1. Math 3033 Wanwisa Smith 1 Base on text book: A Modern Introduction to Probability and Statistics Understanding Why and How By: F.M. Dekking, C. Kraaikamp, H.P.Lopulaa, L.E.Meester Temple University Fall 2009 Instructor: Dr. Longin Jan Latecki Slides by: Wanwisa Smith

2.  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 Wanwisa Smith

3. 15.1 Example: the Old Faithful data 3 Wanwisa Smith

4. A closer look at the ordered data that the two middle elements (the 136 th and 137 th elements in ascending order) are equal to 240, which is much closer to the maximum value 306 than to the minimum value 96. 4 Wanwisa Smith

5. 15.2 Histograms: The term histogram appears to have been used first by Karl Pearson. 5 Wanwisa Smith

6. How to construct the histogram? 6 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 Bi is denoted by ǀ Bi ǀ and is called the bin width. The bins do not necessarily have the same width. We want the area under the histogram on each bin Bi to reflect the number of elements in Bi. 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 Bi is equal to the proportion of elements in Bi: The height of the histogram on bin Bi must be equal to Wanwisa Smith

7. Choice of the bin width 7 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. Wanwisa Smith

8. 15.3 Kernel density estimates 8 The idea behind the construction of the plot is to “put a pile of sand” around each element of the dataset. At places where the elements accumulate, the sand will pile up. The kernel K reflects the shape of the piles of sand, whereas the bandwidth is a tuning parameter that determine how wide the piles of sand will be. Wanwisa Smith

9. A kernel K is a function K:R  R and a kernel K typically satisfies the following conditions. 9 Wanwisa Smith

10. Examples of Kernel Construction 10 Wanwisa Smith

11. Scaling the kernel K 11 Scale the kernel K into the function Then put a scaled kernel around each element xi in the dataset Wanwisa Smith

12. 12 The bandwid th is too small The bandwidth is too big Wanwisa Smith

13. Boundary Kernels 13 Wanwisa Smith

14. 15.4 The empirical distribution function 14 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 of the data. It is denote by Fn and is defined that a point x as the proportion of elements in the dataset that are less than or equal to x: Wanwisa Smith

15. Empirical distribution function Continued 15 Wanwisa Smith

16. Empirical distribution function Continued 16 Wanwisa Smith

17. 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 dataset, which consists of observations of one particular quantity. We often like to investigate whether we can describe the relation between the two variables. A first is to take a look at the data, i.e., to plot the points (Xi, Yi) for i = 1, 2, …,n. Such a plot is called a scatterplot. Wanwisa Smith

18. Example of Scatterplot 18 Wanwisa Smith

19. More Example of Scatterplot 19 Wanwisa Smith

20. 20 Thank you for reading this slide show. I hope you enjoy it.