Joined 8 years ago
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...
