For a spectrum estimation algorithm I need to find the best fitting linear combination of vectors to fit a target spectral distribution. So far, this works relatively well using the lsqlin optimizer in MATLAB.
However, for the final application I would like to approximate/solve this problem for exclusively zeros and ones, meaning Ax=b solved for Boolean x.
Is there any way to parametrize lsqlin or another optimizer function for this purpose?
If the problem is just:
Solve Ax=b for x in {0,1}
then you can use a MIP solver (e.g. Matlab intlinprog). If the problem is over-constrained and you want a least squares solution:
Min w'w
S.t. Ax - b = w
x in {0,1} (binary variable)
w free variable
then you have a MIQP (Mixed Integer Quadratic Programming) problem. There are good solvers for this such as Cplex and Gurobi (callable from Matlab). Also Matlab has a discussion about an approximation scheme using intlinprog. Another idea is to replace the quadratic objective by a sum of absolute values. This can be formulated as linear MIP model.
I am solving a hug optimization problem that takes a lot of time to converge to a solution. This is for the reason that Matlab uses finite difference method for calculating the Gradient of objective functions and nonlinear constraint and also constructing Hessian matrix. But there is an option in fmincon solver that allow you to supply the analytic derivative of functions and constraints.
For this reason I wanted to know how can I calculate the Grad of the namely function which is given here both in mathematical aspect and symbolic math tool. I should note that still I want the gradient of the objective in the vector format. (not by extracting Eq1 in 5 equation.)
Lets assume we have these optimization variables
Pd=[x1 x2 x3 x4]
Now we define these 2 variables based on optimization vector i.e.,Pd
Pdn=[pd(1);mo;Pd(2);0;Pd(4)]
Pgn=[pd(2);Pd(1);m1;Pd(4),Pd(1)]
Now this is the equation that I want to take the gradient from:
Eq1=Sin(Pdn)+Pdn+Pgn.^2
I have the following equation:
I want to do a exponential curve fitting using MATLAB for the above equation, where y = f(u,a). y is my output while (u,a) are my inputs. I want to find the coefficients A,B for a set of provided data.
I know how to do this for simple polynomials by defining states. As an example, if states= (ones(size(u)), u u.^2), this will give me L+Mu+Nu^2, with L, M and N being regression coefficients.
However, this is not the case for the above equation. How could I do this in MATLAB?
Building on what #eigenchris said, simply take the natural logarithm (log in MATLAB) of both sides of the equation. If we do this, we would in fact be linearizing the equation in log space. In other words, given your original equation:
We get:
However, this isn't exactly polynomial regression. This is more of a least squares fitting of your points. Specifically, what you would do is given a set of y and set pair of (u,a) points, you would build a system of equations and solve for this system via least squares. In other words, given the set y = (y_0, y_1, y_2,...y_N), and (u,a) = ((u_0, a_0), (u_1, a_1), ..., (u_N, a_N)), where N is the number of points that you have, you would build your system of equations like so:
This can be written in matrix form:
To solve for A and B, you simply need to find the least-squares solution. You can see that it's in the form of:
Y = AX
To solve for X, we use what is called the pseudoinverse. As such:
X = A^{*} * Y
A^{*} is the pseudoinverse. This can eloquently be done in MATLAB using the \ or mldivide operator. All you have to do is build a vector of y values with the log taken, as well as building the matrix of u and a values. Therefore, if your points (u,a) are stored in U and A respectively, as well as the values of y stored in Y, you would simply do this:
x = [u.^2 a.^3] \ log(y);
x(1) will contain the coefficient for A, while x(2) will contain the coefficient for B. As A. Donda has noted in his answer (which I embarrassingly forgot about), the values of A and B are obtained assuming that the errors with respect to the exact curve you are trying to fit to are normally (Gaussian) distributed with a constant variance. The errors also need to be additive. If this is not the case, then your parameters achieved may not represent the best fit possible.
See this Wikipedia page for more details on what assumptions least-squares fitting takes:
http://en.wikipedia.org/wiki/Least_squares#Least_squares.2C_regression_analysis_and_statistics
One approach is to use a linear regression of log(y) with respect to u² and a³:
Assuming that u, a, and y are column vectors of the same length:
AB = [u .^ 2, a .^ 3] \ log(y)
After this, AB(1) is the fit value for A and AB(2) is the fit value for B. The computation uses Matlab's mldivide operator; an alternative would be to use the pseudo-inverse.
The fit values found this way are Maximum Likelihood estimates of the parameters under the assumption that deviations from the exact equation are constant-variance normally distributed errors additive to A u² + B a³. If the actual source of deviations differs from this, these estimates may not be optimal.
I have a system of dynamic equations that ultimately can be written in the well-known "spring-mass-damper" form:
[M]{q''}+[C]{q'}+[K]{q}={0}
[M], [C], [K]: n-by-n Coefficient Matrices
{q}: n-by-1 Vector of the Degrees of Freedom
(the ' mark represents a time derivative)
I want to find the eigenvalues and eigenvectors of this system. Obviously due to the term [C]{q'}, the standard MATLAB function eig() will not be useful.
Does anyone know of a simple MATLAB routine to determine the eigenvalues, eigenvectors of this system? The system is homogeneous so an efficient eigenvalue analysis should be very feasible, but I'm struggling a bit.
Obviously I can use brute force and a symbolic computing software to find the gigantic characteristic polynomial. But this seems inefficient for me, especially because I'm looping this through the other parts of the code to determine frequencies as a function of other varied parameters.
I have to calculate:
gamma=(I-K*A^-1)*OLS;
where I is the identity matrix, K and A are diagonal matrices of the same size, and OLS is the ordinary least squares estimate of the parameters.
I do this in Matlab using:
gamma=(I-A\K)*OLS;
However I then have to calculate:
gamma2=(I-K^2*A-2)*OLS;
I calculate this in Matlab using:
gamma2=(I+A\K)*(I-A\K)*OLS;
Is this correct?
Also I just want to calculate the variance of the OLS parameters:
The formula is simple enough:
Var(B)=sigma^2*(Delta)^-1;
Where sigma is a constant and Delta is a diagonal matrix containing the eigenvalues.
I tried doing this by:
Var_B=Delta\sigma^2;
But it comes back saying matrix dimensions must agree?
Please can you tell me how to calculate Var(B) in Matlab, as well as confirming whether or not my other calculations are correct.
In general, matrix multiplication does not commute, which makes A^2 - B^2 not equal to (A+B)*(A-B). However your case is special, because you have an identity matrix in the equation. So your method for finding gamma2 is valid.
'Var_B=Delta\sigma^2' is not a valid mldivide expression. See the documentation. Try Var_B=sigma^2*inv(Delta). The function inv returns a matrix inverse. Although this function can also be applied in your expression to find gamma or gamma2, the use of the operator \ is more recommended for better accuracy and faster computation.