Multivariate Analysis: PCA Overview

Questions and Answers

What is the primary goal of Principal Component Analysis (PCA)?

Reduce the dimensionality while preserving variability (correct)

Transform data into non-linear components

Increase noise and redundancy in data

Maximize the dimensionality of a dataset

Which of the following steps is NOT part of the PCA process?

Standardization of data

Computing the variance of original variables

Projecting data onto non-linear dimensions (correct)

Calculating the covariance matrix

What do eigenvalues represent in the context of PCA?

The variance captured by each principal component (correct)

The relationships between different datasets

The original variables in the dataset

The size of the data matrix

Which of the following statements about PCA is true?

PCA transforms original variables into uncorrelated principal components Signup and view all the answers

What is a limitation of PCA?

It assumes linear relationships among variables Signup and view all the answers

In which field is PCA commonly applied for analyzing high-dimensional data?

Image processing Signup and view all the answers

What role do eigenvectors play in PCA?

They define the direction of new component axes Signup and view all the answers

Which of the following is a major advantage of using PCA?

It reduces the number of variables while preserving key information Signup and view all the answers

Study Notes

Multivariate Analysis: Principal Component Analysis (PCA)

Definition

Principal Component Analysis (PCA) is a statistical technique used to reduce the dimensionality of a dataset while preserving as much variability as possible.

Objectives

Simplify data interpretation by reducing the number of variables.
Identify patterns in data and highlight similarities and differences.
Enhance visualization of complex datasets.

Key Concepts

Variables: PCA transforms original variables into new uncorrelated variables called principal components.
Principal Components: These are linear combinations of the original variables, ordered by the amount of variance they explain.
Variance: PCA focuses on maximizing the variance captured in fewer dimensions.

Process

Standardization: Scale the data, especially if variables are measured on different scales.
Covariance Matrix: Compute the covariance matrix to understand relationships between variables.
Eigenvalues and Eigenvectors:
- Eigenvalues indicate the amount of variance captured by each principal component.
- Eigenvectors define the direction of the new component axes.
Selecting Components: Choose a subset of principal components based on eigenvalues (typically those with higher values).
Transformation: Project the original data onto the selected principal components to obtain a reduced representation.

Advantages

Reduces noise and redundancy in data.
Facilitates visualization in 2D or 3D plots.
Helps in identifying key variables that contribute most to variance.

Limitations

PCA is sensitive to outliers, which can distort results.
Assumes linear relationships; may not capture complex patterns effectively.
Interpretation of principal components can be challenging, as they are combinations of original variables.

Applications

Image processing for reducing image dimensions.
Genomics for analyzing high-dimensional biological data.
Market research for identifying customer segments based on multiple attributes.