## Linear vs nonlinear

I’ve run across a lot of ambiguity lately regarding systems described as “nonlinear.” Systems typically have several layers, and are linear at one level and nonlinear at another, and authors are not always clear about which level they’re talking about.

For example, I recently ran across something called a “nonlinear trapezoid filter.” My first instinct was to parse this as

nonlinear (trapezoid filter)

though I didn’t know what that meant. On closer inspection, I think they meant

(nonlinear trapezoid) filter

which is a **linear** filter, formed by multiplying a spectrum by a “nonlinear trapezoid,” a function whose graph looks like a trapezoid except one of the sides is slightly curved.

One of the things that prompted this post was a discussion with a client about Kalman filters and particle filters. You can have linear methods for tracking nonlinear processes, nonlinear methods for tracking nonlinear processes, etc. You have to be careful in this context when you call something “linear” or not.

## Random vs deterministic

There’s a similar ambiguity around whether a system is random or deterministic. Systems are often deterministic at one level and random at another. For example, a bandit design is a **deterministic** rule for conducting an experiment with **random **outcomes. Simulations can be even more confusing because there could be several **alternating layers** of randomness and determinism. People can talk past each other while thinking they’re being clear. They can say something about variance, for example, and other other person nods their head, though they’re thinking about the variance of two different things.

As simple as it sounds, you can often help a team out by asking them to be more explicit when they say something is linear or nonlinear, random or deterministic. In a sense this is nothing new: it helps to be explicit. But I’m saying a little more than that, suggesting a couple particular areas to watch out for, areas where it’s common for people to be vague when they think they’re being clear.

## Related posts

- Nonlinear is not a hypothesis
- Applied category theory (a tool to help see where you’re being vague)

Nice post! I actually wrote something very similar recently as prompted by a question from a student in my PDEs course – https://omaclaren.com/2016/03/29/linear-or-nonlinear-with-respect-to-what/

PS – I also pointed them to your ‘ten life lessons from differential equations’ in my first lecture :-)

How do we make alternating layers of randomness? Would failing to find a factor leave us with a gap of randomness in out deterministic system? I get that the tail would be random.