Time Scale in Least Square Method

Özgür Yeniay,Öznur İşçi,M Niyazi Çankaya,Atilla Göktaş

doi:10.1155/2014/354237

Özgür Yeniay, Öznur İşçi + Show 2 more

Open Access

https://doi.org/10.1155/2014/354237

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Abstract

Study of dynamic equations in time scale is a new area in mathematics. Time scale tries to build a bridge between real numbers and integers. Two derivatives in time scale have been introduced and called as delta and nabla derivative. Delta derivative concept is defined as forward direction, and nabla derivative concept is defined as backward direction. Within the scope of this study, we consider the method of obtaining parameters of regression equation of integer values through time scale. Therefore, we implemented least squares method according to derivative definition of time scale and obtained coefficients related to the model. Here, there exist two coefficients originating from forward and backward jump operators relevant to the same model, which are different from each other. Occurrence of such a situation is equal to total number of values of vertical deviation between regression equations and observation values of forward and backward jump operators divided by two. We also estimated coefficients for the model using ordinary least squares method. As a result, we made an introduction to least squares method on time scale. We think that time scale theory would be a new vision in least square especially when assumptions of linear regression are violated.

Highlights

Theoretical to a high extent, time scale tries to link a bridge between continuous and discrete analysis [1, 2]
Time scale derivative definition is applied to the least squares method
Different β0 and β1 parameter values are obtained, originating from forward and backward jump operators related to the ŷ = β0 + β1x simple linear regression equation

Summary

Introduction

Theoretical to a high extent, time scale tries to link a bridge between continuous and discrete analysis [1, 2]. Time scale derivative definition is applied to the least squares method In this regard, (9) simple linear regression model is considered and β0 and β1, estimators of β0 and β1 coefficients, are obtained in accordance with forward and backward jump operators. Different β0 and β1 parameter values are obtained, originating from forward and backward jump operators related to the ŷ = β0 + β1x simple linear regression equation. The main approach in regression analysis is to minimize the sum of squared (vertical) deviations between actual and estimated values. This is a weighed method for regression. Time scale derivative definition includes normal equations for forward and backward jump operators, as well as β0 and β1 values, which are β0 and β1 estimators of forward and backward jump operators

Time Scale Preliminaries

Figure 1

Simple Linear Regression Analysis

Ordinary Least Square Method

Results

Discussion and Possible