Efficient SRAM failure rate prediction via Gibbs sampling

espanolExact tests, fuzzy p-values, Gibbs sampling, iterval censoring, linear extensions, linear rank statistics, perfect MCMC, proportional hazard model, topological sorting EnglishWe discuss distribution-free exact rank tests from partially ordered data that arise in various biological and other applications where the primary objective is to conduct testing of significance to assess the linear dependence or to compare different groups. The tests here are obtained by treating the usual rank statistics, based on the completely ordered data as “latent” or missing, and conceptualizing the “latent” p-value as the random probability under the null hypothesis of a test statistic that is as extreme, or more extreme, than the latent test statistics based on the completely ordered data. The latent p-value is then predicted by sampling linear extensions or the complete orderings that are consistent with the observed partially ordered data. The sampling methods explored here include importance sampling methods based on randomized topological sorting algorithms, Gibbs sampling methods, random-walk based Metropolis–Hasting sampling methods and random-walk based modern perfect Markov chain Monte Carlo sampling methods. We discuss running times of these sampling methods and their strength and weaknesses. A simulation experiment and three data examples are given. The simulation experiment illustrates how the exact rank tests from partially ordered data work when the desired result is known. The first data example concerns the light preference behavior of fruit flies and tests whether heterogeneity observed in average light-preference behavior can be explained by manipulations in serotonin signaling. The second one is a reanalysis of the lead absorption data in children of employees who worked in a lead battery factory and consolidates the results reported in Rosenbaum [Ann. Statist. 19 (1991) 1091–1097]. The third one reexamines the breast cosmesis data from Finkelstein [Biometrics 42 (1986) 845–854].

A network-on-chip (NoC), a redundancy-rich and thus relatively robust system-chip, is still vulnerable to defects due to its large-scale integration. Thus, it is desirable to analyze the NoC yield in an early design phase. A Monte Carlo (MC) approach was proposed for the NoC yield analysis at the system level; however, it is inefficient due to the requirement of a large number of simulation runs. In this paper, we propose a Gibbs sampling approach, which can efficiently generate failed NoC instances as simulation samples, for yield estimation. This approach significantly reduces the number of required simulation runs for obtaining an accurate yield estimation. Implementation issues, such as initial sample selection, calculation of conditional distributions, and stop criterion, to customize Gibbs sampling for the NoC yield analysis are discussed. Potential optimization opportunities to further improve Gibbs sampling's efficiency are also explored. Compared to the MC approach, our experimental results show that the proposed approach can reduce the simulation runtime by 5×-100× for a high-yield NoC (a failure rate at 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> -10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> ), while achieving the same level of accuracy for yield estimation.

Massively repeated structures such as SRAM cells usually require extremely low failure rate. This brings on a challenging issue for Monte Carlo based statistical yield analysis, as huge amount of samples have to be drawn in order to observe one single failure. Fast Monte Carlo methods, e.g. importance sampling methods, are still quite expensive as the anticipated failure rate is very low. In this paper, a new method is proposed to tackle this issue. The key idea is to improve traditional importance sampling method with an efficient online surrogate model. The proposed method improves the performance for both stages in importance sampling, i.e. finding the distorted probability density function, and the distorted sampling. Experimental results show that the proposed method is 1e2X∼1e5X faster than the standard Monte Carlo approach and achieves 5X∼22X speedup over existing state-of-the-art techniques without sacrificing estimation accuracy.

In this article we develop an estimation method based on the augmented data scheme and EM/SEM (Stochastic EM) algorithms for fitting one-parameter probit (Rasch) IRT (Item Response Theory) models. Instead of using the S steps of the SEM algorithm, that is, instead of simulating values for the unobserved variables (augmented data and the latent traits), we consider the conditional expectations of a set of unobserved variables on the other set of unobserved variables, the current estimates of the parameters and the observed data, based on the full conditional distributions from the Gibbs sampling algorithm. Our method, named the CADEM algorithm (conditional augmented data EM), presents straightforward E steps, which avoid the need to evaluate the usual integrals, also facilitating the M steps, without the need to use numerical methods of optimization. We use the CADEM algorithm to obtain both maximum likelihood estimates and maximum a posteriori estimates of the difficulty parameters for the one-parameter probit (Rasch) model. Also, we obtain estimates for the latent traits, based on conditional expectations. In addition, we show how to calculate the associated standard errors. Some directions are provided to extend our approach to other IRT models. In this respect, we perform a simulation study to compare the estimation methods. The results indicated that our approach is quite comparable to the usual marginal maximum likelihood (MML) and Gibbs sampling methods (GS) in terms of parameter recovery. However, CADEM is as fast as MML and as flexible as GS.

This paper proposes a novel model estimation method, which uses nested Gibbs sampling to develop a mixture-of-mixture model to represent the distribution of the model's components with a mixture model. This model is suitable for analyzing multilevel data comprising frame-wise observations, such as videos and acoustic signals, which are composed of frame-wise observations. Deterministic procedures, such as the expectation–maximization algorithm have been employed to estimate these kinds of models, but this approach often suffers from a large bias when the amount of data is limited. To avoid this problem, we introduce a Markov chain Monte Carlo-based model estimation method. In particular, we aim to identify a suitable sampling method for the mixture-of-mixture models. Gibbs sampling is a possible approach, but this can easily lead to the local optimum problem when each component is represented by a multi-modal distribution. Thus, we propose a novel Gibbs sampling method, called “nested Gibbs sampling,” which represents the lower-level (fine) data structure based on elemental mixture distributions and the higher-level (coarse) data structure based on mixture-of-mixture distributions. We applied this method to a speaker clustering problem and conducted experiments under various conditions. The results demonstrated that the proposed method outperformed conventional sampling-based, variational Bayesian, and hierarchical agglomerative methods.

Estimates of posterior distributions of genetic and phenotypic parameters and functions of them for individual test day milk yield are obtained for 28 873 British Holstein‐Friesian heifers, the progeny of 40 proven and 649 unproven sires, using restricted maximum likelihood (REML) and Gibbs sampling methods with a univariate sire model. Results from the two methods are then compared. It is found that the posterior expectations and REML estimates of the parameters are fairly similar. The marginal posterior expectation and REML estimate of heritability for lactation milk yield are 0.49 and 0.50, respectively. Heritability estimates for individual test day records range from 0.27 to 0.40, whilst the posterior expectations are between 0.28 and 0.42. Generally, heritabilities for test day records were lowest at the start and highest in the second half of the lactation. Gibbs sampling requires substantially more computing than REML estimation, but provides a more informative analysis.

The accuracy of the Gibbs sampling Markov chain monte carlo procedure was examined for estimating item and person ( .) parameters in the one-parameter logistic model. Four datasets were analyzed using the Gibbs sampling method, conditional maximum likelihood, marginal maximum likelihood, and joint maximum likelihood. Maximum likelihood and expected a posteriori. estimation methods were used with marginal maximum likelihood estimation of item parameters. Item parameter estimates from the four methods were almost identical;. estimates from Gibbs sampling were similar to those obtained from the expected a posteriori method.

Gibbs sampling methods are standard tools to perform posterior inference for mixture models. These have been broadly classified into two categories: marginal and conditional methods. While conditional samplers are more widely applicable than marginal ones, they may suffer from slow mixing in infinite mixtures, where some form of truncation, either deterministic or random, is required. In mixtures with random number of components, the exploration of parameter spaces of different dimensions can also be challenging. We tackle these issues by expressing the mixture components in the random order of appearance in an exchangeable sequence directed by the mixing distribution. We derive a sampler that is straightforward to implement for mixing distributions with tractable size-biased ordered weights, and that can be readily adapted to mixture models for which marginal samplers are not available. In infinite mixtures, no form of truncation is necessary. As for finite mixtures with random dimension, a simple updating of the number of components is obtained by a blocking argument, thus, easing challenges found in transdimensional moves via Metropolis-Hastings steps. Additionally, sampling occurs in the space of ordered partitions with blocks labeled in the least element order, which endows the sampler with good mixing properties. The performance of the proposed algorithm is evaluated in a simulation study. Supplementary materials for this article are available online.

In order to solve the problem of inaccurate description of news content features and user interest features in mobile cloud computing, proposed a multimedia text classification algorithm that utilizes multi-tag potential Dirichlet distribution. The algorithm is based on the traditional latent Dirichlet allocation (LDA) model and assumes a linear relationship between user tags and potential topics. Therefore, a relational matrix is introduced in the LDA model to describe the corresponding relationship between the tag and the topic, so that the probability distribution of the tag on the word can be inferred from the probability distribution of the topic on the word. The algorithm first learns the probability distribution table of label words by Gibbs sampling method, then infers the probability distribution of new documents on labels according to the model parameters, so as to realize the purpose of predicting the corresponding multiple labels of documents. In order to improve the ability of the algorithm to deal with massive data, the parallel algorithm has been improved. Since the bottleneck of the algorithm lies mainly in the serial nature of global variable updating and communication, the core idea of our parallelization is that in massive text training, global delay updating and asynchronous communication will not affect the final training results. Experiments show that the proposed algorithm has greatly improved the training efficiency. The classification accuracy is higher than that of Naive Bayesian algorithm and Support Vector Machine (SVM) algorithm proposed in other literatures. The average classification accuracy can achieve at about 95%, and it can be used as a general parallel framework of supervised LDA algorithm.

Summary This paper explains how the Gibbs sampler can be used to perform Bayesian inference on GARCH models. Although the Gibbs sampler is usually based on the analytical knowledge of the full conditional posterior densities, such knowledge is not available in regression models with GARCH errors. We show that the Gibbs sampler can be combined with a unidimensional deterministic integration rule applied to each coordinate of the posterior density. The full conditional densities are evaluated and inverted numerically to obtain random draws of the joint posterior. The method is shown to be feasible and competitive compared with importance sampling and the Metropolis‐Hastings algorithm. It is applied to estimate an asymmetric Student‐GARCH model for the return on a stock exchange index, and to compute predictive option prices on the index. We prove, moreover, that a flat prior on the degrees of freedom parameter leads to an improper posterior density.

Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional probability distribution. The Gibbs sampler, however, is often justifiably criticized for its sometimes slow convergence, especially when it is used to fit highly structured complex models. The recently proposed Partially Collapsed Gibbs (PCG) sampler offers a new strategy for improving the convergence characteristics of a Gibbs sampler. A PCG sampler achieves faster convergence by reducing the conditioning in some or all of the component draws of its parent Gibbs sampler. Although this strategy can significantly improve convergence, it must be implemented with care to be sure that the desired stationary distribution is preserved. In some cases the set of conditional distributions sampled in a PCG sampler may be functionally incompatible and permuting the order of draws can change the stationary distribution of the chain. In this article, we draw an analogy between the PCG sampler and certain efficient EM-type algorithms that helps to explain the computational advantage of PCG samplers and to suggest when they might be used in practice. We go on to illustrate the PCG samplers in three substantial examples drawn from our applied work: a multilevel spectral model commonly used in high-energy astrophysics, a piecewise-constant multivariate time series model, and a joint imputation model for nonnested data. These are all useful highly structured models that involve computational challenges that can be solved using PCG samplers. The examples illustrate not only the computation advantage of PCG samplers but also how they should be constructed to maintain the desired stationary distribution. Supplemental materials for the examples given in this article are available online.

At the multiplexer of an ATM network, the admissible cell loss probability is less than 10−12 for some service categories. Ordinary Monte Carlo (MC) simulation requires considerable computation time to obtain such a small probability. The importance sampling (IS) technique is effective in accclerating the MC simulation. In an IS simulation, the underlying distribution is biased so as to generate more samples in the target event. In order to achieve a fast simulation, it is important to have a simulation distribution that yields an estimate with minimum variance. This is called the optimal simulation distribution. In this paper, we investigate the survivor function P(Q > q) of a stationary queue length Q in ATM queuing and obtain an IS estimate of P(Q > q) at about 10−2 by means of the optimal simulation distribution. Conventional methods [1, 2] for determining the optimal distribution require a lot of computation time. We apply the large deviation theory to this problem to obtain the optimal simulation distribution analytically and carry out an IS simulation. We present some numerical results for M/D/1 and 2-state MMPP/D/1 which demonstrate that our estimates are obtained in a very small computation time and are accurate when compared to MC simulation. © Scripta Technica, Inc. Electron Comm Jpn Pt 1, 80(12): 28–37, 1997

Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asymptotically optimal importance sampling schemes for queueing networks. Using a recently developed game/subsolution approach, we construct simple and efficient state-dependent importance sampling schemes for simulating buffer overflows in stable open Jackson networks. The sampling distributions do not depend on the particular event of interest, and hence overflow probabilities for different events can be estimated simultaneously. A by-product of the analysis is the identification of the minimizing trajectory for the calculus of variation problem that is associated with the sample-path large deviation rate function.

The failure rate of static RAM (SRAM) cells is restricted to be extremely low to ensure sufficient high yield for the entire chip. In addition, multiple performances of interest and influences from peripherals make SRAM failure rate estimation a high-dimensional multiple-failure-region problem. This paper proposes a new method featuring a multistart-point sequential quadratic programming (SQP) framework to extend minimized norm importance sampling (IS) to address this problem. Failure regions in the variation space are first found by the low-discrepancy sampling sequence. Afterward, start points are generated in all identified failure regions and local optimizations based on SQP are invoked from these start points searching for the optimal shift vectors (OSVs). Based on the OSVs, a Gaussian mixture distorted distribution is constructed for IS. To further reduce the computational cost of IS while fully considering the influence of increasing dimensionality, an adaptive model training framework is proposed to keep high efficiency for both low- and high-dimensional problems. The experimental results show that the proposed method can not only approximate failure rate with high accuracy and efficiency in low-dimensional cases but also keep these features in high-dimensional ones.

We develop an efficient importance sampling algorithm for estimating the tail distribution of heavy-tailed compound sums, that is, random variables of the form S M = Z 1 +…+ Z M where the Z i 's are independently and identically distributed (i.i.d.) random variables in R and M is a nonnegative, integer-valued random variable independent of the Z i 's. We construct the first estimator that can be rigorously shown to be strongly efficient only under the assumption that the Z i 's are subexponential and M is light-tailed. Our estimator is based on state-dependent importance sampling and we use Lyapunov-type inequalities to control its second moment. The performance of our estimator is empirically illustrated in various instances involving popular heavy-tailed models.