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

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] \rightarrow R and 0 < p < \infty p-variation is defined as

v_p(f) = \sup \left\{ \sum_{i=1}^m |f(t_i) - f(t_{i-1})|^p : 0=t_0<t_1<\dots<t_m=1, m \geq 1 \right\}

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\left\{ \sum_{i=1}^k |X_{j_i}-X_{j_{i-1}}|^p: 0=j_0<j_1<\dots<j_k=n, \; k=1,2,...,n \right\}

The points 0=t_0<t_1<\dots<t_m=1 (or 0=j_0<j_1<\dots<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] 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.

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).

Concatenate strings

Description

Concatenate Strings

Usage

x %.% y

Arguments

Details

The same result may be achieved with paste, but in some circumstance this function is more user friendly.

Value

A character string of the concatenated values.

See Also

paste

Examples

paste('I ', 'love ', 'R.', sep='')
'I ' %.% 'love ' %.% 'R.'

x = c(2,1,6,7,9)
paste('The length of vector (', paste(x , sep='', collapse =','), ') is ', length(x) , sep='')
'The length of vector (' %.% paste(x , sep='', collapse =',') %.% ') is ' %.% length(x)

Addition of p-variation

Description

Merges two objects of p-variation and effectively recalculates the p-variation of joined sample.

Usage

AddPvar(PV1, PV2, AddIfPossible = TRUE)

Arguments

Details

Note: a short form of AddPvar(PV1, PV2 is PV1 + PV2.

Value

An object of the class pvar. See pvar.

Examples

### creating two pvar objects:
x = rwiener(1000)
PV1 = pvar(x[1:500], 2)
PV2 = pvar(x[500:1000], 2)

layout(matrix(c(1,3,2,3), 2, 2))
plot(PV1)
plot(PV2)
plot(AddPvar(PV1, PV2))
layout(1)

### AddPvar(PV1, PV2) is eqivavalent to PV1 + PV2
IsEqualPvar(AddPvar(PV1, PV2), PV1 + PV2)

Addition of p-variation (in C++)

Description

An internal function(written in C++) that merges two objects of pvar and effectively recalculates the p-variation of joined sample.

Usage

AddPvarC(PV1, PV2, AddIfPossible = TRUE)

Arguments

Details

This is an internal function, therefore, users should not call this function directly (rather use AddPvar or pv1 + pv2).

Value

An object of the class pvar.

Bridge transformation

Description

Transforms data by Bridge transformation.

Usage

BridgeT(x, normalize = TRUE)

Arguments

Details

Let n denotes the length ox x. For each m \in [1,n] bridge transformations BridgeT is defined as

BridgeT(m, x) = \left\{ \sum_{i=1}^m x_i - \frac{m}{n} \sum_{i=1}^n x_i \right\} .

Meanwhile, the transformation with normalization is

BridgeT(m, x) = \frac{1}{\sqrt{n var(x)}} \left\{ \sum_{i=1}^m x_i - \frac{m}{n} \sum_{i=1}^n x_i \right\} .

Value

A numeric vector.

See Also

PvarBreakTest, rbridge

Examples

x <- rnorm(1000)
Bx <- BridgeT(x, FALSE)

op <- par(mfrow=c(2,1),mar=c(4,4,2,1))
plot(cumsum(x), type="l")
plot(Bx, type="l")
par(op)

Change Points of a numeric vector

Description

Finds changes points (i.e. corners) in the numeric vector.

Usage

ChangePoints(x)

Arguments

Details

The end points of the vector will be always included in the results.

Value

The vector of index of change points.

Examples

x <- rwiener(100)
cid <- ChangePoints(x)
plot(x, type="l")
points(time(x)[cid], x[cid], cex=0.5, col=2, pch=19)

Data sets of Monte-Carlo simulations results

Description

The test PvarBreakTest uses quantiles from Monte-Carlo simulations. The results of the simulations are saved in these data sets.

Usage

PvarQuantileDF
MeanCoef
SdCoef

Format

the PvarQuantileDF is a data.frame with fields prob an Qaunt. The field brob represent the probability and Quant gives correspondingly quantile. MeanCoef and SdCoef is a named vector used in functions getMean and getSd.

Details

The distribution of p-variation of BridgeT(x) are unknown, therefore it was approximated form Monte-Carlo simulation based on 140 millions iterations. The data frame PvarQuantile summarize the distribution of normalized statistics. Meanwhile, MeanCoef and SdCoef defines the coefficients of functional form of mean and sd statistics of PvarBreakTest statistics (see getMean).

Author(s)

Vygantas Butkus <Vygantas.Butkus@gmail.com>

Source

Monte-Carlo simulation

Test if two 'pvar' objects are equivalent.

Description

Two pvar objects are considered to be equal if they have the same x, p, value and the same value of x in the points of partition (the index of partitions are not necessary the same). All other tributes like dname or TimeLabel are not important.

Usage

IsEqualPvar(pv1, pv2)

Arguments

Examples

x <- rwiener(100)
pv1 <- pvar(x, 2)
pv2 <- pvar(x[1:50], 2) + pvar(x[50:101], 2)
IsEqualPvar(pv1, pv2)

Structural break test

Description

This function performs structural break test that is based on p-variation.

Usage

PvarBreakTest(x, TimeLabel = as.vector(time(x)), alpha = 0.05,
  FullInfo = TRUE)

## S3 method for class 'PvarBreakTest'
plot(x, main1 = "Data",
  main2 = "Bridge transformation", ylab1 = x$dname,
  ylab2 = "BridgeT(" %.% x$dname %.% ")", sub2 = NULL,
  col.PP = 3, cex.PP = 0.5, col.BP = 2, cex.BP = 1, cex.DP = 0.5,
  ...)

## S3 method for class 'PvarBreakTest'
summary(object, ...)

Arguments

Details

Lets x be a data that should be tested of structural breaks. Then the p-variation of the BridgeT(x) with p=4 is the test's statistics.

The quantiles of H0 distribution is based on Monte-Carlo simulation of 140 millions iterations. The test is reliable then length(x) is between 100 and 10000. The test might work with other lengths too, but it is not tested well. The test will not compute then length(x)<20.

Value

If FullInfo=TRUE then function returns an object of the class PvarBreakTest. It is the list that contains:

Author(s)

Vygantas Butkus <Vygantas.Butkus@gmail.com>

References

The test was proposed by A. Rackaskas. The test is based on the results given in the flowing article

[1] 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.

See Also

Tests statistics is pvar of the data BridgeT(x)(see BridgeT) with (p=4). The critical value and the approximate p-value of the test might by found by functions PvarQuantile and PvarPvalue.

Examples

set.seed(1)
MiuDiff <- 0.3
x <- rnorm(250*4, rep(c(0, MiuDiff, 0, MiuDiff), each=250))

plot(x, pch=19, cex=0.5, main='original data, with several shifts of mean')
k <- 50
moveAvg <- filter(x, rep(1/k, k))
lines(time(x), moveAvg, lwd=2, col=2)
legend('topleft', c('sample', 'moving average (k='%.%k%.%')'),
       lty=c(NA,1), lwd=c(NA, 2), col=1:2, pch=c(19,NA), pt.cex=c(0.7,1)
       ,inset = .03, bg='antiquewhite1')

xtest <- PvarBreakTest(x)
plot(xtest)

Quantiles and probabilities of p-variation

Description

The distribution of p-variation of BridgeT(x) depends on n=length(x). This fact is important for getting appropriate quantiles (or p-value). These functions helps to deal with it.

Usage

PvarQuantile(n, prob = c(0.9, 0.95, 0.99), DF = PvarQuantileDF)

PvarPvalue(n, stat, DF = PvarQuantileDF)

getMean(n, bMean = MeanCoef)

getSd(n, bSd = SdCoef)

NormalisePvar(x, n, bMean = MeanCoef, bSd = SdCoef)

Arguments

Details

The distribution of p-variance is form Monte-Carlo simulation based on 140 millions iterations. The data frame PvarQuantileDF saves the results of Monte-Carlo simulation.

Meanwhile, MeanCoef and SdCoef defines the coefficients of functional form (conditional on n) of mean and sd statistics.

A functional form of mean and sd statistics are the same, namely

f(n) = b_1 + b_2 n^b_2 .

The coefficients (b_1, b_2, b_3) are saved in vectors MeanCoef and SdCoef. Those vectors are estimated with nls function form Monte-Carlo simulation.

Value

Functions PvarQuantile and PvarPvalue returns a corresponding value quantile or the probability. Functions getMean and getSd returns a corresponding value of mean and sd statistics. Function NormalisePvar returns normalize values.

Note

Arguments n, stat and prob might be vectors, but they can't be vectors simultaneously (at least one of then must be a number).

See Also

PvarBreakTest, PvarQuantileDF, NormalisePvar, getMean, getSd

p-variation summation function

Description

It is the sum of absolute differences in the power of p.

Usage

Sum_p(x, p, lag = 1)

Arguments

Details

This is a function that must be maximized by taking a proper subset of x, i.e. if prt is a p-variation partition of sample x, then Sum_p(x[prt], p) == pvar(x, p)$value.

Value

The number equal to sum((abs(diff(x, lag)))^p)

See Also

pvar

Examples

x = rbridge(1000)
pv = pvar(x, 2); pv 
# Sum_p in supreme partition and the value form pvar must match
Sum_p(x[pv$partition], 2)
pv

p-variation calculation

Description

Calculates p-variation of the sample.

Usage

pvar(x, p, TimeLabel = as.vector(time(x)), LSI = 3)

## S3 method for class 'pvar'
summary(object, ...)

## S3 method for class 'pvar'
plot(x, main = "p-variation", ylab = x$dname,
  sub = "p=" %.% round(x$p, 5) %.% ", p-variation: " %.%
  formatC(x$value, 5, format = "f"), col.PP = 2, cex.PP = 0.5, ...)

Arguments

Details

This function is the main function in this package. It calculates the p-variation of the sample. The formal definition is given in pvar-package.

Value

An object of the class pvar. Namely, it is a list that contains

Author(s)

Vygantas Butkus <Vygantas.Butkus@gmail.com>

See Also

IsEqualPvar, AddPvar, PvarBreakTest.

Examples

### randomised data:
x = rbridge(1000)

### the main functions:
pv = pvar(x, 2)
print(pv)
summary(pv)
plot(pv)

### The value of p-variation is    
pv; Sum_p(x[pv$partition], 2)  

### The meaning of supreme partition points:
pv.PP = pvar(x[pv$partition], TimeLabel=time(x)[pv$partition], 2)
pv.PP == pv.PP
op <- par(mfrow = c(2, 1), mar=c(2, 4, 4, 1))
plot(pv, main='pvar with original data')
plot(pv.PP, main='The same pvar without redundant points')
par(op)

p-variation calculation (in C++)

Description

An internal function(written in C++) that calculates p-variation.

Usage

pvarC(x, p, LSI = 3L)

Arguments

Details

This is a waking horse of this packages, nonetheless, users should not call this function directly (rather use pvar).

Value

An object of the class pvar.

Random process generators

Description

Generate a trajectory of random processes.

Usage

rwiener(frequency = 1000, end = 1)

rbridge(frequency = 1000, end = 1)

rcumbin(frequency = 1000, end = 1)

Arguments

Details

rwiener generate Wiener process via partial sums process and rbridge generate Brownian bridge via rwiener. The original code of rwiener and rbridge was written in the package e1071. In this package these functions was modified to include leading zero in the beginning of the sample.

rcumbin generate partial sums process from random variables with values -1, 0, 1.

Value

A time series containing a simulated realization of random processes. The length of time series is frequency+1, since zero is always included in the beginning of the sample.