Package 'mvDFA'

Analyze univariate time series and estimate long memory using Detrended Fluctuations Analysis (DFA; Peng et al., 1995)

Description

Analyze univariate time series and estimate long memory using Detrended Fluctuations Analysis (DFA; Peng et al., 1995)

Usage

DFA(X, steps = 50, brownian = FALSE, degree = 1, verbose = TRUE, cores = 1)

Arguments

X	Univariate time series.
steps	Maximum number of window sizes. These are spread logarithmically. If time series is short and steps is large, fewer window sizes are drawn. Default to 50.
brownian	Indicator whether time series is assumed to be brownian (i.e. variance increases proportional to time)
degree	The maximum order of the detrending polynomial in the segments. This influences the smallest window size minS such that minS = degree + 2.
verbose	Indicator whether additional info should be printed. Default to TRUE.
cores	Number of cores used in computation. Default to 1.

Value

Returns list of Root Mean Squares per window size RMS_s, the window sizes S and the estimated long memory coefficient L - the Hurst Exponent.

References

Peng, C. K., Havlin, S., Stanley, H. E., & Goldberger, A. L. (1995). Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time-series. Chaos, 5, 82–87. <doi:10.1063/1.166141>

Examples

X <- rnorm(500) # generate Gaussian white noise (i.i.d. standard normal variables)
DFA(X = X, steps = 5) # steps = 5 is only for demonstration,
                        # use many steps instead, e.g. steps = 50!

---

Analyze multivariate correlated time series and estimate long memory by the extension of the using univariate Detrended Fluctuations Analysis (DFA; Peng et al., 1995) to multivariate time series: mvDFA

Description

Analyze multivariate correlated time series and estimate long memory by the extension of the using univariate Detrended Fluctuations Analysis (DFA; Peng et al., 1995) to multivariate time series: mvDFA

Usage

mvDFA(
  X,
  steps = 50,
  degree = 1,
  verbose = FALSE,
  cores = 1,
  covlist = FALSE,
  brownian = FALSE
)

Arguments

X	Matrix or data.frame containing the time series in long format.
steps	Maximum number of window sizes. These are spread logarithmically. If time series is short and steps is large, fewer window sizes are drawn. Default to 50. The dimensions (ncol(X)) and the degree influence the smallest possible window size.
degree	The maximum order of the detrending polynomial in the segments. This influences the smallest window size minS such that minS = d + degree + 2, where d is the dimension of the time series.
verbose	Indicator whether additional info should be printed. Default to TRUE.
cores	Number of cores used in computation. Default to 1.
covlist	Indicator whether covariance of the time series per window size should be saved in a list.
brownian	Indicator whether time series are assumed to be brownian (i.e. variance increases proportional to time)

Value

An object of class mvDFA containing long memory coefficients (Hurst exponents) and corresponding further informations:

Ltot	the estimated long memory coefficient for the multivariate time series using the total variance approach
Lgen	the generalized approach
Lfull	the average covariance approach
LmeanUni	average Hurst exponent across all time series
univariate_DFA	univariate Hurst exponents
R2tot	R-squared of total variance approach in regression of log10(RMS) vs log10(S)
R2gen	R-squared of generalized variance approach in regression of log10(RMS) vs log10(S)
R2full	R-squared of covariance approach in regression of log10(RMS) vs log10(S)
R2meanUni	average R-squared across all time series in regression of log10(RMS) vs log10(S)
R2univariate_DFA	R-squares of single time series approach in regression of log10(RMS) vs log10(S)
RMS_tot	a list of Root Mean Squares per window size corresponding to the total variance approach
RMS_gen	a list of Root Mean Squares per window size corresponding to the total generalized approach
Cov_RMS_s	a list of Root Mean Squares per window size corresponding to the covariance approach
S	window sizes used
CovRMS_list	a list of covariance matrices per S may be returned

References

Peng, C. K., Havlin, S., Stanley, H. E., & Goldberger, A. L. (1995). Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time-series. Chaos, 5, 82–87. <doi:10.1063/1.166141>

Examples

Sigma <- matrix(.5, 3, 3); diag(Sigma) <- 1
# generate correlated Gaussian white noise (i.i.d. multivariate normal variables)
X <- mvtnorm::rmvnorm(n = 500, sigma = Sigma)
mvDFA(X = X, steps = 5) # steps = 5 is only for demonstration,
                        # use many steps instead, e.g. steps = 50!

---

print object of class DFA

Description

print object of class DFA

Usage

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

Arguments

x	object of class DFA to print.
...	further parameters.

Value

Truncates the output printed into the console of objects of class DFA, but does not change object itself.

---

print object of class mvDFA

Description

print object of class mvDFA

Usage

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

Arguments

x	object of class DFA to print.
...	further parameters.

Value

Truncates the output printed into the console of objects of class mvDFA, but does not change object itself.

---

Approximate correlated time series with given Hurst Exponent

Description

Approximation of correlated time series with given "Hurst" exponents. Internally longmemo::simFGN0 or longmemo::simFGN.fft are used which simulate Gaussian series by generating fractional ARIMA(0,h,0) models (with $h=H-1/2$, longmemo::FGN0), or fractional Gaussian noise longmemo::FGN.fft. We cautiously note that we use empirical scaling (i.e., the variances are scaled to be 1 in the sample not the population), hence the between sample variance may be underrepresented. We further note that the covariance estimates for correlated time series (not using increments) is unstable.

Usage

simulate_cMTS(
  N,
  H,
  Sigma,
  simulation_process = "FGN0",
  decomposition = "chol",
  cor_increments = TRUE,
  X0 = rep(0, ncol(Sigma))
)

Arguments

N	Length of Times Series
H	Hurst Exponents for d time series. These are then mixed using one of two different decompositions of the given covariance matrix Sigma.
Sigma	Positive semi definite covariance matrix of desired multi-dimensional time series.
simulation_process	The simulation process passed to the longmemo::sim... function. Can either be longmemo::simFGN.fft (using FFT) or longmemo::simFGN0 (using fractional gaussian processes). FGN0 looks more like rnorm, when H=0.5. DEFAULT to "FGN0". Use simulation_process="FGN.fft" to use the FFT based version.
decomposition	Character whether the Cholesky decomposition "chol" (or "cholesky") should be used or whether the eigen decomposition should be used (decomposition = "eigen"). DEFAULT to "chol".
cor_increments	Logical, whether to correlate the increments or the time series themselves. Default to TRUE.
X0	Starting values for the time series if increments are correlated. Default to rep(0, ncol(Sigma)), i.e., the zero vector of required length.

Value

Returns a multivariate correlated time series with covariance matrix Sigma. The Hurst exponents are only approximating the univariate ones, since they result from mixed time series. Uncorrelated time series keep their univariate Hurst exponents H.

Examples

Sigma <- matrix(.5, 3, 3); diag(Sigma) <- c(1,2,3)
data <- simulate_cMTS(N = 10^5, Sigma = Sigma, H = c(.2, .5, .7),
                      cor_increments = TRUE)
cov(data)
cov(apply(data,2,diff))

---

Simulate the Lorenz System with noise

Description

Simulate the Lorenz System with noise

Usage

simulate_Lorenz_noise(
  N = 1000,
  delta_t = NULL,
  tmax = 50,
  X0 = 0,
  Y0 = 1,
  Z0 = 1,
  sdX = NULL,
  sdY = NULL,
  sdZ = NULL,
  sdnoiseX,
  sdnoiseY,
  sdnoiseZ,
  s = 10,
  r = 28,
  b = 8/3,
  naive = FALSE,
  return_time = TRUE
)

Arguments

N	Length of Times Series
delta_t	Step size for time scale. If NULL this is derived using N and tmax. DEFAULT to NULL.
tmax	Upper bound of the time scale. This argument is ignored if delta_t is provided. DEFAULT to 50.
X0	Initial value for X at t=0. DEFAULT to 0.
Y0	Initial value for Y at t=0. DEFAULT to 1.
Z0	Initial value for Z at t=0. DEFAULT to 1.
sdX	Use this argument to rescale the X-coordinate to have the desired standard deviation (exactly). This is ignored if set to NULL. DEFAULT to NULL.
sdY	Use this argument to rescale the Y-coordinate to have the desired standard deviation (exactly). This is ignored if set to NULL. DEFAULT to NULL.
sdZ	Use this argument to rescale the Z-coordinate to have the desired standard deviation (exactly). This is ignored if set to NULL. DEFAULT to NULL.
sdnoiseX	Standard deviation of Gaussian noise of X-coordinate. If set to 0, no noise is created.
sdnoiseY	Standard deviation of Gaussian noise of Y-coordinate. If set to 0, no noise is created.
sdnoiseZ	Standard deviation of Gaussian noise of Z-coordinate. If set to 0, no noise is created.
s	s-parameter of the Lorenz ODE. See Vignette for further details. DEFAULT to 10, which is the original value chosen by Lorenz.
r	r-parameter of the Lorenz ODE. See Vignette for further details. DEFAULT to 28, which is the original value chosen by Lorenz.
b	b-parameter of the Lorenz ODE. See Vignette for further details. DEFAULT to 8/3, which is the original value chosen by Lorenz.
naive	Logical whether naive calculation should be used. DEFAULT to FALSE.
return_time	Logical whether the time-coordinate should be included in the returned data.frame. DEFAULT to TRUE.

Value

Returns a three dimensional time series as data.frame following the Lorenz system (Lorenz, 1963, <doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2>).

References

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of atmospheric sciences, 20(2), 130-141. <doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2>

---

Approximate correlated time series from white, pink and brown noise from independent realization of normal variables

Description

Approximation of correlated time series representing "white", "pink" or "brown" noise from independent realization of normal variates Internally normal variables are simulated using rnorm and then are cumulated for white or brown noise and we use RobPer::TK95 for the generation of pink noise. We cautiously note that we use empirical scaling (i.e., the variances are scaled to be 1 in the sample not the population), hence the between sample variance may be underrepresented. We further note that the covariance estimates for correlated time series (not using increments) is unstable.

Usage

simulate_MTS_mixed_white_pink_brown(
  N,
  Sigma,
  process = "white",
  decomposition = "chol",
  cor_increments = TRUE,
  X0 = rep(0, ncol(Sigma))
)

Arguments

N	Length of multivariate Times Series
Sigma	Positive semi definite covariance matrix the increments of desired multi dimensional time series. The dimensionality of Sigma sets the dimension of the time series. The variance scale the time. If the variances are all 1, then each data point represents one unit of time.
process	Type of process. Can either be "white", "brown" or "pink". Default to "white". If process is a vector, a mixture of the three process is generated, correlated by Sigma.
decomposition	Character whether the Cholesky decomposition "chol" (or "cholesky") should be used or whether the eigen decomposition should be used (decomposition = "eigen"). DEFAULT to "chol".
cor_increments	Logical, whether to correlate the increments or the time series themselves. Default to TRUE.
X0	Starting values for the time series if increments are correlated. Default to rep(0, ncol(Sigma)), i.e., the zero vector of required length.

Value

Returns a multivariate correlated time series with covariance matrix 'Sigma'. The Hurst exponents are only approximating the univariate ones, since they result from mixed time series. Here, a mixture of "white", "pink" and "brown" noise can be chosen from. Uncorrelated time series keep their univariate Hurst exponent 'H'.

Examples

Sigma <- matrix(.5, 3, 3); diag(Sigma) <- c(1,2,3)
data <- simulate_MTS_mixed_white_pink_brown(N = 10^5, Sigma = Sigma,
                                            process = c("white", "pink", "brown"),
                                            cor_increments = FALSE)
cov(data) # unstable covariances
cov(apply(data,2,diff))

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