Sir R-----, my good friend! Come shake the snow from your boots and join me by the hearth for a draught of warming spirits!

And will you also join me in a wager whilst you let the fire chase the chill from your bones?

Fine fellow! Stout fellow!

I have in mind a game that reminds me of my raid upon the vault of Heaven, which I mounted in order to make amends to the Empress for my failure to snatch the Amulet of Yendor from the inner circle of Hell.

We have recently seen how we can define dependencies between random variables with Archimedean copulas which calculate the probability that they each fall below given values by applying a generator function φ to the results of their cumulative distribution functions, or CDFs, for those values, and applying its inverse to their sum.
Like all copulas they are effectively the CDFs of vector valued random variables whose elements are uniformly distributed when considered independently. Whilst those Archimedean CDFs were relatively trivial to implement, we found that their probability density functions, or PDFs, were somewhat more difficult and that the random variables themselves required some not at all obvious mathematical manipulation to get right.
Having done all the hard work implementing the ak.archimedeanCopula, ak.archimedeanCopulaDensity and ak.archimedeanCopulaRnd functions we shall now use them to implement some specific families of Archimedean copulas.

When last they met, the Baron challenged Sir R----- to evade capture whilst moving rooks across and down a chessboard. Beginning with a single rook upon the first file and last rank, the Baron should have advanced it to the second file and thence downwards in rank in response to which Sir R----- should have progressed a rook from beneath the board by as many squares and if by doing so had taken the Baron's would have won the game. If not, Sir R----- could then have chosen either rook, barring one that sits upon the first rank, to move to the next file in the same manner with the Baron responding likewise. With the game continuing in this fashion and ending if either of them were to take a rook moved by the other or if every file had been played upon, the Baron should have had a coin from Sir R----- if he took a piece and Sir R----- one of the Baron's otherwise.

In the last couple of posts we've been taking a look at Archimedean copulas which define the dependency between the elements of vector values of a multivariate random variable by applying a generator function φ to the values of the cumulative distribution functions, or CDFs, of their distributions when considered independently, known as their marginal distributions, and applying the inverse of the generator to the sum of the results to yield the value of the multivariate CDF.
We have seen that the densities of Archimedean copulas are rather trickier to calculate and that making random observations of them is trickier still. Last time we found an algorithm for the latter, albeit with an implementation that had troubling performance and numerical stability issues, and in this post we shall add an improved version to the ak library that addresses those issues.

Over the course of the year my fellow students and I have spent much of our spare time investigating the properties of the set of infinite dimensional vectors associated with the roots of rational numbers by way of the former's elements being the powers to which the latter's prime factors are raised, which we have dubbed ℓ-space.
We proceeded to define functions of such numbers by applying operations of linear algebra to their ℓ-space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.

Last time we took a look at how we could define copulas to represent the dependency between random variables by summing the results of a generator function φ applied to the results of their cumulative distribution functions, or CDFs, and then applying the inverse of that function φ^-1 to that sum.
These are known as Archimedean copulas and are valid whenever φ is strictly decreasing over the interval [0,1], equal to zero when its argument equals one and have n^th derivatives that are non-negative over that interval when n is even and non-positive when it is odd, for n up to the number of random variables.
Whilst such copulas are relatively easy to implement we saw that their densities are a rather trickier job, in contrast to Gaussian copulas where the reverse is true. In this post we shall see how to draw random vectors from Archimedean copulas which is also much more difficult than doing so from Gaussian copulas.

Greetings Sir R-----! Might I suggest that you take one of these spiced beef pies and a mug of mulled cider to stave off this winter chill? And perhaps a wager to fire the blood?

Good man! Good man!

I propose a game that ever puts me in mind of my ill-fated expedition to recover for the glory of the Empress of Russia the priceless Amulet of Yendor from the very depths of Hell.

About a year and a half ago we saw how we could use Gaussian copulas to define dependencies between the elements of a vector valued multivariate random variable whose elements, when considered in isolation, were governed by arbitrary cumulative distribution functions, known as marginals. Whilst Gaussian copulas are quite flexible, they can't represent every possible dependency between those elements and in this post we shall take a look at some others defined by the Archimedean family of copulas.

The Baron's latest wager set Sir R----- the task of surpassing his score before he reached eight points as they each cast an eight sided die, each adding one point to their score should the roll of their die be less than or equal to it. The cost to play for Sir R------ was one coin and he should have had a prize of five coins had he succeeded.

A key observation when figuring the fairness of this wager is that if both Sir R----- and the Baron cast greater than their present score then the state of play remains unchanged. We may therefore ignore such outcomes, provided that we adjust the probabilities of those that we have not to reflect the fact that we have done so.

We have spent a few months looking at how we might interpolate between sets of points (x_i, y_i), where the x_i are known as nodes and the y_i as values, to approximate values of y for values of x between the nodes, either by connecting them with straight lines or with cubic curves.
Last time, in preparation for interpolating between multidimensional vector nodes, we implemented the ak.grid type to store ticks on a set of axes and map their intersections to ak.vector objects to represent such nodes arranged at the corners of hyperdimensional rectangular cuboids.
With this in place we're ready to take a look at one of the simplest multidimensional interpolation schemes; multilinear interpolation.