Package 'ordinalpattern'

Empirical Ordinal Pattern Distribution

Description

Calculates the empirical ordinal pattern distribution.

Usage

countingpatterns(tsx,d=3,block=FALSE,first=TRUE,tiesmethod=c("random","first"),
generalized=FALSE)

## S3 method for class 'patterncounts'
print(x, ...)

Arguments

tsx	numeric vector representing the univariate time series.
d	numeric value determining the length of the ordinal pattern.
block	logical value determining whether patterns are calculated on disjoint blocks or overlapping blocks.
first	logical value indicating which observartions are dropped if block == TRUE and the time series length is no multiple of d.
tiesmethod	character string specifying how ties, that is equal values, are treated if generalized == FALSE, see ‘Details’.
generalized	logical value determining whether classical ordinal patterns or their generalization with regard to ties are considered, see ‘Details’.
x	object of class "patterncounts", which is the output of countingpatterns.
...	further arguments passed to the internal plotting function.

Details

Ordinal patterns, which are defined as sequences of ranks of d subsequent observations, are a useful tool to describe the dependence within or between time series. That sequences of subseqent observations can either move one observation per time or a whole block of d observations. The former is preferred since it uses more information. If one chooses the later, one has to decide whether the first or the last observations are removed in case that the time series length is no multiple of d. With regard to equal values within a window of consecutive observations (ties), the argument tiesmethod determines the approach for computing the respective ordinal patterns. The “first” method is in favor of increasing patterns, whereas the default “random” puts the equal values in random order.

Beside the classical ordinal patterns, one can also consider the generalized version proposed by Schnurr and Fischer (2022), where the ordinal information of stagnation in the case of ties is also included by taking into account a larger set of patterns.

Value

Object of class "patterncounts" containing the following values:

patterncounts	absolute frequencies of ordinal patterns.
allpatterns	list of all ordinal patterns considered.
d	length of the ordinal pattern.
generalized	logical value determining whether classical ordinal patterns or their generalization with regard to ties are considered.
tiesmethod	character string specifying how ties are treated.
block	logical value determining whether patterns are calculated on disjoint blocks or overlapping blocks.

Author(s)

Angelika Silbernagel

References

Schnurr, A. (2014): An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series, Statistical Papers, vol. 55, 919–931.

Schnurr, A., Dehling, H. (2017): Testing for Structural Breaks via Ordinal Pattern Dependence, Journal of the American Statistical Association, vol. 112, 706–720.

Schnurr, A., Fischer, S. (2022): Generalized ordinal patterns allowing for ties and their application in hydrology, Computational Statistics & Data Analysis, vol 171, 107472.

Examples

set.seed(1066)
countingpatterns(rnorm(100))
countingpatterns(rpois(100,1), generalized=TRUE)

---

Changepoint Detection Using Ordinal Patterns

Description

Test for a change in the dependence structure of two time series using ordinal patterns

Usage

patternchange(tsx,tsy,d=3,conf.level,weight=TRUE,weightfun=NULL,bn=log(length(tsx)),
    kernel=function(x){return(max(0,1-abs(x)))})

## S3 method for class 'change'
plot(x, ...)

Arguments

tsx	numeric vector of first univariate time series.
tsy	numeric vector of second univariate time series.
d	numeric value determining the length of ordinal pattern.
conf.level	numerical value indicating the confidence level of the test.
weight	logical value indicating whether one uses weights of the L1 norm or the empirical probability of identical patterns; see details.
weightfun	function which defines the weights given the L1 norm between the patterns if weight=TRUE. If no weight-function is given, the canonical weight function is used; see details.
bn	numerical value determining the bandwidth of the kernel estimator used to estimate the long run variance.
kernel	kernel function for estimating the long run variance.
x	object of class "change"
...	further arguments passed to the internal plotting function (plot).

Details

Given two timeseries tsx and tsy a cusum type statistic tests whether there is a change in the patter dependence or not. The test is based on a comparison of patterns of length d in tsx and tsy. One can either choose the number of identical patterns (weight=FALSE) or a metric that is defined by the weightfun argument to measure the difference between patterns (weight=TRUE). If no (weightfun) is given, the canonical weightfunction is used, which equals 1 if patterns are identical and 0 if the L1 norm of their difference attains the maximal possible value. The value is linear interpolated in between.

The procedure depends on an estimate of the long run variance. Here a kernel estimator is used. A kernel function and a bandwidth can be set using the arguments kernel and bn. If none of them is given, the bartlett kernel with a bandwidth of log(n), where n equals the length of the timeseries, is used.

Value

Object with classes "change" and "htest" containing the following values:

statistic	the value of the test statistic. Under the null the test statistic follows asymptotically a Kolmogorov Smirnov distribution.
p.value	the p-value of the test.
estimate	the estimated time of change.
null.value	the jump height of the at most one change point model, which is under the null hypothesis always 0.
alternative	a character string describing the alternative hypothesis.
method	a characters string describing the test.
trajectory	the cumulative sum on which the tests are based on. Could be used for additional plots.

Author(s)

Alexander Dürre

References

Schnurr, A. (2014): An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series, Statistical Papers, vol. 55, 919–931.

Schnurr, A., Dehling, H. (2017): Testing for Structural Breaks via Ordinal Pattern Dependence, Journal of the American Statistical Association, vol. 112, 706–720.

See Also

Estimation of the pattern dependence is provided by patterndependence.

Examples

set.seed(1066)
a1 <- cbind(rnorm(100),rnorm(100))
a2 <- rmvnorm(100,sigma=matrix(c(1,0.8,0.8,1),ncol=2))
A <- rbind(a1,a2)
testresult <- patternchange(A[,1],A[,2])
plot(testresult)
testresult

---

Ordinal Pattern Dependence

Description

Calculates the ordinal pattern coefficient and related values.

Usage

patterndependence(tsx,tsy,d=3,block=FALSE,first=TRUE,
tiesmethod=c("random","first"),ordinalcor=c("standard","positive","negative"))

## S3 method for class 'pattern'
plot(x, ...)
## S3 method for class 'pattern'
print(x, ...)

Arguments

tsx	numeric vector representing the first univariate time series.
tsy	numeric vector representing the second univariate time series.
d	numeric value determining the length of the ordinal pattern.
block	logical value determining whether patterns are calculated on disjoint blocks or overlapping blocks.
first	logical value indicating which observartions are dropped if block == TRUE and the time series length is no multiple of d.
tiesmethod	character string specifying how ties, that is equal values, are treated, see ‘Details’.
ordinalcor	character string specifying which ordinal pattern coefficient is output, see ‘Details’.
x	object of class "pattern", which is the output of patterndependence.
...	further arguments passed to the internal plotting function.

Details

The standard ordinal pattern coefficient is a non-parametric and robust measure of dependence between two time series. It is based on ordinal patterns, which are defined as sequences of ranks of d subsequent observations. This sequences of subseqent observations can either move one observation per time or a whole block of d observations. The former is preferred since it uses more information. If one chooses the later, one has to decide whether the first or the last observations are removed in case that the time series length is no multiple of d. With regard to equal values within a window of consecutive observations (ties), the argument tiesmethod determines the approach for computing the respective ordinal patterns. The “first” method is in favor of increasing patterns, whereas the default “random” puts the equal values in random order.

Beside the default standard ordinal pattern coefficient, which range from -1 to 1, one can also look at the positive and negative ordinal pattern coefficient, which roughly measures whether there are unusual many identical or opposite patterns in the time series.

The plot function draws both time series and shows the six most frequent coinciding pattern with counts on the right. At the bottom, the location of these coinciding patterns is visualized.

Value

Object of class "pattern" containing the following values:

patterncoef	ordinal pattern coefficient.
numbequal	number of equal ordinal patterns.
numbopposite	number of opposite ordinal patterns.
PatternXz	number of ordinal patterns in first time series.
PatternYz	number or ordinal patterns in second time series.
coding	coding of the ordinal patterns, used in PatternXz and PatternYz.
PatternX	numeric vector representing the time series of patterns in tsx.
PatternY	numeric vector representing the time series of patterns in tsy.
tsx	numeric vector representing the first univariate time series.
tsy	numeric vector representing the second univariate time series.
maxpat	number representing the maximal pattern code.
ordinalcor	character string specifying the type of ordinal pattern coefficient.
tiesmethod	character string specifying how ties are treated.
block	logical value determining whether patterns are calculated on disjoint blocks or overlapping blocks.
d	length of the ordinal pattern.
tablesame	numeric vector representing the number of coinciding patterns, apportioned into different patterns.
tableopposite	numeric vector representing the number of reflected patterns, apportioned into different patterns.
indexsame	logic vector indicating whether patterns in both time series coincide.
indexopposite	logic vector indicating whether patterns in both time series are reflected.

Author(s)

Alexander Dürre, Angelika Silbernagel

References

Schnurr, A. (2014): An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series, Statistical Papers, vol. 55, 919–931.

Schnurr, A., Dehling, H. (2017): Testing for Structural Breaks via Ordinal Pattern Dependence, Journal of the American Statistical Association, vol. 112, 706–720.

Examples

set.seed(1066)
patternobj <- patterndependence(rnorm(100),rnorm(100))
plot(patternobj)

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