approximation methods
I attached image:
(source: piccy.info)
So in this image there is a diagram of the function, which is defined on the given points. For example on points x=1..N.
Another diagram, which was drawn as a semitransparent curve, That is what I want to get from the original diagram, i.e. I want to approximate the original function so that it becomes smooth.
Are there any methods for doing that?
I heard about least squares method, which can be used to approximate a function by straight line or by parabolic function. But I do not need to approximate by parabolic function. I probably need to approximate it by trigonometric function. So are there any methods for doing that? And one idea, is it possible to use the Least squares method for this problem, if we can deduce it for trigonometric functions?
One more question! If I use the discrete Fourier transform and think about the function as a sum of waves, so may be noise has special features by which we can define it and then we can set to zero the corresponding frequency and then perform inverse Fourier transform. So if you think that it is possible, then what can you suggest in order to identify the frequency of noise?
Unfortunately many solutions here presented don't solve the problem and/or they are plain wrong. There are many approaches and they are specifically built to solve conditions and requirements you must be aware of !
a) Approximation theory: If you have a very sharp defined function without errors (given by either definition or data) and you want to trace it exactly as possible, you are using polynominal or rational approximation by Chebyshev or Legendre polynoms, meaning that you approach the function by a polynom or, if periodical, by Fourier series.
b) Interpolation: If you have a function where some points (but not the whole curve!) are given and you need a function to get through this points, you can use several methods:
Newton-Gregory, Newton with divided differences, Lagrange, Hermite, Spline
c) Curve fitting: You have a function with given points and you want to draw a curve with a given (!) function which approximates the curve as closely as possible. There are linear and nonlinear algorithms for this case.
Your drawing implicates:
- It is not remotely like a mathematical function.
- It is not sharply defined by data or function
- You need to fit the curve, not some points.
What do you want and need is
d) Smoothing: Given a curve or datapoints with noise or rapidly changing elements, you only want to see the slow changes over time.
You can do that with LOESS as Jacob suggested (but I find that overkill, especially because choosing a reasonable span needs some experience). For your problem, I simply recommend the running average as suggested by Jim C.
http://en.wikipedia.org/wiki/Running_average
Sorry, cdonner and Orendorff, your proposals are well-minded, but completely wrong because you are using the right tools for the wrong solution.
These guys used a sixth polynominal to fit climate data and embarassed themselves completely.
http://scienceblogs.com/deltoid/2009/01/the_australians_war_on_science_32.php
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