Box-Cox transformation

Definition #

For an observation (x > 0) and power parameter (\lambda), the Box-Cox transform (T_\lambda(x)) is

$$ T_\lambda(x) = \begin{cases} \dfrac{x^\lambda - 1}{\lambda}, & \lambda \ne 0,\\ \log x, & \lambda = 0. \end{cases} $$

• (\lambda = 1) leaves the values unchanged, while (\lambda = 0) corresponds to the natural logarithm.
• The inverse transform is implemented as scipy.special.inv_boxcox.
• Maximum-likelihood estimation of (\lambda) is available via scipy.stats.boxcox_normmax.

Because the expression involves (x^\lambda) and (\log x), all inputs must be strictly positive; add a small constant if necessary.

Worked example #

import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(42)
x = rng.lognormal(mean=1.5, sigma=0.6, size=1_000)
plt.figure(figsize=(6, 4))
plt.hist(x, bins=30, color="steelblue")
plt.title("Original distribution (positive but skewed)")
plt.show()

from scipy.stats import boxcox, boxcox_normmax

lmbda = boxcox_normmax(x)  # maximum-likelihood estimate of lambda
print(f"Estimated lambda: {lmbda:.3f}")

x_trans = boxcox(x, lmbda=lmbda)
plt.figure(figsize=(6, 4))
plt.hist(x_trans, bins=30, color="seagreen")
plt.title("After Box-Cox transformation")
plt.show()

The transformed data are far closer to symmetric, making downstream linear models and distance-based algorithms easier to fit.

Practical tips #

• Fit (\lambda) on the training split only and reuse it for validation/test data to avoid leakage.
• Apply the inverse transform to predictions when you need to report results on the original scale.
• Combine Box-Cox with scaling (StandardScaler) if the model expects zero-mean unit-variance inputs.
• If the feature contains zeros or negatives, shift it by a constant or move to Yeo-Johnson, which is designed for signed data.