Understanding Principal Component Analysis from Scratch

 In today’s world, overflowing with vast amounts of data, there is a growing need for methods to extract essential patterns.

Principal Component Analysis (PCA) is a powerful technique that summarizes complex, high-dimensional data into fewer dimensions, contributing to data organization, visualization, and even noise reduction.

Here, we will explain what principal component analysis is, the situations in which it is used, and the benefits of learning this technique.

1. What is Principal Component Analysis?

Principal Component Analysis is one method of dimensionality reduction. It efficiently represents data by replacing multiple correlated variables with a smaller number of new variables (principal components). 

- Visualizing Data Structure

By projecting high-dimensional data into lower-dimensional spaces (such as 2D or 3D), patterns and correlations hidden within the data become visually easier to understand.

- Feature Extraction and Noise Reduction

By focusing on the major factors of variation (principal components), redundant information and noise are removed, leading to improved accuracy in subsequent analysis and machine learning models.

- Mathematical Background

PCA is based on calculating the eigenvalues and eigenvectors of the data's covariance matrix. The directions with the largest eigenvalues (i.e., the directions with the greatest variation) are selected as the principal components. Data standardization is also a prerequisite; ensuring that each feature's scale is consistent makes it easier to extract true information. 

As such, PCA is not simply a tool for dimensionality reduction but a powerful method for revealing the essential structure of data, and it is used in many fields.

2. In What Situations is it Used?

Due to its flexibility and usefulness, principal component analysis is practiced in a wide range of fields and scenarios. Specific examples include:

- Marketing and Customer Analysis

By aggregating diverse data such as customer purchasing behavior, attributes, and survey results into lower dimensions, characteristics of each segment and potential purchasing patterns can be identified, which is then used to design target strategies.

- Image Processing and Computer Vision

Image data is very high-dimensional, but principal component analysis can extract the main features of the images (e.g., elements for face recognition) and is used as pre-processing for pattern recognition and classification.

- Finance and Economic Data

It is used to analyze multiple economic indicators and market data, identify key risk factors and market fluctuation patterns, and is effective as a method for portfolio management and risk assessment.

- Bioinformatics and the Medical Field

It extracts key points from high-dimensional and complex data such as gene expression data, medical images, and clinical test data, and is utilized as a means of classifying diseases and assisting in diagnosis.

- IoT and Sensor Data Analysis

By extracting the major variation patterns from multiple sensor data, practical applications such as anomaly detection and condition monitoring can be developed.

As these examples demonstrate, PCA serves as a foundation for “simplifying the complexity of data” and demonstrates its power in a wide variety of fields.

3. What are the Benefits of Learning It?

Learning principal component analysis is greatly beneficial in building an important skillset for data science and machine learning. The benefits include:

- Deepening Data Understanding

It fosters a perspective for capturing the essence of high-dimensional data, allowing you to quantitatively determine which variables contribute the most to data variation. This streamlines pre-analysis and feature selection.

- Saving and Improving Computational Resources

By reducing the dimensionality, the computational burden on subsequent machine learning models is reduced, leading to shorter learning times and memory savings. It also contributes to reducing the risk of overfitting the model.

- Promoting Visual Data Analysis

By projecting data into 2D or 3D, you can intuitively confirm the results of clustering and pattern recognition, making it easier to find trends in the data and potential group structures. 

- Improving Applicability

Understanding the principles of PCA provides a bridge to other advanced techniques (e.g., factor analysis, independent component analysis, dimensionality reduction techniques in deep learning) and fosters a foundation for tackling more complex data analysis projects.

- Practicality in Many Fields

Principal component analysis is utilized in various fields such as business, science, engineering, and medicine, so the knowledge and skills you learn will be directly applicable to many real-world scenarios and will be a significant asset to your career.

In Summary

Principal Component Analysis (PCA) is a very useful technique that extracts essential information from complex, high-dimensional data and contributes to visualization, noise reduction, and model efficiency.

Understanding PCA from the basics to the applications is an important step in broadening the scope of data science. First, try implementing PCA with a dataset and experiencing its effects. This will surely improve your analytical skills by discovering the essential patterns behind the data.

If you want to learn principal component analysis, we recommend this book (access here).