COPULA MODELING IN ANALYSIS OF DEPENDENCY OF OIL PALM PRODUCTION AND RAINFALL

Copula is a method that examines the relationship pattern between variables. Copula is characterized as a nonparametric method with several benefits, i.e., it is independent of the assumption of the distribution, accommodates nonlinear relationship among variables, and is convenient in building joint distribution. This study investigates the relationship and prediction analysis using the copula approach. The method is applied to the monthly data of oil palm production and the amount of rainfall. The results show that the model of Frank Copula is the best model for rainfall and oil palm production relationship.


INTRODUCTION
Oil palms are tropical plants that can grow well in the latitude of 12o N to 12o S. The amount of rainfall in the tropical area is considered important to the plant growth and productivity.Some studies show that the amount of rainfall and rainy days affect the productivity of the plants (Prasetyo 2009;Simanjuntak, Sipayung and Irsal 2014).The ideal rainfall for oil palm is 2000-2500 mm per year evenly distributed throughout the year without long dry months.A long dry season could reduce the production of oil palm since fewer minerals in soil are absorbed by the plants (Risza 2014).On the other hand, too much rainfall can cause erosion of topsoil especially in areas with bad topography.
Copula is one method that can be used to describe the relationship between variables without any assumption of the distribution.It can reveal relation of dependency even at extreme points (Anisa and Sutikno 2015).
Copula has an important role when one or both variables have an abnormal marginal distribution or have tail dependencies.The use of the copula in investigating the relationship among variables has been studied by many authors in various fields.Anisa and Sutikno (2015) and Udayani, Sumarjaya, and Susilawati (2016) used copula in the analysis of relationship in agricultural research.Murterio and Lourencio (2007) applied copula in the health care utilization, whereas Zhu, Ghosh and Goodwin (2008) used copula in modeling dependence in the design of the insurance contract.Syahrir (2011) estimated copula parameters in the field of climatology.The Copula approach resulted in better estimates even in the presence of extreme observational data and for conditions that violate normality assumptions.This paper discusses the application of the copula in modeling the relationship between two variables, namely rainfall and palm oil production.The aim of Due to this transformation, Zj always has uniform marginal distribution.The function CX is called a copula and cX is the corresponding copula density.
The definition of the parametric form of the copula function or the copula density allows them to be grouped into families.Two important classes are the Elliptical and Archimedean Copula families.Details about other copula families and their selection can be found in Embrechts et al. (2001).Gaussian Copula or Normal Copula is part of the Elliptical Copula family.The normal copula is derived from the transformation of random variables into standard normal distributions.Normal Copula function is given in the following equation: Commonly used Archimedean Copula are Clayton, Gumbel, and Frank Copula.The general form of Archimedean Copula is as follows: Generator and bivariate copula for each copula in the family of Archimedean are presented in Table 1.Clayton Copula has lower tail dependencies, Frank Copula has no tail dependencies, while Gumbel Copula has upper tail dependencies (Schoelzel and Friedrichs 2008).
One method for estimating copula parameters is the maximum likelihood estimation (MLE).MLE for this copula is obtained by maximizing the log likelihood function (Udayani et. al. 2016).Estimated Copula parameters are presented in Table 2. Test of hypotheses of 00 : H   versus 10 : H   is carried out using the following test statistic: The null hypothesis is rejected when the z statistic is greater than 2 z  of the standard normal distribution (Wang 2010).

RESULT AND DISCUSSION
The data used in this research are secondary data obtained from one estate of PT.Perkebunan Nusantara XIII and Meteorology, Climatology and Geophysics Agency Class II Siantan Station.These secondary data are the monthly production of oil palm plantation in the period of 2010-2015 and monthly rainfall data on the same period.The plot of oil palm production and rainfall are presented in Figure The plot in Figure 1 shows a tendency of an increasing amount of rainfall in the last two years.It is followed by a decrease in the production of oil palm in the corresponding period of time.
Identification of a pattern in the relationship between the oil palm production and the rainfall was carried out by examining the scatter plot of the two variables.In examining the existence of dependency of the two variables, the oil palm data were plotted on the rainfall data.Different time lags of oil palm production were also plotted on the rainfall data.The plots are presented in Figure 2. Figure 2 (a) representing the data at the same period.In Figure 2 (b), (c) and (d) the different time lags of oil palm production were plotted on the rainfall data.This is carried out to inspect the relationship of rainfall to oil palm production in the next months.The scatter plots show that the pattern of oil palm production and the corresponding time lags versus the rainfall are skewed to the right.The bigger the time lag, the sharper the skewness.
The correlation between the time lags of oil palm production and the rainfall data were also calculated.Table 3 presents the Pearson, Spearman and the Tau Kendall correlation coefficients.They are all significant with negative values.However, normality tests on the data showed that the rainfall data are not normally distributed.This suggests that further investigation is to be carried out.With copula approach, the variables need to be transformed into Uniform distribution [0,1].Three types of copula models were compared with the analysis, i.e., Frank, Clayton, and Gaussian Copula.The Comparison was made by means of parameter estimates and the corresponding log likelihood values.The resulting estimates of the parameters and the corresponding log likelihood values are presented in Table 4.
The value of the log likelihood suggests that of the three models, the Frank Copula fit the data very well.The log likelihood of Frank Copula is always bigger than those of others.This occurs in all pairs of variables.The bigger the estimate indicated the stronger the dependency of the two variables.The Frank Copula model of the time lag 2 oil palm production on rainfall data resulted in the highest value of the log likelihood.This suggests that the amount of rainfall could affect the oil palm production in the next two months in this estate.A negative value of the estimate indicates a negative relationship between the two variables.

CONCLUSION
The pattern of rainfall and oil palm production relationships at different time lags mostly follows the Frank Copula model.The strongest relationship between rainfall and oil palm production occurs in lag 2 of oil palm production.This indicates that the amount of rainfall could affect oil palm production in the next two months.A sequence of simulated, variable specific power levels has been selected for ORIGEN2 fuel depletion computation.This simulated power level sequence incorporates reactivity feedback from earlier burnable poison and fuel depletion.

ACKNOWLEDGEMENT
We would like to thank the Faculty of Mathematics and Natural Sciences for providing the research fund.Thanks should also go to PT. Perkebunan Nusantara XIII and Meteorology, Climatology and Geophysics Agency Class II Siantan Station who provided the data for analysis.Travel assistance and support from the Committee of the 3rd International Symposium is highly appreciated.

Fig 1 .
Fig 1. Plot of oil palm production and rainfall over time Indonesian Journal of Physics and Nuclear Applications Volume 3, Number 3, October 2018, p. 89-94 e-ISSN e-ISSN 2550-0570, © FSM UKSW Publicationthe study is to provide an overview of Copula and its application in the field of agriculture.

Table 3 .
Correlation Coefficients between oil palm production and rainfall data

Table 4 .
Estimate of copula parameters of lag production on rainfall