While SEB113 is all about Quantitative Methods in Science it’s been quite notable that the last two weeks have been mathematical rather than statistical. We started the semester with visualisation and then headed into summary statistics, confidence intervals, hypothesis tests and linear regression. We extended what we knew about linear regression into dealing with non-linear regression (using the nls function) and using multiple predictors to explain variation.
In week 7, the non-linear models we used were largely presented without comment as to where they came from, although we did discuss their use to model particular curves. Last semester we planned to do mathematical modelling but Dann Mallet was away and therefore couldn’t deliver a lecture. My undergrad was in mathematical modelling and computational mathematics, so I leapt at the chance to do some mathematical modelling. I figured that the best place to start would be the non-linear models that we dealt with in week 7.
I had to do a bit of digging but I found the differential equations for the asymptotic model, first order compartment model and biexponential model as well as a nice little example of how they’re used. While many students haven’t done derivatives recently I do feel like by the end of the workshops there was a bit more of an understanding and appreciation of the modelling and the mathematics behind it.
I figured that seeing as we weren’t actually doing a differential equations class and could only assume Maths B it would be a bit much to discuss the calculation of the exact answers for anything beyond the exponential growth model. We talked about how these differential equations have exact solutions but that we can’t always find an exact solution so we look to numerical solutions. We worked through how you can take the approximation for the derivative and ignore the limiting case and rearrange the definition to arrive at a very simple numerical technique. Only one of the workshop rooms properly installed the deSolve library but we did have a good discussion with the groups about how the Lotka-Volterra system has no exact solution and that the solution you obtain lies on an orbit in the phase plane (but I didn’t say that was what it was called).
One of my favourite moments in that workshop was when one of a group of students asked about what would happen if the logistic model overshot the carrying capacity steady state. This led into a discussion about discrete time difference equations and the stability of the equilibrium point. The book we used, Barnes and Fulford, has a nice section on the chaotic behaviour of the logistic growth difference model and how as you increase the growth rate you move from smooth evolution that doesn’t overshoot to decaying oscillations around the carrying capacity to non-decaying oscillation between two population levels, to four, eight and then an infinite number of levels (you don’t return to the same population in finite time). That seemed to really tickle them and prompted a discussion about how this sort of behaviour has been observed in real world populations (elephants in a wildlife reserve?).
This week we focussed on linear algebra, compressing three weeks of lectures from last semester into one single lecture. This may have been a little rushed in the delivery, because there were so many topics to deal with, but we basically split it up so the first half was showing how to turn a mathematical problem into a matrix system and visualising the solution. The second half was then a walk through of Gauss-Jordan elimination in order to get a matrix in reduced row echelon form (RREF). There were some very salient questions about why we care about matrices being in RREF and what each of the criteria for RREF actually mean. I thought at the time that it was a bit of a distraction from the main thrust of the class but thinking back to it now I am very glad that the students asked these questions as it showed that they were attempting to engage with and understand the material rather than just dismissing it as unimportant.
I like to try to point out how the various topics in SEB113 are related, so we had a talk about how making a mesh from a continuous domain to arrive at a discrete set of nodes turns our mathematical model into a system of equations that we can solve numerically (week 9). We spent a bit of time in the week 9 lecture and workshops discussing how Euler’s method uses an approximation of a time derivative to approximate the evolution of a system. In week 10 we looked at the equilibrium temperature of a domain with known boundary conditions, which I pointed out was all about discretising the second derivative from our model.
In a similar vein, the solution of an overdetermined system with the normal equations was a callback to the regression we’d done, where an overdetermined system has a non-zero number of degrees of freedom. We use the normal equations to go from an overdetermined system to an exactly solvable one that minimises the sum of squares. I didn’t want to go too deep into the linear algebra of how that works but the workshop exercises featured fitting a non-linear regression with the nls function in R (week 7) and doing it “manually” by solving the normal equations in R after converting it to a linear function. I think the groups that did this one got a bit deeper understanding of how regression works, which I’m definitely happy about. I told a student that I don’t mind if they can’t reproduce a linear regression by coding it manually, so long as they walk away at the end of the semester being able to load some data into R, look at the potential relationships, model them and interpret the model output.
These last few weeks have probably been some of the mathematically most challenging and I think that while not all students necessarily understand the maths that’s going on behind the R functions they at least have a bit of an appreciation for where it comes from and why its useful. I’m quite pleased by that, as it’s helping get the message across that mathematics is a useful tool for solving problems rather than just some abstract thing that “I’m never going to use once I finish high school”. The same goes for statistics. Rather than just being this painful thing that requires manual calculation of margins of error for your experiment (forget that for a joke, no one learns anything useful that way), statistics helps you uncover the patterns in your data, model them and make statements about how meaningful they are. If we can get students seeing regression as more than just “What’s the line of best fit and its R2?” then we’ll have done a good job.