Conditional Expectation Method Research Articles

A Greedy Probabilistic Heuristic for Graph Black-and-White Anticoloring

Given a graph $G$ and positive integers $b$ and $w$, the Black-and-White Coloring problem asks about the existence of a partial vertex-coloring of $G$, with $b$ vertices colored black and $w$ white, such that there is no edge between a black and a white vertex. The problem is known to be NP-complete. In this paper, we deal with the optimization version, mainly for random graphs. Using the method of conditional expectations, we develop a heuristic with a good performance. We also obtain theoretical results on some of the relevant quantities, and compare the performance of the heuristic with that of several others.

Event-Triggered Distributed State Estimation: A Conditional Expectation Method

This paper mainly focuses on distributed filtering for a discrete time-varying system observed by a sensor network, where each sensor can measure some partial state information of the system and communicate with its neighbours. A novel distributed event-triggered communication mechanism is designed to reduce the communication rate among the sensors and guarantee the performance of the filter. With a data scheduler, the sensor is able to decide whether to transmit data to its neighbours. By applying Gaussian approximation, an evaluation of the effect caused by the non-transmission event is derived, which characterizes the tradeoff between communication rate and state estimation performance. Subsequently, a corresponding sub-optimal filtering gain design protocol is proposed. Compared with the literature, the filtering algorithm proposed in this paper is less conservative. Finally, numerical simulation is provided to illustrate the improvement of performance and the robustness of the approximation.

Machine Learning Aided Prediction of Glass-Forming Ability of Metallic Glass

The prediction of the glass-forming ability (GFA) of metallic glasses (MGs) can accelerate the efficiency of their development. In this paper, a dataset was constructed using experimental data collected from the literature and books, and a machine learning-based predictive model was established to predict the GFA. Firstly, a classification model based on the size of the critical diameter (Dmax) was established to determine whether an alloy system could form a glass state, with an accuracy rating of 0.98. Then, regression models were established to predict the crystallization temperature (Tx), glass transition temperature (Tg), and liquidus temperature (Tl) of MGs. The R2 of the prediction model obtained in the test set was greater than 0.89, which showed that the model had good prediction accuracy. The key features used by the regression models were analyzed using variance, correlation, embedding, recursive, and exhaustive methods to select the most important features. Furthermore, to improve the interpretability of the prediction model, feature importance, partial dependence plot (PDP), and individual conditional expectation (ICE) methods were used for visualization analysis, demonstrating how features affect the target variables. Finally, taking Zr-Cu-Ni-Al system MGs as an example, a prediction model was established using a genetic algorithm to optimize the alloy composition for high GFA in the compositional space, achieving the optimal design of alloy composition.

Towards interpreting machine learning models for predicting soil moisture droughts

Determination of the dominant factors which affect soil moisture (SM) predictions for drought analysis is an essential step to assess the reliability of the prediction results. However, artificial intelligence (AI) based drought modelling only provides prediction results without the physical interpretation of the models. Here, we propose an explainable AI (XAI) framework to reveal the modelling of SM drought events. Random forest based site-specific SM prediction models were developed using the data from 30 sites, covering 8 vegetation types. The unity of multiply XAI tools was applied to interpret the site-models both globally (generally) and locally. Globally, the models were interpreted using two methods: permutation importance and accumulated local effect (ALE). On the other hand, for each drought event, the models were interpreted locally via Shapley additive explanations (SHAP), local interpretable model-agnostic explanation (LIME) and individual conditional expectation (ICE) methods. Globally, the dominant features for SM predictions were identified as soil temperature, atmospheric aridity, time variables and latent heat flux. But through local interpretations of the drought events, SM showed a greater reliance on soil temperature, atmospheric aridity and latent heat flux at grass sites, with higher correlation to the time-dependent parameters at the sites located in forests. The temporal variation of the feature which effects the drought events was also demonstrated. The interpretation could shed light on how predictions are made and could promote the application of AI techniques in drought prediction, which may be useful for irrigation and water resource management.

Using the method of conditional expectations to supply an improved starting point for CCLS

Using the method of conditional expectations to supply an improved starting point for CCLS

GO-MOCE: Greedy Order Method of Conditional Expectations for Max Sat

In this paper we present and study a new algorithm for the Maximum Satisfiability (Max Sat) problem. The algorithm is based on the Method of Conditional Expectations (MOCE, also known as Johnson’s Algorithm) and applies a greedy variable ordering to MOCE. Thus, we name it Greedy Order MOCE (GO-MOCE). We also suggest a combination of GO-MOCE with CCLS, a state-of-the-art solver. We refer to this combined solver as GO-MOCE-CCLS.We conduct a comprehensive comparative evaluation of GO-MOCE versus MOCE on random instances and on public competition benchmark instances. We show that GO-MOCE reduces the number of unsatisfied clauses by tens of percents, while keeping the runtime almost the same. The worst case time complexity of GO-MOCE is linear. We also show that GO-MOCE-CCLS improves on CCLS consistently by up to about 80%.We study the asymptotic performance of GO-MOCE. To this end, we introduce three measures for evaluating the asymptotic performance of algorithms for Max Sat. We point out to further possible improvements of GO-MOCE, based on an empirical study of the main quantities managed by GO-MOCE during its execution.

Estimation of missing data in a geophysical series of precipitation

The analysis of dynamic systems is a topic of great interest in the basic sciences since it allows direct inference of the behavior of different systems. The study of physical phenomena provides large databases that, if recorded at regular time intervals, constitute time series. However, time series of geophysical data in many cases present missing data and their estimation requires the application of valid methods that allow estimating reliable information to complete the series since some analysis methods require these series to be complete. Two methods are used in this article to estimate the missing values of the precipitation series in the city of San José de Cúcuta, Colombia, the first one consists of considering the univariate data series and applying an adjustment of the sequential conditional expectation method of forecasting with restrictions, the second one refers to analyze the data series of a nearby station and through multivariate methods establish the cointegration between the series, and then use this as a basis for estimating the missing data in the analysis series. The two methods are recursive, a first estimation of the model is made ignoring the missing data, an initial estimation of the missing data is made, then a new estimation of the model parameters and a new estimation of the missing data is made, the algorithm continues running with the new values replacing the values estimated in the previous phase until the difference of the estimated values between successive iterations is less than a value fixed beforehand. Finally, a comparison is made between the estimates made by the two methods.

A residual load modeling approach for household short-term load forecasting application

The household residual component of total power consumption can be considered as a portion of load demand describing the non-temperature-related factors. This component can be decomposed to irregular and predictable energy demands. The predictable component of the residual load include consumptions which are likely to have a periodic behavior. The modeling of this periodic part of the residual load can help to enhance the overall forecasting accuracy. This paper intends to model the periodic part of the residual component by capturing the behavioral patterns of overestimated and underestimated residuals as the divisions of the main residual component. Accordingly, in order to achieve its ambition, this work proposes an Adaptive Circular Conditional Expectation (ACCE) method on the basis of circular analysis to define the sub-residuals operation schedules. Consequently, an adaptive Linear Model (LM) procedure is employed to predict the residual component demand using the results of the ACCE process at each time window. Subsequently, the predicted residual is utilized to adaptively improve the performance of total electricity demand forecasting. The accuracy of the forecasting results is evaluated using Normalized Mean Absolute Error (NMAE). As a result, the proposed approach of the periodic residual demand modeling in a daily horizon leads to a promising accuracy increase of 23%. Furthermore, the proposed residual modeling method, in combination with the temperature-related component forecasting, can increase the total power consumption prediction performance by 7%. The efficacy of the proposed approach is examined via numerical analysis of real data.

Deterministic Parallel Algorithms for Fooling Polylogarithmic Juntas and the Lovász Local Lemma

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with Luby (1993) and continuing with Berger and Rompel (1991) and Chari et al. (2000), showed that these techniques can be combined to give deterministic parallel algorithms for combinatorial optimization problems involving sums of w -juntas. We improve these algorithms through derandomized variable partitioning, reducing the processor complexity to essentially independent of w and time complexity to linear in w . As a key subroutine, we give a new algorithm to generate a probability space which can fool a given set of neighborhoods. Schulman (1992) gave an NC algorithm to do so for neighborhoods of size w ≤ O (log n ). Our new algorithm is in NC 1 , with essentially optimal time and processor complexity, when w = O (log n ); it remains in NC up to w = polylog( n ). This answers an open problem of Schulman. One major application of these algorithms is an NC algorithm for the Lovász Local Lemma. Previous NC algorithms, including the seminal algorithm of Moser and Tardos (2010) and the work of Chandrasekaran et. al (2013), required that (essentially) the bad-events could span only O (log n ) variables; we relax this to polylog( n ) variables. We use this for an NC 2 algorithm for defective vertex coloring, which works for arbitrary degree graphs.

Deterministic Parallel Algorithms for Bilinear Objective Functions

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial optimization problems, with low time- and processor-complexity. We extend and generalize a technique of Luby for efficiently handling bilinear objective functions. One noteworthy application is an NC algorithm for maximal independent set. On a graph G with m edges and n vertices, this takes $${\tilde{O}}(\log ^2 n)$$ time and $$(m + n) n^{o(1)}$$ processors, nearly matching the best randomized parallel algorithms. Other applications include reduced processor counts for algorithms of Berger (SIAM J Comput 26:1188–1207, 1997) for maximum acyclic subgraph and Gale–Berlekamp switching games. This bilinear factorization also gives better algorithms for problems involving discrepancy. An important application of this is to automata-fooling probability spaces, which are the basis of a notable derandomization technique of Sivakumar (In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp 619–626, 2002). Our method leads to large reduction in processor complexity for a number of derandomization algorithms based on automata-fooling, including set discrepancy and the Johnson–Lindenstrauss Lemma.

Numerical Procedures for Random Differential Equations

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

Utilization of alternating conditional expectation method for pin power reconstruction in CANDU reactor physics analyses

This paper presents a new methodology using the Alternating Conditional Expectation (ACE) algorithm for reconstructing fuel pin powers in the Canada Deuterium Uranium (CANDU) reactor. Variations in assembly powers in neighboring fuel bundles are used as predictor variables to determine the response, e.g. power in each of the fuel pin. When a sufficient number of cases have been evaluated, the relation between the predictor variables and the response can be treated as a multi-variable regression problem. The ACE algorithm which is a non-parametric approach (i.e., no predetermined functional relationship between the predictors and the response is assumed) is then utilized for estimating optimal transformations of the predictors and the response. The geometry used is a two-dimensional, 3-by-3 fuel bundles configuration. Three hundred different configurations, which represent various power distributions, were evaluated to develop these optimal transformations. The optimal transformation is determined for each of 37 pins in a CANDU natural uranium fuel bundle. The quality of these transformations for predicting fuel pin power has been verified against an independent set of 50 two-dimensional, 3-by-3 fuel bundle configurations. The results indicate that the proposed pin power reconstruction algorithm is sufficiently accurate, giving an RMS (root-mean-squared) error of around 0.59% for all pins. The RMS error in predicting the maximum pin power is 0.60%. The development of the proposed methodology continued with the examination of its robustness. The impact of employing different training sets on the overall relative error in predicting the pin power is examined. In addition, the impact of employing different sizes of training sets is also investigated. For each training set size, a bias in predicting the maximum pin power is determined statistically based on the numerical results observed during the study. The results indicate that the methodology is likely to under-predict the maximum pin power; to circumvent this issue a multiplicative bias of 1.0118 needs to be applied, based on evaluating results from various sizes of the training sets.

Decentralized Kalman Filtering with Multilevel Quantized Innovation in Wireless Sensor Networks

Because of low power consumption and limited power supply significance in wireless sensor networks (WSNs), this paper studies the multilevel quantized innovation Kalman filtering (MQI-KF) for decentralized state estimation in WSNs since the MQI-KF can help to save power. In the first place, the common features of the practical low energy consumption WSNs are explored. On this basis, the new quantization scheme is presented. Besides, this paper explores the quantization state estimation by adopting the Bayesian method rather than the traditional iterated conditional expectation method. After that, this paper proposes a new decentralized state estimation algorithm (MQI-KF) for WSNs. Information entropy is analyzed to evaluate the performance of the quantization scheme. Performance analysis and simulations show that the MQI-KF is more efficient than the other decentralized Kalman filtering (KF) algorithms, and the accuracy of its estimation is close to that of the standard KF based on nonquantized measurements. Since the new quantization scheme and algorithm take into consideration the features of real WSNs which are based on the universal network protocol IEEE 802.15.4 standard, they can almost be applied into all practical WSNs with low energy consumption.

基于Copula函数的梯级水库设计洪水地区 组成研究

目前工程设计中常用的同频率地区组成带有一定的人为性，统计理论基础不足。利用Copula函数推导了两种考虑空间相关性的梯级水库设计洪水地区组成，即条件期望组成和最可能组成。以清江流域为例，对三种地区组成进行比较研究，分析了上游水布垭-隔河岩梯级水库的调蓄对高坝洲断面设计洪水的影响。结果表明：同频率组成计算的设计结果系统偏小，可能会增加设计断面的防洪风险。基于Copula函数的两种组成充分考虑了各分区洪水的随机性和空间相关性，设计成果比较合理。受上游梯级水库调蓄的影响，高坝洲断面千年一遇设计洪水的削峰率为40%左右。 The equivalent frequency regional composition method is widely used in practice. However, this method is based on artificial hypothesis and lacks the statistical basis. Two statistical composition methods, i.e. the conditional expectation method and the most likely method are proposed and derived based on Copula function. The Qingjiang cascade reservoirs are selected as the case study and the impact of Shuibuya-Geheyan cascade reservoirs regulation on the design flood of Gaobazhou dam site is analyzed. It is demonstrated that the results of equivalent frequency method are systematically smaller than those of the two proposed methods, which may lead to increasing the hydrological estimation risk. The proposed methods with strong statistical basis can capture the spatial correlation of flood events for each contributing watershed area, and the estimated design flood values are more reasonable than those of the equivalent frequency method. The 1000-year design peak discharge at Gaobazhou dam site has been cut about 40% due to the regulation of upstream cascade reservoirs.

On the statistical and biological behaviors of nonlinear mixed forest models

This study examined both statistical and biological behaviors of two nonlinear mixed models fitted by the first-order (FO) and first-order conditional expectation (FOCE) methods. The simpler FO method was found to alter the mathematical forms of the base models and produce biologically unreasonable predictions for some subjects. This was not the case for the more complex FOCE method. Since the computations for predicting random parameters and for making subject-specific (SS) predictions were different for the two methods, mixing them would potentially produce large prediction biases and unexpected biological behaviors. This was demonstrated on two data sets (basal area–age and height–diameter data sets). For basal area–age data, accurate and precise basal area predictions were produced as long as a consistent method was used for predicting random parameters and for making SS predictions, regardless of the method used for model fitting. Otherwise, less accurate and precise predictions were produced, with some predictions totally against biological expectations. For height–diameter data, both consistent and inconsistent methods produced similar prediction statistics, but this does not mean that inconsistent methods should be adopted. Overall the FOCE method was found to be superior to the FO method in terms of producing biologically more reasonable predictions.

Calculating empirical best linear unbiased predictors (EBLUPs) for nonlinear mixed effects models in Excel/Solver

Nonlinear mixed-effects models have become common in the forestry literature. Calibration of these models for a new subject (one not used in the fitting of the model) involves estimating the values of the of random-effects parameters. Estimators can be obtained by taking a Taylor-series expansion of the nonlinear model around the expected value or the conditional expectation of the random-effects parameters. The conditional expectation method requires an iterative technique to find the estimates, which can be a complicated programming exercise. This note describes a relatively easy way to do the calculations necessary for both the zero expansion and conditional expectation methods in Excel and demonstrates the procedure on a small example.

Comparison of different methods for fitting nonlinear mixed forest models and for making predictions

Nonlinear mixed models have become popular in forestry applications, and various methods have been proposed for fitting such models. However, it is difficult or even confusing to choose which method to use, and there is not much relevant information available, especially in the forestry context. The main objective of this study was to compare three commonly used methods for fitting nonlinear mixed models: the first-order, the first-order conditional expectation, and the adaptive Gaussian quadrature methods. Both the maximum likelihood and restricted maximum likelihood parameter estimation techniques were evaluated. Three types of data common in forestry were used for model fitting and model application. It was found that the first-order conditional expectation method provided more accurate and precise predictions for two models developed from data with more observations per subject. For one model developed on data with fewer observations per subject, the first-order method provided better model predictions. All three models fitted by the first-order method produced some biologically unrealistic predictions, and the problem was more obvious on the data with fewer observations per subject. For all three models fitted by the first-order and first-order conditional expectation methods, the maximum likelihood and restricted maximum likelihood fits and the resulting model predictions were very close.

Polynomial computation of exact correlated equilibrium in compact games

In a landmark paper, Papadimitriou and Roughgarden described a polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample correlated equilibria of concisely-represented games. Recently, Stein, Parrilo and Ozdaglar showed that this algorithm can fail to find an exact correlated equilibrium, but can be easily modified to efficiently compute approximate correlated equilibria. It remained an open problem to determine whether the algorithm can be modified to compute an exact correlated equilibrium. In a new paper, we showed that it can, presenting a variant of the Ellipsoid Against Hope algorithm that guarantees the polynomial-time identification of exact correlated equilibrium. Our new algorithm differs from the original primarily in its use of a separation oracle that produces cuts corresponding to pure-strategy profiles. As a result, we no longer face the numerical precision issues encountered by the original approach, and both the resulting algorithm and its analysis are considerably simplified. Our new separation oracle can be understood as a derandomization of Papadimitriou and Roughgarden's original separation oracle via the method of conditional expectations.

An improved derandomized approximation algorithm for the max-controlled set problem

A vertex i of a graph G=(V,E) is said to be controlled by M ⊆ V if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex i ∈ V is controlled by M. Given a set M ⊆ V and two graphs G 1=(V,E 1) and G 2=(V,E 2) where E 1 ⊆ E 2, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G=(V,E) (i.e., a graph where E 1 ⊆ E ⊆ E 2) such that M is a monopoly in G=(V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G=(V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio \(\frac{1}{2}+ \frac{1+\sqrt{n}}{2n-2}\), where n=|V|>4. (The case n ≤ 4 is solved exactly by considering the parameterized version of the mcsp.) The algoritm is obtained through the use of randomized rounding and derandomization techniques, namely the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.

Estimation of Structural Failure Probability Based on a Quasi Ideal Importance Sampling Simulation Combined with the Conditional Expectation (Construction of a Quasi Ideal Importance Sampling Marginal Probability Density through Recursive Simulation Procedures)

This paper describes a quasi ideal importance sampling simulation combined with the conditional expectation method for the simulation-based structural reliability estimation. A quasi ideal importance sampling joint probability density function is formulated on the basis of the ideal importance sampling concept and the respective marginal probability densities are constructed numerically by recursive simulations and piecewise integrations. The respective quasi ideal importance sampling marginal probability densities are renewed through recursive simulation procedures. Numerical examples to estimate the failure probability of structures with multiple failure modes show that the proposed method gives accurate estimations of the failure probability of structures efficiently.