This time we shall take a look at another family of special functions derived from the beta function B. ]]>

Whilst we didn't originally derive the Cauchy distribution in this way, there are others, known as ratio distributions, that are explicitly constructed in this manner and in this post we shall take a look at one of them. ]]>

An easy way to create rotationally symmetric functions, known as radial basis functions, is to apply univariate functions that are symmetric about zero to the distance between the interpolation's argument and their associated nodes. PDFs are a rich source of such functions and, in fact, the second bell shaped curve that we considered is related to that of the Cauchy distribution, which has some rather interesting properties. ]]>

An alternative approach is to construct a single function that passes through all of the points and, given that

and together are known as the spectral decomposition of

In this post, we shall add it to the

`ak`

library using the `householder`

and `givens`

functions that we have put so much effort into optimising.]]>

The columns of the matrix of transformations

and, since the product of

which is known as the spectral decomposition of

Last time we saw how we could efficiently apply the Householder transformations in-place, replacing the elements of

implying that the columns of

where

From a mathematical perspective the combination of Householder transformations and shifted Givens rotations is particularly appealing, converging on the spectral decomposition after relatively few matrix multiplications, but from an implementation perspective using

`ak.matrix`

multiplication operations is less than satisfactory since it wastefully creates new `ak.matrix`

objects at each step and so in this post we shall start to see how we can do better.
]]>
and, since the transpose of

which is known as the spectral decomposition of

Unfortunately, the way that we used Givens rotations to diagonalise tridiagonal symmetric matrices wasn't particularly efficient and I concluded by stating that it could be significantly improved with a relatively minor change. In this post we shall see what it is and why it works. ]]>

known as the eigenvectors and the eigenvalues respectively, with the vectors typically restricted to those of unit length in which case we can define its spectral decomposition as the product

where the columns of

You may recall that this is a particularly convenient representation of the matrix since we can use it to generalise any scalar function to it with

where

You may also recall that I suggested that there's a more efficient way to find eigensystems and I think that it's high time that we took a look at it. ]]>

A simple way of constructing them is to initially place each datum in its own cluster and then iteratively merge the closest pairs of clusters in each clustering to produce the next one in the sequence, stopping when all of the data belong to a single cluster. We have considered three ways of measuring the distance between pairs of clusters, the average distance between their members, the distance between their closest members and the distance between their farthest members, known as average linkage, single linkage and complete linkage respectively, and implemented a reasonably efficient algorithm for generating hierarchical clusterings defined with them, using a min-heap structure to cache the distances between clusters.

Finally, I claimed that there is a more efficient algorithm for generating single linkage hierarchical clusterings that would make the sorting of clusters by size in our

`ak.clustering`

type too expensive and so last time we implemented the `ak.rawClustering`

type to represent clusterings without sorting their clusters which we shall now use in the implementation of that algorithm.
]]>
`ak.minHeap`

implementation of the min-heap structure to cache the distances between clusters, saving us the expense of recalculating them for clusters that don't change from one step in the hierarchy to the next.Recall that we used three different schemes for calculating the distance between a pair of clusters, the average distance between their members, known as average linkage, the distance between their closest members, known as single linkage, and the distance between their farthest members, known as complete linkage, and that I concluded by noting that our algorithm was about as efficient as possible in general but that there is a much more efficient scheme for single linkage clusterings; efficient enough that sorting the clusters in each clustering by size would be the most costly operation and so in this post we shall implement objects to represent clusterings that don't do that. ]]>

We did this by selecting the closest pairs of clusters in one clustering and merging them to create the next, using one of three different measures of the distance between a pair of clusters; the average distance between their members, the distance between their nearest members and the distance between their farthest members, known as average linkage, single linkage and complete linkage respectively.

Unfortunately our implementation came in at a rather costly

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. ]]>

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. ]]>

This time we shall take a look at a clustering algorithm that uses nearest neighbours to identify clusters, contrasting it with the