Robust geographically weighted regression with least absolute deviation method in case of poverty in Java Island

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1 Robust geographically weighted regression with least absolute deviation method in case of poverty in Java Island Rawyanil Afifah, Yudhie Andriyana, and I. G. N. Mindra Jaya Citation: AIP Conference Proceedings 1827, (2017); View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Implementations of geographically weighted lasso in spatial data with multicollinearity (Case study: Poverty modeling of Java Island) AIP Conference Proceedings 1827, (2017); / Geographically weighted poisson regression semiparametric on modeling of the number of tuberculosis cases (Case study: Bandung city) AIP Conference Proceedings 1827, (2017); / Development of generalized space time autoregressive integrated with ARCH error (GSTARI ARCH) model based on consumer price index phenomenon at several cities in North Sumatera province AIP Conference Proceedings 1827, (2017); / Modeling the human development index and the percentage of poor people using quantile smoothing splines AIP Conference Proceedings 1827, (2017); / Modeling relationship between mean years of schooling and household expenditure at Central Sulawesi using constrained B-splines (COBS) in quantile regression AIP Conference Proceedings 1827, (2017); / Prediction of cadmium pollutant with ordinary point kriging method using Gstat-R AIP Conference Proceedings 1827, (2017); /

2 Robust Geographically Weighted Regression with Least Absolute Deviation Method in Case of Poverty in Java Island Rawyanil Afifah a), Yudhie Andriyana b) and I G N Mindra Jaya c) Department of Statistics, Universitas Padjajaran, Bandung Corresponding author: a) rawy.afifah@gmail.com b) y.andriyana@unpad.ac.id c) jay.komang@gmail.com Abstract. Geographically Weighted Regression (GWR) is a development of an Ordinary Least Squares (OLS) regression which is quite effective in estimating spatial non-stationary data. On the GWR models, regression parameters are generated locally, each observation has a unique regression coefficient. Parameter estimation process in GWR uses Weighted Least Squares (WLS). But when there are outliers in the data, the parameter estimation process with WLS produces estimators which are not efficient. Hence, this study uses a robust method called Least Absolute Deviation (LAD), to estimate the parameters of GWR model in the case of poverty in Java Island. This study concludes that GWR model with LAD method has a better performance. INTRODUCTION Poverty is a multidimensional problem that is faced by most countries, including Indonesia. Poverty is defined as an individual situation of being unable to meet the minimum levels of income, food, clothing, healthcare, shelter, and other essentials [5]. In Indonesia, the poverty rate is published by Statistics Indonesia (Badan Pusat Statistik (BPS)-the official statistics of Indonesia). BPS used the basic need approach to determine the poverty. Poverty is seen as an economic inability to meet the basic needs of food and non-food which is measured from the expenditure side [2]. The data of poverty rate is distributed spatially. According to Anselin [1], spatial data has two effects, spatial dependency and spatial heterogeneity. Spatial dependency described as the observations at one location are depend on the observation at other locations. Observations on adjacent location will tend to have the same characteristics, and will increasingly be different along the distance of the observations. Spatial heterogeneity can be shown by the different influence of explanatory variables on the response variable at each location.the existence of spatial heterogeneity will lead to the homoscedasticity assumption at the classical regression model is not fulfilled. The variance of the model is no longer constant, but different at each observation. Therefore, the regression model was developed to allow the variances of the model to be different for each location by making the local regression coefficients, it means that each location will have its own regression coefficients [3]. A method that accommodates local regression coefficients is Geographically Weighted Regression (GWR). GWR uses point approach where each parameter of the regression model is estimated at any point in the geographic location. The estimation of the regression coefficients uses Weighted Least Squares (WLS) method. WLS procedure estimates the parameter by minimizing the sum squared of errors. As we knew, this procedure is sensitive to the existence of outliers, hence, a robust method to the existence of outliers is needed. One of the methods that can be used in the existence of outliers is Least Absolute Deviation (LAD). LAD is able to overcome the effect that caused by the outlier without detecting the observation that influences the outlier. Statistics and its Applications AIP Conf. Proc. 1827, ; doi: / Published by AIP Publishing /$

3 BASIC CONCEPT Linear Regression Analysis Regression analysis is a model to describe the relationship between response and explanatory variables. Mathematically, linear regression model is given by: y i 0 K k1 x k Where y i is the response variable of i th observation ; 0, 1,..., K are regression coefficient to be estimated or can also be called parameters; x i1, xi2,..., xik are the explanatory variables; i i 1,2,..., nare independent random errors 2 with mean zero and variance ; n is number of observations, and K is number of explanatory variables. The model can be expressed as matrix: ik i where,,, and Estimating the regression coefficient is done by using Ordinary Least Squares (OLS) procedure by minimizing: n n K 2 2 i y i 0 j xik i1 i1 k1 Then we get the OLS estimator of is: T 1 ˆ T X X X Y Where ˆ is ank 11estimated regression coefficients vector; X is nk 1 a matrix of explanatory variables which are 1 at the first column, andy is n1a vector of the response variable. In the next discussion in this paper, this linear regression model will be called global model. Testing the Spatial Heterogeneity Estimating the global model using OLS is depend on some assumptions that have to be fulfilled. The assumptions are normality, nonautocorrelation, homoscedasticity, and multicollinearity. All of this should be fulfilled to get the Best Linear Unbiased Estimator (BLUE). If one of this assumption is not fulfilled, the estimator produced will be no longer BLUE. In spatial data, the assumption of nonautocorrelation and homoscedasticity are almost cannot be fulfilled because of the spatial effect on the spatial data. The dependency effect on the spatial data cannot fulfill the assumption of nonautocorrelation because the observations, in this context are location, are depend on each others. The other effect, heterogeneity, cannot fulfill the assumption of homoscedasticity because of variances of the model are no longer homogeneous.variances of the model will be different depend on the location. To test the spatial heterogeneity, we can use Breusch-Pagan (BP) Test [1]. The hypotheses are expressed as follows: at least one i where

4 and the statistics test is formulated as below: ~ where, and, With Z is a matrix that contains the vectors at each observation. Hypothesis null will be rejected if, where K is number of variables Detecting Outliers Least Squares is the most common method used to estimate parameters of the regression model. This method talks about parameters that can produce sum squared of error as small as possible. It is a problem when the are outliers in the data,which may produce large errors. This large errors of outliers will be doubled because they are squared in finding the estimator. Therefore, it is important to detect whether outliers are existed in the data we used. Boxplot is one of the common method that is used to detect outliers. Boxplot uses the value of Interquartile Range (IQR) to detect outliers. IQR defines as the difference of 1st quartile and 3rd quartile. Outliers are observations that have condition: a. The value less than, or b. The value more than In Exploratory spatial data analysis, [15] transform boxplot into a map that is called boxmap. Boxmap visualize the outliers spatially. Geographically Weighted Regression (GWR) There are many ways to overcome the existence of the heterogeneity effect that caused the homoscedasticity assumption unfulfilled. But, in the case of poverty, we need models that can describe it locally, for each location because overcome the poverty problem locally is easier than globally. Therefore, in this case, we use models which can estimate parameters locally. Fotheringham [3] introduced a model that can estimate the regression parameters locally. This model is called Geographically Weighted Regression (GWR). The model of GWR can be written as follows: y i K ui vi k ui vi xik i, (1) 0, k1 u i, v i is the spatial coordinate of the ith location; x i1, xi2,..., xik yi is the observation value of response variable; are observations of explanatory variables X,...,X at u 1 K i, v i ; k u i, vi k 1,2,...,K coefficients to be estimated and i 1,2 n are unknown regression 2 i,..., are independent random errors with mean zero and variance. The model is estimated by using Weighted Least Squares (WLS) method. The procedure is to minimize the equation: where is the weigth of location j to estimate the parameters in location i.the equation can be written as matrix form below: The regression coefficient will be estimated using this equation: This weighting matrix describes the influence of the neighbors observation through their distances to the ultimate location. The weight will increase as the decreasing of the distances. The matrix of weighting matrix is:

5 The values of the weighting matrix can be computed by using several methods. Kernel function is the most common procedure. The kernel is a function that describes the density of the distances among all locations.there are several Kernel function which is commonly used. One of them is Gaussian Kernel. The formula to compute the weight using Gaussian Kernel is as follow: where is the euclidean distance between location i and location j. h is a bandwidth which is non-negative. Bandwidth is important in GWR to choose the neighborhood that will include in estimation parameter at one location. Bandwidth control the fit of the curve to data and the smoothness of the data. The bigger the bandwidth, the model produced will be smoother. But if the model is too smooth, there will be no differentiation between the model produced and the global model. Therefore, choosing the optimum bandwidth is very important to obtain the best model. One of the criteria that can be used to choose the optimum bandwidth is Cross-Validation which is formulated as follow: (2) is the estimated value of when the ith location ommited from the model. The bandwidth is optimum when the CV score is minimum [3]. Geographically Weighted Regression Using Least Absolute Deviation (Robust GWR) Estimating model using Least Squares procedure is sensitive to the presence of outliers. The outliers can produce a great residual that can make the variance also greater so that the confidence interval will be wider. The impact is that the estimated coefficients of regression will be no longer consistent. Hence,we need an alternative method to estimate the parameters. A more robust method, that can be less influenced onthe final estimates by the outliers. One of the more robust method than least squares is Least Absolute Deviation (LAD). LAD is well suited to longer-tailed error distribution, like Laplace or Cauchy [6]. The concept of this method is no more difficult than the concept of least squares. The concept of LAD just replaces the term of the least squares to. But, it is quite complicated when we use it in the actual calculation. The concept of LAD parameter estimation is to minimize the equation: (3) If the least square has the formula to estimate parameters by differencing the loss function, the LAD method does not have any formula to estimate them because the eq.(3) is not differentiable. So that, the calculation of the LAD method to estimate parameters is solved by using an algorithm. Charnes, Cooper, and Ferguson in 1955 showed the equivalence between LAD and linear programming problem [6]. In 1959, Wagner suggested that the LAD problem can be solved by solving the dual of LAD problem that can be reduced to a problem with a smaller basis. In the context of GWR, the parameters from the eq.(1) at certain location ( ) and shows the weight at that location, can be estimated using LAD method by adopting the eq.(3) as follows: Minimize subject to Let and the decomposition of the residual is

6 Then dan The linear programming solving the LAD problem as follows: Minimize Subject to Technically, the estimation of RGWR will use the rq function at quantreg package of R-language programming. The Estimation at location is by inputing the matrix as the covariate and as the weighting vector. The process repeated at different location, where. In choosing the bandwidth that will be used to estimate the model, it is necessary to use criteria that also robust to the presence of outliers. The CV score like being introduced in eq.(2) is sensitive to outliers. The CV errors of outliers can dominate the whole CV score. In other words, the bandwidth chosen will no longer have an essential effect because the CV score will almost same at every bandwidth given. So that, Zhang [4] use Absolute Value Cross Validation (ACV) criteria that is more robust and can minimize the large errors of the outliers. The ACV score is defined as: (4) Similar to the CV score, the bandwidth chosen is the bandwidth which has the smallest ACV score. METHODOLOGY This study uses the data from Survei Sosial Ekonomi Nasional (SUSENAS), Survei Angkatan Kerja Nasional (SAKERNAS), and Potensi Desa (PODES) in 2015 that was held by BPS. The coverage area used is 119 regions/cities in Java Island. The variables used in this study are: 1. The percentage of the poor 2. The percentage of the head of households educated less that elementary school 3. The percentage of people who suffered health complaints during the past month 4. The percentage of households by the use of protected drinking water 5. Underemployment rate 6. The percentage of per capita consumption for food 7. The Percentage of people employed in informal sectors 8. The percentage of villages in the area of land The steps taken for modeling the poverty rate using robust GWR are described as follows: 1. Prepare the data. The data consists of poverty rate of study area as response variable and the explanatory variables are social and economic variables which are indicated to be the cause of poverty. 2. Test the spatial heterogeneity using BP Test. 3. Detect the existence of outliers. 4. If there are outliers and variances of the model are not constant across the study area, use Robust GWR to establish the poverty modelling. The steps are: a. Calculate the distance matrix using the data. b. Find the optimum bandwidth using ACV criteria. c. Construct the weighting matrix for the ultimate location using the optimum bandwidth. d. Estimate parameters on the model by using linear programming algorithm

RESULT The Java Island consists of six provinces that is divided into 119 regions/cities. It is 129.738 km 2 with density 1121 people per km 2.

7 RESULT The Java Island consists of six provinces that is divided into 119 regions/cities. It is km 2 with density 1121 people per km 2. The highest percentage of poverty in Java is in Sampang Region with 25,69% and the lowest is in Tangerang Selatan City with 1,69%. Figure 1 shows the distribution of percentage of poverty in Java Island. FIGURE 1. Distribution of Percentage of Poverty in Java Island, 2015 The summary of the data can be seen as Table 1 below. TABLE 1. Summary of the Data Variables Min Max Mean Median Std. Dev 1,69 25,69 11,32 11,27 4,90 15,02 85,69 55,48 61,68 18,11 18,05 47,62 33,36 33,24 5,98 38,47 100,00 74,14 74,34 13,06 9,34 68,80 28,42 27,13 11,83 31,07 60,95 47,61 48,68 6,94 16,50 86,12 56,33 60,74 18,34 0,00 100,00 82,58 87,67 19,84 The resulting global regression model is as follows : Testing the spatial heterogeneity using Breucsh-Pagan test conclude that the global model has non-constant variance. It is indicated by statistic as 14,267 with p-value 0,046. Therefore the GWR model can be used to describe the relationship between the percentage of poverty and it explanatory variables. In another hand, detection of outliers using boxmap shows that there are outliers of the global model residual. The outliers are in Tasikmalaya City, Purbalingga Region, Bondowoso Region, Kulon Progo Region, Bantul Region, Gunung Kidul Region, and Sampang Region. The boxmap shows by Figure

FIGURE 2. The Boxmap of Global Regression Model Using the GWmodel package on R 3.

GWR Model Parameter Estimation Summary Parameter Minimum Maximum Mean Median Standard Deviation -0.49 0.30 0.01 0.08 0.21-0.43 0.97 0.18 0.23 0.39-0.20 0.34 0.03-0.02 0.16-0.15 0.18 0.01-0.01 0.10-0.

8 FIGURE 2. The Boxmap of Global Regression Model Using the GWmodel package on R software and fixed gaussian kernel function, the optimum bandwidth obtained for GWR model is 83,39 km with CV score 41,87. The parameter estimation by GWR as follows: TABLE 2. GWR Model Parameter Estimation Summary Parameter Minimum Maximum Mean Median Standard Deviation Like GWR model, the RGWR model also obtained using and fixed gaussian kernel function. Instead of using CV score as the criteria for choosing the bandwidth, RGWR model uses ACV score. The optimum bandwidth obtained is 33,10 km with ACV score 94,21. The parameter estimation by RGWR as follows: TABLE 3. RGWR Model Parameter Estimation Summary Parameter Minimum Maximum Mean Median Standard Deviation -1,26 1,52-0,05-0,06 0,52-2,17 3,83 0,33 0,27 1,05-1,52 1,18 0,04 0,00 0,41-1,56 1,27-0,02 0,00 0,49-2,00 3,20 0,10-0,08 0,75-2,43 1,66-0,03 0,08 0,73-1,28 3,01 0,44 0,52 0,86-3,86 2,42 0,15 0,14 0,

9 The mapping of the percentage of poverty by using GWR and RGWR model to compare visually are shown by Figure 3 below. (a). Percentage of Poverty (b). Estimation of Percentage of Poverty Using RGWR (c). Estimation of Percentage of Poverty Using GWR FIGURE 3. Distribution of Percentage of poverty and Its Estimation Using RGWR and GWR model Based on Figure 3 can be shown that the estimation of percentage of poverty using RGWR model are more similar with actual values than using GWR model. According to the Mean Square Error (MSE) value, RGWR model obtained as 0,29. This value is relatively lower than the value of MSE on the model GWR 3,76. This indicates that the RGWR model is better than GWR to describe the percentage of poverty in Java Island in CONCLUSION From this study, we can conclude that the percentage of poverty in Java Island 2015 is spatially distributed and has spatial heterogeneity effect. The RGWR model can produces the estimate values closer to the actual values than the GWR model. This indicates that the use of robust technique for spatial data containing outliers have a better performance

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