CCST9017 Hidden Order in Daily Life: A Mathematical Perspective. Lecture 8. Statistical Frauds and Benford s Law

Transcription

1 CCST9017 Hidden Order in Daily Life: A Mathematical Perspective Lecture 8 Statistical Frauds and Benford s Law Dr. S. P. Yung (9017) Dr. Z. Hua (9017B) Department of Mathematics, HKU

2 Outline Recall on probabilities rules Simple application Example of Traps Fabricated list Probability of 5 heads or 5 tails in a row from 120 tosses Benford s Law Heuristic Reason for Benford s Law 2

3 Recall on probabilities rules All other possible outcomes of an event A is called A complement, denoted as A C. P(A) denotes the probability that A occurs. Rule 0 : 0 P (A) 1. Rule 1 : P(A) + P(A C ) = 1. Rule 2 : If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). Rule 3 : If A and B are independent, then P(A and B) = P(A) x P(B). Rule 4 : If B is a subset of A, then P(B) P(A). 3

4 Simple application Recall P(A) = # of occurrences of event A Total # of occurrences of all possible events It can be used, say, to find the number of fishes in a lake. Let the number of fishes in the lake is N. We first catch, say 500 fishes from the lake, mark them with labels on their bodies and release them back to the lake. We then catch, say 200 fishes from the lake and find that 100 are marked by us before. Since the probability of catching a marked fish is given both by 500/N and 100/200, so they much be equal and so 500/N = 100/200. Thus we have N = 500*200/100=1000! 4

5 Simple application In general, suppose that the number of fishes in the lake is N and we catch M fishes, mark and release them, then M/N = m/n. If m marked fishes are found from catching n fishes, then N = Mn/m. * More example on Monte Carlo simulation: Computation of pi 5

6 Example of traps Suppose that a drug test is run in a city that has a large population, of which only 1% are drug junkies. Suppose that the drug test is very reliable and has a 98% successful detection rate (i.e., drug junkies test positive with probability 0.98, and non-drug-users test positive with probability 0.02). If a person is chosen randomly and was tested with positive result, that person is two times more likely to be a non-drug-user than a junky! To see this, let s draw a tree diagram: Example taken from The Difficulty of Faking Data, Theodore P. Hill. 6

7 Tree Diagram, Example of Traps Tested Negative Non-Junky 0.02 Tested Positive 0.02 Tested Negative 0.01 Junky 0.98 Tested Positive 7

8 Example of traps From Rule 3, the probability of a non-drug-user tested positive is: 0.99 * 0.02 = The probability of a junky tested positive is: 0.01 * 0.98 = , which is almost half of ! So if tested positive, it is twice likely to be a non-drug-user! This is one of the false-positives" in medical testings. 8

9 Example of traps In some cases, we could even manipulate our conclusion from the data. Suppose the followings are the prices of two fruits and we want to come to the conclusion of whether the fruit prices (a) remain unchanged, (b) increase, or (c) decrease Apples $1 $2 Oranges $2 $1 9

10 Example of traps (I) If we look at the average prices, the answer is (a) remain unchanged Apples $1 $2 Oranges $2 $1 Average Price $1.5 $1.5 10

11 Example of traps (II) If we look at the average price of percentages using the prices at 2000 as 100%, the answer is (b) 25% increase Apples 100% ($1) 200% ($2) Oranges 100% ($2) 50% ($1) Average price of percentages 100% ($1.5) 125% ($1.5) 11

12 Example of traps (III) If we look at the average price of percentages using the prices at 2010 as 100%, the answer is (c) 25% decrease Apples 50% ($1) 100% ($2) Oranges 200% ($2) 100% ($1) Average price of percentages 125% ($1.5) 100% ($1.5) 12

13 Example of traps So to avoid being mis-led in a statistical report, we should not only look at one single index, but also look at others as well, such as the mode (i.e. the value which occurs most frequently), the median (i.e. the value which is in the middle of the population, with 50% below and 50% over it), the mean (arithmetical average),.. etc. 13

14 Fabricated list One of the data sets is obtained by tossing a fair coin 120 times, with Head =1 and Tail =0, and another data set is a fabricated list. Can you distinguish them? Sequence A Sequence B 14

15 Fabricated list RHS is from simulating the process by EXCEL. rand() gives you a random number in (0,1). int(x) will give you the integral part of the number x. For example, int(0.9)=0 and int(1.1)=1. Thus the process of flipping a fair coin can be generated by using =int(2*rand()) in EXCEL. *Random number generation: Computational vs Physical Middle-square method of von Neumann 15

16 Which one is fabricated? We want to compute the probability of getting a successive run (heads or tails) of certain length in a given number of coin tosses. Say, we want to calculate the probability that in n fair coin tosses, a run of r heads or r tails occurs (r =< n). Let (i,j) be the situation that there are still j tosses to go and the last i tosses are of the same outcome, but so far no run of r heads or r tails. X.XH.H-.- or X.XT.T-.- i heads j toss left i tails j toss left 16

17 Which one is fabricated? Let u(i,j) be the probability of getting a run of either r heads or r tails in n fair coin tosses given the current situation is (i,j). Then we have the following recursive relation : u(i,j) =0.5*u(i+1,j-1) + 0.5*u(1,j-1). Toss the coin one more, then j will decrease by one to j-1. There are two possible situations: (i) the outcome is the same as the last i tosses then we are in situation (i+1,j-1) and the probability is 0.5. (ii) the outcome is different from the last i tosses then we have (1,j-1) and the probability is

18 Which one is fabricated? All the different situations are listed in the nodes of the diagram, and we want to find u(i,j) at those nodes. Chart that denotes all the situations: j i 18

19 Probability of 5 heads or 5 tails in a row from 120 tosses We already knew the following boundary conditions: u(i,0)=0 for i=1,2,.,r-1. u(r,j)=1 for j=0,1,2,.,n-r. For the probability of getting 5 heads or 5 tails in 120 fair coin tosses (n=120, r=5), we have u(1,0)=u(2,0)=.=u(4,0)=0. u(5,0)=u(5,1)=.=u(5,115)=1. 19

20 Computing u(i,1), i=1,2,3,4. From u(i,j)=0.5*u(i+1,j-1) + 0.5*u(1,j-1), we have u(4,1)=0.5*u(5,0)+0.5*u(1,0)=1/2 u(3,1)=0.5*u(4,0)+0.5*u(1,0)=0 u(2,1)=0.5*u(3,0)+0.5*u(1,0)=0 u(1,1)=0.5*u(2,0)+0.5*u(1,0)=0 20

21 Computing u(i,2), i=1,2,3,4. u(4,2)=0.5*u(5,1)+0.5*u(1,1)=1/2 u(3,2)=0.5*u(4,1)+0.5*u(1,1)=1/4 u(2,2)=0.5*u(3,1)+0.5*u(1,1)=0 u(1,2)=0.5*u(2,1)+0.5*u(1,1)=0 21

22 Computing u(i,3), i=1,2,3,4. u(4,3)=0.5*u(5,2)+0.5*u(1,2)=1/2 u(3,3)=0.5*u(4,2)+0.5*u(1,2)=1/4 u(2,3)=0.5*u(3,2)+0.5*u(1,2)=1/8 u(1,3)=0.5*u(2,2)+0.5*u(1,2)=0 22

23 Computing u(i,4), i=1,2,3,4. u(4,4)=0.5*u(5,3)+0.5*u(1,3)=1/2 u(3,4)=0.5*u(4,3)+0.5*u(1,3)=1/4 u(2,4)=0.5*u(3,3)+0.5*u(1,3)=1/8 u(1,4)=0.5*u(2,3)+0.5*u(1,3)=1/16 23

24 Computing u(i,5), i=1,2,3,4. u(4,5)=0.5*u(5,4)+0.5*u(1,4)=17/32 u(3,5)=0.5*u(4,4)+0.5*u(1,4)=9/32 u(2,5)=0.5*u(3,4)+0.5*u(1,4)=5/32 u(1,5)=0.5*u(2,4)+0.5*u(1,4)=3/32 and so on for the other cases. 24

25 Which one is fabricated? n The probability of getting 5 heads or 5 tails in 120 tosses is u(1,119)= which is very high; but this is only observed in Sequence B but not found in Sequence A. n The probability of not observing 5 heads or 5 tails in a row is 1- u(1,119)= around 1%. n Most likely, Sequence A is a fake one! 25

26 Benford s Law n n [Small observation, Big discovery] In 1881, a Mathematician Simon Newcomb noticed that the pages of logarithm tables with small initial digits were dirtier than those with larger initial digits, such that 1>2>3>4>5>6>7>8>9. In 1938, a Physicist Frank Benford proposed the Benford s law based on the empirical evident: P(The first significant digit = d) = log 10 (1+1/d) for d=1,2,3,4,5,6,7,8,9. 26

27 Frequency of First Digits, from 1 to 9. Digit Probability

28 Frequency of First Digits, from 1 to 9. From "The First-Digit Phenomenon" by T. P. Hill, American Scientist, July-August [The New York Times, Tuesday, August 4, 1998] 28

29 First Significant Digits in Tax Data 29

30 Detecting Frauds, Benford s Law. n An interesting application of Benford s Law is to help in detecting possible fraud in tax returns. n Empirical research in US has shown that the interest paid and received are very good fit to Benford s Law. So any deviations from the Benford s Law could be possible fraud cases. 30

31 Heuristic Reason for Benford s Law n Suppose in month 0, Hang Seng Index is 100. n We assume that it increases at a rate of 10 % per year (it in general can be any r%). n Let f(1) be the number of years for the index to reach 200 from 100, then we have 100*(1.1) f(1) =200 or f(1)=(log(200)-log(100)) / log(1+1/10) 31

32 Heuristic Reason for Benford s Law n Let f(2) be the number of years for the index to reach 300 from 200, then we have 200*(1.1) f(2) =300 or f(2)=(log(300)-log(200)) / log(1+1/10) n Thus we have for d=1,2,3,4,5,6,7,8,9 f(d) = (log(100(d+1))-log(100d)) /log(1+1/10) n We note that log(100d)=log(100)+log(d) and log(100(d+1))=log(100)+log(d+1). 32

33 Heuristic Reason for Benford s Law n Thus f(d) can be simplified as follows: f(d) = log(1+1/d) / log(1+1/10) n We also note that log(10)=1 and total time of all first digit changings is given by F:=f(1)+f(2)+ +f(9) =log(10)/log(1+1/10). n Therefore the probability of observing d (d=1,2,3,4,5,6,7,8,9) as the first digit is P(first digit=d) = f(d)/f = log(1+1/d). 33

34 Assignment 8 Due date: November 5 (before 1:00pm) Please put your assignment into the assignment box. Please write your tutorial group number on the right hand corner of your assignment. 1. Two checkers are employed for checking (proofreading) a book. Checker I found 30 errors (typos) and Checker II found 25 errors (typos). Suppose among those errors found by Checkers I & II, there are only 15 in common. Please find the total number of errors (typos) in the book. 34

35 Assignment 8 2.In the section Heuristic Reason for Benford s Law, suppose that in month 0, Hang Seng Index is 100 and it increases at a rate of r% (for some r >0) per year. Show that P(first digit = d) = log(1+1/d). 35

36 Reference Darrell Huff and Irving Geis, How to Lie With Statistics, Penguin, S.D.Levitt and S.J.Dubner, Freakonomics, Henk Tijms, Understanding Probability, Cambridge University Press, Ted Hill, The Difficulty of Faking Data, Chance Magazine, (3),