Clustering & Dimensionality Reduction

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Clustering

Unsupervised Learning

We don't have actual labels

We may have groups, but nothing is defined and there are no correct answers

Why use unsupervised Learning?

Find similar data groups (clusters)

Find outliers in data (anomaly detection)

Reduce the amount of data but keep information (dimensionality reduction)

Clustering Applications

Related search results

Related articles on news sites

market segmentation

Social media clustering

Image segmentation

Sequence analysis

High throughput genotyping

Divide data into groups that "fit" together

Two different methods we will be looking at

Partitioning Methods

Heirachical-Based Methods

Partitioning Methods

Partition the samples into k clusters, and each partition forms one cluster

This method often uses the distance between different samples as a partition criterion

K-means Clustering

The algorithm will categorize the items into k groups of similarity

To calculate the similarity, the Euclidean distance is used as a measurement.

The algorithm works in three significant steps:

Initialization centroids

Initialize K random points to be the center of the cluster

Often, three random points from the dataset are chosen as initial values

2. Categorizing samples

Determine for each sample which is the closest centroid

We do this using the Euclidean distance

3. Update centroids

Update centroid location to the mean of all samples assigned to that centroid

Repeat

Categorize Samples

Update Centroids

Categorize Samples

Update Centroids

Categorize Samples

Results after n iterations

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K-means Clustering Characteristics

Easy to use, and results are easily interpretable

Sometimes gets stuck at local optima

Final result is dependent on initialized centroids

Try multiple differently initialized models to find the best model (global optima)

Very sensitive to outliers

Problem with not linearly divisible data

Solution: use another clustering algorithm

e.g., Spectral clustering

All sklearn algorithms with example images: sklearn clustering

Determining the Optimal Number of Clusters

Different strategies exist

Visual inspection of data

Silhouette clustering

Elbow method

Elbow Method

Vary the number of clusters, for example, between 1 and 10, and calculate the within-cluster sum of squared errors (WCSSE), where the error is the Euclidean distance between the sample and its centroid.

The "elbow is the best number of clusters

Hierarchical Clustering

This is an essential family of clustering algorithms that build nested clusters by merging or splitting them successively

Can be represented as a tree/ dendrogram

Root is a cluster that contains all samples

Leaves/terminal nodes are clusters that contains a single sample

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Two Major types of Heirachal Clustering

Agglomerative hierarchical clustering

Combines smaller clusters into larger ones (bottom-up)

Divisive hierarchical clustering

Dividing larger clusters into smaller ones (top-down)

Hierarchical Clustering Advantages

We do not need to know the number of clusters in advance as you can only split, so any times

The structure of the tree can be helpful to (such as a phylogenetic tree)

How do you determine Distances (Links) between clusters?

There are four ways to go about this:

Single linkage

The distance between the closest points of each cluster

Complete Linkage (Maximum Linkage)

The distance between the farthest points of each cluster

Average LInnkage

Minimizes the average of the distance between all observations of pairs of clusters

Ward Linkage (Centroi Linkage)

Similar to K-means but in a hierarchical way

Complexity

Hierarchical clustering is very computationally intensive

Calculations are at least quadratically dependent on the number of data points

Not useable for massive datasets

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Dimensionality Reduction

What is dimensionality reduction?

The process of dimensionality reduction involves reducing the number of random variables or attributes that are being considered.

Applications

Visualizations (not possible with more than 3 features)

Removing noise

Data compression

"The curse of dimensionality"

Some models do not perform well with a lot of different features

Algorithms

Principal component analysis (PCA)

Linear Discriminant Analysis (LDA)

Multidimensional scaling (MDS)

Uniform Manifold Approximation and Projection (UMAP)

t-distributed stochastic neighbor embedding (t-SNE)

Autoencoders (special neural network)

Cheap or easy ways

Missing values Ratio

Low variance filter

These will just tell you if a column is useful or not and remove them

Why would we want to use the above algorithms?

Practical examples

Classification of genes

Face recognition (reduces pixels)

Voice recognition

Handwriting recognition

Compression

(outlier detection)

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Principal Component Analysis

PCA transforms several features into more minorPrincipal Component Analysis (PCA) is a technique that helps to reduce the complexity of a dataset by transforming several interrelated features into a set of new, non-correlated features, also called principal components. This method aims to retain as much information as possible while reducing the number of variables in the dataset. non-correlated features, also called principal components, while preserving as much information as possible.

How to conduct principal component analysis

Determine the axis where there is the most variance.

This axis becomes principal component 1. The perpendicular axis becomes principal

component 2. This can also be scaled up to higher dimensions

The axis is determined based on the smallest SSE.

Rotate to this axis and scale it so that the points are centered around 0,0

Reconstruction of data using PCA

I need a break bro

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What is Multidimensional scaling (MDS)?

Attempts to model similarity or dissimilarity data as distances in a

geometric space. It tries to keep the distance between data similar to

the distance in the higher-order dataset

t-distributed Stochastic Neighbor Embedding

Converts affinities (similarities) to probabilities

Advantages:

• Revealing the structure at many scales on a single map

• Revealing data that lie in multiple, different manifolds or clusters

• Reducing the tendency to crowd points together at the center

Disadvantages:

• Computationally expensive (=slow, several hours on million-sample

datasets vs. seconds for PCA)

• Stochastic and different seeds can give different results

Main purpose:

Visualizing high-dimensional data in 2D or 3D