KOEFISIEN DETERMINASI REGRESI FUZZY SIMETRIS UNTUK PEMILIHAN MODEL TERBAIK

Abstrak: Dalam analisis regresi biasa, indeks yang digunakan untuk membandingkan dekomposisi dari total jumlah kuadrat variabel dependen tegas adalah koefisien determinasi atau nilai adjusted-nya. Dalam konteks regresi fuzzy dengan variabel dependen fuzzy, diperlukan suatu kriteria pemilihan variabel independen yang menghasilkan model terbaik. Dibangun indeks berdasarkan dekomposisi dari total jumlah kuadrat variabel dependen fuzzy. Pada makalah ini dikaji kriteria pemilihan sub model terbaik dengan menggunakan koefisien determinasi dan nilai adjusted-nya. Selanjutnya diberikan simulasi data yang menggambarkan keefektivan kriteria tersebut. Kata kunci: variabel fuzzy simetris, dekomposisi jumlah kuadrat, koefisien determinasi

SOLUSI KUADRAT TERKECIL MODEL REGRESI FUZZY DENGAN VARIABEL DEPENDEN FUZZY SIMETRIS

ABSTRACT: It is known that the regression model used to analyze the crisp data. In the condition that there is  uncertainty  (imprecision,  vagueness)  on  values  of  observed  variables,  then  the  fuzzy  regression model is required. In this paper discussed a fuzzy regression model with symmetrical fuzzy dependent variable and crisp independent variables using the least squares approach. The method used to find the  linear  model  is  to  minimize  the  distance  function  between  observed  fuzzy  variables  with  the estimation dependent variable or the corresponding theoretical value. It is shown that the solution of this  model  is  a  generalization  of  the  classical  regression  model. Further  discussion  is  about  the properties of solution of the model.Keyword: center model, spread model, fuzzy distance, iterative solution

Outlier detection algorithms over fuzzy data with weighted least squares

In the classical leave-one-out procedure for outlier detection in regression analysis, we exclude an observation and then construct a model on the remaining data. If the difference between predicted and observed value is high we declare this value an outlier. As a rule, those procedures utilize single comparison testing. The problem becomes much harder when the observations can be associated with a given degree of membership to an underlying population, and the outlier detection should be generalized to operate over fuzzy data. We present a new approach for outlier detection that operates over fuzzy data using two inter-related algorithms. Due to the way outliers enter the observation sample, they may be of various order of magnitude. To account for this, we divided the outlier detection procedure into cycles. Furthermore, each cycle consists of two phases. In Phase 1, we apply a leave-one-out procedure for each non-outlier in the dataset. In Phase 2, all previously declared outliers are subjected to Benjamini–Hochberg step-up multiple testing procedure controlling the false-discovery rate, and the non-confirmed outliers can return to the dataset. Finally, we construct a regression model over the resulting set of non-outliers. In that way, we ensure that a reliable and high-quality regression model is obtained in Phase 1 because the leave-one-out procedure comparatively easily purges the dubious observations due to the single comparison testing. In the same time, the confirmation of the outlier status in relation to the newly obtained high-quality regression model is much harder due to the multiple testing procedure applied hence only the true outliers remain outside the data sample. The two phases in each cycle are a good trade-off between the desire to construct a high-quality model (i.e., over informative data points) and the desire to use as much data points as possible (thus leaving as much observations as possible in the data sample). The number of cycles is user defined, but the procedures can finalize the analysis in case a cycle with no new outliers is detected. We offer one illustrative example and two other practical case studies (from real-life thrombosis studies) that demonstrate the application and strengths of our algorithms. In the concluding section, we discuss several limitations of our approach and also offer directions for future research

Outlier detection algorithms over fuzzy data with weighted least squares

In the classical leave-one-out procedure for outlier detection in regression analysis, we exclude an observation and then construct a model on the remaining data. If the difference between predicted and observed value is high we declare this value an outlier. As a rule, those procedures utilize single comparison testing. The problem becomes much harder when the observations can be associated with a given degree of membership to an underlying population, and the outlier detection should be generalized to operate over fuzzy data. We present a new approach for outlier detection that operates over fuzzy data using two inter-related algorithms. Due to the way outliers enter the observation sample, they may be of various order of magnitude. To account for this, we divided the outlier detection procedure into cycles. Furthermore, each cycle consists of two phases. In Phase 1, we apply a leave-one-out procedure for each non-outlier in the dataset. In Phase 2, all previously declared outliers are subjected to Benjamini–Hochberg step-up multiple testing procedure controlling the false-discovery rate, and the non-confirmed outliers can return to the dataset. Finally, we construct a regression model over the resulting set of non-outliers. In that way, we ensure that a reliable and high-quality regression model is obtained in Phase 1 because the leave-one-out procedure comparatively easily purges the dubious observations due to the single comparison testing. In the same time, the confirmation of the outlier status in relation to the newly obtained high-quality regression model is much harder due to the multiple testing procedure applied hence only the true outliers remain outside the data sample. The two phases in each cycle are a good trade-off between the desire to construct a high-quality model (i.e., over informative data points) and the desire to use as much data points as possible (thus leaving as much observations as possible in the data sample). The number of cycles is user defined, but the procedures can finalize the analysis in case a cycle with no new outliers is detected. We offer one illustrative example and two other practical case studies (from real-life thrombosis studies) that demonstrate the application and strengths of our algorithms. In the concluding section, we discuss several limitations of our approach and also offer directions for future research

Goodness of fit and variable selection in the fuzzy multiple linear regression

In performing a fuzzy multiple linear regression model, important topics are: to measure the fitting quality of the model and to find the "best" set of input variables that explain the variation in the observed system responses. In this paper, by considering an exploratory approach, to express the quality of fit of a fuzzy linear regression model, a coefficient of multiple determination R-2 for symmetrical fuzzy variable has been suggested. Furthermore, for overcoming the inconveniences of R-2 an adjusted version of R-2 (denoted by (R) over bar (2)) has been defined. For measuring the fitting performances of the estimated model, a fuzzy extension of another goodness of fit measure, the so-called Mallows index (C-p), has been considered. All the proposed fitting measures have been utilized for selecting suitably the input variables of a fuzzy linear regression model. To this purpose, some variable selection procedures based on R-2, (R) over bar (2) and C-p have been suitably extended in a fuzzy framework. To explain the efficacy of the goodness of fit measures and the variable selection criteria some examples are also shown. (c) 2006 Elsevier B.V. All rights reserved

Estimación Borrosa del Riesgo Beta. Análisis Comparativo

Aquesta tesi representa una aportació a la literatura empírica sobre el risc sistemàtic a nivell sectorial en mercats emergents llatinoamericans, en calcular betes borroses, sectorials i individuals, de Xile, Brasil i Mèxic, i comparar el seu comportament amb el de les betes d’alguns països desenvolupats com Estats Units, Regne Unit i Japó. Proposem una representació borrosa del model de mercat que incorpora el càlcul del rendiment d’un actiu expressat a través d’un interval de confiança. D’aquesta manera incorporem en el càlcul de la beta tota la informació disponible de les cotitzacions d’un actiu durant el dia. Com a resultat de l’estimació amb aquest model obtenim un coeficient beta borrós. Comencem l’estudi comparant i avaluant els resultats obtinguts segons s’expressi la rendibilitat dels actius i segons els diferents mètodes d’estimació de la beta, MCO i regressió borrosa lineal de Tanaka i Ishibuchi (1992) millorada amb el mètode de detecció d’outliers de Hung i Yang (2006). Finalment, avancem en l’estudi de la beta borrosa com a indicador del risc sistemàtic. Proposem una classificació dels actius basada en la beta borrosa i verifiquem si dues de les hipòtesis tradicionals de la teoria de carteres es compleixen en un entorn d’incertesa: i) la beta sectorial presenta major estabilitat que la beta individual; ii) com més gran és el període d’estimació, major és l’estabilitat de la beta.Esta tesis representa un aporte a la literatura empírica sobre el riesgo sistemático a nivel sectorial en mercados emergentes latinoamericanos, al calcular betas borrosas, sectoriales e individuales, en Chile, Brasil y Méjico y comparar su comportamiento con él de las betas de algunos países desarrollados como Estados Unidos, Reino Unido y Japón. Proponemos una representación borrosa del modelo de mercado que incorpora el cálculo del rendimiento de un activo expresado a través de un intervalo de confianza. Con ello incorporamos en el cálculo de la beta toda la información disponible de las cotizaciones de un activo durante el día. Como resultado de la estimación con dicho modelo obtenemos un coeficiente beta borroso. Comenzamos el estudio comparando y evaluando los resultados obtenidos según se exprese la rentabilidad de los activos y según los diferentes métodos de estimación de la beta, MCO y regresión borrosa lineal de Tanaka e Ishibuchi (1992) mejorada con el método de detección de outliers de Hung y Yang (2006). Por último, avanzamos en el estudio de la beta borrosa como indicador del riesgo sistemático. Proponemos una nueva clasificación de los activos basada en la beta borrosa y verificamos si dos de las hipótesis tradicionales de la teoría de carteras se cumplen en un entorno de incertidumbre: i) la beta sectorial presenta mayor estabilidad que la beta individual; ii) Cuánto mayor es el período de estimación, mayor es la estabilidad de la beta.This thesis represents a contribution to empirical literature on systematic risk at the sectoral level in Latin American emerging markets, by calculating fuzzy betas, sectoral and individual, in Chile, Brazil and Mexico, and to compare its behavior with that of betas in some developed countries as the United States, the United Kingdom and Japan. We propose a fuzzy representation of the model of market that incorporates the calculation of the return of an asset expressed through a confidence interval. With it we incorporate in the calculation of the beta all the information available of the quotation of an asset during the day. As a result of the estimation with that model we obtain a fuzzy beta coefficient. We begin the study by comparing and evaluating the results obtained according to assets return and to the different beta methods of estimation, MCO and lineal fuzzy regression of Tanaka e Ishibuchi (1992) improved with the detection of outliers model of the Hung and Yung (2006). Finally, we advance in the study of fuzzy beta as an indicator of systematic risk. We propose a classification of assets based on fuzzy beta and we verify if two of the traditional hypotheses of the portfolio theory are met in an uncertainty environment: i) sectoral beta shows greater stability than individual beta; ii) the longer the estimation period is, the greater the stability of the bet