|Posted: August 5, 2007, 6:13 am - IP Logged|
I am curious, how the coefficients are chosen? Thanks.
P.S. Just for grins I wonder if differentiating the equations and plotting against the original may give any valuable data from the intersects? Or possibly shifting the phase by introducing a sine, cosine, or cotangent function into the lower degrees of the polynomials. I say this because the plot shown looks like 2.5 cycles of a dinged decay. Just a thought.
I'd like to redirect on your earlier reply.
Quote: "P.S. Just for grins I wonder if differentiating the equations and plotting against the original may give any valuable data from the intersects? Or possibly shifting the phase by introducing a sine, cosine, or cotangent function into the lower degrees of the polynomials. I say this because the plot shown looks like 2.5 cycles of a dinged decay. Just a thought."
I have made another post related to this one topic, link: 1 to 11 Point - Polynomial Wave Projection Equations. It is an expansion of this topic.
Ideally, you'd only want to use these Equations to project for the next data point. Once a new data point has been selected for the data you're analyzing, the calculations should be done again for the new point giving a new direction based on a new projection. The equations are set up that way to accommodate just the next possible point.
You could apply the differential to a wave in the Wave Matrix, just as a suggestion. Or, even a differential of a differential; it could be interesting.
The idea for this polynomial projection came from my study into the Taylor Series Expansion of the Sine and Cosine functions. Also, in the similarities to the Least Squares Fitting for finding a polynomial curve fit for plotted data. Below are the Taylor Series Expansion for a few terms of the Sine and Cosine functions.
Taylor Series for Cosine and Sine
Cos x = (x0 / 0!) - (x2 / 2!) + (x4 / 4!) - (x6 / 6!) + ...
Sin x = (x1 / 1!) - (x3 / 3!) + (x5 / 5!) - (x7 / 7!) + ...
Adding these two functions leads to an nth degree polynomial.
Cos x + Sin x = (x0 / 0!) + (x1 / 1!) - (x2 / 2!) - (x3 / 3!) + (x4 / 4!) + (x5 / 5!) - (x6 / 6!) - (x7 / 7!) + ...
Both Cosine and Sine can operate at different frequencies (a - Cosine, b - Sine) and magnitudes (A - Cosine, B - Sine). This gives the equation as follows:
A Cos ax + B Sin bx = (A a0 / 0!) x0 + (B b1 / 1!) x1 - (A a2 / 2!) x2 - (B b3 / 3!) x3 + (A a4 / 4!) x4 + (B b5 / 5!) x5 - (A a6 / 6!) x6 - (B b7 / 7!) x7 + ...
The coefficients are then as follows:
a0 = (A a0 / 0!)
a1 = (B b1 / 1!)
a2 = (-A a2 / 2!)
a3 = (-B b3 / 3!)
a4 = (A a4 / 4!)
a5 = (B b5 / 5!)
a6 = (-A a6 / 6!)
a7 = (-B b7 / 7!)
I needed a way to find A, B, a and b. Also, I only needed part of the equation because I would only need to estimate the next point and recalculate after a new data point was recorded. The Least Squares Fitting allowed for a similar polynomial equation to estimate for the a0 to an values rather than try to find A, B, a and b directly.
The plots you are seeing in this topic are just generic. They are to show the basic concept and have no actual data associated with it. The frequencies and magnitudes of actual waves can vary greatly. Tweak the equations to suite your needs.
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