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I find it rather amazing that by plotting the points (0,0),(0.5,0.25),(1,1),(2,4), one can "predict" what the graph will look like. In certain cases, a person may even be able to "sketch"(freehand) the in between values by "connecting the dots", without being told that the function giving the result is $f(x)=x^2$.

Is there any way to get from a set of data to a prediction of what the plot will look like by having just a few data points?

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    Yes and no. With the data point example for $x^2$, there could also be a wildly oscillating function with period $0.5$ hidden in there. When you "connect the dots", you make a certain assumption about the smoothness.2011-05-05
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    @Lagerbaer, If you were to give anyone those points, they would come up with remarkably the same values in between. What is going on? (few if any will draw a oscillatory connection)2011-05-05
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    And, the down-votes start..2011-05-05
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    This isn't a math question. It's a question about human psychology and a phenomenon where a significant number of people will probably assume these points "should be" connected as by $x^2$. Nevertheless, this is close to some basic math-ish questions like "how should we decide what the best function to model this data is", but without additional assumptions, you can't.2011-05-05
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    Is there any way to get from your past life experience a prediction of your future life experience by having lived just a few years?2011-05-05
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    @matt, is it really a question about psychology (note the title about "remarkable") or is there some "error minimization" that is going on in our heads that gives us all (most of us) similar expectations for a smooth curve. (as a side note, I am happy that the question is getting both up and down votes. That to me is a good question)2011-05-05
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    @matt: doing math (thinking mathematically, asking questions, creating proofs, understanding mathematical concepts, curiosity...creativity, sorting through the relevant and not so relevant, computing, conjecturing, problem solving, learning...) can hardly be separated from human psychology. No one can encounter or engage in mathematics void of human psychological factors ...unless, of course, one is a *GOD*...As *pure* aof picture we try to paint of *M*athematics, we know it (and can learn, engage with, and teach math) only through the "lens" of our human perspectives.2011-05-05
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    @picakhu: Yes, I believe so, exactly because your question is as you just said, whether there is some process going on *in our heads* that gives us all mostly the same expectation. I think the answer to that question lies comfortably within the study of psychology or some kind of brain science to decide.2011-05-05
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    @matt, I do not want to disagree with you. I am just questioning it. To me, everything we do is an algorithm. But how we are performing the algorithm is what is astounding. We learn algorithms faster than we can teach computers. Why is that? However my question is more serious. It is, "in this instance, what is the algorithm?"2011-05-06
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    @picakhu I think it's an interesting question. I just think it's a question answered by studying the brain. I don't think you're asking "what methods are there for doing this in general" to which the answer is "look at interpolation" but rather "what is the method the brain is using" to which the answer would be "this particular kind of interpolation". I'm suggesting that you go ask people who study brains to get that. Mathematicians can propose models and study their consequences, but someone is going to have to actually get people together and study them to see what they actually do.2011-05-06
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    @matt, that makes a lot of sense. Thanks :)2011-05-06

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"In engineering and science one often has a number of data points, as obtained by sampling or experimentation, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points."

Actually this is a very difficult task and there are quite a few approaches for it, the easiest one is probably that you assume the function has a certain form like $f(x)=a\exp(b\cdot x)+c$ and then optimize the parameters a,b and c to get the "best fitting" function. Of course the initial guess is very important and can also be automated. Like already mentioned when working in a very abstract context you are never sure if some very odd / oscilating function is hiding between the data points, but one might assume that this is not the case if you have enough data points.

This is a very neat example from Wikipedia that shows that there are also interpolations which you wouldn't probably come up when fitting it from hand but if you optimize it by using splines on a formula level:

Interpolation

Image and quote: http://en.wikipedia.org/wiki/Interpolation

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This is the starting point of a whole theory called interpolation.

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    Thank you for accepting, but I think the other answer deserves it more than mine!2011-05-05
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    I am at a loss of whether I should change to the other answer, primarily because if I go on to wiki, I will see the same thing that the other user has posted. As a matter of kindness, I feel I should not change.2011-05-05
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    @picakhu I extracted the information that I thought is useful for you (introductory paragraph) and added my own explaination. Why should I rewrite something that is already written very well? Does this reduce the utility of the answer, I guess not.2011-05-06