Principal Component Analysis Quiz

What are eigenvalues and eigenvectors in the context of Principal Component Analysis?

• Eigenvalues and eigenvectors are the mathematical constructs used to find the principal components of a dataset in PCA.

• Eigenvalues and eigenvectors are not relevant in the context of Principal Component Analysis.

• Eigenvalues and eigenvectors are the names of the two main components in PCA.

• Eigenvalues and eigenvectors are used to calculate the mean and standard deviation of a dataset in PCA.

How are eigenvalues and eigenvectors used in dimensionality reduction?

• They are used to calculate the mean of the data

• They are used to identify outliers in the data

• They are used to determine the range of the data

• They are used to find the principal components that capture the most variance in the data.

Explain the role of the covariance matrix in Principal Component Analysis.

• It is used to find the mode of the data

• It is used to calculate the mean of the data

• It helps in finding the direction of maximum variance in the data.

• It helps in determining the outliers in the data

Answer explanation

Covariance measures how two variables change together. In the context of PCA, it refers to the measure of the joint variability of two random variables. Specifically, when performing PCA on a dataset, the covariance matrix is calculated. The elements of this matrix represent the covariances between different pairs of features in the data.

PCA aims to find a new set of orthogonal (uncorrelated) variables, called principal components, that capture the maximum variance in the data. The principal components are linear combinations of the original features, and they are determined by the eigenvectors of the covariance matrix.

How does dimensionality reduction help in simplifying data analysis?

• By reducing the number of features or variables in the dataset

• By adding more variables to the dataset

• By making the data more complex

• By increasing the number of features in the dataset

What is an orthogonal transformation and how is it related to Principal Component Analysis?

• An orthogonal transformation only preserves the angles between vectors but not the length, and it is not related to PCA.

• An orthogonal transformation preserves the length of vectors and the angles between them, and it is related to PCA as PCA uses orthogonal transformations to transform the original data into a new set of uncorrelated variables called principal components.

• An orthogonal transformation only preserves the length of vectors but not the angles between them, and it is not related to PCA.

• An orthogonal transformation changes the length of vectors and the angles between them, and it is not related to PCA at all.

Discuss the concept of variance explained in the context of Principal Component Analysis.

• The proportion of the total variance in the data that is accounted for by each principal component

• The amount of error in the data

• The correlation between variables in the data

• The number of principal components in the analysis

What is the significance of a scree plot in Principal Component Analysis?

• It is used to identify outliers in the data

• It helps in determining the correlation between variables

• It is used to calculate the mean of the principal components

• It helps in determining the number of principal components to retain

How does Principal Component Analysis help in reducing the dimensionality of a dataset?

• By increasing the dimensionality of the dataset

• By adding more variables to the dataset

• By shuffling the order of the variables

• By transforming the original variables into a new set of uncorrelated variables