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

When I heard these rules I was reminded of the game of Cram and could see that, just like it, the key to figuring the outcome is to recognise that the Baron could always have kept the remaining draughts in a state of symmetry, thereby ensuring that however Sir R----- had chosen he shall subsequently have been free to make a symmetrically opposing choice. ]]>

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.
]]>
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action. ]]>

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. ]]>

Good man! Good man!

I must say that the contrast between the warmth of this fire and the frost outside brings most vividly to my mind an occasion during my tenure as the Empress's ambassador to the land of Oz; specifically the time that I attended King Quadling Rex's winter masked ball during which his southern palace was overrun by an infestation of Snobbles! ]]>

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.
]]>
Its operation was most satisfactory, which set us to wondering whether we might use its engine to investigate the motions of entirely hypothetical arrangements of heavenly bodies and I should now like to report upon our progress in doing so. ]]>

`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. ]]>

Splendid! Come join me at my table!

I propose a game played as a religious observance by the parishioners of the United Reformed Eighth-day Adventist Church of Cthulhu, the eldritch octopus god that lies dead but dreaming in the drowned city of Hampton-on-Sea.

Several years ago, the Empress directed me to pose as a peasant and infiltrate their temple of Fhtagn in the sleepy village of Saint Reatham on the Hill when it was discovered that Bishop Derleth Miskatonic had been directing his congregation to purchase vast tracts of land in the Ukraine and gift them to the church in return for the promise of being spared when Cthulhu finally wakes and devours mankind. ]]>

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

Upon hearing these rules I recognised that they described the classic probability problem known as Pólya's Urn. I explained to the Baron that it admits a relatively simple expression that governs the likelihood that the bag contains given numbers of black and white tokens at each turn which could be used to figure the probability that he should have triumphed, but I fear that he didn't entirely grasp my point. ]]>

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. ]]>