Is it possible to compute the numerical hessian matrix for this function with respect to W_i,C, epsilon_i easily Matlab? I have computed a hessian by manually take a derivative, but I want to verify if my result is correct.
W = Nx1;
X = NxM;
X_i = Nx1;
y = 1xM;
C = 1x1;
DERIVEST on the file exchange has a function for doing this. There are also tips for doing this eg in Section 18 of this tutorial, or many other places.
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I am tasked to perform a prediction analysis. This requires performing a linear regression on several (~10) predictor variables and coming up with intercepts for all and a constant.
so final equation will be of this format y = c + c1x1 + c2x2 + c3x3....
Now I know that you can use fitlm function in MATLAB that is available with Statistics and Machine Learning Toolbox however at this point I don't know if we will be purchasing it. How do I perform linear regression on them ?
You can use the closed form solution of linear least squares.
C=inv(transpose(X)*X)*transpose(X)*y
In the above, make the first row of X all ones, and the following rows are x1, x2,...
C will contain the corresponding constants. The first entry in C is c.
From: https://www.mathworks.com/help/matlab/data_analysis/linear-regression.html
You can write your predictor variables as a matrix X using X = [ones(length(x1),1),x1,x2,x3,...,xn] and formulating the response variables Y as the equation Y = XB and doing a matrix inverse operation using mldivide as B = X\Y to find your regression coefficients.
I want to fit a curve to my data points (x;y) that will have a formula as such:
1/y = (x^-1)*a + b
At first I wanna do this using Octave but later I have to code this into microcontroller using c.
A quick search on google and matlab documentation don't give an anwesr I can't find a function that do polyfit with elements with negative order.
Is there a special set of function for such operation or do I have to somehow transfer my formula to fit into standard math problem ?
Your unknowns are aand b which are both linear in your problem. So you can use the 1st order polynomial fitting. It is already in the form of a standard math problem. To see just rename
Y = a*X + b
with the known data vectors (or points)
Y = 1/y
X = 1/x
Thats all.
So I'm trying to implement an EM-Algorithm to train a Gaussian Class Conditional model for classifying data. I'm stuck in the M-step at the moment because I can't figure out how to calculate the covariance matrix.
The problem is I have a big data set and using a for loop to go through each point would be way to slow. I also can't use the covariance function cov(), because I need to use a mean which I calculated using this formula(mu symbol one)
Is there a way to adjust cov() to use the mean I want? Or is there another way I could do this without for loops?
Edit: Forgot to explain what the data matrix is like. Its an nx3 where each row is a data point.
It technically needs to work for the general case nxm but n is usually really big(1000 or more) while m is relatively small.
You can calculate your covariance matrix manually. Let data be the matrix containing all your variables (for example, [x y]) and mu your custom mean, proceed as follows:
n = size(data,1);
data_dem = data - (ones(n,1) * mu);
cov_mat = (data_dem.' * data_dem) ./ (n - 1);
Notice that I used the Bessel's Correction (n-1 instead of n) because the Matlab cov function uses it, unless you specify the third argument as 1:
cov_mat = cov(x,y,1);
C = cov(___,w) specifies the normalization weight for any of the
previous syntaxes. When w = 0 (default), C is normalized by the number
of observations-1. When w = 1, it is normalized by the number of
observations.
I have a vector of x and y coordinates drawn from two separate unknown Gaussian distributions. I would like to fit these points to a three dimensional Gauss function and evaluate this function at any x and y.
So far the only manner I've found of doing this is using a Gaussian Mixture model with a maximum of 1 component (see code below) and going into the handle of ezcontour to take the X, Y, and Z data out.
The problems with this method is firstly that its a very ugly roundabout manner of getting this done and secondly the ezcontour command only gives me a grid of 60x60 but I need a much higher resolution.
Does anyone know a more elegant and useful method that will allow me to find the underlying Gauss function and extract its value at any x and y?
Code:
GaussDistribution = fitgmdist([varX varY],1); %Not exactly the intention of fitgmdist, but it gets the job done.
h = ezcontour(#(x,y)pdf(GaussDistributions,[x y]),[-500 -400], [-40 40]);
Gaussian Distribution in general form is like this:
I am not allowed to upload picture but the Formula of gaussian is:
1/((2*pi)^(D/2)*sqrt(det(Sigma)))*exp(-1/2*(x-Mu)*Sigma^-1*(x-Mu)');
where D is the data dimension (for you is 2);
Sigma is covariance matrix;
and Mu is mean of each data vector.
here is an example. In this example a guassian is fitted into two vectors of randomly generated samples from normal distributions with parameters N1(4,7) and N2(-2,4):
Data = [random('norm',4,7,30,1),random('norm',-2,4,30,1)];
X = -25:.2:25;
Y = -25:.2:25;
D = length(Data(1,:));
Mu = mean(Data);
Sigma = cov(Data);
P_Gaussian = zeros(length(X),length(Y));
for i=1:length(X)
for j=1:length(Y)
x = [X(i),Y(j)];
P_Gaussian(i,j) = 1/((2*pi)^(D/2)*sqrt(det(Sigma)))...
*exp(-1/2*(x-Mu)*Sigma^-1*(x-Mu)');
end
end
mesh(P_Gaussian)
run the code in matlab. For the sake of clarity I wrote the code like this it can be written more more efficient from programming point of view.
I'm trying to compute a rather ugly integral using MATLAB. What I'm having problem with though is a part where I multiply a very big number (>10^300) with a very small number (<10^-300). MATLAB returns 'inf' for this even though it should be in the range of 0-0.0005. This is what I have
besselFunction = #(u)besseli(qb,2*sqrt(lambda*(theta + mu)).*u);
exponentFuncion = #(u)exp(-u.*(lambda + theta + mu));
where qb = 5, lambda = 12, theta = 10, mu = 3. And what I want to find is
besselFunction(u)*exponentFunction(u)
for all real values of u. The problem is that whenever u>28 it will be evaluated as 'inf'. I've heared, and tried, to use MATLAB function 'vpa' but it doesn't seem to work well when I want to use functions...
Any tips will be appreciated at this point!
I'd use logarithms.
Let x = Bessel function of u and y = x*exp(-u) (simpler than your equation, but similar).
Since log(v*w) = log(v) + log(w), then log(y) = log(x) + log(exp(-u))
This simplifies to
log(y) = log(x) - u
This will be better behaved numerically.
The other key will be to not evaluate that Bessel function that turns into a large number and passing it to a math function to get the log. Better to write your own that returns the logarithm of the Bessel function directly. Look at a reference like Abramowitz and Stegun to try and find one.
If you are doing an integration, consider using Gauss–Laguerre quadrature instead. The basic idea is that for equations of the form exp(-x)*f(x), the integral from 0 to inf can be approximated as sum(w(X).*f(X)) where the values of X are the zeros of a Laguerre polynomial and W(X) are specific weights (see the Wikipedia article). Sort of like a very advanced Simpson's rule. Since your equation already has an exp(-x) part, it is particularly suited.
To find the roots of the polynomial, there is a function on MATLAB Central called LaguerrePoly, and from there it is pretty straightforward to compute the weights.