Geographically and Temporally Weighted Regression Modeling in Analyzing Factors Affecting the Spread of Dengue Fever in Malang

Geographically and Temporally weighted regression (GTWR) modeling has been developed to evaluate spatial heterogeneity and temporal heterogeneity in factors influencing the spread of dengue fever in Malang city. By using the monthly data in 2012-2015 as the temporal unit of each urban village in Malang and village is considered as a spatial unit. GTWR model is compared with the GWR model using several statistical criteria. GTWR model shows that the relationship between dengue incidence with population density and monthly average temperature significantly affects each Village in Malang.


INTRODUCTION 
Dengue hemorrhagic fever (DHF) is a disease caused by the dengue virus that is found in tropical and subtropical areas. The number of sufferers and the extent of their spreading areas increases with increasing mobility and population density. In Indonesia, the disease transmitted by Aedes aegypti mosquitoes was first discovered in Surabaya and Jakarta in 1968 [1].
For the last three years, patients with dengue fever in Malang continue to increase (Health Office of City Government of Malang, 2016, unpublished). Various programs have been implemented for disease prevention and control, but DHF in Malang remains high. High dengue cases require serious efforts to prevent the spread of DHF endemic disease by analyzing the risk factors associated with dengue virus infection. Many research on the factors that cause the spread of dengue virus in the community were conducted [2].
Previous study used the Geographically Weighted Regression (GWR) model in its analysis; where GWR statistical methods adjusted the global framework with local regression models allowing estimation regression parameters for each spatial unit [3]. And it is reported that rainfall significantly affects the spread of DHF. However, this research is limited to spatial effects and rainfall factors, where the relationship between dengue fever and rainfall varies greatly in each location of observation. As a result, its conclusions only look at the spatial effects and one factor, not paying attention to the influence of time and other factors that can also affect the spread of dengue in the area.
In many cases, data structures are not only owned by geographic location but also related to the time factor. One method for examining spatial and temporal influences is geographically and temporally weighted regression (GTWR). This statistic method is developed from GWR. It is expexted that this model can accommodate spatial influence, and also able to analyze the temporal influence which present in the data simultaneously [4].
The purpose of this study is to extend the GTWR modeling framework in analyzing factors causing dengue extensively by observing the presence of spatial heterogeneity and temporal heterogeneity in the modeling. The GTWR model obtained is compared with the GWR model using several statistical tests. The results of this study will practically be useful for the authorities in regulating the prevention program of DHF infection locally and at certain times.

MATERIAL AND METHOD Study Area
Location of this research is Malang City, East Java, which consists of 57 urban villages. Total area of Malang is 252.10 km 2 which is located on Java island about 7 o 06 '-8 o 07' latitude and 112 o 06 '-112 o 07' longitude [5]. Village is used as a spatial unit in this study, there are 57 villages and the month as a temporal unit, consisting of 48 months.

Data Collection
Data on the number of DHF patients in each Village and each month during the year 2012 -2015 was obtained from the Health Office of Malang city government. Monthly data as a temporal unit in each Village, one Village is defined as one spatial unit. For population density data, the number of Puskesmas (health center) in each Village, as well as the height of the place in each Village obtained from Malang Central Agency on Statistics publication [5][6][7][8]. Data of rainfall and average temperature obtained from Karangploso Agency for Meteorological, Climatological and Geophysics.
Number of DHF patients selected as response variable (Y), population density (X1), Number of health center auxiliary (X2), rainfall (X3), average temperature (X4) and altitude of place from sea level (X5) as predictors variable.

Statistical Methods
The relationship between DHF incidence and factors considered to affect the spread of DHF was analyzed by comparing the results of GWR model and GTWR model. If GTWR model is better than GWR model, the relation between respondent variable and the predictor variables is spatially and temporally different for each Village.
GWR model is a simple model and is usually used in research. The spatial heterogeneity of the relationship between the occurrence of DHF and factors influencing it was analyzed by the GWR model allowing estimation of different regression parameters in each Village [3].
is the location point of the Village is the definition of the longitude and latitude coordinates ; and i  is an error that is assumed to be normal distribution with mean 0 and its variant known.
The GTWR model is formed by using the spatial weight function and the temporal weighting function in estimating its parameters [4].
λ and μ are scale factors used to measure location and time difference. Distance in equation (3) as a basis in weighting data to estimate GTWR model parameters. The closer the distance between the expected points each village, the greater the weight of the data during the estimation. In this study, data weighting is done by using the adaptive Gaussian kernel function Where, h is the constant bandwidth performed by the Incremental Spatial Autocorrelation (ISA) method [9]. The Adaptive Gaussian kernel function is used in forming a weighted matrix.
Each observed data has one weighted matrix in estimating the parameters. Using the algebraic matrix approach, and using the Weighted least square (WLS) method the parameter estimation can be written [4]  Significant test of GTWR model is given on [10]. Where the null hypothesis of test indicates the function of the research location. If the null hypothesis is rejected, then the GTWR model is accepted to describe the data. Meanwhile, tests for the variation of each set of k  for the entire study site will be used in F3 statistics [10]. The identification of the GTWR model will also be shown by coefficient determination (R 2 ). The comparison of the GWR model and the GTWR model will be shown with Root mean square error. According to RMSE the regression model that has the smallest RMSE value is defined as the best model.

RESULT AND DISCUSSION Result of GWR Model
Using the spatial weighting, the parameter estimation ratios (regression coefficients) for the obtained GWR model are shown in Table 1. In this model the value of obtained R 2 = 51.1% and RMSE = 13.13. Further analysis of the model indicates that the error approaches normal distribution.

CONCLUSION
In this paper, a discrete fractional-order Population density and temperature are significant variables on the number of dengue fever occurrences in Malang. Residents living in densely populated locations have a higher risk of dengue transmission. So also if there is an increase in temperature then the risk of transmission of dengue virus will also be greater.