Worst-case Distributions Research Articles

EP-1898: Smooth animations of the probabilistic analog to worst-case dose distributions

EP-1898: Smooth animations of the probabilistic analog to worst-case dose distributions

Worst-case temporal aggregate power ramp distributions for a PAC wind turbine

As a critical component for evaluation of output power variability at wind farms, short duration power ramp distributions are typically evaluated as ensemble estimates across elaborate time-aggregated compilation of on-site field measurements. This paper proposes an algorithmic alternative to the above, by introducing statistical estimates both for probability density function (pdf) and cumulative distribution function (cdf) of power ramps. The estimates are pessimistic as possible real time filtering, attributable to turbine inertia and time-constants, is assumed to be negligible. The proposed algorithm is conveniently implemented on popular spreadsheet software, has limited dependence on source wind statistics, and easily accommodates turbulence conditions given by the IEC 61400-1 standards. Its application is illustrated for the popular Vestas V-90 3 MW turbine, assumed to operate at a site with source wind accordingly specified.

Data-Driven DG Capacity Assessment Method for Active Distribution Networks

This paper proposes a data-driven method based on distributionally robust optimization to determine the maximum penetration level of distributed generation (DG) for active distribution networks. In our method, the uncertain DG outputs and load demands are formulated as stochastic variables following some ambiguous distributions. In addition to the given expectations and variances, the polyhedral uncertainty intervals are employed for the construction of the probability distribution set to restrict possible distributions. Then, we decide the optimal sizes and locations of DG to maximize the total DG hosting capacity under the worst-case probability distributions among this set. Since more information is utilized, our proposed model is expected to be less conservative than the robust optimization method and the traditional distributionally robust method. Using the CVaR (Conditional Value at Risk) reformulation technique and strong duality, we transform the proposed model into an equivalent bilinear matrix inequality problem, and a sequential convex optimization algorithm is applied for solution. Our model guarantees that the probability of security constraints being violated will not exceed a given risk threshold. Besides, the predefined risk level can be tuned to control the conservativeness of our model in a physically meaningful way. The effectiveness and robustness of this proposed method are demonstrated numerically on the two modified IEEE test systems.

Practical Applications of Identifying Economic Regimes: Reducing Downside Risks for University Endowments and Foundations

<b>Overview</b> Institutional investors, particularly nonprofits who rely on their endowments, must protect their capital from worst-case scenarios of major financial crashes, situations that frequently involve contagion—increases in correlation and volatility. In <b><i>Identifying Economic Regimes: Reducing Downside Risks for University Endowments and Foundations</i></b>, authors <b>John Mulvey</b> and <b>Han Liu</b> of <b>Princeton University</b> develop a machine-learning algorithm to identify financial regimes and help investors more accurately model downside risk. They find that a multiregime simulation provides more accurate estimates, and they also demonstrate the advantages of adjustable-spending rules during drawdown periods “In the two-regime approach, you get fatter tails, and the worst-case distributions tend to be more historically accurate than using data from a single regime,” explains Mulvey. <b>TOPICS:</b>Exchanges/markets/clearinghouses, risk management

Worst-case demand distributions in vehicle routing

A recent focal point in research on the vehicle routing problem (VRP) is the issue of robustness in which customer demand is uncertain. In this paper, we conduct a theoretical analysis of the demand distributions whose induced workloads are as undesirable as possible. We study two common variations of VRP in a continuous approximation setting: the first is the VRP with time windows, and the second is the capacitated VRP, in which regular returns to the vehicle’s point of origin are required.

SU-E-T-551: PTV Is the Worst-Case of CTV in Photon Therapy

Purpose: To examine the supposition of the static dose cloud and adequacy of the planning target volume (PTV) dose distribution as the worst-case representation of clinical target volume (CTV) dose distribution for photon therapy in head and neck (H&N) plans. Methods: Five diverse H&N plans clinically delivered at our institution were selected. Isocenter for each plan was shifted positively and negatively in the three cardinal directions by a displacement equal to the PTV expansion on the CTV (3 mm) for a total of six shifted plans per original plan. The perturbed plan dose was recalculated in Eclipse (AAA v11.0.30) using the same, fixed fluence map as the original plan. The dose distributions for all plans were exported from the treatment planning system to determine the worst-case CTV dose distributions for each nominal plan. Two worst-case distributions, cold and hot, were defined by selecting the minimum or maximum dose per voxel from all the perturbed plans. The resulting dose volume histograms (DVH) were examined to evaluate the worst-case CTV and nominal PTV dose distributions. Results: Inspection demonstrates that the CTV DVH in the nominal dose distribution is indeed bounded by the CTV DVHs in the worst-case dose distributions. Furthermore, comparison of the D95% for the worst-case (cold) CTV and nominal PTV distributions by Pearson's chi-square test shows excellent agreement for all plans. Conclusion: The assumption that the nominal dose distribution for PTV represents the worst-case dose distribution for CTV appears valid for the five plans under examination. Although the worst-case dose distributions are unphysical since the dose per voxel is chosen independently, the cold worst-case distribution serves as a lower bound for the worst-case possible CTV coverage. Minor discrepancies between the nominal PTV dose distribution and worst-case CTV dose distribution are expected since the dose cloud is not strictly static. This research was supported by the NCI through grant K25CA168984, by The Lawrence W. and Marilyn W. Matteson Fund for Cancer Research, and by the Fraternal Order of Eagles Cancer Research Fund, the Career Development Award Program at Mayo Clinic.

PTV-based IMPT optimization incorporating planning risk volumes vs robust optimization

Robust optimization leads to intensity-modulated proton therapy (IMPT) plans that are less sensitive to uncertainties and superior in terms of organs-at-risk (OARs) sparing, target dose coverage, and homogeneity compared to planning target volume (PTV)-based optimized plans. Robust optimization incorporates setup and range uncertainties, which implicitly adds margins to both targets and OARs and is also able to compensate for perturbations in dose distributions within targets and OARs caused by uncertainties. In contrast, the traditional PTV-based optimization considers only setup uncertainties and adds a margin only to targets but no margins to the OARs. It also ignores range uncertainty. The purpose of this work is to determine if robustly optimized plans are superior to PTV-based plans simply because the latter do not assign margins to OARs during optimization. The authors retrospectively selected from their institutional database five patients with head and neck (H&N) cancer and one with prostate cancer for this analysis. Using their original images and prescriptions, the authors created new IMPT plans using three methods: PTV-based optimization, optimization based on the PTV and planning risk volumes (PRVs) (i.e., "PTV+PRV-based optimization"), and robust optimization using the "worst-case" dose distribution. The PRVs were generated by uniformly expanding OARs by 3 mm for the H&N cases and 5 mm for the prostate case. The dose-volume histograms (DVHs) from the worst-case dose distributions were used to assess and compare plan quality. Families of DVHs for each uncertainty for all structures of interest were plotted along with the nominal DVHs. The width of the "bands" of DVHs was used to quantify the plan sensitivity to uncertainty. Compared with conventional PTV-based and PTV+PRV-based planning, robust optimization led to a smaller bandwidth for the targets in the face of uncertainties {clinical target volume [CTV] bandwidth: 0.59 [robust], 3.53 [PTV+PRV], and 3.53 [PTV] Gy (RBE)}. It also resulted in higher doses to 95% of the CTV {D(95%): 60.8 [robust] vs 59.3 [PTV+PRV] vs 59.6 [PTV] Gy (RBE)}, smaller D(5%) (doses to 5% of the CTV) minus D(95%) {D(5%) - D(95%): 13.2 [robust] vs 17.5 [PTV+PRV] vs 17.2 [PTV] Gy (RBE)}. At the same time, the robust optimization method irradiated OARs less {maximum dose to 1 cm(3) of the brainstem: 48.3 [robust] vs 48.8 [PTV+PRV] vs 51.2 [PTV] Gy (RBE); mean dose to the oral cavity: 22.3 [robust] vs 22.9 [PTV+PRV] vs 26.1 [PTV] Gy (RBE); maximum dose to 1% of the normal brain: 66.0 [robust] vs 68.0 [PTV+PRV] vs 69.3 [PTV] Gy (RBE)}. For H&N cases studied, OAR sparing in PTV+PRV-based optimization was inferior compared to robust optimization but was superior compared to PTV-based optimization; however, target dose robustness and homogeneity were comparable in the PTV+PRV-based and PTV-based optimizations. The same pattern held for the prostate case. The authors' data suggest that the superiority of robust optimization is not due simply to its inclusion of margins for OARs, but that this is due mainly to the ability of robust optimization to compensate for perturbations in dose distributions within target volumes and normal tissues caused by uncertainties.

Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

We propose a semidefinite optimization (SDP) model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of second-stage random variables belongs to a set of multivariate distributions with known first and second moments. For the minimax stochastic problem with random objective, we provide a tight SDP formulation. The problem with random right-hand side is NP-hard in general. In a special case, the problem can be solved in polynomial time. Explicit constructions of the worst-case distributions are provided. Applications in a production-transportation problem and a single facility minimax distance problem are provided to demonstrate our approach. In our experiments, the performance of minimax solutions is close to that of data-driven solutions under the multivariate normal distribution and better under extremal distributions. The minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity.

Robust hidden Markov LQG problems

For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes’ law. Duality of risk sensitivity to the multiplier version of min–max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I–III, the minimizing players’ worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

Robust congestion pricing under boundedly rational user equilibrium

This paper investigates congestion pricing strategies in static networks with boundedly rational route choice behavior. Under such behavior, users do not necessarily choose a shortest or cheapest route when doing so does not reduce their travel times by a significant amount. A general path-based definition and a more restrictive link-based representation of boundedly rational user equilibrium (BRUE) are presented. The set of BRUE flow distributions is generally non-convex and non-empty. The problems of finding best- and worst-case BRUE flow distributions are formulated and solved as mathematical programs with complementarity constraints. Because alternative tolled BRUE flow distributions exist, our congestion pricing models seek a toll vector or pattern that minimizes the system travel time of the worst-case tolled BRUE flow distribution. As formulated, the models are generalized semi-infinite min–max problems and we propose a heuristic algorithm based on penalization and a cutting-plane scheme to solve them. Numerical examples are presented to illustrate key concepts and results.

On the Relationship Between Robust Minimum Variance Beamformers With Probabilistic and Worst-Case Distortionless Response Constraints

An interesting relationship between the probability-constrained and worst-case optimization based robust minimum variance (MV) beamformers has been discovered. It is shown that both in the cases of circularly symmetric Gaussian and worst-case distributions of the steering vector mismatch, the probability-constrained robust MV beamforming problem can be tightly approximated as a convex second-order cone programming (SOCP) problem. The latter problem is mathematically equivalent to that resulting from the deterministic worst-case approach and, therefore, probability-constrained beamformers can be interpreted and implemented using their deterministic worst-case counterparts. However, an important advantage of the developed probability-constrained MV beamformers with respect to their standard worst-case counterparts is that the former approaches enable to explicitly quantify the parameters of the uncertainty region in terms of the beamformer outage probability.

Block QIM Watermarking Games

While binning is a fundamental approach to blind data embedding and watermarking, an attacker may devise various strategies to reduce the effectiveness of practical binning schemes. The problem analyzed in this paper is design of worst-case noise distributions against L-dimensional lattice quantization index modulation (QIM) watermarking codes. The cost functions considered are 1) probability of error of the maximum-likelihood decoder, and 2) the more tractable Bhattacharyya upper bound on error probability, which is tight at low embedding rates. Both problems are addressed under the following constraints on the attacker's strategy: the noise is independent of the marked signal, blockwise memoryless with block length L, and may not exceed a specified quadratic-distortion level. The embedder's quadratic distortion is limited as well. Three strategies are considered for the embedder: optimization of the lattice inflation parameter (also known as Costa parameter), dithering, and randomized lattice rotation. Critical in this analysis are the symmetry properties of QIM nested lattices and convexity properties of probability of error and related functionals of the noise distribution. We derive the minmax optimal embedding and attack strategies and obtain explicit solutions as well as numerical solutions for the worst-case noise. The role of the attacker's memory is investigated; in particular, we demonstrate the remarkable effectiveness of impulsive-noise attacks as L increases. The formulation proposed in this paper is also used to evaluate the capacity of lattice QIM under worst-noise conditions

Testing the performance of a 2D nearest point algorithm with genetic algorithm generated Gaussian distributions

Genetic algorithms have successfully been used in automatic software testing. Particularly programming errors and inputs that conflict with time constraints can be found. In this paper, the idea of genetic algorithm based software testing is broadened to algorithm performance testing. It is shown how the best and worst case performance of the algorithms can be found effectively. This information can be further utilized when comparing and improving algorithms. In this paper, the proposed test method is introduced and the advantages of using genetic algorithms are discussed. Furthermore, the proposed method is applied to a 2D nearest point algorithm, which is tested by optimizing the parameters of 2D Gaussian distributions using genetic algorithms in order to find the best and worst case distributions and the corresponding performances.

Dynamic pipelining

A truly scalable IP-lookup scheme must address five challenges of scalability, namely: routing-table size, lookup throughput, implementation cost, power dissipation, and routing-table update cost. Though several IP-lookup schemes have been proposed in the past, none of them do well in all the five scalability requirements. Previous schemes pipeline tries by mapping trie levels to pipeline stages. We make the fundamental observation that because this mapping is static and oblivious of the prefix distribution, the schemes do not scale well when worst-case prefix distributions are considered. This paper is the first to meet all the five requirements in the worst case. We propose scalable dynamic pipelining (SDP) which includes three key innovations: (1) We map trie nodes to pipeline stages based on the node height. Because the node height is directly determined by the prefix distribution, the node height succinctly provides sufficient information about the distribution. Our mapping enables us to prove a worst-case per-stage memory bound which is significantly tighter than those of previous schemes. (2) We exploit our mapping to propose a novel scheme for incremental route-updates. In our scheme a route-update requires exactly and only one write dispatched into the pipeline. This route-update cost is obviously the optimum and our scheme achieves the optimum in the worst case. (3) We achieve scalability in throughput by simultaneously pipelining at the data-structure level and the hardware level. SDP naturally scales in power and implementation cost. We not only present a theoretical analysis but also evaluate SDP and a number of previous schemes using detailed hardware simulation. Compared to previous schemes, we show that SDP is the only scheme that scales well in all the five requirements.

Fully Scalable Fault-Tolerant Simulations for BSP and CGM

In this paper we consider general simulations of algorithms designed for fully operational BSP and CGM machines on machines with faulty processors. The BSP (or CGM) machine is a parallel multicomputer consisting of p processors for which a memory of n words is evenly distributed and each processor can send and receive at most h messages in a superstep. The faults are deterministic (i.e., worst-case distributions of faults are considered) and static (i.e., they do not change in the course of computation). We assume that a constant fraction of processors are faulty. We present two fault-tolerant simulation techniques for BSP and CGM: 1.A deterministic simulation that achieves O(1) slowdown for local computations and O((loghp)2) slowdown for communications per superstep, provided that a preprocessing is done that requires O((loghp)2) supersteps and linear (in h) computation per processor in each superstep. 2.A randomized simulation that achieves O(1) slowdown for local computations and O(loghp) slowdown for communications per superstep with high probability, after the same (deterministic) preprocessing as above. Our results are fully scalable over all values of p from Θ(1) to Θ(n). Furthermore, our results imply that if p⩽nε for 0<ε<1 and h=Θ((n/p)δ) for 0<δ⩽1 (which hold in almost all practical BSP and CGM computations), algorithms can be made resilient to a constant fraction of processor faults without any asymptotic slowdown.

Worst-Case Distributions for Performance Evaluation of Proportionally Guided Missiles

Worst-Case Distributions for Performance Evaluation of Proportionally Guided Missiles

Fault-Tolerant Meshes with Small Degree

This paper presents constructions for fault-tolerant, two-dimensional mesh architectures. The constructions are designed to tolerate k faults while maintaining a healthy n by n mesh as a subgraph. They utilize several novel techniques for obtaining trade-offs between the number of spare nodes and the degree of the fault-tolerant network. We consider both worst-case and random fault distributions. In terms of worst-case faults, we give a construction that has constant degree and O(k3) spare nodes. This is the first construction known in which the degree is constant and the number of spare nodes is independent of n. In terms of random faults, we present several new degree-6 and degree-8 constructions and show (both analytically and through simulations) that these constructions can tolerate large numbers of randomly placed faults.

Effects of intravoxel velocity distributions on the accuracy of the phase-mapping method in phase-contrast MR angiography.

The phase-mapping method of phase-contrast magnetic resonance angiography is shown to be based on an implicit assumption that the intravoxel velocity distribution is symmetric about its mean velocity. The effect of asymmetric distributions on the accuracy of quantitative average velocity measurements is determined analytically and verified experimentally. An explicit formulation is developed for the estimated average velocity in a voxel as a function of the true average velocity and the asymmetry of the distribution about the true average velocity. Worst-case distributions are determined for unidirectional and bidirectional flow, and the special case of laminar flow is also investigated. Computer simulations and phantom imaging experiments demonstrate the accuracy of the analysis. For voxels with unidirectional flow, the phase-mapping method produces accurate estimates of average velocity, while results for bidirectional flow indicate possible large errors unless the aliasing velocity is increased, which decreases the signal-to-noise ratio in the resultant velocity map image.

Moment-based minimax stopping functions for sequences of random variables

Minimax-optimal stopping times and minimax (worst-case) distributions are found for the problem of stopping a sequence of uniformly bounded independent random variables, when only the means and/or variances are known, in contrast to the classical setting where the complete joint distributions of the random variables are known. Results are obtained for both the independent and i.i.d. cases, with applications given to the problem of order section in optimal stopping.

Minimax stopping times for I.I.D. Random variables

Minimax optimal stopping times and minimax worst—case distributions are found for the problem of stopping a sequence of uniformly bounded i.i.d. random variables in a cost and a discount model when only the mean and/or the variance (and not the complete distribution) of the random variables is known