In this blog I hope to explore and share an area of research that runs as a thread through most of my thinking and interest. I hope to produce a series of these if the interest is there. I want to take you on a journey from the predictable to less predictable to the damn right strange – and, for those that know me, I am not publishing minutes from one of our meetings or a series of text messages.
If we start with the predictable, the boring, the linear, we see that this action causes that reaction and it is perfectly proportional. The radius of a circle will give its circumference, if your service provider charges you £10 per month plus calls at £0.10 it will be £10 + £0.10 x calls, even if the relationship is not perfectly linear we can match a straight line to it and our model will look similar (for example the relationship between price and demand is unlikely to form a perfect straight line if we survey a large number of people but the general trend will show lower prices equal greater demand).
So far so good. But of course not every reaction is directly proportional (I struggled to think of those examples). If you press the accelerator in your car speed increases but it is not exactly proportional to the amount you press. Wind and rolling resistance means that more ever more pressing is required. It is not a straight line anymore but a curve. The same may be true for revision (if I revise more I do not get an exactly proportional increases in grade – the link between hours’ revision and grade is non-linear). We would call that the ‘law of diminishing returns’ you can call it life. The opposite is also true. In some circumstances the gains achieved outstrip the effort out in. You are probably familiar with the saying ‘money comes to money’, if I ‘teach’ you then you may become bored (diminishing returns); if I teach you to learn for yourself each extra bit of effort by me results in larger chunks of learning for you.
Reality may present us with a combination of curves and lines and more than one thing at a time. Your grade at University may depend on a combination of attendance, IQ, hours available for study outside of class, lecturer’s ability to generate interest, etc. As long as we can measure each of these for a sample of students we can produce a, albeit crude, model. This will look like our service provider model but with more 10 pences. These ‘10s’ are called coefficients or parameters and we will come back to those.
So far we have journeyed from the the simple linear, via non-linear to multiple regression modelling. Throughout all of this there are two big assumptions. First we assume that the relationship is free of interaction effects. We assume that there is not a relationship between the lecturer being interesting and attendance – but clearly there may be. We also there are no feedback loops. Consider for example the relationship between stock price and company value. This should be a perfectly linear rational relationship – it isn’t. Speculators buy stock because it is rising knowing it is overpriced relative to company value but the greater the activity the greater the stock price. This price increase in turn signalling the ‘wisdom’ of the ‘investment’ (gamble).
To recap: more of something gives us exactly proportionally more of something else (minutes and phone bill), or more of something gives us even more of something else but at a slower (or faster rate) (press accelerator to go faster), or maybe it is a mix of these variables (factors that influence students’ grades) but maybe the world isn’t always that simple (feedback loops). Sorry for taking you through all of this but it’s a foundation.
The relationships we have talked about so far have been influenced by parameters, Pi is a parameter in the circumference of a circle – a nice constant one and so was the 10 pence. In the orderly world we have described so far things do what they are told when they are told. Let’s now step through the looking glass.
Picture a car driving along the road, for safety a toy car, if you turn the steer wheel a little bit it turns a little bit, if you turn it a lot it probably turns into the hedge, if you adjust the accelerator it goes faster but not proportionally. It does though behave as expected. Imagine though if a tiny adjustment to that parameter (steering left) resulted in the vehicle travelling in a (seemingly) unpredictable way? We would say its path had become random. But has it? The variables are still all measureable, the parameters known and present. Luckily this does not happen with cars (excluding ice). What we have is a model (already non-linear) that is no longer behaving as expected. Feedback loops are working on the parameters. This is shown in a rather famous chart that can be viewed here.
Moving from the car analogy let’s try populations instead. Assume that some set of variables and parameters control the population of ravens. Variables might be a mixture of habitat, food, gestation, breeding pairs and the parameters the impact changes in these have on populations (the equivalent for values for pi in a circle). After estimating the parameters we can run the model and expect to see increases in ravens (yea!). It is easy to imagine that after a certain point there may be a reduction due to over feeding, for example, and the population may settle into a natural rise and fall – a cycling.
Civilisations may follow a similar pattern with some making comebacks again and again (China) whilst others seem to vanish (Easter Island and the Inca Empire comes to mind, and the perhaps the fall of the Roman Empire though that requires more debate and analysis). Through all of this it remains predictable – a simple up and down or up and out– it does what it’s told. But what if it doesn’t (to borrow my favourite Whinnie the Pooh quote)? What if changing a parameter ever so slightly means we can no longer predict (as in see on a graph – if we enter the data we still get the result) where the next turn will be? This is not randomness but chaos – strangely a more understandable state than randomness despite its connotations. How many times should you tell someone you love them? Too few and it is over, too many and it is annoying but is there a parameter that gives us a golden number of times a day/week/year/decade and what controls that parameter – of course there isn’t. This requirement to get the parameter absolutely perfect is what we call sensitivity to initial conditions. Most models work quite well with crude estimates but we should not be surprised that, on occasions, models fall apart. The reasons for this are manifold but the ones of interest to me are lack of ability to accurately measure parameters and an incomplete understanding of all the interactions (and long may that continue).
We have now moved into the world of Chaos, skipping over complexity which is what I intended to talk about in this blog. I couldn’t predict that. I’m going to have a bit of lunch and come back to complexity this afternoon. If you find Chaos a strange state to live in the frontier town of complexity will really interest you. Translational zones, fringes and shifts are always the most interesting....