Recently in Clustering Category

A Place In The Hierarchy

Last time we implemented the clusterings type to store a set of clustering objects in order to represent hierarchical clusterings, which are sequences of clusterings having the property that if a pair of data are in the same cluster in one clustering then they will be in the same cluster in the next, where clusters are subsets of a set of data that are in some sense similar to each other.
We then went on to define the ak.clade type to represent hierarchical clusterings as trees, so named because that's what they're called in biology when they are used to show the relationships between species and their common ancestors.
Now that we have those structures in place we're ready to see how to create hierarchical clusterings and so in this post we shall start with a simple, general purpose, but admittedly rather inefficient, way to do so.

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We have so far seen a couple of schemes for identifying clusters in sets of data which are subsets whose members are in some sense similar to each other. Specifically, we have looked at the k means and the shared near neighbours algorithms which implicitly define clusters by the closeness of each datum to the average of each cluster and by their closeness to each other respectively.
Note that both of these algorithms use a heuristic, or rule of thumb, to assign data to clusters, but there's another way to construct clusterings; define a heuristic to measure how close to each other a pair of clusters are and then, starting with each datum in a cluster of its own, progressively merge the closest pairs until we end up with a single cluster containing all of the data. This means that we'll end up with a sequence of clusterings and so before we can look at such algorithms we'll need a structure to represent them.

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Nearest And Dearest

Last time we saw how we could use the list of the nearest neighbours of a datum in a set of clustered data to calculate its strengths of association with their clusters. Specifically, we used the k nearest neighbours algorithm to define those strengths as the proportions of its k nearest neighbours that were members of each cluster or with a generalisation of it that assigned weights to the neighbours according to their positions in the list.
This time we shall take a look at a clustering algorithm that uses nearest neighbours to identify clusters, contrasting it with the k means clustering algorithm that we covered about four years ago.

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In The Neighbourhood

A little under four years ago we saw how we could use the k means algorithm to divide sets of data into distinct subsets, known as clusters, whose members are in some sense similar to each other. The interesting thing about clustering is that even though we find it easy to spot clusters, at least in two dimensions, it's incredibly difficult to give them a firm mathematical definition and so clustering algorithms invariably define them implicitly as the subsets identified by this algorithm.
The k means algorithm, for example, does so by first picking k different elements of the data as cluster representatives and then places every element in the cluster whose representative is nearest to it. The cluster representatives are then replaced by the means of the elements assign to it and the process is repeated iteratively until the clusters stop changing.
Now I'd like to introduce some more clustering algorithms but there are a few things that we'll need first.

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Try Another K

In our exploration of the curious subject of cluster analysis, in which the goal is to classify a set of data into subsets of similar data without having a rigorous mathematical definition of what that actually means, we have covered the k means algorithm that implicitly defines a clustering as minimising the sum of squared distances of the members of the clusters from their means and have proposed that we might compare clusterings by the amount of the variance in the data that they account for.
Unfortunately, it turned out that trying to identify the actual number of clusters in the data using the accounted for variance was a rather subjective business and so in this post we shall see if we can do any better.

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A Good K To Try

We have seen how the k means algorithm can classify a set of data into k subsets of mutually similar data with the simple iterative scheme of placing each datum into the cluster whose representative it is closest to and then replacing those representatives with the means of the data in each cluster. Whilst this has a reasonably intuitive implicit definition of similarity it also has the unfortunate problem that we need to know how many clusters there are in the data if we are to have any hope of correctly identifying them.

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Last time we took a first brief look at cluster analysis in which we seek to algorithmically differentiate sets of data into subsets of similar data. The difficulty in doing so stems from the fact that similarity is a rather poorly defined concept and so clustering algorithms typically proceed by updating cluster memberships according to some rule of thumb, or heuristic, that is designed to reflect some intuitive notion of similarity. As a consequence, clusters have the rather unusually circular definition of that which are identified by a clustering algorithm!

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Some Of These Things Are Not Like The Others

Of all of the problems that we might attack with numerical methods, one of my favourites is cluster analysis; the task of dividing a set of data into subsets of similar data. Now I'm fully aware that my having favourite applications of numerical methods might lead you to suspect that I live at the most cringingly awkward end of nerd street, but please bear with me; the reason that I find cluster analysis so appealing, and that I hope you will too, is that it is a rare example of a mathematical problem that we struggle to even properly define.

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Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

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