Research Article Time Scale in Least Square Method

Research ArticleTime Scale in Least Square Method

Oumlzguumlr Yeniay1 Oumlznur EGccedili2 Atilla GoumlktaG2 and M Niyazi Ccedilankaya3

1 Department of Statistics Faculty of Sciences Hacettepe University Beytepe 06800 Ankara Turkey2Department of Statistics Faculty of Sciences Mugla Sıtkı Kocman University 48000 Mugla Turkey3 Department of Statistics Faculty of Sciences Ankara University Besevler 06100 Ankara Turkey

Correspondence should be addressed to Ozgur Yeniay yeniayhacettepeedutr

Received 8 January 2014 Accepted 6 March 2014 Published 3 April 2014

Academic Editor Dumitru Baleanu

Copyright copy 2014 Ozgur Yeniay et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Although theoretical to a high extent time scale tries tolink a bridge between continuous and discrete analysis [1 2]Such calculations provide an integrated structure for theanalysis of difference and differential equations [3ndash5] Thoseequations [6 7] have been applied in dynamic programming[8ndash13] neural network [10 14 15] economic modeling[10 16] population biology [17] quantum calculus [18]geometric analysis [19] real-time communication networks[20] intelligent robotic control [21] adaptive sampling [22]approximation theory [13] financial engineering [12] on timescales and switched linear circuits [23] among others

This study deals with the estimation of parameters ofregression equation by using least squares method throughtime scale

Themain purpose of least squares method is to minimizethe sum of squared vertical deviations For this to find thecoefficients in simple linear regression model 119884 = 120573

0+ 1205731119909 +

120576 partial derivatives related to coefficients of equation 119876 =

sum119899

119894=1120576119894

2= sum119899

119894=1(119910119894minus 1205730minus 1205731119909119894)2 are obtained and the stage

takes place in which normal equations are to be obtained bysetting partial derivatives for each coefficient equal to zeroAnd from normal equations equation 119910

119894= 1205730+ 1205731119909119894+ 119890119894

predictive model for (9) is obtainedIn statistics least squares method is used in accordance

with the known derivative definition when parameters ofregression equation are obtained In this study time scalederivative definition is applied to the least squaresmethod Inthis regard (9) simple linear regression model is consideredand 120573

0and 120573

1 estimators of 120573

0and 120573

1coefficients are

obtained in accordance with forward and backward jumpoperators Different 120573

0and 120573

1parameter values are obtained

originating from forward and backward jump operatorsrelated to the 119910 = 120573

0+1205731119909 simple linear regression equation

In cases when observation values are discrete forward jumpoperator equation 120590(119905) = 119905 + 1 and backward jump operatorequation 120588(119905) = 119905 minus 1 are taken

The main approach in regression analysis is to minimizethe sum of squared (vertical) deviations between actual andestimated values This is a weighed method for regression

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 354237 6 pageshttpdxdoiorg1011552014354237

2 Abstract and Applied Analysis

analysis due to its statistical properties [24] Here the pointwhich should be considered is that since in least squaresmethod sum of squared vertical deviations is of interestanalysis would also operate accordinglyThe issue of applyingfirst and second derivatives should also be taken into con-sideration since the aim is to minimize the sum of squaredvertical deviations

The study consists of two main parts and in the last parttakes place an implementation regarding time scale theoryMain parts discussed are time scale theory and simple linearregression respectively Time scale theory part includes anexplanation of time scale derivative related to and the defi-nitions of forward and backward jump operators The othermain part includes explanations to simple linear regressionmodel being in the form of (9) and to the calculation of 120573

0

and 1205731 estimators of 120573

0and 120573

1 by using the method of least

squares Time scale derivative definition includes normalequations for forward and backward jump operators as wellas1205730and1205731values which are120573

0and1205731estimators of forward

and backward jump operators

2 Time Scale Preliminaries

Bohner and Peterson [1] suggested the concept of time scalein their studies Their purpose was to bring together discreteanalysis and continuous analysis under one model Anyclosed nonempty subset 119879 of real numbers R is called astime scale and is indicated with symbol 119879 ThusR Ζ119873119873

0

real numbers integers natural numbers and positive naturalnumbers respectively are examples of time scale and consid-ered as [0 1] cup [2 3] [0 1] cup 119873 meaning 119871 = 0 and 119871 sube Rmeaning [ ] cup [ ] cup sdot sdot sdot =R

119876R 119876 119862 (0 1) rational numbers irrational numberscomplex numbers and open interval of 0 and 1 respectivelyare not included in time scale Since time scale is closed it canclearly be seen that rational numbers are never a time scale

Delta derivative 119891Δ for function 119891 defined at 119879 is actually

defined as follows(i) If 119891Δ = 119891

1015840 119879 = R then it is a general derivative(ii) If 119891

Δ= Δ119891 119879 = 119885 then it is a forward difference

operator

Definition 1 Given the function119891 119879 rarr 119877 and 119905 isin 119879119870

isingt 0there exists a neighborhood119880 of 119905 (119880 = (119905minus120575 119905+120575)) and 120575 gt 0

such that10038161003816100381610038161003816[119891 (120590 (119905)) minus 119891 (119904)] minus 119891

Δ(119905) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120576 |120590 (119905) minus 119904|

(1)

is defined for all 119904 isin 119880 and called as delta derivative of119891Δ(119905)119891at 119905 [1 25 26]

Definition 2 Given the function 119891 119879 rarr 119877 and 119905 isin 119879119870

isingt

0 there exists a neighborhood 119880 of 119905 (119880 = (119905 minus 120575 119905 + 120575)) and120575 gt 0 such that

10038161003816100381610038161003816[119891 (120588 (119905)) minus 119891 (119904)] minus 119891

nabla(119905) [120588 (119905) minus 119904]

10038161003816100381610038161003816le 120576

1003816100381610038161003816120588 (119905) minus 1199041003816100381610038161003816

(2)

X

Y Observed valueof datum Y

regression lineEstimated

ei

Figure 1 Variance of data from estimated regression model

is defined for all 119904 isin 119880 and called as nabla derivative of119891nabla(119905)119891 at 119905 [16 26]



such that1003816100381610038161003816[119891 (120590 (119905)) minus 119891 (120588 (119905))] minus 119891

119888(119905) [120590 (119905) minus 120588 (119905)]

1003816100381610038161003816

le 1205761003816100381610038161003816120590 (119905) minus 120588 (119905)

1003816100381610038161003816

(3)

is defined for all 119904 isin 119880 and called as center derivative of119891119888(119905)119891 at 119905 [27]

Definition 4 (forward jump operator [1]) Let 119879 be a timescale Then for 119905 isin 119879 forward jump operator 120590 119879 rarr 119879

is defined by

120590 (119905) = min 119904 isin 119879 119904 gt 119905 (4)

For right-graininess function for all 119909 isin 119879 120583120590

119879 rarr [0infin)

is defined as follows

120583120590(119905) = 120590 (119905) minus 119905 (5)

Definition 5 (backward jump operator [1]) Let 119879 be a timescale Then for 119905 isin 119879 backward jump operator 120588 119879 rarr 119879 isdefined by

120588 (119905) = max 119904 isin 119879 119904 lt 119905 (6)

Left-graininess function 120583120588

119879 rarr [0infin) is defined asfollows

120583120588(119905) = 119905 minus 120588 (119905) (7)

3 Simple Linear Regression Analysis

Simple linear regression [28 29] consists of one singleexplanatory variable or independent variable or responsevariable or dependent variable Let us take that a realrelationship exists between 119884 and 119909 and 119884 values observed ateach 119909 level are random Expected value of 119884 at each 119909 levelis defined by the equation

119864 (119884 | 119909) = 1205730+ 1205731119909 (8)

Abstract and Applied Analysis 3

where 1205730is the interceptor and 120573

1is the slope and 120573

0and 120573

1

are unknown regression coefficients Each 119884 observation canbe defined with the model below

In equation

119884 = 1205730+ 1205731119909 + 120576 (9)

120576 is the random error term with zero mean constitution of(unknown) variance 120590

2Take 119899 binary observations of (119909

1 1199101) (1199092 1199102)

(119909119899 119910119899) Figure 1 shows the distribution diagram of observed

values and the estimated regression line Estimators of 1205730

and 1205731would pass the ldquobest linerdquo through the data German

scientist Karl Gauss (1777ndash1855) made a suggestion inestimation of parameters 120573

0and 120573

1in (9) to minimize the

sum of squared vertical deviations in Figure 1This criterion used in estimation of regression parameters

is the least squares method By using (9) 119899 observations in thesample can be defined as follows

119910119894= 1205730+ 1205731119909119894+ 120576119894 119894 = 1 2 119899 (10)

and sum of squared deviations of observations from theactual regression line is as follows

119876 =

119899

sum

119894=1

1205762

119894=

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894)2 (11)

Least square estimators of 1205730and1205731are1205730and 1205731 whichmay

be calculated as follows

120597119876

1205971205730

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) = 0

120597119876

1205971205731

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) 119909119894= 0

(12)

When these two equations are simplified

1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

(13)

These equations may be called as normal equations of leastsquares

Thus estimated regression line will be as follows

119910 = 1205730+ 1205731119909 (14)

Each observation pair is provided with the relation below

119910119894= 1205730+ 1205731119909119894+ 119890119894 119894 = 1 2 119899 (15)

where 119890119894

= 119910119894minus 119910119894 residuals 119910

119894 observation values 119910

119894

estimated 119910119894values

4 Ordinary Least Square Method

41 Normal Equations These include normal situation andformula values of 120573

0and 120573

1estimators for forward and

backward operatorsNormal equations in normal (usual) situations

1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

1205730=

sum119899

119894=1119909119894sum119899

119894=1119910119894minus 119899sum119899

119894=1119910119894119909119894

minus119899sum119899

119894=11199092

119894+ (sum119899

119894=1119909119894)2

1205731=

minussum119899

119894=1119910119894sum119899

119894=11199092

119894+ sum119899

119894=1119909119894sum119899

119894=1119910119894119909119894


119894=11199092

119894+ (sum119899

119894=1119909119894)2

(16)

Time scale normal equations (forward jump operator)

1198991205730+

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894+

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894

1205730119911119894

= minus(119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894minus 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119894

= (119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119909119894119910119894minus 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(17)

Time scale normal equations (backward jump operator)

1198991205730minus

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894minus

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894


1205730119911119892

= (minus119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894+ 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

minus2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119892

= (minus119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119910119894119909119894+ 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(18)

42 Graphs with Data 10-20-30-40-50 of Normal SituationForward and Backward Jump Operator Respective illustra-tions of simple linear regression equations obtained fromsamples of 10 20 30 40 and 50 data for normal situationforward and backward jump operators are shown in Figure 2

Sum of vertical distances between simple linear regres-sion model for normal and forward and backward operatorsand observation values 119884

119894for sample sizes of 119899 = 10 20 30

40 and 50 are provided in Table 1There is a relationship between sample size and sum of

vertical distances between regression line and observationvalues 119884

119894 This is derived from the result of sum of vertical

distances between regression lines equal to half of sample sizeand observation values 119884

119894 Results of implementation can be

seen in Table 1

43 Minimum Test Let (119896119898) 119876(119886 119887) the critical pointmeaning 119876

119886(119896119898) = 0 and 119876

119887(119896 119898) = 0

119871 = 119876119886119886

(119896119898)119876119887119887

(119896119898) minus [119876119886119887

(119896119898)]2 (19)

If 119871 gt 0 and 119876119886119886

(119896119898) gt 0 119876(119896119898) has a minimum valueMinimum test calculated according to normal derivativedefinition is as follows [30]

1205972119876

120597119886119886= 119899

1205972119876

120597119887119887=

119899

sum

119894=1

1199092

119894

1205972119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(20)

30

20

10

0 5 10

x

minus10 minus5

minus10

minus20

minus30

minus40

Figure 2 Simple linear regression equations for normal situationand forward and backward operators

Minimum test calculated according to time scale derivativedefinition for forward and backward jump operators is asfollows

120597nablanabla

120597119886119886=

120597ΔΔ

119876

120597119886119886= 119899

120597nablanabla

120597119887119887=

120597ΔΔ

119876

120597119887119887=

119899

sum

119894=1

1199092

119894

120597nablanabla

120597119886119887=

120597ΔΔ

119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(21)

Since 119871 gt 0 and 119876119886119886

gt 0 always 119876(119896119898) always has aminimum value

5 Results

In statistics least squares method is used in accordancewith the known derivative definition when parameters ofregression equation are obtained In the study time scalederivative definition is applied to the least squaresmethod Inthis regard (9) simple linear regression model is consideredand both 120573

0and 120573


0and 120573

1coefficients are


0and 120573




In cases when observation values are discrete forward jumpoperator equation 120590(119905) = 119905 + 1 and backward jump operatorequation 120588(119905) = 119905 minus 1 are taken [31]

The study includes the analysis of integers 119885 inaccordance with time scale derivative definition Standard


Table 1 Values for results obtained

For 119899 = 10

Simple linear regression equation Sum of vertical deviations Observation 119884119894

Estimation 119884119894

Normal 327 + 101x 0 62 62Fwd 742 minus 059x 5 62 57Bckwd minus089 + 262119909 minus5 62 67

For 119899 = 20



Normal 290 + 109119909 0 155 155Fwd 1126 minus 090119909 10 155 145Bckwd minus546 + 308119909 minus10 155 165

For 119899 = 30




For 119899 = 40




For 119899 = 50




approach in regression analysis is to minimize the sum ofsquared (vertical) deviations between actual and estimatedvalues Here the point which should be considered isthat since in least squares method sum of squaredvertical deviations is of interest analysis would also operateaccordinglyThe issue of applying first and second derivativesshould also be taken into consideration since the aim is tominimize the sum of squared vertical deviations Since 119871 gt 0

and 119876119886119886

gt 0 always 119876(119896119898) always has a minimum valueLeast square method yields results such that sum of

vertical deviations is minimum When least squares methodis used according to time scale derivative definition arelationship emerges between sample size and sum of verticaldistances between regression line and observation values 119884

119894

This is derived from the sum of vertical distances betweenregression lines equal to half of sample size and observa-tion values 119884

119894 To minimize the sum of vertical distances

different 1205730and 120573

1parameter values resulting from forward

and backward jump operators for simple linear regressionequation 119910 = 120573

0+ 1205731119909 have been of interest An alternative

solution is to create a regression equation in time scale As aconclusion time scale derivative definition is applied to thestudy with integers 119885 and suggested solutions are proposedfor the results obtained

6 Discussion and Possible Future Studies

This study is only introducing a very basic derivative conceptfrom the time scale and applying for obtaining the regressionparameters An extension of this study would be determiningthe integer 119885 apart from the value 1 This would be usefulespecially when the actual assumptions of linear regressionmodel have been violated and need a robust estimation oflinear regression line Perhaps it is hard to determine anoptimum of 119885 analytically We would suggest that the studyshould be performed for generated data containing outliersthat is replicated at least 1000 times

It is also possible to extend the number of regressorsfrom a single explanatory variable to multiple ones in thelinear regression model and estimate the parameters usingtime scale of both forward and backward jump operatorsAnalytically speaking it will not be easy to estimate theparameters in a simple form of equation using time scaleIn fact we would be dealing with very complicated eitherformulas ormatrices However it may be worth extending thestudy to a multiple linear regression model

In the meantime taking 119885 to be 1 results with the sumof vertical distances between regression lines equal to half ofsample size (See Table 1 and Figure 2) The 119885 value in bothforward and backward jump operators should be determinedin a way that the estimated regression line from both forward


and backward operator is fairly close to the actual regressionline especially when the assumptions of linear regressionmodel are violated

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[2] W B Powell Approximate Dynamic Programming Solving theCurses of Dimensionality John Wiley amp Sons New York NYUSA 2011

[3] E Girejko A B Malinowska and D F M Torres ldquoThecontingent epiderivative and the calculus of variations on timescalesrdquo Optimization A Journal of Mathematical Programmingand Operations Research vol 61 no 3 pp 251ndash264 2012

[4] N H Du and N T Dieu ldquoThe first attempt on the stochasticcalculus on time scalerdquo Stochastic Analysis and Applications vol29 no 6 pp 1057ndash1080 2011

[5] R Almeida and D F M Torres ldquoIsoperimetric problems ontime scales with nabla derivativesrdquo Journal of Vibration andControl vol 15 no 6 pp 951ndash958 2009

[6] M Bohner and A Peterson Eds Advances in Dynamic Equa-tions on Time Scales Birkhauser Boston Mass USA 2003

[7] M Bohner ldquoCalculus of variations on time scalesrdquo DynamicSystems and Applications vol 13 no 3-4 pp 339ndash349 2004

[8] J Seiffertt S Sanyal and D C Wunsch ldquoHamilton-Jacobi-Bellman equations and approximate dynamic programmingon time scalesrdquo IEEE Transactions on Systems Man andCybernetics B Cybernetics vol 38 no 4 pp 918ndash923 2008

[9] Z Han S Sun and B Shi ldquoOscillation criteria for a class ofsecond-order Emden-Fowler delay dynamic equations on timescalesrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 847ndash858 2007

[10] C C Tisdell and A Zaidi ldquoBasic qualitative and quantitativeresults for solutions to nonlinear dynamic equations on timescales with an application to economic modellingrdquo NonlinearAnalysis Theory Methods amp Applications vol 68 no 11 pp3504ndash3524 2008

[11] C Lizama and J G Mesquita ldquoAlmost automorphic solutionsof dynamic equations on time scalesrdquo Journal of FunctionalAnalysis vol 265 no 10 pp 2267ndash2311 2013

[12] S Sanyal Stochastic Dynamic Equations [PhD dissertation]Missouri University of Science and Technology Rolla MoUSA 2008

[13] Q Sheng M Fadag J Henderson and J M Davis ldquoAnexploration of combineddynamic derivatives on time scales andtheir applicationsrdquoNonlinear Analysis Real World Applicationsvol 7 no 3 pp 395ndash413 2006

[14] X Chen and Q Song ldquoGlobal stability of complex-valuedneural networks with both leakage time delay and discrete timedelay on time scalesrdquo Neurocomputing vol 121 no 9 pp 254ndash264 2013

[15] AChen andFChen ldquoPeriodic solution toBAMneural networkwith delays on time scalesrdquoNeurocomputing vol 73 no 1ndash3 pp274ndash282 2009

[16] F M Atici D C Biles and A Lebedinsky ldquoAn applicationof time scales to economicsrdquo Mathematical and ComputerModelling vol 43 no 7-8 pp 718ndash726 2006

[17] M BohnerM Fan and J Zhang ldquoPeriodicity of scalar dynamicequations and applications to population modelsrdquo Journal ofMathematical Analysis and Applications vol 330 no 1 pp 1ndash92007

[18] M Bohner and T Hudson ldquoEuler-type boundary value prob-lems in quantum calculusrdquo International Journal of AppliedMathematics amp Statistics vol 9 no J07 pp 19ndash23 2007

[19] G Sh Guseinov and E Ozyılmaz ldquoTangent lines of generalizedregular curves parametrized by time scalesrdquo Turkish Journal ofMathematics vol 25 no 4 pp 553ndash562 2001

[20] I Gravagne R J Marks II J Davis and J DaCunha ldquoAppli-cation of time scales to real world time communicationsnetworksrdquo in Proceedings of the American Mathematical SocietyWestern Section Meeting 2004

[21] I A Gravagne J M Davis and R J Marks ldquoHow deterministicmust a real-time controller berdquo in Proceedings of the IEEEIRSRSJ International Conference on Intelligent Robots andSystems (IROS rsquo05) pp 3856ndash3861 Edmonton Canada August2005

[22] I Gravagne J Davis J DaCunha and R J Marks II ldquoBand-width reduction for controller area networks using adaptivesamplingrdquo in Proceedings of the International Conference onRobotics and Automation pp 2ndash6 NewOrleans La USA April2004

[23] R J Marks II I A Gravagne J M Davis and J J DaCunhaldquoNonregressivity in switched linear circuits and mechanicalsystemsrdquoMathematical and Computer Modelling vol 43 no 11-12 pp 1383ndash1392 2006

[24] J S Armstrong Principles of Forecasting A Handbook forResearchers and Practitioners Kluwer Academic DordrechtThe Netherlands 2001

[25] D R Anderson and J Hoffacker ldquoGreenrsquos function for aneven order mixed derivative problem on time scalesrdquo DynamicSystems and Applications vol 12 no 1-2 pp 9ndash22 2003

[26] R AgarwalM Bohner DOrsquoRegan andA Peterson ldquoDynamicequations on time scales a surveyrdquo Journal of Computationaland Applied Mathematics vol 141 no 1-2 pp 1ndash26 2002

[27] Q Sheng and A Wang ldquoA study of the dynamic differenceapproximations on time scalesrdquo International Journal of Differ-ence Equations vol 4 no 1 pp 137ndash153 2009

[28] J Neter M H Kutner C J Nachtsheim and W WassermanApplied Linear Statistical Models McGraw-Hill New York NYUSA 4th edition 1996

[29] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 3rd edition 2002

[30] J R Hass F R Giordano and M D Weir Thomasrsquo CalculusPearson Addison Wesley 10th edition 2004

[31] R P Agarwal andM Bohner ldquoBasic calculus on time scales andsome of its applicationsrdquo Results inMathematics vol 35 no 1-2pp 3ndash22 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


analysis due to its statistical properties [24] Here the pointwhich should be considered is that since in least squaresmethod sum of squared vertical deviations is of interestanalysis would also operate accordinglyThe issue of applyingfirst and second derivatives should also be taken into con-sideration since the aim is to minimize the sum of squaredvertical deviations

The study consists of two main parts and in the last parttakes place an implementation regarding time scale theoryMain parts discussed are time scale theory and simple linearregression respectively Time scale theory part includes anexplanation of time scale derivative related to and the defi-nitions of forward and backward jump operators The othermain part includes explanations to simple linear regressionmodel being in the form of (9) and to the calculation of 120573

0

and 1205731 estimators of 120573

0and 120573

1 by using the method of least

squares Time scale derivative definition includes normalequations for forward and backward jump operators as wellas1205730and1205731values which are120573

0and1205731estimators of forward

and backward jump operators

2 Time Scale Preliminaries

Bohner and Peterson [1] suggested the concept of time scalein their studies Their purpose was to bring together discreteanalysis and continuous analysis under one model Anyclosed nonempty subset 119879 of real numbers R is called astime scale and is indicated with symbol 119879 ThusR Ζ119873119873

0

real numbers integers natural numbers and positive naturalnumbers respectively are examples of time scale and consid-ered as [0 1] cup [2 3] [0 1] cup 119873 meaning 119871 = 0 and 119871 sube Rmeaning [ ] cup [ ] cup sdot sdot sdot =R

119876R 119876 119862 (0 1) rational numbers irrational numberscomplex numbers and open interval of 0 and 1 respectivelyare not included in time scale Since time scale is closed it canclearly be seen that rational numbers are never a time scale

Delta derivative 119891Δ for function 119891 defined at 119879 is actually

defined as follows(i) If 119891Δ = 119891

1015840 119879 = R then it is a general derivative(ii) If 119891

Δ= Δ119891 119879 = 119885 then it is a forward difference

operator



such that10038161003816100381610038161003816[119891 (120590 (119905)) minus 119891 (119904)] minus 119891

Δ(119905) [120590 (119905) minus 119904]

10038161003816100381610038161003816le 120576 |120590 (119905) minus 119904|

(1)

is defined for all 119904 isin 119880 and called as delta derivative of119891Δ(119905)119891at 119905 [1 25 26]

Definition 2 Given the function 119891 119879 rarr 119877 and 119905 isin 119879119870

isingt

0 there exists a neighborhood 119880 of 119905 (119880 = (119905 minus 120575 119905 + 120575)) and120575 gt 0 such that

10038161003816100381610038161003816[119891 (120588 (119905)) minus 119891 (119904)] minus 119891

nabla(119905) [120588 (119905) minus 119904]

10038161003816100381610038161003816le 120576

1003816100381610038161003816120588 (119905) minus 1199041003816100381610038161003816

(2)

X

Y Observed valueof datum Y

regression lineEstimated

ei

Figure 1 Variance of data from estimated regression model

is defined for all 119904 isin 119880 and called as nabla derivative of119891nabla(119905)119891 at 119905 [16 26]



such that1003816100381610038161003816[119891 (120590 (119905)) minus 119891 (120588 (119905))] minus 119891

119888(119905) [120590 (119905) minus 120588 (119905)]

1003816100381610038161003816

le 1205761003816100381610038161003816120590 (119905) minus 120588 (119905)

1003816100381610038161003816

(3)

is defined for all 119904 isin 119880 and called as center derivative of119891119888(119905)119891 at 119905 [27]

Definition 4 (forward jump operator [1]) Let 119879 be a timescale Then for 119905 isin 119879 forward jump operator 120590 119879 rarr 119879

is defined by

120590 (119905) = min 119904 isin 119879 119904 gt 119905 (4)

For right-graininess function for all 119909 isin 119879 120583120590

119879 rarr [0infin)

is defined as follows

120583120590(119905) = 120590 (119905) minus 119905 (5)

Definition 5 (backward jump operator [1]) Let 119879 be a timescale Then for 119905 isin 119879 backward jump operator 120588 119879 rarr 119879 isdefined by

120588 (119905) = max 119904 isin 119879 119904 lt 119905 (6)

Left-graininess function 120583120588

119879 rarr [0infin) is defined asfollows

120583120588(119905) = 119905 minus 120588 (119905) (7)

3 Simple Linear Regression Analysis

Simple linear regression [28 29] consists of one singleexplanatory variable or independent variable or responsevariable or dependent variable Let us take that a realrelationship exists between 119884 and 119909 and 119884 values observed ateach 119909 level are random Expected value of 119884 at each 119909 levelis defined by the equation

119864 (119884 | 119909) = 1205730+ 1205731119909 (8)




0and 120573

1


In equation

119884 = 1205730+ 1205731119909 + 120576 (9)



1 1199101) (1199092 1199102)





0and 120573




119910119894= 1205730+ 1205731119909119894+ 120576119894 119894 = 1 2 119899 (10)


119876 =

119899

sum

119894=1

1205762

119894=

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894)2 (11)



120597119876

1205971205730

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) = 0

120597119876

1205971205731

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) 119909119894= 0

(12)


1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

(13)



119910 = 1205730+ 1205731119909 (14)


119910119894= 1205730+ 1205731119909119894+ 119890119894 119894 = 1 2 119899 (15)

where 119890119894

= 119910119894minus 119910119894 residuals 119910


119894




0and 120573



1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

1205730=

sum119899

119894=1119909119894sum119899

119894=1119910119894minus 119899sum119899

119894=1119910119894119909119894


119894=11199092

119894+ (sum119899

119894=1119909119894)2

1205731=

minussum119899

119894=1119910119894sum119899

119894=11199092

119894+ sum119899

119894=1119909119894sum119899

119894=1119910119894119909119894


119894=11199092

119894+ (sum119899

119894=1119909119894)2

(16)


1198991205730+

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894+

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894

1205730119911119894

= minus(119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894minus 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119894

= (119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119909119894119910119894minus 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(17)


1198991205730minus

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894minus

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894


1205730119911119892

= (minus119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894+ 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

minus2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119892

= (minus119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119910119894119909119894+ 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(18)









seen in Table 1


119886(119896119898) = 0 and 119876

119887(119896 119898) = 0

119871 = 119876119886119886

(119896119898)119876119887119887

(119896119898) minus [119876119886119887

(119896119898)]2 (19)

If 119871 gt 0 and 119876119886119886


1205972119876

120597119886119886= 119899

1205972119876

120597119887119887=

119899

sum

119894=1

1199092

119894

1205972119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(20)

30

20

10

0 5 10

x

minus10 minus5

minus10

minus20

minus30

minus40



120597nablanabla

120597119886119886=

120597ΔΔ

119876

120597119886119886= 119899

120597nablanabla

120597119887119887=

120597ΔΔ

119876

120597119887119887=

119899

sum

119894=1

1199092

119894

120597nablanabla

120597119886119887=

120597ΔΔ

119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(21)

Since 119871 gt 0 and 119876119886119886


5 Results


0and 120573


0and 120573

1coefficients are


0and 120573








For 119899 = 10




For 119899 = 20




For 119899 = 30




For 119899 = 40




For 119899 = 50





and 119876119886119886



119894
















References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0and 120573

1


In equation

119884 = 1205730+ 1205731119909 + 120576 (9)



1 1199101) (1199092 1199102)





0and 120573




119910119894= 1205730+ 1205731119909119894+ 120576119894 119894 = 1 2 119899 (10)


119876 =

119899

sum

119894=1

1205762

119894=

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894)2 (11)



120597119876

1205971205730

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) = 0

120597119876

1205971205731

| 1205730 120573

1= minus2

119899

sum

119894=1

(119910119894minus 1205730minus 1205731119909119894) 119909119894= 0

(12)


1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

(13)



119910 = 1205730+ 1205731119909 (14)


119910119894= 1205730+ 1205731119909119894+ 119890119894 119894 = 1 2 119899 (15)

where 119890119894

= 119910119894minus 119910119894 residuals 119910


119894




0and 120573



1198991205730+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894=

119899

sum

119894=1

119910119894119909119894

1205730=

sum119899

119894=1119909119894sum119899

119894=1119910119894minus 119899sum119899

119894=1119910119894119909119894


119894=11199092

119894+ (sum119899

119894=1119909119894)2

1205731=

minussum119899

119894=1119910119894sum119899

119894=11199092

119894+ sum119899

119894=1119909119894sum119899

119894=1119910119894119909119894


119894=11199092

119894+ (sum119899

119894=1119909119894)2

(16)


1198991205730+

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894+

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894

1205730119911119894

= minus(119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894minus 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119894

= (119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119909119894119910119894minus 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(17)


1198991205730minus

119899

2+ 1205731

119899

sum

119894=1

119909119894=

119899

sum

119894=1

119910119894

1205730

119899

sum

119894=1

119909119894+ 1205731

119899

sum

119894=1

1199092

119894minus

sum119899

119894=11199092

119894

2=

119899

sum

119894=1

119910119894119909119894


1205730119911119892

= (minus119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894+ 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

minus2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119892

= (minus119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119910119894119909119894+ 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(18)









seen in Table 1


119886(119896119898) = 0 and 119876

119887(119896 119898) = 0

119871 = 119876119886119886

(119896119898)119876119887119887

(119896119898) minus [119876119886119887

(119896119898)]2 (19)

If 119871 gt 0 and 119876119886119886


1205972119876

120597119886119886= 119899

1205972119876

120597119887119887=

119899

sum

119894=1

1199092

119894

1205972119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(20)

30

20

10

0 5 10

x

minus10 minus5

minus10

minus20

minus30

minus40



120597nablanabla

120597119886119886=

120597ΔΔ

119876

120597119886119886= 119899

120597nablanabla

120597119887119887=

120597ΔΔ

119876

120597119887119887=

119899

sum

119894=1

1199092

119894

120597nablanabla

120597119886119887=

120597ΔΔ

119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(21)

Since 119871 gt 0 and 119876119886119886


5 Results


0and 120573


0and 120573

1coefficients are


0and 120573








For 119899 = 10




For 119899 = 20




For 119899 = 30




For 119899 = 40




For 119899 = 50





and 119876119886119886



119894
















References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205730119911119892

= (minus119899

119899

sum

119894=1

1199092

119894+

119899

sum

119894=1

119909119894

119899

sum

119894=1

1199092

119894+ 2

119899

sum

119894=1

119909119894

119899

sum

119894=1

119910119894119909119894

minus2

119899

sum

119894=1

119910119894

119899

sum

119894=1

1199092

119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

1205731119911119892

= (minus119899

119899

sum

119894=1

1199092

119894minus 2119899

119899

sum

119894=1

119910119894119909119894+ 119899

119899

sum

119894=1

119909119894

+2

119899

sum

119894=1

119910119894

119899

sum

119894=1

119909119894)

times ((minus119899

119899

sum

119894=1

1199092

119894+ (

119899

sum

119894=1

119909119894)

2

)2)

minus1

(18)









seen in Table 1


119886(119896119898) = 0 and 119876

119887(119896 119898) = 0

119871 = 119876119886119886

(119896119898)119876119887119887

(119896119898) minus [119876119886119887

(119896119898)]2 (19)

If 119871 gt 0 and 119876119886119886


1205972119876

120597119886119886= 119899

1205972119876

120597119887119887=

119899

sum

119894=1

1199092

119894

1205972119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(20)

30

20

10

0 5 10

x

minus10 minus5

minus10

minus20

minus30

minus40



120597nablanabla

120597119886119886=

120597ΔΔ

119876

120597119886119886= 119899

120597nablanabla

120597119887119887=

120597ΔΔ

119876

120597119887119887=

119899

sum

119894=1

1199092

119894

120597nablanabla

120597119886119887=

120597ΔΔ

119876

120597119886119887=

119899

sum

119894=1

119909119894

119871 = 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

= 119899

119899

sum

119894=1

1199092

119894minus (

119899

sum

119894=1

119909119894)

2

=sum119899

119894=1(119909119894minus 119909)2

119899

(21)

Since 119871 gt 0 and 119876119886119886


5 Results


0and 120573


0and 120573

1coefficients are


0and 120573








For 119899 = 10




For 119899 = 20




For 119899 = 30




For 119899 = 40




For 119899 = 50





and 119876119886119886



119894
















References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











For 119899 = 10




For 119899 = 20




For 119899 = 30




For 119899 = 40




For 119899 = 50





and 119876119886119886



119894
















References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Time Scale in Least Square Method

Documents

Transcript of Research Article Time Scale in Least Square Method