$S\alpha S$ The research entitled  “The Codifference Function of Moving Average Processes With Symmetric $\alpha$-stable Innovation ” was conducted by Iqbal Kharisudin under the guidance of Dr.rer.nat., Dedi Rosadi, M.Sc., Dr. Abdurakhman, M.Si., and Dr. Suhartono, M.Sc. in 2018.

The following is the abstract of this research.

ABSTRACT

Most of statistical models require the existence of a second-order moment or based on the distribution with finite variance. We can study the dependency structure of the model based on second-order moment. In the time series modeling with finite variance innovation (eg Gaussian model), the autocorrelation function (ACF) plays an important role. One of the applications of ACF on Box-Jenkins time series modeling is for model identification. In the MA($q$) models, we found that the ACF values is zero after lag q. Such essential properties are used as a basis for identifying the MA process given by time-series data.

In the stable time series modeling, ACF function can not be used, because it requires finite variance ($\sigma <\infty$). Therefore, in the modeling of time series with stable assumptions we use alternative dependency measures, such as coddifference function or covariance function. The covariance function and its application were studied in Gallagher (2000, 2001). Rosadi and Deistler (2011) discussed the generalized form of the sample codifference functions for linear stationary models and its asymptotic property. The special case of iid process has also been discussed. Based on the literature review and the author’s knowledge, the study of samples codifference function and its asymptotic properties for MA($q$) model with symmetric $\alpha$-stable ($S\alpha S$) innovation are still an open problem.

In this research, we examine the properties of codifference function, its estimator, and the asymptotic properties of sample codifference function for MA($q$) $S\alpha S$. We can define the sample autocodifference (ACodF) by using sample normalized codifference function. In this study, we examine the computational simulation and application of sample autocodifference function (ACodF) for the identification of MA($q$) process with Gaussian ($\alpha = 2$) and $S\alpha S$ non Gaussian ($\alpha < 2$) innovation using asymptotic confidence intervals.

The main results of this research are the asymptotically normal distributed properties of the sample autocodifference function (ACodF, $\hat{I}(k)$) of MA($q$) $S\alpha S$ process for $k > q.$ We found that the sample ACodF for MA($q$) $S\alpha S$ process was asymptotically normal distributed with covariance matrix depends on its coefficients. These results are expressed in the theorems and corollary. The obtained results was an extension of the asymptotic property of the sample ACodF for the iid case. We can do inference about ACodF for MA($q$) $S\alpha S$  process based on these properties. For application purposes, we proposed an iterative procedure for the $q$ order identification of MA($q$) $S\alpha S$ process.

A simulation study was conducted to evaluate computationally the asymptotic properties of the sample ACodF. From the simulation, we found that the order identification of the MA(1) and MA(2) processes using the corrected confidence interval bounds $\hat{I}$ of the ACodF can improve the identification performance of the desired order. The model identification increases better than using ACF $\hat{\ro}$, auto-covariation (ACovF) $\hat{\lambda}$, or using ACodF by MA(0) confidence interval bound, especially if the sample size is large enough (no less than 100). Based on simulation study, we obtain these improvements are about $59,5%-86,8%$. These results are consistent with the asymptotic properties of sample ACodF for MA($q$) $S\alpha S$ process. In the last section, we discuss the application of the sample ACodF for MA model identification base on iterative procedure for returns of stock data in Indonesia.

Keywords: dependence measure, infinite variance, symmetric $\alpha$-stable, MA($q$)

The research entitled  “N-soft Sets and Their Decision-Making Algorithms” was conducted by Fatia Fatimah under the guidance of Dr.rer.nat., Dedi Rosadi, S.Si., M.Sc., and Dr. Raden Bagus Fajriya Hakim, S.Si., M.Si. in 2018.

The following is the abstract of this research.

ABSTRACT

Decision-making uncertainty has been applied in various fields. Soft set is one of the theories that can handle the decision-making problems under uncertainty where its parameters as the benchmark of decision making. The parameters may be real numbers, words, sentences, functions and so on. Therefore, it is flexible in representing individual needs. In everyday life, data sets can be found in the form of probabilities or rankings. Unfortunately, there is no literature in soft sets for handling decision-making problems using probabilities or N ary rankings. Thus, this research discusses about decision-making approach using probabilistic soft sets and N-soft sets. The results are as follows. First, probabilistic soft sets is useful for decision making where its probability distributions based on parameters. Seven decision-making algorithms are proposed i.e., Probabilistic Soft Sets-Choice Values, Probabilistic Soft Sets-Minimax, Probabilistic Soft Sets-Opportunity Cost, Probabilistic Soft Sets-Weighted Choice Values, Probabilistic Soft Sets-Weighted Minimax, Probabilistic Soft Sets Weighted Opportunity Costs, and Probabilistic Soft Sets Positive Matrices. Second, dual probabilistic soft sets, as an adaptation of probabilistic soft sets, is necessary for handling decision-making problems where its probability distributions based on objects. Dual Probabilistic Soft Sets Positive Matrices algorithm is presented. Third, N-soft sets can handle decision making with binary or non-binary rankings. Three decision-making algorithms are introduced i.e., Extended Choice Values, Extended Weight Choice Values, and T-Extended Choice Values. They are applied to a real case study. Fourth, graded soft set, a special case of N-soft set, draws a bridge between decision making in soft set and social choice theory. Its decision-making mechanism using choice values coincides with the Borda count in voting.

The research entitled  “Analysis and Design of Input-Output Group Decoupling Problems for Regular Linear Descriptor System with Index One” was conducted by Arman under the guidance of Dr.rer.nat. Ari Suparwanto, M.Si. and Prof. Salmah, M.Si. in 2018.

The following is the abstract of this research.

ABSTRACT

In the control systems, every input is generally controls more than one output and every output can be controlled by more than one input. Such a system is called coupled system. In general, the coupling system is very difficult to control. It is also known that not all coupling systems can be converted into a decoupling system. Therefore it is necessary to design some compensator such that the coupling system can be converted into a decoupling system in the sense that every input controls only one output and every output is controlled by only one input. This problem is called the input-output decoupling problem.

In this research, the input-output decoupling problem for linear descriptor system was studied using geometric approach through controllability subspace. First, the necessary and sufficient condition for the input-output group decoupling of regular linear descriptor systems of index one is established. Furthermore, the characterization of the input-output group decoupling yields an equivalent forms for the input-output group decoupling of linear descriptor system with index one.

Then, based on the standard decomposition form, we discussed the problem of group decoupled structure for slow and fast subsystems using the controllability subspace. This construction presented some results that yield a matrix structure of group decoupled system for regular linear descriptor systems of index one. System with such structure is called the normal form for input-output group decoupling. The normal form of input-output group decoupling can be considered as an input-output decoupling system which is controllable consists of independently controllable subsystems.

Key words: linear descriptor systems, group decoupling, controllable subspaces

The research entitled  “A Characterization of Cut Set on Semilattice-Valued Fuzzy Sets” was conducted by Harina Orpa Lefina Monim under the guidance of Prof. Dr. Sri Wahyuni, MS.  and Dr. Indah Emilia W., M.Si. in 2018.

The following is the abstract of this research.

ABSTRACT

Semilattice $(S,\leq)$ is a partially ordered set which a pair of its elements have infimum or supremum. When it is equipped by an equivalence relation defines as $p\approx_{M} q \iff \uparrow p \cap M = \uparrow q \cap M$ for any $p, q \in S$ where $M\subseteq S, M\neq \emptyset$ and $\uparrow p$ and $\uparrow q$ are principle filters generated by $p,q$ respectively. Semilattice $S$ is partitioned into equivalence classes-$\approx M$. A collection of the classes-$\approx M$ forms a poset under inclusion. We call the collection as a quotient set and denoted $S/ \approx  M$. The quotient set is a poset under inclusion. Generally, the poset of a quotient set is not a semilattice.

On a semilattice valued fuzzy set $\mu: X\rightarrow S$, we replace $S$ by the semilattice $S$ equipped by relation $\approx_{M}$ on $S$ above. For any $p\in S$, a cut set will be defined by $\mu_{p}=\{x\in X \mid p\leq \mu(x)\}$. Then a collection of all of cut sets, denoted $\mu_{S}$, is a poset by inclusion. Inverse function of $\mu$ from $\uparrow p \cap M$, denoted $\mu^{-1}(\uparrow p\cap M)$, will determine a characteristic function for any $p\in S$. The characteristic function of $\mu^{-1}(\uparrow p\cap M)$ along with $p\in S$ is
\begin{equation*}
M_{p} : X\rightarrow {0,p},
\end{equation*}
by definition:
\begin{equation}
M_{p}(x)=\begin{cases}
p, & x\in \mu^{-1}(\uparrow p\cap M)\vspace*{0.2cm}\\
0, & \text{others.}
\end{cases}
\end{equation}

Furthermore, the collection of characteristic function indexed by $p\in S$, $\mathcal{M}_{S}=\{M_{p}\mid p\in S\}$, is a collection of all subsets of $X\neq 0$. We use this collection for a sinthesys theorem on the semillatice valued fuzzy set. We will investigate a sufficient and necessary condition for $\mathcal{M}_{S}$ to be the collection of all cuts sets of $\mu \in S^{X}$. On the other hand we will present a concept of semi closure system and its dual. Then we will present also the isomorphism between the two collections. Using this property we investigate sufficient and necessary conditions for two semilattices pvalued fuzzy sets to have a coincide collection of cuts sets.

The main result is a collection of cut sets of semilattice valued fuzzy sets is the family$\mathcal{M}_{S}=\{M_{p}\mid p\in S\}$. This family of cut sets is a semi closure system on $X$ when $S$ is a complete join semilattice. However, the family of cut sets is dual semi closure system on $X$ when $S$ is complete meet semilattice. Another result is any two of semilattice valued fuzzy sets $ \mu, v\in S^{X}$ have a coincide collection of cut sets.

Keywords: Semilattices, fuzzy sets, cuts, semi closure systems, and dual semi closure systems.

The research entitled “Developing of Mean-Variance Portfolio Modeling Using Robust Estimation and Robust Optimization Method” was conducted by Epha Diana Supandi under the guidance of Prof.Dr. rer. nat. Dedi Rosadi, S.Si., M.Sc. and Dr. Abdurakhman, S.Si., M.Si in 2018.

The following is the abstract of this research.

ABSTRACT

The person who pioneered the basic theory of portfolio selection was Markowitz (1952) who shed light on the concept of mean-variance (MV) in allocating the asset and management of active portfolio. Mean vector and variance-covariance matrix must be discovered at hand as an entry in the procedure of developing optimum portfolio of MV model requiring estimation. There are a number of estimation techniques to apply parameter estimation, all of which are not free from estimation error. As a pivotal input in the making of mean-variance portfolio model, estimation error will impinge on the output of optimum portfolio formation. Some researchers have built robust portfolio, ie the portfolio that can reduce the estimation error of the mean vector and variance-covariance matrix of the portfolio MV’s model. There are two standard approaches in the formation of the optimal robust portfolio through robust estimation and robust optimization approaches The formation of optimum portfolio through robust estimation can be carried out in two stages. The first stage, the estimation of mean vector and covariance matrix constructed by using robust estimators. After the robust estimator came to light, it is being input to the MV portfolio model to attain robust estimation portfolio of MV model. This research selects two robust estimators with high breakdown namely S estimator, Constrained-M (CM) and Fast Minimum Covariance Determinant (FMCD). Unlike robust statistic approach, the theoretical basis of robust optimization is to reduce the sensitivity of optimum portfolio due to uncertain estimation of mean vector and variance-covariance matrix. The parameter input of robust portfolio optimization is considered uncertain situated in the uncertainty set. Afterwards, the optimum solution of this model which is accomplished for the worst solution occurs at the minimum expected return and maximum risk. In the robust optimization, the uncertainty set has a pivotal role for determining parameter. Until now there is no fixed certainty as how to determine uncertainty set accurately. In this study, a new approach is carried out to construct the set of uncertainty for the mean vector and variance-covariance matrix ie using block Bootstrap percentile method. This method is appropriately used because the resampling is performed on return data, therefore the structure of dependencies between data is not lost. The determination of optimum portfolio in the value of the worst cases of the robust optimization becomes one drawback of this method. One of the potential consequences of this approach is a decision strongly influenced by the extreme observations (outlier) in the set of uncertainty. As a result, the portfolio will tend to be too pessimistic and unable to attain optimum result. To overcome these obstacles, then this research also attempts to develop MV portfolio by combining robust optimization with robust estimator. The research leads to unbiased formulation for optimum portfolio of MV model, portfolio model formulation for robust estimation, and development of portfolio model of robust optimization. The research also builds computation program to ease the end-user in utilizing the resulted theory. Afterwards, the resulted portfolio models will be applied in the registered share data as a blue chip share. The last stage of the research is the performance comparison of those portfolios by using in-samples and out-samples analysis.

Keywords: portofolio mean-variance, penduga robust, optimasi robust, Bootstrap

The research entitled  “Optimalisasi Norma Jangkauan Vektor Eigen Atas Aljabar Maks-Plus Interval” was conducted by Siswanto under the guidance of Dr.rer.nat. Ari Suparwanto, M.Si. and Dr. M. Andy Rudhito, S.Pd., M.Si. in 2018.

The following is the abstract of this research.

ABSTRACT

Let R be the set of real numbers and R epsilon: the set of real number union infinite negative. Max-plus algebra is algebraic structure formed from the set R epsilon equipped with maximum and addition operations. A matrix in the size of m times n, whose components belong to R epsilon, called matrix over max-plus algebra. The set of matrices over max-plus algebra is denoted by R epsilon power m times n. Algebraic structure formed from the set of all interval in R epsilon union [infinite negative, infinite negative] equipped with maximum and plus operations is called interval max-plus algebra. A matrix in the size of m times n, whose components belong to all interval in R epsilon union [infinite negative,infinite negative], called matrix over interval max-plus algebra. In this research discussed about optimizing range norm of eigenvector over interval max-plus algebra and optimizing range norm of eigenvector over interval max-plus algebra with prescribed components. This research based on expanssion of max-plus algebra into interval max-plus algebra, optimizing range norm of eigenvector over max-plus algebra and optimizing range norm of eigenvector over max-plus algebra with prescribed components. The discussing initiated how to find eigenvalue. eigenvector and eigenvector space over max-plus algebra the generally matrices included reducible and irreducible matrices. In the last part of this research given application examples of the problem of eigenvalue and eigenvector over interval max-plus algebra in the production system. The concepts related to the main discussion were also investigated in this research. The concepts related, namely how to indicate the existence and uniqueness of solution of linear equation system in interval max-plus algebra by the normalization of linear equation system. Included in the discussion of linear equation system are the concepts about image set of matrices over interval max-plus algebra, strongly reguler matrices and simple image set.

Keywords: Optimalisasi, Norma Jangkauan, Aljabar Maks-Plus Interval

The research entitled  “Integral Henstock-Kurzweil di Dalam Ruang C[Alfa,Beta]” was conducted by Firdaus Ubaidillah under the guidance of Prof. Dr. Soeparna Darmawijaya and Prof. Dr. Ch. Rini Indrati in 2017.

The following is the abstract of this research.

ABSTRACT

This dissertation is the result of research to construct of Henstock-Kurzweil type integral for $C[\alpha,\beta]$ space-valued functions that defined on a closed interval $[f,g]\subseteq C [\alpha, \beta] $, where $C[\alpha,\beta]$ is the collection of all real-valued continuous functions defined on a closed interval $[\alpha,\beta]\subseteq \mathbb$. The concept of this integral follows Henstock-Kurzweil integral concept in the real space that has been known so far. We begin by introducing some fundamental concepts about the space $ C [\alpha, \beta] $, such as some properties of $ C [\alpha, \beta] $ as a Riesz space, convergence of sequences and series, $C[\alpha,\beta]$ space-valued norms and metrics, and neighborhood of a point in $ C [\alpha, \beta] $. Furthermore, we construct calculus on $ C [\alpha, \beta] $ for constructing Henstock-Kurzweil integral in the space $ C [\alpha, \beta] $ such as limit, continuity, derivative of $C [\alpha , \beta]$ space-valued functions, and convergence of a sequence of functions. To construc the Henstock-Kurzweil integral of a $ C [\alpha, \beta] $ space-valued function that defined on a closed interval $ [f, g] \subseteq C [\alpha, \beta] $, we begin by constructing a partition on $ [f, g] $. Furthermore, we define Henstock-Kurzweil integral of a $ C [\alpha, \beta] $ space-valued function that defined on a closed interval $ [f, g]\subseteq C[\alpha,\beta] $. From this definition, we develop some properties of a Henstock-Kurzweil integrable function into theorems. From a Henstock-Kurzweil integrable function, we define a primitive of a Henstock-Kurzweil integrable function. To know some properties of a primitive of a Kurzweil-Henstock integrable function, we discuss the absolute Henstock-Kurzweil integral. In this dissertation, we give a brief discussion to Denjoy integral of $C [\alpha , \beta]$ space-valued functions. Further, we show that Denjoy integral is equivalent with Henstock-Kurzweil integral. To discuss this, we introduce the bounded variation function and the absolute continuous function. Finally, we discuss some convergence theorems of a sequence of Henstock-Kurzweil integrable functions. Our objective here is to prove that the monotone, uniform, controlled and dominated convergences of a sequence of Henstock-Kurzweil integral functions imply the convergence of the sequence formed by its corresponding integrals.

Kata Kunci: Henstock-Kurzweil integral, Henstock-Kurzweil primitive, bounded variation function, absolute continuous function, convergence theorem

The research entitled  “Recurrent Neural Network untuk Peramalan Runtun Waktu dengan Pola Long Memory” was conducted by Walid, S.Pd., M.Si. under the guidance of Prof.Drs. Subanar, Ph.D.,  Prof.Dr.rer.Nat. Dedi Rosadi, M.Si., dan Dr. Suhartono, M.Si. in 2017.

The following is the abstract of this research.

ABSTRACT

In daily practice, modeling of time series was often not only involve the lag or order autoregressive (AR) but also involves a lag or order moving average (MA). This condition occurs in both the linear model which known as the model of autoregressive moving average (ARMA) and the nonlinear models, which is one of its forms is a model of recurrent neural networks (RNN). Feedforward neural networks (FFNN) is one of nonlinear models that can be viewed as a group of highly flexible model that can be used for various applications. Recurrent Neural Network as one of the hybrid models are often used to predict and estimate the issues related to electricity, can be used to describe the cause of the swelling of electrical load which experienced by PLN. In this research will be developed RNN forecasting procedures at the time series with long memory patterns. Considering the application is national electrical load which of course has a different trend with the condition of the electrical load in any country. This research produce the algorithm of time series forecasting which has long memory pattern using FFNN hereinafter referred to the algorithm of fractional integrated feedforward neural networks (FIFFNN). In addition, this research also produce the algorithm of time series forecasting which has long memory pattern using RNN in this case using E-RNN hereinafter referred to the algorithm of integrated fractional recurrent neural networks (FIRNN). The forecasting results of long memory time series using the model of Fractional Integrated Feedforward Neural Network (FIFFNN) showed that the model with the selection of data difference in the range of [-1,1] and the model of Fractional Integrated Feedforward Neural Network (FIFFNN) (24,7,1) provides the smallest MSE value, which is 0.00170185. The forecasting results of long memory time series using models Fractional Integrated Recurrent Neural Network (FIRNN) showed that the model with the selection of data difference in the range of [-1,1] and the model of Fractional Integrated Recurrent Neural Network (FIRNN) (24,6,1) provides the smallest MSE value, which is 0.00149684.

Keywords: ARFIMA, FIFFNN Algorithm, FIRNN Algorithm, FFNN, Long Memory, NN, RNN.

The research entitled  “wavelet Neural Network Modeling With Genetic Algorithm” was conducted by Budi Warsito under the guidance of Prof. Drs. Subanar, Ph.D.,  and Dr. Abdurakhman, M.Si. in 2017.

The following is the abstract of this research.

ABSTRACT

Neural Network (NN) is one of the non-linear model which has been developed a lot in statistical modeling, especially in time series analysis. Many researches have shown the advantages of this model compared to the others. However, there are still some problems in NN modeling for time series, including the handling of the input preprocessing, inconsistency of the estimation results and the selection of optimal architecture. The handling of the input preprocessing problem is carried out through the separation of data based on the original components and the noise using wavelet decomposition. Wavelet coefficients resulting from the decomposition are then become the inputs of NN model.Wavelet transformation which is regarded more appropriate for time series data is Maximal Overlap Discrete Wavelet Transform (MODWT), because each level of the decomposition contained wavelet coefficients and scaling coefficients as many as the length of the data. This advantage reduces the weaknesses of filtering by Discrete Wavelet Transform (DWT) which can not be performed on any sample size. The next problems arising is how to determine the level of decomposition and the number of coefficients in each level. In this study, the Maximal Relevance Minimum Redundancy (mRMR) criteria is applied in MODWT decomposition to obtain the optimal input. The criteria based on the Mutual Information (MI) value has been selected due to consider the redundancy between the input variable itself, as well as its value is based on the relevance of input variables with the target.

The optimal input obtained from the wavelet decomposition, then used as input of NN model. The hybrid model formed is named Wavelet Neural Network (WNN) model. Determination of the number of units in the hidden layer emphasizes aspects of computing, through the selection of multiple architectures from simple to more complex and then choose the best one. On the issues relating to the inconsistency of the parameters (weights) estimation which often reach only a local optimum is done through the use of genetic algorithms. The analysis showed that the addition of a super individual would make a genetic algorithm converges to a global optimum.

There are two main contributions of this research. The first is the establishment procedure of determining the optimal architecture of WNN model through the input selection of wavelet coefficients resulting from MODWT decomposition with mRMR criterion. The second contribution is the analysis of convergence of genetic algorithm in time series modeling. Furthermore, the model formed was applied in time series data, either by generating data from a certain model randomly and the real data in financial fields. The experimental results showed that, in general, the stability of the predicted results with the genetic algorithm is better than the conventional methods, especially in the simulation data generated from AR(2), GARCH and nonlinear models. Analysis in real data at the financial fields also shows the stability of predictions resulting from genetic algorithm, especially for the training data.

Keywords: Time series, MODWT, WNN, mRMR, genetic algorithm

The research entitled  “The Wavelet Radial Basis Model for Forecasting Time Series With Jumps” was conducted by Rukun Santoso under the guidance of Prof. Drs. Subanar, Ph.D., Prof. Dr. Dedi Rosadi, M.Sc., and Dr. Suhartono, M.Sc. in 2017.

The following is the abstract of this research.

ABSTRACT

This dissertation research has resulted new computation mathematics model called as Wavelet Radial Basis (WRB) model. This model can be used for nonlinear time series forecasting, especially when clustering effect was occurred. The model construction consist of three stage, i.e. prepossessing through wavelet transformation, filtering the clustering effect through radial basis function, and estimation of model parameters. If the result of radial basis filtering support the linear regression assumption, then ordinary least squared method can be used for parameters estimation. In other case, an empirical solution becomes an alternative solution. One of the recommended solution can be reached by neural network (NN) method.

In the beginning of research, it was successful to be uncovered the fact that if a time series is stationer and follows an AR(1), then the Haar wavelet coefficients of MODWT which are elected as predictors in the MAR model has auto-correlation less than the auto-correlation of AR(1). Under these circumstances the OLS method may be used to estimate the model parameters, but the statistic test of parameters may not significant. The MAR model able to make one step ahead predictions of time series properly when the auto-correlation in the data is quite high. This capability will decline in the situation of auto-correlation in the data is small enough, heteroskedastic existence or jumps existence. The jumps will lead to change the average process which means that the process is not stationary. The addition of radial basis layers in the MAR model architecture will transform wavelet coefficients which are elected as predictors into a new variables which more homogeneous. This condition becomes a strongly support to the fulfillment of linear model assumptions that allows the use of OLS method to estimate the model parameters. The application of WRB model to the simulation data and observation data has pointed out that the assumption of classical regression model are fulfilled, especially refers to the error normality.

Keywords : wavelet, radial basis, nonlinear time series, WRB model, neural network.