Relate coefficients under linear feature transformation
Company: Databricks
Role: Data Scientist
Category: Statistics & Math
Difficulty: hard
Interview Round: Technical Screen
Suppose you are fitting a linear regression model and you consider two different feature parameterizations.
Original features: `x1`, `x2`.
Transformed features:
- `z1 = x1 + x2`
- `z2 = x1 - x2`
Model A (original features):
- `y = β0 + β1 x1 + β2 x2 + ε`
Model B (transformed features):
- `y = γ0 + γ1 z1 + γ2 z2 + ε`
Questions:
1. What is the relationship between `(β1, β2)` and `(γ1, γ2)`?
2. Are Model A and Model B equivalent (i.e., do they have the same expressive power / can they produce identical fitted values)? Under what assumptions could they behave differently in practice?
Overview: The question evaluates understanding of linear regression and linear-algebraic feature transformations, focusing on how coefficient vectors relate under reparameterization and how interpretability of parameters changes; Category/domain: Statistics & Math, position type: Data Scientist.