Weight Loss Mathematics
I was function spotting the other day (it's like bird watching but you're looking for functions) at the grocery. I saw this array:
and -- immediately -- my first thought was, ƒ(x)!
There is some function which maps from {calories in} × {calories out} to change in weight over time. And it looks very similar to
Some claim that there is more to it than that (hormones, etc) but most agree the above function is a good simple model of weight gain and weight loss.
You can think about someone's weight over time as ƒ(t)
plot
and their choices as interacting with a differential equation
dƒ/dt =asdf
This is a Markovian process because your weight today only depends on your weight yesterday and today's choices. Of course, it's 99% due to your weight yesterday. But that's OK as far as the maths go.
We could take the maths a bit further and recognise there are functions which pipe our choices into the (caloric intake, calories used) figures. For example if I eat a pint of Ben & Jerry's I take in 775 calories over 20 minutes. Imagine a function mapping from quantity × type of food to caloric intake. And if I run hard for twenty minutes (let's say I do back-to-back 6:20 miles) I'll burn 325 calories. (It's much easier to take in calories than to use them up these days -- unless you're poor and/or work with your body.)
Let's step back even further and look at psychology. Again there are some choices that are easier to make and some that are harder to make. One possible model of this is "psychic cost" and "psychic benefit" as a simple plus/minus: you need to have a certain amount of points on the pleasure meter for your self-control to function (you must be happy to go for that half-hour jog).
I don't really believe in that model, though. I think self-control and habituation are more complex. I would look to the yogic model of habituation: every time you take a decision, it becomes easier to take that decision again. So like water cuts ravines by flowing its course so do we habituate ourselves. (I believe this is what psychologists refer to Hebbian learning but don't quote me on that.)
It's also possible to use exercise to get out of a psychic funk (change your mood to the more positive), so the "psychic cost of exercise" doesn't really make sense.
Furthermore I'd guess there are important stylised facts about the timing one can use to "hack" one's willpower and weight-control dynamics. For example maybe a craving lasts for 3 minutes and then passes. (Does it disappear or turn into something else? I would guess the latter. Some integral is conserved across feelings.) There have to be a lot of tricks and nooks within the mental honeycomb of urges, satiety, discipline, habit, and so on. I couldn't express them all here. My point is only that they are mathematisable. Physics and electrical engineering gave us plenty of mathematical concepts for damped oscillators, energy being diverted elsewhere, and so on.
You can go for a 7-billion dimensional space and do the time-concurrent weights of everyone on Earth. That seems kind of boring, though.
The system of ODE's only has x terms (what's the technical word for that) and no y terms because your weight doesn't affect my weight per se.
In that space not everybody is interested in weight loss <link aboubakars>
The Jacobian would show you how easy it is for different people to lose weight (the individual derivative terms)
You could also do a 3-dimensional codomain. Instead of a donut mapping to calories, it maps to fat calories × carb calories × protein calories.
Or expand that codomain and include dimensions for vitamin K, riboflavin, niacin, vitamin B12, and rubidium (hey, who said it has to map to things that you're supposed to eat?)
