Select a retained representation

Understand the idea

Production uses the smallest prefix reaching at least 90% cumulative variance. The threshold is a stated compression criterion, not a universal optimum.

Choose the smallest component prefix meeting a variance criterion.Scroll the diagram horizontally if needed.

What the syntax does

np.searchsorted(cumulative, 0.9): Finds the first zero-based position reaching the variance threshold.
retained <= budget: Checks whether the smallest sufficient prefix fits a component storage limit; do not silently lower the stated variance requirement.

Supplied for this practice

These objects are prepared afresh before your editor runs.

X: Feature dataframe for the declared population, preserving row indices.
X = df.copy()
scaler: Already fitted coordinate system learned from X; transform reuses its statistics without fitting again. Fitted on X; reuse without refitting.
scaled: Standardised features in the same row/column order as X.
scaled = scaler.fit_transform(X)
pca: Already fitted on scaled. PCA fitted on scaled measurements. Fitted on scaled; reuse without refitting.
scores: Supplied scores; its definition below specifies the values and role.
scores = pca.transform(scaled)

Outputs you create

answer: The requested result, with the values, labels and row order described by this task.

PCA48 · first 8 rows
`a`	`b`	`c`	`d`	`e`
103.047	50.8622	22.7157	44.6184	0.906993
89.6002	44.3015	20.2703	40.1253	-0.680116
107.505	53.9521	21.1565	41.7009	0.818161
109.406	54.2501	22.5252	44.9095	1.39291
80.4896	40.0557	14.1714	29.4087	-3.27953
86.9782	44.1387	18.7213	37.5997	-1.70077
101.278	50.4611	18.1185	36.0784	-0.343557
96.8376	48.7875	17.4445	33.8533	-0.987809

Your task

Keep the fewest leading PCA components that explain at least 90% variance.
Store that count in retained and the matching row scores in answer.

answer

Hints

Think: The rule asks for the first cumulative total that reaches the threshold.

Tools: np.cumsum, np.searchsorted and score-column slicing.

Approach: Accumulate variance ratios, locate the first qualifying component count and retain that prefix of scores.

Explained solution

cumulative=np.cumsum(pca.explained_variance_ratio_)
retained=int(np.searchsorted(cumulative,.9)+1)
answer=scores[:,:retained]

Adding one converts a zero-based position into a dimension count; the prefix retains the required variation with the fewest leading axes.