Principal Component Analysis (PCA): A Guide to Dimensionality Reduction

Question 1

Which is the primary objective of PCA?

Accepted Answer

To identify linear combinations of features that explain maximum variance

Answer

To eliminate redundant or correlated features

Answer

To transform non-linear data into linear data

Answer

To cluster data points into meaningful groups

Question 2

What is the computational complexity of PCA?

Accepted Answer

O(n^3)

Answer

O(n^2)

Answer

O(n^4)

Answer

O(n^5)

Question 3

How does Principal Component Analysis (PCA) affect the understandability of data?

Accepted Answer

Enhances interpretability by reducing features while capturing key variations.

Answer

Reduces interpretability by introducing complex components.

Answer

No impact on interpretability, as it only transforms data without altering its meaning.

Answer

Improves interpretability by eliminating noise and outliers.

Question 4

How does PCA reduce data redundancy, and why is this beneficial in practical applications?

Accepted Answer

PCA transforms data into a new set of uncorrelated variables, reducing the number of features needed to represent the data effectively.

Answer

PCA identifies and eliminates duplicate data points, reducing redundancy.

Answer

PCA clusters similar data points, eliminating redundant information.

Answer

PCA removes outliers, which are often redundant and can distort analysis.

Question 5

Which of the following is NOT a benefit of utilizing Principal Component Analysis (PCA)?

Accepted Answer

Increased risk of overfitting

Answer

Diminished computational time

Answer

Improved data visualization

Question 6

Which of the following assumptions must be met for successful application of Principal Component Analysis (PCA)?

Accepted Answer

Data are linearly related

Answer

Data have an equal number of observations for each feature

Answer

Data are purely categorical

Answer

Data are normally distributed

Question 7

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

Accepted Answer

To reduce dimensionality while preserving maximum variance

Answer

To cluster data into distinct groups

Answer

To increase dimensionality to better visualize data

Question 8

Which of the following is a crucial step in implementing PCA?

Accepted Answer

Eigenvalue decomposition

Answer

Linear regression

Answer

K-means clustering

Answer

Forward feature selection

Question 9

What do eigenvalues represent in PCA?

Accepted Answer

Variance explained by each principal component

Answer

Number of clusters in the data

Answer

Centroid of each cluster

Answer

Importance of each feature in the data

Question 10

How does PCA relate to linear regression?

Accepted Answer

PCA can be used to reduce dimensionality before performing linear regression, improving model interpretability

Answer

Linear regression cannot be applied to data after PCA

Answer

PCA is a type of linear regression

Question 11

What is a key advantage of using PCA?

Accepted Answer

Improved model interpretability by reducing the number of features

Answer

Prevents overfitting

Answer

Always increases model accuracy

Answer

Eliminates outliers

Question 12

What fundamental difference distinguishes PCA from Factor Analysis?

Accepted Answer

PCA assumes observed variables are continuous, while Factor Analysis allows for both continuous and categorical variables

Answer

PCA uses eigenvalues, while Factor Analysis uses eigenvectors

Question 13

Which of the following is the fundamental goal of Principal Component Analysis (PCA)?

Accepted Answer

To reduce data dimensionality while retaining as much of its variance as possible.

Answer

To identify exceptions within a dataset.

Answer

To categorize data points into distinct groupings.

Question 14

What mathematical concept serves as the theoretical foundation of PCA?

Accepted Answer

Singular value decomposition (SVD) or eigenvalue decomposition.

Answer

K-means clustering.

Answer

Linear regression.

Question 15

In a dataset with 'n' features, what is the highest possible number of principal components that can be extracted?

Accepted Answer

n

Answer

n + 1

Answer

n - 1

Answer

n / 2

Question 16

Which of the following factors is taken into account when deciding the optimal number of principal components to preserve?

Accepted Answer

Explained variance or eigenvalue thresholds.

Answer

R-squared value.

Answer

The number of data points in the dataset.

Answer

AIC or BIC criteria.

Question 17

What is the impact of data scaling before conducting PCA?

Accepted Answer

It ensures that all features have comparable scales and equally contribute to the analysis.

Answer

It eliminates the mean of the data.

Answer

It transforms the data to conform to a normal distribution.

Question 18

What distinguishes PCA from other dimensionality reduction techniques?

Accepted Answer

It is an orthogonal transformation, preserving the relationships among the original features.

Answer

Greater computational efficiency.

Answer

Applicability to non-linear data.

Question 19

How does PCA relate to linear regression?

Accepted Answer

PCA can be used to reduce the dimensionality of input features for linear regression, potentially improving model performance.

Answer

PCA is a type of linear regression.

Question 20

In PCA, what does "variance explained" by a principal component represent?

Accepted Answer

The proportion of the total variance in the original data that is captured by that component.

Answer

The difference between the largest and smallest eigenvalues.

Answer

The number of observations in the dataset.

Question 21

Which of the following is NOT a benefit of using Principal Component Analysis (PCA)?

Accepted Answer

Increased computational complexity

Answer

Improved data visualization

Answer

Reduced dimensionality

Question 22

What is the primary purpose of the covariance matrix in PCA?

Accepted Answer

To capture the variance and covariance relationships between features

Answer

To identify the most important features

Answer

To reduce the number of features

Question 23

Which of the following is a key underlying assumption of PCA?

Accepted Answer

Linear relationships between features

Answer

Equal variance of all features

Answer

Gaussian distribution of data

Answer

Absence of outliers

Question 24

Which of the following is a common practical application of PCA?

Accepted Answer

Image compression

Answer

Regression analysis

Answer

Decision tree induction

Answer

Clustering

Question 25

Which data standardization technique is commonly used before applying PCA?

Accepted Answer

Z-score normalization

Answer

Log transformation

Answer

Mean normalization

Answer

Min-max normalization

Question 26

How does PCA affect the variance of the transformed data?

Accepted Answer

Variance is preserved along the principal components

Answer

Variance is increased along the first few components

Answer

Variance is reduced along all components

Question 27

Which algorithm is commonly used to perform PCA?

Accepted Answer

Singular Value Decomposition (SVD)

Answer

K-means clustering

Answer

Logistic regression

Answer

Naive Bayes

Question 28

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

Accepted Answer

Reduce data dimensionality while preserving maximum variance.

Answer

Cluster data points based on their similarity.

Answer

Predict continuous target variables from given features.

Question 29

In PCA, what do eigenvectors represent?

Accepted Answer

Directions of greatest variance in the data.

Answer

Weights assigned to each principal component.

Answer

Mean of data points along each dimension.

Answer

Correlation coefficients between feature pairs.

Question 30

How are principal components ordered?

Accepted Answer

Decreasing order of variance explained.

Answer

Increasing order of variance explained.

Answer

Randomly, as order is inconsequential.

Question 31

Which of the following is NOT a typical application of PCA?

Accepted Answer

Predicting the value of a dependent variable.

Answer

Reducing noise in data.

Answer

Visualizing high-dimensional data.

Answer

Feature extraction for machine learning models.

Question 32

What is a fundamental use of a scree plot in PCA?

Accepted Answer

Determining the optimal number of principal components to retain.

Answer

Identifying outliers in the data.

Answer

Visualizing correlation between original features.

Question 33

Why is it important to standardize data before performing PCA?

Accepted Answer

Prevent features with larger scales from dominating the analysis.

Answer

Ensure data is normally distributed.

Answer

Reduce computational cost of the PCA algorithm.

Question 34

Which scenario presents a potential challenge for PCA?

Accepted Answer

Data with highly non-linear relationships between features.

Answer

Data with a large number of samples.

Answer

Data with missing values (appropriately handled).

Question 35

How can PCA be integrated into a machine learning pipeline?

Accepted Answer

As a dimensionality reduction technique before model training.

Answer

To replace feature selection methods entirely.

Answer

As a standalone classification algorithm.

Question 36

What is the connection between PCA and Singular Value Decomposition (SVD)?

Accepted Answer

PCA can be implemented using SVD.

Answer

PCA and SVD are unrelated techniques.

Answer

SVD is a preprocessing step for PCA.

Answer

SVD is a more computationally expensive version of PCA.

Question 37

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

Accepted Answer

To reduce data dimensionality while retaining maximum variance

Answer

To perform feature selection based on information gain

Answer

To identify outliers and noisy data points

Question 38

Which of the following is a potential limitation of PCA?

Accepted Answer

It assumes linear relationships between features

Answer

It can handle missing data without imputation

Answer

It guarantees that the transformed data will be normally distributed

Answer

It is suitable for both numerical and categorical data

Question 39

What is the purpose of the covariance matrix in PCA?

Accepted Answer

To capture the correlations and variances within the data

Answer

To calculate the mean of the data

Answer

To normalize the features

Question 40

Which of the following is NOT an objective of Principal Component Analysis (PCA)?

Accepted Answer

To increase the dimensionality of data

Answer

To reduce the noise in the data

Answer

To identify the most important features in the data

Answer

To make the data more interpretable

Question 41

What is the mathematical process involved in PCA?

Accepted Answer

Eigenvalue decomposition

Answer

Singular value decomposition

Answer

Cluster analysis

Answer

Matrix factorization

Question 42

Which of the following is a measure of the variance captured by a principal component?

Accepted Answer

Eigenvalue

Answer

Correlation

Answer

Covariance

Answer

Standard deviation

Question 43

How many principal components are typically extracted in PCA?

Accepted Answer

As many as there are original features

Answer

The number of features with positive eigenvalues

Answer

Half the number of original features

Answer

A fixed number determined by the user

Question 44

What is a scree plot used for in PCA?

Accepted Answer

Determining the optimal number of principal components to retain

Answer

Identifying outliers

Answer

Visualizing the distribution of data points

Question 45

Which of the following is a limitation of PCA?

Accepted Answer

It assumes linear relationships between features

Answer

It can be computationally expensive for large datasets

Answer

It can only be applied to numeric data

Question 46

What is the difference between PCA and Factor Analysis?

Accepted Answer

PCA identifies features with the highest variance, while Factor Analysis identifies features with the highest common variance

Answer

PCA is used for data reduction, while Factor Analysis is used for data exploration

Question 47

What is the computational complexity of PCA?

Accepted Answer

O(n^3)

Answer

O(log n)

Answer

O(n^2)

Answer

O(n)

Question 48

Which of the following is NOT a benefit of using PCA?

Accepted Answer

It can increase the accuracy of machine learning models

Answer

It can make the data more interpretable

Answer

It can speed up the training of machine learning models