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CIS 2033 1 Based on text book: F.M. Dekking, C. Kraaikamp, H.P.Lopulaa, L.E.Meester. A Modern Introduction to Probability and Statistics Understanding Why and How Instructor: Dr. Longin Jan Latecki

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The set of observations is called a dataset. By exploring the dataset we can gain insight into what probability model suits the phenomenon. To graphically represent univariate datasets, consisting of repeated measurements of one particular quantity, we discuss the classical histogram, the more recently introduced kernel density estimates and the empirical distribution function. To represent a bivariate dataset, which consists of repeated measurements of two quantities, we use the scatterplot. 2 Chapter 15 Exploratory data analysis: graphical summaries

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15.2 Histograms: The term histogram appears to have been used first by Karl Pearson. 3

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Histogram construction and pdf 4 Denote a generic (univariate) dataset of size n by First we divide the range of the data into intervals. These intervals are called bins and denoted by The length of an interval B i is denoted by ǀ B i ǀ and is called the bin width. We want the area under the histogram on each bin B i to reflect the number of elements in B i. Since the total area 1 under the histogram then corresponds to the total number of elements n in the dataset, the area under the histogram on a bin B i is equal to the proportion of elements in B i : The height of the histogram on bin B i must be equal to As we know from Ch. 13.4, the histogram approximates the pdf f, in particular, for a bin centered at point a, B a =(a-h, a+h], we have

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5 The function g in blue is a mixture of two Gaussians. We draw 200 samples from it, which are shown as blue dots. We use the samples to generate the histogram (yellow) and its kernel density estimate f (red). The Matlab script is twoGaussKernelDensity1.mtwoGaussKernelDensity1.m In Matlab: binwidth=0.5; bincenters=[0.5:binwidth:9.5]; hx=hist(x,bincenters)/(200*binwidth);

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Choice of the bin width 6 Consider a histogram with bins of equal width. In that case the bins are of the from where r is some reference point smaller than the minimum of the dataset and b denotes the bin width. Mathematical research, however, has provided some guide- line for a data-based choice for b or m, where s is the sample std:

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15.3 Kernel density estimates 7

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A kernel K is a function K:R R and a kernel K typically satisfies the following conditions. 8

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Examples of Kernel Construction 9

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Scaling the kernel K 10 Scale the kernel K into the function Then put a scaled kernel around each element xi in the dataset

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11 The bandwid th is too small The bandwidth is too big

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12 The function g in blue is a mixture of two Gaussians. We draw 200 samples from it, which are shown as blue dots. We use the samples to generate the histogram (yellow) and its kernel density estimate f (red). The Matlab script is twoGaussKernelDensity1.mtwoGaussKernelDensity1.m

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15.4 The empirical distribution function 13 Another way to graphically represent a dataset is to plot the data in a cumulative manner. This can be done by using the empirical cumulative distribution function.

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Empirical distribution function Continued 14

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Example 15.6. Given is the following information about a histogram, compute the value of the empirical distribution function at point t = 7: By: Wanwisa Smith 15 Because (2 - 0) * 0.245 + (4 - 2) * 0.130 + (7 - 4) * 0.050 + (11 - 7) * 0.020 + (15 - 11) * 0.005 = 1, there are no data points outside the listed bins. Hence

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Relation between histogram and empirical cdf 15.11. Given is a histogram and the empirical distribution function F n of the same dataset. Show that the height of the histogram on a bin (a, b] is equal to By: Wanwisa Smith 16 The height of the histogram on a bin B i = (a, b] is Hence

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15.5 Scatterplot 17 In some situation we might wants to investigate the relationship between two or more variable. In the case of two variables x and y, the dataset consists of pairs of observations: We call such a dataset a bivariate dataset in contrast to the univariate. The plot the points (X i, Y i ) for i = 1, 2, …,n is called a scatterplot.

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