Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9720946, 6 pages

https://doi.org/10.1155/2017/9720946

## Finsler Geometry for Two-Parameter Weibull Distribution Function

^{1}Department of Electrical and Electronics Engineering, Engineering Faculty, Bilecik S.E. University, 11210 Bilecik, Turkey^{2}Department of Computer Engineering, Engineering Faculty, Bilecik S.E. University, 11210 Bilecik, Turkey

Correspondence should be addressed to Emrah Dokur

Received 19 December 2016; Accepted 16 February 2017; Published 9 March 2017

Academic Editor: Qin Yuming

Copyright © 2017 Emrah Dokur et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To construct the geometry in nonflat spaces in order to understand nature has great importance in terms of applied science. Finsler geometry allows accurate modeling and describing ability for asymmetric structures in this application area. In this paper, two-dimensional Finsler space metric function is obtained for Weibull distribution which is used in many applications in this area such as wind speed modeling. The metric definition for two-parameter Weibull probability density function which has shape () and scale () parameters in two-dimensional Finsler space is realized using a different approach by Finsler geometry. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed which can be used in many real world applications. For future studies, it is aimed at proposing more accurate models by using this novel approach than the models which have two-parameter Weibull probability density function, especially used for determination of wind energy potential of a region.

#### 1. Introduction

Two-parameter Weibull function is one of the most used distribution functions for different purposes such as modeling, reliability analysis, life time data analysis, and many applied science areas such as mechanic, biosystem, nuclear, and energy system engineering [1–6]. In the literature studies, it is seen that two-parameter Weibull distribution is mainly used for the determination of wind energy potential in the different regions in the world [7–14]. The reasons that the usage of two-parameter Weibull distribution in this area are very good fit to the wind distribution, flexible structure of distribution, and having two-parameter. Also, after the determination of parameters for an observation height, parameters can be estimated for different height.

Before the installation of a wind energy conversion system in a region, the wind speed potential of that region needs to be determined and modeled. In line with this purpose, the most important problem of modeling by two-parameter Weibull distribution is accuracy estimation of parameters associated with designing optimal model. In accordance with this purpose, many different statistical and empirical methods are developed in the literature [15–19]. Also, different function structures such as Rayleigh, Lognormal, Gamma, and Burr are used for accurately modeling wind speed in the literature [20–22]. Determination of a new distribution function in order to develop correct and accurate model structure has importance for wind speed modeling in different regions and other real world application problems.

The fact that the wind speed and similar distributions have a nonsymmetrical and unstable character brings along many difficulties from the stand point of modeling. In this context, Finsler geometry is a very strong tool than well-known Riemann geometry for modeling physical phenomena that are genuinely asymmetric and/or nonisotropic [23–26].

Finsler metric function whose geodesics have two-parameter family of curve in Finsler space is obtained by Matsumoto [27–29]. In this paper, Finsler metrics which are associated with different parameters, defined in nonnegative real numbers, are derived and they are obtained by Weibull distribution function which has two-parameter curve family. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed in this paper. Calculation of geodesics that have Finsler metrics and novel two-parameter probability and cumulative probability density functions is evaluated for chosen different nonnegative numbers, comparatively. Two-parameter Weibull distribution function structure is presented with an example in Section 2. Definition of Weibull distribution that has two-parameter family of curve and Finsler metrics that are obtained for two-parameter curve family are given in Section 3. In the last section, Finsler metrics that have two-parameter Weibull distribution function family of curve and their geodesics are evaluated for nonnegative different numbers, comparatively. Finally, conclusions are given in Section 5.

#### 2. Two-Parameter Weibull Distribution

Two-parameter Weibull distribution is used in many real world applications. In this section, the structure of the two-parameter Weibull distribution function will be discussed on the real world problem such as the wind speed distribution which has a nonlinear structure in the asymmetric platform.

Two-parameter Weibull distribution is one of the widely used statistical methods in the modeling of wind speed data. The Weibull distribution function is given by the following [30–36]:where is the frequency or probability of occurrence of wind speed , is the Weibull scale parameter with unit equal to the wind speed unit (m/s), and is the unitless Weibull shape parameter. The higher value of indicates that the wind speed is higher, while the value of shows the wind stability [37].

The cumulative Weibull distribution function gives the probability of the wind speed exceeding the value . It is expressed by the following [38, 39]: Probability and cumulative probability density function with sample wind speed data that is Bilecik region in Turkey are shown in Figure 1 for two-parameter Weibull distribution in which scale () and shape () parameters are calculated, 1.9416 and 2.5110, respectively, by maximum likelihood method.