2Electrical and Computer Engineering DepartmentSaurabh PrasadPattern RecognitionChapter 4•Introduction•Density Estimation•Parzen WindowsOutline
3Electrical and Computer Engineering DepartmentSaurabh PrasadPattern RecognitionChapter 4Introduction•All Parametric densities are unimodal (have a single localmaximum), whereas many practical problems involve multi‐modal densities–Exception: GMMs•Nonparametric procedures can be used with arbitrarydistributions and without the assumption that the forms ofthe underlying densities are known•There are two types of nonparametric methods:–Estimating P(x |ωj)–Bypass probability and go directly to a‐posteriori probabilityestimation
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4Electrical and Computer Engineering DepartmentSaurabh PrasadPattern RecognitionChapter 44Density Estimation–Basic idea:–Probability that a vectorxwill fall in region R is:–P is a smoothed (or averaged) version of the density functionp(x). If we have a sample of size n, the probability that k pointsfall inRis then:and the expected value for k is:E(k) = nP(3)∫ℜ=(1)'dx)'x(pP(2))P1(PknPknkk−−⎟⎟⎠⎞⎜⎜⎝⎛=