Type:	Package
Title:	Simulations and Statistical Inference for Linear Fractional Stable Motions
Version:	1.1.2
Maintainer:	Dmitry Otryakhin <d.otryakhin.acad@protonmail.ch>
Author:	Dmitry Otryakhin [aut, cre], Stepan Mazur [aut], Mathias Ljungdahl [ctb]
Description:	Contains functions for simulating the linear fractional stable motion according to the algorithm developed by Mazur and Otryakhin <doi:10.32614/RJ-2020-008> based on the method from Stoev and Taqqu (2004) <doi:10.1142/S0218348X04002379>, as well as functions for estimation of parameters of these processes introduced by Mazur, Otryakhin and Podolskij (2018) <doi:10.48550/arXiv.1802.06373>, and also different related quantities.
License:	GPL-3
URL:	https://gitlab.com/Dmitry_Otryakhin/Tools-for-parameter-estimation-of-the-linear-fractional-stable-motion
Encoding:	UTF-8
RoxygenNote:	7.2.1
Depends:	methods, foreach, doParallel
Imports:	ggplot2, stabledist, reshape2, plyr, Rdpack, Rcpp
Suggests:	elliptic, testthat, stringi
RdMacros:	Rdpack
LinkingTo:	Rcpp
NeedsCompilation:	yes
Packaged:	2022-08-27 10:44:26 UTC; d
Repository:	CRAN
Date/Publication:	2022-08-27 13:20:05 UTC

Parameter estimation procedure in continuous case.

Description

Parameter freq is preserved to allow for investigation of the inference procedure in high frequency case.

Usage

ContinEstim(t1, t2, p, k, path, freq)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
alpha<-0.8; H<-0.8; sigma<-0.3
p<-0.3; k=3; t1=1; t2=2

lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm
ContinEstim(t1,t2,p,k,path=lfsm,freq='L')

High frequency estimation procedure for lfsm.

Description

General estimation procedure for high frequency case when 1/alpha is not a natural number. "Unnecessary" parameter freq is preserved to allow for investigation of the inference procedure in low frequency case

Usage

GenHighEstim(p, p_prime, path, freq, low_bound = 0.01, up_bound = 4)

Arguments

Details

In this algorithm the preliminary estimate of alpha is found via using uniroot function. The latter is given the lower and the upper bounds for alpha via low_bound and up_bound parameters. It is not possible to pass 0 as the lower bound because there are numerical limitations on the alpha estimate, caused by the length of the sample path and by numerical errors. p and p_prime must belong to the interval (0,1/2) (in the notation kept in rlfsm package) The two powers cannot be equal.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
sigma<-0.3
p<-0.2; p_prime<-0.4

#### Continuous case
lfsm<-path(N=N,m=m,M=M,alpha=1.8,H=0.8,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

GenHighEstim(p=p,p_prime=p_prime,path=lfsm,freq="H")

#### H-1/alpha<0 case
lfsm<-path(N=N,m=m,M=M,alpha=0.8,H=0.8,
           sigma=sigma,freq='H',disable_X=FALSE,seed=3)$lfsm

GenHighEstim(p=p,p_prime=p_prime,path=lfsm,freq="H")

Low frequency estimation procedure for lfsm.

Description

General estimation procedure for low frequency case when 1/alpha is not a natural number.

Usage

GenLowEstim(t1, t2, p, path, freq = "L")

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
sigma<-0.3
p<-0.3; k=3; t1=1; t2=2

#### Continuous case
lfsm<-path(N=N,m=m,M=M,alpha=1.8,H=0.8,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

GenLowEstim(t1=t1,t2=t2,p=p,path=lfsm,freq="L")

#### H-1/alpha<0 case
lfsm<-path(N=N,m=m,M=M,alpha=0.8,H=0.8,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

GenLowEstim(t1=t1,t2=t2,p=p,path=lfsm,freq="L")

#### The procedure works also for high frequency case
lfsm<-path(N=N,m=m,M=M,alpha=1.8,H=0.8,
           sigma=sigma,freq='H',disable_X=FALSE,seed=3)$lfsm

GenLowEstim(t1=t1,t2=t2,p=p,path=lfsm,freq="H")

Statistical estimator of H in high/low frequency setting

Description

The statistic is defined as

\widehat{H}_{\textnormal{high}} (p,k)_n:= \frac 1p \log_2 R_{\textnormal{high}} (p,k)_n, \qquad \widehat{H}_{\textnormal{low}} (p,k)_n:= \frac 1p \log_2 R_{\textnormal{low}} (p,k)_n

Usage

H_hat(p, k, path)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Numerical properties of statistical estimators operating on the linear fractional stable motion.

Description

The function is useful, for instance, when one needs to compute standard deviation of \widehat \alpha_{high} estimator given a fixed set of parameters.

Usage

MCestimLFSM(Nmc, s, m, M, alpha, H, sigma, fr, Inference, ...)

Arguments

Details

MCestimLFSM performs Monte-Carlo experiments to compute parameters according to procedure Inference. More specifically, for each element of s it generates Nmc lfsm sample paths with length equal to s[i], performs the statistical inference on each, obtaining the estimates, and then returns their different statistics. It is vital that the estimator returns a list of named parameters (one or several of 'sigma', 'alpha' and 'H'). MCestimLFSM uses the names to lookup the true parameter value and compute its bias.

For sample path generation MCestimLFSM uses a light-weight version of path, path_fast. In order to be applied, function Inference must accept argument 'path' as a sample path.

Value

It returns a list containing the following components:

Examples

#### Set of global parameters ####
m<-25; M<-60
p<-.4; p_prime<-.2; k<-2
t1<-1; t2<-2
NmonteC<-5e1
S<-c(1e2,3e2)
alpha<-1.8; H<-0.8; sigma<-0.3


# How to plot empirical density

theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,ContinEstim,
                     t1=t1,t2=t2,p=p,k=k)
l_plot<-Plot_dens(par_vec=c('sigma','alpha','H'),
                  MC_data=theor_3_1_H_clt, Nnorm=1e7)



# For MCestimLFSM() it is vital that the estimator returns a list of named parameters

H_hat_f <- function(p,k,path) {hh<-H_hat(p,k,path); list(H=hh)}
theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,H_hat_f,
                     p=p,k=k)


# The estimator can return one, two or three of the parameters.

est_1 <- function(path) list(H=1)
theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,est_1)

est_2 <- function(path) list(H=0.8, alpha=1.5)
theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,est_2)

est_3 <- function(path) list(sigma=5, H=0.8, alpha=1.5)
theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,est_3)

Alpha-norm of an arbitrary function

Description

Alpha-norm of an arbitrary function

Usage

Norm_alpha(fun, alpha, ...)

Arguments

Details

fun must accept a vector of values for evaluation. See ?integrate for further details. Most problems with this function appear because of rather high precision. Try to tune rel.tol parameter first.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

Norm_alpha(h_kr,alpha=1.8,k=2,r=1,H=0.8,l=4)

Path array generator

Description

The function takes a list of parameters (alpha, H) and uses expand.grid to obtain all possible combinations of them. Based on each combination, the function simulates an lfsm sample path. It is meant to be used in conjunction with function Plot_list_paths.

Usage

Path_array(N, m, M, l, sigma)

Arguments

Value

The returned value is a data frame containing paths and the corresponding values of alpha, H and frequency.

Examples

l=list(H=c(0.2,0.8),alpha=c(1,1.8), freq="H")
arr<-Path_array(N=300,m=30,M=100,l=l,sigma=0.3)
str(arr)
head(arr)

(alpha,H,sigma)- density plot

Description

Plots the densities of the parameters (alpha,H,sigma) estimated in Monte-Carlo experiment. Works in conjunction with MCestimLFSM function.

Usage

Plot_dens(par_vec = c("alpha", "H", "sigma"), MC_data, Nnorm = 1e+07)

Arguments

See Also

Plot_vb to plot variance- and bias dependencies on n.

Examples

m<-45; M<-60

p<-.4; p_prime<-.2
t1<-1; t2<-2; k<-2

NmonteC<-5e2
S<-c(1e3,1e4)
alpha<-.8; H<-0.8; sigma<-0.3
theor_4_1_clt_new<-MCestimLFSM(s=S,fr='L',Nmc=NmonteC,
                       m=m,M=M,
                       alpha=alpha,H=H,sigma=sigma,
                       GenLowEstim,t1=t1,t2=t2,p=p)
l_plot<-Plot_dens(par_vec=c('sigma','alpha','H'), MC_data=theor_4_1_clt_new, Nnorm=1e7)
l_plot

Rendering of path lattice

Description

Rendering of path lattice

Usage

Plot_list_paths(arr)

Arguments

Examples

l=list(H=c(0.2,0.8),alpha=c(1,1.8), freq="H")
arr<-Path_array(N=300,m=30,M=100,l=l,sigma=0.3)
p<-Plot_list_paths(arr)
p

A function to plot variance- and bias dependencies of estimators on the lengths of sample paths. Works in conjunction with MCestimLFSM function.

Description

A function to plot variance- and bias dependencies of estimators on the lengths of sample paths. Works in conjunction with MCestimLFSM function.

Usage

Plot_vb(data)

Arguments

Value

The function returns a ggplot2 graph.

See Also

Plot_dens

Examples

# Light weight computaions

m<-25; M<-50
alpha<-1.8; H<-0.8; sigma<-0.3
S<-c(1:3)*1e2
p<-.4; p_prime<-.2; t1<-1; t2<-2
k<-2; NmonteC<-50

# Here is the continuous H-1/alpha inference procedure
theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,ContinEstim,
                     t1=t1,t2=t2,p=p,k=k)
Plot_vb(theor_3_1_H_clt)


# More demanding example (it is better to use multicore setup)
# General low frequency inference

m<-45; M<-60
alpha<-0.8; H<-0.8; sigma<-0.3
S<-c(1:15)*1e2
p<-.4; t1<-1; t2<-2
NmonteC<-50

# Here is the continuous H-1/alpha inference procedure
theor_4_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC,
                     m=m,M=M,alpha=alpha,H=H,
                     sigma=sigma,GenLowEstim,
                     t1=t1,t2=t2,p=p)
Plot_vb(theor_4_1_H_clt)

R high /low

Description

Defined as

R_{\textnormal{high}} (p,k)_n := \frac{\sum_{i=2k}^n \left| \Delta_{i,k}^{n,2} X \right|^p} {\sum_{i=k}^n \left| \Delta_{i,k}^{n,1} X \right|^p}, \qquad

R_{\textnormal{low}} (p,k)_n := \frac{\sum_{i=2k}^n \left| \Delta_{i,k}^{2} X \right|^p} {\sum_{i=k}^n \left| \Delta_{i,k}^{1} X \right|^p}

Usage

R_hl(p, k, path)

Arguments

Details

The computation procedure for high- and low frequency cases is the same, since there is no way to control frequency given a sample path.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
alpha<-0.8; H<-0.8; sigma<-0.3
p<-0.3; k=3

lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm
R_hl(p=p,k=k,path=lfsm)

Retrieve statistics(bias, variance) of estimators based on a set of paths

Description

Retrieve statistics(bias, variance) of estimators based on a set of paths

Usage

Retrieve_stats(paths, true_val, Est, ...)

Arguments

Examples

m<-45; M<-60; N<-2^10-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='L';t1=1; t2=2
r=1; k=2; p=0.4

Y<-paths(N_var=10,parallel=TRUE,N=N,m=m,M=M,
         alpha=alpha,H=H,sigma=sigma,freq='L',
         disable_X=FALSE,levy_increments=NULL)

Retrieve_stats(paths=Y,true_val=sigma,Est=sigma_hat,t1=t1,k=2,alpha=alpha,H=H,freq="L")

alpha norm of u*g

Description

alpha norm of u*g

Usage

U_g(g, u, ...)

Arguments

Examples

g<-function(x) exp(-x^2)
g<-function(x) exp(-abs(x))
U_g(g=g,u=4,alpha=1.7)

alpha-norm of u*g + v*h.

Description

alpha-norm of u*g + v*h.

Usage

U_gh(g, h, u, v, ...)

Arguments

Examples

g<-function(x) exp(-x^2)
h<-function(x) exp(-abs(x))
U_gh(g=g, h=h, u=4, v=3, alpha=1.7)

A dependence structure of 2 random variables.

Description

It is used when random variables do not have finite second moments, and thus, the covariance matrix is not defined. For X= \int_{\R} g_s dL_s and Y= \int_{\R} h_s dL_s with \| g \|_{\alpha}, \| h\|_{\alpha}< \infty. Then the measure of dependence is given by U_{g,h}: \R^2 \to \R via

U_{g,h} (u,v)=\exp(- \sigma^{\alpha}{\| ug +vh \|_{\alpha}}^{\alpha} ) - \exp(- \sigma^{\alpha} ({\| ug \|_{\alpha}}^{\alpha} + {\| vh \|_{\alpha}}^{\alpha}))

Usage

U_ghuv(alpha, sigma, g, h, u, v, ...)

Arguments

Examples

g<-function(x) exp(-x^2)
h<-function(x) exp(-abs(x))
U_ghuv(alpha=1.5, sigma=1, g=g, h=h, u=10, v=15,
rel.tol = .Machine$double.eps^0.25, abs.tol=1e-11)

Function a_p.

Description

Computes the corresponding function value from Mazur et al. 2018.

Usage

a_p(p)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Creates the corresponding value from the paper by Stoev and Taqqu (2004).

Description

a_tilda triggers a_tilda_cpp which is written in C++ and essentially performs the computation of the value.

Usage

a_tilda(N, m, M, alpha, H)

Arguments

References

Stoev S, Taqqu MS (2004). “Simulation methods for linear fractional stable motion and FARIMA using the Fast Fourier Transform.” Fractals, 95(1), 95-121. https://doi.org/10.1142/S0218348X04002379.

Statistical estimator for alpha

Description

Defined for the two frequencies as

\widehat \alpha_{high} := \frac{\log | \log \varphi_{high} (t_2; \widehat H_{high} (p,k)_n, k)_n| - \log | \log \varphi_{high} (t_1; \widehat H_{high} (p,k)_n, k)_n|}{\log t_2 - \log t_1}

\widehat \alpha_{low} := \frac{\log | \log \varphi_{low} (t_2;k)_n| - \log | \log \varphi_{low} (t_1; k)_n|}{\log t_2 - \log t_1}

Usage

alpha_hat(t1, t2, k, path, H, freq)

Arguments

Details

The function triggers function phi, thus Hurst parameter is required only in high frequency case. In the low frequency, there is no need to assign H a value because it will not be evaluated.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^14-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='H'
r=1; k=2; p=0.4; t1=1; t2=2

# Estimating alpha in the high frequency case
# using preliminary estimation of H
lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

H_est<-H_hat(p=p,k=k,path=lfsm)
H_est
alpha_est<-alpha_hat(t1=t1,t2=t2,k=k,path=lfsm,H=H_est,freq=freq)
alpha_est

Function h_kr

Description

Function h_{k,r}: R \to R is given by

h_{k,r} (x) = \sum_{j=0}^k (-1)^j {{k}\choose{j}} (x-rj)_+^{H-1/\alpha}, \ \ \ x\in R

Usage

h_kr(k, r, x, H, alpha, l = 0)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

#### Plot h_kr ####
s<-seq(0,10, by=0.01)
h_val<-sapply(s,h_kr, k=5, r=1, H=0.3, alpha=1)
plot(s,h_val)

Higher order increments

Description

Difference of the kth order. Defined as following:

\Delta_{i,k}^{n,r} X:= \sum_{j=0}^k (-1)^j{{k}\choose{j}}X_{(i-rj)/n}, i\geq rk.

Index i here is a coordinate in terms of point_num. Although R uses vector indexes that start from 1, increment has i varying from 0 to N, so that a vector has a length N+1. It is done in order to comply with the notation of the paper. This function doesn't allow for choosing frequency n. The frequency is determined by the path supplied, thus n equals to either the length of the path in high frequency setting or 1 in low frequency setting. increment() gives increments at certain point passed as i, which is a vector here. increments() computes high order increments for the whole sample path. The first function evaluates the formula above, while the second one uses structure diff(diff(...)) because the formula is slower at higher k.

Usage

increment(r, i, k, path)

increments(k, r, path)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
alpha<-0.8; H<-0.8; sigma<-0.3

lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

tryCatch(
increment(r=1,i=length(lfsm),k=length(lfsm)+100,path=lfsm),
error=function(c) 'An error occures when k is larger then the length of the sample path')

increment(r=3,i=50,k=3,path=lfsm)


path=c(1,4,3,6,8,5,3,5,8,5,1,8,6)

r=2; k=3
n <- length(path) - 1
DeltaX = increment(seq(r*k, n), path = path, k = k, r = r)
DeltaX == increments(k=k,r=r,path)

m(-p,k)

Description

defined as m_{p,k} := E[|\Delta_{k,k} X|^p] for positive powers. When p is negative (-p is positive) the equality does not hold.

Usage

m_pk(k, p, alpha, H, sigma)

Arguments

Details

The following identity is used for computations:

m_{-p,k} = \frac{(\sigma \|h_k\|_{\alpha})^{-p}}{a_{-p}} \int_{\R} \exp(-|y|^{\alpha}) |y|^{-1+p} dy = \frac{2(\sigma \|h_k\|_{\alpha})^{-p}}{\alpha a_{-p}} \Gamma(p/\alpha)

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Generator of linear fractional stable motion

Description

The function creates a 1-dimensional LFSM sample path using the numerical algorithm from the paper by Otryakhin and Mazur. The theoretical foundation of the method comes from the article by Stoev and Taqqu. Linear fractional stable motion is defined as

X_t = \int_{\R} \left\{(t-s)_+^{H-1/\alpha} - (-s)_+^{H-1/\alpha} \right\} dL_s

Usage

path(
  N = NULL,
  m,
  M,
  alpha,
  H,
  sigma,
  freq,
  disable_X = FALSE,
  levy_increments = NULL,
  seed = NULL
)

Arguments

Value

It returns a list containing the motion, the underlying Levy motion, the point number of the motions from 0 to N and the corresponding coordinate (which depends on the frequency), the parameters that were used to generate the lfsm, and the predefined frequency.

References

Mazur S, Otryakhin D (2020). “Linear Fractional Stable Motion with the rlfsm R Package.” The R Journal, 12(1), 386–405. doi:10.32614/RJ-2020-008.

Stoev S, Taqqu MS (2004). “Simulation methods for linear fractional stable motion and FARIMA using the Fast Fourier Transform.” Fractals, 95(1), 95-121. https://doi.org/10.1142/S0218348X04002379.

See Also

paths simulates a number of lfsm sample paths.

Examples

# Path generation

m<-256; M<-600; N<-2^10-M
alpha<-1.8; H<-0.8; sigma<-0.3
seed=2

List<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)

# Normalized paths
Norm_lfsm<-List[['lfsm']]/max(abs(List[['lfsm']]))
Norm_oLm<-List[['levy_motion']]/max(abs(List[['levy_motion']]))

# Visualization of the paths
plot(Norm_lfsm, col=2, type="l", ylab="coordinate")
lines(Norm_oLm, col=3)
leg.txt <- c("lfsm", "oLm")
legend("topright",legend = leg.txt, col =c(2,3), pch=1)


# Creating Levy motion
levyIncrems<-path(N=N, m=m, M=M, alpha, H, sigma, freq='L',
                  disable_X=TRUE, levy_increments=NULL, seed=seed)

# Creating lfsm based on the levy motion
  lfsm_full<-path(m=m, M=M, alpha=alpha,
                  H=H, sigma=sigma, freq='L',
                  disable_X=FALSE,
                  levy_increments=levyIncrems$levy_increments,
                  seed=seed)

sum(levyIncrems$levy_increments==
    lfsm_full$levy_increments)==length(lfsm_full$levy_increments)

Generator of a set of lfsm paths.

Description

It is essentially a wrapper for path generator, which exploits the latest to create a matrix with paths in its columns.

Usage

paths(N_var, parallel, seed_list = rep(x = NULL, times = N_var), ...)

Arguments

See Also

path

Examples

m<-45; M<-60; N<-2^10-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='L'
r=1; k=2; p=0.4

Y<-paths(N_var=10,parallel=TRUE,N=N,m=m,M=M,
         alpha=alpha,H=H,sigma=sigma,freq='L',
         disable_X=FALSE,levy_increments=NULL)

Hs<-apply(Y,MARGIN=2,H_hat,p=p,k=k)
hist(Hs)

Phi

Description

Defined as

\varphi_{\textnormal{high}}(t; H,k)_n := V_{\textnormal{high}}(\psi_t; k)_n \qquad \textnormal{and} \qquad \varphi_{\textnormal{low}}(t; k)_n := V_{\textnormal{low}}(\psi_t; k)_n

, where \psi_t(x):=cos(tx)

Usage

phi(t, k, path, H, freq)

Arguments

Details

Hurst parameter is required only in high frequency case. In the low frequency, there is no need to assign H a value because it will not be evaluated.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Inverse alpha estimator

Description

A function from a general estimation procedure which is defined as m^p_-p'_k /m^p'_-p_k, originally proposed in [13].

Usage

phi_of_alpha(p, p_prime, alpha)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Statistic V

Description

Statistic of the form

V_{\textnormal{high}}(f; k,r)_n:=\frac{1}{n}\sum_{i=rk}^n f\left( n^H \Delta_{i,k}^{n,r} X \right),

V_{\textnormal{low}}(f; k,r)_n :=\frac{1}{n}\sum_{i=rk}^n f\left( \Delta_{i,k}^{r} X \right)

Usage

sf(path, f, k, r, H, freq, ...)

Arguments

Details

Hurst parameter is required only in high frequency case. In the low frequency, there is no need to assign H a value because it will not be evaluated.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

See Also

phi computes V statistic with f(.)=cos(t.)

Examples

m<-45; M<-60; N<-2^10-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='L'
r=1; k=2; p=0.4
S<-(1:20)*100

path_lfsm<-function(...){

    List<-path(...)
    List$lfsm

}

Pths<-lapply(X=S,FUN=path_lfsm,
             m=m, M=M, alpha=alpha, sigma=sigma, H=H,
             freq=freq, disable_X = FALSE,
             levy_increments = NULL, seed = NULL)

f_phi<-function(t,x) cos(t*x)
f_pow<-function(x,p) (abs(x))^p

V_cos<-sapply(Pths,FUN=sf,f=f_phi,k=k,r=r,H=H,freq=freq,t=1)
ex<-exp(-(abs(sigma*Norm_alpha(h_kr,alpha=alpha,k=k,r=r,H=H,l=0)$result)^alpha))

 # Illustration of the law of large numbers for phi:
plot(y=V_cos, x=S, ylim = c(0,max(V_cos)+0.1))
abline(h=ex, col='brown')

# Illustration of the law of large numbers for power functions:
Mpk<-m_pk(k=k, p=p, alpha=alpha, H=H, sigma=sigma)

sf_mod<-function(Xpath,...) {

    Path<-unlist(Xpath)
    sf(path=Path,...)
}

V_pow<-sapply(Pths,FUN=sf_mod,f=f_pow,k=k,r=r,H=H,freq=freq,p=p)
plot(y=V_pow, x=S, ylim = c(0,max(V_pow)+0.1))
abline(h=Mpk, col='brown')

Statistical estimator for sigma

Description

Statistical estimator for sigma

Usage

sigma_hat(t1, k, path, alpha, H, freq)

Arguments

Examples

m<-45; M<-60; N<-2^14-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='H'
r=1; k=2; p=0.4; t1=1; t2=2

# Reproducing the work of ContinEstim
# in high frequency case
lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=1)$lfsm

H_est<-H_hat(p=p,k=k,path=lfsm)
H_est
alpha_est<-alpha_hat(t1=t1,t2=t2,k=k,path=lfsm,H=H_est,freq=freq)
alpha_est

sigma_est<-tryCatch(
                    sigma_hat(t1=t1,k=k,path=lfsm,
                    alpha=alpha_est,H=H_est,freq=freq),
                    error=function(c) 'Impossible to compute sigma_est')
sigma_est

Function theta

Description

Function of the form

\theta(g,h)_{p} = a_p^{-2} \int_{\R^2} |xy|^{-1-p}U_{g,h}(x,y) dxdy

Usage

theta(p, alpha, sigma, g, h)

Arguments

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.