Re: [math-fun] Curve-fitting methods ?
A few random thoughts on curve fitting: At a job some years ago in which I had to fit a curve to some data which showed how much hotter something got in response to how much radiant energy, I was criticized for first adding an additional data point representing no increase in temperature with no radiant energy. I added it because it was obviously true, and would help to make the curve more realistic. I was criticized because that experiment wasn't actually performed. When I first learned of Fourier transforms, I dreamed of getting rich by using them to project stock data ahead. I was disappointed when I discovered that all I got were the same sets of historic stock values repeated over and over again. Another learning experience. I had a similar experience after I learned that any N points can be fitted exactly onto an order N-1 polynomial. I was disappointed when I saw that the curve took absurd values, not just when extrapolated, but also when interpolated. One of the most astonishing (to me) calculus theorems is that any smooth curve (in the sense that all of its derivatives are continuous) can only be extended in one way. Any non-zero-length segment of such a curve, no matter how short, uniquely determines the rest of the curve. Also, if you know all the derivatives at any one point, then you know the value at every point. Since it seems plausible that my own motion is smooth, i.e. I never experience infinite anything (distance, velocity, acceleration, jerk, snap, crackle, or pop (yes, those are the official terms for successive derivatives of position)), that implies that all my past and future travels are predetermined. Eventually, especially after studying radio theory, I realized that that's just an instance of the many paradoxes that come from assuming an infinite signal-to-noise ratio. Or from assuming infinite anything. There's *always* noise in any physical signal, even if it's only from quantum fluctuations. A radio receiver is basically a device which fits a curve to a noisy signal in an attempt to reconstruct the intended signal. One of my projects for satisfying extreme skeptics is a way to construct a crude picture of the Earth from years of data on the brightness of the Earth-lit part of the crescent moon as seen from one's backyard. Of course the skeptic would have to trust my software, so there's little point in my writing such software unless I am the skeptic. Still, it's an interesting project. A much easier project is using a table of distances between cities to calculate the radius of the Earth. (Has anyone done that?) Any four cities will suffice, assuming a spherical Earth. With more cities you can tell more about Earth's shape. Or you can just try more subsets of four, and average all of the resulting values of radius to best approximate the correct radius of the spherical Earth. Should all sets of four cities be equally weighted? If not, what should determine their relative weightings, assuming all the distances between cities listed in the table are equally accurate and equally precise?
I'm sure it's been done. I've seen it illustrated in textbooks. Of course cities which are far from all the others provide better estimates, since the proportional distance error is smaller. Brent On 9/23/2018 8:50 PM, Keith F. Lynch wrote:
A much easier project is using a table of distances between cities to calculate the radius of the Earth. (Has anyone done that?) Any four cities will suffice, assuming a spherical Earth. With more cities you can tell more about Earth's shape. Or you can just try more subsets of four, and average all of the resulting values of radius to best approximate the correct radius of the spherical Earth. Should all sets of four cities be equally weighted? If not, what should determine their relative weightings, assuming all the distances between cities listed in the table are equally accurate and equally precise?
On Sun, Sep 23, 2018 at 11:50 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
One of the most astonishing (to me) calculus theorems is that any smooth curve (in the sense that all of its derivatives are continuous) can only be extended in one way. Any non-zero-length segment of such a curve, no matter how short, uniquely determines the rest of the curve. Also, if you know all the derivatives at any one point, then you know the value at every point.
What's most astonishing to me is the fact that calculus classes, while usually not actually making any false statements, seem designed to mislead people into thinking that this statement is true. It's completely false. Consider the function f(x) = 0, defined on [0, 1]. If your claimed theorem was true, the the only smooth extension of this function to the real numbers would be identically 0. But one can extend it for x > 1 with the function f(x) = e^(- 1 / (x-1)^2) and for x < 1 with the function f(x) = -47 e(- (1/x^2)) and the resulting function is infinitely differentiable everywhere, even at 0 and 1. In fact, given an infinitely differentiable function defined on [0,1], and another infinitely differentiable function on [2,3], there is a way (infinitely many ways, in fact), to interpolate between them to produce an infinitely differentiable function on [0, 3] that extends them both. So rather than knowing exactly what a function can do by knowing in on a non-zero length segment, it can literally do anything whatsoever any short finite distance away from that segment, and remain infinitely differentiable. Why do so many people think the false "theorem" is true? One reason is that when we consider functions from the complex numbers to the complex numbers, rather than from the reals to the reals, the theorem is true. The statement that a function from C to C is differentiable is a much stronger statement than the statement that it is differentiable as a function from R^2 to R^2. The latter statement says that the function is well approximated locally by a function in the family of linear functions from R^2 to R^2, a space with real dimension 4. The former statement says that this function is one of the ones that corresponds to multiplication by a complex number, a linear subspace of dimension 2. The statement that a function is differentiable from C to C doesn't just say it's smooth; it says it's conformal, a much more rigid criterion. The fact that the theorem is true in complex analysis has little bearing on the kind of curve-fitting we are talking about here, since there is no meaningful extension of the functions in question to complex arguments and values, and no reason to expect that the resulting function would be complex-analytic if we did extend it. But I don't think confusion with the complex numbers is the main reason people believe this false theorem; it's a widespread misconception even among those who have never considered the idea that calculus can be done with complex numbers. I think the real reason the false theorem is believed is that calculus texts place great emphasis on the notion of the Taylor series of a function, and want to emphasize its importance. Its importance would seem diminished if the texts let out the dirty secret that even if the Taylor series converges, there is no reason to expect that it converges to the original function, rather than to some completely different function, even on a neighborhood of 0. The expression for the remainder term is given, but nothing is done with it, and the texts quickly move on to other topics, without mention of the fact that even if it converges, this remainder need not converge to 0 as the number of terms goes to infinity, even in a neighborhood of 0.
Since it seems plausible that my own motion is smooth, i.e. I never experience infinite anything (distance, velocity, acceleration, jerk, snap, crackle, or pop (yes, those are the official terms for successive derivatives of position)), that implies that all my past and future travels are predetermined.
Only if you think it plausible that your motion is well-defined and smooth for complex values of t, and that there is reason that the resulting function is differentiable in the strong form of being differentiable as a function from C to C, rather than as a function from R^2 to R^2. Andy
I'll bite. Have you constructed this crude picture of Earth, or are you suggesting that it could be done? I would be interested in such a picture, as it is an analogy to the "1 pixel camera" discussed here previously. At 08:50 PM 9/23/2018, Keith F. Lynch wrote:
One of my projects for satisfying extreme skeptics is a way to construct a crude picture of the Earth from years of data on the brightness of the Earth-lit part of the crescent moon as seen from one's backyard. Of course the skeptic would have to trust my software, so there's little point in my writing such software unless I am the skeptic. Still, it's an interesting project.
participants (4)
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Andy Latto -
Brent Meeker -
Henry Baker -
Keith F. Lynch