AN ANALYSIS OF POVERTY IN ITALY THROUGH A FUZZY REGRESSION MODEL S. Montrone, F. Campobasso, P. Perchinunno, A. Fanizzi Un...
Over recent years, and related in particular to the significant contemporary international economic crisis, an increasingl...
1. Different  approaches to the poverty  (absolute, relative, subjective)  INDEX  2. Techniques of the  Fuzzy Set  ( A Fuz...
<ul><li>Traditional distinction between  absolute  and  relative  poverty:  </li></ul><ul><li>the first is understood as t...
2. A FUZZY REGRESSION MODEL Fuzzy regression techniques  can be used to fit fuzzy data into a regression model. Diamond (1...
A FUZZY NUMBERS Modalities of quantitative variables are commonly given as exact single values, although sometimes they ca...
<ul><li>A triangular fuzzy number  for the variable X is characterized by this function  that expresses the   membership d...
<ul><li>The same Author derived the expression of the estimated coefficients in a fuzzy regression model of a dependent va...
<ul><li>Assuming to regress a dependent variable  </li></ul><ul><li>on k independent variables  in a set of n units,  </li...
<ul><li>The estimates of the fuzzy regression coefficients are determined by minimizing the sum of the Diamond’s squared d...
<ul><li>The estimates of the fuzzy regression coefficients are so given by this formula </li></ul><ul><li>   = [ X' X + (...
<ul><li>A fuzzy version of the index R 2 , which may be called  Fuzzy Fit Index (FFI) ,  </li></ul><ul><li>will be used in...
In the present study data are elaborated arising from EU-SILC survey regarding the perception of the Italian families in “...
3.  THE APPLICATION OF A FUZZY REGRESSION MODEL
The present work aims to identify the relationship between several independent variable X i   (expenses for rent or mortga...
In particular, the response categories in terms of  mortgage payments, rent and household costs  are centred on 1, 3 and 5...
<ul><li>The verification of those expenses which determine the degree of difficulty (in terms of getting through to the en...
The estimated regression coefficients for families in rented houses are reported below: The most relevant expenses (in ter...
The estimated regression coefficients for families with mortgage rates are reported below: The most relevant expenses resu...
In this work we propose a Fuzzy Regression Model in order to identify the factors that most influence the perception of po...
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An Analysis of Poverty in Italy through a fuzzy regression model

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An Analysis of Poverty in Italy through a fuzzy regression model
Paola Perchinunno, Francesco Campobasso, Annarita Fanizzi, Silvestro Montrone - Department of Statistical Science, University of Bari

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An Analysis of Poverty in Italy through a fuzzy regression model

  1. 1. AN ANALYSIS OF POVERTY IN ITALY THROUGH A FUZZY REGRESSION MODEL S. Montrone, F. Campobasso, P. Perchinunno, A. Fanizzi Università degli Studi di Bari - Dipartimento di Scienze Statistiche ICCSSA 2011 GEOG-AN-MOD 11 Santander, 20-23 June 2011
  2. 2. Over recent years, and related in particular to the significant contemporary international economic crisis, an increasingly worrying rise in poverty levels has been observed both in Italy, as well as in other countries. INTRODUCTION The present work elaborates data revealed by the EU-SILC survey (2006) regarding the perception of poverty by Italian families, through a fuzzy regression model .
  3. 3. 1. Different approaches to the poverty (absolute, relative, subjective) INDEX 2. Techniques of the Fuzzy Set ( A Fuzzy Regression Model ) 3. The application of the Fuzzy Approach: construction of Eu-Silc indicators and definition of fuzzy numbers 4. Results of the Fuzzy Regression Model
  4. 4. <ul><li>Traditional distinction between absolute and relative poverty: </li></ul><ul><li>the first is understood as the incapacity to reach an objective level of wellbeing; </li></ul><ul><li>the second is based on the assumption that the social condition of an individual, cannot be well defined without taking in to account his context of living. </li></ul>1. DIFFERENT APPROACHES TO THE POVERTY A transversal approach is considered as subjective , through which the poor is defined as his perception in comparison with the rest of society (in terms of perceived wellbeing).
  5. 5. 2. A FUZZY REGRESSION MODEL Fuzzy regression techniques can be used to fit fuzzy data into a regression model. Diamond (1988) treated the simple Fuzzy regression model introducing a metrics into the space of triangular fuzzy numbers . In this work we explicit the expression of the parameters of the model with fuzzy asymmetric intercept in the multiple case, starting from the simple model handled by Diamond.
  6. 6. A FUZZY NUMBERS Modalities of quantitative variables are commonly given as exact single values, although sometimes they cannot be precise (the imprecision of measuring instruments and the continuous nature of some observations). On the other hand qualitative variables are commonly expressed using common linguistic terms (which also represent verbal labels of sets with uncertain borders). The appropriate way to manage such an uncertainty of observations is provided by fuzzy numbers.
  7. 7. <ul><li>A triangular fuzzy number for the variable X is characterized by this function that expresses the membership degree of any possible value of X to </li></ul><ul><li>Diamond (1988) introduced the following metrics onto the space of triangular fuzzy numbers </li></ul><ul><li>: </li></ul>A FUZZY NUMBERS
  8. 8. <ul><li>The same Author derived the expression of the estimated coefficients in a fuzzy regression model of a dependent variable on a single independent variable . </li></ul><ul><li>We generalize this estimation procedure to the case of several independent variables with a fuzzy asymmetric intercept. </li></ul>A FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPT
  9. 9. <ul><li>Assuming to regress a dependent variable </li></ul><ul><li>on k independent variables in a set of n units, </li></ul><ul><li>the linear regression model with a fuzzy intercept is given by this formula: </li></ul><ul><li>where </li></ul>A FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPT
  10. 10. <ul><li>The estimates of the fuzzy regression coefficients are determined by minimizing the sum of the Diamond’s squared distances, between theoretical and empirical values of the dependent variable respect to a, b 1 , .., b k , , </li></ul><ul><li>The function to minimize assumes different expressions according to the signs of the regression coefficients b 1 , .., b k . </li></ul>ESTIMATION OF THE FUZZY REGRESSION MODEL
  11. 11. <ul><li>The estimates of the fuzzy regression coefficients are so given by this formula </li></ul><ul><li> = [ X' X + (X L ' X L + X R ' X R ) ] -1 [ X'y + (X L 'y L +X R 'y R ) ] </li></ul><ul><li>where: </li></ul><ul><li>y is the n-dimensional vector of cores of the dependent variable; </li></ul><ul><li>y L and y R are the n-dimensional vectors of the lower extremes and the upper extremes respectively of the dependent variable; </li></ul><ul><li>X is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent variables and two vectors 0; </li></ul><ul><li>X L is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent variables and two vectors -1 and 0; </li></ul><ul><li>X R is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent variables and two vectors 1 and 0; </li></ul><ul><li> is the vector (a, b 1 , .., b k , γ L , γ R )'. </li></ul>ESTIMATION OF THE FUZZY REGRESSION MODEL
  12. 12. <ul><li>A fuzzy version of the index R 2 , which may be called Fuzzy Fit Index (FFI) , </li></ul><ul><li>will be used in order to evaluate how the model fits data: </li></ul><ul><li>The more this index is next to one, the better the model fits the observed data. </li></ul><ul><li>Finally we propose a stepwise procedure in order to simplify, </li></ul><ul><li>in computational terms, the identification of the most significant independent variables. </li></ul>STEPWISE PROCEDURE
  13. 13. In the present study data are elaborated arising from EU-SILC survey regarding the perception of the Italian families in “getting through to the end of the month”. 3. THE APPLICATION OF THE FUZZY APPROACH It emerges, in particular, that the majority of households surveyed declared themselves to be in a state of hardship (either in great hardship 13.4%, in hardship 19.4% or in some degree of hardship, 40.2%). There are, however, few families (6.0%) declaring that they get through to the end of the month with absolute confidence.
  14. 14. 3. THE APPLICATION OF A FUZZY REGRESSION MODEL
  15. 15. The present work aims to identify the relationship between several independent variable X i (expenses for rent or mortgage payments, for the running of the household and for other debts) and a single dependent variable Y (the difficulty of “getting through to the end of month”). 3. DEFINITION OF FUZZY NUMBERS Furthermore, in order to normalize the data collected, the explanatory variables have been quantified with the same criteria.
  16. 16. In particular, the response categories in terms of mortgage payments, rent and household costs are centred on 1, 3 and 5, whilst the response categories in terms of expenses for other debts are centred on 0, 1, 3 and 5. 3. DEFINITION OF FUZZY NUMBERS
  17. 17. <ul><li>The verification of those expenses which determine the degree of difficulty (in terms of getting through to the end of the month) is conducted: </li></ul><ul><li>firstly, comparing families in rental accommodation against homeowners with a mortgage </li></ul><ul><li>secondly, according to geographical area (north of Italy, centre and south and islands). </li></ul>4.RESULTS OF THE REGRESSION MODEL
  18. 18. The estimated regression coefficients for families in rented houses are reported below: The most relevant expenses (in terms of difficulty in getting through to the end of month) results those relative to rental payments in all geographic area. The “expenses for other debt” are relevant only in the North area. 4.RESULTS OF THE REGRESSION MODEL
  19. 19. The estimated regression coefficients for families with mortgage rates are reported below: The most relevant expenses results those relative to the payment of mortgage rates and differently from the previous model, above all in the south of Italy. Besides, as we can see by the spread values, the variability is higher in this case. 4.RESULTS OF THE REGRESSION MODEL
  20. 20. In this work we propose a Fuzzy Regression Model in order to identify the factors that most influence the perception of poverty by Italian families . 5. CONCLUDING REMARKS A subjective approach to poverty suggests the adoption of a fuzzy regression model, made possible by an initial transformation of data into triangular fuzzy numbers The results of the analysis of poverty levels has showed at what degree the most relevant expenses (in terms of getting through to the end of the month), for Italian families, are those for rent and mortgage in the different geographical areas.

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