I have a system of three nonlinear equations with eight unknowns. I'm currently setting each equation equal to a desired value and then using Matlab's fsolve (a numerical solver) to find a solution. Instead of running fsolve in real-time, I'd like to pre-compute solutions for a specific set of values to which I set the equations equal.
Pursuant that goal, I've run the solver over a set of values and created a 3D matrix (N x N x N) which I've attempted to load into eight Simulink 3-D lookup tables, Direct Lookup Table n-D block, so I can fetch each of the eight solved unknowns. It's my understanding the inputs to this block should work the same way I would reference an element in my 3-D array: table(x,y,z) but I'm constantly getting Simulink table input out-of-range errors. I've confirmed the inputs are within the table size, so I'm not sure what's wrong.
This isn't the most elegant implementation, so I'm open to better solutions. Ideally, I'd like to have a Simulink lookup that takes three inputs and returns a vector of the eight solved unknowns, or even better, can do some type of linear interpolation between the three lookup values to return an approximate solution.
Thanks!
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I have the following problem. I have a N x N real matrix called Z(x; t), where x and t might be vectors in general. I have N_s observations (x_k, Z_k), k=1,..., N_s and I'd like to find the vector of parameters t that better approximates the data in the least square sense, which means I want t that minimizes
S(t) = \sum_{k=1}^{N_s} \sum_{i=1}^{N} \sum_{j=1}^N (Z_{k, i j} - Z(x_k; t))^2
This is in general a non-linear fitting of a matrix function. I'm only finding examples in which one has to fit scalar functions which are not immediately generalizable to a matrix function (nor a vector function). I tried using the scipy.optimize.leastsq function, the package symfit and lmfit, but still I don't manage to find a solution. Eventually, I'm ending up writing my own code...any help is appreciated!
You can do curve-fitting with multi-dimensional data. As far as I am aware, none of the low-level algorithms explicitly support multidimensional data, but they do minimize a one-dimensional array in the least-squares sense. And the fitting methods do not really care about the "independent variable(s)" x except in that they help you calculate the array to be minimized - perhaps to calculate a model function to match to y data.
That is to say: if you can write a function that would take the parameter values and calculate the matrix to be minimized, just flatten that 2-d (on n-d) array to one dimension. The fit will not mind.
I'm currently trying to evaluate a large polynomial (Degree ~10^5, the constants are the same for every input case) and find the values of the polynomial for many different values (I.e, evaluate the same polynomial and find the answer for a large number (~10^5) of values of x).
I tried using horner's method for evaluating every single input, but the program is too slow. Is there a way to optimize for many values?
I have a Function V which depends on two variables v1 and v2 and a parameter-Array p containing 15 Parameters.
I want to Minimize my Function V regarding v1 and v2, but there is no closed expression for my Function, so I can't build and use the Derivatives.
The Problem is the following : For caluclating the Value of my Function I need the Eigenvalues of two 4x4 Matrices (which should be symmetric and real by concept, but sometimes the EigenSolver does not get real Eigenvalues). These Eigenvalues I calculate with the Eigen Package. The entries of the Matrices are given by v1,v2 and p.
There are certain Input Sets for which some of these Eigenvalues become negative. These are Input Sets which I want to ignore for my calculation as they will lead to an complex Function value and my Function is only allowed to have real values.
Is there a way to include this? My first attempt was a Nelder-Mead-Simplex Algorithm using the GSL-Library and an way too high Output value for the Function if one of the Eigenvalues becomes negative, but this doesn't work.
Thanks for any suggestions.
For the Nelder-Mead simplex, you could reject new points as vertices for the simplex, unless they have the desired properties.
Your method to artificially increase the function value for forbidden points is also called penalty or barrier function. You might want to re-design your penalty function.
Another optimization method without derivatives is the Simulated Annealing method. Again, you could modify the method to avoid forbidden points.
What do you mean by "doesn't work"? Does it take too long? Are the resulting function values too high?
Depending on the function evaluation cost, it might be an approach to simply scan a 2D interval, evaluate all width x height function values and drill down in the tile with the lowest function values.
I'm not too familiar with MATLAB or computational mathematics so I was wondering how I might solve an equation involving the sum of squares, where each term involves two vectors- one known and one unknown. This formula is supposed to represent the error and I need to minimize the error. I think I'm supposed to use least squares but I don't know too much about it and I'm wondering what function is best for doing that and what arguments would represent my equation. My teacher also mentioned something about taking derivatives and he formed a matrix using derivatives which confused me even more- am I required to take derivatives?
The problem that you must be trying to solve is
Min u'u = min \sum_i u_i^2, u=y-Xbeta, where u is the error, y is the vector of dependent variables you are trying to explain, X is a matrix of independent variables and beta is the vector you want to estimate.
Since sum u_i^2 is diferentiable (and convex), you can evaluate the minimal of this expression calculating its derivative and making it equal to zero.
If you do that, you find that beta=inv(X'X)X'y. This maybe calculated using the matlab function regress http://www.mathworks.com/help/stats/regress.html or writing this formula in Matlab. However, you should be careful how to evaluate the inverse (X'X) see Most efficient matrix inversion in MATLAB
I run
Y_testing_obtained = classify(X_testing, X_training, Y_training);
and the error I get is
Error using ==> classify at 246
The pooled covariance matrix of TRAINING must be positive definite.
X_training is 1550 x 5 matrix. Can you please tell me what this error means, i.e. why is it appearing, and how to work around it?
Thanks
Explanation: When you run the function classify without specifying the type of discriminant function (as you did), Matlab uses Linear Discriminant Analysis (LDA). Without going into too much details on LDA, the algorithms needs to calculate the covariance matrix of X_testing in order to solve an optimisation problem, and this matrix has to be positive definite (see Wikipedia: Positive-definite matrix). The underlying assumption is that your data is represented by a multivariate probability distribution, which always has a positive definite covariance matrix unless one or more variables are exact linear combinations of the others.
To solve your problem: It is possible that one of your variables is a linear combination of the others. You can try selecting a sensible subset of your variables, or perform Principal Component Analysis (PCA) on the training data and then classify using the first few principal components. Or, you could specify the type of discriminant function and choose one of the two naive Bayes classifiers, for example:
Y_testing_obtained = classify(X_testing, X_training, Y_training, 'diaglinear');
As a side note, you also need to have more observations (rows) than variables (columns), but in your case this is not the problem as you seem to have 1550 observations and 5 variables.
Finally, you can also have a look at the answers posted to a similar question on the Matlab forum.
Try regularizing the data using cvshrink function in Matlab