 Dear Fellow Scholars, this is two-minute papers with Karo Ejona Efehir. 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 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 $52,000 that year. In the next year, 10 times that, and they have kept the study and quite impressive growth ever since. In 2015, they raised almost $4 million for open research. It's amazing. Okay, 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 are 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 four years of data, four 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. 10 to the sixth is 1 million, so this says that we can expect $4.2 million. But let's be more optimistic and do a super-linear extrapolation. Super-linear means that 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. 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's 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 estimation. $3.1 million next year. Sorry guys. A common pitfall in popular media is that the mathematically untrained minds almost always assume 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 nine months, your conclusion would be run because she is 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 yesterday 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 four samples, that means four dots. If I plot these possible functions that we have 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 base growth. You can also imagine your logarithm here. The more dots we have over a greater period of time, 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 extrapolation, always ask, are you sure that the process your 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.