Essential Search Mathematics for SAR Managers & Planners Presented by Dan O’Connor NEWSAR

1. Essential Search Mathematicsfor SAR Managers & PlannersPresented byDan O’ConnorNEWSAR

3. “Windows” CASIE Computer-Aided Search Information Exchange FREE at http://www.wcasie.com

4. “Background” 3 Types of Search Systems “Defective”Probability “Closed” System “Open” System 50% IPP 30% Physical & PsychologicalLimits PhysicalLimits ROW + Segments= 100% POA SA Less Than100% POA or POC No ROW100% POC

5. 1. Theoretical vs. Statistical Search Area (SA) What’s the difference?

6. THEORETICAL Search Area The Straight-Line Distance that a Lost Person could have traveled “in theory” over the Elapsed Time since reported Missing Rate x Time = Distance (as est. of radius)2 mph x 12hrs = 24 milesA radius of 24 miles means a Circular Search Area of 1,810 Square Miles!Equivalent to a 40 mi by 45 mi Area!

7. STATISTICAL Search Area An AREA based on Distances that other Lost Persons have traveled in the PAST. Ideally, these distances traveled are compiled by Lost Person Category (child, elderly, hiker, etc.)Search Managers typically draw Statistical Search Areas based on the MEDIAN (50th Percentile) &75th & 90th & 95th PercentilesMaybe should be called “Potential Search Area”

8. Q. Why are Potential Search Areas Drawn as Circles?

9. A. Because in the Absence of CLUES, we have no idea about the Lost Subject’sDirection of Travel

10. Sources for STATISTICALDistances Traveled . . . Ken Hill (Nova Scotia data) published inthe NASAR MLPI Text & CASIE “Lost Person Behavior,” Robert Koester ISRID Koester & Twardy et al SARSTATISTICS.org (under development) Your OWN or other Local Agency Data

12. CASIE Source Distances Traveled

13. 2. The MEDIAN: the value which divides the Data in Equal Halves. 50% is At or Above the MedianAnd 50% is At or Below the Median“The Median home price in the area is $300,000.” Half sold at or above, half sold at or below.

14. IMPORTANT! The POSITION of the MEDIAN Is NOT the VALUE of the MEDIAN!

15. To find the POSITION of the MEDIAN in a SORTED Dataset use: MEDpos = 0.5 * (n+1) For 99 data points, the POSITION Of the Median = 0.5 * (99+1) = 50

16. 17 SORTED Lost Person Distance Traveled Data Pts

17. The MEDIAN is More Stable, The MEAN is More Variable Consider our 17 Data Points , from 0.5mi to 20.3 miwith Mean=6.2 mi and MEDIAN=4.8 mi . . . If we ADD 2 more data points at 1 mi and 30 mi, the Mean goes to 7.1 mi, but the MEDIAN=4.8! The Mean is sensitive to Outliers – the Median is NOT!

18. The MEDIAN also defines the position of the 50th Percentile Data: 0.5mi 4.8mi 20.3mi Percentile: 0 10 20 30 40 50 60 70 80 90 100 The MEDIAN lives here at the 50th Percentile OR end of the 5th Decile

19. Questions on Radius “r” Why Use the MEDIAN?When to Use the MEDIAN?Why not 75th or 90th Pctile?What should “r” radius be?

20. Which Area is easiest to search? Both represent 50% of cases . . . 20.3 mi 50% 50% 4.8 mi IPP AREA= ? AREA= ?

21. AREA of a Circle = pi * r^2 For r = 4.8, Area = 3.14 * (4.8 * 4.8) = 72 sq units For r = 20.3, Area = 3.14 * (20.3 * 20.3) = 1294 sq unitsArea of Outer Circle (annulus) = 1294 – 72 = 1222 sq units

22. Area of an Annulus in CASIE

23. Which Area is easiest to search? Both represent 50% of cases . . . 50% 20.3 mi 50% 4.8 mi IPP AREA= 72 sq mi AREA= 1222 sq mi

24. Another way to look at it . . . pDEN Probability Density: % Statistical POA per Unit Area 20.3 mi 50% 50% 4.8 mi IPP pDen= 50% / 72 sq mi = 0.69% per sq mi pDen= 50% / 1222 sq mi = 0.041% per sq mi

25. NOTE! CONSENSUS POA is different from Statistical Probability. The Area with the top 50% of cases might be assignedonly 10% POA initially as a Region

26. WHEN to Search Within the Median • RESOURCES Are LIMITED • TIME Is Limited • HIGH Coverage is Required • Increased Urgency for Good Confinement • It’s a 50-50 Tradeoff for a smaller SA

27. Statistical Circles are NOT Limits to the Search Area . . .Go wherever the CLUES Lead!

28. “Background” 3 Types of Search Systems “Defective”Probability “Closed” System “Open” System 50% IPP 30% Physical & PsychologicalLimits PhysicalLimits ROW + Segments= 100% POA SA Less Than100% POA or POC No ROW100% POC

29. 3. Analyzing OWN Agency Data A. Sort and Compute PercentilesB. Compute the “75% Plus” Range of Finds

30. Advantage to “75% Plus” . . . • Uses STANDARD DEVIATION in Data to estimate Variability in LPDT values • Very Robust for SMALL Datasets • “Conservative” way to proceed

31. Sorted Data LP Distance Traveled11 Data Points in Miles 2 2 2 34 7 8 9 1011 26 MED = 7 75th Percentile = 10 (9th Position) MEDpos 0.5 * (11+1) = 0.5 * 12 = 6The Data Value “7” is at the 6th Position in the Dataset

32. For “75% Plus” Compute SampleSTANDARD DEVIATION in Excel by using:+STDEV(data range)then for “75% Plus” range calculate: Mean – (2 * SD) = lower bound Mean + (2 * SD) = upper bound

33. Sorted Data LP Distance Traveled For MEAN=7.63 & SD = 6.975 2 2 2 34 7 8 9 1011 26 Lower = 0.0 Upper = 21.59 75% Plus Range = [Mean – 2*SD to Mean + 2*SD] Reflects VARIABILITY Within the Data; When Lower Bound is NEGATIVE, Use Zero

34. 4. Methods for Creating a Consensus In CASIE there are 3 Methods available:1. MATTSON (numeric POA’s = 100%)2. O’CONNOR (use Verbal Cues)3. PROPORTIONAL (rate relative to Baseline #)

35. MATTSON

36. O’CONNOR

37. PROPORTIONAL

38. Initial POA’s from Proportional Consensus

39. 5. 2-Methods for Updating a Search • Bayes Formula, With ROW • OPOS Summation, Without ROW

40. Bayes Formula, With ROW Based on P(A|B) or “the Probability of A, Given B” The fact that I have searched in B affects theprobability of finding the subject in A. OnceB is searched, the POA of A goes UP. B A

41. Bayes Formula, With ROW BIG SCARY Formula . . . Hard to Do by Hand, especially multiple updates Do It In CASIE or a Spreadsheet!

42. Bayes Formula, With ROWUpdate in CASIE

43. Overall Probability of Success, Without ROW

44. 6. Optimizing Resources • Brute Force, Calculate to Exhaustion(David Lovelock, Retired Math Prof, U of AZ) • Washburn Algorithm(Alan Washburn, Naval Post-Graduate School) • Both require estimating Resource POD

45. Optimizing Resourcesin CASIE go to . . .1. top menu “What If” then “Resource Allocation Advice2. Create a New Table

46. Resource Allocation Table: Estimated POD for Each Resourcein Each Segment of Interest

47. WHY BRUTE FORCE?

48. BRUTE FORCE ADVICE – 3 Scenarios

49. Washburn Algorithm – 1 “Optimal” Scenario

50. 7. The Mathematical Importance of CONFINEMENT At a 1 Mile Radius (5,280 feet), Step ONE FOOT farther and the AREA increases by 33,179 sq ft. About 3/4ths of a Football Field (210’ x 150’) to the 74 Yard Line!