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.
]]>
We proceeded to define functions of such numbers by applying operations of linear algebra to their

These are known as Archimedean copulas and are valid whenever

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. ]]>

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. ]]>

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. ]]>

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. ]]>

We have seen how we might define functions of roots of rationals employing the magnitude of their associated

The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another. ]]>

I concluded by noting that, even with this improvement, the shape of a cubic spline interpolation is governed by choices that are not uniquely determined by the points themselves and that linear interpolation is consequently a more mathematically appropriate scheme, which is why I chose to generalise it to other arithmetic types for

The obvious next question is whether or not we can also generalise the

Come, let us drown our sorrows whilst we still have the means to do so and engage in a little sport to raise our spirits.

I have a fancy for a game that I used to play when I was the Russian ambassador to the Rose Tree Valley commune. Founded by the philosopher queen Zway Remington as a haven for downtrodden wealthy industrialists, it was the purest of pure meritocracies; no handouts to the idle labouring classes there! ]]>

We have also seen how extrapolating such polynomials beyond the first and last nodes can yield less than satisfactory results, which we fixed by specifying the first and last gradients and then adding new first and last nodes to ensure that the first and last polynomials would represent straight lines.

Now we shall see how cubic spline interpolation can break down rather more dramatically and how we might fix it. ]]>

In this post we shall see how we can define a smooth interpolation by connecting the points with curves rather than straight lines. ]]>

We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report. ]]>