Dear Fellow Scholars, this is Two Minute Papers with Károly Zsolnai-Fehér.
What is extrapolation? We hear the term a lot, so let's try to learn what's behind it.
Despite the complicated definitions that are out there, extrapolation basically means continuing
lines. A good example is when we have data for something from the last few days or years,
and would like to have a forecast for the future.
We'll jump right into an example just give me a second to build this up. It's going to
make sense in the end, I promise! So, in many fields of science, it is really difficult
to get research projects funded. Experiment is a cool new startup that is trying to accelerate
progress in research by crowdsourcing it. It doesn't get simpler than this system: scientists
pitch their research project plan, and kindhearted people pledge a one time donation to help
their cause. It is like kickstarter for research.
Some of the newer funded projects include growing food in space, developing an open
protocol for insulin production, and, of course, a mandatory cat project that includes sequencing
the genome of rare mutations.
Crowdfunding research is such a terrific idea, and I tell you, these guys are really doing
it right.
The startup has been founded in 2012, and people pledged 52 thousand dollars that year.
The next year, ten times that, and they have kept a steady and quite impressive growth
ever since. In 2015, they raised almost $4 million dollars for open research. It's AMAZING!
Ok, so a nice extrapolation problem - how much can they expect to raise next year, in 2016?
Before we start, we have to be extremely sure to extrapolate only if we're reasonably sure
about the nature of the trends and that they won't change significantly in the near future.
With that out of the way, let's do a linear extrapolation. Linear means that growth follows
a straight line. So, we put these dots on a paper, and try to connect them with a line.
Now, we take the mathematical description of this line, and substitute something in
it. Since we have 4 years of data, 4 dots, we would be interested in the location of
the fifth point, which is the amount of raised money in 2016. So let's do it!
Ten to the sixth is one million, so this says that we can expect 4.2 million dollars.
Great! But let's be a more optimistic, and do a superlinear extrapolation. Superlinear
means that the the rate of growth is not a straight line, but something that is accelerating
in time. If this assumption is true, we can expect them to raise way more, 7.4 million
dollars.
A bit more pessimistic solution would be a sublinear extrapolation. Sublinear means that
growth slows down in time. This kind of growth is described well with, for instance, the
logarithm function.
This effect is also often called the effect of diminishing returns. A good example of
this is the skill level of Google DeepMind's artificial intelligence program that plays
Go. As we add more and more computational resources, the algorithm gets better and better
at the game, but after a point, there is only so much one can learn, therefore progress
slows down, and eventually gets close to stopping. There are so many examples of this effect
in our lives - if you have some great examples of logarithmic growth, let me know in the
comments section, I'll include the best ones in the video description box.
According to this logarithm, we can expect the company to raise less than the previous
estimations, 3.1 million dollars next year. Sorry guys!
A common pitfall in popular media is that the mathematically untrained minds almost
always assume a linear growth due to its simplicity. This can lead to hilariously wrong results.
If you would extrapolate the size of the belly of a pregnant woman after 9 months. Your conclusion
would be "run, because she's going to explode!" - whereas we know that a baby is going to
be born and she is going to get back in shape. If I had zero wives yesteday and it's my wedding
day today, I will sure as hell have a couple dozen wives by next month! Many things are
inherently non-linear, and doing a simple linear extrapolation often doesn't do justice
to the problem at hand. Bear in mind that there are many different ways to connect a
bunch of dots!
Let's try to find out why we had wildly varying results. This is due to the fact that we only
had 4 samples, that means 4 dots. If I plot these possible functions that we've been talking
about, we get the following.
It seems that the further we go, the more they diverge. However, in this case, if we
have data only between zero and one, for instance, there is very little difference between a
wild, exponential function and a very conservative, square root-based growth (you can also imagine
your logarithm here). The more dots we have, the more we can distinguish the nature of
our growth.
And, an educated mind has to take into consideration that many phenomena are inherently non-linear.
If you catch someone doing a linear exploration, always ask: "Are you sure that the process
you're modeling is indeed linear? And do you have enough data to prove that?"
That's all for today. Thanks for watching and for your generous support, and I'll see you next time!