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

Chapter 4: Algorithm Overview 49Histograms are good representation tools but not efficient density estimates. We discussthe effect of bin width on the histogram with a simple example in Figure 4.5. It isimportant to note the significant change in shape and the density of the estimate.Another method that is an improvement on the histogram used to estimate the densityis the naïve estimator. It is based on the fact that if the random variable y has density fthen1f ( x ) = lim P( x − h < X < x + h ).h→0 2h(4.7)Thus a natural estimator f of the density can be obtained by choosing a small number has shown in Equation 4.6.fˆ ( x )=[number ofx1, x2….xnfalling in (x - h, x + h) ]2 × h × n(4.8)The naïve estimator can also be mathematically expressed as follows1fˆ ( x ) =nni = 11 x − Xw (h hwhere w(x) represents a rectangle function of height 0.5 and width of 2.i)(4.9)It is easy to generalize the naive estimator to overcome its rugged nature of the densityby replacing the weight function w by a kernel function K which satisfies thecondition described in Equation 4.10.Number of bins = 4Number of bins = 7yDensityDensityxyyFigure 4.5: Illustration that shows the effect of bin width on density estimationusing a histogram.