Quiz 3

Transformations

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Python Week 1: the first filter for runtime behavior
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# Transformations ## 6.1 Box-Cox Transformation $$ y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \lambda \neq 0 \\ \ln(y) & \lambda = 0 \end{cases} $$ Find $\lambda$ by maximum likelihood to satisfy normality and homoscedasticity. ## 6.2 Common Transformations Issue Transformation Skewed right $\ln(y...

Transformations

6.1 Box-Cox Transformation

y(λ)={yλ−1λλ≠0ln⁡(y)λ=0y^{(\lambda)} = \begin{cases} \frac{y^\lambda - 1}{\lambda} & \lambda \neq 0 \\ \ln(y) & \lambda = 0 \end{cases}
Find λ\lambda by maximum likelihood to satisfy normality and homoscedasticity.

6.2 Common Transformations

IssueTransformation
Skewed rightln⁡(y)\ln(y) , y\sqrt{y}
Skewed lefty2y^2 , y3y^3
HeteroscedasticityWeighted least squares
Non-linearityPolynomial terms, ln⁡(x)\ln(x)

6.3 Weighted Least Squares

Minimise ∑wi(yi−xiTβ)2\sum w_i (y_i - \mathbf{x}_i^T \beta)^2 where wi=1/σi2w_i = 1/\sigma_i^2.
β^WLS=(XTWX)−1XTWy\hat{\boldsymbol{\beta}}_{WLS} = (\mathbf{X}^T\mathbf{W}\mathbf{X})^{-1}\mathbf{X}^T\mathbf{W}\mathbf{y}
Join Discord PreviousMulticollinearityNextVariable Selection
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