explained_variance_ vs explained_variance_ratio_

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19-10-2024

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Explained Variance vs. Explained Variance Ratio: Decoding PCA's Power

Principal Component Analysis (PCA) is a powerful dimensionality reduction technique that helps us understand the underlying structure of complex datasets. Two key metrics, explained variance and explained variance ratio, play a crucial role in interpreting PCA results. But what exactly do they tell us, and how do they differ? Let's delve deeper.

What is Explained Variance?

Imagine you have a dataset with multiple features, each representing a different aspect of your data. PCA aims to find a set of new features (principal components) that capture the most variance in the original data. Each principal component represents a direction of maximum variance in the dataset.

Explained variance quantifies the amount of variance captured by each principal component. It essentially tells you how much of the total variation in the original data is explained by that particular component.

Example: Let's say we have a dataset with features like height, weight, and age. After performing PCA, we get three principal components. If the first principal component has an explained variance of 70%, it means that 70% of the total variation in the original data is captured by this component.

What is Explained Variance Ratio?

While explained variance gives us the absolute amount of variance captured by each component, explained variance ratio provides a more interpretable perspective. It represents the proportion of the total variance explained by each component.

Example: Continuing the previous example, if the first principal component has an explained variance of 70% and the total variance in the dataset is 100%, then the explained variance ratio for the first component would be 0.7 (70/100).

Why Use Explained Variance Ratio?

Explained variance ratio is crucial for several reasons:

• Relative Importance: It allows us to compare the importance of different principal components. A higher ratio indicates a more influential component in capturing the data's structure.
• Dimensionality Reduction: We can decide how many principal components to retain based on the explained variance ratio. Generally, we aim to keep components with a cumulative explained variance ratio exceeding a certain threshold (often 95% or 99%).
• Insights: The variance ratio provides valuable insights into the relative importance of different features in the original data. For example, a high variance ratio for a particular component suggests that the corresponding feature plays a significant role in explaining the data's variation.

Finding the Explained Variance and Variance Ratio in Python:

import pandas as pd
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

# Load your dataset
data = pd.read_csv("your_dataset.csv")

# Scale the data
scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)

# Perform PCA
pca = PCA()
pca.fit(scaled_data)

# Get explained variance and variance ratio
explained_variance = pca.explained_variance_
explained_variance_ratio = pca.explained_variance_ratio_

Conclusion:

Explained variance and explained variance ratio are crucial metrics for interpreting PCA results. While explained variance quantifies the absolute variance captured, explained variance ratio provides a relative perspective, highlighting the importance of each component. By analyzing these metrics, we gain insights into the underlying structure of our data and can effectively utilize PCA for dimensionality reduction and feature extraction.

Credit: This article is inspired by insights from the following GitHub repository: [link to the repository]

This article aims to provide a comprehensive understanding of explained variance and explained variance ratio, offering practical insights and Python code examples. By leveraging this knowledge, you can effectively analyze PCA results and unlock the power of this dimensionality reduction technique for your data analysis tasks.