occurrences of the considered term.

In the present study, we are not concerned with detecting innovative technologies but with characterizing scientific fields by analysing their related terminologies. So, hereafter we will not speak anymore of Home Technology but of Home Field (H-F). Moreover, we do not consider the categorization of the keywords according to their diffusion degree, but only the determination for each keyword of its rtf Field and its Gini index. These values allow for each keyword either to assign it to a H-F or to discard it if it is not discriminant enough.

Determining the set of H-F can be done either with the help of a pre-defined taxonomy, or with a content analysis approach starting with a term extraction, followed by a validation step operated on the extracted keywords indexing each document and finishing by a clustering splitting the corpus into a given number of H-F.

In our case, we apply a non-hierarchical clustering algorithm, the axial K-means method, coming from the neuronal formalism of Kohonen's self-organizing maps, followed by a principal component analysis in order to represent the obtained clusters on a 2-D map [START_REF] Lelu | Hypertext paradigm in the field of information retrieval: A neural approach[END_REF]. This step is realized by employing an in-house software tool, Stanalyst [START_REF] Polanco | STANALYST: An integrated environment for clustering and mapping analysis on science and technology[END_REF], dedicated to the scientific and technical information analysis. The axial K-means is a variant of the well-known K-means clustering algorithm: it derives half-axes, or "axoïds" maximizing a global inter-axes inertia criterion, instead of deriving cluster centroïds maximizing the inter-class inertia. One can sort the cluster's describers and documents along one of these half-axes as well as project the other terms and documents onto it. These projections on any given axis represent the weight of the describers and the document in the corresponding cluster. In this way, one can derive a fuzzy interpretation of the resulting axes, though the method is a strict clustering technique. This method is fast and can handle very large amounts of data. It is formally related to neural models with unsupervised winner-take-all learning. With that clustering algorithm a document may belong to one or more clusters. We consider that the cluster where the document has the higher weight is its H-F. At this stage, as each publication is allocated to an H-F, it is possible for each one to calculate its HFT and AFT, respectively, the share of its H-F terms and the share of its "abroad terms", i.e. terms assigned to the other H-F. The higher its AFT value, the more interdisciplinary the publication. However this value does not account for the diversity of origins of these imported terms: do they come from a unique H-F or from several? Indeed, for two publications with an equal value of AFT, the one with the greater number of different origins of abroad terms should receive a higher value. Finally, for each journal represented in the corpus with a statistically significant number of articles, we combine the AFT values calculated for all its articles.

Results

The data set is extracted from the PASCAL database that is specifically adapted to the purpose of our approach. It provides broad multidisciplinary coverage of scientific publications and contains nowadays about 20 million bibliographic records from the analysis of the scientific and technical international literature published predominantly in journals and conference proceedings. On the other hand, the PASCAL records benefit from an indexing by both keywords and thematic categories of a classification scheme assigned to each individual publication, either manually by scientific experts or automatically based on a content analysis.

It is this terminology, formed by the indexing keywords that we can also refer to as "terms", that we employ in our analysis, after an assessment step done by a scientific expert. The query operated in this work aims to represent the whole spectrum of disciplines in the PASCAL database by following the magnitude of their representation. The obtained corpus, that comprises 105,254 bibliographic records, is a set of randomly chosen weekly updates of the PASCAL database.

Although each PASCAL record has at least one classification code, we did not exploit this information to define our H-F because the fine-grained classification scheme produces a huge number of disciplines, way too large for our purpose. As indicated previously, to determine our set of H-F without the help of that in-house taxonomy, we decided to unfold the "content analysis + indexing validation + clustering" sequence described in the Methodology section.

The examination of the clusters and their content bring to a final validation of the cluster list corresponding to the list of H-F. That is at that time, for instance, that some clusters the content of which is very close could be merged. At this stage, we have the list of H-F, the list of documents assigned to each H-F and the set of keywords representing the content of each document. So, all the conditions have been met to apply the diffusion model approach to calculate an interdisciplinarity score for each document.

Then, the set of values got by all the documents from the same journal are combined to produce a journal interdisciplinarity indicator. For obvious statistical reasons, only the journals with a significant number of articles in the studied corpus are taken into consideration. Finally, we put the obtained results into perspective.

This work was partially inspired by DBF (Development and Verification of a Bibliometric Model for the Identification of Frontier Research), a Coordination and Support Action of the IDEAS specific programme of the European Research Council (ERC). The authors wish to acknowledge this contribution.


Pascal Cuxac
email: pascal.cuxac@inist.fr

Jean-Charles Lamirel
email: jean-charles.lamirel@loria.fr

Valerie Bonvallot
email: valerie.bonvallot@inist.fr

EFFICIENT SUPERVISED AND SEMI-SUPERVISED APPROACHES FOR AFFILIATIONS DISAMBIGUATION

The disambiguation of named entities is a challenge in many fields such as scientometrics, social networks, record linkage, citation analysis, semantic web...etc. The names ambiguities can arise from misspelling, typographical or OCR mistakes, abbreviations, omissions... Therefore, the search of names of persons or of organizations is difficult as soon as a single name might appear in many different forms. This paper proposes two approaches to disambiguate on the affiliations of authors of scientific papers in bibliographic databases: the first way considers that a training dataset is available, and uses a Naive Bayes model. The second way assumes that there is no learning resource, and uses a semi-supervised approach, mixing soft-clustering and Bayesian learning. The results are encouraging and the approach is already partially applied in a scientific survey department. However, our experiments also highlight that our approach has some limitations: it cannot process efficiently highly unbalanced data. Alternatives solutions are possible for future developments, particularly with the use of a recent clustering algorithm relying on feature maximization.

1.

Introduction

In bibliographic databases, affiliations of authors are of paramount importance. Hence, they permit to the laboratories or institutes to get national and even international visibility, as well as they consequently provide authors with scientific caution. We cannot discuss the issue of affiliations without talking of "Shanghai ranking" which aims at evaluating universities. Our purpose here is not to feed up the controversy (Van Raan 2005) [START_REF] Liu | Academic ranking of world universities using scientometrics:A comment to the Fatal Attraction[END_REF], but to point out that the management of affiliation plays an important role in the calculation of universities "performance". Moed [START_REF] Moed | Citation Analysis in Research Evaluation[END_REF]) reports some problems with author's names and also institutions: "Authors from the same institution, or even from the same department, may not indicate their institutional affiliations in the same way". Depending on the country, it is not always clear how to name a laboratory with respect to its supervisory authorities. The affiliation is also important information to disambiguate author names in bibliographic databases. In this context, Wang points out that: "as the amount of available information increases, problem caused by misspelling, spelling difference, and name or affiliation change also become worse." [START_REF] Wang | A boosted-trees method for name disambiguation[END_REF]. A standardization of data in bibliographic databases is thus necessary to carry out informetrics studies, but it is not a trivial task: the practice, whatever it is intentional or not, of omitting institutional affiliations, or giving incomplete or wrong information is not uncommon [START_REF] Hood | Informetric studies using databases: Opportunities and challenges[END_REF].

Each year, the CNRS (the French National Center of Scientific Research) has to spot its laboratories affiliations among the French ones. Many of them are easy to identify automatically thank to key words extraction but, at the issue of this process, it still remains unidentified affiliations (fig. 1). New methods for disambiguation could help the information analysts to be more efficient when labeling.

Figure 1: An example of CNRS affiliations with the same label

This paper proposes an approach based on Naive Bayes learning method and overlapping clustering. It is structured as follows: section 2 summarizes related works and identifies problems. Section 3 describes our approach firstly with supervised learning method and then with semi-supervised method. Next, section 4 reports experiments and results. Section 5 concludes and discusses future work.

2.

State of the art and discussion

As part of bibliometric analyzes, the authors affiliations can produce statistics by laboratories as well as by institutes or universities. However, such analyses often face with problems of high variability and heterogeneity of naming: a single laboratory name may thus appear in several different ways if the authors use different abbreviations, incomplete or misspelled words (typing mistakes, spelling...). In addition, some universities can have several names (for example University Pierre and Marie Curie = University Paris VI). This problem is known for long but still persists nowadays. In the 1990s, De Bruin et al. [START_REF] De Bruin | The unification of addresses in scientific publications[END_REF]) point out the problem of variability of the author addresses in databases such as SCI (Science Citation Index). They highlight the case of countries like Germany or France where the heterogeneity of data is particularly important. Zitt [START_REF] Zitt | Challenges for scientometric indicators: data demining, knowledge-flow measurements and diversity issues[END_REF] emphasizes the importance of data standardization (author names, affiliations) with special consideration to countries like France where affiliations overlap is important (for example one laboratory can have a University affiliation and a CNRS affiliation). For many bibliometrics analysis, the unification of institutional ad-dresses is an essential task to be carried out prior to any study ( [START_REF] Bourke | Standards issues in a national bibliometric database: The Australian case[END_REF], [START_REF] Osareh | A comparison of Iranian scientific publications in the SCI: 1985-1989 and 1990-1994[END_REF]).

For solving the problem, De Bruin [START_REF] De Bruin | The unification of addresses in scientific publications[END_REF] propose to deal separately with all the words belonging to affiliations and to use in a second step a classification strategy to unify all possible variations of the different words. In a later work [START_REF] De Bruin | Delimitation of scientific subfields using cog nitive words from corporate addresses in scientific publications[END_REF], the same authors use a "single-link clustering" approach to delineate different areas of science on the basis of affiliations. French et al. [START_REF] French | Using Clustering Strategies for Creating Authority Files[END_REF] supply an authority file after a cleaning step (name of country, zip codes, states, expansion of abbreviations, acronyms ...) and then use a clustering based on an "edit distance".Recent approaches also address the problem by the single use of NLP methods, like in [START_REF] Galvez | The unification of institutional addresses applying parametrized finite-state graphs (P-FSG)[END_REF]. The terms data cleaning, data scrubbing, data standardization, data disambiguation, data homogenization and also entity resolution are used to refer to the tasks of transforming source data into clean or normalized data for loading in databases or linking with other data sets or computing statistical indexes (bibliometric analysis for example). If as we have seen, these problems are essential in bibliometrics, but they are also recurrent in many other areas where the heterogeneity of data is an important problem. It can occur within a file or a database, but also when combining information from heterogeneous sources (e.g. record linkage). Erhard Rahm [START_REF] Rahm | Data Cleaning: Problems and Current Approaches[END_REF] classifies data quality problems encountered in data cleaning tasks (fig. 2). In our case, we can assimilate the multi-source to a bibliographic database reporting papers from journals of different publishers. The approach presented by Fellegi and Sunter [START_REF] Fellegi | A Theory for Record Linkage[END_REF] is the reference widely used in record linkage to identify the same entities in different datafiles. It is based on calculating similarity scores between two records. Generalizations of this method for more than two data files have been recently proposed [START_REF] Sadinle | Approaches to Multiple Record Linkage[END_REF] (Sadinle and Frienberg 2012). Ventura [START_REF] Ventura | Methods Matter: Revamping Inventor Disambiguation Algorithms with Classification Models and Labeled Inventor Records[END_REF] mixes disambiguation and record linkage algorithms, using Random Forest, and applies this methodology to a case study of inventors referenced in USPTO patents in the optoelectronics field. In this context of record linkage, Churches [START_REF] Churches | Preparation of name and address data for record linkage using hidden Markov models[END_REF] shows that exploiting probabilistic Hidden Markov models for pre-processing data (names and address) give accurate results with complex data such as residential addresses.

When training data are available many of these studies use metrics like Jaccard, soft-TF-IDF, or mainly edit distance, to measure similarities between data [START_REF] Huang | Efficient name disambiguation for large-scale databases[END_REF]. In his review paper, Bilenko compares the performance of various matching methods and concludes than learned affine edit distance can outperform others, whenever it is used jointly with EM techniques [START_REF] Bilenko | Adaptive name matching in information integration[END_REF]. Morover, probabilistic approaches have been proposed by several authors like e.g. Carayol [START_REF] Carayol | Whos Who in Patents. A Bayesian approach[END_REF], which proposes a Bayesian approach to treat the who's who problem in European patents. He points out the transitivity issue that we'll discuss in the conclusion. Alternatively, a only few authors propose approaches based on unsupervised algorithms. This is the case of Niu (Niu, Wu and Shi 2006) which presents a new method for entity disambiguation using textual information and interobject relationship to evaluate similarity. Entities are author names, and interobject relationship is related to coauthorship network. The novel methodology developed by Aswani for unification of authors names use web mining to get full names and find publications pages [START_REF] Aswani | Mining Information for Instance Unification[END_REF]). As we see, applications are numerous, be it in bibliographic databases, in datawarehouses, in multiple record linkage, in semantic web, or also in semantic digital libraries as shown in [START_REF] Jiang | Affiliation disambiguation for constructing semantic digital libraries[END_REF]). To conclude lets quote the standardization actions conducted through the International Standard Name Identifier (ISNI) and the Virtual International Authority File (VIAF). The aim of ISNI (http://www.isni.org) is to identify international public identities of individuals or communities and to provide tools for disambiguation. VIAF (http://viaf.org/) is a research project of OCLC (Online Computer Library Center) which aims to align lists of authorities (including proper names) to form an international reference database. Do not forget to mention the Oyster software (Open sYSTem Entity Resolution): is an open-source software developed by Talburt in the ERIQ Research Center (Entity Resolution and Information Quality at the University of Arkansas). OYSTER is an entities resolution system using XML scripts [START_REF] Zhou | OYSTER: A Tool for Entity Resolution in Health Information Exchange[END_REF]).

3.

Our approach

We present two different methods for affiliation disambiguation, depending on whether a training set is available or not.

first, a supervised learning approach relying on manually analyzed reference dataset, and in a further step, a semi-supervised approach whose goal is to get rid of a training corpus for cases where no validation data are available.

Supervised learning method

Supervised learning methods allow producing rules from a learning corpus, generalizing what they could learn to the unknown inputs. In the literature there are many methods such as SVM, Rocchio, K-NN, Naive Bayes, HMM or Decision Trees. Our supervised approach is based on a Naive Bayes (NB) algorithm. Let C a set of affiliations classes C = {c 1 ,c 2 , ..., c k }, the problem is to assign to an affiliation, one of these categories.

Using a set of N labeled affiliations {(a i ,c i ), 1 ≤ i ≤ N}, we construct a classification function F : C with A = set of all affiliations. Bayes' formula for a given affiliation a allows to calculate its probability of belonging to a particular class c : Despite of the two main known defects of such method, that are, its ignorance of the order of the words and its ground hypothesis that words are independent conditionally to their class membership, its represents a good alternative for solving our problem. The results obtained by this implementation of Bayes theorem are valid and demonstrated by [START_REF] Hand | Idiots BayesNot So Stupid After All?[END_REF]. Hence, Domingos et al [START_REF] Domingos | Beyond Independence: Conditions for the Optimality of the Simple Bayesian Classifier[END_REF] formerly showed that the misclassification error of NB is minimized as compared to other methods.

( | ) ( | ) ( ) ( )

Semi-supervised method

Whenever no a priori knowledge is available, we applied a semi-supervised methodology. In this case, we firstly process by the use of an overlapping clustering method. We exploited two clustering methods derived of K-means algorithm developed by MacQueen (McQueen 1967):

 the Axial M-Means algorithm (noted AKM) is a clustering method proposed by Lelu, [START_REF] Lelu | Modèles neuronaux pour l'analyse de donnes documentaires et textuelles[END_REF].

Document vectors are standardized using the Hellinger metric, particularly adapted to textual data, and are assigned to the axes of class ("vectors neurons") pointing towards areas of high data density, with degrees of centrality in their classes more or less pronounced according to the principle of mobile centers of Kmeans. This method allows producing clusters presenting specific characteristics: they might overlap because the clustering method allows an object or a variable to belong to more than one cluster; the constituting elements of a cluster, objects and variables, are ranked by decreasing similarity with the cluster ideal type. But in this algorithm, cluster centers are obtained after allocation of each object to one of these centers, and overlap is performed according to its proximity with the cluster.

 the Overlapping K-Means algorithm (noted OKM) developed by Cleuziou [START_REF] Cleuziou | An extended version of the k-means method for overlapping clustering[END_REF]. This method alternatively assumes that all objects must participate in the definition of each center. In the second step, we only retain the major representatives of the classes, which are the documents that have the highest values of projection on the axes representing the classes. These documents will then be used as the training corpus. Then, we calculate the most representative words of each class and we use each of these groups to query the web (via Google). The website that represents the most relevant answer of the search engine for a given class is used to label the class. If necessary, we proceed to a further step of classes merging.

In the last phase, we train the NB method with the corpus defined in the clustering phase and labeled by the class names extracted from the web. The testing process is achieved on the complementary corpus of documents eliminated after the clustering phase. Figure 3 summarizes in a schema, our semi-supervised approach with the three steps: soft clustering, clusters labeling and clusters fusion, and finally the Naive Bayes classification.

4.

Experiments

In this section we present the datasets, the evaluation parameters and our results obtained with three datasets and the two methods presented previously.

Datasets

We used three different datasets: a first dataset with 10 057 French affiliations (noted hereafter A1), a second small dataset of 150 Lorraine affiliations (French region) (noted A2), and a last dataset with 2266 French affiliations extracted from WOS and SCI (noted A3). All those datasets have been preprocessed by splitting the affiliations into words using space as separator (any punctuation, including dash, is firstly removed). Given the difficulty we had to cut affiliations into words in the datasets A1 and A2 (dash, sometimes missing space ...), we have then used a second splitting technique based on n-grams after converting affiliations in string without spaces or punctuation. Supervised learning is applied on dataset A1, and semi-supervised one is applied on the two other datasets.

Measures of performance

The results are evaluated in terms of recall, precision, F-measure, because we know a priori classes of all affiliations (even when we do not use labels for semi-supervised approach).

Recall: R = ( )

Precision: P = ( ) F-measure: F = ( )
where TP, FP, FN, respectively average number of true-positives, average number of false-positives, and average number of false-negatives. Thereafter, we present the average obtained on these parameters for each experiment.

Supervised learning

The dataset A1 was split into training dataset and test dataset successively represented by the words of affiliations and by the n-grams. The figure 5 shows the distribution of the affiliations in the 53 resulting classes (test + train) and thus highlights the fact that the resulting classification is highly unbalanced. The classification results on the dataset A1 are presented in Table 1. Because of the problem of individualization affiliations into words in dataset A1, the results obtained with n-grams appear slightly better, with an optimum of Recall of 0.92. It should also be noted that a systematic lookup of the analysis results of NB which got a very high probability, whilst being in contradiction with the expected (i.e. human labeled) results, permits us to prove that the manual labeling of the test dataset was sometimes wrong (the model was giving the right answer in all that cases!).

For N-grams, we choose N = 7 because this is the value that allows to have an optimum value of the accuracy on the training corpus (accuracy = 93%). For an higher value of N, the increase is not significant.

Semi-supervised classification

The datasets A2 and A3 were used for this experiment. Those datasets are split into train and test datasets with the methodology described in the former section detailing our approach. The figure 6 reports the distribution of affiliations in the resulting classes (A2: 19 classes; A3: 10 classes). It highlights that the smaller dataset (A2) is highly unbalanced, whilst the bigger one (A3) is homogeneous. The exploitation of this two different kinds of distribution is intentional Hence it permits to highlight the impact of data configuration on our results. With this kind of data, K-means approaches produce low quality results (Table 2). It is probably due to the bad representation of data. As we discuss in the last section of this paper, a vector space representation gathering scientific content and the address should certainly improve the clustering results. As detailed in table 3, the results are very good for the dataset A3, but significantly lower for the dataset A2, where recall is average. This is due to the large number of classes as compared to the small size of the dataset and to the strong imbalance of these classes. Indeed, in this case the exploited clustering method is becoming blind to small classes. The use of OKM clustering can slightly improve results, especially in the case of unbalanced data: this is probably due to the different overlapping strategies exploited by the methods, AKM being impacted by the use of a threshold for assignments to clusters. It must be noted that a small number of affiliations cannot be classified by these approaches. These are very short addresses containing only few discriminant information. Whatever the improvements that can be done, these "poor" affiliations may still be rejected.

5.

Conclusions and discussions

The results we obtained with our approaches for affiliations disambiguation are very encouraging, both in the supervised learning context and in the semi-supervised one. Our experiments also allow us to show that our methods provide significant assistance for correcting the results of human labeled affiliations. However, it is clear that we must achieve more experiments to conclude on the overall relevance of the methodology. Hence, there still remain a few weaknesses in our methodology. Those are related to the exploited clustering methods in the case of very unbalanced classes. We thus plan to conduct tests with other clustering methods and to implement data balancing techniques.

The data used here are only affiliations CNRS laboratories (except for the small dataset "Lorraine"), because the study was done for a steering support department of CNRS. Having more diverse data might change the results. For this purpose, we are going to create a benchmark dataset of labeled international addresses which could be accessible by anyone who would test his disambiguation method.

In a near future, we propose to test the IGNGF algorithm developed by Lamirel [START_REF] Lamirel | Variations to incremental growing neural gas algorithm based on label maximization[END_REF] in conjunction with variable selection techniques based on the optimization of the feature F-measure. Another important point would be to exploit a learning method that should be able to learn with character strings of variable length. It would be necessary to develop an automatic method that could highlight the conflicting cases. Future works should also take into account the xml structure to consider separately the cities, street names, the laboratories names... Another way is to consider the scientific content of documents, such as titles and abstracts of articles published. Once these documents indexed each address would be represented by a vector of words (describing research activities) allowing probably a more relevant classification.

As a last resort, study of transitivity can perhaps permit to detect false positive or false negative results and thereby isolate the results to be verified.

6.

Figure 2 :

 2 Figure 2: Data quality in data cleaning tasks figure from (Rahm and Do 2000)

  = probability of c given a, ( | ) = probability of a given c, P(a), P(c) = respectively probability of a and probability of c. If we simplify by assuming that labels are randomly distributed (are not dependent on the length of the affiliation or the position within the affiliation), then the probability of affiliation a given a class c, is ( | ) ∏ ( ) with w i = the i-th word of a. then, by applying the Bayes rule, we can classify an affiliation a in a class c: c = arg max (a c), (c) = arg max (c a)

Figure 3 :

 3 Figure 3: Semi-supervised classification schema.

Figure 4 :

 4 Figure 4: Data sample with various forms of addresses

Figure 5 :

 5 Figure 5: Distribution of affiliations in resulting classes (dataset A1)

Figure 6 :

 6 Figure 6: Distribution of affiliations in resulting classes (datasets A2 (left) and A3 (right))

Table 1 : R, P and F values for dataset A1.

 1

		Recall	Precision	F-measure
	N-gram (N=7)	0.92	0.94	0.93
	Words	0.81	0.88	0.85

Table 2 : Results of AKM and OKM for datasets A2 and A3.

 2

	Dataset	Clustering	Recall	Precision	F-measure
	Dataset A2	AKM OKM	0.44 0.41	0.81 0.87	0.55 0.67
	Dataset A3	AKM OKM	0.40 0.49	0.95 0.92	0.54 0.63

Table 3 : Semi-supervised approach: R, P and F values for datasets A2 and A3.

 3

	Dataset	Semi-sup	Recall	Precision	F-measure
	Dataset A2	with AKM with OKM	0.79 0.69	0.76 0.81	0.73 0.86
	Dataset A3	with AKM with OKM	0.98 0.92	0.97 0.92	0.97 0.91


Camillo De Lellis
email: camillo.delellis@math.uzh.ch

Matteo Focardi
email: focardi@math.unifi.it

Berardo Ruffini
email: berardo.ruffini@sns.it

Ruffini A Berardo

Note

A NOTE ON THE HAUSDORFF DIMENSION OF THE SINGULAR SET FOR MINIMIZERS OF THE MUMFORD-SHAH ENERGY

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In what follows if A = Ω we shall drop the dependence on the set of integration. We refer to the book [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for all the notations and preliminaries on SBV functions and the regularity theory for local minimizers of the Mumford-Shah energy giving precise references when needed.

In this note we provide a simplified proof of the following result due to Ambrosio, Fusco and Hutchinson [2, Theorem 5.6] (established there for quasi-minimizers as well).

Theorem 1. Let u be a local minimizer of the Mumford-Shah energy, i.e. any function u ∈ SBV (Ω) with MS(u) < ∞ and such that MS(u) ≤ MS(w) whenever {w = u} ⊂⊂ Ω.

Let Σ u ⊆ S u be the set of points out of which S u is locally regular, and let

Σ ′ u := x ∈ Σ u : lim ρ↓0 ρ 1-n Bρ(x) |∇u| 2 = 0 . Then, dim H Σ ′ u ≤ n -2.
The main interest in establishing such an estimate on the set Σ ′ u , the so called subset of triplejunctions, is related to the understanding of the Mumford-Shah conjecture (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Chapter 6] for a related discussion, see also [START_REF] De Lellis | Higher integrability of the gradient for minimizers of the 2d Mumford-Shah energy[END_REF]Section 7]).

Indeed, Theorem 1, together with the higher integrability property of the approximate gradients enjoyed by minimizers as established in 2-dimensions by [START_REF] De Lellis | Higher integrability of the gradient for minimizers of the 2d Mumford-Shah energy[END_REF] and more recently in any dimension by [START_REF] De Philippis | Higher integrability for minimizers of the Mumford-Shah functional[END_REF], imply straightforwardly an analogous estimate on the full singular set Σ u . More precisely, in view of [4, Theorem 1] and [START_REF] De Philippis | Higher integrability for minimizers of the Mumford-Shah functional[END_REF]Theorem 1.1] Theorem 13]) showing that the blow-up limits of the jump set S u in points in the regime of small gradients, i.e. in points of Σ ′ u , are minimal Caccioppoli partitions. The original approach in [START_REF] Ambrosio | Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functionals[END_REF], instead, relies on the notion of Almgren's area mimizing sets, for which an interesting but technically demanding analysis of the composition of SBV functions with Lipschitz deformations (not necessarily one-to-one) and a revision of the regularity theory for those sets are needed (cp. with [2, Sections 2, 3 and 4]).

Given [START_REF] De Lellis | Higher integrability of the gradient for minimizers of the 2d Mumford-Shah energy[END_REF]Theorem 13], the regularity theory of minimal Caccioppoli partitions developed in [START_REF] Massari | Regularity properties of optimal segmentations[END_REF][START_REF] Leonardi | Blow-Up of Oriented Boundaries[END_REF][START_REF] Leonardi | Optimal Subdivisions of n-dimensional Domains[END_REF] and standard arguments in geometric measure theory yield the conclusion, thus bypassing the above mentioned technical complications.

We describe briefly the plan of the note: in Section 2 we introduce necessary definitions and recall some well-known facts about Caccioppoli partitions. In Section 3 we prove our main result and comment on some related improvements in a final remark.

Caccioppoli partitions

In what follows Ω ⊂ R n will denote a bounded open set.

Definition 1. A Caccioppoli partition of Ω is a countable partition E = {E i } ∞ i=1
of Ω in sets of (positive Lebesgue measure and) finite perimeter with ∞ i=1 Per(E i , Ω) < ∞. For each Caccioppoli partition E we define its set of interfaces as

J E := i∈N ∂ * E i .
The partition E is said to be minimal if

H n-1 (J E ) ≤ H n-1 (J F )
for all Caccioppoli partitions F for which there exists an open subset

Ω ′ ⊂⊂ Ω with ∞ i=1 L n (F i △E i ) ∩ (Ω \ Ω ′ ) = 0.
Definition 2. Given a Caccioppoli partition E we define its singular set Σ E as the set of points for which the approximate tangent plane to J E does not exist.

A characterization of the singular set Σ E for minimal Caccioppoli partitions in the spirit of εregularity results is provided in the ensuing statement (cp. with [8, Corollary 4.2.4] and [9, Theorem III.6.5] ). Then, there exists a dimensional constant ε = ε(n) > 0 such that

Σ E = x ∈ Ω ∩ J E : inf Bρ(x)⊂⊂Ω e(x, ρ) ≥ ε , (2.1)
where e(x, ρ) denotes the spherical excess of E at the point x ∈ J E at the scale ρ > 0, that is e(x, ρ) := min

ν∈S N-1 1 ρ n-1 Bρ(x)∩J E |ν E (y) -ν| 2 2 dH n-1 (y).
We recall next a result that is probably well-known in literature; we provide the proof for the sake of completeness.

Theorem 3. Let E be a minimal Caccioppoli partition in Ω, then dim H Σ E ≤ n -2.
If, in addition, n = 2, then Σ E is locally finite.

Proof. We apply the abstract version of Federer's reduction argument in [START_REF] Simon | Lectures on geometric measure theory[END_REF]Theorem A.4] with the set of functions F = {χ J E : E is a minimal Caccioppoli partition} endowed with the convergence

χ J E h → χ J E ⇐⇒ lim h↑∞ J E h g dH n-1 = J E g dH n-1 , for all g ∈ C 1 c (Ω).
and singularity map sing

(χ E ) = Σ E .
It is easy to see that condition A.1 (closure under scaling) and A.3(2) hold true. Moreover, the blow-ups of a minimal Caccioppoli partition converge to a minimizing cone (see [START_REF] Leonardi | Blow-Up of Oriented Boundaries[END_REF]Theorem 3.5], or [8, Theorem 4.4.5 (a)]), so that A.2 holds as well. About A.3(1), we notice that the singular set of an hyperplane is empty. Eventually, if a sequence (χ J E h ) h∈N ⊆ F converges to χ J E and (x h ) h∈N converges to x, with x h ∈ Σ E h for all h, then by the continuity of the excess and the characterization in (2.1), x ∈ Σ E , so that condition A.3(3) is satisfied as well.

To conclude, we recall that [13, Theorem A.4] itself ensures that the set Σ E is locally finite being in this setting dim H Σ E = 0.

Proof of the main result

We are now ready to prove the main result of the note following the approach exploited in [2, Theorem 5.6]. To this aim we recall that Ambrosio, Fusco & Pallara (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]) characterized alternatively the singular set Σ u as follows

Σ u = {x ∈ S u : lim inf ρ↓0 (D(x, ρ) + A (x, ρ)) ≥ ε 0 }, (3.1)
where ε 0 is a dimensional constant, and the scaled Dirichlet energy and the scaled mean-flatness are respectively defined as

D(x, ρ) := ρ 1-n Bρ(x) |∇u| 2 dy, A (x, ρ) := ρ -1-n min T ∈Π Su∩Bρ(x) dist 2 (y, T )dH n-1 (y),
with Π the set of all affine (n -1)-planes in R n .

Proof of Theorem 1. We argue by contradiction: suppose that there exists s > n -2 such that H s (Σ ′ u ) > 0. From this, we infer that H s ∞ (Σ ′ u ) > 0, and moreover that for H s -a.e. x ∈ Σ ′ u it holds lim sup

ρ↓0 + H s ∞ (Σ ′ u ∩ B ρ (x)) ρ s ≥ ω s 2 s (3.2)
(see for instance [1, Theorem 2.56 and formula (2.43)] or [9, Lemma III.8.15]). Without loss of generality, suppose that (3.2) holds at x = 0, and consider a sequence ρ h ↓ 0 for which

H s ∞ (Σ ′ u ∩ B ρ h ) ≥ ω s 2 s+1 ρ s h for all h ∈ N. (3.3)
[4, Theorem 13] provides a subsequence, not relabeled for convenience, and a minimal Caccioppoli partition E such that

H N -1 ρ -1 h S u * ⇀ H N -1 J E , and ρ -1 h S u → J E locally Hausdorff. (3.4)
In turn, from the latter we claim that if F is any open cover of Σ E ∩ B 1 , then for some

h 0 ∈ N ρ -1 h Σ ′ u ∩ B 1 ⊆ ∪ F ∈F F for all h ≥ h 0 . (3.5)
Indeed, if this is not the case we can find a sequence x h j ∈ ρ -1

h j Σ ′ u ∩ B 1 converging to some point x 0 / ∈ Σ E . If π E
x 0 is the approximate tangent plane to J E at x 0 , that exists by the very definition of Σ E , then for some ρ 0 we have

ρ 1-n Bρ(x 0 )∩J E dist 2 (y, π E
x 0 )dH n-1 < ε 0 , for all ρ ∈ (0, ρ 0 ).

In turn, from the latter inequality it follows for ρ ∈ (0, ρ 0 ∧ 1)

lim sup j↑∞ Bρ(x h j )∩ρ -1 h j Su dist 2 (y, π E x 0 ) dH n-1 < ε 0 .
Therefore, as x h j ∈ ρ -1 h j Σ ′ u , we get for j large enough lim sup ρ↓0 D(x h j , ρ) + A (x h j , ρ) < ε 0 , a contradiction in view of the characterization of the singular set in (3.1).

To conclude, we note that by (3.5) we get

H s ∞ (Σ E ∩ B 1 ) ≥ lim sup h↑∞ H s ∞ (ρ -1 h Σ ′ u ∩ B 1 );
given this, (3.3) and (3.4) yield that

H s (Σ E ∩ B 1 ) ≥ H s ∞ (Σ E ∩ B 1 ) ≥ lim sup h↑∞ H s ∞ (ρ -1 h Σ ′ u ∩ B 1 ) ≥ ω s 2 s+1 ,
thus contradicting Theorem 3.

Remark 3. In two dimensions we can actually prove that the set Σ ′ u of triple-junctions is at most countable building upon some topological arguments. This claim follows straightforwardly from the compactness result [4, Theorem 13], David's ε-regularity theorem [3, Proposition 60.1], and a direct application of Moore's triod theorem showing that in the plane every system of disjoint triods, i.e. unions of three Jordan arcs that have all one endpoint in common and otherwise disjoint, is at most countable (see [11, Theorem 1] and [12, Proposition 2.18]). Despite this, we are not able to infer that Σ ′ u is locally finite as in the case of minimal Caccioppoli partitions (cp. with Theorem 3). Indeed, if on one hand we can conclude that every convergent sequence (x j ) j∈N ⊂ Σ ′ u has a limit x 0 / ∈ Σ ′ u thanks to [4, Proposition 11 and Lemma 12]; on the other hand, we cannot exclude that the limit point x 0 is a crack-tip, i.e. it belongs to the set Σ u \ Σ ′ u = {x ∈ Σ u : lim inf ρ↓0 D(x, ρ) > 0}. The same considerations above apply in three dimensions as well for points whose blow-up is a T cone, i.e. a cone with vertex the origin constructed upon the 1-skeleton of a regular tetrahedron. The latter claim follows thanks to [4, Theorem 13], the 3-d extension of David's ε-regularity result by Lemenant in [6, Theorem 8], and a suitable extension of Moore's theorem on triods established by Young in [START_REF] Young | A generalization of Moore's theorem on simple triods[END_REF].

Let us finally point out that we employ topological arguments to compensate for monotonicity formulas, that would allow us to exploit Almgren's stratification type results and get, actually, a more precise picture of the set Σ ′ u (cp. with [14, Theorem 3.2]).

1 .

 1 IntroductionConsider the (localized) Mumford-Shah energy on a bounded open subset Ω ⊂ R n given byMS(v, A) = A |∇v| 2 dx + H n-1 (S v ∩ A), for v ∈ SBV (Ω) and A ⊆ Ω open.(1.1)

Theorem 2 .

 2 Let Ω be an open set and E = {E i } i∈N a minimal Caccioppoli partition of Ω.