Recent content by exmachina

Well A and B are two variables that specify (completely) the state of the system. Suppose I've sampled a whole bunch of data points (a,b) s.t. I can generate their PDFs. I can approximate P(B | A=a1) and P(A | B=b1) as well by taking a slice of my dataset, (eg. B= b1+-0.1) and count the...

Suppose I have the marginal probability density functions of two random variables A and B, P(A), and P(B). Suppose I modeled P(A) and P(B) using a mixture model from some dataset D and obtained a closed form pdf for each. I am interested in finding their joint density function P(A and B) and...

double post sorry

Sorry I was way too sloppy in my original post, I have since updated it, I had forgotten an r term.

Edit: LOTS OF TYPOS (sorry guys) Let: f(r) = e^{-(a-r)^2} g(r) = r e^{-(a-r)^2} Where a is some constant Can: \dfrac{ \sum\limits^{r=\infty}_{r=-\infty} g(r) } {\sum\limits^{r=\infty}_{r=-\infty} f(r) } Be simplified?

Is there a closed form expression for finding all the roots of the derivative of a k-component gaussian mixture model?

edit - doh - this obviously implies that f(x) must be equal to 0 (no other solution satisfies: f(x)=f(x)g(x) unless g(x) = 1, which in this case, it isn't)

Edit: I guess in particular, this is the equation I'm trying to maximize, given an input vector: X = (x_1,x_2,...,x_n) Maximize: \begin{equation} \prod_{j=1}^n\sum_{i=1}^k \frac{p_i}{\sqrt{2\pi} \sigma_i} \exp(-\frac{(x_j-\mu_i)^2}{2\sigma_i^2}) Edit: I found a nice paper tackling this...

Yes that is the trivial solution. Perhaps this can be casted as an eigenvalue problem - as it seems to imply that the convolution operator (wrt to the gaussian) may have certain eigenvalues and corresponding eigenfunctions f(x) being one of them

I've arrived at the following equation involving the convolution of two functions: f(x) = \int_{-\infty}^{\infty} f(t) g(t-x) dt = f(x) \ast g(x) Where: g(z) = e^{-z^2/2} In other words, a function convoluted with a Gaussian pdf results in the same function. I've tried taking Fourier...

Interesting, any idea what method they use? Expectation-Maximization?

Hi, I have a dataset of a random variable whose probability density function can be fitted/modelled as a sum of N probability density functions of normal distributions: F_X(x) = p(\mu_1,\sigma_1^2)+p(\mu_2,\sigma_2^2)+\ldots+p({\mu}_x,\sigma_x^2) I am interested in a fitting method can...

The dataset already seemed quite smooth upon an observation.

I have a dataset in two columns X and Y, sorted in ascending values of X. I'm trying to find its numerical derivative, however, the "noise" (it's very hard to see any noise in the dataset itself when plotted), but the noise gets massively amplified to the point where the numerical derivative...

The Wikipedia article regarding Lagrangian Mechanics mentions that we can essentially derive a new set of equations of motion, thought albeit non-linear ODEs, using Lagrangian Mechanics. My question is: how difficult is it usually to solve these non-linear ODEs? What are the usual numerical...