pvar-package | R Documentation |
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.
Package: | pvar |
Type: | Package |
Version: | 2.2.5 |
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 and Maintainer: Vygantas Butkus <Vygantas.Butkus@gmail.com>.
Special thanks to Rimas Norvaisa the supervisor of my studies.
[1] V. Butkus, R. Norvaisa. Lith Math J (2018). https://doi.org/10.1007/s10986-018-9414-3
[2] R. M. Dudley, R. Norvaisa. An Introduction to p-variation and Young Integrals, Cambridge, Mass., 1998.
[3] 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.
[4] 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.
[5] J. Qian. The p-variation of Partial Sum Processes and the Empirical Process. The Annals of Probability, 1998, Vol. 26, No. 3, 1370-1383.
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
).
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