Feature Scaling

Fit numeric transforms on training rows, then apply the same saved parameters to new rows. Use rectangular Python lists with rows as examples and columns as features; assume finite values, no missing entries, and compatible parameter lengths.

Functions to implement

min_max_scale(X)

Return (scaled_X,mins,maxs), computing each column's minimum and maximum from X. A nonconstant column transforms as (x-min)/(max-min). For a constant training column, output 0.0. Empty fitting input returns ([],[],[]).

min_max_transform(X,mins,maxs)

Apply the saved rule without recomputing parameters. If the saved range is zero, output 0.0 even if a new value differs from the training constant. Do not clip: held-out values can produce coordinates below zero or above one. This zero-range rule is the exercise's explicit policy, not a promise of equivalence to every library scaler.

standardize(X)

Return (standardized_X,means,stds). Compute each column's population variance with denominator n, not n-1. The returned stds are effective denominators: use sqrt(variance) when variance is positive and 1.0 when variance is zero. Transform with (x-mean)/stds[j]. Constant training columns become zero but do not have unit variance. Empty fitting input returns ([],[],[]).

standardize_transform(X,means,stds)

Apply (x-means[j])/stds[j] using the saved effective scales, which are positive. Do not refit. A changed value in a previously constant column retains its displacement from the training mean. Both transform functions return [] for no input rows.

Worked examples

scaled, mins, maxs = min_max_scale([[2,7],[4,7],[6,7]])
# scaled = [[0,0],[0.5,0],[1,0]], mins=[2,7], maxs=[6,7]
min_max_transform([[8,9]], mins, maxs)  # [[1.5,0]]

scaled, means, stds = standardize([[1,7],[3,7]])
# scaled = [[-1,0],[1,0]], means=[2,7], stds=[1,1]
standardize_transform([[5,9]], means, stds)  # [[3,2]]

Each feature is scaled independently. Save the returned vectors with the model; refitting them on validation or prediction rows changes the representation.