Description Usage Arguments Details Value Note Author(s) References See Also Examples
Simulate the increasing random walk associated with a real-valued continuous random variable.
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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 |
K |
kernel transition for conditional generations. |
burnin |
burnin parameter. |
sigma |
radius parameter for |
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 |
plot.lsf |
a boolean indicating if the |
print_plot |
if TRUE, print the updated plot after each iteration. This might
be slow; use with a small |
output_dir |
if plots are to be saved in pdf in a given directory. This will
be pasted with ‘_IRW.pdf’. Together with |
plot.lab |
the x and y labels for the plot |
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 μ^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
.
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 |
X |
a matrix containing the final particles. |
y |
the value of |
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. |
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.
Clement WALTER clementwalter@icloud.com
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 \& 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 \& Optimization, 64(2), 171-196.
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