Box-Cox transformation is a widely used mathematical transformation used in statistics and data analysis. It’s primarily used to stabilize variance and make the data more normally distributed, both of which are essential assumptions for many statistical techniques.

The purpose of the Box-Cox transformation is to stabilize variance and make the data closer to a normal distribution. This can be particularly useful for data analysis techniques that assume normally distributed data, such as linear regression. To determine the optimal value of  $\lambda$,  typically search for the value that maximizes the log-likelihood function, which measures how well the transformed data fit a normal distribution. This is often done using numerical optimization techniques.

Yeo-Johnson transformation is a mathematical formula used to transform data in statistics and data analysis. It’s an extension of the Box-Cox transformation and is designed to handle a broader range of data, including both positive and negative values, as well as zero.

The Yeo-Johnson transformation can handle a wider range of data than the original Box-Cox transformation, which is limited to positive data or data with positive shifts. The Yeo-Johnson transformation can be applied to both positive and negative data, making it more versatile in practical applications . Choosing the appropriate value of $\lambda$ is critical to obtaining a meaningful transformation. Typically, this is done through a search for the optimal $\lambda$ that maximizes the log-likelihood function, or by using other criteria such as minimizing the mean squared error.