The Numbers Behind NUMB3RS: Solving Crime with Mathematics

CHAPTER 1

Finding the Hot Zone

Criminal Geographic Profiling

FBI Special Agent Don Eppes looks again at the large street map of Los Angeles spread across the dining-room table of his father’s house. The crosses inked on the map show the locations where, over a period of several months, a brutal serial killer has struck, raping and then murdering a number of young women. Don’s job is to catch the killer before he strikes again. But the investigation has stalled. Don is out of clues, and doesn’t know what to do next.

Can I help? The voice is that of Don’s younger brother, Charlie, a brilliant young professor of mathematics at the nearby university CalSci. Don has always been in awe of his brother’s incredible ability at math, and frankly would welcome any help he can get. But…help from a mathematician?

This case isn’t about numbers, Charlie. The edge in Don’s voice is caused more by frustration than anger, but Charlie seems not to notice, and his reply is totally matter-of-fact but insistent: "Everything is numbers."

Don is not convinced. Sure, he has often heard Charlie say that mathematics is all about patterns—identifying them, analyzing them, making predictions about them. But it didn’t take a math genius to see that the crosses on the map were scattered haphazardly. There was no pattern, no way anyone could predict where the next cross would go—the exact location where the next young girl would be attacked. Maybe it would occur that very evening. If only there were some regularity to the arrangement of the crosses, a pattern that could be captured with a mathematical equation, the way Don remembers from his schooldays that the equation x² + y² = 9 describes a circle.

Looking at the map, even Charlie has to agree there is no way to use math to predict where the killer would strike next. He strolls over to the window and stares out across the garden, the silence of the evening broken only by the continual flick-flick-flick-flick of the automatic sprinkler watering the lawn. Charlie’s eyes see the sprinkler but his mind is far away. He had to admit that Don was probably right. Mathematics could be used to do lots of things, far more than most people realized. But in order to use math, there had to be some sort of pattern.

Flick-flick-flick-flick. The sprinkler continued to do its job. There was the brilliant mathematician in New York who used mathematics to study the way the heart works, helping doctors spot tiny irregularities in a heartbeat before the person has a heart attack.

Flick-flick-flick-flick. There were all those mathematics-based computer programs the banks utilized to track credit card purchases, looking for a sudden change in the pattern that might indicate identity theft or a stolen card.

Flick-flick-flick-flick. Without clever mathematical algorithms, the cell phone in Charlie’s pocket would have been twice as big and a lot heavier.

Flick-flick-flick-flick. In fact, there was scarcely any area of modern life that did not depend, often in a crucial way, on mathematics. But there had to be a pattern, otherwise the math can’t get started.

Flick-flick-flick-flick. For the first time, Charlie notices the sprinkler, and suddenly he knows what to do. He has his answer. He could help solve Don’s case, and the solution has been staring him in the face all along. He just had not realized it.

He drags Don over to the window. We’ve been asking the wrong question, he says. From what you know, there’s no way you can predict where the killer will strike next. He points to the sprinkler. Just like, no matter how much you study where each drop of water hits the grass, there’s no way you can predict where the next drop will land. There’s too much uncertainty. He glances at Don to make sure his older brother is listening. But suppose you could not see the sprinkler, and all you had to go on was the pattern of where all the drops landed. Then, using math, you could work out exactly where the sprinkler must be. You can’t use the pattern of drops to predict forward to the next drop, but you can use it to work backward to the source. It’s the same with your killer.

Don finds it difficult to accept what his brother seems to be suggesting. Charlie, are you telling me you can figure out where the killer lives?

Charlie’s answer is simple: Yes.

Don is still skeptical that Charlie’s idea can really work, but he’s impressed by his brother’s confidence and passion, and so he agrees to let him assist with the investigation.

Charlie’s first step is to learn some basic facts from the science of criminology: first, how do serial killers behave? Here, his years of experience as a mathematician have taught him how to recognize the key factors and ignore all the others, so that a seemingly complex problem can be reduced to one with just a few key variables. Talking with Don and the other agents at the FBI office where his elder brother works, he learns, for instance, that violent serial criminals exhibit certain tendencies in selecting locations. They tend to strike close to their home, but not too close; they always set a buffer zone around their residence where they will not strike, an area that is too close for comfort; outside that comfort zone, the frequency of crime locations decreases as the distance from home increases.

Then, back in his office in the CalSci mathematics department, Charlie gets to work in earnest, feverishly covering his blackboards with mathematical equations and formulas. His goal: to find the mathematical key to determine a hot zone—an area on the map, derived from the crime locations, where the perpetrator is most likely to live.

As always when he works on a difficult mathematical problem, the hours fly by as Charlie tries out many unsuccessful approaches. Then, finally, he has an idea he thinks should work. He erases his previous chalk scribbles one more time and writes this complicated-looking formula on the board:

*

That should do the trick, he says to himself.

The next step is to fine-tune his formula by checking it against examples of past serial crimes Don provides him with. When he inputs the crime locations from those previous cases into his formula, does it accurately predict where the criminals lived? This is the moment of truth, when Charlie will discover whether his mathematics reflects reality. Sometimes it doesn’t, and he learns that when he first decided which factors to take into account and which to ignore, he must have got it wrong. But this time, after Charlie makes a few minor adjustments, the formula seems to work.

The next day, bursting with energy and conviction, Charlie shows up at the FBI offices with a printout of the crime-location map with the hot zone prominently displayed. Just as the equation x² + y² = 9 that Don remembered from his schooldays describes a circle, so that when the equation is fed into a suitably programmed computer it will draw the circle, so too when Charlie fed his new equation into his computer, it also produced a picture. Not a circle this time—Charlie’s equation is much more complicated. What it gave was a series of concentric colored regions drawn on Don’s crime map of Los Angeles, regions that homed in on the hot zone where the killer lives.

Having this map will still leave a lot of work for Don and his colleagues, but finding the killer is no longer like looking for a needle in a haystack. Thanks to Charlie’s mathematics, the haystack has suddenly dwindled to a mere sackful of hay.

Charlie explains to Don and the other FBI agents working the case that the serial criminal has tried not to reveal where he lives, picking victims in what he thinks is a random pattern of locations, but that the mathematical formula nevertheless reveals the truth: a hot zone in which the criminal’s residence is located, to a very high probability. Don and the team decide to investigate men within a certain range of ages, who live in the hot zone, and use surveillance and stealth tactics to obtain DNA evidence from the suspects’ discarded cigarette butts, drinking straws, and the like, which can be matched with DNA from the crime-scene investigations.

Within a few days—and a few heart-stopping moments—they have their man. The case is solved. Don tells his younger brother, "That’s some formula you’ve got there, Charlie."

FACT OR FICTION?

Leaving out a few dramatic twists, the above is what the TV audience saw in the very first episode of NUMB3RS, broadcast on January 23, 2005. Many viewers could not believe that mathematics could help capture a criminal in this way. In fact, that entire first episode was based fairly closely on a real case in which a single mathematical equation was used to identify the hot zone where a criminal lived. It was the very equation, reproduced above, that viewers saw Charlie write on his blackboard.

The real-life mathematician who produced that formula is named Kim Rossmo. The technique of using mathematics to predict where a serial criminal lives, which Rossmo helped to establish, is called geographic profiling.

In the 1980s Rossmo was a young constable on the police force in Vancouver, Canada. What made him unusual for a police officer was his talent for mathematics. Throughout school he had been a math whiz, the kind of student who makes fellow students, and often teachers, a little nervous. The story is told that early in the twelfth grade, bored with the slow pace of his mathematics course, he asked to take the final exam in the second week of the semester. After scoring one hundred percent, he was excused from the remainder of the course.

Similarly bored with the typical slow progress of police investigations involving violent serial criminals, Rossmo decided to go back to school, ending up with a Ph.D. in criminology from Simon Fraser University, the first cop in Canada to get one. His thesis advisers, Paul and Patricia Brantingham, were pioneers in the development of mathematical models (essentially sets of equations that describe a situation) of criminal behavior, particularly those that describe where crimes are most likely to occur based on where a criminal lives, works, and plays. (It was the Brantinghams who noticed the location patterns of serial criminals that TV viewers saw Charlie learning about from Don and his FBI colleagues.)

Rossmo’s interest was a little different from the Brantinghams’. He did not want to study patterns of criminal behavior. As a police officer, he wanted to use actual data about the locations of crimes linked to a single unknown perpetrator as an investigative tool to help the police find the criminal.

Rossmo had some initial successes in re-analyzing old cases, and after receiving his Ph.D. and being promoted to detective inspector, he pursued his interest in developing better mathematical methods to do what he came to call criminal geographic targeting (CGT). Others called the method geographic profiling, since it complemented the well-known technique of psychological profiling used by investigators to find criminals based on their motivations and psychological characteristics. Geographic profiling attempts to locate a likely base of operation for a criminal by analyzing the locations of their crimes.

Rossmo hit upon the key idea behind his seemingly magic formula while riding on a bullet train in Japan one day in 1991. finding himself without a notepad to write on, he scribbled it on a napkin. With later refinements, the formula became the principal element of a computer program Rossmo wrote, called Rigel (pronounced RYE-gel, and named after the star in the constellation Orion, the Hunter). Today, Rossmo sells Rigel, along with training and consultancy, to police and other investigative agencies around the world to help them find criminals.

When Rossmo describes how Rigel works to a law enforcement agency interested in the program, he offers his favorite metaphor—that of determining the location of a rotating lawn sprinkler by analyzing the pattern of the water drops it sprays on the ground. When NUMB3RS cocreators Cheryl Heuton and Nick Falacci were working on their pilot episode, they took Rossmo’s own metaphor as the way Charlie would hit upon the formula and explain the idea to his brother.

Rossmo had some early successes dealing with serial crime investigations in Canada, but what really made him a household name among law enforcement agencies all over North America was the case of the South Side Rapist in Lafayette, Louisiana.

For more than ten years, an unknown assailant, his face wrapped bandit-style in a scarf, had been stalking women in the town and assaulting them. In 1998 the local police, snowed under by thousands of tips and a corresponding number of suspects, brought Rossmo in to help. Using Rigel, Rossmo analyzed the crime-location data and produced a map much like the one Charlie displayed in NUMB3RS, with bands of color indicating the hot zone and its increasingly hot interior rings. The map enabled police to narrow down the hunt to half a square mile and about a dozen suspects. Undercover officers combed the hot zone using the same techniques portrayed in NUMB3RS, to obtain DNA samples of all males of the right age range in the area.

Frustration set in when each of the suspects in the hot zone was cleared by DNA evidence. But then they got lucky. The lead investigator, McCullan Mac Gallien, received an anonymous tip pointing to a very unlikely suspect—a sheriff’s deputy from a nearby department. As just one more tip on top of the mountain he already had, Mac was inclined to just file it, but on a whim he decided to check the deputy’s address. Not even close to the hot zone. Still something niggled him, and he dug a little deeper. And then he hit the jackpot. The deputy had previously lived at another address—right in the hot zone! DNA evidence was collected from a cigarette butt, and it matched that taken from the crime scenes. The deputy was arrested, and Rossmo became an instant celebrity in the crime-fighting world.

Interestingly, when Heuton and Falacci were writing the pilot episode of NUMB3RS, based on this real-life case, they could not resist incorporating the same dramatic twist at the end. When Charlie first applies his formula, no DNA matches are found among the suspects in the hot zone, as happened with Rossmo’s formula in Lafayette. Charlie’s belief in his mathematical analysis is so strong that when Don tells him the search has drawn a blank, he initially refuses to accept this outcome. You must have missed him, he says.

Frustrated and upset, Charlie huddles with Don at their father Alan’s house, and Alan says, I know the problem can’t be the math, Charlie. It must be something else. This remark spurs Don to realize that finding the killer’s residence may be the wrong goal. "If you tried to find me where I live, you would probably fail because I’m almost never there, he notes. I’m usually at work." Charlie seizes on this notion to pursue a different line of attack, modifying his calculations to look for two hot zones, one that might contain the killer’s residence and the other his place of work. This time Charlie’s math works. Don manages to identify and catch the criminal just before he kills another victim.

These days, Rossmo’s company ECRI (Environmental Criminology Research, Inc.) offers the patented computer package Rigel along with training in how to use it effectively to solve crimes. Rossmo himself travels around the world, to Asia, Africa, Europe, and the Middle East, assisting in criminal investigations and giving lectures to police and criminologists. Two years of training, by Rossmo or one of his assistants, is required to learn to adapt the use of the program to the idiosyncrasies of a particular criminal’s behavior.

Rigel does not score a big win every time. For example, Rossmo was called in on the notorious Beltway Sniper case when, during a three-week period in October 2002, ten people were killed and three others critically injured by what turned out to be a pair of serial killers operating in and around the Washington, D.C., area. Rossmo concluded that the sniper’s base was somewhere in the suburbs to the north of Washington, but it turned out that the two killers did not live in the area and moved too often to be located by geographic profiling.

The fact that Rigel does not always work will not come as a surprise to anyone familiar with what happens when you try to apply mathematics to the messy real world of people. Many people come away from their high school experience with mathematics thinking that there is a right way and a wrong way to use math to solve a problem—in too many cases with the teacher’s way being the right one and their own attempts being the wrong one. But this is rarely the case. Mathematics will always give you the correct answer (if you do the math right) when you apply it to very well-defined physical situations, such as calculating how much fuel a jet needs to fly from Los Angeles to New York. (That is, the math will give you the right answer provided you start with accurate data about the total weight of the plane, passengers, and cargo, the prevailing winds, and so forth. Missing a key piece of input data to incorporate into the mathematical equations will almost always result in an inaccurate answer.) But when you apply math to a social problem, such as a crime, things are rarely so clear-cut.

Setting up equations that capture elements of some real-life activity is called constructing a mathematical model. In constructing a physical model of something, say an aircraft to study in a wind tunnel, the important thing is to get everything right, apart from the size and the materials used. In constructing a mathematical model, the idea is to get the appropriate behavior right. For example, to be useful, a mathematical model of the weather should predict rain for days when it rains and predict sunshine on sunny days. Constructing the model in the first place is usually the hard part. Doing the math with the model—i.e., solving the equations that make up the model—is generally much easier, especially when using computers. Mathematical models of the weather often fail because the weather is simply far too complicated (in everyday language, it’s too unpredictable) to be captured by mathematics with great accuracy.

As we shall see in later chapters, there is usually no such thing as one correct way to use mathematics to solve problems in the real world, particularly problems involving people. To try to meet the challenges that confront Charlie in NUMB3RS—locating criminals, tracing the spread of a disease or of counterfeit money, predicting the target selection of terrorists, and so on—a mathematician cannot merely write down an equation and solve it. There is a considerable art to the process of assembling information and data, selecting mathematical variables that describe a situation, and then modeling it with a set of equations. And once a mathematician has constructed a model, there is still the matter of solving it in some way, by approximations or calculations or computer simulations. Every step in the process requires judgment and creativity. No two mathematicians working independently, however brilliant, are likely to produce identical results, if indeed they can produce useful results at all.

It is not surprising, then, that in the field of geographic profiling, Rossmo has competitors. Dr. Grover M. Godwin of the Justice Center at the University of Alaska, author of the book Hunting Serial Predators, has developed a computer package called Predator that uses a branch of mathematical statistics called multivariate analysis to pinpoint a serial killer’s home base by analyzing the locations of crimes, where the victims were last seen, and where the bodies were discovered. Ned Levine, a Houston-based urban planner, developed a program called Crimestat for the National Institute of Justice, a research branch of the U.S. Justice Department. It uses something called spatial statistics to analyze serial-crime data, and it can also be applied to help agents understand such things as patterns of auto accidents or disease outbreaks. And David Canter, a professor of psychology at the University of Liverpool in England, and the director of the Centre for Investigative Psychology there, has developed his own computer program, Dragnet, which he has sometimes offered free to researchers. Canter has pointed out that so far no one has performed a head-to-head comparison of the various math/computer systems for locating serial criminals based on applying them in the same cases, and he has claimed in interviews that in the long run, his program and others will prove to be at least as accurate as Rigel.

ROSSMO’S FORMULA

finally, let’s take a closer look at the formulas Rossmo scribbled down on that paper napkin on the bullet train in Japan back in 1991.

To understand what it means, imagine a grid of little squares superimposed on the map, each square having two numbers that locate it: what row it’s in and what column it’s in, "i and j". The probability, pij, that the killer’s residence is in that square is written on the left side of the equation, and the right side shows how to calculate it. The crime locations are represented by map coordinates, (x1,y1) for the first crime, (x2,y2) for the second crime, and so on. What the formula says is this:

To get the probability pij for the square in row "i, column j of the grid, first calculate how far you have to go to get from the center point (xi,yj) of that square to each crime location (xn,yn). The little n here stands for any one of the crime locations—n=1 means first crime, n=2 means second crime," and so on. The answer to the question of how far you have to go is:

| xi – xn| + | yj – yn|

and this is used in two ways.

Reading from left to right in the formula, the first way is to put that distance in the denominator, with ? in the numerator. The distance is raised to the power f. The choice of what number to use for this f will be based on what works best when the formula is checked against data on past crime patterns. (If you take f = 2, for example, then that part of the formula will resemble the inverse square law that describes the force of gravity.) This part of the formula expresses the idea that the probability of crime locations decreases as the distance increases, once outside of the buffer zone.

The second way the formula uses the traveling distance of each crime involves the buffer zone. In the second fraction, you subtract the distance from 2B, where B is a number that will be chosen to describe the size of the buffer zone, and you use that subtraction result in the second fraction. The subtraction produces smaller answers as the distance increases, so that after raising those answers to another power, g, in the denominator of the second part of the formula, you get larger results.

Together, the first and second parts of the formula perform a