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Nonparametric Density Estimation Riu Baring CIS 8526 Machine Learning Temple University Fall 2007 Christopher M. Bishop, Pattern Recognition and Machine.

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Presentation on theme: "Nonparametric Density Estimation Riu Baring CIS 8526 Machine Learning Temple University Fall 2007 Christopher M. Bishop, Pattern Recognition and Machine."— Presentation transcript:

1 Nonparametric Density Estimation Riu Baring CIS 8526 Machine Learning Temple University Fall 2007 Christopher M. Bishop, Pattern Recognition and Machine Learning, Chapter 2.5 Some slides from http://courses.cs.tamu.edu/rgutier/cpsc689_f07/

2 Overview Density Estimation  Given: a finite set x 1,…,x N  Task: to model the probability distribution p(x) Parametric Distribution  Governed by adaptive parameters Mean and variance – Gaussian Distribution  Need procedure to determine suitable values for the parameters  Discrete rv – binomial and multinomial distributions  Continuous rv – Gaussian distributions

3 Nonparametric Method Attempt to estimate the density directly from the data without making any parametric assumptions about the underlying distribution. Nonparametric Density Estimation

4 Histogram Divide the sample space into a number of bins and approximate the density at the center of each bin by the fraction of points in the training data that fall into the corresponding bin.

5 Histogram Parameter: bin width.

6 Histogram - Drawbacks The discontinuities of the estimate are not due to the underlying density, they are only an artifact of the chosen bin locations  These discontinuities make it very difficult (to the naïve analyst) to grasp the structure of the data A much more serious problem is the curse of dimensionality, since the number of bins grows exponentially with the number of dimensions  In high dimensions we would require a very large number of examples or else most of the bins would be empty

7 Nonparametric DE

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10 Kernel Density Estimator

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12 k-nearest-neighbors To estimate p(x):  Consider small sphere centered on the point x  Allow the radius of the sphere to grow until it contains k data points

13 k-nearest-neighbors Data set comprising N k points in class C k, so that Suppose the sphere has volume, V, and contains k k points from class C k Density Estimate Unconditional density Class Prior Posterior probability of class membership.

14 k-nearest-neighbors To classify new point x  Identify K nearest neighbors from training data  Assign to the class having the largest number of representatives Parameter, K.

15 My thoughts KDE and KNN require the entire training data set to be stored  Leads to expensive computation Tweak “parameters”  KDE: bandwidth, h  KNN: K


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