Clustering | Jason Siu

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Jun 16, 2022 01:21 PM

Clustering algorithm

Given a set of data points with a set of attributes, the algorithm will find groups of objects that are

similar to data points to become a Cluster A
or other data points which are different to Cluster A become other clusters

is an unsupervised algorithm, i.e. its data is unlabelled; learning algorithms try to find patterns within the data or to cluster the data into groups or sets.

That similarity is measured by :

Euclidean Distance (think Pythagoras’ theorem).

Other distance-based measures (for example, Manhattan).

Other measures if the attribute values are not continuous

Why Clustering?

The set of data to be analysed can be reduced.

There are two types of clustering: Diff between two is Partition → unnested; Hierarchical → nested

Partition Clustering (Non-Hierarchical)

Divide the data objects into unique subsets, of which a number of sets are pre-defined

are sequential either building up from separate clusters or breaking down from the one cluster.

In some cases, the exact number of clusters may be known (like marketing customers group), then we can use Partition Clustering.

2. Hierarchical Clustering

Divide the data objects into a set of NESTED clusters

The number of clusters is chosen a priori.

There are 2 types of Hierarchical Clustering

Agglomerative:

At start, every observation is a cluster. Merge the most similar clusters step by step until all observations are in one cluster.

In Agglomerative clustering, there are ways of defining the distance between clusters A and B :

Single Linkage (where finding the minimal clusters' distance)

Complete Linkage (where finding the maximal clusters' distance)

Average Linkage (where finding the Average clusters' distance)

Centroid method

Ward’s Method

Divisive:

At start, all observations are in one cluster. Split step by step until each observation is in its own cluster.
(it is very slow compared to Agglomerative)

K-mean clustering - Lloyd's algorithm

K-Means Clustering Explanation and Visualization - YouTube

Visualizing K-Means Clustering (naftaliharris.com)

FAQ about clustering

How to measure a distance between clusters in Hierarchical Clustering

Illustration

GitHub - kavana-r/RLadies_Hierarchical_Clustering

For agglomerative clustering

Agglomerative Clustering: how it works - YouTube

Single Linkage (aka. nearest neighbour) (MIN)

The minimal distance between two points.

First, calculate all points between clusters,

Then find the minimum distance between two points, (which then becomes Single Linkage)

This is to find the CLOSEST neighbor, which is shown below : — This is to find the ***CLOSEST*** neighbor, which is shown below :

Complete Linkage (aka furthest neighbour)(MAX)

The maximum distance between two points.

First, calculate all points between clusters,

Then, find the maximum distance between two points, (which then becomes a Single Linkage)

Pros and Cons of Single Linkage

This is to find the FURTHEREST neighbor, which is shown below :

Average Linkage (AVG)

Find the distance between two clusters is defined as the average distance between each point in one cluster to every point in the other cluster.(From here)

Hierarchical Agglomerative Clustering [HAC - Average Link] - YouTube

Avg of all pairwise distances :

Centroid Method

distance between centroids (mean) of two clusters

Ward's algorithm

Why

The most suitable method for quantitative variables.

Problem : Variance between distance

Normally the distance between clusters is SAMLL where the variance is tiny. But When the distance is large, the variance is large as well.

Solution :

Ward’s method merges two clusters to minimise within cluster variance.

How

Instead of measuring the distance directly, it analyzes the variance of clusters.

Ward’s method says that the distance between two clusters, A and B, is how much the sum of squares will increase when we merge them.

The difference amongst MIN, MAX and AVG

Youtube : Hierarchical Clustering 3: single-link vs. complete-link

Are there a correct number of clusters? If not, how to determine it

NO! But generally, DO NOT :

choose too many clusters: (homogeneity within clusters)

A firm developing a different marketing strategy for each market segment may not have the resources to develop a large number of unique strategies.

choose too few clusters: (parsimony within clusters)

If you choose the 1-cluster solution there is no point in doing clustering at all.

How can we know when to use Single Linkage or Complete Linkage? (Pros and Cons)

GitHub - kavana-r/RLadies_Hierarchical_Clustering

What is Chaining? (found on Cons of Single Linkage in ETF3500 p.56)

Chaining is a problem that occurs when clusters are not compact and some observations in the same cluster are far away from one another.

What is the difference between the centroid and average linkage method?

In average linkage

Compute the distances between pairs of observations

Average these distances

Illustration

This method involves looking at the distances between all pairs and averages all of these distances

In the centroid method

Average the observations to obtain the centroid of each cluster.

Find the distance between centroids

Illustration

This involves finding the mean vector location for each of the clusters and taking the distance between the two centroids.

What is Stability?

Stability is the height between between cluster 1 and cluster (n-1), in the visualisation of dendrogram

Aka Tolerance level

Changing tolerance (threshold) affect the Stability .

The way to determine Stability is to look at the range of tolerance level.

I will give an illustration below :

the range of tolerance level for cluster 1 > the range of tolerance level for cluster 2. Therefore, cluster 1 is more STABLE than cluster 2.

What is Robust? Is Single Linkage (MIN) Robust? (only in Hierarchical)

Robust means the analysis does not dramatically change, even adding a single observation.

To Single Linkage (MIN) Robust :

In this instance, the new observation was not even an outlier but called inlier.
Methods that are not affected by single observations are often called robust.

What are the linkages besides Single and Complete? Why do we prefer using them to SL or CL?

Average Linkage

Centroid method

Ward’s Method

They should be used because they are robust (i.e. Not sensitive to outliers)

What is centroid?

The center of a cluster

When do we use hierarchical and non-hierarchical cluster techniques

Although both algorithms can be used in analysis, hierarchical are suited to small data sets (since the dendrogram can be more easily interpreted) while non-hierarchical methods are well suited to large data sets.

What are the advantages and disadvantages of hierarchical & non-hierarchical clustering

Hierarchical: are sequential either building up from separate clusters or breaking down from the one cluster.

Non-hierarchical: The number of clusters is chosen a priori.

Advantages of Hierarchical clustering

All possible solutions (with respect to the number of clusters) is provided in a single analysis

It is structured and can be explored using the dendrogram.

Don’t need to assume any particular number of clusters. Any desired number of clusters can be obtained by ‘cutting’ the dendrogram at the proper level

Advantages of K-Means Clustering

Can explore cluster allocations that will never be visited by the path of hierarchical solutions.

With a large number of variables, if K is small, K-Means may be computationally faster than hierarchical clustering.

K-Means might yield tighter clusters than hierarchical clustering

An instance can change cluster (move to another cluster) when the centroids are recomputed.

Disadvantages of Hierarchical clustering

It is not possible to undo the previous step: once the instances have been assigned to a cluster, they can no longer be moved around.

Time complexity: not suitable for large datasets

Very sensitive to outliers

Disadvantages of Hierarchical clustering

Need to assume any particular number of clusters (K-Value), which is hard to predict.

Initial seeds (centers of each cluster) have a strong influence on the final results

Sensitive to scale.

Dendrogram

a visualisation of the Hierarchical Clustering

How to interpret :

Think of the axis with distance (y-axis) as the measuring a 'tolerance level'

If the distance between two clusters is within the tolerance they are merged into one cluster.
As tolerance increases more and more clusters are merged leading to less clusters overall.

Viewing Stability in Dendrogram

What is Stability?

Stability is the height between between cluster 1 and cluster (n-1), in the visualisation of dendrogram

Aka Tolerance level

Changing tolerance (threshold) affect the Stability .

The way to determine Stability is to look at the range of tolerance level.

I will give an illustration below :

Rand Index

What is Rand Index

The probability of picking two observations at random that are in agreement.

Lies between 0 and 1 and higher numbers indicate agreement.

Express what proportion of the cluster assignments are ‘correct’.

Problems : Even if observations are clustered at random, there will still be some agreement due to chance.

Solution — Adjusted Rand Index

Can use adjustedRandIndex() in the package of mclust

Interpretation

0 = if the level of agreement equals the case where clustering is done at random.

1 = if the two clustering solutions are in perfect agreement. (Good)

Can Rand Index be the same, if solution A is a three-cluster solution and solution B is a two-cluster solution. Given that they are done from the same dataset

No.

KNN

Knowing what Partition Clustering, let's explore one of its kind — KNN.

Each cluster is associated with a centroid

Each point is assigned to the cluster with the CLOSEST centroid.

Assumptions

The data is in feature space, which means data in feature space can be measured by distance metrics such as Manhattan, Euclidean etc.

Each of the training data points consists of a set of vectors and class label associated with each vector.

Wishes to have ‘K’ as an odd number in case of 2 class classification.

Here is the steps of KNN :

Select k points (at random) as the initial centroids

Repeat :

2.1. Form k clusters by assigning all points to the CLOSEST centroid

2.2. Re-compute the centroid of each cluster

Until the centroids don’t change

Why and Why not KNN :

Advantage :

Simple to implement

Since the KNN algorithm requires no training before making predictions, new data can be added seamlessly which will not impact the accuracy of the algorithm.

Disadvantage :

It depends on initial values. The problem of which is that it has different :

Sizes

Density

Non-globular shapes

Contain outliers

FAQ of KNN

How does each point find its closest centroid?

To find its centroid, you need to calculate the "distance" between each point and all the centroids,

then you will know which is the closest one.

Is there any Pre-processing for KNN?

Yes, you need to :

• Normalise the data • Eliminate outliers

Is there any Post-processing for KNN ?

• Eliminate small clusters that may represent outliers 埋吾到堆ge 走開

SSE係埋堆的measure，如果太高就分開佢地

－Split ‘loose’ clusters, i.e., clusters with relatively high SSE.

－Merge clusters that are ‘close’ and that have relatively low SSE

Will the initial centroids influence final clusters? if so, how do you cope with that?

Yes, so there are few ways to do :

Multiple runs

Select more than k initial centroids and then select among these initial centroids

Hierarchical Clustering

Produces a set of nested clusters organised as a hierarchical tree

Visualised as a dendrogram

Why and Why not HC?

Advantage

It kinda compensates for what KNN cant do :

Don’t need to assume any particular number of clusters

Any desired number of clusters can be obtained by ‘cutting’ the dendrogram at the proper level

Disadvantage

What is a dendrogram used in it?

A ‘dendrogram’ shows how the clusters are merged hierarchically (looks like tree diagram)

It decomposes data objects into several levels of nested partitions (tree of clusters)
The height represents the distance between clusters
A clustering of the data objects is obtained by cutting the dendrogram at the desired level, and then each connected component forms a cluster