In this post we shall see how it is related to the gamma distribution and implement its various functions in terms of those of the latter. ]]>

Whilst it is most certainly the case that this was more reasonable than assuming entirely random encounters it failed to take into account the fact that folk should have a much greater proclivity to meet with their friends, family and colleagues than with their neighbours and it is upon this deficiency that we have concentrated our most recent efforts. ]]>

Unfortunately all of these approaches require the step length to be fixed and specified in advance, ignoring any information that we might use to adjust it during the iteration in order to better trade off the efficiency and accuracy of the approximation. In this post we shall try to automatically modify the step lengths to yield an optimal, or at least reasonable, balance. ]]>

Might I again tempt you with a wager?

Splendid!

I have in mind a game that always reminds me of my victory upon the turf at Newmarket. Ordinarily I would not participate in a public sporting event such as this since I am at heart a modest man and derive no pleasure in demonstrating my substantial superiority over my fellows. ]]>

In this post we shall see that these are both examples of a general class of algorithms that can be accurate to still greater orders of magnitude. ]]>

To evaluate its worth to Sir R----- we begin with his expected winnings after a single toss of the coin. ]]>

Unfortunately it isn't very accurate, yielding an accumulated error proportional to the step length, and so this time we shall take a look at a way to improve it. ]]>

A fundamental weakness in our model that we have lately sought to address is the presumption that individuals might initiate contact with other members of the population entirely by chance when it is far more likely that they should interact with those in their immediate vicinity. It is upon our first attempt at correcting this deficiency that I should now like to report. ]]>

This was an improvement over an even more rudimentary scheme which instead placed rectangles spanning adjacent values with heights equal to the values of the function at their midpoints to approximate the area. Whilst there really wasn't much point in implementing this since it offers no advantage over the trapezium rule, it is a reasonable first approach to approximating the solutions to another type of problem involving calculus; ordinary differential equations, or ODEs. ]]>

Would you be in the mood for some sporting diversion?

It pleases me to hear so Sir! It pleases me greatly!

I challenge you to a game that reflects the somewhat unique political system adopted by the three sister-queens of Thornborough; Alnitak, Alnilam and Mintaka. Whilst ruling as a triumvirate their constitution requires all three to concur upon any decision, quite unlike any others in antiquity or modernity which, as I'm quite sure that you are aware, require but two. ]]>

Now there is nothing about such distributions, known as mixture distributions, that requires that the components are univariate. Given that copulas are simply multivariate distributions with standard uniformly distributed marginals, being the distributions of each element considered independently of the others, we can use the same technique to create new copulas too. ]]>

It is quite tempting, therefore, to use weighted sums of PDFs to construct new PDFs and in this post we shall see how we can use a simple probabilistic argument to do so. ]]>

Unfortunately for large numbers of dimension the calculation of the approximation will still be relatively expensive and will require a significant amount of memory to store and so in this post we shall take a look at an algorithm that only uses the vector of first partial derivatives. ]]>