Thesis - Instituto de Telecomunicações

5.1. STATISTICAL MODELING 95Weibull Distribution ModelThe Weibull distributionp weibull (x|a, b) =abx (b−1) e (−axb) , (5.3)given a data transformation, can approximately fit several distributions, such as the exponentialdistribution (when b = 1) and the normal distribution (when µ ≫ 0).In order to estimate the parameters from the data we transform the feature vector xbased on the data skewness: skewness = E(x−µ)3 . If skewness < 0 ⇒ we invert the vectorσ 3as x = −x. As a final adjustment we subtract the vector by its minimum: x = x − min(x).Given the normalized data, maximum likelihood estimates for the parameters a and b areobtained.5.1.2 Nonparametric ModelingWhen a priori information on a parametric model does not exist, several approaches use thedata to establish a non-parametric model with very little base assumptions. Examples ofthe construction of classifiers without a parametric model are artificial neural networks [97],kernel density estimation (also called Parzen windows) [183] and mixture models [157, 69].Kernel Density EstimationKernel Density Estimation, (KDE) or Parzen windows method is an algorithm that usesthe training data and creates a Probability Density Function (PDF) via the accumulationof a n-dimensional window function centered at each of the training samples. The PDF isexpressed aŝf h (x) = 1 NN∑W (x − x i ,h), (5.4)i=1and computed via the Kernel Density Estimation (KDE) method. W is the kernel orwindow function, and the most common function is the normal (Gaussian) function. Themultivariate normal function is represented asW (x,h)=( )1(2πh) n f /2 exp − (x)⊤ (x), (5.5)2h