LabVIEW
Non Linear fit vi slows down computation
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Formula parsing is very expensive. You will gain orders of magnitude is speed by using the version with the VI model. In addition, you can of course calculate all partial derivatives from first principles.
If you look at your formula (a0*x^(1/3)+a1*x^(1/2)+a2*x+a3*x^2+a4*x^3), you can see that it is actually a problem that can be solved using general linear fit. Just form the H matrix (based on x) and solve without any iterations near instantly.
See if this can give you some ideas. Fit is well below milliseconds!
(Of course I question the use of such fancy terms for such a simple function segments. Looks way overparametrized)
We are fitting the last N points. The oldest points are discarded while new ones are added.
(You used a chart, which also only has a fixed history length.).
You did not say if you were able to get it working. Did you?
Most of your questions are mathematical in nature and have nothing to do with LabVIEW. You simply need to brush up on your linear algebra.
The approximation is just linear algebra. Your model is very similar to a polynomial, but you have some non-integer coefficients. The function is just a sum of terms where each term is a function of X and multiplied by a coefficient.. These terms are calculated in the FOR loop once because X is irrelevant for the fit quality (but the coefficient depend on the x scaling, of course). Each term can be nonlinear in x as long as all your coefficient a0, a1, etc. are linear.
Where does your model come from? Compared to a simple polynomial fit it is more fragile because due to the fractional exponents, all x need to be greater than zero. In my casual testing, a polynomial fit seems to work better.
There is no need to guess anything. Just follow the wires!
Obviously, you cannot approximate an infinite sine function with any such model. For example if you do a polynomial fit, you need more and more terms as you try to fit more periods. (Have a look at e.g. the Taylor series of sin(x)). You get a good fit with a reasonable number of terms as long as you have a subsection with not too many turning points.
In the attached example, you can see that a 9th order polynomial works great for a sine with more than one period.
If any of the coefficients is near zero, you could omit that term and see how good the fit is afterwards. (For example if the x are chosen in a certain way, the odd or even polynomial coefficients could vanish). Note that the coefficients of these fits are otherwise not really informative and quite meaningless. If you really have a partial section of a sine function, doing a nonlinear fit using a sine model would give you real parameters (amplitude, phase, offset). If you have many periods, an FFT can give you that too.
@mikoborn wrote:
All these methods imply, that I know what the actual function I'm trying to approximate looks like. However I'm searching for a solution to dynamically be able to approximate behavior of datasets that is unknown to me.
I was thinking that a function like a0*sin(x) + a1*ln(x) + "polynomial" would be able to perform such task, if parameters can become zero.
Well, this seems like a contradiction. First you want a "general" solution, and then you come up with a very "specific" function. The question is: What do you actually want???
- If you want to retrieve parameters that have meaning (time constants, amplitudes, etc.), you need a specific parameterized model based on theory.
- If you just want to get rid of noise, you can do polynomials, filtering, convolution with a smoothing kernel, and many other things, even ptbypt solutions, but there won't be any interesting parameters. All you get is a smother approximation of the data. For example your sin(x) term very strongly depends on the x scaling and only makes sense if you know that there is only exactly that known fundamental frequency at exactly that known phase. In your earlier example, the phase constantly changes, so that won't work at all!!! Also, your ln(x) term is restricted to positive x, i.e. not "general" at all!
If your data section of interest resembles a banana (relatively featureless, few turning points), a simple low-order polynomial will always work extremely well. You simply need to be careful that x is in a reasonable range. For example if your x is in the millions, the 0th order term will be 1 and the 9th order term will be 1e54 and your matrix will become very ill conditioned, making a solution impossible.
For some general topics in fitting data, have a look at my group. Also have a look at my NI week talk for a few years ago.
