The word “model” belongs to the vocabulary of every applied mathematician. Model of this, model of that… What do we actually mean when we talk about mathematical modelling?
I hadn’t realised that this word might be confusing until I attended the UK Graduate Modelling Camp 2016 in Oxford. When I was trying to check in to my accommodation at the University of Oxford, I concisely said that I arrived for the modelling camp. The receptionist looked me up and down, only to conclude that I don’t look like a model. Apart from completely destroying my self-esteem (am I too fat? Too short? Too ugly?), he also inspired this article.
When mathematicians think about models, what comes to mind aren’t skinny women in fancy dresses. Neither do we talk about toy cars. Nor about our role models.
A mathematical model is a representation of reality. Scientists observe different phenomena: from DNA replication to the expansion of the universe. The best, universal way to describe such a variety of processes is to use the common language of mathematics. In other words, to write down some equations.
It’s a very delicate task. We know that all these phenomena are complex, so naturally we should include a lot of equations with hundreds of variables. But would such a model be useful? Not at all! Because we want to use mathematical models in practice.
For example, if you go to a doctor, he needs to know how your body will react to a given medicine. This knowledge comes from a computations based on a mathematical model.
Another example would be weather forecasting. The state of weather is described by a massive set of equations (it’s in fact one of the most complicated models that exist). Thanks to that model, we can compute possible future weather conditions.
However, if a model was too complicated, i.e., if it included all possible details, even the fastest computer wouldn’t be able to compute needed values in a reasonable time. This is why mathematicians need to think carefully which variables (unknowns) are essential and which can be omitted. The model isn’t the reality, it’s the description of the real thing. A bit like a globe – it’s not exactly the Earth (for instance, the Earth is flattened at the poles) but gives a lot of relevant information about our planet.
During the International Study Group with Industry in 2016, my group was working on a model of a marketing campaign. After two days of hard work of the whole group, I looked at the whiteboard covered in our notes and counted the parameters we wanted to include. It was 27 of them! I realised that for this particular problem estimation of so many (unknown) parameters wouldn’t be reasonable, so I stood up and rubbed of two thirds of our work. We realised that even though the new model contained less information, it was more reliable.
How do we know if a model is satisfactory? We look at the data. There exists a plethora of statistical methods validating models, I don’t want to describe the details. The idea is that we check if the realisation of the model resembles the real world observations. If they are close enough (in some well-defined, mathematical sense), then we can say that the model is successful.
I believe that the main difference between pure and applied mathematics is that while in the first one we have precisely stated theorems and proofs that always hold (under given conditions), in applications we don’t have a single correct answer to a given problem. Even more, there is no correct answer at all. Because, as a great statistician George Box famously said, “All models are wrong but some are useful”. And the job of applied mathematicians is to create the useful ones.