mathematics

I ran across an interview between Anna Plutinski and population geneticist Warren Ewens.

I cannot say enough about Ewens' book, Mathematical Population Genetics. If you can work through it, you can do population genetics. It doesn't cover every au courant topic, but those will change next week anyway. And it's on Kindle now. Which I suppose probably looks pretty good on the DX, assuming the math displays well -- the book's format is just the right size for it.

Anyway, this interview from 2004 was probably conducted around the time the book was released. It covers pretty much the gamut of his career. I have to select some part to quote for you, so I'll select the passage that would be most likely to come out of my own math in my genetics class:

WE: Of course there is a strong possibility that the neutral theory is assumed not because it is appropriate but because the math of that theory is so very simple compared to the math applying for any selective theory.

AP: Can I follow that up? Do you think that that has lead to models of phylogenetic change that is not very well supported by the evidence?

WE: I think that that is quite possible. However, here we enter into another question. In mathematical population genetics theory you know from the very start that you are making big simplifying assumptions. You are in a very different position from a physicist, who might believe that his mathematical models describe reality exactly. No sensible population geneticist would make any claim along those lines. He or she is forced to simplify, because reality is so complicated that you don’t know it in any detail, and even if you did know it and used math describing it faithfully, the analysis would be impossible to carry through. So simplification is unavoidable. I do not know whether the use of the neutral theory is too much of a simplification and has lead us to incorrect and distorted views about the true evolutionary tree, it’s shape and dimensions, but I suspect that there has been quite a significant distortion.

There is much more at the link, some history of association testing, genetic draft, a lot on Ewens sampling theory, and a touch about his work here in Madison.

Last week's Science included an article by Adrian Cho examining the way that social modelers use math to describe human behavior on a large scale ("Ourselves and our interactions: the ultimate physics problem?"). I'm sort of irritated at the way physics shows up in this. I mean, sure if -- for the purposes of a model -- we can treat people as interacting particles, then that shares a mathematical basis with (some kinds of) physics modeling.

Behind it all lies the assumption that, at least within distinct types, people are like subatomic particles: basically the same. "We like to think that we are unique," says Alessandro Vespignani, a physicist at Indiana University, Bloomington, who works on networks. "But probably for 90% of our social interactions, we are not so unique."

This isn't a very relevant criticism -- some models may assume that every individual is identical, but they need not do so. If there are well-characterized variations in behavior, a model can incorporate them directly. At some level this is what shopping centers do to predict the behavior of teenagers -- do you put the pink cell phones across from Hollister, or the blue ones?

In any event, does that mean that every kind of mathematical model should be called "physics"? In practice, it seems to be people trained in physics who carry out this kind of work:

Forays into "sociophysics" began in the early 1970s. Physicists proposed, for example, that individuals interact to form public opinion much as neighboring atoms make a crystal magnetic by aligning their magnetic fields; researchers analyzed the social phenomenon by adapting the Ising model used to describe such magnetic interactions. In the 1990s, many physicists turned to economics in the controversial subfield of econophysics (see sidebar, p. 408). Now, the movement seems to be gathering momentum, as complex-systems researchers have made solid contributions in the study of traffic, epidemiology, and economics. Some are now tackling more-daunting problems, such as the emergence of social norms.

"The problems are more complicated than most natural scientists assume, but less hopeless than most social scientists think," says Dirk Helbing, a physicist-turned-sociologist at the Swiss Federal Institute of Technology Zürich (ETHZ).

Sadly many traditional disciplines are safe harbors for the math-impaired. Disciplinary fence-building happens for understandable reasons -- not least, that "interdisciplinary initiatives" often cover administrative efforts to cut faculty or increase courseloads. The route to useful new mathematical models may be easier through cross-disciplinary institutes of various kinds, but even these are often subject to a kind of tunnel vision -- the founders of institutes have pretty specific ideas of what they value.

Is there a future in particle models of humans, from an anthropology perspective? There's no doubt in my mind -- several of the high-ranking anthropologists and primatologists I know are deeply interested in network effects, hub/spoke models, and phase transitions. My only hesitation is that the models are being driven mainly by consistency. Models can produce outcomes that look like real social systems, and people who don't dig into the mathematical details can find this consistency very convincing. But consistency is not enough; untested models may be simpler, more realistic, or consistent with broader observations. So we need more people familiar with social systems to dig into the details of these models.

References:

Cho A. 2009. Ourselves and our interactions: the ultimate physics problem? Science 325:406-408. doi:10.1126/science.325_406

The Telegraph has a short article on Mark Pagel's research into reconstructing ancient languages:

Dr Pagel has tracked how words have changed by comparing languages from the Indo-European family, which includes most of the past and present languages of Europe, the Middle East and the Indian sub-continent.

He has been able to track the evolutionary history of Indo-European back using a computer and said that some of the oldest words were well over 10,000 years old even though the original Indo-European language is thought to date back no more than 9,000 years.

"I can say with confidence that there are sounds or words that predate Indo-European," he said. "If you look at 'thou', 'I' and 'who', we can now tell they are probably at least 15,000 to 20,000 years old. The sounds used then for these meanings were probably very similar to those used today."

Pagel is far from alone in reconstructing proto-Indo-European, of course, but he is introducing evolutionary methods to the problem -- called "glottochronology" -- in a unique way.

The press release from IBM is actually more infomative (the project used an IBM supercomputer for its analysis):

Looking to the future, the less frequently certain words are used, the more likely they are to be replaced. Other simple rules have been uncovered - numerals evolve the slowest, then nouns, then verbs, then adjectives. Conjunctions and prepositions such as: ‘and’, ‘or’, ‘but’ and ‘on’, ‘over’, ‘against’ evolve the fastest, some as much as 100 times faster than numerals. ‘Throw’ which is expected to evolve quickly, has a half-life of 900 years, there are 42 unrelated sounds for it across all the languages. In 10,000 years time, it will likely have been replaced in 10 of them – possibly including English, unless of course we all do our part to keep the word in circulation.

“50% of the words we use today would be unrecognisable to our ancestors living 2,500 years ago. If a time-traveller came to us, and told us he wanted to go back to that period, we could arm him with the appropriate phrase book, and hopefully keep him out of trouble” explained Mark Pagel, Professor of Evolutionary Biology at the University of Reading.

I'll have to wait to see the paper. I will be interested to get an idea of some of the dates they are proposing for language families and their relations.

There are still questions that would pose problems for this method. If we have to rely on highly-conserved words to do chronologies of language relations deeper than 5000 years, we are limited to a very small subset of words in any given language. How likely are these conserved words to be borrowed -- causing similarities without recent ancestry? Are the probabilities of change across many languages over a short time (the source of the statistic) really comparable to the probability of change in a few ancient languages over a long time?

UPDATE (2009-02-28): A linguistically-minded reader writes:

Regarding this recent post in your blog, I was surprised to see you actually recommend the press release by IBM and the University of Reading ("The press release from IBM is actually more infomative ..."). Even an amateur linguist like myself can see that it's crap. There's been considerable discussion on Language Log, starting from the BBC interview, where Pagel was induced to make a right fool of himself, linguistically speaking (which doesn't say anything one way or the other about his computer expertise, of course).

You can find the Language Log story here, along with comments. I always withhold judgment on a piece of work until I've read it myself, but I certainly sympathize with those who think the story was poorly reported in the press.

Particularly the one with the headline "Handy phrasebook for Doctor Who".

I'd like to point readers to James Crow's article in the open access Journal of Biology. Titled, "Mayr, mathematics and the study of evolution," it's a brief summary of some of the important results from mathematical genetics -- sort of a follow-on to Haldane's "A defense of beanbag genetics".

Coming fifty years after Haldane's effort, Crow has been able to include a lot more stuff -- in particular the consequences of the mathematical development of neutral theory, and the effects of computers, permutation tests, and molecular clock models in phylogenetics.

I cannot help but quote this passage, which is direct:

Ironically, Mayr himself unwittingly provided an especially compelling argument for mathematical analysis. His theory of “genetic revolutions” assumed that from a well integrated population, genetic drift in a small founder offshoot will sometimes produce a population with a new set of genotypes integrated in a new way. Intuitively, a small founder population seemed a particularly unlikely place to find a new favorable gene combination, and this was indeed shown to be the case in a very detailed mathematical analysis by Barton and Charlesworth [5]. If Mayr had had more respect for mathematical population genetics, he never would have made what most theorists regard as the mistake of proposing that small founder populations are a likely source of major evolutionary changes by genetic drift (Crow 2009:13.2).

Lest you think this is an argument against the role of chance, Crow later describes the more au courant view of speciation:

Until recently, mathematical theory had contributed little to the study of speciation. Mayr emphasized allopatric speciation and the prevailing model, due to Dobzhansky and Muller [9], prevailed. Recent mathematical studies [10] support it and favor the view that speciation genes correspond to normal genes, selected for their effects within the species. Furthermore, there is evidence that these genes evolve rapidly. Thus, hybrid incompatibility is a by-product of ordinary selection in geographically isolated populations (Crow 2009:13.4).

This model of speciation recognizes chance and contingency, but not mainly from stochastic fluctuations in allele frequency (drift). Instead we have the stochastic processes of mutation and environmental change and the (possibly complex) epistatic interactions among selected alleles.

There's more in the essay. Crow does refer to human evolution -- the out of Africa scenario and Neandertal genetics make appearances not entirely to my taste, but he notes that selective sweeps -- dear to my heart -- are an important feature of the recent landscape of mathematical genetics as well.

Crow could have included a number of other mathematical developments, particularly the Price equation, Hamilton's contributions, and Maynard Smith's "evolutionarily stable strategies", all of which share his theme of the mathematical derivation coming first, and the non-mathematical descriptive formulations only coming later.

References:

Crow JF. 2009. Mayr, mathematics and the study of evolution. J Biol 8:13. doi:10.1186/jbiol117

I'm reading through Peter Turchin's 1998 book, Quantitative Analysis of Movement, for a project I'm working on. I found that his second chapter gives a very nice introduction to the reasons why biology depends on formal mathematical models. This is a topic I often review in my courses, so I'll quote some of his discussion.

He lists six objectives for model-building on pp. 33-35, each with some explanatory text. This amounts to a paragraph or so for each reason; I'm only giving one or two sentences of each, with much omitted.

Formal statement of the problem ...The necessity of stating the assumptions of the model is another benefit. A mathematical description of a problem forces one to be very clear about what the different variables and parameters in the model are, and how they are interrelated.

Identifying knowledge gaps ...It may turn out that good quantitative data are available to estimate some functions and parameters but not others, immediately suggesting a focus for the empirical program. When there are many gaps, one has to decide which parameters need to be estimated precisely, and for which parameters ``guesstimates'' will do....

Gaining theoretical insights There is a large class of models that are never intended to be directly confronted with data.... The purpose of such models is to gain insights into possible causal interconnections between various factors and, in general, extend our intuition...

Quantitative tests of theory ...A qualitative prediction allows one to test the theory that generated it, but it does not provide a very strong test. Because there are only a few possible outcomes in a qualitative situation (e.g., factor X will either increase, stay the same, or decrease), the probability that the ``correct'' outcome will happen by chance is correspondingly high. A quantitative prediction, on the other hand, can be a much stronger test of the theory, because it will not only say that X will increase, but how much...

Interpreting the data Sometimes an investigator is motivated not by a desire to test general theory, but by the necessity of measuring some specific quantity [that would be impossible to measure directly]...

Forecasting and prediction ...Forecasting is weaker than prediction, and uses the knowledge of the past behavior of the system to forecast its future state. Forecasting does not necessarily require an in-depth understanding of the system's dynamics, and can be done at the phenomenological level. However, forecasting will most likely fail if the system's dynamics change. I use prediction in its strongest sense: that is, to predict a situation that was not encountered in the past. For example, it may be necessary to generate predictions about how a system's behavior will change as a result of a certain human intervention. Prediction, in general, requires a mechanistic understanding of the system....

I especially appreciate the point about quantitative tests --- one that has eluded many paleontologists who are content with categorical statements that are essentially untestable, because they only assert that something should happen ``regularly'' or ``more often'' than something else.

Also, the final point, about forecasting and prediction, is valuable -- although perhaps idiosyncratic, as I have not seen that distinction made elsewhere. Still, it applies far beyond theoretical biology and into historical science generally. If we consider our state of knowledge about climate change in response to human activity, clearly this is an example where the distinction between forecasting and prediction is relevant. We can have confidence in a prediction only if it entails a suitable understanding of the mechanisms of change in the system, whereas forecasting is accurate only to the extent that we can depend on a uniformitarian assumption -- that the conditions observed in the past followed the same mechanistic relations that will be relevant to the future.

I tend to lecture about genetic models, for which there is a great value in simplicity (point 3), but which may require quite complicated extensions to handle reasonable biological populations (point 2). In that connection, some reasonable people go to extremes of interpretation -- sometimes claiming that the data necessitate some assumption on the basis of a very simplified model, and in other cases claiming that no model can apply to the complex history of the population. It is our task (my task) to determine which factors are important and conceivably affect results, and which will always be too weak to influence the interpretation of the data (point 1). And the end will often be to discover evidence for values in past human populations for which we have no direct means of estimating aside from genetic variation (point 5).

References:

Turchin P. 1998. Quantitative analysis of movement. Sinauer Associates, Sunderland MA.

In last week's Science, Stanislas Dehaene and colleagues describe the relation of cultural invention to "universal intuition" about mathematical logic:

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education (Dehaene et al. 2008:1217).

The idea is that children in Western societies have to learn that a number line is a linear representation; they begin by compressing the space devoted to large numbers:

When asked to point toward the correct location for a spoken number word onto a line segment labeled with 0 at left and 100 at right, even kindergarteners understand the task and behave nonrandomly, systematically placing smaller numbers at left and larger numbers at right. They do not distribute the numbers evenly, however, and instead devote more space to small numbers, imposing a compressed logarithmic mapping. For instance, they might place number 10 near the middle of the 0-to-100 segment. This compressive response fits nicely with animal and infant studies that demonstrate that numerical perception obeys Weber's law, a ubiquitous psychophysical law whereby increasingly larger quantities are represented with proportionally greater imprecision, compatible with a logarithmic internal representation with fixed noise (7, 20, 21). A shift from logarithmic to linear mapping occurs later in development, between first and fourth grade, depending on experience and the range of numbers tested (17-19).

They note that there's a problem testing these ideas in Western children, who are surrounded throughout their development by numbers -- in books, "elevators" and other places. Most of these numbers are small ones -- especially one through ten -- so they might naturally accentuate the ones they know.

They found when testing the Mundurucu that both adults and children tended to compress the high end of the number scale, even testing numbers between one and ten. This compression is logarithmic -- they accentuate contrasts between small numbers disproportionately. It makes sense logically -- we care more about detailed contrasts between small numbers than large numbers. They don't give an idea of which logarithm people are using; and in fact it may be different ones for different people. The important fact is the small number/large number contrast.

Dehaene and colleagues attribute this scaling to mapping at the neural level:

What are the sources of this universal logarithmic mapping? Research on the brain mechanisms of numerosity perception have revealed a compressed numerosity code, whereby individual neurons in the parietal and prefrontal cortex exhibit a Gaussian tuning curve on a logarithmic axis of number (27). As first noted by Gustav Fechner, such a constant imprecision on a logarithmic scale can explain Weber's law -- the fact that larger numbers require a proportional larger difference in order to remain equally discriminable. Indeed, a recent model suggests that the tuning properties of number neurons can account for many details of elementary mental arithmetic in humans and animals (21). In the final analysis, the logarithmic code may have been selected during evolution for its compactness: Like an engineer's slide rule, a log scale provides a compact neural representation of several orders of magnitude with fixed relative precision.

From that perspective, the Western conception of the number line appears as a very distinctive invention, capable of adjusting the logarithmic encoding to arrive at faster and more accurate mathematical conclusions about large numbers. The authors speculate that addition and subtraction (which display invariance between large and small numbers) and experience with measurement underlay the development of the linear concept in Western children.

References:

Dehaene S, Izard V, Spelke E, Pica P. 2008. Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science 320:1217-1220. doi:10.1126/science.1156540