Box-Cox transformation vs Yeo-Johnson transformation: A comprehensive comparison

📚 Introduction to Data Transformation

Data transformation is a crucial step in statistical analysis and machine learning. It involves altering the scale of your data to make it more suitable for modeling. Two popular methods for achieving normality are the Box-Cox and Yeo-Johnson transformations. Let's dive into what they are and how they compare.

🧮 Box-Cox Transformation

The Box-Cox transformation is a power transformation technique used to stabilize variance and normalize data. It involves finding the optimal $\lambda$ value to transform the data $x$ as follows:

• 📐 If $\lambda \neq 0$: $x^{(\lambda)} = \frac{x^{\lambda} - 1}{\lambda}$
• 📈 If $\lambda = 0$: $x^{(\lambda)} = \log(x)$

Important Note: The Box-Cox transformation can only be applied to strictly positive data.

📊 Yeo-Johnson Transformation

The Yeo-Johnson transformation is a more flexible power transformation that can handle both positive and non-positive data. It also involves finding the optimal $\lambda$ value, but applies a different formula based on whether the data is positive or non-positive:

• ✅ If $x \geq 0$: $x^{(\lambda)} = \frac{(x + 1)^{\lambda} - 1}{\lambda}$ if $\lambda \neq 0$, and $x^{(\lambda)} = \log(x + 1)$ if $\lambda = 0$
• ❌ If $x < 0$: $x^{(\lambda)} = -\frac{(-x + 1)^{2 - \lambda} - 1}{2 - \lambda}$ if $\lambda \neq 2$, and $x^{(\lambda)} = -\log(-x + 1)$ if $\lambda = 2$

🆚 Comparison Table

🔑 Key Takeaways

• ✔️ Data Type: If your data contains non-positive values, use the Yeo-Johnson transformation.
• 🧪 Flexibility: Yeo-Johnson is more flexible due to its ability to handle different data types.
• 💡 Simplicity: Box-Cox is simpler if you are only dealing with positive data.
• 📈 Normality: Both aim to achieve normality but Yeo-Johnson is usually a safer bet when you have mixed data.