To the Graduate Council: I am submitting herewith a thesis written by ...

More documents

Recommendations

Info

Chapter 4: Algorithm Overview 51Method used to compute densityPlot of the density functionHistogramsThe Naïve EstimatorThe Kernel Density EstimatorThe Nearest NeighborhoodMethodFigure 4.6: Different methods used to estimate the density of the same dataset.Adapted from [Silverman, 1986].
Chapter 4: Algorithm Overview 52There is a plethora of research in automating the process of bandwidth estimation thatwill give us the best estimate of the density as possible. Recollecting Equation 4.9, theparameters that influence the density estimate are the kernel function, density span andthe kernel bandwidth. We ignore the effect of kernel function and density spanbecause of the assumption that we have represented the digitized surface with enoughpoints to represent a continuous surface. We hence assume that our dataset is too largeto react to the effect of the different kernel functions listed in Table 4.1. We havedecided to use the Gaussian kernel for our implementation for its continuity though theEpanechnikov kernel is considered to be the most efficient of kernel functions.We have performed a simple experiment on a normally distributed pseudo randomdata set with zero mean and unit variance at 512 points to study the effect of thebandwidth parameter on density estimation. In Figure 4.7, the red curves represent theground truth Gaussian density function and the blue ones represent the estimateddensity. Figures 4.7(a-f) portray the amount of smoothing that the bandwidthparameter imposes on the estimated density. Figure 4.7 illustrates the importance of thebin width parameter in density estimation with a simple example. Though the figuresrepresent the density of the same dataset, we are able to anticipate the instability of itsinformation measure.The paper [Turlach, 1996] is an excellent survey on bandwidth selection in kerneldensity estimation. The books [Silverman, 1986] and [Wand, 1995] are classics in thefield of kernel estimation and kernel smoothing and have detailed descriptions of kernelTable 4.1: Kernel functions.KernelK(u)Uniform I (| u | ≤ 1 )Triangle ( 1−| u |) I (| u | ≤ 1)Epanechnikov ( 1 − u )I(| u | ≤ 1)4123 215 21635 2( 1−u )3 I(| u | ≤ 1322Triweight ( 1 − u ) I(| u | ≤ 1 )Quartic )− u2Gaussian e ( )12 ππ π uCosinus cos( )I (| u | ≤ 1 )422
Page 1 and 2:
To the Graduate Council:I am submit
Page 3 and 4:
Acknowledgementsii“Experience is
Page 5 and 6:
Contents1 INTRODUCTION.............
Page 7 and 8:
List of TablesviTable 2.1:Qualitati
Page 9 and 10: 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.
Page 21 and 22: Chapter 2: Literature Review 12Bimb
Page 23 and 24: Chapter 2: Literature Review 14is f
Page 25 and 26: Chapter 2: Literature Review 16foun
Page 27 and 28: Chapter 2: Literature Review 18Kort
Page 29 and 30: Chapter 2: Literature Review 20boun
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
Page 51 and 52: Chapter 4: Algorithm Overview 42Tri
Page 53 and 54: Chapter 4: Algorithm Overview 44whe
Page 55 and 56: Chapter 4: Algorithm Overview 46The
Page 57 and 58: Chapter 4: Algorithm Overview 48of
Page 59: Chapter 4: Algorithm Overview 50∞
Page 63 and 64: Chapter 4: Algorithm Overview 54smo
Page 65 and 66: Chapter 4: Algorithm Overview 564.2
Page 67 and 68: Chapter 4: Algorithm Overview 58Wit
Page 69 and 70: Chapter 5: Analysis and Results 60W
Page 71 and 72: Chapter 5: Analysis and Results 62S
Page 73 and 74: Chapter 5: Analysis and Results 64F
Page 75 and 76: Chapter 5: Analysis and Results 66t
Page 77 and 78: Chapter 5: Analysis and Results 68A
Page 79 and 80: Chapter 5: Analysis and Results 705
Page 81 and 82: Chapter 5: Analysis and Results 72S
Page 83 and 84: Chapter 5: Analysis and Results 74W
Page 85 and 86: Chapter 5: Analysis and Results 76C
Page 87 and 88: Chapter 5: Analysis and Results 78T
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
show all

To the Graduate Council: I am submitting herewith a thesis written by ...

Create successful ePaper yourself

Delete template?

Save as template?