A function to compute Abbe values

Description

Calculates "Abbe" values.

Usage

Abbe(x)

Arguments

Details

"Abbe" values quantify the degree of smoothness of a time series. Decreases to zero for very smooth time series, tends to unity for purely white noise. Following Tarnopolski et al. Physica A 461 (2016) 662-673.

Value

Abbe value

Author(s)

Sebastian Sippel

References

Tarnopolski et al. (2016), Physica A 461, 662-673.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
Abbe(x)

A function to compute Horizontal Visibility Graphs and associated statistics

Description

Calculates a Horizontal Visibility Graph

Usage

HVG(x, meth, maxL, rho)

Arguments

Details

Horizontal Visibility Graphs map a time series into a complex network. Following Luque, B., Lacasa, L., Ballesteros, F. and Luque, J., 2009. Horizontal visibility graphs: Exact results for random time series. Physical Review E, 80(4), p.046103. ATTENTION: This function is still in development and needs further testing!

Value

A list that contains the adjacency matrix, degree distribution, and further HVG-based statistics.

Author(s)

Sebastian Sippel

References

Luque, B., Lacasa, L., Ballesteros, F. and Luque, J., 2009. Physical Review E, 80(4), p.046103.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^2)
HVG(x, meth = "HVG", maxL = 10^9, rho = NA)

A function to compute the MPR-complexity

Description

The function computes the MPR complexity, i.e. a generalized (global) complexity measure based on the Jenson-Shannon divergence.

Usage

MPR_complexity(opd)

Arguments

Details

Generalized complexity measures combine an information measure (i.e. entropy) with the distance of the distribution from the uniform distribution ("disequilibrium"). As a global measure, MPR-complexity is insensitive to the permutation coding scheme.

Value

The normalized MPR complexity measure in the range [0, 1].

Author(s)

Sebastian Sippel

References

Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
MPR_complexity(opd)

A function to compute Turning points

Description

Calculates Turning point values.

Usage

Turning_point(x)

Arguments

Details

Turning point of a time series. Following Tarnopolski et al. Physica A 461 (2016) 662-673.

Value

Turning point value

Author(s)

Sebastian Sippel

References

Tarnopolski et al. (2016), Physica A 461, 662-673.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
Turning_point(x)

A function to create new pattern-coding schemes for the Fisher Information.

Description

Adjusts and reorders a pattern ordering matrix.

Usage

adjust_pattern(pattern_matrix, adjustment)

Arguments

Details

This function reorders permutations based on "jumps" or based on "bitflips".

Value

A numeric matrix that contains the permutation matrix.

Author(s)

Sebastian Sippel

References

Sebastian Sippel, 2014. Master Thesis. University of Bayreuth.

A function to compute Renyi complexity

Description

Renyi complexity

Usage

complexity_Renyi(opd, alpha)

Arguments

Details

This function calculates the Renyi complexity as described in Jauregui et al., Physica A, 498 74-85, 2018.

Value

The Renyi complexity value.

Author(s)

Sebastian Sippel

References

Jauregui et al., Physica A, 498 74-85, 2018.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
complexity_Renyi(opd = opd, alpha = 0.5)

A (low-level) function to compute the Fisher-information

Description

The function computes the Fisher information, i.e. a local information measure based on two different discretizations.

Usage

fis(opd, discretization)

Arguments

Details

The Fisher information is a local information and complexity measure, computed based on the ordinal pattern distribution. The Fisher information is based on local gradients, hence it is sensitive to the permutation coding scheme. Options for discretization: 'Olivares.2012' or 'Ferri.2009', following Fisher Information discretization schemes in the respective publications.

Value

The normalized Fisher information measure in the range [0, 1].

Author(s)

Sebastian Sippel

References

Olivares, F., Plastino, A. and Rosso, O.A., 2012. Ambiguities in Bandt-Pompe's methodology for local entropic quantifiers. Physica A: Statistical Mechanics and its Applications, 391(8), pp.2518-2526. Ferri, G.L., Pennini, F. and Plastino, A., 2009. LMC-complexity and various chaotic regimes. Physics Letters A, 373(26), pp.2210-2214.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
fis(opd = opd)

A function to generate the Lehmer permutation ordering.

Description

Generates all permutations of a given embedding dimension, ordered according to the Lehmer coding scheme.

Usage

generate_lehmerperm_matrix(ndemb)

Arguments

Details

This function converts ranks to indices and back.

Value

A numeric matrix that contains the Lehmer permutation pattern.

Author(s)

Sebastian Sippel

References

http://www.keithschwarz.com/interesting/code/?dir=factoradic-permutation

A function to compute global information and complexity measures for time series

Description

This is a high-level function that calculates global complexity measures directly from a given time series or ordinal pattern distribution.

Usage

global_complexity(x = NA, opd = NA, ndemb)

Arguments

Details

This function calculates the following global measures of complexity and information:

• Permutation Entropy (PE, cf. Bandt and Pompe, 2002)

• Permutation Statistical complexity (MPR complexity, cf. Martin, Plastino and Rosso, 2006)

• Number of "forbiden patterns" (cf. Amigo 2010)

Value

A named vector containing the three global complexity measures.

Author(s)

Sebastian Sippel

References

Bandt, C. and Pompe, B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17), p.174102. Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462. Amigo, J., 2010. Permutation complexity in dynamical systems: ordinal patterns, permutation entropy and all that. Springer Science & Business Media.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
global_complexity(x = x, ndemb = 6)
# or:
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
global_complexity(opd = opd, ndemb = 6)

Distance measure between ordinal pattern distributions: Hellinger distance

Description

Compute the Hellinger Distance

Usage

hellinger_distance(p, q)

Arguments

Details

This function returns a distance measure.

Value

A vector of length 1.

Author(s)

Sebastian Sippel

References

none

Examples

p = ordinal_pattern_distribution(rnorm(10000), ndemb = 5)
q = ordinal_pattern_distribution(arima.sim(model=list(ar=0.9), n= 10000), ndemb = 5)
hellinger_distance(p=p, q = q)

A function to generate a time series from the Henon Map

Description

Generates a time series from the Henon map

Usage

henon_map(N, a, b, startx="rand", starty="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

Schuster, H.G., 1988. Deterministic chaos. An Introduction.

Examples

henon_map(N = 10^4, a=1.4, b=0.3)

Generalized disequilibrium measure for ordinal pattern distributions based on the Jensen-Shannon Divergence

Description

Computes a normalized form of the Jensen-Shannon Divergence

Usage

jensen_shannon_divergence(p, q="unif")

Arguments

Details

This function returns a distance measure.

Value

A vector of length 1.

Author(s)

Sebastian Sippel

References

Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Examples

p = ordinal_pattern_distribution(rnorm(10000), ndemb = 5)
q = ordinal_pattern_distribution(arima.sim(model=list(ar=0.9), n= 10000), ndemb = 5)
jensen_shannon_divergence(p = p, q = q)

Limit curves in the Entropy-Complexity plane

Description

Compute the limit curves in the Entropy Complexity plane

Usage

limit_curves(ndemb, fun = "min")

Arguments

Details

This function returns the respective limit curve.

Value

A list with two entries

Author(s)

Sebastian Sippel

References

none

A function to generate a time series from the logistic map

Description

Generates a time series from the logistic map

Usage

logistic_map(N, r, start="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature, 261(5560), pp.459-467.

Examples

logistic_map(N = 10^4, r=4)

Maximum curve of time-causal entropy-complexity plane at ndemb=3

Description

Maximum curve of time-causal entropy-complexity plane at ndemb=3

Usage

maxd3

Format

A data frame with 494 rows and 2 columns:

x

x-values of minimum curve if ndemb==3

y

y-values of minimum curve if ndemb==3

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Maximum curve of time-causal entropy-complexity plane at ndemb=4

Description

Maximum curve of time-causal entropy-complexity plane at ndemb=4

Usage

maxd4

Format

A data frame with 2139 rows and 2 columns:

x

x-values of minimum curve if ndemb==4

y

y-values of minimum curve if ndemb==4

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Maximum curve of time-causal entropy-complexity plane at ndemb=5

Description

Maximum curve of time-causal entropy-complexity plane at ndemb=5

Usage

maxd5

Format

A data frame with 4151 rows and 2 columns:

x

x-values of minimum curve if ndemb==5

y

y-values of minimum curve if ndemb==5

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Maximum curve of time-causal entropy-complexity plane at ndemb=6

Description

Maximum curve of time-causal entropy-complexity plane at ndemb=6

Usage

maxd6

Format

A data frame with 3438 rows and 2 columns:

x

x-values of minimum curve if ndemb==6

y

y-values of minimum curve if ndemb==6

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

A function to compute Mean information gain (MIG) and Fluctuation complexity (FC)

Description

Calculates MIG and FC

Usage

migfc(x, L)

Arguments

Details

MIG and FC are based on a median partitioning of the time series Following Hauhs, M. and Lange, H., 2008. Classification of runoff in headwater catchments: A physical problem?. Geography Compass, 2(1), pp.235-254. ATTENTION: This function is still in development and needs further testing!

Value

A list containing MIG, FC and transition matrices.

Author(s)

Sebastian Sippel

References

Hauhs, M. and Lange, H., (2008) Geography Compass, 2(1), pp.235-254.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
migfc(x, L=4)

Minimum curve of time-causal entropy-complexity plane at ndemb=3

Description

Minimum curve of time-causal entropy-complexity plane at ndemb=3

Usage

mind3

Format

A data frame with 500 rows and 2 columns:

x

x-values of minimum curve if ndemb==3

y

y-values of minimum curve if ndemb==3

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Minimum curve of time-causal entropy-complexity plane at ndemb=4

Description

Minimum curve of time-causal entropy-complexity plane at ndemb=4

Usage

mind4

Format

A data frame with 500 rows and 2 columns:

x

x-values of minimum curve if ndemb==4

y

y-values of minimum curve if ndemb==4

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Minimum curve of time-causal entropy-complexity plane at ndemb=5

Description

Minimum curve of time-causal entropy-complexity plane at ndemb=5

Usage

mind5

Format

A data frame with 500 rows and 2 columns:

x

x-values of minimum curve if ndemb==5

y

y-values of minimum curve if ndemb==5

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

Minimum curve of time-causal entropy-complexity plane at ndemb=6

Description

Minimum curve of time-causal entropy-complexity plane at ndemb=6

Usage

mind6

Format

A data frame with 500 rows and 2 columns:

x

x-values of minimum curve if ndemb==6

y

y-values of minimum curve if ndemb==6

...

Source

Computed based on Martin, M.T., Plastino, A. and Rosso, O.A., 2006. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and its Applications, 369(2), pp.439-462.

A function to compute bitflip statistics and time series

Description

Computation of bitflip statistics of a time series

Usage

nbitflips(x, ndemb)

Arguments

Details

This function returns a histogram and time series of the number of bitflips occurring in the associated ordinal patterns. NA values are allowed, and any pattern that contains at least one NA value will be ignored. WARNING: Can be slow with very long time series (n > 10^7).

Value

A list with two entries is returned.

Author(s)

Sebastian Sippel

References

Sippel, S., 2014. Evaluating the carbon dynamics of biogeochemical models using statistical complexity measures. Master Thesis, University of Bayreuth.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
nbitflips(x = x, ndemb = 6)

A function to compute ordinal pattern statistics

Description

Computation of the ordinal patterns of a time series (see e.g. Bandt and Pompe 2002)

Usage

ordinal_pattern_distribution(x, ndemb)

Arguments

Details

This function returns the distribution of ordinal patterns using the Keller coding scheme, detailed in Physica A 356 (2005) 114-120. NA values are allowed, and any pattern that contains at least one NA value will be ignored. (Fast) C routines are used for computing ordinal patterns.

Value

A character vector of length factorial(ndemb) is returned.

Author(s)

Sebastian Sippel

References

Bandt, C. and Pompe, B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17), p.174102.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
ordinal_pattern_distribution(x = x, ndemb = 6)

A function to compute time series of ordinal patterns

Description

Computation of the ordinal patterns of a time series (see e.g. Bandt and Pompe 2002)

Usage

ordinal_pattern_time_series(x, ndemb)

Arguments

Details

This function returns the distribution of ordinal patterns using the Keller coding scheme, detailed in Physica A 356 (2005) 114-120. NA values are allowed, and any pattern that contains at least one NA value will be ignored. (Fast) C routines are used for computing ordinal patterns.

Value

A character vector of length(x) is returned.

Author(s)

Sebastian Sippel

References

Bandt, C. and Pompe, B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17), p.174102.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
ordinal_pattern_time_series(x = x, ndemb = 6)

A function to compute the permutation entropy

Description

Computation of the permutation entropy of a time series based on its ordinal pattern distribution (see Bandt and Pompe 2002). Permutation entropy is a global information measure, hence insensitive to the permutation ordering scheme.

Usage

permutation_entropy(opd)

Arguments

Details

This function calculates the permutation entropy as described in Bandt and Pompe 2002.

Value

The normalized permutation entropy as a numeric value in the range [0,1].

Author(s)

Sebastian Sippel

References

Bandt, C. and Pompe, B., 2002. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17), p.174102.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
permutation_entropy(opd)

A function to compute Renyi entropy

Description

Renyi permutation entropy

Usage

permutation_entropy_Renyi(opd, alpha)

Arguments

Details

This function calculates the Renyi entropy as described in Jauregui et al., Physica A, 498 74-85, 2018.

Value

The Renyi entropy value.

Author(s)

Sebastian Sippel

References

Jauregui et al., Physica A, 498 74-85, 2018.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
permutation_entropy_Renyi(opd = opd, alpha = 0.5)

A function to compute q-log permutation entropy

Description

q-log permutation entropy

Usage

permutation_entropy_qlog(opd, q)

Arguments

Details

This function calculates the q-log permutation entropy as described in Ribeiro et al. 2017.

Value

The q-log permutation entropy value.

Author(s)

Sebastian Sippel

References

Ribeiro et al. 2017, https://arxiv.org/abs/1705.04779.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
permutation_entropy_qlog(opd = opd, q = 1)

A function to generate k-noise

Description

Generates samples of power law noise.

Usage

powernoise(k, N)

Arguments

Details

Generates samples of power law noise. The power spectrum of the signal scales as f^(-k). The R function uses fft(), similarly to the knoise_fft Matlab function.

Value

A named list with three entries is returned. x - N x 1 vector of power law samples

Author(s)

Sebastian Sippel and Holger Lange

Examples

powernoise_series = powernoise(k=2, N=10000)

A function to compute q-log complexity

Description

q-log complexity

Usage

q_complexity(opd, q)

Arguments

Details

This function calculates the q-log complexity as described in Ribeiro et al. 2017.

Value

The q-log complexity value.

Author(s)

Sebastian Sippel

References

Ribeiro et al. 2017, https://arxiv.org/abs/1705.04779.

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
opd = ordinal_pattern_distribution(x = x, ndemb = 6)
q_complexity(opd = opd, q = 1)

A function to generate a time series from the Quadratic map

Description

Generates a time series from the Quadradtic map

Usage

quadratic_map(N, k, start="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

Grebogi, C., Ott, E. and Yorke, J.A., 1983. Crises, sudden changes in chaotic attractors, and transient chaos. Physica D: Nonlinear Phenomena, 7(1-3), pp.181-200.

Examples

quadratic_map(N = 10^4, k=1.4)

A function to convert a "ranks-based" permutation notation to an "index-based" permutation scheme.

Description

Converts permutations denoted by ranks to permutations denoted by indices and back.

Usage

rank_to_permutation(pattern, permutation.notation)

Arguments

Details

This function converts ranks to indices and back.

Value

A numeric vector, which contains the transformed permutation.

Author(s)

Sebastian Sippel

References

Sebastian Sippel (2014). Master Thesis. University of Bayreuth.

A function to generate a time series from the Schuster Map

Description

Generates a time series from the Schuster map

Usage

schuster_map(N, z, start="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

Schuster, H.G., 1988. Deterministic chaos. An Introduction.

Examples

schuster_map(N = 10^4, z=2)

A function to generate a time series from the logistic map

Description

Generates a time series from the Skew-Tent map

Usage

skew_tent_map(N, a, start="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

Schuster, H.G., 1988. Deterministic chaos. An Introduction.

Examples

skew_tent_map(N = 10^4, a=0.1847)

A function to generate a time series from the logistic map

Description

Generates a time series from the logistic map

Usage

tent_map(N, mu, start="rand", disregard_N=0)

Arguments

Value

A vector of length N

Author(s)

Sebastian Sippel

References

Feldman, D.P., McTague, C.S. and Crutchfield, J.P., 2008. The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(4), p.043106.

Examples

tent_map(N = 10^4, mu=1.8)

A function to generate a vector from an index-transformation vector from a permutation coding scheme

Description

Generates a position vector to change the ordinal pattern distribution in the default permutation coding scheme (i.e. generated by ordinal_pattern_distribution(x, ndemb)) into a user-specified coding scheme. This is a required input for the function changePermCodingOPD.

Usage

transformPermCoding(target_pattern, ndemb)

Arguments

Details

This function returns a character vector to transform the output of ordinal_pattern_distribution (permutation coding as of Keller and Sinn, 2005) into a user-specified permutation coding scheme. For example, pattern #5 in "lehmerperm" (ndemb = 5) is given by the ranks c(0, 1, 4, 2, 3). This corresponds to pattern #41 in the (original) Keller coding scheme, as given by transformPermCoding(target_pattern = "lehmerperm", ndemb = 5)[5].

Value

A numeric vector of length factorial(ndemb), which contains the positions of the corresponding patterns in the Keller Coding scheme.

Author(s)

Sebastian Sippel

References

Olivares, F., Plastino, A. and Rosso, O.A., 2012. Ambiguities in Bandt-Pompe's methodology for local entropic quantifiers. Physica A: Statistical Mechanics and its Applications, 391(8), pp.2518-2526.

Examples

transformPermCoding(target_pattern = "lehmerperm", ndemb = 4)

A function to compute weighted ordinal pattern statistics

Description

Computation of weighted ordinal patterns of a time series. Weights can be generated by a user-specified function (e.g. variance-weighted, see Fadlallah et al 2013).

Usage

weighted_ordinal_pattern_distribution(x, ndemb)

Arguments

Details

This function returns the distribution of weighted ordinal patterns using the Keller coding scheme, detailed in Physica A 356 (2005) 114-120. NA values are allowed. The function uses old and slow R routines and is only maintained for comparability. For faster routines, see weighted_ordinal_pattern_distribution.

Value

A character vector of length factorial(ndemb) is returned.

Author(s)

Sebastian Sippel

References

Fadlallah, B., Chen, B., Keil, A. and Principe, J., 2013. Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information. Physical Review E, 87(2), p.022911.

See Also

weighted_ordinal_pattern_distribution

Examples

x = arima.sim(model=list(ar = 0.3), n = 10^4)
weighted_ordinal_pattern_distribution(x = x, ndemb = 6)