Vivien Kana Zeumo
email: vivien.kana-zeumo@dauphine.fr

Alexis Tsoukiàs
email: tsoukias@lamsade.dauphine.fr

Blaise Somé
email: some@univ-ouaga.bf

A new methodology for multidimensional poverty measurement based on the capability approach

Keywords: Meaningful measurement, Capabilities approach, Policy making, Decision aiding

In this paper, we propose a methodology based on the use of clustering techniques derived from data analysis and multi-attribute decision analysis methods aiming at purposeful multidimensional poverty measurement. The issue of meaningfulness is thus analysed both from a theoretical point of view (measurement theory) and from an operational one (policy effectiveness). Through this new methodology of multidimensional poverty measurement, we aim at providing a contribution to methodological knowledge insisting on the necessity to build "meaningful measurements" for policy making and policy implementation. On the other hand, "meaningful measurements" appear as a contribution to the operationalisation of Sen's capabilities approach. Our standpoint underlines the necessity to consider the problem of poverty measurement as a decision problem and to tackle its measurement issue with that in mind.

Introduction

The review of the literature on poverty measurement (see [START_REF] Kana | A survey on multidimensional poverty measurement: a decision aiding perspective[END_REF] allows us to conclude that measuring poverty is not a representation of an objective situation, it is rather an instrument for pursuing a policy. People may feel poor and not be identified as such. People may be identified as poor and not feel as such. Indeed, poverty is an evolutive, multidimensional, fuzzy and non-objective situation which does not contain anything of numerical, but only the sensation of those who are suffering. We are more or less poor and in many different ways.

Many authors (see [START_REF] Nussbaum | Nature, function and capability: Aristotle on political distribution[END_REF][START_REF] Nussbaum | Women and Human Development: the Capabilities Approach[END_REF][START_REF] Fusco | La Contribution des Analyses Multidimensionnelles à la Compréhension et à la Mesure du Concept de Pauvreté: Application Empirique au Panel Communautaire des Ménages[END_REF][START_REF] Alkire | Valuing Freedoms: Sen's Capability Approach and Poverty Reduction[END_REF][START_REF] Bertin | Pauvreté non Monétaire: une Analyse des Intéractions Appliquée à la Guinée[END_REF][START_REF] Kana | A survey on multidimensional poverty measurement: a decision aiding perspective[END_REF] agree that Sen's capability approach (see [START_REF] Sen | Commodities and Capabilities[END_REF] is appropriated as tool aiming at assessing how welfare is distributed among a given population. The reason is that allows to highlight the diversity of relationships between people and goods (commodities), the complex relationships of individuals between themselves (social relations) and of individuals with their environment (institutions, norms, cultures). The strong argument for the capability approach is based on the postulate that commodities (goods or individual resources) are insufficient to evaluate and describe in a faithful way, the welfare of people. As an example, two people can aspire to different things in terms of welfare, while owning the same resources equivalent to, let's say, $3500 U.S. This is why [START_REF] Sen | Commodities and Capabilities[END_REF] introduces a broad distinction between a person's interests and their fulfilment, respectively called "well-being" and "advantage". Sen argued that "well-being is concerned with a person's achievement: how 'well' is his or her 'being' ? 'Advantage' refers to the real opportunities that the person has, especially compared with others". This postulate considers the commodities a mean for improving the quality of life of individuals and advocates to focus on how these individuals will use their resources. This led Sen to develop a broad discussion about the distinctions between commodities, characteristics, functionings and capabilities.

Sen's capabilities approach allows to take into account the notion of freedom that has a person to achieve a certain level of well-being and the assumption of human diversity in the process of poverty measurement. Therefore, while trying to measure poverty we need to take into account several different dimensions of uncertainty. We must select and validate the space of functionings that individuals are able to "do" (doing) or aspire to "be" (being) through their commodities and their characteristics. The choice and validation of the space of functionings can be done in an efficient and realistic way only within a decision aiding setting.

This paper shows how we can process the information that is required to implement the capability approach in a way useful for policy design, policy implementation and the assessment of poverty reduction initiatives. We present a new methodology which operationalises Sen's capabilities approach through the development of meaningful multidimensional poverty measurements. The general methodology shown in Figure (1) outlines the different stages allowing to derive the meaningful measurements. The stage concerning supervised learning (schematised as the part appearing in Figure (1) with "double line") will be not developed in this paper, but in a forthcoming one. It will only be briefly presented in this paper. The paper is organised as follows: Section 2 introduces basic notions concerning our motivations, the concept of meaningful measurement, the concept of capability and the setting. Section 3 shows how we transform the data table into a proximity matrix, while Section 4 develops the clustering process aiming at partitioning the data set. In Section 5 we show how to construct meaningful measurements, while in Section 6 we show how to translate meaningful measurements into concrete actions in terms of policies, programmes and projects for implementation. Section 7 presents a brief discussion about some main advantages of our methodology. In Section 8 we introduce some properties that our suggested measurements satisfy. Concluding comments are given in Section 9. In order to fully understand our methodology, we will develop an illustrative example at the end of each section.

Database

• • • • • • x j • • • • • • Ω x i • • • • • • x j i • • • • • • . . . . . . . . .
client can be represented, for instance, by the World Bank, the European Union, the UNDP1 , the WHO2 or the NEPAD3 with a specific poverty reduction policy aiming to support specific categories of citizens through precise actions such as facilitated access to credit, land re-distribution, water supply enhancement programmes, health research programmes, education aid programmes. Our client is faced to several major problems:

• Know what the situation is and measure it: There are different types of poverty which imply different perspectives between policy maker and subjects. Income is not always representative and the cutting off thresholds are arguable. Measuring poverty has to be an instrument of pursuing a policy. Hence, in order to design interventions best adapted to a given reality, we firstly need to understand the factors and causes determining the present situation. This calls at replying to questions of the type: which elements describe better the specific conditions of the observed population with respect to the precise policy to be pursued? which elements better characterize the perception of the interested population as members of a specific category? How can we measure it? People being differently poor, how can we construct measurements reflecting different categories of poverty?

• Dealing with different poverties: It is misleading to talk only about "poor" and "not poor", at least as far as a multidimensional perspective of poverty is considered. What we observe in reality are different types of poverty. Various different, though related, questions can be asked: What is the underlying problem that has to be dealt with in priority? What specific objectives are to be pursued in confronting these different poverties? Who are eligible for some policy measure? Who is expected to benefit from such policies? How they should benefit? Is that specific policy efficient? Is this specific policy appropriate for the target group? What is the cost for implementing such a policy? Why? What does it mean fighting poverty?

• Dealing with several different dimensions of uncertainty: Mostly, poverty databases are very large and are formed by mixed variables. Then, the heterogeneous information has to be considered. The challenge consists to identify undiscovered groupings of individuals and establish hidden relationships between them. It is therefore an operation aimed at extracting relevant information from data. This calls at replying to questions of the type: which information are readily available and relevant? Are they useful for drawing rational conclusions and recommendations? How easy is to assess the missing information?

• Predicting the consequences and valuing the outcomes: Sometimes, policies can be unsuccessful and ineffective without any positive impact in the medium or long-term. This can be due to several reasons such as uncertainties or missing information. Since a policy is considered as a set of actions (or alternatives) that our client would like to undertake in a given region, it is crucial to explore all alternatives of each policy in order to analyse the consequences of the various possible policies which have to be pursued in order to improve the living conditions of households. This leads to assess the effectiveness of various possible policies by putting the best evidence at the heart of research such as to determine whether a particular policy will produce a positive impact on welfare of people in the future when that policy will be translate into concrete actions. This involves replying to questions of the type: What are the potential alternatives? What are their consequences? If outcomes are undeterminable or uncertain, what can we decide about a compromise?

To sum up, considering this problem situation, an analyst (or policy maker), a population and some knowledge about it, we are looking for understanding: How to identify the different types of poverty? What are the population needs? Once we have classified the population into different classes of households (which are in reality the target of some policy), how may we derive adequate policies to help "poor people" to get out this situation? How to design and identify potential alternatives or actions such as to highlight preferable alternatives which are more important with respect to certain objectives and less important with respect to others? How to decide to which alternative policy each individual/household has to be subjected? How to monitor and assess such policies?

To tackle this problem situation, in a way allowing to provide answers to such questions, we have introduced in this paper the concept of meaningful multidimensional poverty measurement (MDPM) by combining the capability approach with decision aiding methodology [START_REF] Tsoukiàs | From decision theory to decision aiding methodology[END_REF].

What is a meaningful measurement?

A rather complete definition of the term 'measurement' has been given by [START_REF] Mari | Epistemology of measurement[END_REF] who argued that "measurement is a specific kind of evaluation, i.e. it is an operation aimed at associating an information entity, the result of measurement, with the state of the system under measurement in reference to a given quantity, the measurand". We believe that a measurement of poverty should be considered as a set of operations allowing to build a bridge (field of subjective human experiences) between the physical world (field of physical things) and the informational world (field of objective knowledge). The concept of meaningfulness comes from measurement theory (see [START_REF] Suppes | Measurement empirical meaningfulness and three-valued logic[END_REF][START_REF] Krantz | Foundation of Measurement: Additive and Polynomial Representations[END_REF]. Fred [START_REF] Roberts | Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences[END_REF] presented 'meaningfulness' as an essential condition for a measurement to be well-defined in the meaning of correctness, completeness and rationality. Roberts's standpoint is clearly in the same line of definition given by [START_REF] Stevens | On the theory of scales of measurement[END_REF] according to which "measurement, in the broadest sense, is defined as the assignment of numerals to objects or events according to rules".

However, in the case of poverty, a measurement is not only performed in order to assign numbers to individuals or households, but it has to help decision makers make well-informed decisions about a particular policy in such a way to design and identify the preferable alternatives with respect to same complex policy issues. It is a decision aiding process run in a suitable way aiming at selecting appropriate policies, laying out the alternatives (or actions), predicting the consequences and valuing the negative and positive outcomes when that policy is being translated into concrete actions, i.e. in terms of strategies, programmes and projects for implementation. This standpoint imposes to define the framework of decision aiding and implies two essential requirements that a poverty measurement has to satisfy beyond measurement meaningfulness: operationality and legitimation. A poverty measurement is operational if it can be used efficiently to recognise actors drawn from some universe it denotes and if it can help decision makers to elaborate well-informed interventions. Otherwise, a poverty measurement is legitimate if it takes into account how a final recommendation is presented, implemented and perceived by the other actors besides its precise contents. Note that operationality and legitimation have not been defined explicitly in this paper, so the definitions given here are based on our retrospective analysis and reconstruction (see [START_REF] Bouyssou | Evaluation and Decision Models: a Critical Perspective[END_REF][START_REF] Tsoukiàs | On the concept of decision aiding process: an operational perspective[END_REF].

Therefore, in the field of poverty or welfare, a measurement is "meaningful" if it complies to three conditions:

Theoretical soundness: poverty measurement needs to be theoretically sound, in the sense that the concepts used to construct it are in adequacy with measurement theory;

Operational completeness: poverty measurement needs to be operationally complete, in the sense that it is useful for policy making, policy implementation and it helps decision makers to make well-informed decisions about policies, programmes or projects.

Legitimation: poverty measurement needs to be legitimated in the sense that, it should reflect the perception of the society, the stakeholders and actors.

Thus, we can define a meaningful multidimensional poverty measurement (MDPM) as follows:

Definition 2.1. A MDPM is a meaningful measurement derived from a decision aiding process aiming to improve people's capabilities and their living standards.

Note that, our postulate concerning the MDPMs is in adequation with the following three positions and Sen's capability approach sketched at the standpoint of its operationalization:

(P1): Measurements are inherent properties of the measured things (see [START_REF] Mari | Epistemology of measurement[END_REF] (P2): Measurements are results of operations that preserve the relations observed among measured things (see [START_REF] Mari | Epistemology of measurement[END_REF][START_REF] Roberts | Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences[END_REF] (P3): Measurements are results of a decision aiding process (see [START_REF] Bouyssou | Evaluation and Decision Models: a Critical Perspective[END_REF][START_REF] Tsoukiàs | On the concept of decision aiding process: an operational perspective[END_REF]

Endowments, commodities, functionings and capabilities

The origins of the capabilities approach can be found in a series of papers critiquing traditional welfare economics, written by Sen in the early 1980s (see [START_REF] Sen | Famines as failures of exchange entitlements[END_REF][START_REF] Sen | Starvation and exchange entitlements: A general approach and its application to the great Bengal famine[END_REF][START_REF] Sen | Poverty and Famines: an Essay on Entitlement and Deprivation[END_REF][START_REF] Sen | Equality of what? The Tanner lecture on human values[END_REF][START_REF] Sen | Commodities and Capabilities[END_REF][START_REF] Sen | Capability and Well-being[END_REF] where he developed the concepts of endowment, commodity, functioning, and capability for assessing the well-being of individuals. As stated by [START_REF] Alkire | Valuing Freedoms: Sen's Capability Approach and Poverty Reduction[END_REF], "the goal of both human development and poverty reduction should be to expand the capability that people have to enjoy 'valuable beings and doings'. They should have access to the positive resources they need in order to have these capabilities. And they should be able to make choices that matter to them. The capabilities approach appears as the most influential recent attempt for valuing a person's achievement ('well-being' ), the real opportunities that this person has ('advantage' ) and the quantity of 'happiness' generated in this person's life. These purposes are tackled in this paper through four core concepts introduced in Sen's capability approach: endowments, commodities, functionings, capabilities.

Initially, each individual legally owns a combination of resources (called 'endowments set' or more simply 'endowments' ). These 'resources' include both tangible assets such as land, equipment, animals, etc., and intangibles assets such as natural talents or qualities, labour power, physical abilities, knowledge, skills, etc. Thus, each individual can use the resources of his endowment set to produce and to legally obtain the set of all possible combinations of goods and services that are exchanged within the society. We call this 'commodities set' or more simply 'commodities'. Note that, depending on his tastes and preferences, an individual can choose to enjoy only one of such possible combinations. For example4 , a farmer may use his land, labour, and other resources to produce the food he wants; a labourer may 'exchange' his labour power to secure his food; a fisherman may first use his labour, equipment and fishing boat to 'produce' a catch of fish and then exchange it to get the rice he wants; an unemployed person may use his resource of 'citizenship of a welfare state' to claim a 'transfer' of state funds in the form of unemployment benefit. In general, the set of exchange resources can be seen as a mapping from a given person's endowment vectors (the set of all possible endowments set) to availability sets of commodity vectors (the set of all possible commodities set). Many authors (see [START_REF] Sen | Famines as failures of exchange entitlements[END_REF][START_REF] Sen | Starvation and exchange entitlements: A general approach and its application to the great Bengal famine[END_REF][START_REF] Osmani | The entitlement approach to famine: An assessment[END_REF] have proposed several procedures for valuing exchange resources. For example, [START_REF] Osmani | The entitlement approach to famine: An assessment[END_REF] suggests to consider: "for the farmer, the input-output ratios in farm production; for the labourer, the ratio between money wage and the price of food i.e., the real wage rate; for the fisherman, both the input-output ratio in fishing and the relative price of fish and rice; and for the unemployed person, the rate of unemployment benefit".

The commodities aren't necessarily translated into well-being but they are rather conceptualised in terms of a person's characteristics. For example, the possession of a vehicle allows the owner to benefit of all properties of the vehicle, which can be used to satisfy mobility, to transport goods, to obtain happiness from travelling and for rental purpose in order to generate income. As mentioned by [START_REF] Sen | Commodities and Capabilities[END_REF], the 'characteristics of commodities' don't tell us what an individual will do with such commodities. For example, a person can possess a vehicle and mechanical skills but, be "poorly transported" because there is no gas station in his village. This latter standpoint led Sen to consider the 'functionings' of individuals. A functioning is the achievement of an individual i.e. what he succeeds in doing and being with his commodities and their characteristics given his personal characteristics as well as the social and environmental factors beyond his control. For example, owning a vehicle (commodity), considering the characteristic 'mobility', some individuals may achieve the functioning 'adequately transported', while others, considering the same commodity and the same characteristic (but without gas station in their village) may achieve the functioning 'poorly transported'. Note that, a functioning has to be distinguished from the commodities which are used to achieved those functionings. For instance, to 'be adequately transported' must be distinguished from the fact to own a vehicle.

The totality of all the alternative functioning vectors the person can choose from, given the contingent circumstances, reflects the person's capabilities (see [START_REF] Sen | Commodities and Capabilities[END_REF]. A 'capability' reflects a person's ability to achieve, through choice, a given functioning among the various alternative functioning bundles. For example, a person may have the ability to buy a vehicle and to have the capability to be 'well transported', but he may choose not to do so and continue to get to work by foot or by bike.

The following diagram shows the relationship among the three basic concepts:

Endowments -→ Commodities -→ Functionings -→ Capabilities
In order to well understand the remaining sections, we introduce the following definitions:

Definition 2.2. A household is a basic socio-economic unit in which different people, related or not, are living in the same house or concession under the authority of a person named chief of household, putting their commodities and their characteristics together to improve their abilities in doing and being.

Definition 2.3. A cluster is a set of households which are similar or alike in term of distribution of commodities.

Definition 2.4. A variable is a characteristic allowing to describe each household.

Definition 2.5. A commodity is a good or service that a household declares to legally attain with his endowments.

The setting

The basic unit of analysis is a household. However, in practice, our methodology can be conducted both at an individual level such as a person, and at a collective level such as a group or class using the standard device of assuming a 'representative individual'. Initially, each household x i is endowed by a set of resources. These resources are transformed in commodities which can be exchanged within the society. In this paper, we will not deal with the question aiming to know how the endowments are transformed in commodities. We suppose that the input data table represents the set of commodity vectors of households which was determined from a 'Household Living Standards Survey'. Thus, a household's commodities vector can be defined as being the set of all goods and services that he declares to legally attain with his endowments.

We consider a multidimensional distribution for a population Ω of n households on a n × m data table (1).

X 1 . . . X j . . . X m Ω =

x 1 . . .

x i . . . x n        x 1 1 • • • x j 1 • • • x m 1 . . . . . . . . . x 1 i . . . x j i • • • x m i . . . . . . . . . x 1 n . . . x j n • • • x m n        (1)
where x j i is the ith commodity of household in the jth variable, X = X 1 × • • • × X m and Ω ⊂ X. The interpretation is as follows: we have households x i , i ∈ {1, 2, . . . , n}, evaluated on a set J = {1, 2, . . . , m} of variables. The set X j gathers all possible levels that a household can possibly take on the jth variable (j ∈ J ). The set X j is the jth set of evaluations on the n households. The household's commodities vector x i is the set of evaluations of the ith household on the m variables and x i ∈ X. The set X is the set of all possible commodities vectors of evaluations on the m variables.

As is customary in data analysis, we consider two characteristics of data: data type and data scale. Data type refers to the degree of quantization in the data and data scale indicates the relative significance of numbers. The input data can be typed as binary (e.g. "yes-no"), as discrete and as continuous. The input data can also be scaled as qualitative (nominal and ordinal) scales and quantitative (interval and ratio) scales. The possible values of a qualitative (nominal or discrete) variable are called the "modalities of the variable".

Remark 2.1. In the case of continuous or quantitative variables, the mean µ j on the jth variable is given by µ j = n i=1 x j i and the standard deviation σ j on the jth variable is given by σ

2 j = n i=1 (x j i -µ j ) 2 n-1 .

Transformation of the data table into a proximity matrix

Initially, we have to select properly the features (selection of individuals and selection of variables) on which clustering is to be performed so as to encode as much information as possible concerning the task of our interest. Depending on the structure of the data table, it may be necessary to standardise some variables before computing the proximity matrix.

Standardization of variables

Sometimes, it is useful to delete the effects of origin and scale in the measurement of variables through the standardisation of variables. [START_REF] Milligan | A study of standardization of variables in cluster analysis[END_REF] argues that "a methodological problem in applied clustering, involves the decision of whether or not to standardize the input variables prior to the computation of an Euclidean distance dissimilarity measure". Standardization of variables is mostly recommended in those cases where the dissimilarity measure, such as Euclidean distances, is sensitive to differences in the magnitudes or scales of the input variables. It helps to adjust the magnitude of the scores and the relative weighting of the variables (see [START_REF] Anderberg | Cluster analysis for applications[END_REF]Milligan and Cooper, 1988, for applications). For empirical reasons, we considered two procedures for standardization of variables: standardization z-score and standardizing by variable ranges.

The standardization z-score transforms each continuous or quantitative variable to zero mean and unit variance as follows:

Ω • • • , x j , • • • := x j -µ j σ j (2)
The standardization by variable ranges converts each ordinal variable to range from 0 (minimum value) to 1 (maximum value).

Ω • • • , x j , • • • := x j -min{x j } max{x j } -min{x j }
(3)

Computation of proximity matrix

Let x i refer to the ith household belonging to the set Ω = {x 1 , x 2 , . . . , x n } sample of n households from the universe U (the whole population). Clustering methods require that a measure of proximity (alikeness or affinity) be established between pairs of households. Thus, this step aims to transform the data table into a proximity matrix. A proximity matrix [δ(x i , x k )] accumulates the pairwise indices of similarity (or indices of dissimilarity) in a matrix in which each row and column represents a household. The more the ith and kth household resemble one another, the smaller a dissimilarity index, the larger a similarity index. We assume that an index of proximity is a measure of the dissimilarity δ(x i , x k ) between the ith and kth household if it satisfies the following three conditions for all x i , x k ∈ Ω:

(i) δ(x i , x k ) ≥ 0; (ii) δ(x i , x i ) = 0; (iii) δ(x i , x k ) = δ(x k , x i ).
(4)

The third condition rules out asymmetric indices of proximity and, taken in conjunction with (ii), implies that a proximity matrix is fully specified by providing the n(n -1)/2 values in its lower triangle. Let J 1 be the set of quantitative variables, J 2 the set of ordinal variables and J 3 the set of nominal variables such that J = J 1 ∪ J 2 ∪ J 3 and J 1 ∩ J 2 ∩ J 3 = ∅. For any nonempty subset J 1 (resp. J 2 and J 3 ) of the set of variables J , we denote by X J 1 (resp. X J 2 and

X J 3 ) the "block data table" X J 1 = j∈J 1 X j (resp. X J 2 = j∈J 3 X j and X J 3 = j∈J 3 X j ). Definition 3.1. A "block data table" X J 1 (resp. X J 2 or X J 3
) is the sub-data table of X representing all possible commodities vectors of evaluations on the sub-set of variables J 1 (resp. J 2 and J 3 ).

For the sub-set X J 1 , we use the euclidean distance in order to compute a dissimilarity index between two households as follows:

δ 1 (x i , x k ) = j∈J 1 w ikj x j i -x j k 2 1/2 j∈J 2 w ikj (5)
w ikj being equal to 1 or 0, depending upon the comparaison being valid for the jth variable. For the sub-set X J 2 , we compute a dissimilarity index using Gower's dissimilarity index (see [START_REF] Gower | A general coefficient of similarity and some of its properties[END_REF] given by the following equation:

δ 2 (x i , x k ) = j∈J 2 w ikj S ikj j∈J 2 w ikj with S ikj = |x j i -x j k | (6)
For the sub-set X J 3 , we compute a dissimilarity index using Gower's dissimilarity index (see [START_REF] Gower | A general coefficient of similarity and some of its properties[END_REF]) applied on nominal scales:

δ 3 (x i , x k ) = 1 - j∈J 3 w ikj S ikj j∈J 3 w ikj with S ikj = 1 if x j i = x j k 0 if x j i = x j k (7)
S ikj is the contribution of the jth variable similarity measure. Combining the equations ( 5), ( 6) and ( 7), we obtain the proximity matrix [∆(x i , x k )] Ω which is given by the following equation:

∆(x i , x k ) = 3 l=1 δ l (x i , x k ) 1/ε with ε ≥ 1 (8)
where ∆(x i , x k ) is the cumulated dissimilarity index between two households x i and x k defined on J = J 1 ∪ J 2 ∪ J 3 . Note that, each δ l is the dissimilarity index defined on J l and ε is a sensibility parameter. Note also that, this way to compute a proximity matrix has been adopted in order to avoid the coding of the original characteristics of the data. However, it is possible to transform the data table through coding techniques (see [START_REF] Diday | Éléments d'analyse des données[END_REF] and to study the effects of a such transformations in the final recommendation.

Clustering

Clustering (see [START_REF] Diday | Éléments d'analyse des données[END_REF][START_REF] Jain | Algorithms for Clustering[END_REF][START_REF] Berkhin | Survey of clustering data mining techniques[END_REF], for more details) is used in order to organise data into clusters in such a way that each cluster consists of households that are similar in term of commodities distribution between themselves and dissimilar to households of other clusters.

Clustering algorithms selection

The literature [START_REF] Jain | Algorithms for Clustering[END_REF][START_REF] Jain | Data clustering: a review[END_REF][START_REF] Berkhin | Survey of clustering data mining techniques[END_REF] in cluster analysis proposes several clustering methods and algorithms among which we can make a choice. Traditionally, clustering methods and algorithms are broadly divided in hierarchical and partitioning ones. Hierarchical clustering methods transform a proximity matrix into a nested sequence of clusters, whereas partitional clustering methods generate a single partition of the data table in an attempt to recover natural clusters present in data. Hierarchical clustering algorithms produce a nested series of partitions based on a criterion for merging or splitting clusters based on dissimilarity. Partitional clustering algorithms identify the partition that optimizes (usually locally) a clustering criterion. Hierarchical clustering methods generally require only the proximity matrix among objects and allow to a data analyst to visualise how objects are being merged into clusters, whereas partitional clustering methods expect the data in the form of a data table whose variables are of the same type.

In this paper, we have chosen a hierarchical clustering method based on Ward's method, also known as the minimum variance method. It is one of the most widespread hierarchical clustering methods which is distinct from all other ones because it uses an analysis of variance to evaluate the distances between clusters. In short, this method attempts to minimize the Sum of Squares of any two (hypothetical) clusters that can be formed at each step (see [START_REF] Ward | Hierarchical grouping to optimize an objective function[END_REF], for more details concerning this method). In general, this method is regarded as very efficient, however, it tends to create clusters of small size.

Cluster validation

This step deals with deciding the best number of clusters that fits the data set and to discuss about the problem of cluster validation. A common question in clustering is "how many clusters are there in my data?" and the search for a response to this question led to procedures evaluating the results of a clustering algorithm, also known under the term of cluster validity. Cluster validation refers to a set of procedures allowing to evaluate the results of a classification in a quantitative way. In general terms, there are three approaches to investigate cluster validity: the external approach, the internal approach and the relative approach. The external approach (based on external criteria) evaluates the results of a clustering algorithm based on a pre-specified structure which is imposed on a data set and should reflect our intuition about the clustering structure of the data set. The internal approach (based on internal criteria) evaluates the results of a clustering algorithm in terms of quantities that involve the vectors of the data set themselves (e.g. proximity matrix). The relative approach (based on relative criteria) evaluates the clustering structure by comparing it to other clustering schemes, resulting by the same algorithm but with different parameter values. In general, we need to establish a stopping rule necessary in order to decide about the correct number of clusters in a data set. This stopping rule can be defined in a probabilistic sense or not, depending on the type of approach that we have chosen. The clustering literature proposes several methods aiming at determining the correct number of cluster and at examining clusters' validity. For empirical reasons we have chosen two of them: the [START_REF] Calinski | A dendrite method for cluster analysis[END_REF] index and the multiscale bootstrapping technique developed by [START_REF] Suzuki | Pvclust: an R package for assessing the uncertainty in hierarchical clustering[END_REF]. The [START_REF] Calinski | A dendrite method for cluster analysis[END_REF] index is one of the most widespread in the clustering literature providing in general the correct number of clusters [START_REF] Milligan | An examination of procedures for determining the number of clusters in a data set[END_REF]. The maximum value of this index indicates the correct number of clusters in the data. The multiscale bootstrapping technique is defined in a probabilistic sense and provides p-values for hierarchical clustering based on multiscale bootstrap resampling. Clusters that are highly supported by the data will have large p-values (see [START_REF] Jain | Algorithms for Clustering[END_REF][START_REF] Gordon | Classification. 2nd Edition[END_REF][START_REF] Halkidi | On Clustering Validation Techniques[END_REF][START_REF] Berkhin | Survey of clustering data mining techniques[END_REF], for more details).

Cluster visualisation

We have used multidimensional scaling (MDS) in order to visualise the proximity matrix. MDS is a set of related statistical techniques often used in information visualization for exploring similarities or dissimilarities in data. A MDS algorithm starts with a proximity matrix between objects, then assigns a location to each object in a q-dimensional space, where q ≥ 2 is specified a priori. In practice, objects are represented as points in a usually two-dimensional space, such that the distances between the points match the observed dissimilarities as closely as possible (see Figure 2). See [START_REF] Kruskal | Multidimensional Scaling[END_REF]; [START_REF] Cox | Multidimensional Scaling. 2nd Edition[END_REF]; [START_REF] Groenen | Multidimensional scaling[END_REF]; [START_REF] Borg | Modern Multidimensional Scaling: Theory and Applications[END_REF] for more details.

Modal-valued matrix and modal-valued criterion

Cluster description aims to transform the large data table into a summary table in order to gain initial knowledge. Such exercise allows to identify the 'relative importance' of a variable within a given cluster. For this purpose, we introduce two core concepts: modalvalued matrix and modal-valued criterion. The modal-valued matrix can be considered as a summary table of the large data table where the jth column represents the variable X j , j ∈ J and the hth row denotes the cluster h ∈ {1, . . . , µ}. The intersection of the hth row and the jth column denotes the modal-valued criterion which is the description of the variable X j within the cluster L h .

Formally, let L = {L 1 , . . . , L µ } be the set of clusters obtained after clustering the population Ω. We consider a multidimensional cluster distribution for a population Ω of n households with µ × m modal-valued matrix :

X ≡ [X 1 , . . . , X j , . . . , X m ] =        X 1 (L 1 ) • • • X j (L 1 ) • • • X m (L 1 ) . . . . . . . . . X 1 (L h ) . . . X j (L h ) • • • X m (L h ) . . . . . . . . . X 1 (L µ ) . . . X j (L µ ) • • • X m (L µ )        =        X 1 . . . X h . . . X µ        (9)
X j (L h ) denotes the modal-valued criterion of cluster h ∈ {1, 2, . . . , µ} in variable j ∈ J defined as follows:

X j (L h ) = (β j1 , π h j1 ); • • • ; (β js j , π h js j ) ( 10
)
where {β jk : k = 1, . . . , s j } is a set of modalities (or states) of L h over the domain D j of X j , j = 1, . . . , m; π h jk is a non-negative measure associated with β jk using equation ( 11) and s j is the number of values actually taken by D j of X j .

π h jk = |{x i ∈ L h : x j i = β jk }| |{L h }| , k = 1, . . . , s j (11)
Note that the modality β jk can be finite or infinite in number, quantitative or categorical in value; and the measure π h jk represents the probability, the proportion or the frequency of value β jk within cluster L h . Intuitively, π h jk represents the 'relative importance' of the modality β jk within the cluster L h i.e. the number of times that a modality β jk occurs within the variable X j of cluster L h . Definition 4.1. A modal-valued criterion X j (L h ) is a fuzzy subset of the set of modalities (or states) {β jk : k = 1, . . . , s j } of a cluster L h over the domain D j of X j , j ∈ J defined by:

X j (L h ) = (β jk , π h jk ) : k = 1, . . . , s j (12)
where π h jk is the weight of the modality β jk on variable j ∈ J associated with cluster L h , h ∈ {1 . . . , µ}.

The modal-valued matrix of X, written X, is the set of all subsets of X such as X = X 1 × • • • × X m and X j = {β jk : k = 1, . . . , s j } denotes the subset of modalities (or states) over the domain D j of X j . In the particular case where X j ⊆ R, we set X j ⊆ R. X h , called 'modal-valued cluster ', represents the description of cluster L h on the modal-valued matrix X given by equation ( 13):

X h = X 1 (L h ); • • • ; X m (L h ) (13)

An illustrative example

Let us consider for instance the data table (see Table 1) obtained from a survey of standards of living of households. For this example, we consider 12 households Ω = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 } evaluated on 7 variables (one quantitative variable and 6 qualitative variables):

Variables Households X 1 X 2 X 3 X 4 X 5 X 6 X 7 x 1 250 N S N F A W x 2 4500 Y V Y V N Be x 3 1500 Y A Y F S U x 4 200 N S N F A W x 5 800 N S N F S W x 6 5000 Y V Y V N Be x 7 2500 Y A Y A S U x 8 600 N S N F A W x 9 2000 Y A Y A S Be x 10 6500 Y V Y V N U x 11 1000 N A N F A W x 12 2800 Y A Y A S Be Table 1: Illustrative example X j
Description of variables X Monthly income for basic needs of households in euros. X Is there a room equipped for cooking? Yes(Y), No(N). X Type of housing? Villa(V), Apartment Building(A), Single Individual House(S). X Owner of a vehicle, car or lorry? Yes(Y), No(N). X Duration for reaching the nearest public transport (in minutes)?

[0; 14] =Very Close(V), [15; 29] =Acceptably Close(A), [45; 59] =Far(F). X Has had problems to meet food needs? Never(N), Sometimes(S), Always(A). X Economic situation? Worse Now(W), Unchanged(U), Better Now(Be).

We conducted hierarchical cluster analysis via multiscale bootstrap (number of bootstrap 1000; see [START_REF] Suzuki | Pvclust: an R package for assessing the uncertainty in hierarchical clustering[END_REF])) using Ward's method [START_REF] Ward | Hierarchical grouping to optimize an objective function[END_REF] and correlationbased dissimilarity matrix. The best number of clusters is three. This result may confirmed by [START_REF] Calinski | A dendrite method for cluster analysis[END_REF] index which allows to compare the homogeneity of partitions. The visualisation of clusters is possible through the dendrogram of households and multidimensional scaling as shown on Figure 2. Considering the illustrative example (see Table 1), we obtain clusters

L 1 = {x 1 , x 4 , x 5 , x 8 , x 11 }, L 2 = {x 2 , x 6 , x 10 }, and L 3 = {x 3 , x 7 , x 9 , x 12 }. The multidimensional distribution
for this population of 12 households is given by the following data table:

X = [x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ] =                     250.00 N S N F A W 200.00 N S N F A W 800.00 N S N F S W 600.00 N S N F A W 1000.00 N A N F A W 4500.00 Y V Y V N Be 5000.00 Y V Y V N Be 6500.00 Y V Y V N U 1500.00 Y A Y F S U 2500.00 Y A Y A S U 2000.00 Y A Y A S Be 2800.00 Y A Y A S Be                     =                     x 1 x 4 x 5 x 8 x 11 x 2 x 6 x 10 x 3 x 7 x 9 x 12                     (14)
and its modal-valued matrix is given by:

X =   X 1 (L 1 ) X 2 (L 1 ) X 3 (L 1 ) X 4 (L 1 ) X 5 (L 1 ) X 6 (L 1 ) X 7 (L 1 ) X 1 (L 2 ) X 2 (L 2 ) X 3 (L 2 ) X 4 (L 2 ) X 5 (L 2 ) X 6 (L 2 ) X 7 (L 2 ) X 1 (L 3 ) X 2 (L 3 ) X 3 (L 3 ) X 4 (L 3 ) X 5 (L 3 ) X 6 (L 3 ) X 7 (L 3 )   ≡   X 1 X 2 X 3   ( 15
)
where each modal-valued criterion

X j (L h ) is: X 1 (L 1 ) = ([200; 1000], 1) , X 2 (L 1 ) = (N, 1.00);
(Y, 0.00) , X 3 (L 1 ) = (A, 0.20); (S, 0.80); (V, 0.00) , X 4 (L 1 ) = (N, 1.00); (Y, 0.00) , X 5 (L 1 ) = (A, 0.00); (F, 1.00); (V, 0.00) , X 6 (L 1 ) = (A, 0.80); (N, 0.00); (S, 0.20) , X 7 (L 1 ) = (Be, 0.00); (U, 0.00); (W, 1.00) ; X 1 (L 2 ) = ([4500; 6500], 1) , X 2 (L 2 ) = (N, 0.00); (Y, 1.00) , X 3 (L 2 ) = (A, 0.00); (S, 0.00); (V, 1.00) , X 4 (L 2 ) = (N, 0.00); (Y, 1.00) , X 5 (L 2 ) = (A, 0.00); (F, 0.00); (V, 1.00) , X 6 (L 2 ) = (A, 0.00); (N, 1.00); (S, 0.00) , X 7 (L 2 ) = (Be, 0.67); (U, 0.33); (W, 0.00) ; X 1 (L 3 ) = ([1500; 2800], 1) , X 2 (L 3 ) = (N, 0.00); (Y, 1.00) , X 3 (L 3 ) = (A, 1.00); (S, 0.00); (V, 0.00) , X 4 (L 3 ) = (N, 0.00); (Y, 1.00) , X 5 (L 3 ) = (A, 0.75); (F, 0.25); (V, 0.00) , X 6 (L 3 ) = (A, 0.00); (N, 0.00); (S, 1.00) , X 7 (L 3 ) = (Be, 0.50); (U, 0.50); (W, 0.00) .

For example, X 7 (L 2 ) should be read as: "considering the variable X 7 describing the 'economic situation' of a household, 67% of households legally own the commodity 'Be' and 33% the commodity 'U', while there is no household with the commodity 'W' within cluster L 2 ".

Construction of meaningful measurements

Modalities characterisation

Sometimes the frequency π h jk of modality β jk within cluster L h , such as defined in equation (21), may not accurately reflect how strong is the statement: ''this modality is more relevant in this cluster than in all other clusters". This is why we need to characterize the modalities in a probabilistic sense by using hypothesis testing. Hypothesis testing is an essential part of statistical inference aiming at determining the probability that a given hypothesis is true. The characterisation used in this paper has been inspired from the principle of statistical characterisation introduced by [START_REF] Morineau | Note sur la caractérisation statistique d'une classe et les valeurs tests[END_REF]. The idea consists at testing if the modality β jk of modal-value criterion X j (L h ) is a relevant characteristic of the cluster L h through the probability that the hypothesis "the modality β jk is 'significantly' more abundant in the cluster L h than in the population of Ω" is true. Thus, we set the null hypothesis H 0 of random draw (without replacement) of n h households among the n households of population. H 0 ensures that the frequencies π h jk and p h jk are nearly equal with respect to random fluctuations; where p h jk is:

p h jk = |{x i ∈ Ω : x j i = β jk }| |{Ω}| , k = 1, . . . , s j (16)
Intuitively, p h jk represents the 'relative importance' of the modality β jk within the whole population i.e. the number of times that a modality β jk occurs within the variable X j of population Ω.

Let N be a random variable such that N = n h jk with n h jk = n h • π h jk . Under hypothesis H 0 , N follows a hypergeometric distribution, N ∼ H(n, n jk , n h ), with a mean given by

E h (N) = n h • p h jk , a standard deviation σ 2 h (N) = n h • n -n h n -1 • p h jk • 1 -p h jk and n jk = n • p h jk .
Hence, the degree of significance ρ(β jk ) under H 0 is given by the following equation:

ρ(β jk ) = Prob H 0 {N > β jk } = Prob H 0 {t h (N) > t h (β jk )} ( 17
)
where

t h (N) = N -E h (N ) σ h (N )
such that t h (N) follows a gaussian distribution N (0; 1). In practice, a hypergeometric distribution can be approximated by a gaussian distribution when the number of households in the clusters is sufficiently high (over 30 households for instance).

For each modal-valued criterion X j (L h ) and each modality β jk , we define the relevance index ξ(β jk ) by equation ( 18) as follows:

ξ(β jk ) = ρ(β jk ) • π h jk + λ h • ζ h jk ( 18
)
where ζ h jk is given by the following equation ( 19) and λ h =

n k n . ζ h jk = |{x i ∈ L h : x j i = β jk }| |{x i ∈ Ω : x j i = β jk }| = n h jk n jk , k = 1, . . . , s j (19)
We denote n jk (n h jk respectively) the number of times that the modality β jk occurs within the jth variable of population Ω (of cluster L h respectively). ξ(β jk ) is nomalised between 0 to 1 interval through equation ( 20).

ξ(β jk ) = ξ(β jk ) max {j,k} ξ(β jk ) (20) Definition 5.1. A modal-valued characterisation Y j (L h
) is a fuzzy subset of the set of modalities (or states) {β jk : k = 1, . . . , s j } of a cluster L h over the domain D j of X j , j ∈ J defined by:

Y j (L h ) = β jk , ξ(β jk ) : k = 1, . . . , s j (21)
where ξ(β jk ) is the relevance index of the modality β jk on variable j ∈ J associated with cluster L h , h ∈ {1 . . . , µ}.

The relevance index ξ(β jk ) appears as a mean showing on a scale between 0 and 1, the level with which the assertion "ξ(β jk ) is more relevant than ξ(β j ′ k ′ )" is valid. Intuitively, ξ(β jk ) measures the 'degree of connection' of the modality β jk with the cluster L h considering all other clusters and the whole population i.e. the 'connectedness' between the modality β jk within the variable X j and a given cluster L h (as in the case of one causing the other or sharing features with it).

Remark 5.1. The relevance index is a numerical representation which allows comparisons between modalities, but it cannot be treated as a probability because it does not satisfy the 'countable additivity' property. We need these relevance indexes to establish the relation between commodities and welfare dimension levels as we show it in the following sections.

Specification of preferential information

The decision maker has to specify his preferential information according to policies, programmes or projects that he intends to undertake in a given region. In this paper, we assume that the problem involves at least five types of preferential information (PI): preference information concerning the importance of the variables, preference information concerning welfare dimensions, preference information concerning welfare dimension levels, preference information concerning the ranking of modalities and preference information concerning the specification of priority households and concrete actions to undertake.

Welfare dimensions

The decision maker in collaboration with the analyst establishes the set of "welfare dimensions". Welfare dimensions allow to specify and describe the totality of functioning dimensions that the decision maker would like to deal with during the decision process aiming at finding an appropriate final recommendation for improving the living standards of households. This concept is based on the idea that the way through which individuals or households perceive their position or their state within a society is an important aspect to be considered when designing policies which concern them. However, in order to assess the input of policies, we need an analytic structure and for this purpose we introduce the concept of "welfare dimension". Definition 5.2. A welfare dimension describes the set of all possible functionings of a household under the form of achievable levels of welfare on this dimension.

Formally, let J be a set of variables. For any non empty subset J u of J we denote by F Ju (resp. F -J u ) the set j∈Ju X j (resp.

j / ∈Ju X j ) such that J = J 1 ∪ • • • ∪ J p and J 1 ∩ • • • ∩ J p = ∅.
F Ju is a potential welfare dimension describing J u functioning and F = {F Ju : u = 1, . . . , p} denotes the set of all potential welfare dimensions. We will write

F Ju (L h ) if it is related to cluster L h .
Remark that, each F Ju is perceived as the set of all commodities related to the set j∈Ju X j within the whole population Ω and F Ju (L h ) is perceived as the set of all commodities related to the set j∈Ju X j within the cluster L h .

Welfare dimension levels

The decision maker has to define a set of levels associated to each welfare dimension. A welfare dimension level allows, in a more general case, to distinguish the possible intensity degrees within a given welfare dimension. In the case of poverty, it allows to distinguish the degree of necessity of households in term of policy intervention on this welfare dimension. For instance, 'Bad', 'Average' and 'Good' can be three levels associated to the welfare dimension 'Mobility' or 'Social Integration'; 'Low', 'Acceptably Low', 'Relatively Low', 'High' and 'Very High' can be five levels associated to the welfare dimensions 'Air Pollution' or 'Risk'.

Formally, for the uth potential welfare dimension F Ju , we obtain a set of levels

F Ju = {ξ u v : v = 1, . . . , t u } ( 22
)
where u ∈ {1, . . . , p}, ξ u v denotes the vth level of the uth potential welfare dimension F Ju and t u is the number of possible level(s) actually taken by F Ju .

Construction of capabilities

5.3.1. Computing of indexes C h (S Ju ) and C h (U Ju )
For the purpose of constructing the capabilities set associated to each cluster, we introduce two indexes: the index C h (S Ju ) and the index C h (U Ju ). We denote by M = β jk : j ∈ J and k ∈ {1, . . . , s j } the set of all modalities where s j is the number of values actually taken by X j . The idea consists to directly ask the decision maker to partition the set of all modalities in three subsets S, N, U . We interpret the subset S as containing Satisfactory modalities, N contains Neutral modalities while U contains Unsatisfactory ones. Formally, S = β jk : β jk > β jk ′ with k = k ′ and k, k ′ ∈ {1, . . . , s j }, j ∈ J , U = β jk : β jk < β jk ′ with k = k ′ and k, k ′ ∈ {1, . . . , s j }, j ∈ J and N = β jk : not(β jk > β jk ′ ) and not(β jk < β jk ′ ) with k = k ′ and k, k ′ ∈ {1, . . . , s j }, j ∈ J .

Hence, for each F Ju (with u = 1 . . . , p) we set S Ju = {β jk : j ∈ J u and β jk ∈ S} and U Ju = {β jk : j ∈ J u and β jk ∈ U}. Remark that S = 1≤u≤p S Ju and U = 1≤u≤p U Ju . Then, the twofold partition S, U allows us to evaluate and pairwise rank the clusters according to a welfare dimension level. The evaluation of the cluster L h is given by the strongness of its Satisfactory modalities (see equation 23) and the weakness of its Unsatisfactory modalities (see equation 24).

C h (S Ju ) = j∈Ju w j k∈S Ju ξ(β jk ) (23) C h (U Ju ) = j∈Ju w j k∈U Ju ξ(β jk ) ( 24
)
where h = 1, . . . µ, u = 1, . . . , p and w j denote the jth positive weight representing the importance of the variable X j . Intuitively, C h (S Ju ) (respectively, C h (U Ju )) measures the 'degree of connection' (respectively, 'degree of disconnection' ) of the higher (respectively, the lower) welfare dimension level of F Ju (L h ) with the cluster L h considering all other clusters and the whole population.

Functioning vectors

A 'functioning vector' is a collection of attainable levels on each welfare dimension. A 'component' of functioning vector is also called 'achievement level' on a particular welfare dimension. Formally, we end getting the lth functioning vector of cluster L h :

F l (L h ) = F J 1 (L h ); • • • ; F Jp (L h ) l ( 25
)
where h ∈ {1, . . . , µ}, l ∈ {1, . . . , l h } and l h represents the number of functioning vectors actually taken by L h .

Remark 5.2. Trivially, if each potential welfare dimension J u has N u welfare dimension levels (with u = 1, . . . , p), the number of all possible functioning vectors N s.t.:

N = N 1 × N 2 × • • • × N p (26)

Functioning lattice

A functioning lattice is the visualisation of the totality of functioning vectors partially ordered by natural dominance. Any further relation is established by the client following his own private preferences. In general, the functioning vectors are built independently from the clusters. However, given a cluster we may establish the functioning threshold which denotes the 'functioning vectors frontier' within the functioning lattice.

Construction of functioning thresholds

An obvious numerical representation amounts to associate a real number to each subset C h of the set of all possible functioning vectors in such a way that the comparison between these numbers faithfully reflects the preference relation on the various potential welfare dimensions where ≻ refers to strict preference relation (asymmetric part) and ∼ indifference relation (symmetric part). Note that, C a C b can be interpreted as "the standard living (or welfare) offered to households belonging to cluster L a is considered to be at least as preferable as the standard living (or welfare) offered to households belonging to cluster L b ". Hence, we associate to each MDPM defined on J u , noted by M h (J u ), a sequence of alternatives which concretely represents the appropriate interventions to the households belonging to cluster L h . M h is then given by the following equation:

M h (J u ) = ϑ u C h (S Ju ), C h (U Ju ) (27)
where ϑ u is the uth real-valued function which allows to aggregate C h (S Ju ) and C h (U Ju ). This leads to numerically recoding the value judgment between C h (S Ju ) and C h (U Ju ) on the various potential welfare dimensions F Ju , in such a way that the subsets C a and C b can simply be compared taking the sum of these functions as follows:

C a C b ⇐⇒ p u=1 M a (J u ) ≥ p u=1 M b (J u ) (28) ⇐⇒ p u=1 ϑ u C a (S Ju ), C a (U Ju ) ≥ p u=1 ϑ u C b (S Ju ), C b (U Ju ) (29)
We now construct a relation judgement R h (J u ) which is a formal mechanism for linking a specific functioning vector ξ u v , v = 1, . . . , t u to the relevant potential welfare dimension F Ju , u = 1, . . . , p. The decision maker (or the client) has to define the attainable levels of each potential welfare dimension F Ju and the cut-offs ε vu . We suppose that the attainable levels of each potential welfare dimension F Ju are ordered i.e.

ξ u 1 > ξ u 2 > • • • > ξ u tu , for all u ∈ {1, . . . , p}. R h (J u ) =          ξ u 1 , if M h (J u ) ≥ ε 1u ξ u 2 , if M h (J u ) ≥ ε 2u . . . . . . ξ u tu , if M h (J u ) ≥ ε tuu (30)
M h (J u ) is given by equation ( 27) and

M h (ε u ) ≥ ε 1u > ε 2u > • • • > ε tuu ≥ -M h (ε u ) with φ h [J u ] = min C h (S Ju ); C h (U Ju ) ; ψ h [J u ] = C h (S Ju )+C h (U Ju ) and M h (ε u ) = 1- φ h [J u ] ψ h [J u
] for all h ∈ {1, . . . , µ} and u ∈ {1, . . . , p}.

Hence, given each cluster, we may establish the sup-functioning vectors. We call these functioning thresholds. Formally, the functioning threshold F sup (L h ) is defined as follows:

F sup (L h ) = R h (J 1 ), R h (J 2 ), . . . , R h (J p ) (31)

Feasible capabilities

A 'feasible capability' is a subset of all possible functioning vectors compatible with a given set of commodities. Formally, from each lth functioning vector of cluster L h denoted by F l (L h ), we define the subset C h of the set of all possible functioning vectors. C h represents the feasible capability of all households within L h as follows.

C h = F l (L h ) ∈ C : F l (L h ) ≤ F sup (L h ) ( 32
)
where h ∈ {1, . . . , µ} and C denotes the non-empty set of all possible functioning vectors. Remark that, l ∈ {1, . . . , l h } and l h represents the number of functioning vectors actually taken by L h . Thus, we introduce the following definition.

Definition 5.3. A "core poor" is a household who has a 'bad score' on all conflictual commodities taken into account simultaneously to evaluate its 'feasible capability' in the society in which he lives, whose measurement of his poverty is in conformity with his selfperception of situation.

Note that, each feasible capability associated to a given cluster reflects the 'ability' of households within this cluster to achieve a subset of functionings among the various alternatives functioning bundles.

Extended capabilities

While generally commodities are uniquely associated to one specific welfare dimension, some commodities (such as income) are instead associated to the whole set of welfare dimensions. We denote them as 'generic commodities'.

Definition 5.4. A generic commodity is a commodity which can improve any welfare dimension.

Feasible capabilities can be extended using the generic commodities which allow to increase some welfare dimension levels of some of the functioning vectors. Thus, feasible capabilities are transformed in order to obtain what we call 'extended capabilities'.

Formally, let ξ u 1 > ξ u 2 > • • • > ξ u tu be the ordered attainable levels for all u ∈ {1, . . . , p}. Let F sup (L h ) be the functioning threshold of feasible capability C h such that we have

F sup (L h ) = ξ 1 k 1 , ξ 2 k 2 , . . . , ξ p kp . Hence, ξ u ku ∈ {ξ u 1 , ξ u 2 , • • • , ξ u tu } for all k u ∈ {1, .
. . , t u } and u ∈ {1, . . . , p}. We set ξ u ku = max v∈{1,...,tu} ξ u v . A extended capability C h is defined as follows for all h ∈ {1, . . . , µ}:

Ext(C h ) = C h = {F l (L h ) ∈ C : F l (L h ) ≤ F sup (L h ) with l = 1, . . . , l h } (33) such that F sup (L h ) = F sup (L h ) ∪ Q 1 k 1 , Q 2 k 2 , . . . , Q p kp (34) = ξ 1 k 1 , ξ 2 k 2 , . . . , ξ p kp ∪ Q 1 k 1 , Q 2 k 2 , . . . , Q p kp (35) = Ξ 1 k 1 , Ξ 2 k 2 , . . . , Ξ p kp ( 36
)
where

Ξ u ku = ξ u ku ∪ Q u ku , if Q u ku = ∅; ξ u
ku , else for all u ∈ {1, . . . , p}.

Q u ku = ξ v kv : ξ v kv > ξ u ku and k v ∈ {1, . . . , t u }\{k u } . In practice, we can use evidence-based policy (EBP) to tackle this problem. This implies the use of data collection on welfare for experimenting, quantitative and qualitative analysing, the use of poverty knowledge, expert knowledge, existing national and international research, existing statistics, stakeholder skills to judge how and the extent to which generic commodities can be used to increase some welfare dimension levels of some of the functioning vectors. Intuitively, F sup (L h ) represents the maximum 'amount' of functioning vector that a household in L h can achieve under extended capability C h .

An illustrative example 5.5. Modalities characterisation

Considering our previous illustrative example (Table 1) where (λ 1 , λ 2 , λ 3 ) = (0.42, 0.25, 0.33) for each cluster L 1 , L 2 and L 3 , we obtain the Table (2) using the R software5 and each modalvalued characterisation Y j (L h ) with ξ(β jk ) is nomalised between 0 to 1 using equation ( 20) is: Y 2 (L 1 ) = (N, 1.00); (Y, 0.00) , Y 3 (L 1 ) = (A, 0.06); (S, 0.85); (V, 0.00) , Y 4 (L 1 ) = (N, 1.00); (Y, 0.00) , Y 5 (L 1 ) = (A, 0.00); (F, 0.95); (V, 0.00) , Y 6 (L 1 ) = (A, 0.85); (N, 0.00); (S, 0.06) , Y 7 (L 1 ) = (Be, 0.00); (U, 0.00); (W, 1.00) ; Y 2 (L 2 ) = (N, 0.00); (Y, 0.76) , Y 3 (L 2 ) = (A, 0.00);

(S, 0.00); (V, 1.00) , Y 4 (L 2 ) = (N, 0.00); (Y, 0.76) , Y 5 (L 2 ) = (A, 0.00); (F, 0.00); (V, 1.00) , Y 6 (L 2 ) = (A, 0.00); (N, 1.00); (S, 0.00) , Y 7 (L 2 ) = (Be, 0.51); (U, 0.17); (W, 0.00) ; Y 2 (L 3 ) = (N, 0.00); (Y, 0.89) , Y 3 (L 3 ) = (A, 1.00); (S, 0.00); (V, 0.00) , Y 4 (L 3 ) = (N, 0.00); (Y, 0.89) , Y 5 (L 3 ) = (A, 0.85); (F, 0.05); (V, 0.00) , Y 6 (L 3 ) = (A, 0.00); (N, 0.00); (S, 1.00) , Y 7 (L 3 ) = (Be, 0.37); (U, 0.48); (W, 0.00) . Remark that there is a distinction between the modalvalued criterion X j (L h ) and the modal-valued characterisation Y j (L h ). Consider for example X 7 (L 3 ) and Y 7 (L 3 ). X 7 (L 3 ) shows that: "for the variable X 7 describing the 'economic situation' of a household, 50% of households legally own the commodity 'Be' and the commodity 'U' within cluster L 2 ", whereas Y 7 (L 3 ) shows that: "for the variable X 7 describing the 'economic situation' of a household, the degree of connection of the commodity 'U' (equal to 0.48) is greater than the degree of connection of the commodity 'Be' (equal to 0.37) within cluster L 3 ." Then, the commodity 'U' (equal to 0.48) is more connected to cluster L 3 than the commodity 'Be' (equal to 0.37) in the same cluster. The notion of relevance is not revealed by the modal-valued criterion X j (L h ) but only by the modal-valued characterisation Y j (L h ).

F Ju Housing (J 1 ) Mobility (J 2 ) Nutrition (J 3 ) X X 2 X 3 X 4 X 5 X 6 X 7 β jk N Y A S V N Y A F V A N S
Be U W ρ(β jk ) 1.00 0.00 0.03 0.99 0.16 1.00 0.00 0.16 0.99 0.16 0.99 0.16 0.03 0.07 0.16 1.00

ζ 1 jk 1.00 0.00 0.20 1.00 0.00 1.00 0.00 0.00 0.83 0.00 1.00 0.00 0.20 0.00 0.00 1.00 π 1 jk 1.00 0.00 0.20 0.80 0.00 1.00 0.00 0.00 1.00 0.00 0.80 0.00 0.20 0.00 0.00 1.00 p 1 jk 0.42 0.58 0.42 0.33 0.25 0.42 0.58 0.25 0.50 0.25 0.33 0.25 0.42 0.33 0.25 0.42 ξ(β jk ) 1.00 0.00 0.06 0.85 0.00 1.00 0.00 0.00 0.95 0.00 0.85 0.00 0.06 0.00 0.00 1.00 ρ(β jk ) 0.16 0.84 0.16 0.26 1.00 0.16 0.84 0.38 0.09 1.00 0.26 1.00 0.16 0.76 0.38 0.16

ζ 2 jk 0.00 0.43 0.00 0.00 1.00 0.00 0.43 0.00 0.00 1.00 0.00 1.00 0.00 0.50 0.33 0.00 π 2 jk 0.00 1.00 0.00 0.00 1.00 0.00 1.00 0.00 0.00 1.00 0.00 1.00 0.00 0.67 0.33 0.00 p 2 jk 0.42 0.58 0.42 0.33 0.25 0.42 0.58 0.25 0.50 0.25 0.33 0.25 0.42 0.33 0.25 0.42 ξ(β jk ) 0.00 0.76 0.00 0.00 1.00 0.00 0.76 0.00 0.00 1.00 0.00 1.00 0.00 0.51 0.17 0.00 ρ(β jk ) 0.07 0.93 0.99 0.14 0.26 0.07 0.93 0.98 0.03 0.26 0.14 0.26 0.99 0.59 0.76 0.07 ζ 3 jk 0.00 0.57 0.80 0.00 0.00 0.00 0.57 1.00 0.17 0.00 0.00 0.00 0.80 0.50 0.67 0.00 π 3 jk 0.00 1.00 1.00 0.00 0.00 0.00 1.00 0.75 0.25 0.00 0.00 0.00 1.00 0.50 0.50 0.00 p 3 jk 0.42 0.58 0.42 0.33 0.25 0.42 0.58 0.25 0.50 0.25 0.33 0.25 0.42 0.33 0.25 0.42 ξ(β jk ) 0.00 0.89 1.00 0.00 0.00 0.00 0.89 0.85 0.05 0.00 0.00 0.00 1.00 0.37 0.48 0.00

n jk 5 7 5 4 3 5 7 3 6 3 4 3 5 4 3 5
Table 2: Computation of some indexes of modality characterisation 5.5.1. Welfare dimensions According to the illustrative example, we consider F = Housing, Mobility, Nutrition . We associate subsets of commodities to dimensions of welfare. For example, we associate X 2 and X 3 to "Housing", X 4 and X 5 to "Mobility", X 6 and X 7 to "Nutrition".

Remark 5.3. In the illustrative example the unique generic commodity is income given by criterion X 1 .

Welfare dimension levels

Considering our example, we can associate the three levels 'Bad', 'Average' and 'Good' to all welfare dimensions in such a way that

F J 1 (L h ) = F J 2 (L h ) = F J 2 (L h ) = {G, A, B},
where F J 1 = 'Housing', F J 2 = 'Mobility', F J 3 = 'Nutrition'.

Specification of subsets S and U

Considering the illustrative example, the decision maker can split the set of Satisfactory modalities (and the set of Unsatisfactory modalities respectively) of potential dimensions of welfare J 1 , J 2 and J 3 (called 'Housing', 'Mobility' and 'Nutrition' respectively) as follows: S J 1 = {X 2 .Y; X 3 .V; X 3 .A}, S J 2 = {X 4 .Y; X 5 .V; X 5 .A}, S J 3 = {X 6 .N; X 7 .Be}, and U J 1 = {X 2 .N; X 3 .S}, U J 2 = {X 4 .N; X 5 .A; X 5 .F}, U J 3 = {X 6 .A; X 6 .S; X 7 .W} respectively. The Neutral modality is given by N = {X 7 .U} but for empirical reasons we add it to S J 3 and U J 3 .

Construction of functionings thresholds

In the example (see Table 4), we consider that ϑ u is given by equation ( 37).

M h (J u ) = ϑ u C h (S Ju ), C h (U Ju ) = C h (S Ju ) -C h (U Ju ) max h=1,...,µ C h (S Ju ), C h (U Ju ) (37)
The cut-off F sup (L h ) in Figure 3 represents the 'functioning threshold' that we have to determine. Using equation ( 23) and equation ( 24), we compute C h (S Ju ) and C h (U Ju ) for each cluster L h , h ∈ {1, 2, 3} and each welfare dimension u ∈ {1, 2, 3} with w j = 1. These are provided by Table 3.

Housing (J 1 ) Mobility (J 2 ) Nutrition (J 3 ) Clusters S J 1 U J 1 S J 2 U J 2 S J 3 U J 3 C 1 0.
(U Ju ) R h (J u ) =    Good, M h (J u ) ≥ ε 1u Average, M h (J u ) ≥ ε 2u Bad, M h (J u ) ≥ ε 3u (38)
M h (J u ) is given by equation ( 37) and

-M h (ε u ) ≤ ε 3u < ε 2u < ε 1u ≤ M h (ε u ) with φ h [J u ] = min C h (S Ju ); C h (U Ju ) ; ψ h [J u ] = C h (S Ju )+C h (U Ju ) and M h (ε u ) = 1- φ h [J u ] ψ h [J u ] .
With ε 1u = 0.75 and ε 2u = 0.50, we compute the meaningful measurements given by Table 4.

Thus we obtain

F sup (L 1 ) = B, B, B , F sup (L 2 ) = G, G, G and F sup (L 3 ) = G, B, B . J 1 ε 1 J 2 ε 2 J 3 ε 3 M 1 -
0.95 0.97 -1.00 1.00 -1.00 1.00 M 2 0.93 1.00 0.91 1.00 0.79 0.91 M 3 1.00 1.00 0.43 0.66 -0.33 0.64

Functioning vectors

Considering our example where we have associated the three levels 'Bad', 'Average' and 'Good' to all welfare dimensions s.t.

F J 1 (L h ) = F J 2 (L h ) = F J 2 (L h ) = {G, A, B} for F J 1 = 'Housing', F J 2 = 'Mobility', F J 3 = 'Nutrition
', the lth functioning vector associated to cluster h can be given by F l (L h ) = A; B; G and an achievement level on welfare dimension 'Nutrition' is 'G'.

Functioning lattice

The totality of all possible functioning vectors can be visualized through the functioning lattice given by Figure (3) which shows the functioning vectors from the 'best' ones (given by G, G, G ) to the worst (given by B, B, B ). If we suppose for example that the symbol → denotes preference relation , G, A, G → G, B, A can be interpreted as "the standard living (or welfare) offered by G, A, G is considered to be at least as preferable as that the standard living (or welfare) offered by G, B, A ". The notation introduced in Figure (3) refers to 'natural' dominance existing between functionings. However, some cases (such as A, G, B vs G, B, A ) can be conflictual and difficult to interpret in term of preference relation. In this situation, the client has to use his own private preferences to decide.

Feasible capabilities

The feasible capabilities are given in Table 5. A quick analysis of households from each cluster shows that the 'worst' feasible capabilities C 1 (F sup (L 1 ) = B, B, B ) are characterised by the fact that the heads of households are in the majority described by Unsatisfactory modalities U J 1 , U J 2 and U J 3 . They are those we have called 'core poor' (see definition 5.3) in this paper. The 'best' ones C 2 (F sup (L 2 ) = G, G, G ) are characterised by the fact that the heads of households are in the majority described by the Satisfactory modalities S J 1 , S J 2 and S J 3 . The feasible capabilities C 3 (F sup (L 3 ) = G, B, B ) is characterised by households which are in majority described by Satisfactory and Unsatisfactory modalities S J 1 , U J 2 and U J 3 . Thus, we look for the presence and absence of modalities in each cluster. It is important to note that, the words 'worst' and 'best' do not contain anything of numerical but only reflect the household's well-being and advantage.

Extended capabilities

The feasible capabilities are transformed to extended capabilities using the generic commodities (such as income on the illustrative example) which allow to increase some of the attainable functionings. This raises two main questions: How to increase some of the attainable functionings while avoiding contradictions and inconsistencies? How to deal with legitimacy and rationality of such a procedure? 25

G, G, G G, G, A G, A, G A, G, G G, G, B G, B, G B, G, G G, A, A A, G, A A, A, G B, B, B A, B, B B, A, B B, B, A A, A, B A, B, A B, A, A G, B, B B, G, B B, B, G F sup (L h )
Figure 3: Functioning lattice (from the example) The analyst can compute the attainable levels of each welfare dimension which he can improve (or increase) using the preferential information provided by the decision maker. Considering the set of feasible capabilities shown in Table ( 5), we note that it is possible to improve the feasible capability C 3 on the welfare dimension J 2 representing 'Mobility'. In fact, a quick observation shows that M 3 (J 3 ) < M 3 (J 2 ) < ε v = 0.50. Thus, one can be tempted to conclude that "the standard living (or welfare) offered by C 1 and C 3 on the welfare dimension J 2 is 'equivalent to' the standard living (or welfare) offered by C 1 and C 3 on the welfare dimension J 3 ". The modal-valued matrix given by equation ( 15 4 for the generic commodity 'income' belonging in interval [1500, 2800]), we can intuitively increase the attainable level of welfare dimension J 2 of L 3 from B to A. For feasible capability C 3 , we have

C 1 = B, B, B C 2 = G, G, G ; G, G, A ; G, A, G ; A, G, G ; • • • B, B, B C 3 = G, B, B ; A, B, B ; B, B, B
) shows that X 1 (L 1 ) = [200, 1000] while X 1 (L 3 ) = [1500, 2800]. As M 3 (J 2 ) > M 3 (J 3 ) and X 1 (L 3 ) > X 1 (L 1 ) (i.e M 3 (J 2 ) > M 3 (J 3 ) in Table
F sup (L 3 ) = ξ 1 k 1 , ξ 2 k 2 , ξ 3 k 3 ∪ Q 1 k 1 , Q 2 k 2 , Q 3 k 3 (39) = Ξ 1 k 1 , Ξ 2 k 2 , Ξ 3 k 3 (40)
where

Ξ 1 k 1 = ξ 1 k 1 = G and Q 1 k 1 = ∅; Ξ 2 k 2 = ξ 2 k 2 ∪ Q 2 k 2 = {B} ∪ {A} = A and Q 1 k 1 = {A}; Ξ 3 k 3 = ξ 3 k 3 = B and Q 3 k 3 = ∅.
Thus, the extended capability of the feasible capability C 3 is given in Table 6. Note that a such transformation from feasible capabilities to extended capabilities depends on preferential information provided by the decision maker and scientific evidence from empirical experiences.

C 1 = B, B, B C 2 = G, G, G ; G, G, A ; G, A, G ; A, G, G ; • • • B, B, B C 3 = G, A, B ; G, B, B ; A, A, B ; A, B, B ; B, A, B ; B, B, B

Taking action

The reader should note that meaningful measurement is carried out before policy making and policy implementation, since the analysis of the positive and negative consequences of potential policies is expected to be done exactly in order to supply the appropriate information to decision maker. These information are crucial for the decision makers in the sense that they help them to select adequate policies for each cluster of households. In other words, the reason to being of meaningful measurement is related to its capacity to help in designing and identifying of preferable alternatives with respect to complex policy issues. The meaningful measurement allows to make analytics among alternatives. Analytics, in our context, implies highlighting the implicit values and judgments in adequation with the preference information of the decision maker and identifying clusters which are in relation with them. Clusters which preclude implication for alternative policy choices are not worth to be considered.

Suppose that the decision maker establishes a set of actions, let say: house improvement programme (such as sanitation infrastructure supply, water supply, power supply), nutrition improvement programme, support for micro credit, etc. Considering our illustrative example, analytics imply to conclude that the policy established by the decision maker only concerns poorly housed and poorly nourished households i.e. L 1 , and poorly nourished and acceptably transported households i.e. L 2 . Remark that the households from L 2 are subject to any particular improvement programme. Thus, identifying the clusters which are in relation with the policy established by the decision maker can lead to the following policy implementation: households from L 1 are subject to alternative (i), households from L 3 are subject to alternative (ii) and households from L 2 are subject to alternative (iii) as specified bellow:

(i) poorly housed and poorly nourished households are subject to a specific house improvement programme (such as sanitation infrastructure supply, water supply, power supply) and a nutrition improvement programme totally in charge of government.

(ii) poorly nourished and acceptably transported households are subject to a specific nutrition improvement programme totally in charge of government if they are also poorly housed, but they have to contribute up to 70% of costs if they are acceptably or well housed.

(iii) well-housed, well-transported and well nourished households are not subject to any particular policy but they can benefit of taxe reduction if they financially contribute or support the government in applying concrete actions in (i) and (ii).

Discussion

The capability approach developed by [START_REF] Sen | Commodities and Capabilities[END_REF] has generated much criticism in the literature (see [START_REF] Navarro | Development and quality of life: a critique of Amartya Sen's development as freedom[END_REF][START_REF] Pogge | Can the capability approach be justified?[END_REF][START_REF] Nussbaum | Nature, function and capability: Aristotle on political distribution[END_REF]. The most virulent are those of [START_REF] Bénicourt | Contre Amartya Sen[END_REF][START_REF] Bénicourt | Amartya Sen: une nouvelle ère pour le développement?[END_REF]. In general, critics are focused on the ambiguities of Sen's capability theory and the doubts about its operationalisation. The issue that generates debate is: how to assess people's capabilities? The new methodology developed in this paper proposes a answer to this crucial question. It shows how we can assess people's capabilities in order to make well-informed decisions about policies, programs and projects. The methodology is based on two stages:

The first stage consists to an unsupervised classification of population samples who aims at finding a convenient and valid segmentation of the population in classes with homogenous socio-economic commodities. To achieve this goal, we use a statistical classification technique for discovering whether the households of a population fall into different groups by making quantitative comparisons of such commodities. This stage implies also the construction of indexes and the assessment of household's capabilities of each cluster, and the association of specific actions to each group identified. Then, groups and the policies associated, are at the basis of the preferences expressed by the policy makers.

The second stage aims at formulating an assignment procedure of 'new' households to one or several clusters by examining the commodities vector of each household and by referring to diagnosis clusters, admissibility indices, eligibility indices and rejection indices that we have broadly developed in [START_REF] Kana | Construction d'Indices: Mesure Multidimensionnelle de la Pauvreté[END_REF]. We suppose that clusters are not ordered. They are rather described/characterised by one or several 'central' modalities grouped into subsets such as each subset, called 'diagnosis cluster', be associated to their corresponding cluster. Hence, we proceed to assign the new households to specific clusters using diagnosis of clusters. To achieve this goal, we define an algorithm based on the idea of supervised classification methods. We have not developed this stage in this paper, but the reader can see more details in [START_REF] Kana | Construction d'Indices: Mesure Multidimensionnelle de la Pauvreté[END_REF].

Specifically, the meaningful multidimensional poverty measurements (MDPMs) developed in this paper are an instrument for policy making and policy implementation, for regular monitoring and diagnosis of social problems. This instrument can be also used to control and evaluate policies for social inclusion. The resulting meaningful measurements are theoretically sound since it respects the nature of the information manipulated, are operationally complete because they are designed to fit the policy requirements of decision makers and produce legitimate results because reflect both the ground reality and the policy makers values. This methodology proposes meaningful measurements satisfying three core competencies: cognitive competence, analytical competence, predicting the consequences and valuing the outcomes competence. • Analytical competence: the analytical stage aims at understanding the commodities determining this situation. It leads to the description and characterization of different classes which have been discovered during the cognitive stage. The analytical stage consists at identifying and defining the modal-valued criterion X j (L h ) and the modal-valued characterisation Y j (L h ) within each cluster. This also involves an analysis of data whose primary purpose is the highlighting of existing relationships between clusters, between the commodities that characterize them, and between clusters and variables which are generally difficult to detect in very large databases. For instance, considering the ASSL 2007 database of Burkina Faso (see Annexe 10), the 'worst' feasible capabilities (e.g. C 3 ; C 2 ; C 4 ) are characterised by the fact that the head of household is Farmer, while the 'best' ones (e.g C 7 ; C 8 ; C 1 ) are characterised by the fact that the head of household is Public Sector Employees. This is crucial since it allows to understand each class from the household's commodities perspective, identifying possible deprivations and targeting the classes to which we may tailor adequate policies (e.g. the representation (41) in Annexe 10).

• Predicting the consequences and valuing the outcomes competence: since a policy is considered as a set of actions (or alternatives) that our client would like to undertake in a given region, it is crucial to explore all alternatives of each policy in order to analyse the consequences of the various possible policies which have to be pursued in order to improve the living conditions of households. The predicting the consequences and valuing the outcomes stage aims at associating a set of appropriate actions to classes in such a way to allow to assess the effectiveness of the current policies and to determine whether the situation is changing or not ('monitoring' ). This leads to assess the effectiveness of various possible policies evaluating whether a particular policy will produce a positive impact on the welfare of the people in the future when that policy will be translated into concrete actions. Considering for instance the ASSL 29 2007 database of Burkina Faso (see Annexe 10) and suppose that the decision maker establishes a set of actions, let say: house improvement programme (such as sanitation infrastructure supply, water supply, power supply), education programme, nutrition improvement programme, health improvement programme, mobility improvement programme, support micro credit. The classes of households L 2 , L 4 (with respectively the extended capabilities C 2 , C 4 ) may benefit from all improvement programmes, while the classe of households L 3 (with extended capabilities C 3 ) does not need the micro credit support.

Some properties and axioms

One can show that C h , h = 1, . . . , µ (see equation 32) intuitively satisfies the following mathematical properties and axioms developed by [START_REF] Xu | Functioning, capability and the standard of living -an axiomatic approach[END_REF]; [START_REF] Echávarri | Ranking profiles of capability sets[END_REF].

Let C be the non-empty set of all possible functioning vectors that are Property 8.1. Non-degenerate: an extended capability C h ⊂ C is non-degenerate if and only if there exists

F l (L h ) = F J 1 (L h ); • • • ; F Jp (L h ) l ∈ C h such that J u (L h ) > ǫ u
for all u = 1, . . . , p, where ǫ u denotes the smallest functioning that we have called 'nullfunctioning'.

The property 8.1 means that decision maker can impose a lower limit ('smallest functioning' ) according to policies that he wants to translate into concrete actions, i.e. in terms of plans, programmes and projects for implementation.

Property 8.2. Comprehensive: an extended capability C h ⊂ C is comprehensive if and only if, for all

F(L h 1 ) = F J 1 (L h 1 ); • • • ; F Jp (L h 1 ) , F(L h 2 ) = F J 1 (L h 2 ); • • • ; F Jp (L h 2 ) ∈ C with h 1 , h 2 = 1, . . . , µ; if F Ju (L h 1 ) ≥ F Ju (L h 2 )
for all u = 1, . . . , p, and

F(L h 1 ) ∈ C h , then F(L h 2 ) ∈ C h .
The property 8.2 means that as much a functioning vector is improved on a particular welfare dimension is high, as much its extended capability is improved on this dimension. For all C r , C s ⊂ C, we say that the extended capability C s lies entirely in the extended capability The binary relation over C satisfies:

C r if for all F b (L s ) ∈ C s , there exists F a (L r ) ∈ C r such that F a (L r ) > F b (L s ),
Axiom 8.1. Monotonicity iff: ∀ C r , C s ⊂ C, if C s ⊆ C r then C r C s , and if C s ⊂ C r , then C r ≻ C s .
The property of Monotonicity argues that if the extended capability C s is a subset of the extended capability C r , the standard of living offered by C r is at least as preferable as the standard of living offered by C s , and if C s lies entirely in C r , then C r offers at least as higher level of the standard of living than C s . In other words, the households belonging to cluster L r related to extended capability C r have as least as higher level of opportunities than the households belong to cluster L s . This property highlights the fact that it is always desirable to have a more large extended capability and reflects the degree of freedom of household's cluster to achieve their actual living considering the extent of their opportunities.

Proof.

Let C r , C s ⊂ C such that C s ⊆ C r . Then for all F b (L s ) ∈ C s , there exists F a (L r ) ∈ C r such that F a (L r ) ≥ F b (L s ). Taking for example F a (L r ) = F sup (L r ), we have F sup (L r ) ≥ F b (L s ) for all F b (L s ) ∈ C s . Thus, F Ju (L r ) ≥ F Ju (L s
) for all u ∈ {1, . . . , p} and by equation ( 28) we obtain p u=1 M r (J u ) ≥ p u=1 M s (J u ). Therefore, C r C s . Besides, C s ⊂ C r implies C s lies entirely in C r i.e. F sup (L r ) > F sup (L s ) and C r ≻ C s Axiom 8.2. Desirability of each functioning iff: ∀ C r ⊂ C and for all u = 1, . . . , p, ∃ F Ju with ǫ u < F Ju < max

J u ′ F J u ′ ∈ F l s.t. F l ∈ C r such that C r ≻ C r ∩ {F z ∈ C s.t. F Jv ∈ F z and F Jv ≤ F Ju } for all v = 1, . . . , p.
This axiom specifies that, the standard living offered by a extended capability decreases when we take off all functioning vectors that have more of a certain amount of functioning F Jv . This means each functioning vector of a extended capability is important in assessing the standard living offered by the extended capabilities.

Proof.

Let C r ∈ C and F sup (L r ) ∈ C its functioning threshold. We set

F a = F Ju ∈ F l : ǫ u < F Ju < max J u ′ {F J u ′ ∈ F l s.t. F l ∈ C r }, 1 ≤ u ≤ p and C s = {F z ∈ C : F Jv ≤ F Ju s.t. F Jv ∈ F z , F Ju ∈ F a , and 1 ≤ v ≤ p . Then, we have F sup (L s ) = F a < F sup (L r ). This implies that C r ∩ C s ⊂ C r . Hence, C r ≻ C r ∩ C s Axiom 8.3. Dominance iff: for all positive integer N, all C r , C 1 , . . . , C N ⊂ C s.t. C s = ∪ N q=1 C q and C s ⊂ C, if C r ≻ C 1 C 2 . . . C N , then C r ≻ C s .
Dominance requires that when the extended capability C r offers at least as higher level of the standard of living than each and every C q (with q ∈ {1, . . . , N}), and the standards of living offered by those C q are comparable, then C r offers at least as higher of the standard of living than ∪ N q=1 C q . This axiom places an implicit importance on the household's actual living when ranking his standard of living: as far as the freedom aspect of the living standard is concerned, we may have the situation in which putting all C q together may increase the freedom of choice already offered by each of them.

Proof.

Let C r , C 1 , . . . , C N ⊂ C such that C s = ∪ N q=1 C q and C s ⊂ C. Because C r ≻ C 1 C 2 . . . C N , the functioning threshold F sup (L s ) ∈ C s such that F sup (L s ) = F sup (L 1 ) with F sup (L 1 ) ∈ C 1 . Since C r ≻ C 1 , we trivially have C r ≻ C s
Axiom 8.4. Continuity iff: for all C r ⊂ C and a sequence of extended capabilities { C q } q=1,...,+∞ such that C q ⊂ C for all q and C = lim

q→+∞ C q ⊂ C, if C q C r for all q, then C C r .
Continuity requires that the standard of living ranking should not reverse the ranking suddenly. This means that if every extended capability C q in the sequence { C q } q=1,...,+∞ offers at least as preferable standard of living as the extended capability C r , then the limit of this sequence should also offer at least as high a standard of living C r .

Proof.

Considering the extended capability C r ⊂ C and a sequence { C q } q=1,...,+∞ , C = lim q→+∞ C q C q by definition. Hence, if C q C r for all q, we trivially have C C r

Conclusion

We have introduced a new methodology for multidimensional poverty measurement based on Sen's capability approach under a decision aiding perspective. Initially, our methodology starts to select properly the features (selection of individuals and selection of variable) on which clustering is to be performed so as to encode as much information as possible concerning the task of our interest. The input data table representing the set of commodity vectors of households is organised into clusters in such a way that each cluster consists of households that are similar in term of commodities distribution between themselves and dissimilar to households of the others clusters.

Thus, we establish the welfare dimensions and the welfare dimension levels from which we construct the functionings and the feasible capability corresponding to each cluster. Each feasible capability associated to a given cluster reflects the ability of households within this cluster to achieve a subset of functionings among the various alternative functioning bundles. Depending on generic commodities, the feasible capability of a cluster can be improved in such a way to obtain an extended capability. This allows to take into account in the meaningful multidimensional poverty measurement (MDPM) process some commodities such as income, occupational status, marital status, age, size of household, etc. Specifically, MDPM is an instrument that helps a decision maker in making well-informed decisions about policies, programmes and projects by putting the best available evidence from research at the heart of policy development and implementation. Thus, it allows decision makers to target resources and design policies more effectively. This instrument is also useful for regular monitoring and diagnosis of social problems and a tool that allows us to control and evaluate policies for social inclusion. Our methodology allows both to improve how knowledge about standard of people's living and to operationalise Sen's capability approach.

Let us also mention that this methodology may be used in order to establish rigorous ways to define and measure what costs and outcomes are. It is also a way of promoting the transparency in the management of resources through cost-effectiveness, cost-benefit, and cost-utility of policies, programmes or projects, and the consideration of evidence on usefulness of taking actions. Moreover, our methodology may be extended to more complex situations to highlight a vivid picture of different types of poverty, both across countries, regions and the world and within countries by ethnic group, urban/rural location, or other key household features. year 2007 (ASSL) from the Burkina Faso's National Institute of Statistics and Demography (INSD6 ). Burkina Faso implementation of this survey follows standardised guidelines and receives technical assistance, in terms of Unified Questionnaire of Basic Indicators of Wellbeing (QUIBB 2007 in french), the sampling procedure and training of the enumerators, so that within the survey there is greater homogeneity and comparability than between other national multi-topic household surveys. The annual survey monitoring the living conditions of households was conducted from the perspective of a better understanding of poverty in Burkina Faso and for better tracking its manifestations. It aims at providing useful data to refine the analysis within the various sectoral and thematic groups of institutional arrangements for monitoring of the implementation of the Strategic Framework to Fight against Poverty. It will allow all stakeholders of the Poverty Reduction Strategy Papers (PRSP) to obtain information to determine the trends of poverty in Burkina Faso by updating the indicators. An indicator can be defined as a measurement that helps us to understand where we are, where we are going and how far we are from the goal. It allows to summarize the characteristics of systems or highlight what is happening in a system. Note that we have made the hypothesis in this paper that the variables describe the commodities of households.

The survey dataset (1255 households on 48 variables) used in order to illustrate our methodology is a national representative sample of households living in Ouagadougou (the capital of Burkina Faso). The ASSL provides information on the dimensions of poverty that we have grouped in six potential welfare dimension characterizing a cluster L h of the population Ω as follows: nutrition, education, water and sanitation, housing, health and transportation. Note that, the decision about the number of potential welfare dimension and the choice of variables to associate to each of them depend on the decision maker (or policy maker) and its interest field. The choices we did in this paper, concerning the six potential welfare dimensions, are just for illustrative purposes.

Remark 10.1. This survey has been done in the framework of the Millennium Development Goals (MDG). The MDG and targets come from the Millennium Declaration, signed by 189 countries, including 147 heads of State and Government, in September 2000 (see UNDP, 2003).

• Nutrition: ASSL provides nutritional information for each household member. We use three dimensions of poverty to identify whether a household is deprived or vulnerable in term of nutrition (poorly nourished) by measuring directly whether a household has had problems satisfying his basic needs and his economic situation.

• Education: ASSL provides information on the years of education and access to school for each household member. Years of schooling capture the level of knowledge and understanding of each household. Like the first one, it doesn't capture the quality of education nor the level of skills but, we have considered it in this paper as a relative good indicator of functionings.

• Water and Sanitation (WS): The factor water and sanitation uses six dimensions of poverty which, in combination, represent the deprivation situation of each cluster of households in terms of access to clean drinking water and to adequate sanitation. It includes two standard MDG indicators (clean drinking water and improved sanitation) which provide some rudimentary indications of the quality of water and sanitation services for the households.

• Housing: The factor housing uses twenty-one dimensions of poverty which, in combination, represent the housing poverty situation of each cluster. It includes one standard MDG indicators (the use of clean cooking fuel) and two non-MDG indicators (electricity and flooring material). Both of them provide some rudimentary indications of the quality of housing for each cluster.

• Health: The factor health uses one dimension of poverty which evaluates access to health service no matter what mode of transportation is required to access it.

• Mobility: The final factor covers the ownership of some consumer goods for transportation such as bicycle, motorbike, car and the access to public service transportation.

Remark that, each F Ju , u ∈ {1, 2, 3, 4, 5, 6} is perceived as the set of all commodities related to the set j∈Ju X j within the whole population Ω and F Ju (L h ) is perceived as the set of all commodities related to the set j∈Ju X j within the cluster L h .

The common problem of missing data can happen during cluster analysis. To solve this problem, we only deal with valid values. Thus, this methodology can be also applied to others datasets such that the Demographic and Health Surveys (DHS) and World Health Surveys (WHS). Our choice for ASSL 2007 was only motivated by the needs of the illustration.

Application

For a broad illustration of our methodology, let us consider the survey dataset of standard of living of 1255 households evaluated on 48 variables related to six potential dimensions of welfare (see section 10.1). We conducted hierarchical cluster analysis via multiscale bootstrap (number of bootstrap 1000; (see [START_REF] Suzuki | Pvclust: an R package for assessing the uncertainty in hierarchical clustering[END_REF])) using the Ward method [START_REF] Ward | Hierarchical grouping to optimize an objective function[END_REF]) and a correlation-based dissimilarity matrix. The best number of clusters was obtained for eight clusters. This result can be also confirmed by [START_REF] Calinski | A dendrite method for cluster analysis[END_REF] index which allows to compare the homogeneity of partitions. Using the R software with (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 ) = (0.063, 0.272, 0.224, 0.164, 0.09, 0.118, 0.028, 0.042) and L = {L 1 , L 2 , L 3 , L 4 , L 5 , L 6 , L 7 , L 8 }, we have obtained the following tables.

From equation (28), we obtain the ordered representation (41) which shows that the feasible capability C 4 is the poorest ones and then the highest priority in term of policies intervention or policies implementation. From Table 8, we can derive the functioning thresholds F sup (L h ) which are given by the first functioning vector of the feasible capabilities in Table 9. For instance, we have J 1 ε 1 J 2 ε 2 J 3 ε 3 J 4 ε 4 J 5 ε 5 J 6 ε 6 M 1 0.84 0.96 0.78 0.95 0.85 0.93 0.80 0.93 0.65 0.90 0.75 0.89 M 2 -0.78 0.85 -0.43 0.74 -0.06 0.53 -0.82 0.85 -0.36 0.70 -0.41 0.68 M 3 -0.35 0.66 0.29 0.63 0.29 0.64 -0.34 0.64 0.10 0.56 -0.13 0.57 M 4 -0.89 0.90 -0.73 0.91 -0.11 0.56 -0.72 0.91 -0.06 0.55 -0.44 0.76 M 5 0.64 0.88 0.68 0.93 0.68 0.84 0.73 0.90 0.84 0.97 0.61 0.83 M 6 0.38 0.73 0.45 0.79 0.70 0.85 0.48 0.75 0.84 0.94 0.42 0.74 M 7 0.94 0.95 0.97 0.97 0.85 0.87 0.84 0.95 1.00 1.00 0.99 0.99 M 8 0.92 0.97 0.89 0.98 0.55 0.83 0.96 0.98 0.86 0.99 0.90 0.98 However, a quick analysis of households from each cluster shows that the 'worst' feasible capabilities (e.g. C 3 ; C 2 ; C 4 ) are characterised by the fact that the head of household is Farmer and the 'best' ones (e.g C 7 ; C 8 ; C 1 ) are characterised by the fact that the head of household is Public Sector Employees. It is important to note that, the words 'worst' and 'best' do not contain anything of numerical but only reflect the household's well-being and advantage.

C 7 ≻ C 8 ≻ C 1 ≻ C 5 ≻ C 6 ≻ C 3 ≻ C 2 ≻ C 4 (
Consider now the generic commodity 'income'7 and the feasible capabilities C 5 and C 6 (see Table 9) that we would like to increase some of the attainable levels of functioning vector. A quick observation shows that the functioning thresholds F sup (L 5 ) = F sup (L 6 ). One can be tempted to conclude that "the standard living (or welfare) offered by C 5 is considered to be equivalent to the standard living (or welfare) offered by C 6 ". However, Table 8 shows that only M 5 (J 5 ) and M 6 (J 5 ) are equal in terms of meaningful measurement. Furthermore, Table 11 shows that X 2 (L 5 ) = [90000, 110000], while X 2 (L 6 ) = [63000,90000]. Thus, we can intuitively increase the attainable levels of welfare dimension J 3 from A to G, i.e where M 5 (J 3 ) < M 6 (J 3 ) in Table 8. The extended capabilities of the feasible capabilities C 5 and C 6 are then given by the sets described in 42 and 43. [12, 15] [1.75, 5] [3.5, 6.5] [0.1, 2.5] [9, 11] [6.3, 9] [25, 60] [17, 22.5

Figure 2 :

 2 Figure 2: Visualisation of households by clusters

  with a ∈ {1, . . . , l r } and b ∈ {1, . . . , l s }. Thus, C s lies entirely in C r iff F sup (L r ) > F sup (L s ) and C r ≻ C s . The binary relation defines over C satisfies reflexivity i.e. for all C r ⊂ C, C r C r and transitivity i.e. for all C r , C s , C z ⊂ C, if C r C s and C s C z then C r C z . When satisfies completeness, i.e for all C r , C s ⊂ C with C r = C s , C r C s or C s C r , reflexivity and transitivity, is called an ordering.

  F sup (L 1 ) = G, G, G, G, A, G and F sup (L 2 ) = F sup (L 4 ) = B, B, B, B, B, B , F sup (L 3 ) = B, A, A, B, B, B , F sup (L 5 ) = F sup (L 6 ) = A, A, A, A, G, A , F sup (L 7 ) = G, G, G, G, G, G , F sup (L 8 ) = G, G, A, G, G, G .

C 1 =

 1 G, G, G, G, A, G ; G, G, G, G, A, A ; G, G, G, A, A, A ; • • • B, B, B, B, B, B C 2 = B, B, B, B, B, B C 3 = B, A, A, B, B, B ; B, B, A, B, B, B ; B, A, B, B, B, B ; B, B, B, B, B, B C 4 = B, B, B, B, B, B C 5 = A,A, A, A, G, A ; A, A, A, A, A, A ; A, A, A, A, A, B; • • • B, B, B, B, B, B C 6 = A,A, A, A, G, A ; A, A, A, A, A, A ; A, A, A, A, A, B; • • • B, B, B, B, B, B C 7 = G, G, G, G, G, G ; G, G, G, G, A, G ; G, G, G, G, G, A ; • • • B, B, B, B, B, B C 8 = G, G, A, G, G, G ; G, G, A, G, G, A ; G, G, A, A, G, G ; • • • B, B, B, B, B, B

C 5 =

 5 A, A, G, A, G, A ; A, A, G, A, A, A ; A, A, A, A, G, A ; • • • • • • • • • (42) C 6 = A, A, A, A, G, A ; A, A, A, A, A, A ; A, A, A, A, A, B ; • • • • • • • • • (43)Assuming that the decision maker has imposed some levels of control in such a way that the extended capabilities of the feasible capabilities C 7 and C 8 are given byC 7 = G, G, G, G, G, G ; G, G, G, A, G, G ; G, G, A, G, G, G and C 8 = G, G, A, G, G, G ; G, G, A, G, G, A ; G, G, A, A, G, G .Thus, the extended capabilities are given as follows:C = G, G, G, G, A, G ; G, G, G, G, A, A ; • • • B, B,B, B, B, B C = B, B, B, B, B, B C = B, A, A, B, B, B ; B, B, A, B, B, B ; B, A, B, B, B, B ; B, B, B, B, B, B C = B, B, B, B, B, BC = A, A, G, A, G, A ; A, A, G, A, A, A ; A, A, A, A, G, A ; • • • • • • • • • C = A, A, A, A, G, A ; A, A, A, A, A, A ; A, A, A, A, A, B ; • • • • • • • • • C = G, G, G, G, G, G ; G, G, G, A, G, G ; G, G, A, G, G, G C = G, G, A, G, G, G ; G, G, A, G, G, A ; G, G, A, A, G, G

Table 3 :

 3 Evaluation of C h (S Ju ) and C h

Table 4 :

 4 Example of meaningful measurements

Table 5 :

 5 The set of feasible capabilities

Table 6 :

 6 The set of extended capabilities

•

  Cognitive competence: the cognitive stage allows to understand what the situation is and to identify (using clustering) different types of poverty within the society to which we may tailor adequate policies. Since it makes more sense to have different classes of households being differently poor, the data table is organised into classes in such a way that each classe consists of households that are similar in term of commodities distribution between themselves and dissimilar to households of other classes. It is likely to have more effective policies if these are correctly targeted. For instance, the ASSL 2007 database of Burkina Faso (seeAnnexe 10) shows eight classes of people differently poor given by L = {L 1 , L 2 , L 3 , L 4 , L 5 , L 6 , L 7 , L 8 } highlighting that class L 4 (with feasible capability C 4 ) is the poorest one i.e. the highest priority in terms of policies intervention or policies implementation.

Table 7 :

 7 J 1 U J 1 S J 2 U J 2 S J 3 U J 3 S J 4 U J 4 S J 5 U J 5 S J 6 U J 6 Evaluation of C h (S Ju ) and C h (U Ju )

	41)

Table 8 :

 8 Example of meaningful measurements

Table 9 :

 9 The set of feasible capabilities

Table 10 :

 10 The set of extended capabilities

	C 1	C 2	C 3	C 4	C 5	C 6	C 7	C 8
		X

2 : Income in CFA (Scale 1/10000)

Table 11 :

 11 Description of some generic commodities

	]

UNDP: United Nations Development Programme.

WHO: World Health Organization.

NEPAD: New Partnership for Africa's Development.

This example taken from[START_REF] Osmani | The entitlement approach to famine: An assessment[END_REF] research paper is a perfect illustration of how the endowments can be transformed into commodities.

Free downloadable on http://www.r-project.org (seeR Development Core Team, 2011)

INSD (Institut National de la Statistique et de la Démographie) in French.

Note that 'income' in ASSL 2007 is given in CFA franc. The CFA franc is the name of two currencies used in Africa: 1 euro = 655.957 CFA.