pca test questions and answers

Lemon

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31-01-2025

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Principal Component Analysis (PCA) is a powerful dimensionality reduction technique used extensively in data science and machine learning. Understanding PCA is crucial for anyone working with high-dimensional datasets. This guide provides a range of PCA test questions and answers, covering fundamental concepts to more advanced applications. We'll explore both theoretical underpinnings and practical considerations.

Fundamental Concepts of PCA

Q1: What is Principal Component Analysis (PCA)?

A1: PCA is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. The first principal component accounts for the largest possible variance in the data, the second component accounts for the largest possible remaining variance, and so on. Essentially, PCA aims to reduce the dimensionality of the data while preserving as much information as possible.

Q2: What are the main assumptions of PCA?

A2: PCA assumes that:

• Linearity: The relationships between variables are linear. Nonlinear relationships may require transformations before applying PCA.
• Data Scaling: Variables should be standardized (e.g., z-score normalization) to prevent variables with larger scales from dominating the analysis.
• No Outliers: Outliers can significantly influence the principal components. Identifying and handling outliers is crucial.
• Sufficient Sample Size: A sufficiently large sample size is needed for reliable PCA results.

Q3: Explain the difference between eigenvalues and eigenvectors in the context of PCA.

A3: Eigenvectors represent the directions of the principal components, indicating the orientation of the maximum variance in the data. Eigenvalues represent the magnitude of the variance explained by each corresponding eigenvector (principal component). Larger eigenvalues correspond to principal components that capture more variance in the data.

Intermediate PCA Concepts and Applications

Q4: How do you determine the optimal number of principal components to retain?

A4: Several methods exist:

• Scree Plot: A scree plot graphs the eigenvalues in descending order. The "elbow" point in the plot suggests the optimal number of components to retain, as it represents the point where adding more components yields diminishing returns in variance explained.
• Variance Explained: Calculate the cumulative variance explained by the principal components. Retain enough components to explain a sufficient percentage of the total variance (e.g., 95%).
• Kaiser Criterion: Retain components with eigenvalues greater than 1.

Q5: What is the role of standardization (or normalization) in PCA?

A5: Standardization ensures that all variables contribute equally to the analysis, preventing variables with larger scales from disproportionately influencing the principal components. If variables are not standardized, the PCA results will be biased towards variables with larger ranges.

Q6: How can PCA be used for feature selection?

A6: PCA can indirectly aid in feature selection. By identifying the principal components that explain most of the variance, you can examine the loadings (contributions) of the original variables to these components. Variables with high loadings on important principal components are considered more relevant and can be retained, while variables with low loadings on all principal components may be less important and potentially removed.

Advanced PCA Topics

Q7: What are some limitations of PCA?

A7:

• Linearity Assumption: PCA struggles with nonlinear relationships between variables.
• Sensitivity to Outliers: Outliers can strongly influence the principal components.
• Interpretability: Principal components may be difficult to interpret, especially in high-dimensional datasets.
• Data Loss: While PCA aims to minimize information loss, some information is always lost when reducing dimensionality.

Q8: How does PCA differ from other dimensionality reduction techniques like t-SNE or UMAP?

A8: PCA is a linear dimensionality reduction technique, while t-SNE and UMAP are nonlinear. PCA focuses on preserving variance, whereas t-SNE and UMAP prioritize preserving local neighborhood structures in the data. t-SNE and UMAP are often better for visualization, while PCA is more suitable for situations where interpretability and computational efficiency are crucial.

Q9: Describe a real-world application of PCA.

A9: PCA has numerous applications, including:

• Image compression: Reducing the dimensionality of image data to store and transmit images more efficiently.
• Face recognition: Extracting relevant features from facial images for identification.
• Gene expression analysis: Reducing the dimensionality of gene expression data to identify patterns and clusters of genes.
• Financial modeling: Reducing the number of variables in portfolio optimization and risk management.

This guide provides a foundation for understanding PCA. Further exploration through practical application and advanced literature will enhance your mastery of this powerful technique. Remember to always consider the context of your data and the specific goals of your analysis when applying PCA.