Rmfrac: Simulation and analysis of multifractional processes

Description

A collection of tools for simulating, analysing and visualising multifractional processes and time series. The package includes estimation techniques for the Hurst function, Local Fractal Dimension and several other geometric statistics. It provides highly customisable plotting functions for simulated realisations, user-provided time series and their statistics.

Features

• Simulation of Brownian motion, fractional Brownian motion, fractional Gaussian noise, Brownian bridge and fractional Brownian bridge.

• Simulation of Gaussian Haar-based multifractional process (GHBMP).

• Estimation of Hurst function and Local Fractal Dimension.

• Customisable plotting functions for GHBMP and user provided time series with estimates of Hurst function and Local Fractal Dimension.

• Estimation and visualisation of geometric statistics using realisations of stochastic processes and time series. Clustering based on the Hurst function. Estimating sojourn measure, excursion area, etc.

• An interactive Shiny application that provides options to explore and visualise the core functionalities of the package through simulations and user-provided time series.

Author(s)

• Andriy Olenko ORCID a.olenko@latrobe.edu.au

• Nemini Samarakoon ORCID neminisamarakoon95@gmail.com

See Also

Useful links:

• https://github.com/Nemini-S/Rmfrac

• Report bugs at https://github.com/Nemini-S/Rmfrac/issues

Simulation of Brownian bridge

Description

This function simulates a realisation of the Brownian bridge over the time interval [0,t_end] which has the initial value x_start and terminates at x_end with N time steps.

Usage

Bbridge(x_end, t_end, x_start = 0, N = 1000, plot = FALSE)

Arguments

Value

A data frame where the first column is t and second column is simulated values of the realisation of Brownian bridge.

References

Bianchi, S., Frezza, M., Pianese, A., Palazzo, A.M. (2022). Modelling H-Volatility with Fractional Brownian Bridge. In: Corazza, M., Perna, C., Pizzi, C., Sibillo, M. (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance. MAF 2022. Springer, Cham. doi:10.1007/978-3-030-99638-3_16.

See Also

Bm, FBm, FBbridge, FGn, GHBMP

Examples

Bbridge(x_end=2,t_end=1,plot=TRUE)

Simulation of Brownian motion

Description

This function simulates a realisation of the Brownian motion over the time interval [t_start,t_end] with N time steps and initial value x_start.

Usage

Bm(x_start = 0, t_start = 0, t_end = 1, N = 1000, plot = FALSE)

Arguments

Value

A data frame where the first column is t and second column is simulated values of the realisation of Brownian motion with added constant mean.

See Also

GHBMP, FBm, FGn, Bbridge, FBbridge

Examples

Bm(t_end = 2, plot = TRUE)

Simulation of fractional Brownian bridge

Description

This function simulates a realisation of the fractional Brownian bridge for a provided Hurst parameter over the time interval [0,t_end], which has the initial value x_start and terminates at x_end with N time steps.

Usage

FBbridge(H, x_end, t_end, x_start = 0, N = 1000, plot = FALSE)

Arguments

Value

A data frame where the first column is t and second column is simulated values of the realisation of fractional Brownian bridge.

References

Bianchi, S., Frezza, M., Pianese, A., Palazzo, A.M. (2022). Modelling H-Volatility with Fractional Brownian Bridge. In: Corazza, M., Perna, C., Pizzi, C., Sibillo, M. (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance. MAF 2022. Springer, Cham. doi:10.1007/978-3-030-99638-3_16.

See Also

FBm, FGn, Bm, GHBMP, Bbridge

Examples

FBbridge(H = 0.5, x_end = 2, t_end = 1,plot = TRUE)

Simulation of fractional Brownian motion

Description

This function simulates a realisation of the fractional Brownian motion over the time interval [t_start,t_end] for a provided Hurst parameter, which has the initial value x_start.

Usage

FBm(H, x_start = 0, t_start = 0, t_end = 1, N = 1000, plot = FALSE)

Arguments

Value

A data frame where the first column is t and second column is simulated values of the realisation of fractional Brownian motion with added constant mean.

References

Banna, O., Mishura, Y., Ralchenko, K., & Shklyar, S. (2019). Fractional Brownian motion: Approximations and Projections. John Wiley & Sons. doi:10.1002/9781119476771.app3.

See Also

FGn, Bm, GHBMP, Bbridge, FBbridge

Examples

FBm(H = 0.5, plot = TRUE)

Simulation of fractional Gaussian noise

Description

This function simulates a realisation of the fractional Gaussian noise over the time interval [t_start,t_end] for a provided Hurst parameter.

Usage

FGn(H, t_start = 0, t_end = 1, N = 1000, plot = FALSE)

Arguments

Value

A data frame where the first column is t and second column is simulated values of the realisation of fractional Gaussian noise.

References

Banna, O., Mishura, Y., Ralchenko, K., & Shklyar, S. (2019). Fractional Brownian motion: Approximations and Projections. John Wiley & Sons. doi:10.1002/9781119476771.app3.

See Also

FBm, Bm, GHBMP, Bbridge, FBbridge

Examples

FGn(H=0.5,plot=TRUE)

Simulation of Gaussian Haar-based multifractional processes

Description

This function simulates a realisation of a Gaussian Haar-based multifractional process at any time point or time sequence on the interval [0,1].

Usage

GHBMP(t, H, J = 15, num.cores = availableCores(omit = 1))

Arguments

Details

The following formula defined in Ayache, A., Olenko, A. & Samarakoon, N. (2025) was used in simulating Gaussian Haar-based multifractional process.

X(t) := \sum_{j=0}^{+\infty} \sum_{k=0}^{2^{j}-1}\left(\int_{0}^{1} (t-s)_{+}^{H_{j}(k/{2^j})-{1}/{2}} h_{j,k}(s)ds \right)\varepsilon_{j,k},

where

\int_{0}^{1} (t-s)_{+}^{H_{j,k}-\frac{1}{2}} h_{j,k} (s) ds = 2^{-j H_{j,k}} h^{[H_{j,k}]} (2^jt-k)

with h^{[\lambda]} (x) = \int_{\mathbb{R}} (x-s)_{+}^{\lambda-\frac{1}{2}} h(s) ds. h is the Haar mother wavelet, j and k are positive integers, t is time, H is the Hurst function and \varepsilon_{j,k} is a sequence of independent \mathcal{N}(0,1) Gaussian random variables. For simulations, the truncated version of this formula with first summation up to J is used.

Value

A data frame of class "mp" where the first column is time moments t and second column is simulated values of X(t).

Note

See Examples for the usage of constant, time-varying, piecewise or step Hurst functions.

References

Ayache, A., Olenko, A. and Samarakoon, N. (2025). On Construction, Properties and Simulation of Haar-Based Multifractional Processes. doi:10.48550/arXiv.2503.07286. (submitted).

See Also

Hurst, plot.mp, Bm, FBm, FGn, Bbridge, FBbridge

Examples

#Constant Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.4 + 0*t)}
GHBMP(t, H)

#Linear Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.2 + 0.45*t)}
GHBMP(t, H)

#Oscillating Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
GHBMP(t, H)

#Piecewise Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(x) {
ifelse(x >= 0 & x <= 0.8, 0.375 * x + 0.2,
      ifelse(x > 0.8 & x <= 1,-1.5 * x + 1.7, NA))
}
GHBMP(t, H)

Creates objects of class H_LFD

Description

For user provided time series creates objects of class "H_LFD" with the Hurst function estimated using Hurst, local fractal dimension estimated using LFD and smoothed estimated Hurst function and LFD added.

Usage

H_LFD(X, N = 100, Q = 2, L = 2)

Arguments

Value

The return from H_LFD is an object list of class "H_LFD" with the following components:

Raw_Hurst_estimates

A data frame of where the first column is a time sequence and second column is estimated values of the Hurst function.

Smoothed_Hurst_estimates

A data frame of where the first column is a time sequence and second column is smoothed estimates of the Hurst function. Smoothed using the loess method.

Raw_LFD_estimates

A data frame of where the first column is a time sequence and second column is Local fractal dimension estimates.

Smoothed_LFD_estimates

A data frame of where the first column is a time sequence and second column is smoothed estimates of Local fractal dimension. Smoothed using the loess method.

Data

User provided time series.

Note

Since these are estimators of local characteristics, reliable results can only be obtained when a sufficiently large number of points is used.

See Also

plot.H_LFD, Hurst, LFD

Examples

TS <- data.frame("t"=seq(0,1,length = 1000),"X(t)" = rnorm(1000))
Object <- H_LFD(TS)
#Plot of time series, estimated and smoothed Hurst and LFD estimates
plot(Object)

Statistical estimation of the Hurst function

Description

This function computes statistical estimates for the Hurst function.

Usage

Hurst(X, N = 100, Q = 2, L = 2)

Arguments

Details

Statistical estimation of the Hurst function is done based on the results of Ayache, A., & Bouly, F. (2023). The estimator is built through generalized quadratic variations of the process associated with its increments.

Value

A data frame of where the first column is a time sequence and second column is estimated values of the Hurst function.

Note

Since these are estimators of local characteristics, reliable results can only be obtained when a sufficiently large number of points is used.

References

Ayache, A. and Bouly, F. (2023). Uniformly and strongly consistent estimation for the random Hurst function of a multifractional process. Latin American Journal of Probability and Mathematical Statistics, 20(2):1587–1614. doi:10.30757/alea.v20-60.

See Also

LFD, H_LFD, plot.mp, plot_tsest, plot.H_LFD

Examples

#Hurst function of a multifractional process simulated using GHBMP function
T <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X <- GHBMP(T, H)
Hurst(X)


#Hurst function of a fractional Browian motion simulated using FBm
X <- FBm(H = 0.5, x_start = 0, t_start = 0, t_end = 2, N = 1000)
Hurst(X)

Estimation of the local fractal dimension

Description

This function computes the estimates for the local fractal dimension.

Usage

LFD(X, N = 100, Q = 2, L = 2)

Arguments

Details

The formula \widehat{LFD} = 2-\widehat{H}(t) is used to compute the estimated local fractal dimension, where \widehat{H}(t) is the estimated Hurst function.

Value

A data frame where the first column is a time sequence and the second column is estimated values of the local fractal dimension.

Note

Since these are estimators of local characteristics, reliable results can only be obtained when a sufficiently large number of points is used.

References

Gneiting, T., and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review, 46(2):269-282. doi:10.1137/S0036144501394387.

See Also

Hurst, H_LFD, plot.mp, plot_tsest, plot.H_LFD

Examples

#LFD of a multifractional process simulated using GHBMP function
T <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X <- GHBMP(T, H)
LFD(X)


#LFD of a fractional Browian motion simulated using FBm
X <- FBm(H = 0.5, x_start = 0, t_start = 0, t_end = 2, N = 1000)
LFD(X)

Relative strength index

Description

This function computes the Relative Strength Index (RSI) for a time series.

Usage

RS_Index(X, period = 14, plot = FALSE, overbought = 70, oversold = 30)

Arguments

Details

To compute the RSI,

\text{RSI} = 100 \dfrac{\text{Average\_gain}}{\text{Average\_gain}+\text{Average\_loss}}

formula is used. Average gain and average loss are computed using the Wilders's smoothing method.

Value

A list or vector of the RSI values.

References

Wilder, J. W. (1978). New concepts in technical trading systems. Greensboro, NC.

Examples

X <- c(74.44,74.19,74.25,73.65,74.37,74.73,75.15,75.46,75.88,76.78,
            75.81,76.53,75.11,76.28,76.68,76.08,76.53,76.11,76.42,75.58,
            75.44,75.46,74.98)
RS_Index(X, plot = TRUE)

Estimated maximum of a time series

Description

This function computes the maximum of a time series for the provided time interval or its sub-interval.

Usage

X_max(X, subI = NULL, plot = FALSE, vline = FALSE, hline = FALSE)

Arguments

Value

A list of numeric vector(s). The first element in the vector is the corresponding t value and second the maximum of the time series.

See Also

X_min

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t, "X(t)" = rnorm(100))
X_max(TS, subI = c(0.5, 0.8), plot = TRUE)

Estimated minimum of a time series

Description

This function computes the minimum of a time series for the provided time interval or its sub-interval.

Usage

X_min(X, subI = NULL, plot = FALSE, vline = FALSE, hline = FALSE)

Arguments

Value

A list of numeric vector(s). The first element in the vector is the corresponding t value and second the minimum of the time series.

See Also

X_max

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t, "X(t)" = rnorm(100))
X_min(TS, subI = c(0.2, 0.8), plot = TRUE)

Covariance of Gaussian Haar-based multifractional processes

Description

Computes the theoretical covariance matrix of a Gaussian Haar-based multifractional process.

Usage

cov_GHBMP(
  t,
  H,
  J = 8,
  theta = NULL,
  plot = FALSE,
  num.cores = availableCores(omit = 1)
)

Arguments

Details

To make it comparable with the empirical covariance function the same smoothing parameter theta can be used if needed.

Value

An m \times m matrix, where m is the length of t.

References

Ayache, A., Olenko, A. and Samarakoon, N. (2025). On Construction, Properties and Simulation of Haar-Based Multifractional Processes. doi:10.48550/arXiv.2503.07286. (submitted).

See Also

GHBMP, est_cov

Examples

t <- seq(0, 1, by = 0.01)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}

#Smoothed covariance function
cov_GHBMP(t, H, theta = 0.1, plot = TRUE)

#Non-smoothed covariance function
cov_GHBMP(t, H, plot = TRUE)

Estimated crossing times

Description

Computes the estimated t value(s), in which a time series crosses a specific constant level for the provided time interval or its sub-interval.

Usage

cross_T(X, A, subI = NULL, plot = FALSE, vline = FALSE)

Arguments

Value

The estimated crossing times at a given level.

See Also

cross_rate, cross_mean

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t, "X(t)" = rnorm(100))
cross_T(TS, 0.1, subI = c(0.2, 0.8), plot = TRUE, vline = TRUE)

Mean time between crossings

Description

Computes the mean duration between crossings of a time series at a specified constant level for the provided time interval or its sub-interval.

Usage

cross_mean(X, A, subI = NULL, plot = FALSE)

Arguments

Value

The estimated mean time between crossings.

See Also

cross_T, cross_rate

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t, "X(t)" = rnorm(100))
cross_mean(TS, 0.1, subI = c(0.2, 0.8), plot = TRUE)

Crossing rate

Description

Computes the rate at which a time series crosses a specific constant level for the provided time interval or its sub-interval.

Usage

cross_rate(X, A, subI = NULL, plot = FALSE)

Arguments

Value

The crossing rate, which gives average number of crossings per time unit.

See Also

cross_T, cross_mean

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t, "X(t)" = rnorm(100))
cross_rate(TS, 0.1, subI = c(0.2, 0.8), plot = TRUE)

Empirical covariance function

Description

Computes the empirical covariance function of a process, for each pair of time points in the time sequence using M realisations of the process.

Usage

est_cov(X, theta = 0.1, plot = FALSE)

Arguments

Details

The smoothing parameter theta can help to better visualise changes between neighbour estimated values.

Value

An m \times m matrix, where m is the number of time points. Each element represents the estimated value of covariance function for the corresponding time points. Time points are arranged in ascending order.

See Also

cov_GHBMP

Examples

#Matrix of empirical covariance estimates of the GHBMP with Hurst function H.
t <- seq(0, 1, by = (1/2)^8)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
#Only 5 realisations of GHBMP are used in this example to reduce the computational time.
X.t <- replicate(5, GHBMP(t, H), simplify = FALSE)
X <- do.call(rbind, lapply(X.t, function(df) df[, 2]))
Data <- data.frame(t, t(X))
cov.mat <- est_cov(Data, theta = 0.2, plot = TRUE)
cov.mat

Excursion area

Description

Computes the excursion area where a time series X(t) is greater or lower than the constant level A for the provided time interval or its sub-interval.

Usage

exc_Area(X, A, N = 10000, level = "greater", subI = NULL, plot = FALSE)

Arguments

Value

Excursion area.

See Also

sojourn

Examples

t <- seq(0, 1, length = 1000)
TS <- data.frame("t" = t, "X(t)" = rnorm(1000))
exc_Area(TS, 0.8, level = 'lower', subI = c(0.5, 0.8), plot = TRUE)

Hierarchical clustering

Description

This function performs hierarchical clustering of realisations based on the estimated Hurst functions.

Usage

hclust_hurst(
  X.t,
  k = NULL,
  h = NULL,
  dist.method = "euclidean",
  method = "complete",
  dendrogram = FALSE,
  N = 100,
  Q = 2,
  L = 2
)

Arguments

Details

The Hurst function of each realisation is estimated using the function Hurst and the smoothed Hurst estimates are used for the cluster analysis. The distances between smoothed Hurst estimates are computed by the dist.method provided and passed into the hclust for hierarchical clustering.

Value

An object list of class "hc_hurst" with print and plot methods. The list has following components:

cluster_info

A data frame indicating the cluster number and distance to cluster center from each smoothed estimated Hurst function (item). Distance is obtained from the dist.method.

cluster

A vector with cluster number of each item.

cluster_sizes

Number of items in each cluster.

centers

A data frame of cluster centers. Center obtained as the average of each smoothed estimated Hurst function in the cluster. Columns denote time points in which estimates were obtained. Row names denote cluster numbers.

smoothed_Hurst_estimates

A data frame of smoothed Hurst estimates. Columns denote time points in which estimates were obtained. Rows denote estimates for each realisation.

raw_Hurst_estimates

A list of data frames of raw Hurst estimates.

call

Information about the input parameters used.

See Also

print.hc_hurst, plot.hc_hurst, kmeans_hurst

Examples

#Simulation of multifractional processes
t <- seq(0, 1, by = (1/2)^10)
H1 <- function(t) {return(0.1 + 0*t)}
H2 <- function(t) {return(0.2 + 0.45*t)}
H3 <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X.list.1 <- replicate(3, GHBMP(t,H1),simplify = FALSE)
X.list.2 <- replicate(3, GHBMP(t,H2),simplify = FALSE)
X.list.3 <- replicate(3, GHBMP(t,H3),simplify = FALSE)
X.list <- c(X.list.1, X.list.2, X.list.3)

#Hierarchical clustering based on k = 3 clusters with dendrogram plotted
HC <- hclust_hurst(X.list, k=3, dendrogram = TRUE)
print(HC)

#Plot of smoothed Hurst functions in each cluster with cluster centers
plot(HC,type = "ec")

K-means clustering

Description

This function performs k-means clustering of realisations based on the estimated Hurst functions.

Usage

kmeans_hurst(X.t, k, ..., N = 100, Q = 2, L = 2)

Arguments

Details

The Hurst function of each realisation is estimated using Hurst. The smoothed Hurst estimates are used for k-means clustering in kmeans. The Hartigan and Wong algorithm is used as the default k-means clustering algorithm.

Value

An object list of class "k_hurst" with print and plot methods. The list has following components:

cluster_info

A data frame indicating the cluster number and euclidean distance to cluster center of each smoothed estimated Hurst function (item)

cluster

A vector of cluster number of each item.

cluster_sizes

Number of item in each cluster.

centers

A data frame of cluster centers. Center obtained as the average of each smoothed estimated Hurst function in the cluster. Columns denote time points in which estimates were obtained. Row names denote cluster numbers.

smoothed_Hurst_estimates

A data frame of smoothed Hurst estimates. Columns denote time points in which estimates were obtained. Rows denote estimates for each realisation.

raw_Hurst_estimates

A list of data frames of raw Hurst estimates.

call

Information about the input parameters used.

See Also

print.k_hurst, plot.k_hurst, hclust_hurst

Examples

#Simulation of multifractional processes
t <- seq(0, 1, by = (1/2)^10)
H1 <- function(t) {return(0.1 + 0*t)}
H2 <- function(t) {return(0.2 + 0.45*t)}
H3 <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X.list.1 <- replicate(3, GHBMP(t, H1),simplify = FALSE)
X.list.2 <- replicate(3, GHBMP(t, H2),simplify = FALSE)
X.list.3 <- replicate(3, GHBMP(t, H3),simplify = FALSE)
X.list <- c(X.list.1, X.list.2, X.list.3)

#K-means clustering based on k = 3 clusters
KC <- kmeans_hurst(X.list, k = 3)
print(KC)

#Plot of smoothed Hurst functions in each cluster with cluster centers
plot(KC, type = "ec")

Longest increasing/decreasing streak

Description

Computes the time span of the longest increasing or decreasing streak(s) of a time series for the provided time interval or its sub-interval.

Usage

long_streak(X, direction = "increasing", subI = NULL, plot = FALSE)

Arguments

Value

A data frame with one row for each longest streak, containing the time span and the corresponding values of X(t) at the streak endpoints.

See Also

mean_streak

Examples

t <- seq(0, 1, length = 100)
TS <- data.frame("t" = t,"X(t)" = rnorm(100))
long_streak(TS, direction = 'decreasing', subI = c(0.2, 0.8), plot = TRUE)

Mean time span of increasing/decreasing streaks

Description

Computes the mean time span of the increasing/decreasing streaks for the provided time interval or its sub-interval.

Usage

mean_streak(X, direction = "increasing", subI = NULL, plot = FALSE)

Arguments

Value

Mean time span of the increasing/decreasing streaks.

See Also

long_streak

Examples

t <- seq(0, 1 ,length = 100)
TS <- data.frame("t" = t,"X(t)" = rnorm(100))
mean_streak(TS, direction = 'decreasing', subI = c(0.2,0.8), plot = TRUE)

Plot the estimated Hurst functions and local fractal dimension estimates for objects of class H_LFD

Description

Creates a plot of the user provided time series with the Hurst function estimated using Hurst, the smoothed estimated Hurst function and local fractal dimension estimated using LFD and smoothed estimates of local fractal dimension for objects of class "H_LFD".

Usage

## S3 method for class 'H_LFD'
plot(
  x,
  H_Est = TRUE,
  H_Smooth_Est = TRUE,
  LFD_Est = TRUE,
  LFD_Smooth_Est = TRUE,
  ...
)

Arguments

Details

Compared to plot_tsest, the function's argument is a "H_LFD" object, not a time series.

Value

A ggplot object which is used to plot the time series with theoretical, raw and smoothed estimates of Hurst function and raw and smoothed estimates of local fractal dimension.

See Also

H_LFD, Hurst, LFD, plot_tsest

Examples

TS <- data.frame("t" = seq(0, 1, length = 1000), "X(t)" = rnorm(1000))
Object <- H_LFD(TS)
#Plot of time series, estimated and smoothed Hurst and LFD estimates
plot(Object)

Plot smoothed Hurst functions in each cluster with cluster centers

Description

Creates a plot of the smoothed Hurst functions of realisations of processes (or time series) separately in each cluster with cluster centers using the return from hclust_hurst. Options to plot only estimates, only centers or both are available.

Usage

## S3 method for class 'hc_hurst'
plot(x, type = "estimates", ...)

Arguments

Value

A ggplot object which is used to plot the relevant type of plot: "estimates", "centers" or "ec".

See Also

hclust_hurst

Examples

#Simulation of multifractional processes
t <- seq(0, 1, by = (1/2)^10)
H1 <- function(t) {return(0.1 + 0*t)}
H2 <- function(t) {return(0.2 + 0.45*t)}
H3 <- function(t) {return(0.5 - 0.4*sin(6*3.14*t))}
X.list.1 <- replicate(3, GHBMP(t,H1), simplify = FALSE)
X.list.2 <- replicate(3, GHBMP(t,H2), simplify = FALSE)
X.list.3 <- replicate(3, GHBMP(t,H3), simplify = FALSE)
X.list <- c(X.list.1, X.list.2, X.list.3)

#Hierarchical clustering based on k=3 clusters with dendrogram plotted
HC<- hclust_hurst(X.list, k = 3, dendrogram = TRUE)
print(HC)

#Plot of smoothed Hurst functions in each cluster with cluster centers
plot(HC, type = "ec")

Plot smoothed Hurst functions in each cluster with cluster centers

Description

Creates a plot of the smoothed Hurst functions of realisations of processes (or time series) separately in each cluster with cluster centers using the return from kmeans_hurst. Options to plot only estimates, only centers or both are available.

Usage

## S3 method for class 'k_hurst'
plot(x, type = "estimates", ...)

Arguments

Value

A ggplot object which is used to plot the relevant type of plot: "estimates", "centers" or "ec".

See Also

kmeans_hurst

Examples

#Simulation of multifractional processes
t <- seq(0, 1, by = (1/2)^10)
H1 <- function(t) {return(0.1 + 0*t)}
H2 <- function(t) {return(0.2 + 0.45*t)}
H3 <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X.list.1 <- replicate(3, GHBMP(t, H1), simplify = FALSE)
X.list.2 <- replicate(3, GHBMP(t, H2), simplify = FALSE)
X.list.3 <- replicate(3, GHBMP(t, H3), simplify = FALSE)
X.list <- c(X.list.1, X.list.2, X.list.3)

#K-means clustering based on k=3 clusters
KC <- kmeans_hurst(X.list, k = 3)
print(KC)

#Plot of smoothed Hurst functions in each cluster with cluster centers
plot(KC, type = "ec")

Plot Gaussian Haar-based multifractional processes with their theoretical and estimated Hurst functions and local fractal dimension

Description

Creates a plot of the Gaussian Haar-based multifractional process simulated by using GHBMP with theoretical Hurst function (if provided), Hurst function estimated using Hurst, the smoothed estimated Hurst function and local fractal dimension estimated using LFD and smoothed estimates of local fractal dimension.

Usage

## S3 method for class 'mp'
plot(
  x,
  H = NULL,
  H_Est = TRUE,
  H_Smooth_Est = TRUE,
  LFD_Est = TRUE,
  LFD_Smooth_Est = TRUE,
  N = 100,
  Q = 2,
  L = 2,
  ...
)

Arguments

Value

A ggplot object which is used to plot the multifractional process with theoretical, raw and smoothed estimates of Hurst function and raw and smoothed estimates of local fractal dimension.

See Also

GHBMP, Hurst, LFD

Examples

#Simulation of the multifractional process and estimation of the Hurst function
T <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
X <- GHBMP(T, H)

#Plot of process, theoretical Hurst function, estimated and smoothed Hurst and LFD estimates
plot(X, H = H)

#Plot of process, estimated and smoothed Hurst and LFD estimates
plot(X)

Plot the estimated Hurst functions and local fractal dimension estimates for a user provided time series

Description

Creates a plot of the user provided time series with the Hurst function estimated using Hurst, the smoothed estimated Hurst function and local fractal dimension estimated using LFD and smoothed estimates of local fractal dimension.

Usage

plot_tsest(
  X,
  H_Est = TRUE,
  H_Smooth_Est = TRUE,
  LFD_Est = TRUE,
  LFD_Smooth_Est = TRUE,
  N = 100,
  Q = 2,
  L = 2
)

Arguments

Details

Compared to plot.H_LFD the function's first argument is a time series, not H_LFD object.

Value

A ggplot object which is used to plot the time series with raw and smoothed estimates of Hurst function and local fractal dimension.

See Also

Hurst, LFD, plot.H_LFD

Examples

TS <- data.frame("t" = seq(0, 1, length = 1000), "X(t)" = rnorm(1000))
#Plot of time series, estimated and smoothed Hurst and LFD estimates
plot_tsest(TS)

Print method for "hc_hurst" class objects

Description

Prints the results of hierarchical clustering of realisations of processes.

Usage

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

Arguments

Value

Prints an object of class "hc_hurst".

See Also

hclust_hurst

Print method for "k_hurst" class objects

Description

Prints the results of k-means clustering of realisations of processes.

Usage

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

Arguments

Value

Prints an object of class "k_hurst".

See Also

kmeans_hurst

Shiny app to visualise and analyse processes

Description

Launches a Shiny app to visualise and analyse realisations of Brownian motion (see Bm), Brownian bridge (see Bbridge), fractional Brownian motion (see FBm), fractional Brownian bridge (see FBbridge), fractional Gaussian noise (see FGn), Gaussian Haar-based multifractional processes (see GHBMP) and user-provided time series data.

Usage

shinyapp_sim()

Details

For Input time series, provide a .csv file that has headers of its columns. It must have two columns: the first column should contain the numeric time sequence t, and the second the corresponding time series values X(t). To get reliable estimation results for time series, it is recommended to use at least 500 time points. Make sure there are no extra header rows or footnotes.

Value

An interactive Shiny app with the following user interface controls.

GHBMP simulation

Hurst function

Input the Hurst function in terms of t. The default is set to 0.5 + 0*t.

Time sequence

Input the time sequence which belongs to the interval [0,1]. The default is set to seq(0, 1, by = (1/2)^10).

J

Input or select a positive integer. For large J could be rather time consuming. Default is set to 15.

Hurst function and LFD estimation

Number of sub-intervals for estimation

Default is set to 100.

Q

Input or select an integer greater than or equal to 2. Default is set to 2.

L

Input or select an integer greater than or equal to 2. Default is set to 2.

Plot

Choose the required from: Theoretical Hurst function, Raw estimate of Hurst function, Smoothed estimate of Hurst function, Raw estimate of Local Fractal Dimension, Smoothed estimate of Local Fractal Dimension.

Excursion set and area

Number of time steps

Input the number of steps the time interval needs to be split into.

Constant level

Input the constant level.

Compare to level

Greater, Lower.

Plot

Select: Excursion set, Excursion area.

Longest streak

Longest streak to plot

Select: Increasing, Decreasing.

Maximum and minimum

Plot

Select: Maximum, Minimum.

See Also

Bm, FBm, FGn, Bbridge, FBbridge, GHBMP, Hurst, LFD, sojourn, exc_Area long_streak, X_max, X_min

Examples

if (interactive()) {
  shinyapp_sim()
}

Estimated sojourn measure

Description

Computes the estimated sojourn measure for a time series X(t) greater or lower than the constant level A for the provided time interval or its sub-interval.

Usage

sojourn(X, A, N = 10000, level = "greater", subI = NULL, plot = FALSE)

Arguments

Value

Estimated sojourn measure.

See Also

exc_Area

Examples

t <- seq(0, 1, length = 1000)
TS <- data.frame("t" = t,"X(t)" = rnorm(1000))
sojourn(TS, 0.8, level = 'lower',subI = c(0.5, 0.8), plot = TRUE)