Control Variate Technique Research Articles

CureAuxSP: An R package for estimating mixture cure models with auxiliary survival probabilities

Background and Objective:There is a rising interest in exploiting aggregate information from external medical studies to enhance the statistical analysis of a modestly sized internal dataset. Currently available software packages for analyzing survival data with a cure fraction ignore the potentially available auxiliary information. This paper aims at filling this gap by developing a new R package CureAuxSP that can include subgroup survival probabilities extracted outside into an interested internal survival dataset. Methods:The newly developed R package CureAuxSP provides an efficient approach for information synthesis under the mixture cure models, including Cox proportional hazards mixture cure model and the accelerated failure time mixture cure model as special cases. It focuses on synthesizing subgroup survival probabilities at multiple time points and the underlying method development lies in the control variate technique. Evaluation of homogeneity assumption based on a test statistic can be automatically carried out by our package and if heterogeneity does exist, the original outputs can be further refined adaptively. Results:The R package CureAuxSP provides a main function SMC.AxuSP() that helps us adaptively incorporate external subgroup survival probabilities into the analysis of an internal survival data. We also provide another function Print.SMC.AuxSP() for printing the results with a better presentation. Detailed usages are described, and implementations are illustrated with numerical examples, including a simulated dataset with a well-designed data generating process and a real breast cancer dataset. Substantial efficiency gain can be observed by our results. Conclusions:Our R package CureAuxSP can make the wide applications of utilizing auxiliary information possible. It is anticipated that the performance of mixture cure models can be improved for the survival data with a cure fraction, especially for those with small sample sizes.

Mathematical and Physical Characteristics of the Phase Spectrum of Earthquake Ground Motions

This study presents a rigorous investigation into the mathematical and physical properties inherent in the Fourier phase spectrum of earthquake ground motions. This exploration includes a detailed examination of the probability distribution of phase angles and differences, elucidated through two novel numerical experiments utilizing the reduction ad absurdum approach. Moreover, the study scrutinizes the physical attributes of earthquake ground motion’s phase spectrum, employing the circular frequency-dependent phase derivative as a key analytical factor. In a novel approach, the research delves into the relationship between circular frequency-dependent phase derivatives and Fourier amplitudes, shedding light on essential connections within earthquake phenomena, particularly addressing non-stationarity. Expanding the scope, the study comprehensively examines the influence of source, propagation path, and site on both the phase spectrum and accelerogram. Employing the control variate technique facilitates this analysis, providing valuable insights into the underlying physical mechanisms governing earthquake wave behavior. The findings highlight the temporal properties of the phase spectrum, attributing its complexity to the temporal heterogeneity in energy release during the fault rupture and dispersion of earthquake waves. This novel approach not only enhances the understanding of earthquake dynamics, but also underscores the significance of considering temporal variations in earthquake events.

Efficient auxiliary information synthesis for cure rate model

Abstract We propose a new auxiliary information synthesis method to utilize subgroup survival information at multiple time points under the semi-parametric mixture cure rate model. After summarizing the auxiliary information via estimating equations, a control variate technique is adopted to reduce the variance efficiently, together with a test statistic to check the homogeneity assumption. Revision using penalization is further considered to adaptively accommodate potential population heterogeneity. Our methods can be adjusted when the uncertainty is not negligible. We establish asymptotic properties of our proposed estimators, and demonstrate their practical performances through extensive simulations and an invasive breast cancer study.

ANALISIS VALUE AT RISK PORTOFOLIO SAHAM LQ45 DENGAN METODE SIMULASI MONTE CARLO CONTROL VARIATES

Value at Risk (VaR) with the Monte Carlo (MC) simulation is an estimate of the maximum loss over a given period of time and with a specific degree of confidence. MC VaR uses the Control Variates (CV) technique which is one of the reduction techniques in the MC method to improve the efficiency of VaR estimation. This study also aims to analyze the risk of the LQ45 indexed stock portofolio with Monte Carlo Control Variates (MCCV) VaR. In addition, this study compares MCCV VaR with standar MC VaR. The closing prices of the shares of PT Bank Negara Indonesia Tbk (BBNI) and PT Bank Central Asia Tbk (BBCA) were the source of the data for this study. The 95% confidence level is used for this study to estimate one-day MCCV VaR. The results obtained show that MCCV is able to reduce the variance of the estimate more quickly than the standar MC VaR. Thus, MCCV VaR is more efficient than the standard MC VaR.

Stein’s Method Meets Computational Statistics: A Review of Some Recent Developments

Stein’s method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical statistics, Stein’s method has led to significant advances in computational statistics in recent years. The goal of this survey is to bring together some of these recent developments, and in doing so, to stimulate further research into the successful field of Stein’s method and statistics. The topics we discuss include tools to benchmark and compare sampling methods such as approximate Markov chain Monte Carlo, deterministic alternatives to sampling methods, control variate techniques, parameter estimation and goodness-of-fit testing.

Commodity Asian option pricing and simulation in a 4-factor model with jump clusters

Mean reversion, stochastic volatility, convenience yield and presence of jump clustering are well documented salient features of commodity markets, where Asian options are very popular. We propose a model which takes into account all these stylized features. We first state our model under the historical measure, then, after introducing a structure preserving change of measure, we provide a risk-neutral version of the same model and we show how to price geometric and arithmetic Asian options. To this end, we derive semi-closed formulas for the geometric Asian options price and develop a computationally efficient simulation scheme for the price process, allowing to price the arithmetic counterparts using control variate technique. Finally, we propose a simple econometric experiment to document presence of jump clusters in commodity prices and evaluate the performances of the proposed simulation scheme on some parameter sets calibrated on real data.

An adaptive artificial neural network for reliability analyses of complex engineering systems

Reliability analyses of engineering systems require methods capable of reducing the number of function calls and response variances to calculate the failure probability. This paper is aimed to employ a combination of the Monte Carlo simulation (MCS), artificial neural network (ANN) and control variate technique (CVT) to improve the efficiency and accuracy of the reliability analyses of various engineering systems. The ANN used in this paper is of the multilayer perceptron type. The paper uses the design of experiments (DoE), importance sampling and uniform sampling methods to generate the samples. The combination of the MCS, ANN model, DoEs, and CVT is used for the reliability analyses and significantly improved the results in terms of efficiency. Various numerical examples have shown the superiority of the improved ANN model using CVT in reducing the number of calls of the limit state function and revealed that the obtained results are not sensitive to the number of hidden layer neurons. The reduced number of required simulations and computation costs has enabled the proposed method to be applicable to reliability analyses in complex engineering systems such as structures made by functionally graded materials.

Option pricing with the control variate technique beyond Monte Carlo simulation

Although mostly used alongside Monte Carlo simulation, the control-variate (CV) technique can be applied to other numerical algorithms in option pricing. This paper studies the conditions under which a numerical method (simulation-based or not) can benefit from the CV technique and what approximators can serve as CVs. We demonstrate the ideas with Carr and Madan’s Fourier transform-based algorithm, convolution-based pricing algorithms, and classic binomial trees. Numerical results are provided to show that the CV-enhanced versions are more efficient than the original algorithms.

Unsupervised neural network-enabled spatial-temporal analytics for data authenticity under environmental smart reporting system

Environment, social and governance (ESG) disclosures required on listed companies have aroused considerable interest in both academia and industry for sustainable development and investing. However, the authenticity and credibility of the ESG report exposed to the public remain in doubt due to black box-like reporting processes with massive human involvements. In this study, a framework of environmental smart reporting system (BI-ESRS) based on the blockchain and Internet of Things (IoT) technologies is developed to automate the acquisition of environment-related data and make the reporting reliable and traceable. In addition, we evaluate the authenticity of the data collected from IoT devices, considering human-made counterfeits on measuring instruments for greenwashing. It is anticipated to stimulate companies to submit high-quality data without fake. Specifically, an unsupervised neural network-enabled spatial-temporal analytics (UN-STA) method is devised to achieve anomalies detection and index the data with an authenticity rate. An artificial neural network, self-organizing map (SOM), is applied to construct the prediction model. The received signal strength indicator (RSSI) of Bluetooth low energy (BLE), the vibration amplitude of smart instruments and data uploading interval constitute the input vector for competitive learning. Finally, an experimental simulation is carried out to demonstrate the implementation of the proposed system and method, and their effectiveness has also been testified. Moreover, the sensitivity of the SOM model over the three factors has been analyzed by applying the control variate technique. This work is expected to serve as a reference for practitioners to satisfy similar requirements in the industry and inspire new ideas for scholars.

A comparison of variance reduction techniques in the simulation of a retrial M/M/1/K queue

Retrial queueing systems play a significant role in the modeling and performance evaluation of various dynamic stochastic systems, such as computer networks, airlines, communication and financial systems. This article presents a comprehensive assessment on the applicability of three popular variance reduction techniques (VRTs) in the performance evaluation of an M/M/1/K retrial queue based on shortest job first served (SJFS) retrial policy. The M/M/1/K retrial queue with SJFS retrial policy is not currently analytically tractable. Therefore, the credibility of such queueing systems is heavily dependent on simulations and accurate output data analysis. We seek to identify the best possible VRT so that a system performance evaluator can focus on optimizing the accuracy of relevant performance metrics under study. A comparative applicability analysis of three VRTs, i.e., antithetic variates, importance sampling, and control variates, on the retrial M/M/1/K queueing model has been presented. Our analysis showed a significant level of reduction in terms of variance and confidence interval size of the final estimates for the control variates technique. Furthermore, it has been found that unlike the other VRTs, the control variates technique shows a consistent trend in variance reductions as the number of retrials increases.

A Monte-Carlo approach for pricing arithmetic Asian rainbow options under the mixed fractional Brownian motion

We derive a closed-form solution for pricing geometric Asian rainbow options under the mixed geometric fractional Brownian motion (FBM). In particular, the number of underlying assets is allowed to be arbitrary, and fully correlated fractional Brownian motions are taken into account. The analytical solution obtained is used as a control variate for Monte Carlo based computations of the price of arithmetic Asian rainbow options. Numerical experiments are presented in which options on two, three, four and ten underlying assets are considered. Results reveal that the proposed control variate technique is very effective to reduce the variance of the Monte Carlo estimator and yields a reliable approximation of the Asian rainbow option price.

Semi-exact control functionals from Sard’s method

Summary A novel control variate technique is proposed for the post-processing of Markov chain Monte Carlo output, based on both Stein’s method and an approach to numerical integration due to Sard. The resulting estimators of posterior expected quantities of interest are proven to be polynomially exact in the Gaussian context, while empirical results suggest that the estimators approximate a Gaussian cubature method near the Bernstein–von Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results across a selection of Bayesian inference tasks are presented.

Brownian Path Generation and Polynomial Chaos

We introduce a new Brownian path generation method using a second degree polynomial chaos expansion, and use a conjugate gradient algorithm to find the orthogonal transformation that minimizes the mean truncation dimension of this polynomial. We combine the new path generation method with a control variate technique based on the same second degree polynomial chaos expansion, and apply the method to estimation of equity option prices under the Black--Scholes framework and interest rate option prices under the Libor Market Model. We use numerical results to compare the new method with some other path generation methods from the literature.

Quasi-Stationary Monte Carlo and The Scale Algorithm

SummaryThis paper introduces a class of Monte Carlo algorithms which are based on the simulation of a Markov process whose quasi-stationary distribution coincides with a distribution of interest. This differs fundamentally from, say, current Markov chain Monte Carlo methods which simulate a Markov chain whose stationary distribution is the target. We show how to approximate distributions of interest by carefully combining sequential Monte Carlo methods with methodology for the exact simulation of diffusions. The methodology introduced here is particularly promising in that it is applicable to the same class of problems as gradient-based Markov chain Monte Carlo algorithms but entirely circumvents the need to conduct Metropolis–Hastings type accept–reject steps while retaining exactness: the paper gives theoretical guarantees ensuring that the algorithm has the correct limiting target distribution. Furthermore, this methodology is highly amenable to ‘big data’ problems. By employing a modification to existing naive subsampling and control variate techniques it is possible to obtain an algorithm which is still exact but has sublinear iterative cost as a function of data size.

SESC: A new subset simulation method for rare-events estimation

A sequential space conversion (SESC) method is proposed to solve complex and high dimensional rare event problems. While the conventional Subset Simulation (SubSim) formulation is based on the Bayes theorem, that of the SESC is derived from the control variate technique. This approach first estimates a fast imprecise failure probability and then improves the estimation using refining terms. It designs a set of scaled limit state functions similar to the original one but with higher failure probabilities, then uses the set as the control variates, and, finally, conducts the Markov chain Monte Carlo samples toward the important failure region. Hence, unlike the conventional SubSim, the SESC performance does not depend on the geometry of the performance function away from the limit state surface. The reliability analysis of complex and high dimensional problems that involve several counterexamples of subset simulations shows that the proposed method is capable of solving problems with complex/misleading performance functions that cannot be solved with conventional SubSim or other existing approaches.

A refined subset simulation for the reliability analysis using the subset control variate

The reliability analysis of an engineering system with a small failure probability (Pf) and a complex-geometry performance function is a major challenge in the probabilistic engineering mechanics for which the subset simulation (SS) is a promising addressing algorithm. However, SS is potentially problematic when solving complex problems. This has triggered an interest in improving and reformulating the SS. This paper presents the subset control variate (SCV) technique, a novel approach to reformulate the conventional SS, and provides the statistical properties such as coefficient of variation (c.o.v) of the estimate of Pf using SCV. SCV enhances and generalizes the original SS. It improves the conventional SS formulation and can employ different sampling approaches in the SS to properly find the most probable failure domain with complex and misleading geometry. The proposed method’s capabilities are compared with those of the conventional SS and examined by solving several numerical and practical problems with challenging performance functions. The results, validated by the Monte Carlo Simulation (MCS), show that the SCV improved the SS robustness for solving highly nonlinear problems involving misleading performance functions and its c.o.v is less than that of the original SS for all the numerical examples considered in this study.

Efficient Simulation of Ruin Probabilities When Claims are Mixtures of Heavy and Light Tails

We consider the classical Cramér-Lundberg risk model with claim sizes that are mixtures of phase-type and subexponential variables. Exploiting a specific geometric compound representation, we propose control variate techniques to efficiently simulate the ruin probability in this situation. The resulting estimators perform well for both small and large initial capital. We quantify the variance reduction as well as the efficiency gain of our method over another fast standard technique based on the classical Pollaczek-Khinchine formula. We provide a numerical example to illustrate the performance, and show that for more time-consuming conditional Monte Carlo techniques, the new series representation also does not compare unfavorably to the one based on the Pollaczek-Khinchine formula.

Control variates for efficient long-term extreme analysis of mooring lines

Predicting the long-term extreme response is a major challenge for compliant offshore structures such as mooring lines. The structure is exposed to long-term uncertainties from varying environmental conditions and short-term uncertainties from random waves. The “all sea states approach” is rigorous but computationally prohibitive, since each sea state necessitates time domain simulations to capture the nonlinearities and non-Gaussian characteristics of the dynamic system. This paper presents a new approach for accurate long-term extreme analysis of mooring lines. The short-term conditional failure probability is estimated based on extrapolation of crossing rates. Monte Carlo simulation (MCS) is used to perform the integration over the long-term variables, which include the significant wave height, average period, and the speed/direction of wind and current. The classical control variates technique is invoked to reduce the variance of the MCS estimator. The difficulty of obtaining a good control function for complex problems is resolved through a new auto-control variates scheme, wherein the control is formulated as a surrogate model constructed from existing MCS scatter. Since variance reduction is effected by post-processing, multiple stress locations can be evaluated from the same dynamic simulations. The proposed method is implemented on a turret-moored vessel, and the predicted failure probability is compared against subset simulation, in which all the short- and long-term uncertainties are incorporated as random variables and no arbitrary assumptions are made. The proposed method is found to be in excellent agreement with subset simulation, and the variance reduction is considerable, resulting in a speedup of roughly an order of magnitude or more.

Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of one-dimensional Brownian diffusion with applications in finance

The exponential timestepping Euler algorithm with a boundary test is adapted to simulate an expected of a function of exit time, such as the expected payoff of barrier options under the constant elasticity of variance (CEV) model. However, this method suffers from a high Monte Carlo (MC) statistical error due to its exponentially large exit times with unbounded samples. To reduce this kind of error efficiently and to speed up the MC simulation, we combine such an algorithm with an effective variance reduction technique called the control variate method. We call the resulting algorithm the improved Exp algorithm for abbreviation. In regard to the examples we consider in this paper for the restricted CEV process, we found that the variance of the improved Exp algorithm is about six times smaller than that of the Jansons and Lythe original method for the down-and-out call barrier option. It is also about eight times smaller for the up-and-out put barrier option, indicating that the gain in efficiency is significant without significant increase in simulation time.

The CV Makes the Difference – Control Variates for Neural Networks

We consider the application of a control variate technique for Deep Learning. In analogy to applications for Monte Carlo simulation or Fourier integration methods, this technique improves the quality of deep learning applied to option pricing problems. Many well known approximation methods are limited for practical applications but can be used as a control variate. For instance approximation formulas for SABR or the Black-Scholes price when pricing options in the Heston model. The neural network is only applied to calculate the difference to an accurate numerical method. In this way we increase the accuracy of applying neural nets since a large portion of the price is already mimicked by the control variate. This may result in a higher acceptance of such numerical techniques for financial applications.