R: Increasing Randow Walk

IRW {mistral}

R Documentation

Increasing Randow Walk

Description

Simulate the increasing random walk associated with a real-valued continuous random variable.

Usage

IRW(
  dimension,
  lsf,
  N = 10,
  q = Inf,
  Nevent = Inf,
  X,
  y = lsf(X),
  K,
  burnin = 20,
  sigma = 0.3,
  last.return = TRUE,
  use.potential = TRUE,
  plot = FALSE,
  plot.lsf = FALSE,
  print_plot = FALSE,
  output_dir = NULL,
  plot.lab = c("x_1", "x_2")
)

Arguments

`dimension`	dimension of the input space.
`lsf`	limit state function.
`N`	number of particules.
`q`	level until which the randow walk is to be generated.
`Nevent`	the number of desired events.
`X`	to start with some given particles.
`y`	value of the `lsf` on X.
`K`	kernel transition for conditional generations.
`burnin`	burnin parameter.
`sigma`	radius parameter for `K`.
`last.return`	if the last event should be returned.
`use.potential`	tu use a ‘potential’ matrix to select starting point not directly related to the sample to be moved with the MH algorithm.
`plot`	if `TRUE`, the algorithm plots the evolution of the particles. This requieres to evaluate the `lsf` on a grid and is only for visual purpose.
`plot.lsf`	a boolean indicating if the `lsf` should be added to the plot. This requires the evaluation of the `lsf` over a grid and consequently should be used only for illustation purposes.
`print_plot`	if TRUE, print the updated plot after each iteration. This might be slow; use with a small `N`. Otherwise it only prints the final plot.
`output_dir`	if plots are to be saved in pdf in a given directory. This will be pasted with ‘_IRW.pdf’. Together with `print_plot==TRUE` this will produce a pdf with a plot at each iteration, enabling ‘video’ reconstitution of the algorithm.
`plot.lab`	the x and y labels for the plot

Details

This function lets generate the increasing random walk associated with a continous real-valued random variable of the form Y = lsf(X) where X is vectorial random variable.

This random walk can be associated with a Poisson process with parameter N and hence the number of iterations before a given threshold q is directly related to P[ lsf(X) > q]. It is the core tool of algorithms such as nested sampling, Last Particle Algorithm or Tootsie Pop Algorithm.

Bascially for N = 1, it generates a sample Y = lsf(X) and iteratively regenerates greater than the found value: Y_{n+1} \sim \mu^Y( \cdot \mid Y > Y_n. This regeneration step is done with a Metropolis-Hastings algorithm and that is why it is usefull to consider generating several chains all together (N > 1).

The algorithm stops when it has simulated the required number of events Nevent or when it has reached the sought threshold q.

Value

An object of class list containing the following data:

`L`	the events of the random walk.
`M`	the total number of iterations.
`Ncall`	the total number of calls to the `lsf`.
`X`	a matrix containing the final particles.
`y`	the value of `lsf` on `X`.
`q`	the threshold considered when generating the random walk.
`Nevent`	the target number of events when generating the random walk.
`Nwmoves`	the number of rejected transitions, ie when the proposed point was not stricly greater/lower than the current state.
`acceptance`	a vector containing the acceptance rate for each use of the MH algorithm.

Note

Problem is supposed to be defined in the standard space. If not, use UtoX to do so. Furthermore, each time a set of vector is defined as a matrix, ‘nrow’ = dimension and ‘ncol’ = number of vector to be consistent with as.matrix transformation of a vector.

Algorithm calls lsf(X) (where X is a matrix as defined previously) and expects a vector in return. This allows the user to optimise the computation of a batch of points, either by vectorial computation, or by the use of external codes (optimised C or C++ codes for example) and/or parallel computation; see examples in MonteCarlo.

Author(s)

Clement WALTER clementwalter@icloud.com

References

C. Walter:
Moving Particles: a parallel optimal Multilevel Splitting method with application in quantiles estimation and meta-model based algorithms
Structural Safety, 55, 10-25.
C. Walter:
Point Process-based Monte Carlo estimation
Statistics and Computing, in press, 1-18.
arXiv preprint arXiv:1412.6368.
J. Skilling:
Nested sampling for general Bayesian computation
Bayesian Analysis, 1(4), 833-859.
M. Huber and S. Schott:
Using TPA for Bayesian inference
Bayesian Statistics 9, 9, 257.
A. Guyader, N. Hengartner and E. Matzner-Lober:
Simulation and estimation of extreme quantiles and extreme probabilities
Applied Mathematics and Optimization, 64(2), 171-196.

Examples

# Get faililng samples for the kiureghian limit state function
# Failure is defined as lsf(X) < 0 so we have to invert the lsf
lsf <- function(x) -1*kiureghian(x)
## Not run: 
fail.samp <- IRW(2, lsf, q = 0, N = 10, plot = TRUE)

## End(Not run)

[Package mistral version 2.2.2 Index]