Songklanakarin Journal of Science and Technology SJST R1 Sifriyani

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1 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Development Of Nonparametric Geographically Weighted Regression Using Truncated Spline Approach Journal: Songklanakarin Journal of Science and Technology Manuscript ID SJST--00.R Manuscript Type: Original Article Date Submitted by the Author: -May- Complete List of Authors: Sifriyani, Sifriyani; Universitas Gadjah Mada, Mathematics; Universitas Mulawarman, Mathematics Budiantara, I Nyoman; Institut Teknologi Sepuluh Nopember, Statistika Haryatmi, Sri; Universitas Gadjah Mada, Mathematics Gunardi, Gunardi; Universitas Gadjah Mada, Mathematics Keyword: Nonparametric Geographically Weighted Regression, Truncated Spline, Spatial Data, Unbiased Estimation

2 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Development Of Nonparametric Geographically Weighted Regression Using Truncated Spline Approach Sifriyani,,*, Budiantara, Haryatmi, Gunardi Departement of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta Indonesia. Departement of Statistics, Faculty of Mathematics and Natural Sciences, Institut Tekhnologi Sepuluh Nopember, Surabaya Indonesia. Abstract Departement of Mathematics, Faculty of Mathematics and Natural Sciences, Mulawarman University, Samarinda Indonesia. *) Corresponding Author: sifriyani@mail.ugm.ac.id Telp: + Nonparametric geographically weighted regression with truncated spline approach is a new method of statistical science. it is used to solve the problems of regression analysis of spatial data if the regression curve is unknown. This method is the development of nonparametric regression with truncated spline function approach to the analysis of spatial data. Spline truncated approach can be a solution for solving the modeling problem of spatial data analysis if the data pattern between the response and the predictor variables is unknown or regression curve is not known. This study focused on finding the estimators of the model Nonparametric Geographically Weighted Regression by Maximum Likelihood Estimator (MLE) and then these estimators are investigated the unbiased property. The results showed Nonparametric geographically weighted regression with truncated spline approach can be used in spatial data to solve problems regression curve that can not be identified. Keywords: Nonparametric Geographically Weighted Regression, Truncated Spline, Spatial Data, Unbiased Estimation.

3 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of Introduction The development in science and technology has been rapid over the years, and the observation of the signs in nature led to unsual patterns, therefore it is difficult to predict the behaviour of nature. In the past decade, the start and end of dry and rainy season in various geographical areas could easily be predicted, hence farmers were able to prepare when to harvest and plant rice, however in the recent years it has become difficult. Examples of events that shape the unsual and irregular patterns of nature are the issue of the percentage of poverty, underdevelopment, literacy rates, ignorance that is growing and uneven development in each area along with other variables. For more than a century, geographers, economists, city planners, business strategy experts, regional scientists and other social scientists have tried to explain the "why" and "where" of implemented activities. This encourages the proliferation of research on the influence of spatial effect which is able to explain the effect of events caused by their geographical influence. Some methods of spatial analysis that has been developed is geographically weighted regression or known as Geographically Weighted Regression (GWR). GWR was first introduced by Fotheringham in. The response variables in GWR model contains predictor variables that each regression coefficient depends on the location where the data is observed. The development of researches on GWR was as follows. A new model of GWR which is Geographically Weighted Poisson Regression Models was found (Nakaya, Fotheringham, Brunsdon, & Charltron, 0). A model that incorporates regression model global and GWR, known as Mixed Geographically Weighted Regression Models, was produced (Chang-Lin Mei, Ning Wang, & Wen-Xiu Zhang, 0). Another GWR model known as Spatio-Temporal (Demsar,

4 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Fotheringham, Brunsdon, & Charltron, 0), was also developed in the field of spatially related time series. Furthermore, a research on Geographically and Temporally Weighted Regression Model (Huang, Wu, & Barry, 0) and research on spatial panel data with Geographically Weighted Regression Panel method (Yu, 0) were examined Multivariate Geographically Weighted Regression (MGWR). This study discusses the estimation of the model using weighted maximum likelihood method (Harini, Purhadi, Mashuri, & Sunaryo, 0). Geographically and Temporally Weighted Likelihood Regression model (Wrenn & Sam, ) and Geographically Weighted Regression with Spline Approach (Sifriyani, Haryatmi, I. N. Budiantara, & Gunardi, ). The methods which were developed in this study are still in the form of linear and assumed that the data pattern is known. In fact, when modeling the data the question is, it true that all relationships between predictor variables and the response variables form a known regression curve. In reality, of course not all the data pattern of relationships has a known regression curve so it is necessary to use nonparametric regression analysis for solving this problem. A good model should be viewed from various aspects and put a proper modeling issues in its portion. The difference between the environmental characteristics and geographic location of observation, effects the observations to have different variations or there are different influences of predictor variables on the response variables for each observation location. How to solve when predictor variables have variable patterns which are not following a specific pattern on response variables and regression curve is not known. in this case, the GWR models have not been able to solve this problem so the authors developed a nonparametric

5 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of regression model that is GWR otherwise known as nonparametric geographically weighted regression. This paper is organized as follows; Section : Introduction, Section : Model Nonparametric Geographically Weighted Regression Using Truncated Spline Approach, Section : Result And Discussion and Section : Conclution.. Nonparametric Geographically Weighted Regression Models Using Truncated Spline Approach. Nonparametric geographically weighted regression models with truncated spline approach is the development of nonparametric regression for spatial data which parameter estimator is local to each observation location. Spline truncated approach used to solve the problems of spatial analysis regression curve is unknown. In the regression model assumptions used are normally distributed error with zero mean and variance, at each location,. Location coordinates, is one of the important factors in determining the weighting used for estimating the parameters of the model. Given the data,,,, and the relationship between,,,, and assumed to follow a nonparametric regression model as follows: =,,,,+, =,,, with as the response variable and,,,, is a function of the unknown regression curve shape which is assumed to be additive. If,,,, is approached by a multivariable spline function, it can be written as follows: = +

6 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Given = then approximated by a truncated spline functions for each location,, defined as follows: =, +, and truncated function +, =, h 0, < h Mathematically the relationship between the response variable and the predictor variables,,,, on the -th location for nonparametric geographically weighted regression models with truncated spline approach, can be expressed as follows: =, +, +, + Equation () is a nonparametric geographically weighted regression models with truncated spline approach -order with number of areas. The components in equation () are described as follows: is respon variable to locations all where =,,,. is predictor variables on the - that area with =,,,. is point knots on the h-th at the components of predictor variables with h=,,,., is parameter regression on the - that the predictor variables and the -th area., is parameter regression of the function truncated, this parameter is a parameter to +h, at which point knots h-th and predictor variables. Equation () can be expressed by: with = + = β, + δ, +

7 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of =, = ( u, v ) ( u, v ) ( u, v ) ( u, v ) ( u, v ) ( u, v ) % β i i = β0 i i β i i β i i L βl i i L βlm i i ( u, v ) ( u, v ) ( u, v ) ( u, v ) ( u, v ) % δ i i = δm+ i i δm+ i i L δ lm+ i i L δ lm+ r i i and the predictor variables defined by the matrix m m m x x x L x x x L x L xl xl L x l m m m x x x L x x x L x L xl xl L xl = M M M M O M M M O M O M M O M m m m x n x n x n L x n xn xn L x L xln x L x ln ln = ( x K) L ( x Kr) L ( xl K) L ( xl Kr) ( x K ) L ( x K ) L ( x K ) L ( x K ) m m m m m m m m + r + l + l r + M O M O M O M ( x K ) L ( x K ) L ( x K ) L ( x K ) m m m m n + n r + n r + ln r +. Result And Discussion.. Estimation Model In nonparametric geographically weighted regression models with truncated spline approach, the assumptions used are error i-th observation of identical, independent and normally distributed with zero mean and variance,, where the parameter β,, δ, and, to are unknown. This shows that every geographical location has different parameter values. If the parameter value is constant at any geographical location, the nonparametric geographically weighted regression models with truncated spline approach is the same as usual nonparametric regression model. That means that each geographical location has the same model. Theorem. T T

8 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani If the regression model () with an error normally distributed with zero mean and variance, was given Maximum Likelihood Estimator (MLE) is used to obtain estimator β,,δ, and as follows. β, = δ, = with = β, + δ, = =,,,, =,,, u i,v i = + Proof. Regression model given in equation (), Having obtained the joint density function of,,, then to estimate the nonparametrik geographically weighted regression models with truncated spline approach on location ke-, given the likelihood function on the location ke- is as following: β,,δ,,, =, exp,, +, +, nonparametric geographically weighted regression models with truncated spline approach, it takes a geographic weighting. Where data at each location of the observations by weighting which varies depending on the size of influence between the observation location. The closer the distance between the observation location, the greater the influence of an observation location. Suppose the weighting for the location and location is, then to get the parameter estimator In this research to gain estimator parameter β,,δ,,, of

9 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of β,,δ,,, first providing j ke- weighting on the following observations: =, +, +, + =,, will follow a normal distribution with mean zero and variance, and ~0,,. The weighted likelihood function is β,,δ,,, =, exp,,, +, +, then performed the operation to facilitate the natural logarithm mathematical operations in order to obtain ln equation as follows: ln β,,δ,,, with = = ln ln,, 0, +, + +h, h + == =h= = β, δ,, β, δ, Estimation parameter β,,δ, and, obtained by maximizing ln L shape equation (). Estimator β, will be obtained based on the following derivatives: T β, δ,, β, δ, β, = β,

10 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani β,, +β,, δ, +β,, β, = β,, +, δ, +, β, =0,, δ,, β, =0, β, =,, δ, β, =,,,, δ, So that the parameter estimator β, is β, =,,,, δ, because there are n observations location, then using equation () obtained estimator β, is β, =,,, δ, Furthermore, to obtain estimator δ,, can be obtained by maximizing the form ln L equation () and conducted operations against the derivative δ,. β, δ,, β, δ, δ, = δ, δ,, +δ,, β, +δ,, δ, = δ,, +, β, +, δ, =0, δ, =,, β, δ, =,,,, β, So that the parameter estimator δ, is δ, =,,,, β,

11 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page 0 of because there are n observations location, then using equation () obtained estimator δ, is δ, =,,, β, 0 Estimator β, in equation () still contains estimator δ,. Similarly estimator δ, in equation (0) still contains estimator β,. In order to obtain the form of independent estimator it is necessary to substitution technique. To obtain the estimator β, free of δ, the substitution of the equation () to the equation (0) as follows: δ, =,,,, β,. =,, +,,,, +,, δ, =,, +,,,, + +,,,, δ,. Part containing estimator δ, grouped in a segment as follows δ,,,,, δ, =,, +,,,,. With a bit of elaboration of the following equation: I,,,, δ, = 0

12 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani ,, +,,,,. Part that does not load estimator δ, moved to the right δ, = I,,,,,, +,,,,. If defined =I,,,, By replacing the existing components in equation () with R defined on similarities (), equation () can be written as follows: δ, =,, +,,,,. Then the estimator obtained δ ui,v i δ, =,, +,,,, =,,,, = thus obtained estimator δ, is δ, = with hat matrix is =,,,,

13 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of Next to gain estimator β, free of δ, the substitution of the equation (0) to the equation (). The obtained estimator β, is β, =,,,, = Based on the description above estimator β, can be written as follows: β, = With the hat matrix is =,,,, Next to determine the estimator function of nonparametric geographically weighted regression with truncated spline approach in equation (), can be substituted β, and δ, value in the following equation: = β, + δ, = + = + = The matrix = + is a hat matrix containing knots point K for nonparametric geographically weighted regression models with truncated spline approach...unbiased Estimator,,, and. Unbiased estimator β, is given in lemma as follows: Lemma. If β, is a estimator of nonparametrik geographically weighted regression with truncated spline approach that follows the equation (), then β, is an unbiased

14 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani estimator for β,. Proof: After the obtained estimator β,, then shown an unbiased nature of the estimator β, in the following manner: Eβ, =E. =E,,,,. =,, +,,,, E. () Furthermore E is substituted into the equation (), in order to obtain the following equation: Eβ, =,, +,,,, β, + δ,. With a bit of mathematical elaboration and there are some components of the matrix is equal to, was obtained Eβ, = β, +,, δ, +,,,, β, +,, δ,. Furthermore, the elaboration of mathematical is obtained: Eβ, = β, +,,,, β,. =I,,,, β,.

15 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of Furthermore substitutable matrix S, is obtained: Eβ, =I,,,, I,,,, β,. Thus obtained Eβ, = β, From the results obtained have proved β, is an unbiased estimator β, The unbiased estimator δ, and estimator is given in lemma. and lemma, Proof of lemma and lemma in Appendix. Lemma. If δ, is estimator of nonparametrik geographically weighted regression models with truncated spline approach that follows the equation (), then δ, is an unbiased estimator for δ,. Lemma. If is the estimator function of nonparametrik geographically weighted regression models with truncated spline approach that follows the equation (), then is an unbiased estimator for... Discussion Analysis of the data on the crude birth rate with five predictor variables i.e. the percentage of family head education (x : completed / not completed for the primary school), the percentage of working status of family head (x : having work / no work), the percentage of family head who married at - years old (x ), the number of rough marriages (x ) and the number of migrant (x ). Selection of the optimum knot points of the nonparametric regression methods and nonparametric geographically weighted

16 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani regression methods with truncated spline approach is respectively given in Table and Table. Estimation using the Nonparametric Regression Geographically Weighted Regression With Truncated Spline Approach is as follows: = Nonparametric Geographically Weighted Regression with Truncated Spline Approach with knot points that has.% of R. This may explain the crude birth rate of.%.. Conclution Estimation of nonparametric geographically weighted regression using truncated spline approach was successfully formulated. It was found that:. Nonparametrik geographically weighted regression models using truncated spline approach is =+ with = β, + δ,, obtained estimator β, = with matrix hat is =,,,,. and estimator δ, = with matrix hat is =,,,,. Regression curve Estimator is = β, + δ, =. with matrix hat = + is a matrix that has a point knots.. Result of estimators β,, δ, and from nonparametric geographically weighted regression models using truncated spline approach has been shown as

17 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of unbiased estimator.. Application of nonparametric geographically weighted regression models with truncated spline approach of the crude birth rate data in areas in eastern Java, have resulted GCV values smaller than when using a nonparametric regression method. It concluded that using the nonparametric geographically weighted regression models with truncated spline approach to model the effect of the crude birth rate was better and more appropriately than the nonparametric regression models with truncated spline approach. References Chang-Lin, M., Ning, W., & Wen-Xiu, Z. (0). Testing the Importance of the Explanatory Variables in a Mixed Geographically Weighted Regression Model. Environment and Planning A, (), -. Demsar, U., Fotheringham, A. S., & Charlton, M. (0). Exploring the spatio-temporal dynamics of geographical processes with geograhically weighted regression and geovisual analytics. Inference,(),. Fotheringham, A. S., Brundson, C. & Charlton, M. (0).Geographically Weighted Regression : The Analysis of Spatially Varying Relationships, John Wiley & Sons Ltd, England. Harini, S., Purhadi, Mashuri, M., & Sunaryo, S. (0). Linear Model Parameter Estimator of Spatial Multivariate Using Restricted Maximum Likelihood Estimation. Journal of Mathematics and Technology,, -. Huang, B., Wu, B., & Barry, M. (0). Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. International Journal Geographical Inference Science, (), 0. Nakaya, T., Fotheringham, A. S., Brunsdon, C., & Charltron, M. (0). Geographically Weighted Poisson Regression for disease association mapping. Journal Statistics Medicine, (), -.

18 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Sifriyani, Haryatmi, I. N. Budiantara, & Gunardi. (). Geographically Weighted Regression with Spline Approach. Far East Journal of Mathematical Sciences, 0(), -. Wrenn, D. H., & Sam, A. G. (). Geographically and temporally weighted likelihood regression : Exploring the spatiotemporal determinants of land use change. Regional Science and Urban Economics,, 0-. Yu, D. (0). Exploring spatiotemporally varying regressed relationships: the Lampiran geographically weighted panel regression analysis. Proceedings of the Joint International Conference on Theory, Data Handling and Modeling in GeoSpatial Information Science,. Proof of Lemma. After the obtained estimator δ,, then shown an unbiased nature of the estimator δ, in the following manner: Eδ, =E. =E,,,,. =E,, +,,,,. =,, +,,,, E. Furthermore E is substituted into the equation (), in order to obtain the following equation: Eδ, =,, +,,,, β, + δ,. With a bit of mathematical elaboration: ()

19 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of Eδ, =,, β, + +,, δ, +,,,, β, +,,,, δ,. () There are several components of the matrix is equal to so that the equation () can be simplified as follows: Eδ, =,, β, + δ, +,, β, +,,,, δ,. With a bit of mathematical elaboration: Eδ, = δ, +,,,, δ,. Tribe containing δ, are grouped on the right, so that the equation () can be simplified as follows: Eδ, = I,,,, δ,. Substitutable matrix obtained Eδ, =I,,,, I,,,, δ,. Thus obtained Eδ, =δ, From the above results proved that δ, is an unbiased estimator δ, Proof of Lemma.

20 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Having obtained estimator models, will further shown no bias nature of the estimator in the following way: E =E β, + δ, =E + = E + E To prove the nature of unbias in the estimator, first completed component E. E =,,,,. =,, +,,,, E. Furthermore substitutable value of E (Y) is obtained: E =,, +,,,, β, + δ,. Based on the operating results of mathematics is obtained: E =,, β, + +,, δ, +,,,, β, +,,,, δ,. There matrix components,, = and,, = thus obtained E = β, +,, δ, +,,,, β, +,, δ,. with matrix mathematics operations obtained:

21 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of E= β, +,,,, β, + +,, δ, +,, δ,. The equation is simplified as follows: E= β, +,,,, β,. Thus obtained E : E= I,,,, β,. Substitutable matrix S obtained E= I,,,, I,,,, β,. Thus obtained E= β, Having obtained the value of the component E, then resolved component E as follows: E = =,,,, E. =,, +,,,, E. Furthermore substitutable value of E is obtained: E =,, +,,,, β, + δ,. Based on the operating results of mathematics is obtained:

22 Page of Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani E =,, β, + +,, δ, +,,,, β,,,,, δ,. There matrix components,, = and,, = thus obtained: E =,, β, + δ, +,, β,,,,, δ,. The equation is simplified as follows: E = δ, +,,,, δ,. Thus obtained E: E = I,,,, δ,. Substitutable matrix obtained: E =I,,,, I,,,, δ,. Thus obtained E = δ, Based on the equation () and equation (0) then the result of the expectation is E = + = β, + δ, = So it proved not biased nature of the model estimator

23 Songklanakarin Journal of Science and Technology SJST--00.R Sifriyani Page of Table. Optimum knot point of Nonparametric Regression method With Truncated Spline Approach The number of knots GCV points Table. Optimum knot point of Nonparametric Regression Geographically Weighted Regression With Truncated Spline Approach The number of knots GCV points

Geographically weighted regression approach for origin-destination flows

Geographically weighted regression approach for origin-destination flows Kazuki Tamesue 1 and Morito Tsutsumi 2 1 Graduate School of Information and Engineering, University of Tsukuba 1-1-1 Tennodai, Tsukuba,