ANALYSIS OF GENERALIZED SPACE-TIME AUTOREGRESSIVE MODEL THROUGH KERNEL FUNCTION APPROACH (CASE STUDY: LOG GAMMA RAY DATA ON MULTIPLE COAL DRILLHOLE LOCATIONS) - - NIM: 30114009 , YUNDARI
The first order of generalized space-time autoregressive written by GSTAR(1;1) model is a fairly popular space-time model in Indonesia. This model is consistent with Indonesia's heterogeneous geographical conditions. Time and location dependences were required in this modeling. Time dependence was associated with value of observation in the present and the past, whereas location dependence was expressed by a spatial weight matrix at each locations. The spatial weight matrix represents a proximity relationship based on both distance and observation value. If it based on the distance then weight matrix was fixed, but if it based on observation value then weight matrix is random. Until now GSTAR modeling still uses a fixed spatial weight matrix, therefore subjectivity is very likely to occur depending on each researcher. This dissertation introduces a new method that more standard in contruction of spatial weight matrix. The method was used kernel function approach with domain of observation values. Therefore the function of random variable gives a random variable as well. Consequently, weight matrix obtained was random. This is where it derives a statistical characteristics of the GSTAR (1;1) process that contains kernel weight matrix. The first is the mean / expectations (1st moment) of the GSTAR (1; 1) process. The obtained result was the same as GSTAR process with the fixed weight matrix that it was equal to zero. Furthermore, because the resulted mean was zero, the second characteristic was the variance or 2nd moment of the GSTAR (1: 1) process .The mean and variance obtained were independent of time. In other words, the assumption of GSTAR (1;1) process that stationary (weak) was fulfilled. The stationary properties of GSTAR (1;1) model parameter also needs to be assessed. The method used inverse of the autocovariance matrix (IAcM). The difference in definition of IAcM with kernel weight matrix and the specified weight matrix is coefficient (N-1). Because they were still linear with the coefficient multiplication, then properties of IAcM with specified weight matrix also applied to kernel weight matrix. The next characteristic was invertibility of the GSTAR (1;1) model. Similar to autoregressive (AR) model, the GSTAR (1;1) model can be represents the generalized space-time moving average (GSTMA) model with infinite time-order, written GSTMA ( ∞ ,1). The interesting thing was obtained that if using kernel weight matrix then time order of the resulting GSTMA model becomes finite. Therefore, the GSTAR (1;1) model is equivalent to the GSTMA (n;1) model with n finite. Generally, in terms of estimation of GSTAR (1;1) process was used parametric method i.e. the least squares with spatial weight matrix through kernel function Application of GSTAR (1;1) modeling with kernel weight matrix was applied to logging data i.e. Gamma ray log (GR). Based on the principle of superposition on the rock stratigraphy, that it is the more getting upper a layer of rock then relative age of the rock was getting younger. Otherwise the layer of rock is getting lower then relative age of the rock was getting older. The age of this rock layer was expressed by layer of rock at a certain depth. By discretizing a continuous depth, this particular depth interval was represented as an index of time parameters in the GSTAR (1; 1) process. The GR log data used for verification of result of coal drilling detects all rock layers from bottom to top following the depth of the borehole. This GR log data serves to detect the type of rock layer based on its radioactive content. In the coal exploration, GR log was used to distinguish coal layer with very low radioactive element content to clay stone (mudstone) with high radioactive content. These drilling stage requires substantial costs to determine the potential of coal resources that lie beneath the surface. Thus the result of this study was expected to predict the existence of coal layer based on GR data log modeling. With random variable in the form of GR log value, time in the form of difference of rock layer thickness which have been discretized and presence of spatial dependence from several coal drilling holes, it was expected that GSTAR (1; 1) modeling can be applied.