I have fitted a Gaussian Mixture Model to the multiple joint probability density functions. How can I obtain the conditional probability density function (i.e.,p(x|y)) from this mixture model (NXN matrix) in Matlab?
Based on Bayes rule, you can write down formula p(x|y)=p(x,y)/p(y). If you are able to obtain probability value p(y) for some given y, you can plug it in directly into Bayes formula. Otherwise you can go on and express each gaussian of the mixture as conditional gaussian with parameters (P stands for covariance matrices, mu stands for means):
mu_x|y = mu_x + P_xy P_yy^-1 (y - mu_y)
P_x|y = P_xx + P_xy P_yy^-1 P_yx
Related
Suppose I am finding the power spectral density of data like such:
x = winter_data.values #measured at frequency 1Hz
f, Sxx = sp.signal.welch(x1, fs=1, window='hanning', nperseg=N, noverlap = N / 2)
I want to test that Parseval's theorem works on these data sets. Since welch returns the power spectral density, should we not have
np.trapz(x**2, dx=1)
and
len(x1)*np.trapz(Sxx, f)
equal to eachother? Or is my definition of power spectral density incorrect? (np.trapz() is just used to calculate the integrals). I always thought that power spectral density was defined as
S_xx(f) = (1/T)|X(f)|^2
Currently I am not getting them equal.
Assuming I have already built a Gaussian Mixture Model using the fitgmdist function and want to map the multivariate distributions into a subspace with a smaller dimension without having to recreate the model how do I go about it?
In MATLAB terms, I have a GMM, gmm_goal, with gmm_goal.NumComponents = K and gmm_goal.NumVariables = N and want to reduce N to a number n < N.
If code isn't available, an explanation or mathematical derivation will do.
The parameters of the Gaussian Mixture Model effected by the transformation into a subspace are the mean and variance of the Gaussian distributions that form the GMM.
Assuming a linear transformation of your data points x:
y = A*x + b
Because of linearity of expectation, we can calculate the new mean and variance of the subspace from the old ones:
mean_new = A*mean + b
variance_new = A*variance*A'
I want to calculate weighted kernels (for using in a SVM classifier) in Matlab but I'm currently compeletely confused.
I would like to implement the following weighted RBF and Sigmoid kernel:
x and y are vectors of size n, gamma and b are constants and w is a vector of size n with weights.
The problem now is that the fitcsvm method from Matlab need two matrices as input, i.e. K(X,Y). For example the not weighted RBF and sigmoid kernel can be computed as follows:
K_rbf = exp(-gamma .* pdist2(X,Y,'euclidean').^2)
K_sigmoid = tanh(gamma*X*Y' + b);
X and Y are matrices where the rows are the data points (vectors).
How can I compute the above weighted kernels efficiently in Matlab?
Simply scale your input by the weights before passing to the kernel equations. Lets assume you have a vector w of weights (of size of the input problem), you have your data in rows of X, and features are columns. Multiply it with broadcasting over rows (for example using bsxfun) with w. Thats all. Do not do the same to Y though, just multiply one of the matrices. This is true for every such "weighted" kernel based on scalar product (like sigmoid); for distance based (like RBF) you want to scale both by sqrt of w.
Short proofs:
scalar based
f(<wx, y>) = f(w<x, y>) (linearity of scalar product)
distance based
f(||sqrt(w)x - sqrt(w)y||^2) = f(SUM_i (sqrt(w_i)(x_i - y_i))^2)
= f(SUM_i w_i (x_i - y_i)^2)
I am not doing signal processing. But in my area, I will use the spectral density of a matrix of data. I get quite confused at a very detailed level.
%matrix H is given.
corr=xcorr2(H); %get the correlation
spec=fft2(corr); % Wiener-Khinchin Theorem
In matlab, xcorr2 will calculate the correlation function of this matrix. The lag will range from -N+1 to N-1. So if size of matrix H is N by N, then size of corr will be 2N-1 by 2N-1. For discretized data, I should use corr or half of corr?
Another problem is I think Wiener-Khinchin Theorem is basically for continuous function. I have always thought that Discretized FT is an approximation to Continuous FT, or you can say it is a tool to calculate Continuous FT. If you use matlab build in function 'fft', you should divide the final result by \delta x.
Any kind soul who knows this area well there to share some matlab code with me?
Basically, approximating a continuous FT by a Discretized FT is the same as approximating an integral by a finite sum.
We will first discuss the 1D case, then we'll discuss the 2D case.
Let's look at the Wiener-Kinchin theorem (for example here).
It states that :
"For the discrete-time case, the power spectral density of the function with discrete values x[n], is :
where
Is the autocorrelation function of x[n]."
1) You can see already that the sum is taken from -infty to +infty in the calculation of S(f)
2) Now considering the Matlab fft - You can see (command 'edit fft' in Matlab), that it is defined as :
X(k) = sum_{n=1}^N x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
which is exactly what you want to be done in order to calculate the power spectral density for a frequency f.
Note that, for continuous functions, S(f) will be a continuous function. For Discretized function, S(f) will be discrete.
Now that we know all that, it can easily be extended to the 2D case. Indeed, the structure of fft2 matches the structure of the right hand side of the Wiener-Kinchin Theorem for the 2D case.
Though, it will be necessary to divide your result by NxM, where N is the number of sample points in x and M is the number of sample points in y.
I want to compute the parameters mu and lambda for the Inverse Gaussian Distribution given the CDF.
By 'given the CDF' I mean that I have given the data AND the (estimated) quantile for the data I.e.
Quantile - Value
0.01 - 10
0.5 - 12
0.7 - 13
Now I want to find out the inverse gaussian distribution for this data so that I can e.g. Look up the quantile for value 11 based on my distribution.
How can I find out the values mu and lambda?
The only solution I can think of is using Gradient descent to find the best mu and lambda using RMSE as an error measure.
Isn't there a better solution?
Comment: Matlab's MLE-Algorithm is not an option, since it does not use the quantile data.
As all you really want to do is estimate the quantiles of the distribution at unknown values and you have a lot of data points you can simply interpolate the values you want to lookup.
quantile_estimate = interp1(values, quantiles, value_of_interest);
According to #mpiktas here I implemented a gradient descent algorithm for estimating my mu and lambda:
Make initial guess using MLE
Learn mu and lambda using gradient descent with RMSE as error measure.
The following article explains in detail how to compute quantiles (the inverse CDF) for the inverse Gaussian distribution:
Giner, G, and Smyth, GK (2016). statmod: probability calculations for the inverse Gaussian distribution. R Journal. http://arxiv.org/abs/1603.06687
Code for the R language is contained in the R package statmod available from CRAN. For example:
> library(statmod)
> qinvgauss(0.01, lower.tail=FALSE)
[1] 4.98
computes the 0.01 upper tail quantile of the standard IG distribution.