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Lecture 7. Kernel Smoothing Methods

Lecture 7. Kernel Smoothing Methods. Instructed by Jinzhu Jia. Outline. Description of Kernel One-dimensional Kernel smoothing Selecting the Width of the Kernel Local Regression in R p Kernel Density Estimation and Classification. Nearest Neighbor Smoothing.

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Lecture 7. Kernel Smoothing Methods

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  1. Lecture 7. Kernel Smoothing Methods Instructed by Jinzhu Jia

  2. Outline • Description of Kernel • One-dimensional Kernel smoothing • Selecting the Width of the Kernel • Local Regression in Rp • Kernel Density Estimation and Classification

  3. Nearest Neighbor Smoothing Average according to uniform weights on all k neighbors

  4. Nearest Neighbor Smoothing • Properties: • Approximates E(Y|X) • Not continuous • To overcome discontinuity: assign weights that die off smoothly with distance from the target point.

  5. Epanechnikov Kernel

  6. Kernel • Similarity to the target ``x0” • Larger implies lower variance but high bias

  7. Examples • Example: EpanechnikovKernel – • K-NN: neighborhood sizereplaces • Tri-cube:

  8. Local Linear Regression

  9. Local linear regression

  10. Local linear regression • Make a first order bias correction

  11. Local Polynomial Regression

  12. Local Polynomial Regression • The price of bias reduction? Variance!

  13. Selecting the Width of the Kernel • controls the width of the local region • Epanechnikov or tri-cube: is the radius of the support region • KNN: is the number of neighbors • Gaussian: is the standard deviation

  14. Selecting the Width of the Kernel • A natural bias-variance tradeoff as we change the width of the averaging window (take local average as an exmaple) • If the window is narrow, variance will be bigger and bias is smaller • If the window is wide, variance will be smaller and bias is big, because some x will be far away from x0 • So CV, AIC, BIC can all be used to select a good

  15. Local Regression in

  16. Local Regression in • Boundary effects? • Less useful in high dimensions • Difficult for visualization

  17. Data Visualization

  18. Structured Local Regreesion Models in • When p/n is big, local regression is not helpful, unless we have some structural information.

  19. Structured Local Regreesion Models in • Structured regression functions • One-dimensional local regression for each stage

  20. Varying Coefficient Models • Divide the p predictors into two sets: • and Z • We assume the conditional linear model:

  21. Varying Coefficient Models: an example

  22. Local Likelihood and Other Models • From global to local: • Example: local logistic regression model.

  23. Kernel Density Estimation and Classification • Kernel Density Estimation • Smooth version • One choice of the kernel: histogram

  24. Kernel Density Estimation and Classification

  25. Naïve Bayes Classifier • Assumption: given class G=j, features are independent

  26. Mixture Models for Density Estimation and Classification • Gaussian mixture model: • EM algorithm is used for the parameter estimatation

  27. Homework • Due May 9 • ESLII_print 5, pp216. Exercise: 6.2, 6.12

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