Package 'ambit'

Title: Simulation and Estimation of Ambit Processes
Description: Simulation and estimation tools for various types of ambit processes, including trawl processes and weighted trawl processes.
Authors: Almut E. D. Veraart [aut, cre, cph]
Maintainer: Almut E. D. Veraart <[email protected]>
License: GPL-3
Version: 0.1.2
Built: 2025-02-17 03:03:26 UTC
Source: https://github.com/cran/ambit

Help Index

Autocorrelation function of the exponential trawl function

Description

This function computes the autocorrelation function associated with the exponential trawl function.

Usage

acf_Exp(x, lambda)

Arguments

x

The argument (lag) at which the autocorrelation function associated with the exponential trawl function will be evaluated
lambda

parameter in the exponential trawl

Details

The trawl function is parametrised by the parameter λ>0\lambda > 0 as follows:

g(x)=eλx, for x≤0.g(x) = e^{\lambda x}, \mbox{ for } x \le 0.

Its autocorrelation function is given by:

r(x)=e−λx, for x≥0.r(x) = e^{-\lambda x}, \mbox{ for } x \ge 0.

Value

The autocorrelation function of the exponential trawl function evaluated at x

Examples

acf_Exp(1,0.1)

Autocorrelation function of the long memory trawl function

Description

This function computes the autocorrelation function associated with the long memory trawl function.

Usage

acf_LM(x, alpha, H)

Arguments

x

The argument (lag) at which the autocorrelation function associated with the long memory trawl function will be evaluated
alpha

parameter in the long memory trawl
H

parameter in the long memory trawl

Details

The trawl function is parametrised by the two parameters H>1H> 1 and α>0\alpha > 0 as follows:

g(x)=(1−x/α)−H, for x≤0.g(x) = (1-x/\alpha)^{-H}, \mbox{ for } x \le 0.

Its autocorrelation function is given by

r(x)=(1+x/α)(1−H), for x≥0.r(x)=(1+x/\alpha)^{(1-H)}, \mbox{ for } x \ge 0.

Value

The autocorrelation function of the long memory trawl function evaluated at x

Examples

acf_LM(1,0.3,1.5)

Autocorrelation function of the supIG trawl function

Description

This function computes the autocorrelation function associated with the supIG trawl function.

Usage

acf_supIG(x, delta, gamma)

Arguments

x

The argument (lag) at which the autocorrelation function associated with the supIG trawl function will be evaluated
delta

parameter in the supIG trawl
gamma

parameter in the supIG trawl

Details

The trawl function is parametrised by the two parameters δ≥0\delta \geq 0 and γ≥0\gamma \geq 0 as follows:

g(x)=(1−2xγ−2)−1/2exp⁡(δγ(1−(1−2xγ−2)1/2)), for x≤0.g(x) = (1-2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1-2x\gamma^{-2})^{1/2})), \mbox{ for } x \le 0.

It is assumed that δ\delta and γ\gamma are not simultaneously equal to zero. Its autocorrelation function is given by:

r(x)=exp⁡(δγ(1−1+2x/γ2)), for x≥0.r(x) = \exp(\delta\gamma (1-\sqrt{1+2 x/\gamma^2})), \mbox{ for } x \ge 0.

Value

The autocorrelation function of the supIG trawl function evaluated at x

Examples

acf_supIG(1,0.3,0.1)

Add slices and return vector of the sums of slices

Description

Add slices and return vector of the sums of slices

Usage

AddSlices_Rcpp(slicematrix)

Arguments

slicematrix

A matrix of slices.

Value

Returns the vector of the sums of the slices

Add slices and return vector of the weighted sums of slices

Description

Add slices and return vector of the weighted sums of slices

Usage

AddWeightedSlices_Rcpp(slicematrix, weightvector)

Arguments

slicematrix

A matrix of slices.
weightvector

A vector of weights.

Value

Returns the vector of the weighted sums of the slices

Computing the true asymptotic variance in the CLT of the trawl estimation

Description

This function computes the theoretical asymptotic variance appearing in the CLT of the trawl process for a given trawl function and fourth cumulant.

Usage

asymptotic_variance(t, c4, varlevyseed = 1, trawlfct, trawlfct_par)

Arguments

t

Time point at which the asymptotic variance is computed
c4

The fourth cumulant of the Levy seed of the trawl process
varlevyseed

The variance of the Levy seed of the trawl process, the default is 1
trawlfct

The trawl function for which the asymptotic variance will be computed (Exp, supIG or LM)
trawlfct_par

The parameter vector of the trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)

Details

As derived in Sauri and Veraart (2022), the asymptotic variance in the central limit theorem for the trawl function estimation is given by

σa2(t)=c4(L′)a(t)+2{∫0∞a(s)2ds+∫0ta(t−s)a(t+s)ds−∫t∞a(s−t)a(t+s)ds},\sigma_{a}^{2}(t)=c_{4}(L')a(t)+2\{ \int_{0}^{\infty}a(s)^{2}ds+ \int_{0}^{t}a(t-s)a(t+s)ds-\int_{t}^{\infty}a(s-t)a(t+s)ds\},

for t>0t>0. The integrals in the above formula are approximated numerically.

Value

The function returns σa2(t)\sigma_{a}^{2}(t).

Examples

#Compute the asymptotic variance at time t for an exponential trawl with
#parameter 2; here we assume that the fourth cumulant equals 1.
av<-asymptotic_variance(t=1, c4=1, varlevyseed=1, trawlfct="Exp", trawlfct_par=2)
#Print the av
av$v
#Print the four components of the asymptotic variance separately
av$v1
av$v2
av$v3
av$v4

#Note that v=v1+v2+v3+v4
av$v
av$v1+av$v2+av$v3+av$v4

Estimating the asymptotic variance in the trawl function CLT

Description

This function estimates the asymptotic variance which appears in the CLT for the trawl function estimation.

Usage

asymptotic_variance_est(t, c4, varlevyseed = 1, Delta, avector, N = NULL)

Arguments

t

The time point at which to compute the asymptotic variance
c4

The fourth cumulant of the Levy seed of the trawl process
varlevyseed

The variance of the Levy seed of the trawl process, the default is 1
Delta

The width Delta of the observation grid
avector

The vector (a^(0),a^(Δn),...,a^((n−1)Δn))(\hat a(0), \hat a(\Delta_n), ..., \hat a((n-1)\Delta_n))
N

The optional parameter to specify the upper bound NnN_n in the computations of the estimators

Details

As derived in Sauri and Veraart (2022), the estimated asymptotic variance is given by

σ^a2(t)=v^1(t)+v^2(t)+v^3(t)+v^4(t),\hat \sigma^2_a(t)=\hat v_1(t)+\hat v_2(t)+\hat v_3(t)+\hat v_4(t),

where

v^1(t):=c4(L′)^a^(t)=RQna^(t)/a^(0),\hat{v}_{1}(t):=\widehat{c_{4}(L')}\hat{a}(t)=RQ_n\hat{a}(t)/ \hat{a}(0),

for

RQn:=12nΔn∑k=0n−2(X(k+1)Δn−XkΔn)4,RQ_n:=\frac{1}{\sqrt{2 n\Delta_{n}}} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^4,

and

v^2(t):=2∑l=0Nna^2(lΔn)Δn,\hat{v}_{2}(t):=2\sum_{l=0}^{N_{n}}\hat{a}^{2}(l\Delta_{n}) \Delta_{n},

v^3(t):=2∑l=0min⁡{i,n−1−i}a^((i−l)Δn)a^((i+l)Δn)Δn,\hat{v}_{3}(t):=2\sum_{l=0}^{\min\{i,n-1-i\}}\hat{a}((i-l)\Delta_{n}) \hat{a}((i+l)\Delta_{n})\Delta_{n},

v^4(t):=−2∑l=iNn−ia^((l−i)Δn)a^((i+l)Δn)Δn.\hat{v}_{4}(t):=-2\sum_{l=i}^{N_{n}-i}\hat{a}((l-i)\Delta_{n}) \hat{a}((i+l)\Delta_{n})\Delta_{n}.

Value

The estimated asymptotic variance v^=σ^a2(t)\hat v=\hat \sigma_a^2(t) and its components v^1,v^2,v^3,v^4\hat v_1, \hat v_2, \hat v_3, \hat v_4.

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data <- sim_weighted_trawl(my_n, my_delta,
                               "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl function
my_lag <- 100+1
trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat

#Estimate the fourth cumulant of the trawl process
c4_est <- c4est(Poi_data, my_delta)

asymptotic_variance_est(t=1, c4=c4_est, varlevyseed=1,
                        Delta=my_delta, avector=trawl)$v

Estimating the fourth cumulant of the trawl process

Description

This function estimates the fourth cumulant of the trawl process.

Usage

c4est(data, Delta)

Arguments

data

The data set used to estimate the fourth cumulant
Delta

The width Delta of the observation grid

Details

According to Sauri and Veraart (2022), estimator based on X0,XΔn,…,X(n−1)ΔnX_0, X_{\Delta_n}, \ldots, X_{(n-1)\Delta_n} is given by

c^4(L′)=RQn/a^(0),\hat c_4(L')=RQ_n/\hat a(0),

where

RQn:=12nΔn∑k=0n−2(X(k+1)Δn−XkΔn)4,RQ_n:=\frac{1}{\sqrt{2 n\Delta_{n}}} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^4,

and

a^(0)=12Δnn∑k=0n−2(X(k+1)Δn−XkΔn)2.\hat a(0)=\frac{1}{2\Delta_{n}n} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^{2}.

Value

The function returns the estimated fourth cumulant of the Levy seed: c^4(L′)\hat c_4(L').

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Estimate the fourth cumulant of the trawl process
c4est(Poi_data, my_delta)

Nonparametric estimation of the trawl set Leb(A)

Description

This function estimates the size of the trawl set given by Leb(A).

Usage

LebA_est(data, Delta, biascor = FALSE)

Arguments

data

Data to be used in the trawl function estimation.
Delta

Width of the grid on which we observe the data
biascor

A binary variable determining whether a bias correction should be computed, the default is FALSE

Details

Estimation of the trawl function using the methodology proposed in Sauri and Veraart (2022).

Value

The estimated Lebesgue measure of the trawl set

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 5000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(1726)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl set without bias correction
LebA1 <-LebA_est(Poi_data, my_delta)
LebA1

#Estimate the trawl set with bias correction
LebA2 <-LebA_est(Poi_data, my_delta, biascor=TRUE)
LebA2

#Note that Leb(A)=1/my_lambda for an exponential trawl

Nonparametric estimation of the trawl (sub-) sets Leb(A), Leb(A intersection A_h), Leb(A setdifference A_h)

Description

This function estimates Leb(A), Leb(A intersection A_h), Leb(A\ A_h).

Usage

LebA_slice_est(data, Delta, h, biascor = FALSE)

Arguments

data

Data to be used in the trawl function estimation.
Delta

Width of the grid on which we observe the data
h

Time point used in A intersection A_h and the setdifference A setdifference A_h
biascor

A binary variable determining whether a bias correction should be computed, the default is FALSE

Details

Estimation of the trawl function using the methodology proposed in Sauri and Veraart (2022).

Value

LebA

LebAintersection

LebAsetdifference

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 5000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(1726)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl set and its two slices at time h=2 without bias correction
est1 <- LebA_slice_est(Poi_data, my_delta, h=2)
est1$LebA
est1$LebAintersection
est1$LebAsetdifference

#Estimate the trawl set and its two slices at time h=2 without bias correction
est2 <- LebA_slice_est(Poi_data, my_delta, h=2, biascor=TRUE)
est2$LebA
est2$LebAintersection
est2$LebAsetdifference

#Note that Leb(A)=1/my_lambda for an exponential trawl

Nonparametric estimation of the ratios Leb(A intersection A_h)/Leb(A), Leb(A setdifference A_h)/Leb(A)

Description

This function estimates the ratios Leb(A intersection A_h)/Leb(A), Leb(A\ A_h)/Leb(A).

Usage

LebA_slice_ratio_est_acfbased(data, Delta, h)

Arguments

data

Data to be used in the trawl function estimation.
Delta

Width of the grid on which we observe the data
h

Time point used in A intersection A_h and the setdifference A setdifference A_h

Details

Estimation of the trawl function using the methodology proposed in Sauri and Veraart (2022) which is based on the empirical acf.

Value

LebAintersection_ratio: LebAintersection/LebA

LebAsetdifference_ratio: LebAsetdifference/LebA

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 5000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(1726)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl set and its two slices at time h=0.5
est <- LebA_slice_ratio_est_acfbased(Poi_data, my_delta, h=0.5)
#Print the ratio LebAintersection/LebA
est$LebAintersection_ratio
#Print the ratio LebAsetdifference/LebA
est$LebAsetdifference_ratio

my_mse

Description

Returns the mean absolute error between two vectors

Usage

my_mae(x, y)

Arguments

x

vector
y

vector

Value

Mean absolute error between the two vectors x and y

Examples

#Simulate two vectors of i.i.d.~standard normal data
set.seed(456)
x <- rnorm(100)
y <- rnorm(100)
#Compute the mean absolute error between both vectors
my_mae(x,y)

my_mse

Description

Returns the mean squared error between two vectors

Usage

my_mse(x, y)

Arguments

x

vector
y

vector

Value

Mean square error between the two vectors x and y

Examples

#Simulate two vectors of i.i.d.~standard normal data
set.seed(456)
x <- rnorm(100)
y <- rnorm(100)
#Compute the mean squared error between both vectors
my_mse(x,y)

my_results

Description

Returns summary statistics

Usage

my_results(x, sd = 1, digits = 3)

Arguments

x

data
sd

Optional parameter giving the standard deviation of the normal distribution used for computing the coverage probabilities
digits

Optional parameter to how many digits the results should be rounded, the default is three.

Details

This functions returns the sample mean, sample standard deviation and the coverage probabilities at level 75%, 80%, 85%, 90%, 95%, 99% compared to the standard normal quantiles.

Value

The vector of the sample mean, sample standard deviation and the coverage probabilities at level 75%, 80%, 85%, 90%, 95%, 99% compared to the standard normal quantiles.

Examples

#Simulate i.i.d.~standard normal data
set.seed(456)
data <- rnorm(10000)
#Display the sample mean, standard deviation and coverage probabilities:
my_results(data)

Nonparametric estimation of the trawl function

Description

This function implements the nonparametric trawl estimation proposed in Sauri and Veraart (2022).

Usage

nonpar_trawlest(data, Delta, lag = 100)

Arguments

data

Data to be used in the trawl function estimation.
Delta

Width of the grid on which we observe the data
lag

The lag until which the trawl function should be estimated

Details

Estimation of the trawl function using the methodology proposed in Sauri and Veraart (2022). Suppose the data is observed on the grid 0, Delta, 2Delta, ..., (n-1)Delta. Given the path contained in data, the function returns the lag-dimensional vector

(a^(0),a^(Δ),…,a^((lag−1)Δ)).(\hat a(0), \hat a(\Delta), \ldots, \hat a((lag-1) \Delta)).

In the case when lag=n, the n-1 dimensional vector

(a^(0),a^(Δ),…,a^((n−2)Δ))(\hat a(0), \hat a(\Delta), \ldots, \hat a((n-2) \Delta))

is returned.

Value

ahat Returns the lag-dimensional vector (a^(0),a^(Δ),…,a^((lag−1)Δ)).(\hat a(0), \hat a(\Delta), \ldots, \hat a((lag-1) \Delta)). Here, a^(0)\hat a(0) is estimated based on the realised variance estimator.

a0_alt Returns the alternative estimator of a(0) using the same methodology as the one used for t>0. Note that this is not the recommended estimator to use, but can be used for comparison purposes.

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 5000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(1726)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl function
my_lag <- 100+1
PoiEx_trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat

#Plot the estimated trawl function and superimpose the true one
l_seq <- seq(from = 0,to = (my_lag-1), by = 1)
esttrawlfct.data <- base::data.frame(l=l_seq[1:31],
                               value=PoiEx_trawl[1:31])
p1 <- ggplot2::ggplot(esttrawlfct.data, ggplot2::aes(x=l,y=value))+
  ggplot2::geom_point(size=3)+
  ggplot2::geom_function(fun = function(x) acf_Exp(x*my_delta,my_lambda), colour="red", size=1.5)+
  ggplot2::xlab("l")+
  ggplot2::ylab(latex2exp::TeX("$\\hat{a}(\\cdot)$ for Poisson trawl process"))
p1

Computing the scaled realised quarticity

Description

This function computes the scaled realised quarticity of a time series for a given width of the observation grid.

Usage

rq(data, Delta)

Arguments

data

The data set used to compute the scaled realised quarticity
Delta

The width Delta of the observation grid

Details

According to Sauri and Veraart (2022), the scaled realised quarticity for X0,XΔn,…,X(n−1)ΔnX_0, X_{\Delta_n}, \ldots, X_{(n-1)\Delta_n} is given by

RQn:=12nΔn∑k=0n−2(X(k+1)Δn−XkΔn)4.RQ_n:=\frac{1}{\sqrt{2 n\Delta_{n}}} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^4.

Value

The function returns the scaled realised quarticity RQ_n.

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path

#Compute the scaled realised quarticity
rq(Poi_data, my_delta)

Simulation of a weighted trawl process

Description

This function simulates a weighted trawl process for various choices of the trawl function and the marginal distribution.

Usage

sim_weighted_trawl(
  n,
  Delta,
  trawlfct,
  trawlfct_par,
  distr,
  distr_par,
  kernelfct = NULL
)

Arguments

n

number of grid points to be simulated (excluding the starting value)
Delta

grid-width
trawlfct

the trawl function a used in the simulation (Exp, supIG or LM)
trawlfct_par

parameter vector of trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)
distr

marginal distribution. Choose from "Gamma" (Gamma), "Gauss" (Gaussian), "Cauchy" (Cauchy), "NIG" (Normal Inverse Gaussian), Poi" (Poisson), "NegBin" (Negative Binomial)
distr_par

parameters of the marginal distribution: (Gamma: shape, scale; Gauss: mu, sigma (i.e. the second parameter is the standard deviation, not the variance); Cauchy: l, s; NIG: alpha, beta, delta, mu; Poi: v, NegBin: m, theta)
kernelfct

the kernel function p used in the ambit process

Details

This functions simulates a sample path from a weighted trawl process given by

Yt=∫(−∞,t]×(−∞,∞)p(t−s)I(0,a(t−s))(x)L(dx,ds),Y_t =\int_{(-\infty,t]\times (-\infty, \infty)} p(t-s)I_{(0,a(t-s))}(x)L(dx,ds),

for t≥0t \ge 0, and returns Y0,YΔ,…,YnΔY_0, Y_{\Delta}, \ldots, Y_{n\Delta}.

Value

path Simulated path

slice_sizes slice sizes used

S_matrix Matrix of all slices

kernelweights kernel weights used

Examples

#Simulation of a Gaussian trawl process with exponential trawl function
n<-2000
Delta<-0.1
trawlfct="Exp"
trawlfct_par <-0.5
distr<-"Gauss"
distr_par<-c(0,1) #mean 0, std 1
set.seed(233)
path <- sim_weighted_trawl(n, Delta, trawlfct, trawlfct_par, distr, distr_par)$path
#Plot the path
library(ggplot2)
df <- data.frame(time = seq(0,n,1), value=path)
p <- ggplot(df, aes(x=time, y=path))+
  geom_line()+
  xlab("l")+
  ylab("Trawl process")
  p

Simulation of a weighted trawl process with generic trawl function

Description

This function simulates a weighted trawl process for a generic trawl function and various choices the marginal distribution. The specific trawl function to be used can be supplied directly by the user.

Usage

sim_weighted_trawl_gen(
  n,
  Delta,
  trawlfct_gen,
  distr,
  distr_par,
  kernelfct = NULL
)

Arguments

n

number of grid points to be simulated (excluding the starting value)
Delta

grid-width
trawlfct_gen

the trawl function a used in the simulation
distr

marginal distribution. Choose from "Gamma" (Gamma), "Gauss" (Gaussian), "Cauchy" (Cauchy), "NIG" (Normal Inverse Gaussian), Poi" (Poisson), "NegBin" (Negative Binomial)
distr_par

parameters of the marginal distribution: (Gamma: shape, scale; Gauss: mu, sigma (i.e. the second parameter is the standard deviation, not the variance); Cauchy: l, s; NIG: alpha, beta, delta, mu; Poi: v, NegBin: m, theta)
kernelfct

the kernel function p used in the ambit process

Details

This functions simulates a sample path from a weighted trawl process given by

Yt=∫(−∞,t]×(−∞,∞)p(t−s)I(0,a(t−s))(x)L(dx,ds),Y_t =\int_{(-\infty,t]\times (-\infty, \infty)} p(t-s)I_{(0,a(t-s))}(x)L(dx,ds),

for t≥0t \ge 0, and returns Y0,YΔ,…,YnΔY_0, Y_{\Delta}, \ldots, Y_{n\Delta}. The user needs to ensure that trawlfct_gen is a monotonic function.

Value

path Simulated path

slice_sizes slice sizes used

S_matrix Matrix of all slices

kernelweights kernel weights used

Examples

#Simulation of a Gaussian trawl process with exponential trawl function
n<-2000
Delta<-0.1

trawlfct_par <-0.5
distr<-"Gauss"
distr_par<-c(0,1) #mean 0, std 1
set.seed(233)

a <- function(x){exp(-trawlfct_par*x)}
path <- sim_weighted_trawl_gen(n, Delta, a,
                           distr, distr_par)$path
#Plot the path
library(ggplot2)
df <- data.frame(time = seq(0,n,1), value=path)
p <- ggplot(df, aes(x=time, y=path))+
  geom_line()+
  xlab("l")+
  ylab("Trawl process")
p

Computing the infeasible test statistic from the trawl function estimation CLT

Description

This function computes the infeasible test statistic appearing in the CLT for the trawl function estimation.

Usage

test_asymnorm(ahat, n, Delta, k, c4, varlevyseed = 1, trawlfct, trawlfct_par)

Arguments

ahat

The term a^(kΔn)\hat a(k \Delta_n) in the CLT
n

The number n of observations in the sample
Delta

The width Delta of the observation grid
k

The time point in 0,1,…,n−10, 1, \ldots, n-1; the test statistic will be computed for the time point k∗Δnk*\Delta_n.
c4

The fourth cumulant of the Levy seed of the trawl process
varlevyseed

The variance of the Levy seed of the trawl process, the default is 1
trawlfct

The trawl function for which the asymptotic variance will be computed (Exp, supIG or LM)
trawlfct_par

The parameter vector of the trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)

Details

As derived in Sauri and Veraart (2022), the infeasible test statistic is given by

nΔnσa2(kΔn)(a^(kΔn)−a(kΔn)),\frac{\sqrt{n\Delta_{n}}}{\sqrt{\sigma_{a}^2(k \Delta_n)}} \left(\hat{a}(k\Delta_n)-a(k \Delta_n)\right),

for k∈{0,1,…,n−1}k \in \{0, 1, \ldots, n-1\}.

Value

The function returns the infeasible test statistic specified above.

Examples

test_asymnorm(ahat=0.9, n=5000, Delta=0.1, k=1, c4=1, varlevyseed=1,
              trawlfct="Exp", trawlfct_par=0.1)

Computing the feasible statistic of the trawl function CLT

Description

This function computes the feasible statistics associated with the CLT for the trawl function estimation.

Usage

test_asymnorm_est(
  data,
  Delta,
  trawlfct,
  trawlfct_par,
  biascor = FALSE,
  k = NULL
)

Arguments

data

The data set based on observations of X0,XΔn,…,X(n−1)ΔnX_0, X_{\Delta_n}, \ldots, X_{(n-1)\Delta_n}
Delta

The width Delta of the observation grid
trawlfct

The trawl function for which the asymptotic variance will be computed (Exp, supIG or LM)
trawlfct_par

The parameter vector of the trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)
biascor

A binary variable determining whether a bias correction should be computed, the default is FALSE
k

The optional parameter specifying the time point in 0,1,…,n−10, 1, \ldots, n-1; the test statistic will be computed for the time point kΔnk \Delta_n.

Details

As derived in Sauri and Veraart (2022), the feasible statistic, for t>0t>0, is given by

T(t)n:=nΔnσa2(t)^(a^(t)−a(t)−bias(t)).T(t)_n:=\frac{\sqrt{n\Delta_{n}}}{\sqrt{\widehat{\sigma_{a}^2(t)}}} \left(\hat{a}(t)-a(t)-bias(t)\right).

For t=0t=0, we have

T(t)n:=nΔnRQn(a^(0)−a(0)−bias(0)),T(t)_n:=\frac{\sqrt{n\Delta_{n}}}{\sqrt{RQ_n}} \left(\hat{a}(0)-a(0)-bias(0)\right),

where

RQn:=12nΔn∑k=0n−2(X(k+1)Δn−XkΔn)4.RQ_n:=\frac{1}{\sqrt{2 n\Delta_{n}}} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^4.

We set bias(t)=0bias(t)=0 in the case when biascor==FALSE and bias(t)=0.5∗Δ∗a^′(t)bias(t)=0.5 * \Delta * \hat a'(t) otherwise.

Value

The function returns the vector of the feasible statistics (T(0)n,T((Δ)n,…,T((n−2)Δn))(T(0)_n, T((\Delta)_n, \ldots, T((n-2)\Delta_n)) if no bias correction is required and (T(0)n,T((Δ)n,…,T((n−3)Δn))(T(0)_n, T((\Delta)_n, \ldots, T((n-3)\Delta_n)) if bias correction is required if k is not provided, otherwise it returns the value T(kΔn)nT(k \Delta_n)_n. If the estimated asymptotic variance is <= 0, the value of the test statistic is set to 999.

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data <- sim_weighted_trawl(my_n, my_delta,
                               "Exp", my_lambda, "Poi", my_v)$path

#Compute the test statistic for time t=0
##Either one can use:
test_asymnorm_est(Poi_data, my_delta,
                  trawlfct="Exp", trawlfct_par=my_lambda)[1]
#or:
test_asymnorm_est(Poi_data, my_delta,
                  trawlfct="Exp", trawlfct_par=my_lambda, k=0)

Computing the feasible statistic of the trawl function CLT

Description

This function computes the feasible test statistic appearing in the CLT for the trawl function estimation.

Usage

test_asymnorm_est_dev(
  ahat,
  n,
  Delta,
  k,
  c4,
  varlevyseed = 1,
  trawlfct,
  trawlfct_par,
  avector
)

Arguments

ahat

The estimated trawl function at time t: a^(t)\hat{a}(t)
n

The number of observations in the data set
Delta

The width Delta of the observation grid
k

The time point in 0,1,…,n−10, 1, \ldots, n-1; the test statistic will be computed for the time point k∗Δnk * \Delta_n.
c4

The fourth cumulant of the Levy seed of the trawl process
varlevyseed

The variance of the Levy seed of the trawl process, the default is 1
trawlfct

The trawl function for which the asymptotic variance will be computed (Exp, supIG or LM)
trawlfct_par

The parameter vector of the trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)
avector

The vector (a^(0),a^(Deltan),...,a^((n−1)Δn))(\hat a(0), \hat a(Delta_n), ..., \hat a((n-1)\Delta_n))

Details

As derived in Sauri and Veraart (2022), the feasible statistic is given by

T(kΔn)n:=nΔnσa2(Δn)^(a^(Δn)−a(Δn))T(k \Delta_n)_n:=\frac{\sqrt{n\Delta_{n}}}{ \sqrt{\widehat{\sigma_{a}^2( \Delta_n)}}} \left(\hat{a}( \Delta_n)-a( \Delta_n)\right)

.

Value

The function returns the feasible statistic T(Δn)nT( \Delta_n)_n if the estimated asymptotic variance is positive and 999 otherwise.

Estimating the derivative of the trawl function using the empirical derivative

Description

This function estimates the derivative of the trawl function using the empirical derivative of the trawl function.

Usage

trawl_deriv(data, Delta, lag = 100)

Arguments

data

The data set used to compute the derivative of the trawl function
Delta

The width Delta of the observation grid
lag

The lag until which the trawl function should be estimated

Details

According to Sauri and Veraart (2022), the derivative of the trawl function can be estimated based on observations X0,XΔn,…,X(n−1)ΔnX_0, X_{\Delta_n}, \ldots, X_{(n-1)\Delta_n} by

a^(t)=1Δn(a^(t+Δn)−a^(Δn)),\widehat a(t)=\frac{1}{\Delta_{n}} (\hat a(t+\Delta_n)-\hat a(\Delta_n)),

for Δnl≤t<(l+1)Δn\Delta_nl\leq t < (l+1)\Delta_n.

Value

The function returns the lag-dimensional vector (a^′(0),a^′(Δ),…,a^′((lag−1)Δ)).(\hat a'(0), \hat a'(\Delta), \ldots, \hat a'((lag-1) \Delta)).

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data <- sim_weighted_trawl(my_n, my_delta,
                               "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl function
my_lag <- 100+1
trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat

#Estimate the derivative of the trawl function
trawl_deriv <- trawl_deriv(Poi_data, my_delta, lag=100)

Estimating the derivative of the trawl function

Description

This function estimates the derivative of the trawl function using the modified version proposed in Sauri and Veraart (2022).

Usage

trawl_deriv_mod(data, Delta, lag = 100)

Arguments

data

The data set used to compute the derivative of the trawl function
Delta

The width Delta of the observation grid
lag

The lag until which the trawl function should be estimated

Details

According to Sauri and Veraart (2022), the derivative of the trawl function can be estimated based on observations X0,XΔn,…,X(n−1)ΔnX_0, X_{\Delta_n}, \ldots, X_{(n-1)\Delta_n} by

a^(t)=1nΔn2∑k=l+1n−2(X(k+1)Δn−XkΔn)(X(k−l+1)Δn−X(k−l)Δn),\widehat a(t)=\frac{1}{\sqrt{ n\Delta_{n}^2}} \sum_{k=l+1}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n}) (X_{(k-l+1)\Delta_n}-X_{(k-l)\Delta_n}),

for Δnl≤t<(l+1)Δn\Delta_nl\leq t < (l+1)\Delta_n.

Value

The function returns the lag-dimensional vector (a^′(0),a^′(Δ),…,a^′((lag−1)Δ)).(\hat a'(0), \hat a'(\Delta), \ldots, \hat a'((lag-1) \Delta)).

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data <- sim_weighted_trawl(my_n, my_delta,
                               "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl function
my_lag <- 100+1
trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat

#Estimate the derivative of the trawl function
trawl_deriv <- trawl_deriv_mod(Poi_data, my_delta, lag=100)

Evaluates the exponential trawl function

Description

Evaluates the exponential trawl function

Usage

trawl_Exp(x, lambda)

Arguments

x

the argument at which the exponential trawl function will be evaluated
lambda

the parameter λ\lambda in the exponential trawl

Details

The trawl function is parametrised by parameter λ>0\lambda > 0 as follows:

g(x)=eλx, for x≤0.g(x) = e^{\lambda x}, \mbox{ for } x \le 0.

Value

The exponential trawl function evaluated at x

Examples

trawl_Exp(-1,0.5)

Evaluates the long memory trawl function

Description

Evaluates the long memory trawl function

Usage

trawl_LM(x, alpha, H)

Arguments

x

the argument at which the supOU/long memory trawl function will be evaluated
alpha

the parameter α\alpha in the long memory trawl
H

the parameter HH in the long memory trawl

Details

The trawl function is parametrised by the two parameters H>1H> 1 and α>0\alpha > 0 as follows:

g(x)=(1−x/α)−H, for x≤0.g(x) = (1-x/\alpha)^{-H}, \mbox{ for } x \le 0.

If H∈(1,2]H \in (1,2], then the resulting trawl process has long memory, for H>2H>2, it has short memory.

Value

the long memory trawl function evaluated at x

Examples

trawl_LM(-1,0.5, 1.5)

Evaluates the supIG trawl function

Description

Evaluates the supIG trawl function

Usage

trawl_supIG(x, delta, gamma)

Arguments

x

the argument at which the supIG trawl function will be evaluated
delta

the parameter δ\delta in the supIG trawl
gamma

the parameter γ\gamma in the supIG trawl

Details

The trawl function is parametrised by the two parameters δ≥0\delta \geq 0 and γ≥0\gamma \geq 0 as follows:

gd(x)=(1−2xγ−2)−1/2exp⁡(δγ(1−(1−2xγ−2)1/2)), for x≤0.gd(x) = (1-2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1-2x\gamma^{-2})^{1/2})), \mbox{ for } x \le 0.

It is assumed that δ\delta and γ\gamma are not simultaneously equal to zero.

Value

The supIG trawl function evaluated at x

Examples

trawl_supIG(-1,0.5,0.2)