Detailing Poverty Incidence through Fractals: Which of the Gross National Product or Multidimensional Poverty Index Explain Poverty Incidence Better?

This paper attempts to explain poverty incidence of the 97 countries using fractal analysis. Gross National Product (GNP) and Multidimensional Poverty Index (MPI) of each country were used as poverty indicators. Fractal dimensions were obtained, compared and analyzed. The three variables have fractal characteristics of ruggedness and self-similarity. Results revealed that the ruggedness of poverty incidence across the countries is due to the ruggedness of the MPI, that is, the deprivation to basic services such as health, education and standard of living affects the quality of living. Thus, MPIs explain poverty incidence more precisely. With this finding, implications to policymakers to alleviate poverty can be addressed.


Introduction
Poverty is a global issue. What causes poverty and solution to alleviate deprivations remain a huge concern not only in the Philippines but all over the world. About half of the world (over three billion) people live less than $2.50 a day (Shah, 2013). Poverty and its consequences are of central importance to public policy makers. However, the cause of poverty remains an elusive problem.
Attempts have been made to explain the complexities of poverty through different mathematical models. Chattopadhyayet (2006) made a theoretical analysis on the changes in poverty with respect to the 'global' mean and variance of the income distribution using Indian survey data. When income obeys a log-normal distribution, a rising mean income indicates a reduction in poverty while an increase in the variance of the income distribution increases poverty. This altruistic view for a developing economy, however, is acceptable once the poverty index is found to follow a pareto distribution. Here although a rising mean income indicates reduction in poverty due to the presence of an inflexion point in the poverty function, there is a critical value of the variance below which poverty decreases with increasing variance; while beyond this value, poverty undergoes a steep increase followed by a decrease with respect to higher variance. Hence, the pareto poverty function satisfies all three standard axioms of a poverty index with inflexion point as the poverty line.
While Coromaldi and Zoli (2012) derived indicators of multiple derivation by applying a particular multivariate statistical technique, the nonlinear component analysis overcomes traditional limit of many of the used methodologies for poverty measurement. Second, on the basis of the aforementioned indicators, they provide an accurate identification of the poor in Italy by analyzing both as a distinct phenomenon of poverty in different life domains and as a single multidimensional concept. The main determinant of poverty in Italy is investigated by estimating logit regressions and an ordered probit model. Vijayakumarf and Olga (2012) found and analyzed the significant determinant of the incidence of poverty in the estate sector of Sri Lanka where the highest level of chronic poverty and unemployment exist. The Ordinary Least Squares (OLS) regression analysis indicates that variables such as industrial employment, education, access to market and infrastructure significantly and negatively affect the poverty incidence of the estate sector. Agricultural employment has a negative impact but not significant. Analysis with the Durbin-Watson stat confirms that there is no autocorrelation between the variables Tzavidis and Salvati (2007) used Mquantile models in deriving small area estimates of poverty and inequality. Unlike traditional random effect models, M-quantile models do not depend on strong distributional assumptions and automatically provide outlier robust inference.
In the Philippines, Huelgas (2011) used the classical regression model and the spatial lag model in estimating city and municipal poverty incidence. These models provided estimates for cities and municipalities with no direct estimates as well as to present estimates with improved precision for cities and municipalities with unreliable direct estimates. The study used five factors namely, the employment of the household head, the education of the household male members, age structure of the household members, the materials of which the house is made of and a proxy measure of the community's progress. In classical regression model, results show the importance of education to the members of the household particularly that of male members, and the development and infrastructure of the community where the household resides. Moreover, educational attainment of the household members has the highest effect on poverty. Also, the identity link performs better than the logit or probit link in terms of a lower mean absolute percentage error. In the spatial regression model, the 25-km distance threshold provides the optimum model with relatively lower mean absolute percentage error and root mean square error compared to other threshold distance matrices. Lastly, the spatial lag model using distance matrix was found to be superior than classical regression model by virtue of lower mean absolute percentage error, lower root mean square error and estimates with lower standard errors and coefficients of variation.
In 2010, the Oxford Poverty and Human Development Initiative (OPHI) of Oxford University and the Human Development Report Office of the United Nations Development Programme (UNDP) launched a new poverty measure that gives a multidimensional picture of people living in poverty, the Multidimensional Poverty Index (MPI). MPI identifies deprivations across health, education and living standards and shows the number of people who are multidimensionally poor and the deprivations that they face at the household level. It uses 10 indicators across dimensions. Figure 1 shows the indicators.
A natural phenomenon that demonstrates a repeating pattern is called fractal. Poverty incidence in a large scale exhibits this reiterating form. In this paper, the authors used fractal analysis to determine the root cause of incidence of poverty more precisely. Particularly, this method will assess the fractal characteristics of poverty incidence data. Results of this paper will hopefully serve as basis for policymakers

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to formulate policies to address poverty incidence.

Conceptual Framework
Poverty is a phenomenon that has caught attention of the scientists in the different fields. The complexity of the phenomenon demands a corresponding nonlinear complex and dynamical approach embodied in emerging discipline of fractals.
The incidence of poverty across geographic boundaries can be described as a rugged landscape of poor living among the rich. High poverty incidence is noted when the disparities between income of rich and poor are high. One might therefore surmise that the poverty incidence is closely related to the phenomenon of the distribution of wealth among nations. But some theorists claim that wealth alone cannot explain the rugged environments which host the poor among the rich. This study explores the other factor that could explain the observed irregular and rugged behavior of poverty across the globe.

Objectives
The paper identified the indicators of poverty incidence using fractal analysis. Particularly, this method assessed the fractal characteristics of poverty incidence data.

Basic Concepts
Fractal Statistics is concerned with data irregularities repeated at different scales generalizing the concept of variances. When the variances are too large such that the coefficient of variation (CV) is greater than 1, and if there are more lower values than higher values, then the data are fractals.

Self-Similarity at Various Scales
, k is any real number. . (1) The only self-similar function in one variable is the monomial function , k is any real number. . (2) From the class of self-similar functions, a subclass that gives larger weights to the lower values is obtained, that is, the function where , . ( Given f (x) in (3), we want to convert this into a probability distribution such that . (4) Using (4), , solve for a. Then , >1 (5)

Quantitative Model for Fractal Statistics
A random variable x is said to behave in a fractal distribution if it obeys a power-law: , and where is called the fractal dimension of the distribution.
The model in (6) has two parameters, θ and , which are both unknown. To To estimate the value of θ , we get the minimum of the data, that is, To estimate the value of ë, we use the Maximum Likelihood Estimator (MLE).

Maximum Likelihood Estimator of λ
Given the observations, from the fractal distribution f (x) in (6), the likelihood function L is obtained.
Taking the logarithm of L, The MLE of is the value that maximizes the (8) or (9) by taking its derivative with respect to and equate to zero: Solving for the estimate of λ
Weights denoted by are assigned to ( ) , The ℎ percentile of the distribution obeys the rule Solving for , (13) gives Letting , = 1 , then S is called the scale of fractal spectrum. The fractal spectrum is the visual and graphical representation of the charges in fractal dimensions as function of scales S.

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Poverty Incidence (in percentage) with $1.25/ day and $2.00/day income, respectively. At this point, if we observe a CV <1, then the amount of fluctuations around the mean is bounded, otherwise, unbounded. From this observation, determine which of the variables (GNP and MPI) has likely influenced the PI of these countries considering the computed fractal dimensions is closest to the computed fractal dimension of the PI. Fractal Spectra of the three variables were then obtained by plotting their respective fractal dimensions against their scales and were analyzed. Correlations among variables were also computed based on the computed fractal dimensions.

Results and Discussions
Fractal Characteristics of the Variable Table 2 shows the results obtained from the different measures of the variables MPI, PI and GNP. To determine whether the variables MPI, PI and GNP behave as fractals or not, coefficient of variations and fractal dimensions were considered. It is observed that the three variables have fractal characteristics. Further, it is observed that the fractal dimension of the MPI is closer to the fractal dimension of the PI which implies that this variable defines poverty incidence better. Looking at their respective CVs it is noted the amount of fluctuations around the mean of the MPI and GNP are unbounded and thus explains the large variations of data except for the PI, which is bounded. Thus, fractal characteristics are evident in these variables.
Looking at the given data, the irregularities or ruggedness of income distribution, in terms of GNP of the different countries was observed. There are more countries having low GNP than high GNP. These low-income countries likewise have greater poverty incidence compared to high-income countries.
Among those with low GNP, there are more having even lower income than higher income. The same is true for the high-income countries, there are more having lower income than higher income. Also, fluctuations of poverty incidence in poor countries have the same fluctuations of poverty incidence in not poor countries. Thus, this exhibits a selfsimilarity property.
Also, majority of the poor countries having high MPI have high poverty incidence, and some countries with low MPI have low poverty incidence. Huelgas (2011) and Vijayakumarf and Olga (2012) both revealed that employment and education were some factors that significantly affect the poverty incidence. These factors were also integrated in the MPI. Hence, MPI could explain the rugged and irregular behavior of poverty across the countries.  Table 3 presents the fractal dimension of the PI at the áth percentile while Figure 2 shows the spectrum of the PI. We notice that there are three (3) scales at different ranges. These scales will tell us where a certain country belongs.

Fractal Spectra of the Variables
Fractal spectra of the three variables were then plotted by their respective fractal dimensions against their scales and were analyzed. These are on the poverty incidence, MPI and GNP. The lowest scale (scale 3) for which the fractal dimension is greater than 1.19 is composed of countries which have lesser poverty incidence but greater variability. These are mostly developed countries. The second scale for which fractal dimension is between 1.09 and 1.19 refers to the developing countries including the Philippines. The highest scale (scale 1), having the fractal dimension less than 1.09 is the poorest countries called the least developed countries. This is the group of countries that have been classified by the UN as "least developed" in terms of their low gross national income (GNI), their weak human assets and their high degree of economic vulnerability. Most of these are African countries. Appendix A shows the list of countries belonging to the three different scales under Poverty Incidence.

On Multidimensional Poverty Index
Fractal dimensions are again computed and graphed, as presented in Table 5 and Figure 3, respectively.

Detailing Poverty Incidence through Fractals
As shown in Figure 3, four (4) different scales of MPI fractal spectrum are obtained. The lowest scale (scale 1) with fractal dimension greater than 1.17 is the least developed countries which include African countries. These are the most deprived countries in terms of basic services as indicated by their higher MPIs. The highest scale (scale 4) with fractal dimension lower than 1.14 is the countries with lower MPIs. Some of these are developed countries. Scale 3 is composed of group of developing countries (not so poor) while scale 2 is a group of developing countries including few least developed countries. The jump observed between scales 2 and 3 is due to countries which belong to the poorest countries but with not so high MPI. This implies that these countries may be poor in terms of its income but may have more access to some basic services than those of some developing countries. Appendix 2 shows the summary of countries at different scales of MPI. Table 6 indicates the fractal dimension of the countries in terms of their respective GNP. As observed in Figure 4, the fractal spectrum  of GNP has four (4) scales. Countries with very low gross national income are countries belonging to scale 4. Majority are African countries. The Philippines belongs to Scale 2, which is a group of average -income or developing countries. Developed countries or high-income countries belong to scale 1. Appendix 3 gives the list of these countries. Figure 5 shows clearly the variable that defines poverty incidence better. The spectrum of MPI fits more likely to the spectrum of PI than the spectrum of GNP. This indicates that the ruggedness of poverty incidence across the countries is due to the ruggedness of the MPI. Deprivation of the countries to basic services such as health, education and standard of living affects the quality of living.

On Gross National Product
Furthermore, the percentage of poor countries under PI spectrum is compared to the poor countries in MPI and GNP spectrum. Appendix 4 gives the list of poor countries in each variable. Forty-four percent (44%) of the 79 countries are poor in terms of MPI, 35% are poor in terms of poverty incidence and 17% are poor in terms of GNP. Seventeen percent (17%) of these have low-income and were deprived of basic services.
Moreover, 14% of the countries have high poverty incidence at the same time have low income. Thirty four percent (34%) have high poverty incidence at the same time most deprived in terms of basic services. All poor countries in terms of GNP are also poor in terms of MPI, however, 27% are poor in terms of MPI but not in GNP. This implies that wealth alone cannot explain the rugged behavior of poverty. The ruggedness of MPI could be closely related to the ruggedness in poverty incidence.

Correlation Analysis
Correlations among variables are also computed based on the computed fractal dimensions. This is to determine which of the two variables, the GNP or the MPI, explains poverty more precisely. Results show the regression equation between the PI and MPI, and PI and GNP, respectively as follows:  Tables 7 and 8 further show the significant difference between the fractional dimensions of PI and MPI, versus PI GNP. Results confirmed that MPI is closely related to poverty incidence, but not GNP. In fact, 42.8% of the variation in poverty incidence can be explained by the variation in MPI. However, only 0.2% can be explained by GNP. Hence, MPI explains poverty incidence more precisely. This is consistent with our analysis using spectrum and fractal dimensions.

Conclusion
Based on the results and discussions, poverty incidence of the 97 countries can be explained better by MPI. MPI looks at the deprivation of basic services at the household level while GNP focuses on the income of every citizen of a country. In other words, GNP is directly proportional to the quality of living while MPI is inversely proportional to the standard of living. Thus, the policymakers should focus on programs and projects for poverty alleviation particularly on the basic services.