p-variation calculation and application

Description

This package deals with p-variation for the sample (i.e. the sequence of data values). It gives opportunity to calculate the p-variation for the sample – this is the main purpose of this package. Nonetheless, it could be used to calculate p-variation for arbitrary piecewise monotonic function as well. Moreover, the package includes one example of practical application of the p-variation.

Details

Package: pvar
Type: Package
Version: 2.2
Date: 2016-05-17
License: GPL-2
Institution: Vilnius University Faculty of Mathematics and Informatics

This package is about p-variation. It deals with p-variation of a finite sample data values. To be precise, lets star with the definitions. Originally p-variation is defined for a functions.

For a function f:[0,1] -> R and 0 < p < ∞ p-variation is defined as

v_p(f) = sup { ∑ |f(t_i) - f(t_{i-1})|^p : 0=t_0<t_1<…<t_m=1, m>=1}

Analogically, for a sequences of values X_0, X_1,..., X_n, the p-variation is defined as

v_p({X_i}_{i=0}^n) = max { ∑ |X_{j_i}-X_{j_{i-1}}|^p :0=j_0<j_1<…<j_k=n, \; k=1,2,…,n }

The points 0=t_0<t_1<…<t_m=1 (or 0=j_0<j_1<…<j_k=n) that achieves the maximums is called a supreme partition (or just a partition for short).

There are two main functions that this package is all about, namely it is pvar and PvarBreakTest. The main function in this package is pvar. It calculates the p-variation and the partition. And the function PvarBreakTest is one of the examples of p-variation applications. It performs structural break test of vector x that exams whether there are multiple shifts in mean inside vector x.

All other functions are loaded only for supporting and illustrating purposes.

Author(s)

Author and Maintainer: Vygantas Butkus <Vygantas.Butkus@gmail.com>.

Special thanks to Rimas Norvaisa the supervisor of my studies.

References

[1] R. M. Dudley, R. Norvaisa. An Introduction to p-variation and Young Integrals, Cambridge, Mass., 1998.

[2] R. M. Dudley, R. Norvaisa. Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Springer Berlin Heidelberg, Print ISBN 978-3-540-65975-4, Lecture Notes in Mathematics Vol. 1703, 1999.

[3] R. Norvaisa, A. Rackauskas. Convergence in law of partial sum processes in p-variation norm. Lth. Math. J., 2008., Vol. 48, No. 2, 212-227.

[4] J. Qian. The p-variation of Partial Sum Processes and the Empirical Process. The Annals of Probability, 1998, Vol. 26, No. 3, 1370-1383.

See Also

The main function is pvar - it finds p-variation and the partition that maximizes Sum_p function.

Other important functions is PvarBreakTest it performs structural break test of vector x by calculating p-variations of BridgeT(x) (see BridgeT).