Santa Fe, New Mexico
SUDDEN infant death syndrome is a merciless child-killer. A baby born with SIDS seems perfectly healthy. Like other infants, it laughs and cries and grasps at shiny toys. It crawls on its hands and knees. And its tiny heart beats away with the same healthy rhythm of other babies. But each year in the US alone some 7000 of these apparently healthy babies suddenly die in their sleep. A parent may put an infant to bed and return to find it dead only minutes later.
Time after time, emergency medics struggle to save the lives of babies threatened not by asphyxiation or poisoning but by this more mysterious killer. Many of these lives could be saved, however. For when help does arrive in time, it is a relatively simple matter to revive the child. The problem lies not in treatment, but in detection. Is there any way to recognise an infant at risk from SIDS?
Right now, the answer is unfortunately no. Yet some doctors believe that, whatever the cause of SIDS, it shows itself in a strange tendency for some infants鈥� heartbeats to descend into a deadly pattern of regularity. When looked at closely, a healthy heart beats in a complex, irregular rythm as it responds flexibly to myriad incoming signals from the brain, muscles and digestive organs. Too much order spells trouble. Might it be possible, then, to monitor an infant鈥檚 heartbeat and spot when its pattern begins to become less irregular or 鈥渞andom鈥� than it should?
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Mathematics is seldom a matter of life and death. But in this case it is. Unfortunately, here medical practice runs into an old and extremely subtle mathematical conundrum-what, exactly, does it mean for something to be random? We all think we know what 鈥渞andom鈥� means. Roughly speaking, something is random if it is unpredictable, or has no definite pattern. But making this definition precise is notoriously difficult. Is there an objective way to tell if a signal-or a heartbeat-is random or if it instead harbours some concealed order? And is there a way to measure its 鈥渄egree of randomness鈥�?
For the past three decades, the best mathematical theory of randomness was enshrined in a beautiful but difficult set of ideas known as algorithmic information theory. But now a mathematician, spurred on by the problem of SIDS, has developed a far more practical conception of randomness. It may not settle any deep philosophical quandaries, but this new notion is so easy to apply that it is already being used not only by doctors, but also by scientists, engineers and statisticians to put numbers on the degrees of irregularity of everything from heartbeats and stock markets to the 鈥渞andom鈥� sequences of digits that are used to encrypt sensitive information.
The older definition of randomness was born in the mid-1960s when Gregory Chaitin, then an undergraduate at the City University of New York, and the Soviet mathematician Andrei Kolmogorov independently hit upon an ingenious idea. Every number like 
or the square root of 2 can be expressed in decimal form as an infinite sequence of digits. For irrational numbers such as these-which cannot be expressed as the ratio of two integers-the strings of digits never repeat.
But are these sequences truly random? To find an answer, Chaitin and Kolmogorov first focused on the idea of the 鈥渃omplexity of a number鈥�, which they defined as the length of the shortest computer program that will generate that number.
The length of a program depends, of course, on the language used to write it, so Chaitin and Kolmogorov defined a program鈥檚 length as the number of 0s and 1s needed to write it down in binary form-just as computers store programs in memory. The idea is that the shortest program for generating a number will contain only the information that is absolutely essential for calculating the number-which should reflect the number鈥檚 complexity.
Never the same
How complex is , for example? In the 19th century, the Swiss mathematician Jacques Bernoulli derived a beautiful formula involving which reads: 1 + 1/22 + 1/32 + 1/42 +鈥� = 2/6. Based on this formula, it鈥檚 not difficult to write down a program in ten lines or so to generate as many digits of as you like. So is not terribly complex, even though its non-repeating decimal expression, 3.14159265鈥�, suggests that it might be. Nevertheless, an infinitely long number like certainly is more complex than a simple number of finite length like 47, because we can always use a program like 鈥淧RINT 47鈥� to generate the latter.
Using the notion of program length, Chaitin and Kolmogorov came up with a complexity-based definition of a random number-that is, a number represented by a random sequence of digits. It runs: 鈥淎 number is random if the shortest program for calculating its digits is not appreciably shorter than the number itself.鈥� In other words, a number is random if no one can come up with a simple rule for generating it. With this definition, a number such as is not random, since its digits can be generated using a simple rule, such as a program of fixed length based on Bernoulli鈥檚 formula.
So if a number as apparently complicated as isn鈥檛 random, do random numbers really exist? Yes. Surprisingly, almost all numbers are random. For the British mathematician Alan Turing showed that only an exceedingly tiny fraction of all numbers can be generated by programs significantly shorter than the numbers themselves. So truly random numbers are typical. It seems then, that for the purposes of running random number generators for gambling or for any other purpose, you could simply choose a typical number between, say, 0 and 1, and use its digits as a random sequence.
But there鈥檚 a problem. Let鈥檚 say we want to check that our number really is random-just to be sure-before we use it. Unfortunately, Chaitin and Kolmogorov鈥檚 algorithmic approach doesn鈥檛 help us much, because it鈥檚 just plain hard to know what is the length of the shortest program that will produce that number. And without that information, you鈥檙e in trouble trying to use maximum complexity as a test for randomness.
Practically useless
This means that the entire apparatus of algorithmic information theory is useless when it comes to generating random sequences for practical use, or to inspect a real world sequence, such as a baby鈥檚 heartbeat, to see if it is truly random or whether it contains some hidden pattern. The method is also not much help if you want to compare the degrees of randomness of different numbers. Is more random than, say, 
2? You won鈥檛 get any easy answers from algorithmic information theory.
But all these troubles may now have been erased by the work of Steve Pincus, a freelance mathematician from Princeton, New Jersey. Pincus鈥檚 wife, Mary Jane Minkin, is a professor of obstetrics and gynaecology at Yale University, and in her clinical practice works on the front line of the battle against SIDS. Hoping to provide a valuable weapon for physicians such as his wife, Pincus set out a few years ago to find a new way to measure the irregularity of a signal.
The result is a new measure of randomness that Pincus calls 鈥渁pproximate entropy鈥�, which he is continuing to develop and apply with mathematician Burton Singer of Princeton University and Rudolf Kalman of the Swiss Federal Institute of Technology in Zurich. Instead of dealing directly with the very subtle notion of how difficult it is to generate the sequence of digits in a number, approximate entropy focuses on the string of digits themselves and attempts to measure their 鈥渋rregularity鈥�.
Repeating patterns
A string of digits can always be expressed in binary form as a string of 0s and 1s, the trick then is to see if it is regular or irregular. Clearly, a string of all 1s or all 0s is very regular, but everything else will be somewhere in between. Approximate entropy measures the degree of irregularity by looking at how often the digits 0 and 1, or pairs of digits 00, 01, 10 or 11, appear in the overall sequence. Intuitively, in a highly irregular sequence, 00, 01, 10 and 11 should appear with about the same frequency. On the other hand, a sequence with many more instances of 01 than 00 isn鈥檛 very random. (see Diagram).FIG-mg20965102.GIF
The 鈥渆ntropy鈥� in the phrase 鈥渁pproximate entropy鈥� is borrowed from physics. The entropy of, say, some air held in a balloon is a measure of the number of different ways the air molecules can be distributed, in terms of their positions and velocities, while still giving rise to the same large-scale properties. Roughly speaking, a high entropy state, for example when the gas has reached equilibrium, will have the air molecules spread out over all the available states. They will be distributed more or less uniformly within the balloon and will be moving in random directions.
Similarly, the digits in a string with high approximate entropy will be spread out uniformly between 0s and 1s. Also, pairs of digits in the string should be spread uniformly between the possibilities 00, 01, 10 and 11. The same goes for digits taken in blocks of three and four, and so on. Another way to think of approximate entropy is as a measure of how predictable the elements of the sequence are. For example, if the block of numbers 0101 is more often followed by the digit 1 than by 0, then the sequence has some measure of predictability. But if 0101 is equally likely to be followed by 0 or 1, then it is unpredictable. Approximate entropy measures the average unpredictability over blocks of digits of all lengths.
As an amusing illustration of the idea, Pincus and Kalman calculated the degree of randomness of the digit sequences of four of the most commonly used numbers in mathematics: 2, 3, and e, the base of the natural logarithm. Writing out what these numbers look like when expressed as strings of 0s and 1s, they discovered that the string for is the most unpredictable, while that for 2 is next, followed by the strings for e and 3.
At first glance, this is a curious result. It says that is pretty much a random sequence, but we saw earlier that the complexity of is not very great, so according to algorithmic information theory it is far from random. This disagreement is interesting, but perhaps not so significant. It merely serves to point up the differing aims of the two approaches. One, based on complexity, tries to determine how easily computable and 鈥渟imple鈥� a number is, while the other merely measures in practical terms how irregular its digits are. (See 鈥淩ooting out randomness鈥�)
But the real power of approximate entropy lies not so much in its implications for number theory, but in its practical usefulness, in looking for patterns in heartbeats, for example. To apply it to SIDS, Pincus has teamed up with physicians Theodore Cummins and Gabriel Haddad of Yale to develop a method that may help identify infants at risk.
Their idea is to use approximate entropy to screen infants for a tendency to show episodes of unusually regular heartbeat. Such infants could be fitted with monitors to detect periods of extreme regularity, and alert medical personnel or parents. In an early study, the researchers looked at infants who, because of nonfatal SIDS episodes earlier in their lives, are known to suffer from the condition. The results showed that, in comparison with normal infants, the heartbeats of the SIDS infants frequently lapsed into periods of enhanced regularity, as measured by low approximate entropy. The technique doesn鈥檛 yet offer foolproof detection, but through further refinement could become a powerful medical tool to help save infants from SIDS.
But approximate entropy isn鈥檛 applicable only to infants. In another medical application, Pincus has worked with specialists from the University of Virginia, in Charlottesville to study of the variations of testosterone levels in healthy men of various ages. Testosterone is important in determining levels of aggression and libido in men. But while it is known that most men lose sexual interest with age, previous studies of testosterone levels have found no trend toward lower concentrations in the blood of older men.
But testosterone levels aren鈥檛 constant. They fluctuate over periods of hours and even minutes. So Pincus and his colleagues carefully studied the fluctuating levels of testosterone in men鈥檚 blood by measuring it every 2.5 minutes for up to 7 hours. Analysing the resulting time series in terms of approximate entropy, they found that the levels in the older men-although on average no different from those in the younger men-fluctuate more irregularly.
Loss of control
From a biological point of view, these results seem to imply that in the ageing male, the system of hormonal regulation begins to lose its ability to function precisely. As men age, levels become less tied to bodily conditions and needs, but fluctuate up and down with increasing randomness.
Does the change from regular functioning to random functioning occur gradually over years, or suddenly over a period of months? No one yet knows. But the important thing is that such questions can now be asked with confidence, because we know that hormonal regulatory systems function less well in older men that they do in younger.
Pincus has other projects under way as well, looking at patterns of regularity and irregularity in stock markets to see if some periods are more predictable than others. Approximate entropy should also help statisticians in designing techniques for generating truly random samples, for political polling, for example. So randomness has come a long way, from the theoretical realms of a mathematical conundrum to a practical tool. One day it may even be used to save lives. Who said that mathematics is irrelevant to the real world?
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Rooting out randomness
Is really more random than e? It鈥檚 hard to tell. When considered as a base- 2 sequence, the approximate entropy method shows that is more random than e. But numbers can be represented just as well in base 10, or in any other base. If is really more random that e, then approximate entropy should give the same result no matter which base the calculation is done in. But mathematician Steve Pincus and his colleagues made a similar calculation in base 10 and found that, when looked at this way, the sequences for and e show the same degree of randomness. The degree of irregularity of these numbers can be altered even more strikingly by using less familiar ways of representing numbers as strings of digits. For example, numbers like 2 and 3 can also be expressed using something called a continued fraction. As depicted in the Diagram, 2 can be written as 1 plus a fraction, where that fraction is 1 over 2 plus another fraction, and so on forever. The number that results from such a crazy expression is divided by the string of digits that appears along the lower leftmost edge. In this case for 2, the string is {1,2,2,2,鈥. Since this string is perfectly regular, it has an approximate entropy of 0. So, simply by changing our way of representing the number, the answer of how random a number like 2 is can be changed from 鈥渧ery much鈥� to 鈥渘ot at all鈥�.FIG-mg20965101.GIF
