Linear algebra and sequential importance sampling for network reliability

Publication in the conference proceedings of EUSIPCO, Toulouse, France, 2002

Sequential importance sampling for structural reliability analysis

Applications of graphical models often require the use of approximate inference, such as sequential importance sampling (SIS), for estimation of the model distribution given partial evidence, i.e., the target distribution. However, when SIS proposal and target distributions are dissimilar, such procedures lead to biased estimates or require a prohibitive number of samples. We introduce ReBaSIS, a method that better approximates the target distribution by sampling variable by variable from existing importance samplers and accepting or rejecting each proposed assignment in the sequence: a choice made based on anticipating upcoming evidence. We relate the per-variable proposal and model distributions by expected weight ratios of sequence completions and show that we can learn accurate models of optimal acceptance probabilities from local samples. In a continuous-time domain, our method improves upon previous importance samplers by transforming an SIS problem into a machine learning one.

Damage detection of civil infrastructure has gained significant momentum in the last decade due to aging and being more prone to failure of infrastructure. New advances in sensing technology have made it possible to detect damage in giant infrastructure more effectively. As the size of structure increases, the number of installed sensors on the structure increases significantly. This increase in the number of sensors makes it more challenging to collect and interpret the data. Therefore, utilizing the minimum number of sensors at any time of detection while maintaining an effective monitoring system is of great interest. In this paper, we will develop an innovative on/off sensors methodology using the sequential importance sampling technique to choose the sensors to be turned on at any given time while tracking the potential damage. This method will be used on a cable stayed bridge over Mississippi River called Luling Bridge. The finite element (FE) model of the bridge is developed to introduce the damage at different girder locations in the bridge. This model will be used to show the effectiveness of the proposed sequential importance sampling method in providing effective monitoring with minimal number of turned activated sensors at any given time.

In 2005, Chen et al. introduced a sequential importance sampling (SIS) procedure to analyze zero-one two-way tables with given fixed marginal sums (row and column sums) via the conditional Poisson (CP) distribution. They showed that compared with Monte Carlo Markov chain (MCMC)-based approaches, their importance sampling method is more efficient in terms of running time and also provides an easy and accurate estimate of the total number of contingency tables with fixed marginal sums. In this paper, we extend their result to zero-one multi-way ( $$d$$ -way, $$d \ge 2$$ ) contingency tables under the no $$d$$ -way interaction model, i.e., with fixed $$d-1$$ marginal sums. Also, we show by simulations that the SIS procedure with CP distribution to estimate the number of zero-one three-way tables under the no three-way interaction model given marginal sums works very well even with some rejections. We also applied our method to Samson’s monks data set.

The authors introduce a method of calculating the coefficients of the reliability polynomial of a network when the probabilities of link failure are independent and identical, having a value 1-p. This method involves calculation of the simple paths in the network, forming all combinations of their unions, classifying the unions, and counting and appropriately summing up the numbers of members of these classes. The method is then extended to the calculation of the expected number of operational paths in a network, defined as the connectivity polynomial. The relation of the reliability polynomial to the probabilistic capacity of a network is shown. >

Based on the concept of sequential importance sampling (SIS) and the use of Bayesian theory, particle filter is particularly useful in dealing with nonlinear and non-Gaussian problems. In this paper, a new particle filter is proposed that uses a divided difference filter to generate the importance proposal distribution is proposed. The proposal distribution integrates the latest measurements into system state transition density so it can match the posterior density well. The simulation results show that the new particle filter performs superior to the generic particle filter and other particle filters such as the extended Kalman particle filter and the unscented particle filter.

Demodulation of data transmitted over time-varying channels with a free running hidden Markov state, like the phase noise channel or the fading channel, requires that the receiver tracks the hidden channel state. The tracking technique adopted in the paper is based on non-data-aided sequential importance sampling, also known as particle filtering.The paper proposes a new particle filtering framework for data communication receivers based on an importance distribution such that each individual particle becomes a decision-directed Kalman filter relying upon its local symbol-by-symbol hard decisions. In this framework, different particles are left free to take different sequences of decisions. This leaves to the receiver the possibility of exploring different sequences of transmitted modulation symbols. The weight of the particle will be high for those particles that took in the past the correct sequence of decisions, while will be low for those particles that took wrong decisions. In the resampling procedure, particles with high weight will survive, while particles with low weight will be terminated, leaving space to the birth of new particles resampled from the surviving ones.The crucial point in importance sampling is the choice of the importance distribution and the main novelty of the paper is the proposal of an importance distribution such that the particles of the particle filter become decision-directed Kalman filters. One important benefit brought by our proposed method is that, being non-data-aided, it does not need pilot symbols, thus allowing to preserve the transmission rate. A significant application example, presented and developed in the paper, is constituted by MIMO systems affected by phase noise, where the channel state vector consists of many parameters.

Sequential importance sampling offers an alternative way to approximately evaluate the permanent. It is a stochastic algorithm which seems to work in practice but has eluded analysis. This paper offers examples where the analysis can be carried out and the first general bounds for the sample size required. This uses a novel importance sampling proof of Bregman’s inequality due to Lovasz.

Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or particle) representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

This paper discusses new directions of research to detect and diagnose Gaussian and non-Gaussian faults a new nonlinear observer (NOFS) based on Fuzzy and Sequential Important Sampling (FSIS) filter for each unknown states of the plant depending on the diagnosed. The idea based on expanding the size of freedom for the dynamic states of the observers. Therefore, NOFS has been designed and implemented to be robust nonlinear observer against the colored noise and non-Gaussian noise.

We consider the problem of computing the coefficients of the reliability polynomial (RP) for a random graph with reliable vertices and unreliable edges. To speed up the computation, we use the meaning of RP coefficients in one of its representations and prove vector relations over vectors of binomial coefficients.

Modern probabilistic programming languages aim to formalize and automate key aspects of probabilistic modeling and inference. Many languages provide constructs for programmable inference that enable developers to improve inference speed and accuracy by tailoring an algorithm for use with a particular model or dataset. Unfortunately, it is easy to use these constructs to write unsound programs that appear to run correctly but produce incorrect results. To address this problem, we present a denotational semantics for programmable inference in higher-order probabilistic programming languages, along with a type system that ensures that well-typed inference programs are sound by construction. A central insight is that the type of a probabilistic expression can track the space of its possible execution traces, not just the type of value that it returns, as these traces are often the objects that inference algorithms manipulate. We use our semantics and type system to establish soundness properties of custom inference programs that use constructs for variational, sequential Monte Carlo, importance sampling, and Markov chain Monte Carlo inference.

We study efficient importance sampling techniques for particle filtering (PF) when either (a) the observation likelihood (OL) is frequently multimodal or heavy-tailed, or (b) the state space dimension is large or both. When the OL is multimodal, but the state transition pdf (STP) is narrow enough, the optimal importance density is usually unimodal. Under this assumption, many techniques have been proposed. But when the STP is broad, this assumption does not hold. We study how existing techniques can be generalized to situations where the optimal importance density is multimodal, but is unimodal conditioned on a part of the state vector. Sufficient conditions to test for the unimodality of this conditional posterior are derived. Our result is directly extendable to testing for unimodality of any posterior. The number of particles N to accurately track using a PF increases with state space dimension, thus making any regular PF impractical for large dimensional tracking problems. But in most such problems, most of the state change occurs in only a few dimensions, while the change in the rest of the dimensions is small. Using this property, we propose to replace importance sampling from a large part of the state space (whose conditional posterior is narrow enough) by just tracking the mode of the conditional posterior. This introduces some extra error, but it also greatly reduces the importance sampling dimension. The net effect is much smaller error for a given N, especially when the available N is small. An important class of large dimensional problems with multimodal OL is tracking spatially varying physical quantities such as temperature or pressure in a large area using a network of sensors which may be nonlinear and/or may have nonnegligible failure probabilities. Improved performance of our proposed algorithms over existing PFs is demonstrated for this problem.

Critical events that occur rarely in biological processes are of great importance, but are challenging to study using Monte Carlo simulation. By introducing biases to reaction selection and reaction rates, weighted stochastic simulation algorithms based on importance sampling allow rare events to be sampled more effectively. However, existing methods do not address the important issue of barrier crossing, which often arises from multistable networks and systems with complex probability landscape. In addition, the proliferation of parameters and the associated computing cost pose significant problems. Here we introduce a general theoretical framework for obtaining optimized biases in sampling individual reactions for estimating probabilities of rare events. We further describe a practical algorithm called adaptively biased sequential importance sampling (ABSIS) method for efficient probability estimation. By adopting a look-ahead strategy and by enumerating short paths from the current state, we estimate the reaction-specific and state-specific forward and backward moving probabilities of the system, which are then used to bias reaction selections. The ABSIS algorithm can automatically detect barrier-crossing regions, and can adjust bias adaptively at different steps of the sampling process, with bias determined by the outcome of exhaustively generated short paths. In addition, there are only two bias parameters to be determined, regardless of the number of the reactions and the complexity of the network. We have applied the ABSIS method to four biochemical networks: the birth-death process, the reversible isomerization, the bistable Schlögl model, and the enzymatic futile cycle model. For comparison, we have also applied the finite buffer discrete chemical master equation (dCME) method recently developed to obtain exact numerical solutions of the underlying discrete chemical master equations of these problems. This allows us to assess sampling results objectively by comparing simulation results with true answers. Overall, ABSIS can accurately and efficiently estimate rare event probabilities for all examples, often with smaller variance than other importance sampling algorithms. The ABSIS method is general and can be applied to study rare events of other stochastic networks with complex probability landscape.