Efficient computation of p-variation in metric spaces.

Let p≥1. For a discrete time path X₀,...,X_N in a metric space with distance d, its p-variation is

sup Σ_k d(X_nk, X_nk-1)^p,

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