epidemic model marauder commuter center of mass anchor point taking into account the curvature of Earth regression analysis psychological factors

Transcription

1 Our motivation begins with Dr. John Snow s battle to end the cholera epidemic in Great Britain. Thus, we can think of a series of crimes as an epidemic model. For a disease that spreads from a central location, we can interpret this as a criminal who commits crime within the convex hull of crime locations. We call this the marauder scheme. Similarly, we can interpret a disease that spreads from person to person as a criminal who moves from location to location. We call this the commuter scheme. This is the basis of our two schemes we chose to simulate. Finally, we emphasize that each scheme should be evaluated on a case-by-case basis. For the marauder scheme, we apply physical models in classical mechanics to find the center of mass via double sums and double integration. Finding the center of mass is our approach to estimate the central location where the criminal resides, also known as an anchor point. To assess this model, we use data from a real-life serial murder case, namely, that of Harold Fredrick Shipman. With qualitative information, we estimated the location of the crime scenes by finding the longitude and latitude coordinates of each city and street to get quantitative information. We then compute the center of mass using increments of five locations to see if the center of mass converges to the actual location of the criminal s hideout. Along with time-series and statistical analyses, we found that the distance from the anchor point to the crime scene is normally distributed with a mean approximately 1.5 kilometers away from the anchor point. These distances are calculated using the geographical coordinates, taking into account the curvature of Earth. To improve our model using additional evidence, we can take into account dummy crime locations to shift the center of mass more accurately. For the commuter scheme, we assume the criminal will move in one general direction so that we can estimate the criminal s location by regression analysis. Furthermore, we take into account the criminals psychological factors (such as the lolita complex) to help us estimate patterns in movement, and how frequently victims will be victimized. Calculations and simulations are aided by the used of software such as Excel, Maple, MATLAB, as well as online applets.

2 Team # 6715 Page 1 of 17 Table of Contents 1.1 Executive Summary: An Epidemic Model Introduction Main Results Simplifying Assumptions Case I: Shipman, the Marauder Center of Mass and Double Sums/Integrals Time-Series and Statistical Analysis Case I Simulation Results Remarks 3.2 Case II: Bundy, the Commuter Spatial View of Crime Time-Series and Regression Analysis 4.1 Model Assessment Pros Cons 4.2 Conclusions 11 References 12 Appendix A Case I Data 13 Appendix B Case II Data 15 Appendix C Programming Code

3 Team # 6715 Page 2 of Executive Summary: An Epidemic Model Imagine an infectious flu, such as H1N1. Without a doubt, this virus has killed many people due to germs lurking on doorknobs, stair handrails, and many facilities we use in everyday life. But what if the source of a strain of H1N1 was located on a door handle of a particular school? Students and teachers are at risk of contracting the virus. In the worst case scenario, there will be deaths. If we first assume that the only way of contracting H1N1 is by contact with the door handle, we can collect data on where the victim died to find patterns. Intuitively, we should be able to find that the victims are within the school s radius by seeing where all the incidents occur. On the other hand, if we now assume that virus can spread from person to person, the H1N1 virus is sure to spread and travel away from its source. In a sense, these two cases can be though of as a serial killer, either located idly at a central crime scene, or always on the move away from authorities. This ideal motivates our following discussion Introduction In our epidemic model, we estimate the crime locations and estimate the area where an anchor point occurs. We define an anchor point is a central location of criminal activity; this could be the criminal s home, workplace, or even their favorite food joint. With the motivation of the classical mechanics in physics, the center of mass can be thought of as the origin of criminal activities, anchored within the convex hull of the crime locations. The convex hull is simply the enclosed area within the crime scenes. Assuming that the crime is carried out by one person or a group of people having similar anchor points, we can estimate these anchor points. However, to do this accurately, we only consider criminal cases where there are many offences (or serial crime). Thus, our model is only effective when we have many data points, so we will also assume that a serial crime has at least three crime locations. There are many ways for a serial crime to occur, and this includes having many anchor points where the crime originates. For instance, there may be many criminal activity located in clusters around anchor points over time. Finding the center of mass using the longitude and latitude coordinates fail when we consider all points of criminal activity because we can only estimate the area central to the crimes. If there were three origins of crime distanced apart like a triangle then the center of mass would be located at the triangle s center. To avoid this problem, we find the center of mass for each cluster of crime to estimate where the criminal is hiding out. -2-

4 Team # 6715 Page 3 of 17 Criminals who are located inside the convex hull are called marauder. Finding the center of mass also fails when we have moving criminals who do not reside in a central location of crime. In other words, the criminal is continuously on the move, and does not remain near the scene of the crime. Each crime will be progressively distance the criminal from the previous crime. We will define this type of criminal as a commuter. To analyze these types of criminals, we use a time-series analysis and compute a regression curve. We can assume that commuters will generally travel in one direction after evading previous crimes. It should be noted that our model works for these two schemes criminal schemes: the commuter and the marauder. We have considered other schemes, such as international criminals with easy access to aircrafts or boats. Finding the center of mass for international criminals may result in pinpointing the middle of the ocean. However, to fix this problem, we reduce this situation to the marauder scheme and consider local cases by clustering crimes which take place relatively close to each other Main Results Overall, these models are sufficient to estimate anchor points for crime, and the models become more accurate with either more crime data. To evaluate models more accurately, we must consider the serial crime on a case-by-case basis for clues, such as the criminal s background history. To integrate evidence for criminals who have a particular history, say high school female teen-rape, we would plot dummy points at middle and high schools within the convex hull to get a better location for the center of mass. If crime should indeed occur at one of the dummy points, we can narrow the convex hull to get an idea of where the criminal is anchored. This will make our model more reliable. Generally, we can use dummy points in suspected crime scenes to simulate how the criminal will act. In the following analysis, we collect data from real serial murder cases of Harold Frederick Shipman and Ted Bundy. After plotting the data, we decided to use a fraction of the data at a time to predict the future crime locations. As a result we were able to find that, as the number of victims increased, our estimate gets closer and closer to the actual location of the criminal. We use a variety of software to get results, such as Microsoft Excel, Maple, MATLAB, and online applets to calculate distances using geographical coordinates. See References section. -3-

5 Team # 6715 Page 4 of Simplifying Assumptions We have developed two schemes for modeling serial crime activity: The Marauder Scheme and the Commuter Scheme. For international crime, we reduce this problem into many Marauder schemes. In both cases, we assume that many crimes take place. The number of crimes that defines many depends on a case-by-case basis since some crimes are more predictable than others due to the criminal s background. Also, we assume that the criminal will continue committing crimes forever. General Assumptions: The crime region will be rectangular region unless otherwise stated The criminal will continue to commit crimes until caught Victims will be chosen randomly, and are equally likely to be victimized Psychological factors of the criminal cannot be account, but can be used to strengthen our estimate. Assumptions for the Marauder Scheme: Crimes in which many anchor points can be identified will be separated into as many clusters a there are anchor points (such as international crime) Assumptions for the Commuter Scheme: Each successive crime will take place farther away from the previous one The criminal will continue to move in one general direction o the criminal does not return to the scene of the crime. *most of the reference were used as general knowledge of the subject* -4-

6 Team # 6715 Page 5 of Case I: Shipman, the Marauder Center of Mass and Double Sums/Integrals By understanding the physics of classical mechanics, we can apply the concept of center of mass by either using a double sum (for discrete cases) or double integrals (for continuous cases). Our problem can be thought of as a system of particles spaced around a specific center. Using a two-dimensional space, we ignore land altitude and use Earth s longitude and latitude as our coordinate system. To find the center of mass, we use the following formulae[7]: i ximi x cm = m and i i i yimi y cm = m for the discrete case; (1) i i x cm = V ρ x dv V ρ dv and y cm = V ρ y dv V ρ dv for the continuous case. (2) In this case, m and ρ are constant since we assumed that crimes are equally likely to occur at any point in the specified region. See Appendix A for the data used[3]. Although we believe this problem is discrete, we will simplify our calculations by turning this into a continuous problem Time-Series and Statistical Analysis We will find the center of mass for Shipman s location starting with five data points of the victims. Then, we continue calculating the center of mass with ten victim points, fifteen victim points, and so on, until we used up all of the data points[3]. See Appendix B for the programming code to calculate the center of mass. We predicted that the more data points we use, our center of mass will converge to the actual location. In any case, we are sure to expect the criminal s anchor point to be within the convex hull[5]. -5-

7 Team # 6715 Page 6 of 17 FIGURE 1 Actual location of Shipman and his victims relative to their longitude and latitude coordinates. A few outlier points are left out. From the data, we have plotted distance versus time graphs and found the regression line. Furthermore, we plotted the histogram for the Harold Fredrick Shipman case and found that the majority of his murders are located approximately 1.5 kilometers from his anchor point (his home). It appears that the distance between the killer and the victim is normally distributed. FIGURE 2 The histogram displays the number of deaths by Shipman and how far away from his home it occurred. -6-

8 Team # 6715 Page 7 of Case I Simulation Results It appears that, as more information of new victims becomes available, the estimated center of mass converges to the actual location of the criminal. However, our goal is to catch the criminal sooner and minimize the number of victims y = x R 2 = Distace from Actual Anchor Number of Data Points FIGURE 3 A plot of the estimated center of mass versus the number of data points Remarks Mathematical models alone cannot be used to predict the location of a criminal with past data. We acknowledge that many other factors come into play, as well as random actions which may distort the models. In FIGURE 4 and FIGURE 5, we show Shipman s killing pattern and it could also be predicted intuitively. For example, Shipman could have started killing and become obsessed with it. But as authorities caught on and suspected Shipman, he could have laid low for a while to suppress authorities suspicions. After his quiet phase, we would then kill some more people, and then lay low as necessary. This psychological factor can also be used to estimate when Shipman will strike again. -7-

9 Team # 6715 Page 8 of 17 FIGURE 4 This plot shows Shipman s killings over the years. FIGURE 5 This plot shows Shipman s killing cycle over the years. -8-

10 Team # 6715 Page 9 of Case II: Bundy, the Commuter Spatial View of Crime A sample of Ted Bundy s criminal activities is displayed in FIGURE 6 below. FIGURE 6 This plot displays the locations of Bundy s criminal activities. His initial activities starts from the southeast region and end in the northwest region. As stated before, we will assume the criminal will move in one general direction after committing crime. -9-

11 Team # 6715 Page 10 of Time-Series and Regression Analysis Like a disease spreading from city to city, we can think of Bundy s movements like a disease. Although disease can spread in many directions at once, we can consider one direction at a time to trace the source and predict where it will go next. Ted Bundy's Locations 2000 y = x R 2 = Distace from Inital Crime Time from Inital Crime (days) FIGURE 7 The plot displays the number of deaths by Bundy and how far away from his previous crime. Bundy is traveling northwest from his initial crime. As a result, we can conclude that commuter type criminals tend to travel farther and farter from the initial crime scene. Assuming the criminal will continue to commit crime, it should be easy to track down which direction the criminal is headed. -10-

12 Team # 6715 Page 11 of Model Assessment Pros Simple and easy to understand Estimates a rough area of criminal activity Model can be used in a variety of cases Cons Too many assumptions Not extremely accurate Does not consider altitude of a region Psychological factors cannot be formulated into the models Needs more victims to find criminal Minimizing victims is an issue 4.3 Conclusions The more data we have, the more accurate our estimate. However, we would like to minimize the number of victims to do this. By analyzing each serial crime case-by-case, we can find a good number of clues to find the criminal. We don t believe there is a general model that can fit all crime. Previous research also suggest that current software were only able to get around 30% accurate in their models[5]. We believe this is due to psychological factors, and random events that may occur to change peoples courses of action each day. But with these models, it will provide more logical guess to where the criminal can be. -11-

13 Team # 6715 Page 12 of 17 References [1] O Leary, M. (2008). The Mathematics of Geographic Profiling. < Accessed February 19, [2] Branca, P. (2003). Geographic Profiling An examination of the predictive potential of serial armed robberies. < Accessed February 19, [3] BBC News UK. (2004). Shipman s 215 Victims. < Accessed February 19, [4] Mohler, G.O. & Short, M.B. (2009). Geographical profiling from kinetic models of criminal behavior. < Accessed February 19, [5] Paulsen, D.J. (2004). Geographical Profiling: Hype or Hope? Preliminary Results into the Accuracy of Geographic Profiling Software. < en.pdf> Accessed February 19, [6] Gorr, L.W. (2004). Framework for Validating Geographic Profiling using Samples of Solved Serial Crimes. < Accessed February 19, [7] Fowles, G.R. & Cassiday, G.L. (2006). Analytical Mechanics 7 th Edition. [8] Huerta, R. & Tsimring, L.S. (2002). Contact tracing and epidemics control in social networks. Accessed February 19, [9] [10]

14 Team # 6715 Page 13 of 17 Appendix A Data for Case I Raw Data Distance away from Anchor Number of days after first crime FIGURE 7 Because the data set is large[3], we summarize it into this figure. -13-

15 Team # 6715 Page 14 of 17 TABLE 1 Number of Data Points The center of mass is estimated for each increment of 5 victims data. Distance from Anchor Latitude Longitude Error for Latitude Error for Longitude E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

16 Team # 6715 Page 15 of 17 Appendix B Data for Case II TABLE 2 A sample of data of Ted Bundy s Victims. Name of Victims Date Disappeared Place Disappeared Latitude Longitude Anne Marie Burr Aug-61 Tacoma, WA Lonnie Trumbell 23-Jun-66 Seattle, WA Rita Jolly Jun-73 West Linn, Oregon Vicki Lynn Hollar Aug-73 Eugene, Oregon Katherine Merry Divine 25-Nov-73 Seattle WA Lynda Ann Healy 31-Jan-74 Seattle, WA Donna Gail Manson 12-Mar-74 Evergreen State College Susan Elaine Rancourt 17-Apr-74 Central WA State College Roberta Kathleen Parks 6-May-74 OSU in Corvalles, Oregon Brenda Baker 25-May-74 home in Redmond, WA Brenda Carol Ball 31-May-74 Burien, WA Georgeann Hawkins 10-Jun-74 Seattle, WA Denise Naslund 14-Jul-74 Lake Sammamish State park, WA Janice Ott 14-Jul-74 Lake Sammamish State park, WA Melissa Smith 18-Oct-74 Midvale, Utah Laura Aimes 31-Oct-74 Lehi, Utah Julie Cunningham 15-Mar-75 Vail, Colorado Denise Oliverson 6-Apr-75 Grand Junction, Colorado Melanie Cooley 15-Apr-75 Nederland, Colorado Shelly Robertson 1-Jul-75 Golden, Colorado Debi Kent 8-Nov-75 Bountiful, Utah Caryn Campbell 12-Jan-76 Wildwood Inn, Aspen, Colorado Lisa Levy 14-Jan-78 Tallahassee, Florida Kimberly Leach 9-Feb-78 Lake City, Florida

17 Team # 6715 Page 16 of 17 Appendix C Programming Code The following procedure (written in MAPLE) was used to calculate the center of mass: -16-

18 Team # 6715 Page 17 of 17 > AnchorPoint:=center(Y); -17-