Archive for inversion

composition versus inversion

Posted in Books, Kids, R, Statistics with tags , , , , , , , on March 31, 2021 by xi'an

While trying to convey to an OP on X validated why the inversion method was not always the panacea in pseudo-random generation, I took the example of a mixture of K exponential distributions when K is very large, in order to impress (?) upon said OP that solving F(x)=u for such a closed-form cdf F was very costly even when using a state-of-the-art (?) inversion algorithm, like uniroot, since each step involves adding the K terms in the cdf. Selecting the component from the cumulative distribution function on the component proves to be quite fast since using the rather crude

x=rexp(1,lambda[1+sum(runif(1)>wes)])

brings a 100-fold improvement over

Q = function(u) uniroot((function(x) F(x) - u), lower = 0, 
    upper = qexp(.999,rate=min(la)))[1] #numerical tail quantile
x=Q(runif(1))

when K=10⁵, as shown by a benchmark call

         test elapsed
1       compo   0.057
2      Newton  45.736
3     uniroot   5.814

where Newton denotes a simple-minded Newton inversion. I wonder if there is a faster way to select the component in the mixture. Using a while loop starting from the most likely components proves to be much slower. And accept-reject solutions are invariably slow or fail to work with such a large number of components. Devroye’s Bible has a section (XIV.7.5) on simulating sums of variates from an infinite mixture distribution, but, for once,  nothing really helpful. And another section (IV.5) on series methods, where again I could not find a direct connection.

[Astrostat summer school] fogrise [jatp]

Posted in Kids, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , on October 11, 2017 by xi'an

Les Diablerets skyline [jatp]

Posted in Kids, Mountains, pictures, Statistics, Travel with tags , , , , , , , on February 12, 2017 by xi'an

The short course I gave in Les Diablerets, Switzerland, was highly enjoyable, at least for me!, as it gave me the opportunity to present an  overview of the field, just before our workshop in Banff and to stay in a fantastic skiing area for four days! While I found out that my limited skiing skills are gone even more limited, the constant fresh snow falling during my stay and the very small number of people on the slopes made the outdoor highly enjoyable, the more because the temperatures were quite tolerable. The above picture was taken on the only morning it did not snow, with a nice cloud inversion over the valley separating France and Switzerland.