模式识别讲稿2010-lecture2.ppt

Normal Distribution – Commonly used form for mathematically tractable. Appropriate if is continuous valued, The univariate normal density (Gaussian) For which the mean and variance are (first moment) (second moment) : standard deviation is denoted as 95% of the samples fall in If , Then , is always positive semi-definite but for mathemat- ical simplicity, we assume positive definite, i.e., measure the scatter of samples By excluding the condition of , we restrict our samples to be such that no component of has zero variance and no two component of are identical for all samples. is the variance of is the covariance of and which =0 if and are statistically independent. If all for , reduces to a product of the univariate normal densities for the components , i.e., The bell-shaped distribution is characterized by and , is where the cluster centre is and controls the symmetry and dispersion of the shape. The larger is, The more disperse is the cluster. The principal axis of these hyper-ellipsoids are the eigenvectors of and the length of these axis for an ellipsoid are proportional to the associated eigenvalues. Mahalanobis distance from to : Which is a metric that considers the cluster distribution Such metric can be used for classification if and Discriminant Functions for the Normal Density Consider some simple special cases. Case I ━ The components are statistically independent of each other and each component has the same variance . and Ignoring these terms independent