Computes the correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

Description

Computes the correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

Usage

BivLSD_Cor(p1, p2)

Arguments

Value

Correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

Computes the covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

Description

Computes the covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

Usage

BivLSD_Cov(p1, p2)

Arguments

Value

Covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

Computes the correlation of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Description

Computes the correlation of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Usage

BivModLSD_Cor(delta, p1, p2)

Arguments

Value

Covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Computes the covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Description

Computes the covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Usage

BivModLSD_Cov(delta, p1, p2)

Arguments

Value

Covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Simulates from the bivariate logarithmic series distribution

Description

Simulates from the bivariate logarithmic series distribution

Usage

Bivariate_LSDsim(N, p1, p2)

Arguments

Details

The probability mass function of a random vector X=(X_1,X_2)' following the bivariate logarithmic series distribution with parameters 0<p_1, p_2<1 with p:=p_1+p_2<1 is given by

P(X_1=x_1,X_2=x_2)=\frac{\Gamma(x_1+x_2)}{x_1!x_2!} \frac{p_1^{x_1}p_2^{x_2}}{(-\log(1-p))},

for x_1,x_2=0,1,2,\dots such that x_1+x_2>0. The simulation proceeds in two steps: First, X_1 is simulated from the modified logarithmic distribution with parameters \tilde p_1=p_1/(1-p_2) and \delta_1=\log(1-p_2)/\log(1-p). Then we simulate X_2 conditional on X_1. We note that X_2|X_1=x_1 follows the logarithmic series distribution with parameter p_2 when x_1=0, and the negative binomial distribution with parameters (x_1,p_2) when x_1>0.

Value

An N \times 2 matrix with N simulated values from the bivariate logarithmic series distribution

Simulates from the bivariate negative binomial distribution

Description

Simulates from the bivariate negative binomial distribution

Usage

Bivariate_NBsim(N, kappa, p1, p2)

Arguments

Details

A random vector {\bf X}=(X_1,X_2)' is said to follow the bivariate negative binomial distribution with parameters \kappa, p_1, p_2 if its probability mass function is given by

P({\bf X}={\bf x})=\frac{\Gamma(x_1+x_2+\kappa)}{x_1!x_2! \Gamma(\kappa)}p_1^{x_1}p_2^{x_2}(1-p_1-p_2)^{\kappa},

where, for i=1,2, x_i\in\{0,1,\dots\}, 0<p_i<1 such that p_1+p_2<1 and \kappa>0.

Value

An N\times 2 matrix with N simulated values from the bivariate negative binomial distribution

Computes the mean of the logarithmic series distribution

Description

Computes the mean of the logarithmic series distribution

Usage

LSD_Mean(p)

Arguments

Details

A random variable X has logarithmic series distribution with parameter 0<p<1 if

P(X=x)=(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.

Value

Mean of the logarithmic series distribution

Computes the variance of the logarithmic series distribution

Description

Computes the variance of the logarithmic series distribution

Usage

LSD_Var(p)

Arguments

Details

A random variable X has logarithmic series distribution with parameter 0<p<1 if

P(X=x)=(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.

Value

Variance of the logarithmic series distribution

Computes the mean of the modified logarithmic series distribution

Description

Computes the mean of the modified logarithmic series distribution

Usage

ModLSD_Mean(delta, p)

Arguments

Details

A random variable X has modified logarithmic series distribution with parameters 0 \le \delta <1 and 0<p<1 if P(X=0)=(1-\delta) and

P(X=x)=(1-\delta)(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.

Value

Mean of the modified logarithmic series distribution

Computes the variance of the modified logarithmic series distribution

Description

Computes the variance of the modified logarithmic series distribution

Usage

ModLSD_Var(delta, p)

Arguments

Details

A random variable X has modified logarithmic series distribution with parameters 0\le \delta <1 and 0<p<1 if P(X=0)=(1-\delta) and

P(X=x)=(1-\delta)(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.

Value

Mean of the modified logarithmic series distribution

Simulates from the trivariate logarithmic series distribution

Description

Simulates from the trivariate logarithmic series distribution

Usage

Trivariate_LSDsim(N, p1, p2, p3)

Arguments

Details

The probability mass function of a random vector X=(X_1,X_2,X_3)' following the trivariate logarithmic series distribution with parameters 0<p_1, p_2, p_3<1 with p:=p_1+p_2+p_3<1 is given by

P(X_1=x_1,X_2=x_2,X_3=x_3)=\frac{\Gamma(x_1+x_2+x_3)}{x_1!x_2!x_3!} \frac{p_1^{x_1}p_2^{x_2}p_3^{x_3}}{(-\log(1-p))},

for x_1,x_2,x_3=0,1,2,\dots such that x_1+x_2+x_3>0.

The simulation proceeds in two steps: First, X_1 is simulated from the modified logarithmic distribution with parameters \tilde p_1=p_1/(1-p_2-p_3) and \delta_1=\log(1-p_2-p_3)/\log(1-p). Then we simulate (X_2,X_3)' conditional on X_1. We note that (X_2,X_3)'|X_1=x_1 follows the bivariate logarithmic series distribution with parameters (p_2,p_3) when x_1=0, and the bivariate negative binomial distribution with parameters (x_1,p_2,p_3) when x_1>0.

Value

An N \times 3 matrix with N simulated values from the trivariate logarithmic series distribution

Autocorrelation function of the double exponential trawl function

Description

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

Usage

acf_DExp(x, w, lambda1, lambda2)

Arguments

Details

The trawl function is parametrised by parameters 0\le w\le 1 and \lambda_1, \lambda_2 > 0 as follows:

g(x) = w e^{\lambda_1 x}+(1-w) e^{\lambda_2 x}, \mbox{ for } x \le 0.

Its autocorrelation function is given by:

r(x) = (w e^{-\lambda_1 x}/\lambda_1+(1-w) e^{-\lambda_2 x}/\lambda_2)/c, \mbox{ for } x \ge 0,

where

c = w/\lambda_1+(1-w)/\lambda_2.

Value

The autocorrelation function of the double exponential trawl function evaluated at x

Examples

acf_DExp(1,0.3,0.1,2)

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

Details

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

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

Its autocorrelation function is given by:

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

Details

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

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

Its autocorrelation function is given by

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

Details

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

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 \delta and \gamma are not simultaneously equal to zero. Its autocorrelation function is given by:

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)

Fits the trawl function consisting of the weighted sum of two exponential functions

Description

Fits the trawl function consisting of the weighted sum of two exponential functions

Usage

fit_DExptrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

Details

The trawl function is parametrised by the three parameters 0\leq w \leq 1 and \lambda_1,\lambda_2 > 0 as follows:

g(x) = we^{\lambda_1 x}+(1-w)e^{\lambda_2 x}, \mbox{ for } x \le 0.

The Lebesgue measure of the corresponding trawl set is given by w/\lambda_1+(1-w)/\lambda_2.

Value

w: the weight parameter (restricted to be in [0,0.5] for identifiability reasons)

lambda1: the first memory parameter (denoted by \lambda_1 above)

lambda2: the second memory parameter (denoted by \lambda_2 above)

LM: The Lebesgue measure of the trawl set associated with the double exponential trawl

Fits an exponential trawl function to equidistant time series data

Description

Fits an exponential trawl function to equidistant time series data

Usage

fit_Exptrawl(x, Delta = 1, plotacf = FALSE, lags = 100)

Arguments

Details

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

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

The Lebesgue measure of the corresponding trawl set is given by 1/\lambda.

Value

lambda: the memory parameter \lambda in the exponential trawl

LM: the Lebesgue measure of the trawl set associated with the exponential trawl, i.e. 1/\lambda.

Fits a long memory trawl function to equidistant univariate time series data

Description

Fits a long memory trawl function to equidistant univariate time series data

Usage

fit_LMtrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

Details

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

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

The Lebesgue measure of the corresponding trawl set is given by \alpha/(1-H).

Value

alpha: parameter in the long memory trawl

H: parameter in the long memory trawl

LM: The Lebesgue measure of the trawl set associated with the long memory trawl

Fist a negative binomial distribution as marginal law

Description

Fist a negative binomial distribution as marginal law

Usage

fit_marginalNB(x, LM, plotdiag = FALSE)

Arguments

Details

The moment estimator for the parameters of the negative binomial distribution are given by

\hat \theta = 1-\mbox{E}(X)/\mbox{Var}(X),

and

\hat m = \mbox{E}(X)(1-\hat \theta)/(\widehat{ \mbox{LM}} \hat \theta).

Value

m: parameter in the negative binomial marginal distribution

theta: parameter in the negative binomial marginal distribution

a: Here a=\theta/(1-\theta). This is given for an alternative parametrisation of the negative binomial marginal distribution

Fits a Poisson distribution as marginal law

Description

Fits a Poisson distribution as marginal law

Usage

fit_marginalPoisson(x, LM, plotdiag = FALSE)

Arguments

Details

The moment estimator for the Poisson rate parameter is given by

\hat v = \mbox{E}(X)/\widehat{ \mbox{LM}}.

Value

v: the rate parameter in the Poisson marginal distribution

Fits a supIG trawl function to equidistant univariate time series data

Description

Fits a supIG trawl function to equidistant univariate time series data

Usage

fit_supIGtrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

Details

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

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 \delta and \gamma are not simultaneously equal to zero. The Lebesgue measure of the corresponding trawl set is given by \gamma/\delta.

Value

delta: parameter in the supIG trawl

gamma: parameter in the supIG trawl

LM: The Lebesgue measure of the trawl set associated with the supIG trawl

Finds the intersection of two trawl sets

Description

Finds the intersection of two trawl sets

Usage

fit_trawl_intersection(
  fct1 = base::c("Exp", "DExp", "supIG", "LM"),
  fct2 = base::c("Exp", "DExp", "supIG", "LM"),
  lambda11 = 0,
  lambda12 = 0,
  w1 = 0,
  delta1 = 0,
  gamma1 = 0,
  alpha1 = 0,
  H1 = 0,
  lambda21 = 0,
  lambda22 = 0,
  w2 = 0,
  delta2 = 0,
  gamma2 = 0,
  alpha2 = 0,
  H2 = 0,
  LM1,
  LM2,
  plotdiag = FALSE
)

Arguments

Details

Computes R_{12}(0)=\mbox{Leb}(A_1 \cap A_2) based on two trawl functions g_1 and g_2.

Value

The Lebesgue measure of the intersection of the two trawl sets

Finds the intersection of two exponential trawl sets

Description

Finds the intersection of two exponential trawl sets

Usage

fit_trawl_intersection_Exp(lambda1, lambda2, LM1, LM2, plotdiag = FALSE)

Arguments

Details

Computes R_{12}(0)=\mbox{Leb}(A_1 \cap A_2) based on two trawl functions g_1 and g_2.

Value

The Lebesgue measure of the intersection of the two trawl sets

Finds the intersection of two long memory (LM) trawl sets

Description

Finds the intersection of two long memory (LM) trawl sets

Usage

fit_trawl_intersection_LM(alpha1, H1, alpha2, H2, LM1, LM2, plotdiag = FALSE)

Arguments

Details

Computes R_{12}(0)=\mbox{Leb}(A_1 \cap A_2) based on two trawl functions g_1 and g_2.

Value

the Lebesgue measure of the intersection of the two trawl sets

Plots the bivariate histogram of two time series together with the univariate histograms

Description

Plots the bivariate histogram of two time series together with the univariate histograms

Usage

plot_2and1hist(x, y)

Arguments

Details

This function plots the bivariate histogram of two time series together with the univariate histograms

Value

no return value

Plots the bivariate histogram of two time series together with the univariate histograms using ggplot2

Description

Plots the bivariate histogram of two time series together with the univariate histograms using ggplot2

Usage

plot_2and1hist_gg(x, y, bivbins = 50, xbins = 30, ybins = 30)

Arguments

Details

This function plots the bivariate histogram of two time series together with the univariate histograms

Value

no return value

Simulates a bivariate trawl process

Description

Simulates a bivariate trawl process

Usage

sim_BivariateTrawl(
  t,
  Delta = 1,
  burnin = 10,
  marginal = base::c("Poi", "NegBin"),
  dependencetype = base::c("fullydep", "dep"),
  trawl1 = base::c("Exp", "DExp", "supIG", "LM"),
  trawl2 = base::c("Exp", "DExp", "supIG", "LM"),
  v1 = 0,
  v2 = 0,
  v12 = 0,
  kappa1 = 0,
  kappa2 = 0,
  kappa12 = 0,
  a1 = 0,
  a2 = 0,
  lambda11 = 0,
  lambda12 = 0,
  w1 = 0,
  delta1 = 0,
  gamma1 = 0,
  alpha1 = 0,
  H1 = 0,
  lambda21 = 0,
  lambda22 = 0,
  w2 = 0,
  delta2 = 0,
  gamma2 = 0,
  alpha2 = 0,
  H2 = 0
)

Arguments

Details

This function simulates a bivariate trawl process with either Poisson or negative binomial marginal law. For the trawl function there are currently four choices: exponential, double-exponential, supIG or long memory. More details on the precise simulation algorithm is available in the vignette.

Simulates a univariate trawl process

Description

Simulates a univariate trawl process

Usage

sim_UnivariateTrawl(
  t,
  Delta = 1,
  burnin = 10,
  marginal = base::c("Poi", "NegBin"),
  trawl = base::c("Exp", "DExp", "supIG", "LM"),
  v = 0,
  m = 0,
  theta = 0,
  lambda1 = 0,
  lambda2 = 0,
  w = 0,
  delta = 0,
  gamma = 0,
  alpha = 0,
  H = 0
)

Arguments

Details

This function simulates a univariate trawl process with either Poisson or negative binomial marginal law. For the trawl function there are currently four choices: exponential, double-exponential, supIG or long memory. More details on the precise simulation algorithm is available in the vignette.

Evaluates the double exponential trawl function

Description

Evaluates the double exponential trawl function

Usage

trawl_DExp(x, w, lambda1, lambda2)

Arguments

Details

The trawl function is parametrised by parameters 0\leq w\leq 1 and \lambda_1, \lambda_2 > 0 as follows:

g(x) = w e^{\lambda_1 x}+(1-w) e^{\lambda_2 xz}, \mbox{ for } x \le 0.

Value

The double exponential trawl function evaluated at x

Evaluates the exponential trawl function

Description

Evaluates the exponential trawl function

Usage

trawl_Exp(x, lambda)

Arguments

Details

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

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

Value

The exponential trawl function evaluated at x

Evaluates the long memory trawl function

Description

Evaluates the long memory trawl function

Usage

trawl_LM(x, alpha, H)

Arguments

Details

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

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

Value

the long memory trawl function evaluated at x

Evaluates the supIG trawl function

Description

Evaluates the supIG trawl function

Usage

trawl_supIG(x, delta, gamma)

Arguments

Details

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

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 \delta and \gamma are not simultaneously equal to zero.

Value

The supIG trawl function evaluated at x