Preface to First Edition - lib

ANALYSIS USING R 1478.3 Analysis Using RThe R function density can be used to calculate kernel density estimatorswith a variety of kernels (window argument). We can illustrate the function’suse by applying it to the geyser data to calculate three density estimates ofthe data and plot each on a histogram of the data, using the code displayedwith Figure 8.4. The hist function places an ordinary histogram of the geyserdata in each of the three plotting regions (lines 4, 10, 17). Then, the densityfunction with three different kernels (lines 8, 14, 21, with a Gaussian kernelbeing the default in line 8) is plotted in addition. The rug statement simplyplaces the observations in vertical bars onto the x-axis. All three densityestimates show that the waiting times between eruptions have a distinctlybimodal form, which we will investigate further in Subsection 8.3.1.For the bivariate star data in Table 8.2 we can estimate the bivariate densityusing the bkde2D function from package KernSmooth (Wand and Ripley,2009). The resulting estimate can then be displayed as a contour plot (usingcontour) or as a perspective plot (using persp). The resulting contour plotis shown in Figure 8.5, and the perspective plot in 8.6. Both clearly show thepresence of two separated classes of stars.8.3.1 A Parametric Density Estimate for the Old Faithful DataIn the previous section we considered the non-parametric kernel density estimatorsfor the Old Faithful data. The estimators showed the clear bimodalityof the data and in this section this will be investigated further by fitting aparametric model based on a two-component normal mixture model. Suchmodels are members of the class of finite mixture distributions described ingreat detail in McLachlan and Peel (2000). The two-component normal mixturedistribution was first considered by Karl Pearson over 100 years ago(Pearson, 1894) and is given explicitly byf(x) = pφ(x,µ 1 , σ 2 1) + (1 − p)φ(x,µ 2 , σ 2 2)where φ(x,µ,σ 2 ) denotes a normal density with mean µ and variance σ 2 .This distribution has five parameters to estimate, the mixing proportion, p,and the mean and variance of each component normal distribution. Pearsonheroically attempted this by the method of moments, which required solvinga polynomial equation of the 9 th degree. Nowadays the preferred estimationapproach is maximum likelihood. The following R code contains a function tocalculate the relevant log-likelihood and then uses the optimiser optim to findvalues of the five parameters that minimise the negative log-likelihood.R> logL