K-Means Clustering Concepts
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What is the time complexity of the k-means algorithm?

  • O(t^2k)
  • O(tkn) (correct)
  • O(kn^2)
  • O(k + n)

    • Which is NOT a weakness of the k-means algorithm?

  • Sensitive to initial seeds
  • Requires the user to specify k
  • Can only cluster numerical data (correct)
  • Sensitive to outliers

    • What is a common method to handle outliers in k-means clustering?

  • Remove distant data points (correct)
  • Increase the number of clusters
  • Use a different clustering algorithm
  • Expand the dataset size

    • Which of the following describes k-means as an algorithm?

    <p>It is considered a linear algorithm.</p> Signup and view all the answers

    What happens when k-means clustering is applied to categorical data?

    <p>A different algorithm must be used.</p> Signup and view all the answers

    What does the term 'SSE' refer to in the context of k-means?

    <p>Sum of Squared Errors</p> Signup and view all the answers

    Why is k-means sensitive to initial seeds?

    <p>It may lead to different convergence points.</p> Signup and view all the answers

    Which of the following is a strength of k-means clustering?

    <p>It is efficient with a small k and t.</p> Signup and view all the answers

    What is the primary focus of the single link method in clustering?

    <p>Distance between the two closest data points in two clusters</p> Signup and view all the answers

    Which of the following statements about complete link clustering is true?

    <p>It is sensitive to outliers due to its use of the furthest points.</p> Signup and view all the answers

    What is a potential drawback of using the single link method?

    <p>It can lead to long trailing clusters due to noisy data points.</p> Signup and view all the answers

    How does the average link method differ from complete link clustering?

    <p>It computes the average distance from all pairwise data points.</p> Signup and view all the answers

    What does the centroid method rely on for measuring the distance between two clusters?

    <p>The distance between the centroids of the clusters.</p> Signup and view all the answers

    What is a common characteristic of clusters formed by average and complete linkage methods?

    <p>They tend to yield more balanced clusters.</p> Signup and view all the answers

    What role do distance functions play in clustering?

    <p>They are key to defining the relationships between clusters.</p> Signup and view all the answers

    Which clustering method is likely to result in clusters that reflect a more compact and spherical shape?

    <p>Complete link</p> Signup and view all the answers

    What does the loading vector φ1 represent in PCA?

    <p>The direction where data has the most variance</p> Signup and view all the answers

    How does the second principal component Z2 relate to the first principal component Z1?

    <p>Z2 is uncorrelated and orthogonal to Z1.</p> Signup and view all the answers

    What method can be used to solve for the first principal component loading vector?

    <p>Eigen decomposition</p> Signup and view all the answers

    In PCA, the total number of principal components is limited to which of the following?

    <p>min(n - 1, p)</p> Signup and view all the answers

    What do the projected values of the principal component scores represent?

    <p>Data points projected onto the direction defined by φ1</p> Signup and view all the answers

    How are the variances of the principal components related to singular values?

    <p>They are proportional to the squares of the singular values</p> Signup and view all the answers

    Which process constrains the direction φ2 in PCA?

    <p>Ensuring orthogonality to the direction φ1</p> Signup and view all the answers

    Which dataset contains the number of arrests per 100,000 residents in the USA for several crimes?

    <p>USAarrests</p> Signup and view all the answers

    What is one significant limitation of the k-means algorithm?

    <p>It is not suitable for discovering clusters that are not hyper-ellipsoids.</p> Signup and view all the answers

    Why is the k-means algorithm still widely used despite its weaknesses?

    <p>It is simple, efficient, and performs well on various data types.</p> Signup and view all the answers

    In the context of cluster representation, why might centroids be inadequate?

    <p>They do not represent irregularly shaped clusters well.</p> Signup and view all the answers

    What method is used when clustering categorical data, particularly in text clustering?

    <p>Applying k-modes clustering to find frequent values.</p> Signup and view all the answers

    What approach can be used to evaluate different clustering algorithms?

    <p>Recognizing that there is no definitive way to know the correct clusters.</p> Signup and view all the answers

    What representation is typically considered effective for hyper-spherical clusters?

    <p>The centroid along with the cluster's spread.</p> Signup and view all the answers

    Why might k-means clusters be deemed more useful in specific applications?

    <p>They provide a measure of simplicity and ease of implementation.</p> Signup and view all the answers

    What distinguishes irregular shape clusters from hyper-ellipsoidal clusters?

    <p>Irregular clusters cannot be represented by centroids.</p> Signup and view all the answers

    What is the purpose of constraining the loadings in PCA?

    <p>To prevent arbitrarily large variance</p> Signup and view all the answers

    What does the first principal component represent in the context of PCA?

    <p>The direction along which the observations vary the most</p> Signup and view all the answers

    When computing principal components, what assumption is made about the variables in the data set?

    <p>Each variable must have a mean of zero</p> Signup and view all the answers

    What does the term 'principal component scores' refer to?

    <p>The linear combinations of the original variables</p> Signup and view all the answers

    Which of these is true about the second principal component in PCA visualization?

    <p>It is orthogonal to the first principal component</p> Signup and view all the answers

    In principal component analysis, how is the constraint on the loadings expressed mathematically?

    <p>$\ rac{1}{n} \sum_{j=1}^{p} \phi^2_{j1} = 1$</p> Signup and view all the answers

    What do the dashed black line segments in PCA representation indicate?

    <p>The variance of observations from the first principal component</p> Signup and view all the answers

    What is necessary for a variable to have maximum sample variance in PCA?

    <p>It must be a linear combination of original variables</p> Signup and view all the answers

    What is the primary purpose of principal component analysis (PCA) in relation to observations?

    <p>To find the hyperplane closest to the observations.</p> Signup and view all the answers

    What property does the first principal component loading vector have?

    <p>It defines the line in p-dimensional space closest to the observations.</p> Signup and view all the answers

    Why is scaling of variables important in PCA?

    <p>To prevent one variable from dominating due to its scale.</p> Signup and view all the answers

    What does the Proportion Variance Explained (PVE) indicate in PCA?

    <p>The strength of each principal component in explaining the data variance.</p> Signup and view all the answers

    What cumulative proportion of variance is explained by the first two principal components together?

    <p>87.0%</p> Signup and view all the answers

    If the variables have the same units, what is the approach regarding scaling?

    <p>Scaling the variables is unnecessary.</p> Signup and view all the answers

    What statistical representation is used to examine the significance of the PCA components?

    <p>The cumulative Proportion Variance Explained (PVE).</p> Signup and view all the answers

    How much variance does the second principal component explain in the data?

    <p>24.7%</p> Signup and view all the answers

    Study Notes

    Introduction to Machine Learning AI 305: Unsupervised Learning - Clustering

    • Clustering is a technique used to group similar data points together into clusters.
    • Dissimilar data points are grouped into different clusters.
    • Clustering is often considered a type of unsupervised learning task.

    Supervised vs. Unsupervised Learning

    • Supervised learning involves learning from labeled data, where each data point is associated with a target class.
    • Unsupervised learning, as in clustering, does not involve pre-labeled classes; instead, it aims to discover inherent patterns or structures within the data.

    Clustering

    • Clustering is used to find similarity groups in data.
    • The goal of clustering is to group similar data instances together and separate dissimilar data instances.
    • It is often used as an unsupervised learning method.

    Illustration

    • A data set can have multiple natural clusters or groups of data points.

    What is Clustering For?

    • Example 1: Grouping people by size for clothing. To create "small", "medium", "large" sizing for T-shirts.
    • Example 2: Targeted marketing, identifying subgroups of people, advertising, and product purchasing. Example 3: Organizing text documents (content). This helps make a hierarchical structure for topics/hierarchy.
    • Clustering has applications in various fields, including areas like medicine, psychology, botany, sociology, biology, archeology, marketing, insurance, and libraries.

    Aspects of Clustering

    • A clustering algorithm's quality depends on the
    • algorithm used, the method of determining similarity, and the application.

    K-means Clustering

    • K-means is a partitional clustering algorithm that groups data points into k clusters.
    • Each cluster has a center called a centroid.
    • k is specified by the user.

    K-means Algorithm

    • Randomly select k data points to be initial centroids.
    • Assign each data point to the nearest centroid.
    • Recompute the centroids based on the current cluster members.
    • Repeat until no or minimal data re-assignment occurs to ensure convergence.

    Stopping/Convergence Criteria

    • There are a few methods for recognizing convergence in K-means like:
    • No (or minimum) change in data point assignments to different clusters
    • No (or minimal) change in the centroid positions
    • Minimal decrease in the sum of squared errors (SSE).

    An Example Illustration

    • The algorithm involves multiple iterations to converge on an answer. The result represents a cluster grouping.

    Strengths of K-means

    • Simple and easy to implement.
    • Efficient with a time complexity of O(tkn), typically linear.
    • k-means is the most common clustering algorithm.

    Weaknesses of K-means

    • The algorithm's success relies on properly identifying the ideal k value.
    • The algorithm can be sensitive to outliers.
    • The algorithm's result is sensitive to the initial choice of centroids.
    • Not suitable for discovering clusters that are not hyperellipsoids/hyper-spheres.

    Selecting the k-value

    • Determining k is an important decision.
    • Multiple plots can aid visualizations for understanding the different groupings and the relationships between the variables.

    Weaknesses of K-means: Handling Outliers

    • Removal or random selection methods to reduce outlier influence.

    Weaknesses of K-means: Handling Initial Seeds

    • Variation of random starting points may be necessary for different or improved results.

    Common Ways to Represent Clusters

    • Centroids (averages) for the cluster.
    • Compute radius and standard deviation to determine extent and spread.

    Using Classification

    • Assign a label or classification to every point within a cluster using a supervised learning model.

    Use Frequent Values to Represent Clusters

    • Useful for clustering categorical data.

    Clusters of Arbitrary Shapes

    • Difficult to represent using centroids alone.
    • Centroids may not be able to adequately represent irregular shapes.

    Hierarchical Clustering

    • An alternative to K-means.
    • It does not require pre-specifying the number of clusters (k).
    • Uses a hierarchical structure (tree, Dendrogram) to group data based on relationships.

    Types of Hierarchical Clustering

    • Agglomerative (bottom-up): Starts with individual data points as clusters, and merges the closest clusters iteratively until reaching a single cluster.
    • Divisive (top-down): Starts with a single cluster, and recursively divides clusters into smaller ones until each data point forms its own cluster.

    Agglomerative Clustering Algorithm

    • Each data point starts as its own initial cluster.
    • Progressively merge the clusters based on smallest distance between them.
    • Continue until there is only one large cluster.

    Measuring the Distance of Two Clusters (Agglomerative)

    • The algorithm uses different methods (single link, complete link, average link, centroid) to measure distance between cluster sets.
    • Uses the closest distance points to determine cluster distance.
    • Uses the furthest distance points to determine cluster distance.
    • A compromise/average of distance between points in different clusters rather than furthest or nearest.
    • Centroid method uses distance between cluster centroids for evaluation.

    Distance Functions

    • "Similarity" and "dissimilarity" measurements are critical to clustering.
    • Different types of distance functions are available for different types of data (numerical, nominal) and applications.

    Distance Functions for Numerical Attributes

    • Euclidean distance: Standard distance calculation.
    • Manhattan distance: Absolute differences between data points.
    • Minkowski distance: Generalization of Euclidean and Manhattan distances.
    • Weighted Euclidean distance: Allows varying weights to different dimensions.
    • Squared Euclidean distance: Places greater weight on points far apart.
    • Chebychev distance: Considers the maximum difference in attributes to determine distance.

    How to Choose a Clustering Algorithm

    • No one-size-fits-all answer.
    • Trial and error methods.
    • Consideration of the data's structure/distribution, data standardization/preprocessing, and distance functions used.

    PCA (Principal Components Analysis): Introduction

    • PCA is a dimensionality reduction technique.
    • Used for visualization or pre-processing prior to supervised methods.
    • Creates new variables with maximal variation, uncorrelated with each other to capture most variance in data and reduce dimensionality.

    PCA: Details

    • First principal component corresponds to maximum variance of data.
    • Loading vectors represent linear combinations.
    • Normalized to ensure equal weighting.
    • The components must be uncorrelated to reduce overlap and improve interpretability.

    PCA: Example

    • Illustrates data representation using principal components in two dimensions.

    PCA: Further components

    • Subsequent components explain less variance.
    • Uncorrelated with previous components (and each other).

    Computing Initial Principal Components

    • Calculate using either singular value decomposition or an eigen-decomposition calculation.

    Geometry of PCA

    • Vectors produced by PCA show maximal variations.

    Interpretation of Example

    • The first principal component in the example dataset is primarily influenced by population size and ad spending.

    Scaling of Variables in PCA

    • Variables with different units/spread are standardized for equal weight in the principal component calculations.

    Proportions of Variance Explained

    • Useful for understanding the strength and relative importance of different dimensions/principal components.
    • Total variance is the sum of variances of the principal components.

    Summary of PCA

    • Simplifies and organizes data using fewer, uncorrelated variables/dimensions.

    How many Principal Components Should Be Used?

    • No single answer; need careful consideration of the variance explained.
    • "Scree plot" can help identify the "elbow point" suggesting a reasonable limit and the important components.

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    AI 305 Unsupervised Learning: Clustering (PDF)

    Description

    Test your knowledge of the k-means clustering algorithm through this quiz. Explore key concepts such as time complexity, handling outliers, and the strengths and weaknesses of this popular algorithm. Perfect for students and professionals alike looking to solidify their understanding of k-means.

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