LO 4.2.2.C

Learning Objective: Define standardizing the predictors.

Review:

The ridge regression coeﬃcient estimates can change substantially when multiplying a given predictor by a constant. Therefore, it is best to apply ridge regression after standardizing the predictors (i.e., all predictors on the same scale), using the following formula

$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>~</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msqrt><mstyle displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced><mrow><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mi>j</mi></msub></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mover><msub><mi>σ</mi><mi>j</mi></msub><mo>^</mo></mover></mfrac></math>$

where

xji is the i-ith value of the jth predictor Xj

n is the number of instances

is the mean of the instances xji

is the estimate of the standard deviation of instances xji