Efficient computation of p-variation in metric spaces.
Let p≥1. For a discrete time path X0,...,XN in a metric space with distance d, its p-variation is
sup Σk d(Xnk, Xnk-1)p,
where the supremum is taken over all increasing subsequences nk of 0,...,N. Sometimes one takes p-th root of the sum, but here we don't.There is a very efficient algorithm of computing p-variation for real-valued processes, see paper by Vygantas Butkus and Rimas Norvaiša. But it does not work for example in R2. Here we rectify this, at least partially. We provide a short C++ function which computes p-variation in a general metric space, and is sufficiently fast for paths with millions of points.
In addition, we improve the speed of the one-dimensional algorithm by a factor of 2 to 3.
Below is a minimal working example. Details and important notes are in p_var.h.
#include <iostream>
#include <vector>
#include <array>
// p_var is a one file header-only library
#include "p_var.h"
using p_var_ns::p_var;
// define a type for data points, in this case R^2
typedef std::array<double, 2> vecR2;
// define a distance function, here L^1
double dist(vecR2 a, vecR2 b) {
return std::abs(b[0]-a[0]) + std::abs(b[1]-a[1]);
}
int main () {
// create a path
std::vector<vecR2> path({{0,0}, {1,1}});
auto pv = p_var(path, 3, dist);
// pv.value is the p-variation, and
// pv.points is vector<size_t> with the maximising subsequence
std::cout << "3-variation wrt L^1 distance: " << pv.value << std::endl;
std::cout << "3-variation wrt Euclidean distance: " << p_var(path, 3).value << std::endl;
return 0;
}
Our method is fast on data such as simulated Brownian paths, with complexity of perhaps N log(N). But its worst case complexity is N2. This happens in pathological situations such as monotone paths in R. The one-dimensional method works with such paths much faster.
- Alexey Korepanov
- Terry Lyons
- Pavel Zorin-Kranich
