Variance Explained Ratio

easyPRODimensionality ReductionMachine Learning 2

Compute the explained variance ratio from eigenvalues.

Given a list of eigenvalues (sorted descending), compute: $\text{ratio}_i = \frac{\lambda_i}{\sum_{j} \lambda_j}$

Also compute the cumulative explained variance ratio: entry $i$ is the running sum of the already rounded ratios up to and including $i$ , rounded again to 4 decimal places.

Return a tuple (ratios, cumulative) where both are lists rounded to 4 decimal places.

Example

Input

eigenvalues = [4.0, 2.0, 1.0, 0.5]

Output

([0.5333, 0.2667, 0.1333, 0.0667], [0.5333, 0.8, 0.9333, 1.0])

Reasoning

eigenvalues holds $\lambda_1 \ge \lambda_2 \ge \dots$ , the variance captured along each principal direction. Here the total variance is $4.0 + 2.0 + 1.0 + 0.5 = 7.5$ .
Each ratio is one component's share of that total, rounded to 4 decimal places: $\frac{4.0}{7.5} = 0.5\overline{3} \to 0.5333$ , $\frac{2.0}{7.5} \to 0.2667$ , $\frac{1.0}{7.5} \to 0.1333$ , $\frac{0.5}{7.5} \to 0.0667$ .
Cumulative entry $i$ is the share explained by the first $i$ components together: $0.5333$ , then $0.5333 + 0.2667 = 0.8$ , $0.8 + 0.1333 = 0.9333$ , $0.9333 + 0.0667 = 1.0$ .
Both lists have one entry per eigenvalue. The ratios shrink because the eigenvalues are sorted descending, and the cumulative list climbs to $1$ because all components together explain all the variance.
Notice the running total adds the already rounded ratios. With three equal eigenvalues each ratio is $0.3333$ , so the last cumulative value is $0.9999$ , not $1.0$ .
Reading the result: the first two components already explain $80\%$ of the variance, so keeping 2 of 4 loses only $20\%$ .

Constraints

eigenvalues: list of non-negative floats (sorted descending)
Return (ratios, cumulative) rounded to 4 decimal places
cumulative[i] = sum(ratios[0:i+1])

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Background Knowledge

The explained variance ratio is a concept used in dimensionality reduction techniques, such as Principal Component Analysis (PCA). In PCA, the goal is to reduce the number of features in a dataset while retaining most of the information. This is achieved by transforming the original features into new features, called principal components, which are ordered by their variance. The eigenvalues of the covariance matrix of the dataset represent the variance explained by each principal component. The explained variance ratio is a measure of the proportion of variance explained by each principal component.

The cumulative explained variance ratio is an extension of the explained variance ratio, where we calculate the cumulative sum of the explained variance ratios. This gives us an idea of how much variance is explained by the top principal components. For example, if the cumulative explained variance ratio for the top 2 principal components is 0.8, it means that these 2 components explain 80% of the total variance in the dataset.

In the context of this problem, we are given a list of eigenvalues (sorted in descending order) and asked to compute the explained variance ratio and the cumulative explained variance ratio. The eigenvalues represent the variance explained by each principal component, and we need to calculate the proportion of variance explained by each component and the cumulative sum of these proportions.

Algorithm/Approach

The general approach to solve this problem involves the following steps:

Compute the sum of all eigenvalues
Calculate the explained variance ratio for each eigenvalue by dividing it by the sum of all eigenvalues
Calculate the cumulative explained variance ratio by taking the cumulative sum of the explained variance ratios

This approach involves basic arithmetic operations and cumulative sum calculations, which can be implemented using simple loops or vectorized operations.

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solution.py

Python 3.13

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