Forensics and CS

1. Forensics and CS Philip Chan

2. CSI: Crime Scene Investigation • www.cbs.com/shows/csi/ • high tech forensics tools • DNA profiling • Use as evidence in court cases

3. DNA • Deoxyribonucleic Acid • Each person is unique in DNA (except for twins) • DNA samples can be collected at crime scenes • About .1% of human DNA varies from person to person

4. Forensics Analysis • Focus on loci (locations) of the DNA • Values at the those loci (DNA profile) are recorded for comparing DNA samples. • Two DNA profiles from the same person have matching values at all loci. • More or fewer loci are more accurate in identification? • Tradeoffs? • FBI uses 13 core loci • http://www.cstl.nist.gov/biotech/strbase/fbicore.htm

5. We do not want to wrongly accuse someone • How can we find out how likely another person has the same DNA profile? • How many people are in the world? • How low the probability needs to be so that a DNA profile is unique in the world? • Low probability doesn’t mean impossible • Just very unlikely

6. Review of basic probability • Joint probability of two independent events • P(A,B) = ?

7. Review of basic probability • Joint probability of two independent events • P(A,B) = P(A) * P(B) • Independent events mean knowing one event does not provide information about the other events • P(Die1=1, Die2=1) • = P(Die1=1) * P(Die2=1) • = 1/6 * 1/6 = 1/36.

8. Enumerating the events 36 events, each is equally likely, so 1/36

9. Joint probability • P(Die1=even, Die2=6) = ?

10. Joint probability • P(Die1=even, Die2=6) • = 1/2 * 1/6 = 1/12 • P(Die1=1, Die2=5, Die3=4) = ?

11. Joint probability • P(Die1=even, Die2=6) • = 1/2 * 1/6 = 1/12 • P(Die1=1, Die2=5, Die3=4) • = (1/6)3 = 1/216

12. DNA profile probability • How to estimate?

13. DNA profile probability • How to estimate? • Assuming loci are independent • P(Locus1=value1, Locus2=value2, ...) • = P(Locus1=value1) * P(Locus2=value2) * ...

14. DNA profile probability • How to estimate? • Assuming loci are independent • P(Locus1=value1, Locus2=value2, ...) • = P(Locus1=value1) * P(Locus2=value2) * ... • How to estimate P(Locus1=value1)?

15. DNA profile probability • How to estimate? • Assuming loci are independent • P(Locus1=value1, Locus2=value2, ...) • = P(Locus1=value1) * P(Locus2=value2) * ... • How to estimate P(Locus1=value1)? • a random sample of size N from the population and • find out how many people out of N have value1 at Locus1

16. Database of DNA profiles

17. Problem Formulation • Given • A sample profile (e.g. collected from the crime scene) • A database of known profiles • Find • The probability of the sample profile if it matches a known profile in the database

18. Breaking Down the Problem • Find • The probability of the sample profile if it matches a known profile in the database • What are the subproblems?

19. Breaking Down the Problem • Find • The probability of the sample profile if it matches a known profile in the database • What are the subproblems? • Subproblem 1 • Find whether the sample profile matches • 1a: ? • 1b: ? • Subproblem 2 • Calculate the probability of the profile

20. Breaking Down the Problem • Find • The probability of the sample profile if it matches a known profile in the database • What are the subproblems? • Subproblem 1 • Find whether the sample profile matches • 1a: check entries in the database • 1b: check if all 13 loci match in each entry • Subproblem 2 • Calculate the probability of the profile

21. Simpler Problem for 1a (very common) • Given • an array of integers (e.g. student IDs) • an integer (e.g. an ID) • Find • whether the integer is in the array int[] directory; // student id’s int id; // to be found boolean found; // true if id is in directory

22. Linear/Sequential Search • Check one by one • Stop if you find it • Stop if you run out of items to check • Not found

23. Number of Checks (speed of algorithm) • Consider N items in the array • Best-case scenario • When does it occur? How many checks?

24. Number of Checks (speed of algorithm) • Consider N items in the array • Best-case scenario • When does it occur? How many checks? • First item;1 check • Worst-case scenario • When does it occur? How many checks?

25. Number of Checks (speed of algorithm) • Consider N items in the array • Best-case scenario • When does it occur? How many checks? • First item;1 check • Worst-case scenario • When does it occur? How many checks? • Last item or not there; N checks • Average-case scenario • Average of all cases • (1 + 2 + … + N) / N =

26. Can we do better? Faster algorithm? • What if the array is sorted, items are in an order • E.g. a phone book

27. Binary Search • Check the item at midpoint • If found, done • Otherwise, eliminate half and repeat

28. Breaking down the problem • While more items and not found • What are the two subproblems?

29. Breaking down the problem • While more items and not found • Eliminate half of the items • Find the mid point

30. Number of checks (Speed of algorithm) • Best-case scenario • When does it occur? How many checks?

31. Number of checks (Speed of algorithm) • Best-case scenario • When does it occur? How many checks? • In the middle; 1 check

32. Number of checks (Speed of algorithm) • Best-case scenario • When does it occur? How many checks? • In the middle; 1 check • Worst-case scenario • When does it occur? How many checks?

33. Number of checks (Speed of algorithm) • Best-case scenario • When does it occur? How many checks? • In the middle; 1 check • Worst-case scenario • When does it occur? How many checks? • Dividing into two halves, half has only one item • ? checks

34. Number of checks (Speed of algorithm) • Best-case scenario • When does it occur? How many checks? • In the middle; 1 check • Worst-case scenario • When does it occur? How many checks? • Dividing into two halves, half has only one item • ? checks

35. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1

36. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1

37. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1

38. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1

39. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1

40. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … any pattern?

41. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2k) + k

42. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2k) + k N/2k gets smaller and eventually becomes 1

43. Number of checks (Speed of algorithm) T(1) = 1 T(N) = T(N/2) + 1 = [ T(N/4) + 1 ] + 1 = [ [ T(N/8) + 1] + 1] + 1 = … = T(N/2k) + k • N/2k gets smaller and eventually becomes 1 • solve for k

44. Number of Checks (Speed of Algorithm) • N/2k = 1 N = 2k k = ?

45. Number of Checks (Speed of Algorithm) • N/2k = 1 N = 2k k = log2N

46. Number of Checks (Speed of Algorithm) • N/2k = 1 N = 2k k = log2N • T(N) = T(N/2k) + k = T(1) + log2N = ? + log2N

47. Number of Checks (Speed of Algorithm) • N/2k = 1 N = 2k k = log2N • T(N) = T(N/2k) + k = T(1) + log2N = 1 + log2N

48. Sorting (arranging the items in adesired order) • How is the phone book arranged? • Why? • Why not arranged by numbers?

49. Sorting (arranging the items in adesired order) • How is the phone book arranged? • Why? • Why not arranged by numbers? • Order • Alphabetical • Low to high numbers • DNA profile with 13 loci?

50. Sorting • Imagine you have a thousand numbers in an array • How would you systemically sort them?