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

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