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Kernel Regression

  • A kernel is a density function that satisfies the following 3 properties
    • Symmetric: K(−x)=K(x)K(-x) = K(x)
    • Sum to 1: âˆŦK(x)dx=1\int{K(x)dx=1}
    • Non negative: K(x)>=∀xK(x)>=\forall{x}
  • Using the kernel function, we can map the feature variables to a higher dimension, then the problem can be easily classified or predicted
  • For example, a cluster of points on a 2D plane might not be able to be classified with a line; but when mapped to a 3D plane it might be able to be classified with a hyperplane
  • Polynomial kernel is a commonly used function where it maps the feature to the various higher powered combinations
    • X=(x1,x2)X = (x_1, x_2)
    • ϕ(X)=(x1,x2)2=(x12+2x1x2+x22)\phi{(X)} = (x_1, x_2)^2 = (x^2_1 + 2x_1x_2 + x^2_2)