CHAPTERNonparametricMethods.pptVIP

Lecture Notes for E Alpayd?n 2004 Introduction to Machine Learning ? The MIT Press (V1.1) CHAPTER 8:Nonparametric Methods Alpaydin transparencies significantly modified, extended and changed by Ch. Eick Last updated: March 4, 2011 Eick/Alpaydin: Non-Parametric Density Estimation * Non-Parametric Density Estimation Goal is to obtain a density function: /wiki/Probability_density_function Parametric (single global model), semiparametric (small number of local models) Nonparametric: Similar inputs have similar outputs Keep the training data;“let the data speak for itself” Given x, find a small number of closest training instances and interpolate from these Aka lazy/memory-based/case-based/instance-based learning Histograms Histogram Usually shows the distribution of values of a single variable Divide the values into bins and show a bar plot of the number of objects in each bin. The height of each bar indicates the number of objects Shape of histogram depends on the number of bins Example: Petal Width (10 and 20 bins, respectively) Lecture Notes for E Alpayd?n 2004 Introduction to Machine Learning ? The MIT Press (V1.1) * Density Estimation Given the training set X={xt}t drawn iid from p(x) Divide data into bins of size h, stating from origin xo Histogram: Naive estimator: or Typo corrected on March 5 2011 Lecture Notes for E Alpayd?n 2004 Introduction to Machine Learning ? The MIT Press (V1.1) * Origin 0; h(1)=4/16 h(1.25)=1/8 Lecture Notes for E Alpayd?n 2004 Introduction to Machine Learning ? The MIT Press (V1.1) * h(2)=2/2*8=0.125 * Gaussian Kernel Estimator Kernel function, e.g., Gaussian kernel: Kernel estimator (Parzen windows): Gaussian Influence Functions in general: Influence of xt on x; h determines how quickly influence decreases as distance between xt and x increases; h is called “width” of kernel. Eick/Alpaydin: Non-Parametric Density Estimation Query point * Example: Kernel Density Estimation D={x1,x2,x3,x4} fDGaussian(x)= influ