Physics and Biology

The Goldsboro Incident really happened! Nonlinearity, mathematical and statistical models


Opening up the paper today, I was pleased to see this story on the front page of the Guardian, about the Goldsboro incident in November 1961. Why pleased? Well for years the Goldsboro incident has been my analogy of choice for explaining the difference between linearity and nonlinearity, based on an interpretation of nonlinearity inspired by George Sugihara on physical vs. biological noise. I’ve always prefaced this analogy by saying that it was unconfirmed but useful – and now it appears to be true! So what happened in Goldsboro? From the companion piece in the Guardian:

The document, obtained by the investigative journalist Eric Schlosser under the Freedom of Information Act, gives the first conclusive evidence that the US was narrowly spared a disaster of monumental proportions when two Mark 39 hydrogen bombs were accidentally dropped over Goldsboro, North Carolina on 23 January 1961. The bombs fell to earth after a B-52 bomber broke up in mid-air, and one of the devices behaved precisely as a nuclear weapon was designed to behave in warfare: its parachute opened, its trigger mechanisms engaged, and only one low-voltage switch prevented untold carnage.


The conventional interpretation of nonlinearity. Doubling the input either more than doubles (e.g. oversteer in a car) or less than doubles (e.g. understeer) the response.

Our formal understanding of nonlinearity is based on the idea that, if we consider a response to an input, doubling the input will result, if there is a linear response, in a doubling of the response. Thus if I press the accelerator on my car twice as hard, I might expect to travel (approximately) twice as fast. In a nonlinear response, the return is either more than or less than twice.   However, an alternative understanding of nonlinearity is illustrated by the Goldsboro Incident, where the difference between 5 of 6 safeties failing, and 6 of 6, is the difference between an incident quietly swept under the rug for 50 years, and a monumental disaster.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

This interpretation of nonlinearity can be viewed in terms of the difference between multiplication and addition. We are quite good at predicting additive phenomena; the problem is, we are are less proficient when it comes to multiplication. The recent story of the death of four year old Daniel Pelka (and this is a type of story repeated with tragic Sisyphean regularity) is a case in point. How could this happen? How could so many safety checks fail? How could so many people miss the warning signs? The truth of the matter is likely to be that there are many, many more cases where “the system” almost fails, but with no observable consequence. Overburdened, pressurised staff, sometimes under motivated or under pressure not to raise alarms unnecessarily, may cut corners or make mistakes far more often than we are aware. It is also likely true that because there is no immediate consequence to these actions (the effect of nonlinearity) the potential for disaster is missed. The question may in fact not be, why does this happen, but why does it not happen more often?

And this leads us to the concept of extrapolation and mathematical and statistical models. Statistical models are fantastically valuable tools for rigorously describing relationships in data. However they are fundamentally ontological in nature; that is, built to classify rather than to explain mechanisms, and thus the ultimate arbiter of the quality of a statistical model is the fit to the data. Of course, in designing the statistical model and in interpreting it, a good scientist will be aware of the existence of these underlying mechanisms. This awareness will drive both experimental design and observation, and the interpretation of the statistics. However, these considerations lie outside the statistical model itself. In contrast, mathematical models should be phenomenological, i.e. built to directly describe the often nonlinear relationships between variables, and therefore they are better suited to extrapolate or predict away from the data, rather than interpolate. What is often not understood, is that even very good mathematical models may give an inferior fit to the statistical within close bounds of the data – the aim is not to develop the best fit to the data, but to better be able to predict what may occur, when moving farther away from known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Of course, this is at best a caricature of both mathematical and statistical models, with modern quantitative sciences using in various ways combinations of both of them. Nevertheless there is a fundamental difference in models that aim to describe, and models that aim to explain, a difference that must be considered when evaluating the interpretation of any model.

Nonlinearity is a critical concept in ecology, evolution and epidemiology. The emergence of new pathogens is one example of this. For example, in a paper a few years ago, Nim Pathy and Angela McLean used a theoretical model, to ask whether or not a pathogen (in this case, avian influenza) that has caused hundreds of cases but with little transmission indicates that the species barrier cannot be crossed. Another way of looking at this question is to ask which is worse, 4 introductions of avian flu into humans from birds, or a single introduction, where a chain of 4 infections in humans occurs but the disease then fails? Extrapolation from currently observed data requires an insight into the underlying mechanisms that drive the phenomenon to be understood (in this case, the emergence of a new human pathogen). What Pathy and McLean showed using nonlinear mathematical models, was that a lack of demonstrated transmission cannot rule the possibility of adaptability, regardless of how many zoonoses have occurred – thus even when we think we are safe, we are not necessarily so.

Of course, while I am (unsurprisingly) a keen proponent of the use of mathematical models, it must always be kept in mind that prophecy is difficult, and the biblical admonition against following false prophets reflects the popularity of trying to predict the future, the frequency of our failures, and the ease with which we can be led into following those predictions, especially when espoused by recognised experts.

The Fantastic Mr. Feynman and the Value of Pictures

So I’m back. Its been over a week now since my first post. I was hesitating to add anything while canvassing the members of the Boyd Orr ‘strategic board’ for their views on the blog, and based on their responses have had time to think more usefully about what this blog is going to be about (more in another post).

The night before last I was watching The Fantastic Mr. Feynman on BBC Two, which followed the life story of the famous Richard Feynman, who among other things transformed the field of quantum electrodynamics (for which he got a Nobel Prize). He was also one of the best communicators of science ever; his ‘Feynman lectures’ represent a gold standard for introductory physics courses, one which I wish my lecturers paid more attention to. Those old enough to remember the ‘Challenger Disaster’ might remember that he is also famous for providing the key to understanding why the space shuttle had crashed. The programme was somewhat on the ‘hagiographic’ side, but even given that, well worth a watch. One of the many inspirational quotes I picked up, and very relevant to us as any kind of scientist:

“First you guess. Don’t laugh, this is the most important step. Then you compute the consequences. Compare the consequences to experience. If it disagrees with experience, the guess is wrong. In that simple statement is the key to science. It doesn’t matter how beautiful your guess is or how smart you are or what your name is. If it disagrees with experiment, it’s wrong. That’s all there is to it.” Quote taken from ‘The Fantastic Mr. Feynman’, airing on BBC Two on May 12th, 2013′ based on a previously aired television programme).

To my mind, this is what distinguishes physics (and by extension, the other “hard” sciences) from mathematics, at least in its purest form. It could be argued that mathematics does not, or at the very least, certainly does not have to, bear the same relationship to data as the sciences do. Or at least, the relationship may be more akin to the relationship between poetry and reality – it refers to it, and informs to it, but is under no constraints to conform to it. As Feynman so much more eloquently says, science, on the other hand, is nothing without reference to the data.

The programme also got me thinking a bit, about the value of pictorial representations in research. One of Feynman’s key contributions were the representation of subatomic particle interactions using simple pictures called (oddly enough) ‘Feynman diagrams’ – by integrating in simple sketches movement in both space and time, Feynman was able to take some very complex mathematical ideas and give them a simple, intuitive representation. I’d like to be able to say that, when presented these as a graduate student, I immediately saw their value, but alas I was initially quite resistant to the idea; I’d worked hard understanding all those equations! However, I did realise (eventually) what a beautiful way those diagrams were about how to think about the problem.

And that brings me back to the Boyd Orr Centre. First, Feynman’s point that the data are the key, and our models (both quantitative and conceptual) exist in reference to those data, is very much central to the Centre; we’ve a lot of very talented quantitative scientists and a consistent theme in their science is that a slightly ‘rough around the edges’ approach that says something useful about the data we are trying to understand, is more relevant to us than a more elegant, less ‘useful’ theory. Second, that understanding and communication are paramount – to paraphrase another Feynman quote, if we cannot explain our ideas in a well written undergraduate lecture, then we haven’t really understood it. It is this approach that helps our work be relevant not just to the science, but to the informed and intelligent practitioner of population health. It is these two aspects – relevance and communication – that lie at the heart of the Boyd Orr Centre, and coincidentally make it an inspiring place to do science.