Regrets in life - and how math can prevent them

I don't like regretting past events. But I don't like dreaming about things that aren't certain, either. A dream, or more precisely, an assumption that something will come true makes you imagine many steps in the future.

But so why do I mention math in the title? Well, dreaming is an attempt for predicting future. And this is not any different from extrapolation. Read below to see how simple it actually is.

An imaginary situation

Let's say I'm looking for a new house. I make a list of what are absolutely necessary requirements, then those that would be nice to have but can go without. I look through many offers, still haven't found the right one but I let my mind wander. I imagine how I put my art tools and materials in the the second bedroom on the upper floor. I can sit there by the window, enjoy some morning sunlight and relax with my hobby. Oh, and the flowers in the yard will be in particular arrangement. And I really want mushrooms in my yard! It'd be so fun to go out every morning and evening and take a photo of them to watch how they grow.

Then... reality strikes. I haven't found a good house yet. Well, actually, buying a house might be a bad idea right now. I guess I should stop searching. Too bad because the gear for my hobbies barely fits the shelves any more. The nice future that I thought was imminently on the way didn't come true. And it won't. Well, that's disappointing.

So what does math have to do with that?

In mathematics, when on has some set of data, one could apply methods for approximating a model (called regression). A good model is able to predict values in between the existing ones. But what happens if we try to extend it further and predict how a curve would look like if we just had a few more data points? Sure, let's try. It works. How about a bit further away from our data? Kind of works. Let's go for a bit more... oops... the predicted value is totally wrong.

Example

I have a set of data plotted as blue points. I use regression to find a curve that fits the points. This is a trendline, plotted as a red dashed line. Regression produces a mathematical model that allows us to approximate any value even if we haven't measured it. Great! Wait... there's a limitation. We can only trust it for the data we fitted. We cannot be sure if this hold if I were to ask for a value outside of the interval.

Alright, so what if we explore the region right around the borders? That should be okay? It is. Here I have plotted more data points. The ones in blue were simply taken out of this whole set of points.

So, I made a mathematical fit only according to the blue subset of data. Fine. I could use my model and extend a bit the trendline. I still get a pretty good match. But what if I wanted to extend the trendline on the left side, i.e., the small values on the horizontal axis?

Oops... my model is completely wrong.

So, if I go back to my issues with regret. I was assuming that I will buy the house because at a certain moment of time it looked possible. But there's a difference between possible and certain. In fact, buying the house is like those data points that without further knowledge I couldn't predict accurately.

Remember this next time you hear somebody say that they don't need math after high school.