Evaluating probabilistic models with uncertain model parameters

Probabilistic models are commonly used to evaluate quality attributes, such as reliability, availability, safety and performance of software-intensive systems. The accuracy of the evaluation results depends on a number of system properties which have to be estimated, such as environmental factors or system usage. Researchers have tackled this problem by including uncertainties in the probabilistic models and solving them analytically or with simulations. The input parameters are commonly assumed to be normally distributed. Accordingly, reporting the mean and variances of the resulting attributes is usually considered sufficient. However, many of the uncertain factors do not follow normal distributions, and analytical methods to derive objective uncertainties become impractical with increasing complexity of the probabilistic models. In this work, we introduce a simulation-based approach which uses Discrete Time Markov Chains and probabilistic model checking to accommodate a diverse set of parameter range distributions. The number of simulation runs automatically regulates to the desired significance level and reports the desired percentiles of the values which ultimately characterises a specific quality attribute of the system. We include a case study which illustrates the flexibility of this approach using the evaluation of several probabilistic properties.     

Contextual classification post-processing of LANDSAT data using a probabilistic relaxation model

A probabilistic relaxation model is used to improve maximum likelihood classifications of LANDSAT data of arid and urban areas in and around Ai Jahra, Kuwait. The problems of urban pixels, the role of compatibility coefficients and the iterations of the model are presented and discussed.     

Numerical vs. statistical probabilistic model checking

Numerical analysis based on uniformisation and statistical techniques based on sampling and simulation are two distinct approaches for transient analysis of stochastic systems. We compare the two solution techniques when applied to the verification of time-bounded until formulae in the temporal stochastic logic CSL, both theoretically and through empirical evaluation on a set of case studies. Our study differs from most previous comparisons of numerical and statistical approaches in that CSL model checking is a hypothesis-testing problem rather than a parameter-estimation problem. We can therefore rely on highly efficient sequential acceptance sampling tests, which enables statistical solution techniques to quickly return a result with some uncertainty. We also propose a novel combination of the two solution techniques for verifying CSL queries with nested probabilistic operators.     

The distance of probabilistic fuzzy sets for classification

The probabilistic fuzzy set (PFS) is designed for handling the uncertainties with both stochastic and fuzzy nature. In this paper, the concept of the distance between probabilistic fuzzy sets is introduced and its metric definition is conducted, which may be finite or continuous. And some related distances are discussed. The proposed distance considers the random perturbation in progress by introducing the distance of probability distribution, thus it improves the ability to handle random uncertainties, and some inadequacy of the distance of probability distribution is remedied. Finally, a PFS-based distance classifier is proposed to discuss the classification problem, the numerical experiment shows the superiority of this proposed distance in fuzzy and stochastic circumstance.     

Mining fuzzy association rules from uncertain data

Association rule mining is an important data analysis method that can discover associations within data. There are numerous previous studies that focus on finding fuzzy association rules from precise and certain data. Unfortunately, real-world data tends to be uncertain due to human errors, instrument errors, recording errors, and so on. Therefore, a question arising immediately is how we can mine fuzzy association rules from uncertain data. To this end, this paper proposes a representation scheme to represent uncertain data. This representation is based on possibility distributions because the possibility theory establishes a close connection between the concepts of similarity and uncertainty, providing an excellent framework for handling uncertain data. Then, we develop an algorithm to mine fuzzy association rules from uncertain data represented by possibility distributions. Experimental results from the survey data show that the proposed approach can discover interesting and valuable patterns with high certainty.     

Spatially enabled customer segmentation using a data classification method with uncertain predicates

Spatial attributes are important factors for predicting customer behavior. However, thorough studies on this subject have never been carried out. This paper presents a new idea that incorporates spatial predicates describing the spatial relationships between customer locations and surrounding objects into customer attributes. More specifically, we developed two algorithms in order to achieve spatially enabled customer segmentation. First, a novel filtration algorithm is proposed that can select more relevant predicates from the huge amounts of spatial predicates than existing filtration algorithms. Second, since spatial predicates fundamentally involve some uncertainties, a rough set-based spatial data classification algorithm is developed to handle the uncertainties and therefore provide effective spatial data classification. A series of experiments were conducted and the results indicate that our proposed methods are superior to existing methods for data classification.     

Design of a two-stage fuzzy classification model

This paper proposes a novel two-stage fuzzy classification model established by the fuzzy feature extraction agent (FFEA) and the fuzzy classification unit (FCU). At first, we propose a FFEA to validly extraction the feature variables from the original database. And then, the FCU, which is the main determination of the classification result, is developed to generate the if–then rules automatically. In fact, both the FFEA and FCU are fuzzy models themselves. In order to obtain better classification results, we utilize the genetic algorithms (GAs) and adaptive grade mechanism (AGM) to tune the FFEA and FCU, respectively, to improve the performance of the proposed fuzzy classification model. In this model, GAs are used to determine the distribution of the fuzzy sets for each feature variable of the FFEA, and the AGM is developed to regulate the confidence grade of the principal if–then rule of the FCU. Finally, the well-known Iris, Wine, and Glass databases are exploited to test the performances. Computer simulation results demonstrate that the proposed fuzzy classification model can provide a sufficiently high classification rate in comparison with other models in the literature.     

A novel Bayesian classification for uncertain data

Data uncertainty can be caused by numerous factors such as measurement precision limitations, network latency, data staleness and sampling errors. When mining knowledge from emerging applications such as sensor networks or location based services, data uncertainty should be handled cautiously to avoid erroneous results. In this paper, we apply probabilistic and statistical theory on uncertain data and develop a novel method to calculate conditional probabilities of Bayes theorem. Based on that, we propose a novel Bayesian classification algorithm for uncertain data. The experimental results show that the proposed method classifies uncertain data with potentially higher accuracies than the Naive Bayesian approach. It also has a more stable performance than the existing extended Naive Bayesian method.     

Credit scoring analysis using a fuzzy probabilistic rough set model

Credit scoring analysis is an important activity, especially nowadays after a huge number of defaults has been one of the main causes of the financial crisis. Among the many different tools used to model credit risk, the recent development of rough set models has proved effective. The original development of rough set theory has been widely generalized and combined with other approaches to uncertain reasoning, especially probability and fuzzy set theories. Since coherent conditional probability assessments cope well with the problem of unifying these different approaches, a merging of fuzzy rough set theory with this subjectivist approach is proposed. Specifically, expert partial probabilistic evaluations are encompassed inside a gradual decision rule structure, with coherence of the conclusion as a guideline. In line with Bayesian rough set models, credibility degrees of multiple premises are introduced through conditional probability assessments. Nonetheless, discernibility with this method remains too fine. Therefore, the basic partition is coarsened by equivalence classes based on the arity of positively, negatively and neutrally related criteria. A membership function, which grades the likelihood of default, is introduced by a peculiar choice of t-norms and t-conorms. To build and test the model, real data related to a sample of firms are used.     

A probabilistic spatial data model

Spatial information in autonomous robot tasks is uncertain due to measurement errors, the dynamic nature of the world, and an incompletely known environment. We present a probabilistic spatial data model capable of describing relevant spatial data, such as object location, shape, composition, and other parameters, in the presence of uncertainty. Uncertain spatial information is modeled through continuous probability distributions on values of attributes. The data model is designed to support our visual tracking and navigation prototype.     

A robust design criterion for interpretable fuzzy models with uncertain data

We believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation (H/sup 2/ and H/sup /spl infin// filtering techniques), because of the fact that not only linear parameters (consequents), but also the nonlinear parameters (membership functions) attempt to identify the uncertain behavior of the unknown system. However, the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system's behavior is optimal in some sense. This study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data. We consider online identification of an interpretable fuzzy model, based on the robust solution of a regularized least-squares fuzzy parameters estimation problem. The aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties. The study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors. A time-domain feedback analysis of the proposed identification approach is carried out with emphasis on stability, robustness, and steady-state issues. The simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods.     

Enhanced probabilistic neural network with data imputation capabilities for machine-fault classification

This paper presents the expectation–maximization (EM) variant of probabilistic neural network (PNN) as a step toward creating an autonomous and deterministic PNN. In the real world, faulty reading sensors can happen and will create input vectors with missing features yet they should not be discarded. To overcome this, regularized EM is put in place as a preprocessing step to impute the missing values. The problem faced by users when using random initialization is that they have to define the number of clusters through trial and error, which makes it stochastic in nature. Global k-means is used to autonomously find the number of clusters using a selection criterion and deterministically provide the number of clusters needed to train the model. In addition, fast Global k-means will be tested as an alternative to Global k-means to help reduce computational time. Tests are conducted on both homoscedastic and heteroscedastic PNNs. Benchmark medical datasets and also vibration data collected from a US Navy CH-46E helicopter aft gearbox known as Westland were used. The tests’ results fully support the usage of fast Global k-means and regularized EM as preprocessing steps to aid the EM-trained PNN.     

Efficient processing of probabilistic reverse nearest neighbor queries over uncertain data

Reverse nearest neighbor (RNN) search is very crucial in many real applications. In particular, given a database and a query object, an RNN query retrieves all the data objects in the database that have the query object as their nearest neighbors. Often, due to limitation of measurement devices, environmental disturbance, or characteristics of applications (for example, monitoring moving objects), data obtained from the real world are uncertain (imprecise). Therefore, previous approaches proposed for answering an RNN query over exact (precise) database cannot be directly applied to the uncertain scenario. In this paper, we re-define the RNN query in the context of uncertain databases, namely probabilistic reverse nearest neighbor (PRNN) query, which obtains data objects with probabilities of being RNNs greater than or equal to a user-specified threshold. Since the retrieval of a PRNN query requires accessing all the objects in the database, which is quite costly, we also propose an effective pruning method, called geometric pruning (GP), that significantly reduces the PRNN search space yet without introducing any false dismissals. Furthermore, we present an efficient PRNN query procedure that seamlessly integrates our pruning method. Extensive experiments have demonstrated the efficiency and effectiveness of our proposed GP-based PRNN query processing approach, under various experimental settings.     

Robust Pareto active suspension design for vehicle vibration model with probabilistic uncertain parameters

Using the robust design of a vehicle vibration model considering uncertainties can elaborately show the effects of those unsure values on the performance of such a model. In this paper, probabilistic metrics, instead of deterministic metrics, are used for a robust Pareto multi-objective optimum design of five-degree of freedom vehicle vibration model having parameters with probabilistic uncertainties. In order to achieve an optimum robust design against probabilistic uncertainties existing in reality, a multi-objective uniform-diversity genetic algorithm (MUGA) in conjunction with Monte Carlo simulation is used for Pareto optimum robust design of a vehicle vibration model with ten conflicting objective functions. The robustness of the design obtained using such a probabilistic approach is shown and compared with that of the design obtained using deterministic approach.     

Link classification with probabilistic graphs

The need to deal with the inherent uncertainty in real-world relational or networked data leads to the proposal of new probabilistic models, such as probabilistic graphs. Every edge in a probabilistic graph is associated with a probability whose value represents the likelihood of its existence, or the strength of the relation between the entities it connects. The aim of this paper is to propose two machine learning techniques for the link classification problem in relational data exploiting the probabilistic graph representation. Both the proposed methods will exploit a language-constrained reachability method to infer the probability of possible hidden relationships that may exists between two nodes in a probabilistic graph. Each hidden relationships between two nodes may be viewed as a feature (or a factor), and its corresponding probability as its weight, while an observed relationship is considered as a positive instance for its corresponding link label. Given a training set of observed links, the first learning approach is to use a propositionalization technique adopting a L2-regularized Logistic Regression to learn a model able to predict unobserved link labels. Since in some cases the edges’ probability may be not known in advance or they could not be precisely defined for a classification task, the second xposed approach is to exploit the inference method and to use a mean squared technique to learn the edges’ probabilities. Both the proposed methods have been evaluated on real world data sets and the corresponding results proved their validity.     

Supervised texture classification using a probabilistic neuralnetwork and constraint satisfaction model

The texture classification problem is projected as a constraint satisfaction problem. The focus is on the use of a probabilistic neural network (PNN) for representing the distribution of feature vectors of each texture class in order to generate a feature-label interaction constraint. This distribution of features for each class is assumed as a Gaussian mixture model. The feature-label interactions and a set of label-label interactions are represented on a constraint satisfaction neural network. A stochastic relaxation strategy is used to obtain an optimal classification of textures in an image. The advantage of this approach is that all classes in an image are determined simultaneously, similar to human perception of textures in an image.     

Robust supervised classification with mixture models: Learning from data with uncertain labels

In the supervised classification framework, human supervision is required for labeling a set of learning data which are then used for building the classifier. However, in many applications, human supervision is either imprecise, difficult or expensive. In this paper, the problem of learning a supervised multi-class classifier from data with uncertain labels is considered and a model-based classification method is proposed to solve it. The idea of the proposed method is to confront an unsupervised modeling of the data with the supervised information carried by the labels of the learning data in order to detect inconsistencies. The method is able afterward to build a robust classifier taking into account the detected inconsistencies into the labels. Experiments on artificial and real data are provided to highlight the main features of the proposed method as well as an application to object recognition under weak supervision.     

Extending fuzzy and probabilistic clustering to very large data sets

Approximating clusters in very large (VL=unloadable) data sets has been considered from many angles. The proposed approach has three basic steps: (i) progressive sampling of the VL data, terminated when a sample passes a statistical goodness of fit test; (ii) clustering the sample with a literal (or exact) algorithm; and (iii) non-iterative extension of the literal clusters to the remainder of the data set. Extension accelerates clustering on all (loadable) data sets. More importantly, extension provides feasibility—a way to find (approximate) clusters—for data sets that are too large to be loaded into the primary memory of a single computer. A good generalized sampling and extension scheme should be effective for acceleration and feasibility using any extensible clustering algorithm. A general method for progressive sampling in VL sets of feature vectors is developed, and examples are given that show how to extend the literal fuzzy (c-means) and probabilistic (expectation-maximization) clustering algorithms onto VL data. The fuzzy extension is called the generalized extensible fast fuzzy c-means (geFFCM) algorithm and is illustrated using several experiments with mixtures of five-dimensional normal distributions.     

Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability

This paper presents a new hybrid reliability model which contains randomness, fuzziness and non-probabilistic uncertainty based on the structural fuzzy random reliability and non-probabilistic set-based models. By solving the non-probabilistic set-based reliability problem and analyzing the reliability with fuzziness and randomness, the structural hybrid reliability can be obtained. The presented hybrid model has broad applicability which can handle either linear or non-linear state functions. A comparison among the presented hybrid model, probabilistic and non-probabilistic models, and the conventional probabilistic model is made through two typical numerical examples. The results show that the presented hybrid model, which may ensure structural security, is effective and practical.     

Simulation of random fuzzy variables: an empirical approach tostatistical/probabilistic studies with fuzzy experimental data

In this paper, we simulate different types of random fuzzy variables to get some conclusions concerning fuzzy-valued random variables. This simulation was carried out to illustrate certain limit results formalizing the convergence of the arithmetic mean of sampled fuzzy data to the population mean (or an expected value of the random fuzzy variable), like the well-known strong law of large numbers and the law of iterated logarithm. Since the theoretical results have recently been proved, this simulation analysis represents a complementary study in which we can determine sample sizes providing us with suitable approximations. Finally, future directions regarding statistical inference with fuzzy data and other applications are commented