Skew Transforms

The problem, from first principles

03-feature-scaling puts features on a common scale, but scaling does not change a distribution's shape. Incomes, prices, counts and response times pile up near zero with a long tail of huge values, and in that shape a handful of extreme rows dominate the mean, the variance and any linear model's fit. A transform that compresses the tail, a logarithm or a power, pulls such a feature closer to symmetric and makes the extreme rows ordinary again. This question measures skewness and builds the transforms that reduce it.

From theory to code

Implement skewness(x), which measures how lopsided a sample is, then log1p_transform(x) and its inverse inverse_log1p(z), then box_cox(x, lam), a family of power transforms controlled by one number, then best_box_cox_lambda(x, candidates), which picks the candidate that makes the sample most symmetric. The signatures and docstrings are already in the editor.

Constraints

• x is a 1D float array. skewness returns the population skewness, the mean of cubed standardized values, mean(((x - mean) / std) ** 3) with std the population standard deviation. It returns 0.0 when std is zero.
• log1p_transform(x) returns log(1 + x) and raises ValueError if any value is negative. inverse_log1p(z) returns exp(z) - 1. Use np.log1p and np.expm1.
• box_cox(x, lam) requires every value to be strictly positive and raises ValueError otherwise. It returns (x ** lam - 1) / lam for lam != 0 and log(x) for lam == 0.
• best_box_cox_lambda(x, candidates) returns the value in candidates whose Box-Cox transform has the smallest absolute skewness. On a tie it returns the earliest candidate in the list.
• Inputs must not be modified.

Hints