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Questions and Answers
What is a primary reason for conducting Principal Component Analysis (PCA)?
What is a primary reason for conducting Principal Component Analysis (PCA)?
Which statement accurately describes the relationship between maximizing variance and minimizing reconstruction error in PCA?
Which statement accurately describes the relationship between maximizing variance and minimizing reconstruction error in PCA?
What is the role of the covariance matrix in PCA?
What is the role of the covariance matrix in PCA?
When using PCA, how is the weight vector 'w' selected?
When using PCA, how is the weight vector 'w' selected?
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What is the primary limitation of K-means clustering that may affect the choice of the number of clusters?
What is the primary limitation of K-means clustering that may affect the choice of the number of clusters?
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Which technique could be considered an alternative to K-means clustering?
Which technique could be considered an alternative to K-means clustering?
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In PCA, if the eigenvector with the largest eigenvalue is chosen, what does this vector represent?
In PCA, if the eigenvector with the largest eigenvalue is chosen, what does this vector represent?
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Why is it important to center all features before conducting PCA?
Why is it important to center all features before conducting PCA?
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What is the preferred NumPy function for computing eigenvalues and eigenvectors of a symmetric matrix due to its numerical stability?
What is the preferred NumPy function for computing eigenvalues and eigenvectors of a symmetric matrix due to its numerical stability?
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In the context of N-dimensional data, how many principal components (PCs) are there available to capture variance?
In the context of N-dimensional data, how many principal components (PCs) are there available to capture variance?
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Why is it important to center the data when performing PCA?
Why is it important to center the data when performing PCA?
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What will the covariance matrix become when expressed in the eigenvector basis?
What will the covariance matrix become when expressed in the eigenvector basis?
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What is a common rule of thumb for selecting the number of principal components in PCA?
What is a common rule of thumb for selecting the number of principal components in PCA?
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What is a significant drawback of the K-means clustering algorithm?
What is a significant drawback of the K-means clustering algorithm?
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What is a key characteristic of the covariance matrix for three dimensions, specifically regarding its diagonal?
What is a key characteristic of the covariance matrix for three dimensions, specifically regarding its diagonal?
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Which method is commonly used to determine the optimal number of clusters (K) in K-means clustering?
Which method is commonly used to determine the optimal number of clusters (K) in K-means clustering?
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What must be considered when standardizing features in PCA?
What must be considered when standardizing features in PCA?
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If the eigenvalues of a covariance matrix are $S^2 / N$, what mathematical decomposition does this represent?
If the eigenvalues of a covariance matrix are $S^2 / N$, what mathematical decomposition does this represent?
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Which of the following clustering algorithms can effectively handle non-spherical cluster shapes?
Which of the following clustering algorithms can effectively handle non-spherical cluster shapes?
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What is one of the primary techniques of dimensionality reduction?
What is one of the primary techniques of dimensionality reduction?
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What effect does choosing a very small epsilon value have when using DBSCAN?
What effect does choosing a very small epsilon value have when using DBSCAN?
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Which statement is true regarding the characteristics of K-means clustering?
Which statement is true regarding the characteristics of K-means clustering?
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What is a primary focus of dimensionality reduction techniques?
What is a primary focus of dimensionality reduction techniques?
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Which clustering algorithm creates a hierarchy tree to demonstrate relationships within a dataset?
Which clustering algorithm creates a hierarchy tree to demonstrate relationships within a dataset?
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Study Notes
Data Exploration
- Data exploratory provides insights
- Helps save computation and memory
- Can be used as preprocessing step to reduce overfitting
- Data visualization in 2-3 dimensions is possible
Principle Component Analysis (PCA)
- It is a linear dimensionality reduction technique
- Projects data onto a lower dimensional space
- Turns X into Xw, where w is a unit vector
- Aims to maximize variance and minimize reconstruction error
- PCA achieves both objectives simultaneously
PCA vs. Regression
- PCA aims to minimize projection error, while regression aims to minimize the residual error between actual values and predicted values
PCA Cost Function
- Maximizes the variance of projected data
- The variance is represented as 𝑤𝑤 𝑇𝑇 Cw, where C is the covariance matrix and w is the direction of projection
- The objective is to find the direction w that maximizes 𝑤𝑤 𝑇𝑇 Cw with the constraint ||w|| = 1
Maximizing 𝑤𝑤 𝑇𝑇 C w
- Solved using Lagrange multiplier method
- Introduces a Lagrange multiplier 𝜆𝜆 to enforce the constraint
- Solving 𝜕𝜕𝐽𝐽 / 𝜕𝜕𝑤𝑤 = 0 leads to Cw = 𝜆𝜆𝑤𝑤
- Thus, w is an eigenvector of the covariance matrix C, and 𝜆𝜆 is the corresponding eigenvalue
- To maximize 𝑤𝑤 𝑇𝑇 Cw, the eigenvector with the largest eigenvalue should be chosen
Implementing PCA with NumPy
- Two main functions are used:
- numpy.linalg.eig()
- numpy.linalg.eigh() (preferred for symmetric matrices due to its numerical stability and efficiency)
Covariance Matrix
- Represents covariance between dimensions as a matrix
- Diagonal elements represent the variances of each dimension
- Off-diagonal elements represent covariances between dimensions
- Covariance matrix is symmetric about the diagonal
- N-dimensional data results in an NxN covariance matrix
Spectral Theorem
- A symmetric n x n matrix has n orthogonal eigenvectors
- Projections onto eigenvectors are uncorrelated
- This makes the covariance matrix diagonal in the eigenvector basis
Total Variance of N-dimensional Data
- There are N principal components (PCs) for an N-dimensional dataset
- Each PC captures a portion of the total variance in the dataset
- PC1 captures the largest variance
- Subsequent PCs capture decreasing amounts of variance
Relationship to SVD
- The singular value decomposition (SVD) of X = US𝑉𝑉 𝑇𝑇 can be used to compute the covariance matrix
- 𝐶𝐶 = VS𝑆 2 𝑉𝑉 𝑇𝑇, which is the eigen decomposition of the covariance matrix
- This holds true only if X is centered
Feature Scaling in PCA
- Feature scaling is crucial when features are on different scales
- Standardizing features makes C the correlation matrix
Picking the Number of Components
- Rules of thumb for selecting the number of PCs:
- Look for an "elbow" in the scree plot
- Capture a specified percentage (e.g., 90%) of the total variance
- Assess the explained variance ratio for each PC
Feature Selection
- PCA is a feature transformation method, not feature selection.
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Description
This quiz delves into key concepts of data exploration and Principal Component Analysis (PCA). You will learn how PCA serves as a dimensionality reduction technique and the differences between PCA and regression. Test your understanding of cost functions and maximizing projections in PCA.
