Posts about interesting extensions I’ve found to some statistical algorithms for use in computer graphics, physics and applied maths.

Major focus on quasirandom sampling.

martin @ robertsanalytics.com https://www.linkedin.com/in/martinroberts/

It would be great if someone who has Wikipedia edit privileges, can edit the Wikipedia article at [1] to describe/link how the Chatarjee's correlation coefficient solves many of the known limitation of Pearson's correlation coefficient. ;)

[1] https://en.wikipedia.org/wiki/Pearson_correlation_coefficien...

(especially second top-left diagram)

Another very useful consequence of being order-based, is that this new coefficient is much more robust to noise/outliers than the canonical correlation coefficient.

So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube.

2.Only one boundary can be included

It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge. Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus.

thus the the 2 boundaries acutally map to the same point, so you can only include one of them. In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1.

3) the mapping from the circle to the surface of the sphere is described here https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_p...

the entire top edge of the square maps to the north pole, and the entire bottom edge maps to the south pole.

4.) t0 is the first point, t3 is the 4-th point.

Hope that helps!

Regarding the two-variable function mod(x,b). Typically this is written as x (mod b) in maths, and as x%b in computing.

It is generally well known that for positive integers x and b, the output of this function is the remainder when x is divided by b.

However, what is less well-known is that if b=1, then the convention is that:

x (mod 1) = x%1 = fractional part of x.

For example, Python, Excel both implement this special convention.

As I describe in the article, different methods produce similar but slightly different solutions. An optimal solution for one objective function, may not be the optimal for a different objective function. I then give details about how the solution that optimizes volume of the convex hull is different to the solution that optimizes for packing distance, etc.

Icosahedron: 12 points, 20 faces (and 30 edges)

Dodecahedron: 20 points, 12 faces (and 30 edges)

"Two physicists prove that there are not just four ways to tie a tie, but a further eighty-one. ‘The 85 Ways to Tie a Tie’ unravels the history of ties, the story of the discovery of the new knots and some very elegant mathematics in action."

https://www.amazon.in/85-Ways-Tie-Science-Aesthetics/dp/1841...

http://extremelearning.com.au/trigonometry-visualised/

Congrats on your half marathon. It is in some ways not that much a jump from 21km to 100km runs.

Regarding my personal path, although I have always been generally fit and healthy, I went from no running as an adult to doing a single half marathon, to doing a 100 mile ultra 12 months later (that is, I skipped the build-up through marathons).

Being a geek I initially (obsessively) focused on two quantitive things: heart rate and cadence.

The key to ultras is that that your energies (both mental and physical) should be focused on progressively running farther, rather than getting progressively faster. And thus absolutely key to ultra trianing is the regular long slow run. And this is where Maffetone plays a crucial role because it forces you, no matter how excited or energised you are on a particular training day, to go slow and steady.

Plus it has the benefit that you can track your improvment accurately because the constant max heart rate forces every training run to be equal effort. (Without this restriction, if you are timing yourself during training the temptation is just to push yourself harder, which often leads to injury or inefficient training.)

For what it is worth, after a year or two of running by heart-rate, I generally don't anymore. Intimately linked your cardio system is your respiratory system, and so the complementary version of Maffetone's system is to limit your running based on a maximum respiratory rate. With both a physiological and pragmatic foundation, an optimal rate is to run no faster than you can breath through your nose. (Even if you can't or don't like breathing through your nose, every ten minutes or so, just try running for a whole minute through your nose. If you run out of breath and begin panting then you are running too fast.) I still wear a heart rate monitor when I run, and so i can tell you that the pace based on Maffetone's is very similar to the pace dictated by my 'breathe through your nose' system. So pick whatever works best for you. ;)

Furthermore, I focused on cadence. Read about how a running cadence of 180+ steps per minute can minimise overstriding and thus injury. The debate of heel-first, mid-foot or balls-of-feet running debate fades away if you get your cadence right. These days, there are a million running and/or music apps specifically designed to beep at 180bpm. (And if you picking a 3/4 musical time signature it means you will more equally develop both sides of your body). You don't necessarily need to have them play for the entire duration of your training run, but it is important (and interesting) to see that when people are tired their cadence drops. So it is good to trian yourself that when you are tired to shorten your stride-length and not your cadence -- which is opposite to what most people do.

Ultramarathon training doesnt' require you to be fast, it just requires that you have done lots of distance in training. A cadence of 180 is higher than most people run, but is really good for all sorts of running esp long distnace running as it minimises deceleration on your ankles, knees and hips and thus maximises efficiency and minimises injury. It's like riding a bike in a very low gear.

A useful rule-of-thumb is that to have a good chance of completing a long-distance race of X km, you should be doing at least X km of training per week, for at least 4 weeks prior to the race.

Finally there is the saying, "The first third of the race you run on your legs, the middle in your head, and the last by your heart."

That is, fitness and physical training gets you the base, but there is a lot more of a mental strength for ultras than many expect. Although whatver training system you choose might look like they are focused on developing your physical base, it is just as much about building your mental strength to run for such a long time.

Hope that helps, and Good luck!