for a coincidence

Last night in Vancouver, we were walking back to Chinatown under an expressway, in a rather uninspiring section of town. Waiting at a cross-light with another couple on the other side. As we crossed the street I glanced at the man and noticed his Chamonix North Face tee-shirt. He happened to do the same and… noticed my identical Chamonix North Face tee-shirt! We shared a laugh at this (huge?) coincidence and continued on our respective ways. (He was not taking part in MCMskiii in case this seems a likely explanation!)

Байкало-Амурская магистраль/БАМ

While in Chamonix last month I dropped by the Guérin editions bookstore, always full of tantalising books on climbing and mountaineering, travelling and travellers. I managed to escape with only two small books, one on a young climber stuck for 100 hours at the top of Aiguilles Vertes [not far from my last ice-climb!] and the other one a railway trip along the Baïkal-Amour Mainline (BAM), which goes from the Baïkal Lake to Sovietskaïa Gavan, north of the Trans-Siberian line. The book is not the ultimate travel book as most of the pages are about historical features surrounding this line, first and foremost the constant reminder that Gulag prisoners were relentlessly exploited to build this line, which follows a macabre route along Siberian camps. The trip finishes not at the end of the BAM line or in Vladivostok, but on Sakhaline Island, which was a penitential colony from the mid-1800’s, as covered by Anton Tchekov in a statistical study and a short story, The Murder… (Comments about characters crossed throughout the trip are rarely to the benefit of these characters.) While I do not make this travel book or the places it crosses sound particularly exciting, it still carries with it the inducing whiff of faraway places, which makes me wish I could see Lake Baïkal or Vladivostok one day in the future, if not travel the entire line. And it also brought back memories of Corto Maltese in Siberia, which remains one of my favourites…

a faint memory of ice

 

During the past week of vacations in Chamonix, I spent some days down-hill skiing (which I find increasingly boring!), X-country skiing (way better), swimming (indoors!) and running, but the highlight (and the number one reason for going there!) was an ice cascade climb with a local guide, Sylvain (from the mythical Compagnie des Guides de Chamonix). There were very options due to the avalanche high risk and Sylvain picked a route called Déferlante at the top of Les Grands Montets cabin stop and next to the end of a small icefield, Glacier d’Argentière. We went there quite early to catch the first cabin up, along a whole horde of badasss skiers and snowboarders, and reached the top of the route by foot first, a wee bit after 9 pm. A second guide and a client appeared before we were ready to abseil down, and two more groups would appear later. On touring skis.

Chamonix snapshot #3 [jatp]

Chamonix snapshot #2 [jatp]

uniform on the sphere [or not]

While looking at X validated questions, I came upon this comment that simulating a uniform distribution on a d-dimensional unit sphere does not proceed from generating angles at random on (0,2π) and computing spherical coordinates… Which I must confess would have been my initial suggestion! This is obvious, nonetheless, when computing the Jacobian of the spherical coordinate transform, which involves powers of the sines of the angles, in a decreasing sequence from d-1 to zero. This means that the angles should be simulated according to their respective sine-power densities. However, except for the d=3 case, where simulating from the density sin(φ) is straightforward by inverse cdf, i.e. φ=acos(1-2u), the cdfs for the higher powers are combinations of sines and cosines, and as such are not easily inverted. Take for instance the eighth power:

F⁸(φ)=(840 φ – 672 sin(2 φ) + 168 sin(4 φ) – 32 sin(6 φ) + 3 sin(8 φ))/3072

While the densities are bounded by sin(φ), up to a constant, and hence an accept-reject can be easily derived, the efficiency decreases with the dimension according to the respective ratio of the Wallis’ integrals, unsurprisingly. A quick check for d=4 shows that the Normal simulation+projection-by-division-by-its-norm is faster.

Puzzling a bit further about this while running, I wondered at the simultaneous simulations from sin(φ), sin(φ)², sin(φ)³, &tc., but cannot see a faster way to recycle simulations from sin(φ). Points (φ,u) located in-between two adjacent power curves are acceptable simulations from the corresponding upper curve but they need be augmented by points (φ,u) under the lower curve to constitute a representative sample. In the end, this amounts to multiplying simulations from the highest power density as many times as there are powers. No gain in sight… Sigh!

However, a few days later, while enjoying the sunset over Mont Blanc(!), I figured out that there exists a direct and efficient way to simulate from these powers of the sine function. Indeed, when looking at the density of cos(φ), it happens to be the signed root of a Beta(½,(d-1)/2), which avoids the accept-reject step. Presumably this is well-known, but I have not seen this proposal associated with the uniform distribution on the sphere.

off to Chamonix!