Probability Density Profiles A New Perspective on Subject Behavior Profiles by Martin Colwell SAR Technology Inc. © 2007

Illustrate the Implementation of PDEN GIS Probability Density Profiles A New Perspective on Subject Behavior Profiles Review traditional Probability Profile Plots Demonstrate their typical use in SAR planning Determine their strengths & weaknesses Consider a new method for determining Subject Probability Profiles Describe the theory of PDEN Profiles Create real-data PDEN Profiles Describe the mathematical basis for PDEN Profiles Describe the benefits of PDEN Profiles Demonstrate the generation of PDEN Profile Maps

Traditional Probability Profile Plots Mean (Average): The middle value of all the distances found from the PLS (or IPP). Utilize the standard mathematical terms of Median and Mean (Average) distances the subjects were found from the Point Last Seen (or IPP). Median: The distance at which there are an even number of cases on either side.

Traditional Probability Profile Plots The original distribution of the distances the subjects were found from the PLS. MedianMean Median: The distance at which there are an even number of cases on either side. Mean (Average): The middle value of all the distances found from the PLS.

Traditional Probability Profile Plots Probability Band (Annulus): Defines a distance either side of the median in which a percentage of the subjects were found (eg 25%, 50%, 75% 100%) Median

Traditional Probability Profile Plots Probability Zones: Define a circular zone, from the PLS, in which a percentage of the subjects were found (eg 25%, 50%, 75% 100% POA zones) 25% POA Zone 50% POA Zone

Traditional Probability Profile Plots 75% POA Zone Probability Bands from the MedianProbability Zone from the PLS Typically display subject location POA’s within specific zones or bands

Traditional Probability Profile Plots Disadvantages: Do not take into account either the direction of the subject or the increasing size of the area with distance from the Point Last Seen. Distribution Plots Advantages: Illustrate the linear frequency of actual distances the subjects were found from the Point Last Seen.

Weakness of Traditional POA Probability Profile Plots Probability bands do not take into account the smaller search area inside, compared to outside, the mean. Median and Mean (average) do not describe the actual distribution of the subjects. There may be no subjects found at the median! Do not portray uniformly dispersed or clustered locations either side of the mean. Do not indicate the probable direction of travel. Probability bands do not describe whether subjects were found inside or outside of the mean.

- Displayed as Probability Density Maps in true GIS Applications Probability Density Profiles Probability Density Profiles of Missing Persons - Based on the original subject Distance & Direction Information - Uses an Optimized Bandwidth Procedure to extract useful information - Creates Weibull Type 2 (Raleigh) Distribution Profiles Utilizes Polar (Directional) Coordinate Data - Creates a matrix of Cartesian (XYZ) Grid Data (With thousands of XYZ data points) - Displayed as Two Dimensional & Three Dimensional Probability Density Maps - Generates ESRI GIS Shape and TIFF Files - Can be Geo-Referenced to GeoTiff Files

Probability Density Profiles – Probability Density Profiles – Uniform Distribution

Probability Density Profiles – Probability Density Profiles – Single Cluster Distribution

Probability Density Profiles – Probability Density Profiles – Two Cluster Distribution

Probability Density Profile – Probability Density Profile – Children 1-6 years old

Narrow Optimal Broad

Probability Density Profiles – Probability Density Profiles – Resemble Weibull Distributions Naturally occurring distributions, with mathematically described scale & shape Shape=2

Probability Density Profiles Funding Profile Over Time Probability density: v(t) = 2adte -at 2 - The Weibull distribution models the value’s buildup, peak and taper.

Probability Density Profiles - The Weibull distribution describes the curves buildup, peak and taper. Probability Density = 2adle -al 2 L = Distance d = Scale factor a = Shape parameter The Weibull distribution curve can have it’s shape described in mathematical terms. A general PDEN profile, using the distribution formula, could then be applied to other subject types

Probability Density Profiles - PDEN Profile Prediction

Probability Density Profile – Probability Density Profile – Children 1-6 years old Demonstrating 360 o Radial probability density profile from the PLS Median 0.95km

Probability Density Profile – Probability Density Profile – Children 1-6 years old Demonstrating Directional probability density profile from the PLS

Probability Density Profile – Probability Density Profile – Children 1-6 years old Demonstrating Directional probability density profile from the PLS Expected Direction of Travel 75% POA 50% POA 25% POA

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old Conversion of Radial PDEN Data to X,Y,Z Grid for GIS Display GIS Surface plot

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old PDEN Data in an X,Y,Z Grid format for GIS Display 360 radial data points converts to 6,708 X,Y,Z GIS data points

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old PDEN Data in X,Y,Z Grid format for GIS Display Probability Density Profile displayed as a 3D GIS surface plot PLS

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old Probability Density Profile displayed as a 2D GIS contour map PLS Expected Direction of Travel km With geo-referencing, PDEN contour profiles can be displayed as map layers over a GIS base map. High PDEN region Low PDEN region 2D PDEN Contour map, showing the highest PDEN region in dark red, lowest PDEN region in pale blue. The distance from the central PLS is displayed on X & Y scalebars.

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old 2D Probability Density Profile - GIS Contour Map Layer Associated geo- referencing data displays the PDEN contour map in it’s true geographical location and to the correct map scale. High PDEN region Low PDEN region Expected Direction of Travel PLS 2 km The 2D PDEN Contour map is displayed as a geo- referenced map layer above a street map layer, defining the high PDEN search region on the street map.

Probability Density Profiles – Probability Density Profiles – Children 1-6 years old 2D Probability Density Profiles – As GIS Map Layer The 2D PDEN Contour and Surface Maps can be overlayed on GIS map layers for defining the high probability density areas. 2D and 3D Probability Density Profile Maps can be created in a variety of formats to easily visualize the high probability density areas.

Probability Density Profiles Hikers – North America 50% POA 25% POA Median 2.4km 75% POA Median 2.4km Preliminary Conclusions Children, Hikers and Hunters PDEN profiles have been examined so far. PDEN Map Profiles provide more useful radial and directional planning data than traditional subject POA profiles. Need to re-process the original subject distribution datasets using the Optimized Bandwidth Procedure. ‘Template’ profiles could be created for other subject types with limited datasets. Create a data library of 2D & 3D Subject PDEN Profile maps. Distribute PDEN Profile Maps in various formats – GIS Shape files, Geotiffs, JPG’s, DXF files etc. Weibull distributions appear to occur for many subject types.

Probability Density Profiles Traditional POA Subject Profile PDEN Map Subject Profile Median 2.4km 25% POA 50% POA 75% POA Interim Recommendations For Subject Profile Planning Based on a preliminary examination of the PDEN Profile Maps, compared to traditional Subject POA Profiles. Search radially outwards from the PLS towards the median. Almost all of the high PDEN region is between the PLS and well before the median. The median region may have a fairly low PDEN. Search outwards beyond the median only after searching from the PLS towards the median. Remember that 50% of subjects are still found beyond the median!