Forensics and CS Philip Chan. CSI: Crime Scene Investigation high tech forensics tools DNA profiling Use.

Forensics and CS

Philip Chan

CSI: Crime Scene Investigation

www.cbs.com/shows/csi/

high tech forensics tools

DNA profiling Use as evidence in court cases

http://www.cbs.com/shows/csi/



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

Forensics Analysis

Focus on loci (locations) of the DNA Values at the those loci (DNA profile) are

recorded for comparing DNA samples.

Forensics Analysis


recorded for comparing DNA samples. Two DNA profiles from the same person have

matching values at all loci.

Forensics Analysis



matching values at all loci. More or fewer loci are more accurate in

identification? Tradeoffs?

Forensics Analysis



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

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?




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

Review of basic probability

Joint probability of two independent events P(A,B) = ?

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.

Enumerating the events

1 2 3 4 5 6

1 1,1 1,2 …

2

3

4

5

6

36 events, each is equally likely, so 1/36

Joint probability

P(Die1=even, Die2=6) = ?

Joint probability

P(Die1=even, Die2=6) = 1/2 * 1/6 = 1/12

P(Die1=1, Die2=5, Die3=4) = ?

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

DNA profile probability

How to estimate?


How to estimate?

Assuming loci are independent P(Locus1=value1, Locus2=value2, ...) = P(Locus1=value1) * P(Locus2=value2) * ...


How to estimate?


How to estimate P(Locus1=value1)?


How to estimate?


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

Database of DNA profiles

Id Locus1 Locus2 Locus3 … Locus13

A5212

A6921

…

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

Breaking Down the Problem

Find The probability of the sample profile if it matches a

known profile in the database

What are the subproblems?




What are the subproblems? Subproblem 1

Find whether the sample profile matches 1a: ? 1b: ?

Subproblem 2 Calculate the probability of the profile




What are the subproblems? Subproblem 1

Find whether the sample profile matches 1a: check entries in the database

1b: check loci in each entry

Subproblem 2 Calculate the probability of the profile

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

(e.g. whether you can enter your dorm)

int[] directory; // student id’sint id; // to be found

Linear Search

Linear/Sequential Search

Check one by one Stop if you find it Stop if you run out of items to check

Not found

Number of Checks (speed of algorithm) Consider N items in the array

Best-case scenario When does it occur? How many checks?


Best-case scenario When does it occur? How many checks? First item;1 check

Worst-case scenario When does it occur? How many checks?


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 = [N(N+1)/2] / N = (N+1)/2

Matching DNA profiles

Each profile has 13 loci Do we always need to check all 13 loci to

decide if a match occurs or not?

Can we do better? Faster algorithm? What if the array is sorted, items are in an

order E.g. a phone book

Binary Search

Binary Search

1. Check the item at midpoint

2. If found, done

3. Otherwise, eliminate half and repeat 1 and 2

Breaking down the problem

While more items and not found in the mid point What are the two subproblems?


While more items and not found in the mid point Eliminate half of the items Determine the mid point

Number of checks (Speed of algorithm) Best-case scenario

When does it occur? How many checks?


When does it occur? How many checks? In the middle; 1 check



Worst-case scenario When does it occur? How many checks?



Worst-case scenario When does it occur? How many checks? Dividing into two halves, half has only one

item ? checks

Number of checks (Speed of algorithm)T(1) = 1

T(N) = T(N/2) + 1


T(N) = T(N/2) + 1


T(N) = T(N/2) + 1

= [ T(N/4) + 1 ] + 1


T(N) = T(N/2) + 1

= [ T(N/4) + 1 ] + 1


T(N) = T(N/2) + 1

= [ T(N/4) + 1 ] + 1

= [ [ T(N/8) + 1] + 1] + 1


T(N) = T(N/2) + 1

= [ T(N/4) + 1 ] + 1

= [ [ T(N/8) + 1] + 1] + 1

= … any pattern?


T(N) = T(N/2) + 1

= [ T(N/4) + 1 ] + 1

= [ [ T(N/8) + 1] + 1] + 1

= …

= T(N/2k) + k


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


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

Number of Checks (Speed of Algorithm) N/2k = 1

N = 2k

k = ?


N = 2k

k = log2N


N = 2k

k = log2N

T(N) = T(N/2k) + k

= T(1) + log2N

= ? + log2N


N = 2k

k = log2N

T(N) = T(N/2k) + k

= T(1) + log2N

= 1 + log2N

N (Linear search) vs log N + 1 (Binary search)

N

100 7.6

1,000 11.0

10,000 14.3

100,000 17.6

1,000,000 20.9

10,000,000 24.3

100,000,000 27.6

Before using Binary Search

The array needs to be sorted (in order)

Sorting

Sorting (arranging the items in adesired order) How is the phone book arranged?

Why? Why not arranged by numbers?

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?

Sorting

Imagine you have a thousand numbers in an array

How would you systemically sort them?

Selection Sort (ascending)

Find/select the smallest item Swap the smallest item with the first item

Selection Sort (ascending)

Find/select the smallest item Swap the smallest item with the first item Find/select the second smallest item Swap the second smallest item with the

second item …

Example

6 7 2 5 1

Example

6 7 2 5 1

Example

6 7 2 5 1

1 7 2 5 6

Example

6 7 2 5 1

1 7 2 5 6

Example

6 7 2 5 1

1 7 2 5 6

1 2 7 5 6

Example

6 7 2 5 1

1 7 2 5 6

1 2 7 5 6

Example

6 7 2 5 1

1 7 2 5 6

1 2 7 5 6

1 2 5 7 6

Example

6 7 2 5 1

1 7 2 5 6

1 2 7 5 6

1 2 5 7 6

Example

6 7 2 5 1

1 7 2 5 6

1 2 7 5 6

1 2 5 7 6

1 2 5 6 7


Get all the items in ascending order Get one item at the wanted position/index

What are the two subproblems?



Find the smallest item



1. Find the smallest item

2. Swap the smallest item with the item at the wanted position

Algorithm Summary (Selection Sort) For each “desired” position

Between the “desired” position and the end Find the smallest item

Swap the smallest item with the item at the “desired” position

Number of comparisons (Speed of Algorithm) Consider counting

Number of comparisons between array items


Number of comparisons between array items Best-case scenario (least # of comparisons)

When does it occur? How many comparisons?




Worst-case scenario (most # of comparisons) When does it occur? How many

comparisons?




Worst-case scenario (most # of comparisons) When does it occur? How many

comparisons? Same number of comparisons

For all cases (ie best case = worst case)

Number of comparisons (Speed of Algorithm) To find the smallest item

How many comparisons?


How many comparisons? N-1

To find the second smallest item How many comparisons?



To find the second smallest item How many comparisons? N-2

… Total # of comparisons?




… Total # of comparisons

(N-1) + (N-2) + … + 1




… Total # of comparisons

(N-1) + (N-2) + … + 1 N(N-1)/2 = (N2 – N)/2

Selection Sort

Not the fastest sorting algorithm Learn faster algorithms in more advanced

courses.

Revisiting Binary Search

Binary Search

While more items and not found in the mid point1. Eliminate half of the items

2. Determine the mid point

Eliminate half of the array

How to specify the focus region? Hint: index/position

Eliminate half of the array

How to specify the focus region? Hint: index/position

Start and end

How to determine if the region has items (is not empty)? with start and end

How to determine if the region has items (is not empty)? with start and end

Start <= end

How do we adjust start and end?


What are the two different cases?


What are the two different cases? Item is before the middle item

Item is after the middle item



Start: End:




Start: no change End: position before the mid point





Item is after the middle item Start: End:




Item is after the middle item Start: position after the mid point End: no change

How to determine the mid point?

with start and end?

How to determine the mid point?

with start and end (start + end) / 2

Integer division will eliminate the fractional part

Algorithm Summary

1. Initialize start, end, and mid point (I)

2. While region has items and item is not at the mid point ( C )a) Eliminate half of the items by adjusting start

or end (U)

b) Update the mid point (U)

3. If region has items Position is mid point

else Position is -1

Overall Summary

Overall Summary

DNA samples from crime scene Identify people using known DNA profiles

If there is a match estimate probability of DNA profile

Matching a sample to known DNA profiles Linear/sequential search [N checks] Binary search [log2N + 1 checks]

Faster but needs sorted data/profiles Selection Sort [(N2 – N)/2 comparisons]

Forensics and CS Philip Chan. CSI: Crime Scene Investigation high tech forensics tools DNA profiling Use.

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