6,436 research outputs found
Multilevel nested simulation for efficient risk estimation
We investigate the problem of computing a nested expectation of the form where is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by [M. Broadie, Y. Du, and C. C. Moallemi, Manag. Sci., 57 (2011), pp. 1172--1194]. We propose and analyze an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an complexity to achieve a root mean-squared error . The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), the latter being achieved with complexity.
Read More: https://epubs.siam.org/doi/10.1137/18M117318
Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI
In this paper we develop a very efficient approach to the Monte Carlo
estimation of the expected value of partial perfect information (EVPPI) that
measures the average benefit of knowing the value of a subset of uncertain
parameters involved in a decision model. The calculation of EVPPI is inherently
a nested expectation problem, with an outer expectation with respect to one
random variable and an inner conditional expectation with respect to the
other random variable . We tackle this problem by using a Multilevel Monte
Carlo (MLMC) method (Giles 2008) in which the number of inner samples for
increases geometrically with level, so that the accuracy of estimating the
inner conditional expectation improves and the cost also increases with level.
We construct an antithetic MLMC estimator and provide sufficient assumptions on
a decision model under which the antithetic property of the estimator is well
exploited, and consequently a root-mean-square accuracy of can be
achieved at a cost of . Numerical results confirm the
considerable computational savings compared to the standard, nested Monte Carlo
method for some simple testcases and a more realistic medical application
Multilevel Richardson-Romberg extrapolation
We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which
combines the higher order bias cancellation of the Multistep Richardson-Romberg
method introduced in [Pa07] and the variance control resulting from the
stratification introduced in the Multilevel Monte Carlo (MLMC) method (see
[Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of
diffusion processes, the root mean squared error (RMSE) can
be achieved with our MLRR estimator with a global complexity of
instead of with the standard MLMC method, at least when the weak
error of the biased implemented estimator
can be expanded at any order in and . The MLRR estimator is then halfway between a regular MLMC
and a virtual unbiased Monte Carlo. When the strong error , , the gain of MLRR over MLMC becomes even
more striking. We carry out numerical simulations to compare these estimators
in two settings: vanilla and path-dependent option pricing by Monte Carlo
simulation and the less classical Nested Monte Carlo simulation.Comment: 38 page
Multilevel Weighted Support Vector Machine for Classification on Healthcare Data with Missing Values
This work is motivated by the needs of predictive analytics on healthcare
data as represented by Electronic Medical Records. Such data is invariably
problematic: noisy, with missing entries, with imbalance in classes of
interests, leading to serious bias in predictive modeling. Since standard data
mining methods often produce poor performance measures, we argue for
development of specialized techniques of data-preprocessing and classification.
In this paper, we propose a new method to simultaneously classify large
datasets and reduce the effects of missing values. It is based on a multilevel
framework of the cost-sensitive SVM and the expected maximization imputation
method for missing values, which relies on iterated regression analyses. We
compare classification results of multilevel SVM-based algorithms on public
benchmark datasets with imbalanced classes and missing values as well as real
data in health applications, and show that our multilevel SVM-based method
produces fast, and more accurate and robust classification results.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0625
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