Machine Learning & Data Science Statistics Time Series Analysis Box Cox Transform

The Box-Cox transform is a statistical technique used to stabilise variance (see Machine-Learning-&-Data-Science-Statistics-Time-Series-Analysis-Stationarity) and make data more closely resemble a Normal Distribution, which is often required for certain statistical models and tests. It transforms non-normal dependent variables into a normal shape by applying a power transformation.

The Box-Cox transformation is defined as:

$$y(\lambda) = \begin{cases} \frac{y^{\lambda} - 1}{\lambda}, & \text{if } \lambda \neq 0, \ \log(y), & \text{if } \lambda = 0. \end{cases}​$$ where:

• $y$ is the original data,
• $\lambda$ is the transformation parameter that determines the nature of the transformation.

Key Points:

1. Parameter $\lambda$:
   • The parameter $\lambda$ controls the transformation. The value of $\lambda$ is typically chosen by maximising the likelihood of a model (e.g., maximising the log-likelihood in a regression setting).
   • Common values of $\lambda$ :
     • $\lambda = 1: No transformation (linear),
     • $\lambda = 0$ : Logarithmic transformation,
     • $\lambda = 0.5$ : Square root transformation,
     • $\lambda = -1$ : Reciprocal transformation.
2. Purpose:
   • The primary aim of the Box-Cox transformation is to make the data distribution more normal (Gaussian), which is often a requirement for many statistical tests and models (like linear regression).
   • It can also be used to stabilise variance, which can improve the performance of certain models.
3. Limitations:
   • It only works on positive data since the log transformation (used when $\lambda = 0$ ) is undefined for zero or negative values.

In practice, the optimal value of $\lambda$ is typically estimated from the data to achieve the best possible transformation.