The traditional multi-dimensional measures have failed to properly project  the vulnerability of human-beings towards poverty. Some of the reasons behind this inability may be the failure of the existing measures to recognise the graduality inside the concept of poverty and the disparities within the household in wealth distribution. So this work wants to develop a measure to estimate the vulnerability of households in becoming poor in a multidimensional perspective through incorporating the intra-household disparities and graduality within the causal factors. Dimensional decomposition of the developed vulnerability measure is also under the purview of this work. To estimate the vulnerability and dimensional influences with the help of artificial intelligence an integrated mathematical framework is developed

Sen, Sugata

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveDecomposition of intra-householddisparity sensitive fuzzymulti-dimensional poverty index: Astudy of vulnerability through MachineLearningSen, Sugata28 April 2019Online at https://mpra.ub.uni-muenchen.de/93550/MPRA Paper No. 93550, posted 02 May 2019 13:48 UTC1  DECOMPOSITION OF INTRA-HOUSEHOLD  DISPARITY SENSITIVE  FUZZY MULTI-DIMENSIONAL POVERTY INDEX: A STUDY OF VULNERABILITY THROUGH  MACHINE LEARNING  Dr. Sugata Sen1 1. Introduction The traditional multi-dimensional measures have failed to properly project  the vulnerability of human-beings towards poverty. Some of the reasons behind this inability may be the failure of the existing measures to recognise the graduality inside the concept of poverty and the disparities within the household in wealth distribution. So this work wants to develop a measure to estimate the vulnerability of households in becoming poor in a multidimensional perspective through incorporating the intra-household disparities and graduality within the causal factors. Dimensional decomposition of the developed vulnerability measure is also under the purview of this work. To estimate the vulnerability and dimensional influences with the help of artificial intelligence an integrated mathematical framework is developed.  2. Review of literature One of the major impedement of the well accepted Multi-dimensional Poverty Indices (Alkire & Foster, Counting and Multidimensional Poverty Measurement, 2009) (Alkire & Foster, 2011) (Alkire, Kanagaratnam, & Suppa, The Global Multidimensional Poverty Index (MPI):2018 Revision, 2018) is that they have tried to distinguish the poor from the non-poor through the classical Boolean logic. But the idea of poverty suffers from vagueness and naturally cannot be defined through a well defined cut-off. So discussing the multidimensional poverty through ordinary proposition is not correct (Qizilbash, 2006).                                                           1 Associate Professor of Economics, Panskura Banamali College, INDIA 721152, sensugata@gmail.com, M: +91 7501730545 2  The graduality within a vague concept can well be represented through fuzzy logic (Zadeh, 1965). Cerioli and Zani first attemted to use the fuzzy logic on the measurement of multidimentional poverty (MP) (Cerioli & Zani, 1989). They have estimated the strength of poverty in each dimension through a membership function. Then the strength of all dimensions are added and normalised to get the household level multidimentional poverty.  Their idea is improved subsequently through a voluminous research (Chelli & Lemmi, 1995)  (Martinetti, 2006)  (Betti, Cheli, Lemmi, & Verma, 2006)  (Chakravarty, 2006).  Another important drawback of the conventional multidimentional poverty indices is that all of these measures have accepted that the poverty status of the individuals within a household are equall. Thus these indices have accepted the household as a homogenous unit and such that have failed to capture the intra-household differences. But it is an established fact that the different members within a family enjoys varied endowment as well as bargaining power (Agarwal, 1997) (Duflo, 2003). So computing poverty measures taking households as the basic unit leads to improper estimation. The earliest work to put importance on the individuals instead of the household was carried out by Haddad and Kanbur (Haddad & Kanbur, 1990). Vijaya et. al. (Vijaya, Lahoti, & Swaminathan, 2014) and Klasen et. al. (Klasen & Lahoti, 2016) have developed individual sensitive measures in the multi-dimensional framework. Apart from measuring the composite effect of the multi-dimensional poverty a large volume of research appeared on the decomposition of composite index. Using the properties of sub-group decomposability Alkire et. al. have decomposed Alkire-Foster Adjusted Headcount Ratio (Alkire, Roche, & Vaz, 2017). Deutch and Silber have tried to decompose the fuzzy multidimentional poverty index through Shapley method to find the importantance of the causal factors (Deutsch & Silber, 2006). The Shapley Value Decomposition determines the average of the marginal contributions of a factor under different combinations. To that respect, the concerned factor is first withdrawn from the model and the rest of the factors are permuted to form different distributions. Gradually, the withdrawn factor is added to each of the combination and the marginal contribution of the added factor in a specific distribution is counted. The average of marginal contributions of the stated factor from all the distributions is the influence of that very factor on the composite variation (Shorrocks, 2013). 3  To decompose the multi-dimensional poverty index machine learning can be used. Machine Learning(ML) is a technique of data analytics that instructs computer to learn from experience. Machine Learnings algorithms use computational methods to “learn” information directly from data without depending on a pre-set equation as a model (Kubat, 2017) (Theobald, 2017). Shapley Value Machine Learning can successfully implement Shapley Decomposition of MP. A framework called SHapley Additive exPlanation (SHAP) executes this through Local Interpretable Machine-agnostic Explanation (LIME) algorithms (Lundberg & Lee, 2017). So to improve the poverty estimation incorporation of fuzzy logic and intra-household disparities are needed. Dimensional decomposition  of the improved estimate can examine the influence of individual causal factors on the vulnerability of the households to become poor. Shapley Value Machine Learning can play an important role in this decomposition. Thus the specific objectives of this study are the following.  3. Objectives  Measuring the vulnerability to become poor multi-dimentionaly  incorporating the ideas of graduality and intra-household disparities.  Development of appropriate machine learning process with the help of artificial importance to examine the dimensional influence on household level vulnerability.  4. Findings Let there are n individual, expressed as i=1,2,..n and k dimensions expressed as j=1,2,…k. The performance level of each individual on each dimension can be expressed as a n×k real valued non-negative matrix. Each row vector yi={   }interprets the performance vector of ith individual.  Let, z is a vector of dimensional thresholds when z={zj}. Then matrix gi0={1,0} can be created where gij0 =1, when yij<zj  and gij0 =0, when yij≥zj   4  gij0 is 1 if the ith individual is poor in the jth dimension and otherwise. Now let ci=|g ij0| and vector c shows the number of dimensions where each individual is lying below the established dimensional thresholds. If d is the aggregative cut-off to become poor then ith individual will become multi-dimensionally poor when ci ≥ d where 1 < d < k. This identification has failed to consider the vagueness of poverty and the existence of intra-household disparities. To rectify this let us assume that each of the households consists of q individuals where q is a positive integer.  Let the grade of membership to the poor set of the qth member of the ith household in a specific dimension is expressed through the dimension specific individual membership function     (i) = 1 if  0 ≤     ≤        and    (i) = 0 if     ˃      An individual is definitely poor if his achievement in a particular dimension j is from 0 upto    . On the other hand if individual achievement is above      then the individual is not poor on dimension j. For individual achievement between     and      the membership function thakes on values in [0,1]. More clearly it can be interpreted that if     (i) = 0 if the ith individual is certainly not poor in the jth dimension.     (i) = 1 if the ith individual completely belongs to the poor set corresponding to jth dimension.  0 <    (i) < 1 if the ith individual shows a partial membership to the poor set p of jth dimension. The strength of membership of all the individuals of a particular household in a particular dimension can be added and deflated by the number of household members to get the collective strength of household membership in a particular dimension. Thus the individual sensitive grade of membership of ith household in jth dimension can be represented as        =∑      The grade of membership of the ith individual to the multi-dimensional poor set M can be defined as 5        =  ∑             Where 0 ≤       ≤ 1 and       is the vulnerability of ith household to become multi-dimensionally poor. If weight    is assigned to the jth dimension then it can be written that       =  ∑           ∑         This work is interested to examine the dimensional influences on the household level vulnerability through necessary decomposition. Let the household level vulnerability is   . It is quite natural that the desired value of    is 0. Thus, the difference between desired and observed vulnerability is   . To decompose    Shapley value decomposition has been used. This method calculates the average of marginal contributions of each dimension to the level of vulnerability. To find the contribution of jth dimension we would find different combination of K-1 dimensions. So, the total no of combinations without the jth dimension is – (K-1)C1 + (K-1)C2 + (K-1)C3 + (K-1)C4 + …. + (K-1)CK-1  = ∑               = θ Subsequently, the jth dimension is added to each of θ combinations to find the marginal effect of jth dimension from that particular combination. Naturally, we would get θ marginal contributions. According to Shapley decomposition, the contribution of jth dimension is the average of all the marginal contributions of jth dimension. Let, the marginal contribution of jth dimension from sth combination is  s. So, the set of marginal contributions of the jth dimension is- Ηj = ( 1j,  2j, …,     ) Then average of marginal contribution of the jth dimension is     j =     ∑       sj As we have    contributions under each dimension then it can be said that these marginal contributions have been generated through    functions. The polinomial form of sth combination under the jth dimension can be chosen from the set of ψ alternative polinomials    . To that respect the machine learning 6  process will find the observed value of the composite dependent variable through a particular polinomial from set     and compare that observed value with the expected value to find the error. Learning from the successive errors within     the machine learning process will choose that polinomial from      which will minimise the error. In this way    functional forms can be determined under jth dimension. From these θ functions we can find the    incremental influences. The average of the    incremental influences will deliver    j.  This estimation of dimensional contributions can be executed through Local Interpretable Machine-agnostic Explanation (LIME) algorithms. LIME deliberately perturbs a combination by accepting input variables from the neighbourhood and counts the effect of that perturbation on the output. Finally the relevance of the particular input is determined through the average of deviation in the output due to the perturbations. Following this logic in our decomposition model LIME will first find    functions from the θψ alternatives through perturbation. After that the algorithm proceeds to estimate the dimensional contribution. Technically LIME initiates the process to locate min      = min       –             →       →     where u≠ v and u,v=1,2….. .    is the neighbourhood of sth functional form under jth dimension.       is the expected value and       is the observed value of the multi-dimensional poverty from a particular polinomial related to    .      is the chosen polynomial from    . This process will continue for all the combinations under    to find    incremental influences. Finally the average of the    incremental incluences determines     j =     ∑       sj 5. Conclusion The mathematical framework developed under the current work will deliver a better estimate of vulnerability as well as dimensional influences to household level multi-dimensional poverty through accommodating fuzzy logic and intra-household disparities. The use of artificial importance as demonstrated within the model will make the model easy to execute. This estimation of vulnerability and decomposition will help the planners to locate the role of different dimensions behind the vulnerability of human beings to become poor more precisely.   7  Works Cited  Agarwal, B. (1997). Bargaining and Gender Relations: Within and Beyond the Household. Feminist Economics, 3(1), 1-51. Alkire, a., & Foster, J. (2011). Counting and multidimensional poverty measurement. Journal of Public Economics, 476-487. Alkire, S., & Foster, J. (2009). Counting and Multidimensional Poverty Measurement. Oxford: OPHI. Alkire, S., Kanagaratnam, U., & Suppa, N. (2018). The Global Multidimensional Poverty Index (MPI):2018 Revision. Oxford: OPHI. Alkire, S., Roche, J. M., & Vaz, A. (2017). Changes Over Tme in Multidimensional Poverty: Methodology and Results for 34 Countries. World Development, 232-249. Betti, G., Cheli, B., Lemmi, A., & Verma, V. (2006). Multidimentional and Longitudinal Poverty: an Integrated Fuzzy Approach. In A. L. Betti, FUZZY SET APPROACH TO MULTIDIMENTIONAL POVERTY MEASUREMENT (pp. 114-137). New York: Springer. Bourguignon, F., & Chakravarty, S. R. (2003). The Measurement of Multidimensional Poverty. The Journal of Economic Inequality, 25-49. Carroll, C. D. (1994, February 1). How does Future Income Affect Current Consumption? The Quarterly Journal of Economics, 109(1), 111-147. Cerioli, A., & Zani, S. (1989). A Fuzzy Approach To The Measurement of Poverty. Income and Wealth Distribution, Inequality and Poverty (pp. 272-284). Italy: Springer. Chakravarty, S. R. (2006). An Axiomatic Approach to Multidimentional Poverty Measurement via Fuzzy Sets. In A. Lemmi, & G. Betti, FUZZY SET APPROACH TO MULTIDIMENTIONAL POVERTY MEASUREMENT (pp. 49-72). New York: Springer. Chelli, B., & Lemmi, A. (1995). "Totally" Fuzzy and Relative Approach to the Multidimentional Analysis of Poverty. Economic Notes, 115-134. Chopra, R. (2018). Machine Learning. New Delhi: Khanna Book Publishing. Cowan, R. S. (1976, January). "Industrial Revolution" in the Home: Household Technology and Social Change in the 20th. Technology and Culture, 17, 1-23. Deutsch, J., & Silber, J. (2006). The "uzzy St" Approach to Multidimensional Poverty Analysis: Using the Shapley Decomposition to Analyze the Determinants of Poverty in Israel. In A. L. Betti, FUZZY SET APPROACH TO MULTIDIMENSIONAL POVERTY MEASUREMENT (pp. 155-174). New York: Springer. Duflo, E. (2003). Grandmothers and Granddaughters: Old Age Pension and Intrahousehold Allocation in South Africa. World Bank Economic Review, 17(1), 1-25. 8  Haddad, L., & Kanbur, R. (1990). How Serious is the Neglect of Intra-Household Inequality ? The Economic Journal, 866-881. Klasen, S., & Lahoti, R. (2016, March 11). How Serious is the Neglect of Intra-Household Inequality in Multi-dimensional Poverty Indices ? Retrieved March 23, 2019, from SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2742083 Kubat, M. (2017). An Introduction to Machine Learning. Cham: Springer. Lundberg, S. M., & Lee, S.-I. (2017). A Unified Approach to Interpreting Model. Retrieved March 19, 2019, from NIPS Proceedings: https://computer.org/csdl/proceedings-article/icpr/1994/00576879/12OmNx965uE Martinetti, E. C. (2006). Capability Approach and Fuzzy Set Theory: Description, Aggregation and Inference Issues. In A. L. Betti, FUZZY SET APPROACH TO MULTIDIMENSIONAL POVERTY MEASUREMENT (pp. 93-113). New York: Springer. Munshi, K., & Rosenzweig, M. (2016). Networks and Misallocation: Insurance, Migration, and the Rural-Urban Wage Gap. American Economic Review, 106(1), 46-98. Qizilbash, M. (2006). Philosophical Accounts of Vagueness, Fuzzy Poverty Measures and Multidimensionality. In A. Lemmi, & G. Betti, FUZZY SET APPROACH TO MULTIDIMENSIONAL POVERTY MEASUREMENT (pp. 9-28). New York: Springer. Ravallion, M. (2016). THE ECONOMICS OF POVERTY - HISTORY, MEASUREMENT AND POLICY. New Delhi: Oxford University Press. Sen, A. (1976). Poverty: An Ordinal Approach to Measurement . Econometrica, 219-231. Shorrocks, A. F. (2013). Decomposition Procedures for Distributional Analysis: A Unified Framework Based on the Shapley Value. The Journal of Economic Ineuality, 99-126. Srinivasaraghavan, A., & Joseph, V. (2019). Machine Learning . Hoboken: Wiley. Theobald, O. (2017). Machine Learning For Absolute Beginners: A Plain English Introduction (Second Edition). Seattle: Scatterplot Press. Vargha, L., Gál, R. I., & Crosby-Nagy, M. O. (2017, March 23). Household production and consumption over the life cycle: National Time Transfer Accounts in 14 European countries. DEMOGRAPHIC RESEARCH, 36(32), 905-944. Vijaya, R. M., Lahoti, R., & Swaminathan, H. (2014). Moving from the Household to the Individual: Multidimensional Poverty Analysis. World Development, 59, 70-81. Zadeh, L. A. (1965, June). Fuzzy Sets. Information and Control, 8(3), 338-353.   

Decomposition of intra-household disparity sensitive fuzzy multi-dimensional poverty index: A study of vulnerability through Machine Learning

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Decomposition of intra-household
disparity sensitive fuzzy
multi-dimensional poverty index: A
study of vulnerability through Machine
Learning
Sugata Sen
28 April 2019
Online at https://mpra.ub.uni-muenchen.de/93550/
MPRA Paper No. 93550, posted 2 May 2019 13:48 UTC
1 
 
DECOMPOSITION OF INTRA-HOUSEHOLD  
DISPARITY SENSITIVE  
FUZZY MULTI-DIMENSIONAL POVERTY INDEX: 
A STUDY OF VULNERABILITY THROUGH  
MACHINE LEARNING 
 
Dr. Sugata Sen
1
 
1. Introduction 
The traditional multi-dimensional measures have failed to properly project  the 
vulnerability of human-beings towards poverty. Some of the reasons behind this 
inability may be the failure of the existing measures to recognise the graduality 
inside the concept of poverty and the disparities within the household in wealth 
distribution. So this work wants to develop a measure to estimate the 
vulnerability of households in becoming poor in a multidimensional perspective 
through incorporating the intra-household disparities and graduality within the 
causal factors. Dimensional decomposition of the developed vulnerability 
measure is also under the purview of this work. To estimate the vulnerability 
and dimensional influences with the help of artificial intelligence an integrated 
mathematical framework is developed.  
2. Review of literature 
One of the major impedement of the well accepted Multi-dimensional 
Poverty Indices (Alkire & Foster, Counting and Multidimensional Poverty 
Measurement, 2009) (Alkire & Foster, 2011) (Alkire, Kanagaratnam, & Suppa, 
The Global Multidimensional Poverty Index (MPI):2018 Revision, 2018) is that 
they have tried to distinguish the poor from the non-poor through the classical 
Boolean logic. But the idea of poverty suffers from vagueness and naturally 
cannot be defined through a well defined cut-off. So discussing the 
multidimensional poverty through ordinary proposition is not correct 
(Qizilbash, 2006). 
                                                          
1
 Associate Professor of Economics, Panskura Banamali College, INDIA 721152, sensugata@gmail.com, M: +91 
7501730545 
2 
 
The graduality within a vague concept can well be represented through fuzzy 
logic (Zadeh, 1965). Cerioli and Zani first attemted to use the fuzzy logic on the 
measurement of multidimentional poverty (MP) (Cerioli & Zani, 1989). They 
have estimated the strength of poverty in each dimension through a membership 
function. Then the strength of all dimensions are added and normalised to get 
the household level multidimentional poverty.  Their idea is improved 
subsequently through a voluminous research (Chelli & Lemmi, 1995)  
(Martinetti, 2006)  (Betti, Cheli, Lemmi, & Verma, 2006)  (Chakravarty, 2006).  
Another important drawback of the conventional multidimentional poverty 
indices is that all of these measures have accepted that the poverty status of the 
individuals within a household are equall. Thus these indices have accepted the 
household as a homogenous unit and such that have failed to capture the intra-
household differences. But it is an established fact that the different members 
within a family enjoys varied endowment as well as bargaining power 
(Agarwal, 1997) (Duflo, 2003). So computing poverty measures taking 
households as the basic unit leads to improper estimation. The earliest work to 
put importance on the individuals instead of the household was carried out by 
Haddad and Kanbur (Haddad & Kanbur, 1990). Vijaya et. al. (Vijaya, Lahoti, & 
Swaminathan, 2014) and Klasen et. al. (Klasen & Lahoti, 2016) have developed 
individual sensitive measures in the multi-dimensional framework. 
Apart from measuring the composite effect of the multi-dimensional poverty 
a large volume of research appeared on the decomposition of composite index. 
Using the properties of sub-group decomposability Alkire et. al. have 
decomposed Alkire-Foster Adjusted Headcount Ratio (Alkire, Roche, & Vaz, 
2017). Deutch and Silber have tried to decompose the fuzzy multidimentional 
poverty index through Shapley method to find the importantance of the causal 
factors (Deutsch & Silber, 2006). The Shapley Value Decomposition 
determines the average of the marginal contributions of a factor under different 
combinations. To that respect, the concerned factor is first withdrawn from the 
model and the rest of the factors are permuted to form different distributions. 
Gradually, the withdrawn factor is added to each of the combination and the 
marginal contribution of the added factor in a specific distribution is counted. 
The average of marginal contributions of the stated factor from all the 
distributions is the influence of that very factor on the composite variation 
(Shorrocks, 2013). 
3 
 
To decompose the multi-dimensional poverty index machine learning can be 
used. Machine Learning(ML) is a technique of data analytics that instructs 
computer to learn from experience. Machine Learnings algorithms use 
computational methods to “learn” information directly from data without 
depending on a pre-set equation as a model (Kubat, 2017) (Theobald, 2017). 
Shapley Value Machine Learning can successfully implement Shapley 
Decomposition of MP. A framework called SHapley Additive exPlanation 
(SHAP) executes this through Local Interpretable Machine-agnostic 
Explanation (LIME) algorithms (Lundberg & Lee, 2017). 
So to improve the poverty estimation incorporation of fuzzy logic and 
intra-household disparities are needed. Dimensional decomposition  of the 
improved estimate can examine the influence of individual causal factors on the 
vulnerability of the households to become poor. Shapley Value Machine 
Learning can play an important role in this decomposition. Thus the specific 
objectives of this study are the following.  
3. Objectives 
 Measuring the vulnerability to become poor multi-dimentionaly  
incorporating the ideas of graduality and intra-household 
disparities. 
 Development of appropriate machine learning process with the 
help of artificial importance to examine the dimensional influence 
on household level vulnerability. 
 
4. Findings 
Let there are n individual, expressed as i=1,2,..n and k dimensions 
expressed as j=1,2,…k. The performance level of each individual on each 
dimension can be expressed as a n×k real valued non-negative matrix. Each row 
vector yi={   }interprets the performance vector of ith individual. 
 Let, z is a vector of dimensional thresholds when z={zj}. 
Then matrix gi
0
={1,0} can be created where 
gij
0 
=1, when yij<zj  and 
gij
0 
=0, when yij≥zj   
4 
 
gij
0 
is 1 if the ith individual is poor in the jth dimension and otherwise. 
Now let ci=|g ij
0
| and vector c shows the number of dimensions where each 
individual is lying below the established dimensional thresholds. If d is the 
aggregative cut-off to become poor then ith individual will become multi-
dimensionally poor when ci ≥ d where 1 < d < k. 
This identification has failed to consider the vagueness of poverty and the 
existence of intra-household disparities. To rectify this let us assume that each 
of the households consists of q individuals where q is a positive integer.  
Let the grade of membership to the poor set of the qth member of the ith 
household in a specific dimension is expressed through the dimension specific 
individual membership function  
  
 
(i) = 1 if  0 ≤   
 
 ≤        and 
  
 
(i) = 0 if   
 
 ˃      
An individual is definitely poor if his achievement in a particular 
dimension j is from 0 upto    . On the other hand if individual achievement is 
above      then the individual is not poor on dimension j. For individual 
achievement between     and      the membership function thakes on values in 
[0,1]. More clearly it can be interpreted that if 
   
 
(i) = 0 if the ith individual is certainly not poor in the jth 
dimension. 
   
 
(i) = 1 if the ith individual completely belongs to the poor set 
corresponding to jth dimension. 
 0 <   
 
(i) < 1 if the ith individual shows a partial membership to the 
poor set p of jth dimension. 
The strength of membership of all the individuals of a particular household in a 
particular dimension can be added and deflated by the number of household 
members to get the collective strength of household membership in a particular 
dimension. Thus the individual sensitive grade of membership of ith household 
in jth dimension can be represented as 
       =
∑   
 
 
 
The grade of membership of the ith individual to the multi-dimensional poor set 
M can be defined as 
5 
 
      =  
∑       
 
   
 
 
Where 0 ≤       ≤ 1 and       is the vulnerability of ith household to become 
multi-dimensionally poor. If weight    is assigned to the jth dimension then it 
can be written that 
      =  
∑       
 
   
∑    
 
   
 
This work is interested to examine the dimensional influences on the household 
level vulnerability through necessary decomposition. Let the household level 
vulnerability is   . It is quite natural that the desired value of    is 0. Thus, the 
difference between desired and observed vulnerability is   . To decompose    
Shapley value decomposition has been used. This method calculates the average 
of marginal contributions of each dimension to the level of vulnerability. To 
find the contribution of j
th
 dimension we would find different combination of K-
1 dimensions. So, the total no of combinations without the jth dimension is – 
(K-1)
C1 + 
(K-1)
C2 + 
(K-1)
C3 + 
(K-1)
C4 + …. + 
(K-1)
CK-1  
= ∑        
   
    
= θ 
Subsequently, the j
th 
dimension is added to each of θ combinations to find 
the marginal effect of j
th
 dimension from that particular combination. Naturally, 
we would get θ marginal contributions. According to Shapley decomposition, 
the contribution of j
th
 dimension is the average of all the marginal contributions 
of jth dimension. Let, the marginal contribution of j
th
 dimension from s
th
 
combination is  s. So, the set of marginal contributions of the j
th
 dimension is- 
Ηj = ( 1
j
,  2
j, …,   
 
 ) 
Then average of marginal contribution of the jth dimension is  
   j = 
 
  
 ∑       s
j
 
As we have    contributions under each dimension then it can be said that 
these marginal contributions have been generated through    functions. The 
polinomial form of sth combination under the jth dimension can be chosen from 
the set of ψ alternative polinomials    . To that respect the machine learning 
6 
 
process will find the observed value of the composite dependent variable 
through a particular polinomial from set     and compare that observed value 
with the expected value to find the error. Learning from the successive errors 
within     the machine learning process will choose that polinomial from      
which will minimise the error. In this way    functional forms can be 
determined under jth dimension. From these θ functions we can find the    
incremental influences. The average of the    incremental influences will 
deliver    j.  
This estimation of dimensional contributions can be executed through 
Local Interpretable Machine-agnostic Explanation (LIME) algorithms. LIME 
deliberately perturbs a combination by accepting input variables from the 
neighbourhood and counts the effect of that perturbation on the output. Finally 
the relevance of the particular input is determined through the average of 
deviation in the output due to the perturbations. Following this logic in our 
decomposition model LIME will first find    functions from the θψ alternatives 
through perturbation. After that the algorithm proceeds to estimate the 
dimensional contribution. Technically LIME initiates the process to locate 
min      = min       –             →     
  →   
 
 
where u≠ v and u,v=1,2….. .    is the neighbourhood of sth functional form 
under jth dimension.       is the expected value and       is the observed value 
of the multi-dimensional poverty from a particular polinomial related to    .    
  
is the chosen polynomial from    . This process will continue for all the 
combinations under    to find    incremental influences. Finally the average of 
the    incremental incluences determines  
   j = 
 
  
 ∑       s
j
 
5. Conclusion 
The mathematical framework developed under the current work will deliver a 
better estimate of vulnerability as well as dimensional influences to household 
level multi-dimensional poverty through accommodating fuzzy logic and intra-
household disparities. The use of artificial importance as demonstrated within 
the model will make the model easy to execute. This estimation of vulnerability 
and decomposition will help the planners to locate the role of different 
dimensions behind the vulnerability of human beings to become poor more 
precisely.  
 
7 
 
Works Cited 
 
Agarwal, B. (1997). Bargaining and Gender Relations: Within and Beyond the Household. Feminist 
Economics, 3(1), 1-51. 
Alkire, a., & Foster, J. (2011). Counting and multidimensional poverty measurement. Journal of Public 
Economics, 476-487. 
Alkire, S., & Foster, J. (2009). Counting and Multidimensional Poverty Measurement. Oxford: OPHI. 
Alkire, S., Kanagaratnam, U., & Suppa, N. (2018). The Global Multidimensional Poverty Index 
(MPI):2018 Revision. Oxford: OPHI. 
Alkire, S., Roche, J. M., & Vaz, A. (2017). Changes Over Tme in Multidimensional Poverty: 
Methodology and Results for 34 Countries. World Development, 232-249. 
Betti, G., Cheli, B., Lemmi, A., & Verma, V. (2006). Multidimentional and Longitudinal Poverty: an 
Integrated Fuzzy Approach. In A. L. Betti, FUZZY SET APPROACH TO MULTIDIMENTIONAL 
POVERTY MEASUREMENT (pp. 114-137). New York: Springer. 
Bourguignon, F., & Chakravarty, S. R. (2003). The Measurement of Multidimensional Poverty. The 
Journal of Economic Inequality, 25-49. 
Carroll, C. D. (1994, February 1). How does Future Income Affect Current Consumption? The 
Quarterly Journal of Economics, 109(1), 111-147. 
Cerioli, A., & Zani, S. (1989). A Fuzzy Approach To The Measurement of Poverty. Income and Wealth 
Distribution, Inequality and Poverty (pp. 272-284). Italy: Springer. 
Chakravarty, S. R. (2006). An Axiomatic Approach to Multidimentional Poverty Measurement via 
Fuzzy Sets. In A. Lemmi, & G. Betti, FUZZY SET APPROACH TO MULTIDIMENTIONAL POVERTY 
MEASUREMENT (pp. 49-72). New York: Springer. 
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Decomposition of intra-household disparity sensitive fuzzy multi-dimensional poverty index: A study of vulnerability through Machine Learning

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