| Title: | Simulate and Visualize Kendall Random Walks and Related Distributions |
| Version: | 0.9.4 |
| Description: | Kendall random walks are a continuous-space Markov chains generated by the Kendall generalized convolution. This package provides tools for simulating these random walks and studying distributions related to them. For more information about Kendall random walks see Jasiulis-Gołdyn (2014) <doi:10.48550/arXiv.1412.0220>. |
| Depends: | R (≥ 3.3) |
| License: | MIT + file LICENSE |
| Encoding: | UTF-8 |
| LazyData: | true |
| Imports: | ggplot2, dplyr, tibble, actuar |
| RoxygenNote: | 6.1.0 |
| Suggests: | knitr, rmarkdown, testthat, covr |
| VignetteBuilder: | knitr |
| NeedsCompilation: | no |
| Packaged: | 2018-08-04 12:21:11 UTC; mtst |
| Author: | Mateusz Staniak [aut, cre] |
| Maintainer: | Mateusz Staniak <mateusz.staniak@math.uni.wroc.pl> |
| Repository: | CRAN |
| Date/Publication: | 2018-08-04 21:40:02 UTC |
kendallRandomWalks: explore and visualize Kendall random walks.
Description
Kendall random walks are Markov processes generated by the Kendall convolution. This package helps simulate and visualize these random walks and associated distributions. It also provides function to fit these distributions to data.
Important functions
simulate_kendall_rw simulates Kendall random walks.
transform_kendall_rw applies scaling and shift to simulated Kendall r.w-s.
ladder_moment estimates the distribution of first ladder moment
by simulating Kendall random walks.
ladder_height estimates the distribution of first ladder height
by simulating Kendall random walks.
ladder_moment_pmf computes the PMF of the distribution of first
ladder moment.
g_function Finds the value of G(t) numerically.
pkend, dkend, qkend, rkend
give CDF, PDF, quantile function and random numbers from stable Kendall distribution.
Helper function
Description
Helper function
Usage
Qn(x, y, alpha)
Arguments
x |
numeric |
y |
numeric |
alpha |
numeric, parameter of Kendall random walk |
Value
0 or 1 with probability depending on x, y, alpha
Helper function
Description
Helper function
Usage
U(x, y)
Arguments
x |
numeric |
y |
numeric |
Value
sign of the argument whose abs. val. is bigger
Helper function: min/max
Description
Helper function: min/max
Usage
Z(x, y)
Arguments
x |
numeric |
y |
numeric |
Value
min of arguments divided by max of arguments
PDF of Kendall stable distribution
Description
PDF of Kendall stable distribution
Usage
dkend(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function that returns values of the PDF
Examples
dKend <- dkend(function(x) 1)
# Step distribution: delta_{1}
dKendall <- dKend(1:10, 0.5)
# Values of PDF for arguments 1:10 and alpha = 0.5
Fit alpha parameter using MLE for distribution with one parameter
Description
Fit alpha parameter using MLE for distribution with one parameter
Usage
estimate_stable_alpha(data)
Arguments
data |
Numeric vector |
Value
list with optimal parameter and loglikelihood value
Fit stable Kendall distribution for given data and m_alpha function.
Description
Fit stable Kendall distribution for given data and m_alpha function.
Usage
fit_kendall(data)
Arguments
data |
Numeric vector of observation to which the distribution will be fitted. |
Value
fitted quantiles
Function for fitting stable Kendall distribution separately to two parts of data
Description
Function for fitting stable Kendall distribution separately to two parts of data
Usage
fit_separate(data, separation_point)
Arguments
data |
Numeric vector. Observation to which the distribution will be fitted. |
separation_point |
Order above which data (quantiles) will be separated. |
Value
List of class kendall_fit with estimated and theoretical quantiles and estimated parameters.
Fit stable Kendall distribution with one parameter (alpha)
Description
Fit stable Kendall distribution with one parameter (alpha)
Usage
fit_stable_alpha(data)
Arguments
data |
Numeric vector of data. |
Value
list of type kendall_fit
Gradient of minus loglikelihood for stable Kendall distribution with 3 parameters.
Description
Gradient of minus loglikelihood for stable Kendall distribution with 3 parameters.
Usage
full_loglik_gradient(data)
Arguments
data |
numeric vector of observation. |
Value
Function of one argument of length 3 (alpha, location, scale).
Negative loglikelihood for stable Kendall distr. with 3 parameters.
Description
Negative loglikelihood for stable Kendall distr. with 3 parameters.
Usage
full_minus_loglik(data)
Arguments
data |
Dataset for which the loglikelihood will be calculated. |
Value
numeric, value of loglikelihood
Function G(t) - Williamson transform taken at point 1/t.
Description
This function return only approximated values. To check their precisions use g_function_single function with an argument of length 1.
Usage
g_function(t, alpha, density)
Arguments
t |
Argument to the function. |
alpha |
Value of the alpha parameter. |
density |
Density function of the step distribution. |
Value
Object of class "integrate"
Examples
g_function(1:5, 0.75, dnorm)
Function G(t) - Williamson transform taken at point 1/t.
Description
This function return the whole "integrate" object, so precision of the approximation can be checked.
Usage
g_function_single(t, alpha, density)
Arguments
t |
Argument to the function. |
alpha |
Value of the alpha parameter. |
density |
Density function of the step distribution. |
Value
Object of class "integrate"
Examples
g_function_single(5, 0.26, dnorm)
Log-likelihood for stable kendall distribution with m_alpha = 1
Description
Log-likelihood for stable kendall distribution with m_alpha = 1
Usage
kendall_loglik(alpha, x)
Arguments
alpha |
alpha parameter of the Kendall random walk |
x |
numeric vector of observations |
Value
numeric
Estimate the distribution of first ladder height for given level
Description
NA is returned if the level wasn't crossed. Printing the resulting object will give summary of the estimated distribution and information whether level wasn't crossed in some simulations. This information can be used to pick the right trajectory length for the given level.
Usage
ladder_height(simulations, level)
Arguments
simulations |
kendall_simulation object |
level |
Positive numeric |
Value
tibble
Examples
{
kendall_rw <- simulate_kendall_rw(100, 100, runif, 0.5)
estim_ladder <- ladder_height(kendall_rw, 1000)
estim_ladder
}
Estimate the distribution of first ladder moment for given level
Description
NA is returned if the level wasn't crossed. Printing the resulting object will give summary of the estimated distribution and information whether level wasn't crossed in some simulations. This information can be used to pick the right trajectory length for the given level.
Usage
ladder_moment(simulations, level)
Arguments
simulations |
kendall_simulation object |
level |
Positive numeric |
Value
tibble
Examples
{
kendall_rw <- simulate_kendall_rw(100, 100, runif, 0.5)
estim_ladder <- ladder_moment(kendall_rw, 1000)
estim_ladder
}
Distribution of the first ladder moment.
Description
Distribution of the first ladder moment.
Usage
ladder_moment_pmf(n, level, alpha, step_cdf, step_pdf)
Arguments
n |
Argument to the PDF. |
level |
Level a to be crossed. |
alpha |
Alpha parameter of Kendall random walk. |
step_cdf |
CDF of the step distribution. |
step_pdf |
PDF of the step distribution. |
Value
Value of PMF of the distribution of first ladder moment
Examples
prob <- ladder_moment_pmf(10, 1000, 0.5, pnorm, dnorm)
prob
Mutate each trajectory.
Description
Mutate each trajectory.
Usage
mutate_kendall_rw(simulations, mutate_function, df = T)
Arguments
simulations |
Object of class kendall_simulation. |
mutate_function |
Function that will be applied to each trajectory. |
df |
If TRUE, a d.f will be returned, if FALSE, simulations in the kendall_simulation object passed in simulations argument will be replaced by the result of mutate_function. |
Value
data frame or a list (of class kendall_simulation)
CDF of Kendall stable distribution
Description
CDF of Kendall stable distribution
Usage
pkend(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function function giving values of CDF of Kendall stable distribution
Examples
pKend <- pkend(function(x) 1)
# Step distribution: delta_{1}
pKendall <- pKend(1:10, 0.5)
# Values of CDF for arguments 1:10 and alpha = 0.5
CDF of symmetrical Kendall stable distribution
Description
CDF of symmetrical Kendall stable distribution
Usage
pkendSym(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function function giving values of CDF of Kendall stable distribution
Examples
pKend <- pkendSym(function(x) 1)
# Step distribution: delta_{1}
pKendall <- pKend(1:10, 0.5)
# Values of CDF for arguments 1:10 and alpha = 0.5
Generic function for plotting results of ladder_moment function.
Description
Generic function for plotting results of ladder_moment function.
Usage
## S3 method for class 'kendall_barrier_crossing'
plot(x, ...)
Arguments
x |
kendall_barrier_crossing object |
... |
Additional arguments |
Value
ggplot2
QQ-plot for the result of fitting stable Kendall distribtion.
Description
QQ-plot for the result of fitting stable Kendall distribtion.
Usage
## S3 method for class 'kendall_fit'
plot(x, ...)
Arguments
x |
List returned by fit_separate or fit_kendall function. |
... |
Aditional arguments. |
Value
ggplot2 object
Generic function that draws simulated trajectories of Kendall random walk
Description
Generic function that draws simulated trajectories of Kendall random walk
Usage
## S3 method for class 'kendall_simulation'
plot(x, max_x = NULL, max_id = NULL,
level = NULL, ...)
Arguments
x |
object returned by normalising_sequences function. |
max_x |
maximum value on x axis. |
max_id |
Number of trajectories to plot. If NULL, all paths will be plotted. |
level |
Y-axis value which will be marked (level to be crossed). |
... |
Other arguments |
Value
ggplot2 object
Plot summary of Kendall random walk simulations.
Description
Plot summary of Kendall random walk simulations.
Usage
## S3 method for class 'kendall_summary'
plot(x, ..., type = "density")
Arguments
x |
Object of class kendall_summary |
... |
Optional arguments, currently ignored |
type |
Type of the plot: density, histogram or boxplot |
Value
ggplot2 object
Generic function for printing result of ladder_moment function
Description
Generic function for printing result of ladder_moment function
Usage
## S3 method for class 'kendall_barrier_crossing'
print(x, ...)
Arguments
x |
kendall_barrier_crossing object |
... |
Additional arguments |
Value
invisible x
Generic function that prints information about simulated Kendall random walk
Description
Generic function that prints information about simulated Kendall random walk
Usage
## S3 method for class 'kendall_simulation'
print(x, ...)
Arguments
x |
Object returned by simulate_kendall_rw or transform_kendall_rw function. |
... |
Other arguments. |
Print summary of Kendall random walk simulations.
Description
Print summary of Kendall random walk simulations.
Usage
## S3 method for class 'kendall_summary'
print(x, ...)
Arguments
x |
Object of type kendall_summary |
... |
Optional parameters, currently ignored |
Quantiles of Kendall stable distribution
Description
Quantiles of Kendall stable distribution
Usage
qkend(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function function returning quantiles of given orders
Examples
qKend <- qkend(function(x) 1)
# Step distribution: delta_{1}
qKendall <- qKend(c(0.1, 0.9), 0.5)
# Quantiles of order 0.1 and 0.9 for alpha = 0.5
Quantiles of symmetrical Kendall stable distribution
Description
Quantiles of symmetrical Kendall stable distribution
Usage
qkendSym(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function function returning quantiles of given orders
Examples
qKend <- qkendSym(function(x) 1)
# Step distribution: delta_{1}
qKendall <- qKend(c(0.1, 0.9), 0.5)
# Quantiles of order 0.1 and 0.9 for alpha = 0.5
Pseudo-random number from Kendall stable distribution
Description
Pseudo-random number from Kendall stable distribution
Usage
rkend(m_alpha)
Arguments
m_alpha |
function giving moments of order alpha of step dist. |
Value
function return n numbers genereted from Kendall stable dist.
Examples
rKend <- rkend(function(x) 1)
# Step distribution: delta_{1}
rKendall <- rKend(10, 0.5)
# Ten random number from stable Kendall distribution with alpha = 0.5
Simulate one trajectory ofa Kendall random walk
Description
Simulate one trajectory ofa Kendall random walk
Usage
simulateOneTrajectory(trajectory_length, step_dist, alpha,
symmetric = FALSE, ...)
Arguments
trajectory_length |
Number of samples to simulate. |
step_dist |
Function that returns random numbers from step distribution. |
alpha |
Alpha parameter of the random walk |
symmetric |
If TRUE, random walk on the whole real line will be simulated. |
... |
Additional parameters to step distribution. |
Value
Generated path of the random walk.
Simulate multiple trajectories of Kendall random walk
Description
Object returned by this has print and plot methods.
Usage
simulate_kendall_rw(number_of_simulations, trajectory_length, step_dist,
alpha, symmetric = FALSE, ...)
Arguments
number_of_simulations |
number of trajectories to generate. |
trajectory_length |
length of trajectories. |
step_dist |
function returning random numbers from step dist. |
alpha |
alpha parameter. |
symmetric |
If TRUE, random walk on the whole real line will be simulated. |
... |
parameters for step distribution. |
Value
Object of class kendall_simulation. It is a list that consists of
simulation |
Tibble with simulation id and simulated values, |
step_distribution |
Name of the step distribution, |
alpha |
Value of alpha parameter, |
is_symmetric |
Logical value indicating if this is a symmetric Kendall R.W. |
Examples
kendall_simulations <- simulate_kendall_rw(10, 1000, runif, 0.5)
# Kendall R.W. on positive half-line with uniform step distribution - 10 trajectories.
only_simulations <- kendall_simulations$simulation # tibble with simulated values
kendall_simulations
Calculate some characteristic for every simulated instance.
Description
Calculate some characteristic for every simulated instance.
Usage
summarise_kendall_rw(simulations, summary_function)
Arguments
simulations |
Object of class kendall_simulation. |
summary_function |
Function that will be applied to each trajectory. |
Value
data frame of class "kendall_summary".
Transforming (scaling and shifting) Kendall random walks
Description
If one trajectory has length n, an_seq and bn_seq arguments should be sequnces of length n. Object returned by this function has plot and print methods.
Usage
transform_kendall_rw(simulations, an_seq = 1, bn_seq = 0)
Arguments
simulations |
tibble returned by simulation function |
an_seq |
sequence that the trajectories will be multiplied by |
bn_seq |
sequence that will be substracted from scaled trajectory |
Value
List like in simulate_kendall_rw function after transforming trajectories.
Examples
kendall_simulations <- simulate_kendall_rw(10, 1000, runif, 0.5)
scaled_kendall <- transform_kendall_rw(kendall_simulations, (1:1000)^(-2))
scaled_kendall # kendall random walked scaled by the sequence n^(-1/alpha)
scaled_data <- scaled_kendall$simulation # simulated values
plot(scaled_kendall)