Factor Analysis and PCA Overview

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Questions and Answers

What is the primary purpose of Factor Analysis (FA)?

To achieve a higher rank parameterization of data

To specify a joint density model using fewer parameters (correct)

To create highly overlapping clusters

To increase the number of parameters in a model

Which of the following is a key characteristic of Partitional Clustering?

Objects can belong to multiple clusters

Each object belongs to exactly one cluster (correct)

Clusters are organized in a hierarchical structure

Clusters can be circular only

In which type of clustering are clusters organized in a tree structure?

Hierarchical clustering (correct)

Partitional clustering

Agglomerative clustering

Divisive clustering

Which of the following metrics is commonly used to measure similarity in clustering?

Euclidean distance Signup and view all the answers

What type of clustering allows for arbitrary-shaped clusters?

Both partitional and hierarchical clustering Signup and view all the answers

In relation to clustering, what does unidentifiability refer to?

The inability to distinguish between clusters Signup and view all the answers

Which is a common technique to choose the number of latent dimensions in PCA?

Elbow method Signup and view all the answers

How does hierarchical clustering differentiate itself from partitional clustering?

Hierarchical clustering forms nested clusters Signup and view all the answers

What is purity in the context of clustering algorithms?

A score that ranges between 0 (bad) and 1 (good) based on cluster accuracy. Signup and view all the answers

How is similarity measured for nominal variables in clustering?

By using a binary measure to check if values are equal. Signup and view all the answers

What does the Rand index evaluate in clustering methods?

The degree of overlap between two clustering solutions. Signup and view all the answers

In hierarchical clustering, which type is characterized by progressively merging clusters?

Agglomerative clustering. Signup and view all the answers

What stopping criterion is mentioned in the dissimilarity analysis steps?

When the difference between dissimilarity to two groups becomes negative. Signup and view all the answers

What is a significant challenge when using latent variable models?

They are harder to fit than models without latent variables. Signup and view all the answers

What is the primary benefit of using the EM algorithm with latent variable models?

It maximizes the incomplete-data log likelihood. Signup and view all the answers

How does the K-means algorithm differ from the EM algorithm in clustering?

K-means performs hard assignments, while EM uses soft assignments based on posterior probabilities. Signup and view all the answers

In latent variable modeling, what does the posterior distribution p(Z|X, θ) represent?

The state of knowledge of the latent variables given incomplete data. Signup and view all the answers

What condition must be met for the EM algorithm to be a valid procedure?

Data values must be missing at random. Signup and view all the answers

What issue do mixture models face with discrete latent variables?

They are restricted by one-hot encoding. Signup and view all the answers

What is a primary assumption of the EM algorithm regarding missing data?

The reason for missingness should not relate to unobserved values. Signup and view all the answers

What is a characteristic of mixtures of Gaussians in relation to latent variables?

They can capture complex data distributions using latent variables. Signup and view all the answers

What is the primary goal of image segmentation in the context of K-means clustering?

To partition an image into regions with homogeneous visual appearance. Signup and view all the answers

In K-means clustering, what role do the centers µ_k play?

They provide the RGB intensity values for each pixel in a segment. Signup and view all the answers

How does the choice of K affect the output of the K-means clustering algorithm in terms of data compression?

Smaller values of K lead to higher compression rates but poorer image quality. Signup and view all the answers

What does the Expectation-Maximization (EM) algorithm accomplish in the context of latent variables?

It allows estimation of latent variables by iterating over the data to increase likelihood. Signup and view all the answers

Which of the following best describes hierarchical clustering?

It builds clusters in a top-down or bottom-up approach without predefined K. Signup and view all the answers

What is measured to evaluate the output of clustering methods?

The dissimilarity between points within the same cluster. Signup and view all the answers

What is the purpose of using latent variables in mixtures of Gaussians?

To express more complex relationships between variables. Signup and view all the answers

Which clustering method allows for explicitly measuring dissimilarity between data points?

Both K-means and hierarchical clustering can measure dissimilarity. Signup and view all the answers

Study Notes

Factor Analysis

Factor Analysis (FA) is a statistical method that specifies a joint density model for data using fewer parameters.
It infers the latent factors representing unobserved variables that influence the observed variables.
There are challenges with unidentifiability in factor analysis, leading to multiple possible solutions.
FA employs latent variables to express complex marginal distributions over observed variables using tractable joint distributions.
The Expectation-Maximization (EM) algorithm is used to estimate the maximum likelihood estimator in this context.

Principal Components Analysis (PCA)

PCA is a dimensionality reduction technique that aims to identify the principal components of a dataset, which capture maximum variance.
It provides a linear transformation that projects the original data onto a lower-dimensional space.
The principal components are orthogonal and are ordered in descending order of variance explained.
The classical PCA theorem states that finding principal components is equivalent to maximizing the variance of the projected data.
PCA involves finding the eigenvectors and eigenvalues of the covariance matrix of the data.

Singular Value Decomposition (SVD)

SVD is used to perform PCA.
It decomposes a matrix into three matrices: U, Σ, and V where U and V are orthogonal matrices, and Σ is a diagonal matrix containing singular values.
The singular values represent the variance of the data along each principal component direction.

Choosing the Number of Latent Dimensions

Choosing the number of latent dimensions is a critical aspect of factor analysis and PCA, impacting model complexity and interpretability.
Common methods for choosing the number of latent dimensions include:
- Scree plot: Visualizes the variance explained by each component.
- Elbow method: Identifies a sharp decrease in variance explained, indicating an optimal cutoff.
- Information criteria: Use metrics like AIC or BIC to trade off model complexity and fit.

Clustering

Clustering is a technique for grouping data points such that points within a group are similar and dissimilar from points in other groups.
It falls under unsupervised learning, where no target variable is available.
The goal of clustering is to identify natural groupings or patterns within the data.
Cluster shapes can be arbitrary, not necessarily circular.
There are many clustering algorithms designed to detect arbitrary shapes.
Clustering evaluates similarity using metrics like Euclidean distance, Cosine similarity, and Manhattan distance.

Types of Clustering

Partitional clustering: Each data point belongs to only one cluster.
Hierarchical clustering: Creates a nested set of clusters organized in a tree structure.

K-Means Clustering

K-means clustering is a partitional clustering algorithm that aims to partition data into K clusters.
It uses an iterative process to assign data points to clusters based on their proximity to cluster centroids.
The goal is to minimize the sum of squared distances between data points and their assigned centroids.
It involves a two-step process:
- Assignment step: Assigning each data point to the nearest centroid.
- Update step: Updating the centroids based on the assigned data points.
The K-means algorithm works well for data with distinct clusters and can be accelerated using the Robbins-Monro procedure.
Potential challenges with K-means include:
- Depending on the initial centroid locations, the algorithm can converge to local optima.
- Sensitivity to outliers.

Applications of K-Means

Image segmentation: K-means clustering can be used to partition images into regions based on color or other pixel characteristics.
Data compression: Applications in vector quantization, where a smaller number of K centroids represent the entire dataset, leading to reduced storage.

Mixtures of Gaussians

Mixture of Gaussians is a probabilistic model used for clustering data.
It assumes that the data is generated from a mixture of Gaussian distributions, each representing a different cluster.
It is a flexible model than can handle clusters with different shapes and sizes.
The model involves latent variables, which represent the cluster membership probabilities for each data point.
The goal of the model is to estimate the parameters of the Gaussian distributions (mean, variance) and the mixture weights.

Maximum Likelihood in Mixtures of Gaussians

The maximum likelihood approach is used to estimate the model parameters by maximizing the likelihood of the observed data given the model.
The Expectation-Maximization (EM) algorithm is commonly used to find the maximum likelihood estimates.

EM for Gaussian Mixtures

The EM algorithm alternates between two steps:
- Expectation (E) step: Calculates the expected values of the latent variables given the current parameter estimates.
- Maximization (M) step: Updates the parameter estimates by maximizing the expected complete-data log likelihood.

Alternative View of EM

EM algorithm can be viewed as a way to maximize the incomplete-data log likelihood by iteratively improving the lower bound on the likelihood.
This perspective highlights the key steps:
- Finding a lower bound on the log likelihood using Jensen's inequality.
- Maximizing the lower bound in the M-step, which improves the lower bound and the log likelihood.

Relation to K-Means

K-means algorithm corresponds to a special case of EM for Gaussian mixtures.
Key differences:
- K-means uses hard assignment of data points to clusters, whereas EM uses a soft assignment based on posterior cluster membership probabilities.
- K-means assumes same variance for all clusters.
K-means is a faster but less flexible approach compared to EM for Gaussian mixtures.

Factor Analysis (Continued)

Factor analysis extends beyond mixture models by allowing for continuous and correlated latent variables.
It aims to explain a set of observed variables using a smaller number of underlying factors.
Correlation among observed variables arises from their dependence on the common underlying factors.
Similar to mixtures of Gaussians, factor analysis utilizes latent variables to model complex dependencies and simplify the observed data generation process.

Evaluating Clustering Output

Evaluating the quality of a clustering solution is essential for assessing its performance.
Common evaluation metrics include:
- Purity: Measures how well clusters correspond to known class labels.
- Rand index: Measures the agreement between the clustering and a known ground truth.
- Mutual information: Quantifies the shared information between the clustering and the ground truth.

Hierarchical Clustering

Hierarchical clustering creates a hierarchy of clusters, representing nested structures.
Two main approaches:
- Agglomerative clustering: Starts with each data point as a separate cluster and iteratively merges clusters until a single large cluster is formed.
- Divisive clustering: Starts with a single cluster containing all data points and iteratively divides the clusters until each data point is in its own cluster.

Agglomerative Clustering

Involves the following steps:
- Begin with each data point in a separate cluster.
- Iteratively merge the two closest clusters until a desired number of clusters is obtained.
- The distance between clusters can be defined using various linkage criteria:
  - Single link: Uses the minimum distance between points in two clusters.
  - Complete link: Uses the maximum distance between points in two clusters.
  - Average link: Uses the average distance between all pairs of points in the two clusters.

Divisive Clustering

Involves the following steps:
- Begin with a single cluster containing all data points.
- Iteratively split the cluster into two most dissimilar sub-clusters.
- This process continues until each data point is in its own cluster.
- The key challenge is to determine how to split the cluster based on some measure of dissimilarity.

Dissimilarity Analysis Steps

1. Compute a dissimilarity matrix between all pairs of objects.
1. Create a graph G with vertices representing objects and edges representing object relationships
1. Construct a minimum spanning tree whose edges are the most dissimilar ones.
1. Iteratively move objects between the clusters until a stopping criterion is met.
1. Stop the process when the difference in dissimilarity between the two clusters, G and H, becomes negative.

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Description

Explore the concepts of Factor Analysis and Principal Components Analysis in this quiz. Understand how these statistical methods function in reducing dimensionality and uncovering latent variables. Test your knowledge of key techniques such as the Expectation-Maximization algorithm and principal component extraction.