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lecture_03

# lecture_03 - Financial Data Mining FSRM588 Lecture 03...

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Financial Data Mining FSRM588 Lecture 03: Penalized Least Squares and Penalized Likelihood Department of Statistics & Biostatistics Rutgers University September 17 2013

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Lasso Lasso solves the following optimization problem, ˆ β lasso = arg max β R p +1 - 1 2 N N X i =1 y i - β 0 - p X j =1 x ij β j 2 - λ p X j =1 | β j | for some λ 0 .
Penalized Least Squares We can use some other penalty on the parameters β j as well, and consider the following general penalized least squares problem ˆ β = arg max β R p +1 - 1 2 N N X i =1 y i - β 0 - p X j =1 x ij β j 2 - p X j =1 p λ ( | β j | ) , where p λ ( · ) is the penalty function. Best subset selection: p λ ( t ) = ( λ 2 / 2) I ( t 6 = 0) . Lasso: p λ ( t ) = λ | t | . Hard thresholding: p λ ( t ) = 1 2 [ λ 2 - ( λ - t ) 2 + ] . Elastic net: p λ ( t ) = λ at 2 + (1 - a ) | t | with 0 a 1 . Bridge regression: p λ ( t ) = λ | t | q for some 0 < q 2 . Ridge regression: p λ ( t ) = λ | t | 2 . SCAD: to be introduced.

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L 1 and SCAD 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Penalty Functions t penalty L1 SCAD Hard 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 Derivatives t derivative L1 SCAD Hard
Centering Usually the intercept is not penalized, so we center y and x j first and consider the following optimization problem ˆ β = arg max β R p - 1 2 N N X i =1 y i - p X j =1 x ij β j 2 - p X j =1 p λ ( | β j | ) , with the implicit assumption that 1 0 x j = 0 , 1 j p and 1 0 y = 0 .

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Canonical Model Consider the special case by assuming the columns of the input matrix N - 1 / 2 X are orthonormal, i.e. X 0 X /N = I . Let ˆ β ls j = N - 1 x 0 j y , , ˆ y ls = XX 0 y , the penalized least squares can be rewritten as ˆ β = arg max β R p - 1 2 N y - ˆ y ls 2 + p X j =1 - 1 2 ˆ β ls j - β j 2 - p λ ( | β j | ) , so that the optimization problem becomes maximizing for each β j , ˆ β j = arg max β j R - 1 2 ˆ β ls j - β j 2 + p λ ( | β j | ) . (1)
Desired Properties Lasso estimates are ˆ β lasso j = sign( ˆ β ls j )( | ˆ β ls j | - λ ) + .

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