What is the algorithm that Excel uses to calculate a 2nd-order polynomial regression (curve fitting)? Is there sample code or pseudo-code available?
I found a solution that returns the same formula that Excel gives:
Put together an augmented matrix of values used in a Least-Squares Parabola. See the sum equations in http://www.efunda.com/math/leastsquares/lstsqr2dcurve.cfm
Use Gaussian elimination to solve the matrix. Here is C# code that will do that http://www.codeproject.com/Tips/388179/Linear-Equation-Solver-Gaussian-Elimination-Csharp
After running that, the left-over values in the matrix (M) will equal the coefficients given in Excel.
Maybe I can find the R^2 somehow, but I don't need it for my purposes.
The polynomial trendlines in charts use least squares based on a QR decomposition method like the LINEST worksheet function ( http://support.microsoft.com/kb/828533 ). A second order or quadratic trend for given (x,y) data could be calculated using =LINEST(y,x^{1,2}).
You can call worksheet formulas from C# using the Worksheet.Evaluate method.
It depends, because there are a lot of ways to do such a thing depending on the data you supply and how important it is to have the curve pass through those points.
I'm guessing that you have many more points than you do coefficients in the polynomial (e.g. more than three points for a 2nd order curve).
If that's true, then the best you can do is least square fitting, which calculates the coefficients that minimize the mean square error between all the points and the resulting curve.
Since this is second order, my recommendation would be just create the damn second order terms and do a linear regression.
Ex. If you are doing z~second_order(x,y), it is equivalent to doing z~first_order(x,y,x^2,y^2, xy).
Related
I am given the following problem.
I have a Set of functions which are linear combinations of the following functions (f1,f2,f3....fn) and a noisy dataset of pairs (x,y). I want to find a function from my set which approximates the dataset the best.
They key to finding the solution is to find coefficients a1,a2...an so that the resulting function f=a1*f1...an*fn approximates y well given the input x. If the data wasnt noisy, I could just choose 5 points and solve the resulting system of equations but I dont think this would work well with noisy data.
How would one find the coefficients ?
(I am asking for an algorithm and not for a program, for example matlab, that does the job for me)
In presence of noise you need to find some approximation solution, that minimizes discrepancies with ideal solution.
Such best fit problems are usually solved by optimization algorithms.
Widely used one is Levenberg–Marquardt algorithm.
Some weeks ago I've implemented a simple block matching stereo algorithm but the results had been bad. So I've searched on the Internet to find better algorithms. There I found the semi global matching (SGM), published by Heiko Hirschmueller. It gets one of the best results in relation to its processing time.
I've implemented the algorithm and got really good results (compared to simple block matching) as you can see here:
I've reprojected the 2D points to 3D by using the calculated disparity values with the following result
At the end of SGM I have an array with aggregated costs for each pixel. The disparity is equivalent to the index with the lowest cost value.
The problem is, that searching for the minimum only returns discrete values. This results in individually layers in the point-cloud. In other words: Round surfaces are cut into many layers (see point cloud).
Heiko mentioned in his paper, that it would be easy to get sub-pixel accuracy by fitting a polynomial function into the cost array and take the lowest point as disparity.
The problem is not bound to stereo vision, so in other words the task is the following:
given: An array of values, representing a polynomial function.
wanted: The lowest point of the polynomial function.
I don't have any idea how to do this. I need a fast algorithm, because I have to run this code for every pixel in the Image
For example: 500x500 Pixel with 60-200 costs each => Algorithm has to run 15000000-50000000 times!!).
I don't need a real time solution! My current SGM implementation (L2R and R2L matching, no cuda or multi-threading yet) takes about 20 seconds to process an image with 500x500 pixels ;).
I don't ask for libraries! I try to implement my own independent computer vision library :).
Thank you for your help!
With kind regards,
Andreas
Finding the exact lowest point in a general polynomial is a hard problem, since it is equivalent to finding the root of the derivative of the polynomial. In particular, if your polynomial is of degree 6, the derivative is a quintic polynomial, which is known not to be solvable by radical. You therefore need to either: fit the function using restricted families for which computing the roots of the derivatives e.g. the integrals of prod_i(x-ri)p(q) where deg(p)<=4, OR
using an iterative method to find an APPROXIMATE minimum, (newton's method, gradient descent).
I have implemented fm-elim in c using matrix.
I am wondering if following modification to the original algorithm is allowed or not?
In the original version of the algorithm one takes a row with positive coeff. in front of xr and subtracts it with the one with the negative coeff to create new equations. Matrix can grow in size.
see page 32-33 http://fileadmin.cs.lth.se/cs/Education/EDAF15/F07.pdf
But is it allowed to choose one equation with negative coeff and use gauss elim to eliminate the xr.
I tried solving some small system and it seems to give me correct answer but I don't know if this method is correct or not.
With this method my matrix won't grow in size. This way I will be doing ordinary gauss elimination.
I have a series of points representing values of a function, an example is below:
The values for X and Y can be real (non-integers). The function is monotonic, non-decreasing.
I want to be able to interpolate / assess the value of the function for any X (e.g. 1.5), so that a continuous function line would look like the following:
This is a standard interpolation problem, so I used Lagrange interpolation so far. It's quite simple and gives good enough results.
The problem with interpolation is that it also interpolates the values that are given as input, so the end results are for the input data will be different (e.g x=1, x=2)
Is there an algorithm that can guarantee that all the input values will have the same value after the interpolation? Linear interpolation is one solution, but it's linear the distances between X's don't have to be even (the graph is ugly then).
Please forgive my english / math language, I am not a native speaker.
The Lagrange interpolating polynomial in fact passes through all the n points, http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html. Although, for the 1d problem, cubic splines is a preferred interpolator.
If you rather want to fit a model, e.g., a linear, quadratic, or a cubic polynomial, or another function, to your data than I think you could still put the constraints on the coefficients to ensure the approximating function passes through some selected points. Begin by choosing the model, and then solve the Least Squares fitting problem.
I have some experimental data and I would like to fit them to obtain my parameters using the least-square method (Levenberg-Marquardt).
I am using two non-linear equations and I am using some computational programs (Origin and Matlab).
The first is:
y=A+B*(((2*pi*x)^2+Alfa4^2)*((2*pi*x)^2+Alfa5^2))/(((2*pi*x)^2+Alfa1^2)*((2*pi*x)^2+Alfa2^2)*((2*pi*x)^2+Alfa3^2));
Non-linear equation with the parameters (Alfa1,Alfa2,Alfa3,Alfa4,Alfa5)
And the second fitting equation is:
y=((T2^2+Lc^2*(2*pi*x)^2)/(((2*pi*x)^2*(Lc^2*(2*pi*x)^2+T8))+T6^2))*A1+G;
Rational function, i.e. quadratic function on the numerator and a 4th polynomial function on the denominator
I want to fit using this two equations, but I dont know how to do it. If someone want the experimental data I can post here.
Thank you very much,
Eduardo