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Chapter 4: Algorithm Overview 53(a)(b)(c)(d)(e)(f)Figure 4.7: Effect of bandwidth parameter on kernel density. (a) KDE for h =0.01. (b) KDE for h = 0.1. (c) KDE for h = 0.3. (d) KDE for h = 0.5. (e) KDE for h =0.328(optimal). (f) KDE for h = 1.
Chapter 4: Algorithm Overview 54smoothing as applied to uni-variate and multi-variate datasets. Papers that discuss theinformation bound bandwidth selection methods are [Wu and Lin, 1996] and [Jones etal., 1996].The bandwidth selection for the process of density estimation is important to assert theaccuracy of the density estimate. The choice of the bandwidth at least theoretically canbe derived to minimize the mean integrated square error between the actual densityand the computed density. Some methods that are used for this purpose are Distribution Scale Methods, Cross Validation Methods, Plug-In Methods, and Bootstrap methods.In the next few paragraphs, we will very briefly discuss the rationale behind theseobjective methods for bandwidth selection. Assume that f is the actual density of thedata and fˆ is the estimated density. The process of bandwidth selection is aimed atminimizing the integrated mean square error between the actual and the estimateddensity. The integrated mean square error is defined as the expected value of theintegrated square error and is given by Equation 4.13.MSE{f ( x;h )}122= n− {( K h * f )( x ) − ( K h * f ) ( x )} + {( K h * f )( x ) − f ( x )} . (4.13)The Mean Integrated Square Error ( MISE ) is the integral of the mean squared errorthat can be simplified as shown below.2MISE{MISE{ fˆ ( x,h )}12=MSE{ fˆ ( x;h )}dx = E{fˆ ( x;h ) − f ( x )} dx22{(Kh* f )( x ) −(Kh* f ) ( x )}dx +{(Kh* f )( x ) −fˆ (.;h )} = n− f ( x )} dx = −− + − − + 2 (4.14)MISE is the sum of the integrated square bias and the integrated variance and henceminimization of that error is effectively the tradeoff between the bias and variance.The closed form solution that is derived for the optimal bandwidth by minimizing theMISE is the h opt in Equation 4.15.h opt= n(2K(z)dz 222z K(z)dz)( f "( x)dx)1/ 5(4.15)
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To the Graduate Council:I am submit
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Acknowledgementsii“Experience is
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Contents1 INTRODUCTION.............
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List of TablesviTable 2.1:Qualitati
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viiiFigure 5.2: Curvature analysis
Page 11 and 12: Chapter 1: Introduction 21.1 Motiva
Page 13 and 14: Chapter 1: Introduction 41.2 Propos
Page 15 and 16: Chapter 1: Introduction 6We then pe
Page 17 and 18: Chapter 2: Literature Review 8Shape
Page 19 and 20: Chapter 2: Literature Review 102.2.
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Page 31 and 32: Chapter 2: Literature Review 22Gots
Page 33 and 34: Chapter 2: Literature Review 24Tabl
Page 35 and 36: Chapter 3: Data Collection and Mode
Page 49 and 50: Chapter 4: Algorithm Overview 404.1
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Page 59 and 60: Chapter 4: Algorithm Overview 50∞
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Page 69 and 70: Chapter 5: Analysis and Results 60W
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Page 89 and 90: Chapter 5: Analysis and Results 80(
Page 91 and 92: Chapter 6: Conclusions 82complexity
Page 93 and 94: Bibliography 84BIBLIOGRAPHY
Page 95 and 96: Bibliography 86[Biermann, 2001][Bel
Page 97 and 98: Bibliography 88[Cyr and Kimia, 2001
Page 99 and 100: Bibliography 90[Gourley, 1998][Gosh
Page 101 and 102: Bibliography 92[Joshi and Chang, 19
Page 103 and 104: Bibliography 94[Meyer, 2002][Morse,
Page 105 and 106: Bibliography 96[Safar et al., 2000]
Page 107 and 108: Bibliography 98parts,” IEEE Trans
Page 109: Vita 100VITASreenivas Rangan Sukuma
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