## Chianti

Posted in pictures, Running, Travel, Wines with tags , , , , , on May 29, 2015 by xi'an

## Toscana [#3]

Posted in Mountains, pictures, Running, Travel, Wines with tags , , , , , on May 28, 2015 by xi'an

## Bruce Lindsay (March 7, 1947 — May 5, 2015)

Posted in Books, Running, Statistics, Travel, University life with tags , , , , , , , , , , , on May 22, 2015 by xi'an

## Cauchy Distribution: Evil or Angel?

Posted in Books, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on May 19, 2015 by xi'an

Natesh Pillai and Xiao-Li Meng just arXived a short paper that solves the Cauchy conjecture of Drton and Xiao [I mentioned last year at JSM], namely that, when considering two normal vectors with generic variance matrix S, a weighted average of the ratios X/Y remains Cauchy(0,1), just as in the iid S=I case. Even when the weights are random. The fascinating side of this now resolved (!) conjecture is that the correlation between the terms does not seem to matter. Pushing the correlation to one [assuming it is meaningful, which is a suspension of belief!, since there is no standard correlation for Cauchy variates] leads to a paradox: all terms are equal and yet… it works: we recover a single term, which again is Cauchy(0,1). All that remains thus to prove is that it stays Cauchy(0,1) between those two extremes, a weird kind of intermediary values theorem!

Actually, Natesh and XL further prove an inverse χ² theorem: the inverse of the normal vector, renormalised into a quadratic form is an inverse χ² no matter what its covariance matrix. The proof of this amazing theorem relies on a spherical representation of the bivariate Gaussian (also underlying the Box-Müller algorithm). The angles are then jointly distributed as

$\exp\{-\sum_{i,j}\alpha_{ij}\cos(\theta_i-\theta_j)\}$

and from there follows the argument that conditional on the differences between the θ’s, all ratios are Cauchy distributed. Hence the conclusion!

A question that stems from reading this version of the paper is whether this property extends to other formats of non-independent Cauchy variates. Somewhat connected to my recent post about generating correlated variates from arbitrary distributions: using the inverse cdf transform of a Gaussian copula shows this is possibly the case: the following code is meaningless in that the empirical correlation has no connection with a “true” correlation, but nonetheless the experiment seems of interest…

> ro=.999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> cor(x[,1]/x[,2],y[,1]/y[,2])
[1] -0.1351967
> ro=.99999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> cor(x[,1]/x[,2],y[,1]/y[,2])
[1] 0.8622714
> ro=1-1e-5;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y)))
> cor(x=z,y=w)
[1] 0.9999732
> ks.test((z+w)/2,"pcauchy")

One-sample Kolmogorov-Smirnov test

data:  (z + w)/2
D = 0.0068, p-value = 0.3203
alternative hypothesis: two-sided
> ro=1-1e-3;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2)
> z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y)))
> cor(x=z,y=w)
[1] 0.9920858
> ks.test((z+w)/2,"pcauchy")

One-sample Kolmogorov-Smirnov test

data:  (z + w)/2
D = 0.0036, p-value = 0.9574
alternative hypothesis: two-sided


## bikes vs cars

Posted in Kids, pictures, Running, Travel with tags , , , , , , , on May 9, 2015 by xi'an

Trailer for a film by Frederik Gertten about the poor situation of cyclists in most cities. Don’t miss Rob Ford, infamous ex-mayor of Toronto, and his justification for closing bike lanes in the city, comparing cycling to swimming with sharks… and siding with the sharks.

## moonlight

Posted in pictures, Running with tags , on May 4, 2015 by xi'an

## the 39 steps

Posted in Books, Kids, Mountains, Running, Travel with tags , , , , , , , on May 3, 2015 by xi'an

I had never read this classic that inspired Hitchcock’s 39 steps (which I neither watched before).  The setting of the book is slightly different from the film: it takes place in England and Scotland a few weeks before the First  World War. German spies are trying to kill a prominent Greek politician [no connection with the current Euro-crisis intended!] and learn about cooperative plans between France and Britain. The book involves no woman character (contrary to the film, where it adds a comical if artificial level). As in Rogue Male, most of the story is about an unlikely if athletic hero getting into the way of those spies and being pursued through the countryside by those spies. Even though the hunt has some intense moments, it lacks the psychological depth of Rogue Male, while the central notion that those spies are so good that they can play other persons’ roles without being recognised is implausible to the extreme, a feature reminding me of the Blake & Mortimer cartoons which may have been inspired by this type of books. Especially The Francis Blake Affair. (Trivia: John Buchan ended up Governor General of Canada.)