Distributivity and conditional distributivity equations for $${\varvec{S}}$$-uninorms over $${\varvec{T}}$$-uninorms

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible,i.e.whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a “pseudo-Gibbs sampler”. We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage.

For conditional tail inferences from multivariate distributions, we desire models with positive dependence for which conditional distributions of one variable given others have extreme value indices that can be functions of the values of conditioning variables. That is, the tails of the conditional distributions behave like Pareto distributions with varying tail parameter. It is shown in Arnold (Stat Probab Lett 5:263–266, 1987) and Arnold et al. (Stat Probab Lett 17:361–368, 1993) in the bivariate case that if all conditional distributions are (generalized) Pareto and the tail parameters are non-constant, then tractable solutions have limited range of dependence. To obtain models with our desired properties, we specify one set of conditional Pareto distributions and one marginal distribution: for example, (a) FY |X(⋅|x) is Pareto with tail parameter decreasing in x and/or scale parameter is increasing in x, (b) no specification is made for FX|Y(⋅|x), and (c) X has a distribution with regularly varying upper tail. Based on this construction, we study the following properties: (1) relation of concordance ordering to the tail parameter function, (2) relations of conditional extreme value indices to marginal extreme value indices, and (3) tail dependence.

Suppose a multi-Gaussian process is observed at some set of sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center of mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means.

Consider the sample X1, X2, ..., XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F, and let Fu be their conditional excess distribution function F. We define the ordered sample by . Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F , and large u ,Fu is well approximated by the Generalized Pareto Distribution.The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds.The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.

Transferring knowledge from one domain to another is challenging due to a number of reasons. Since both conditional and marginal distribution of the training data and test data are non-identical, model trained in one domain, when directly applied to a different domain, is usually low in accuracy. For many applications with large feature sets, such as text document, sequence data, medical data, image data of different resolutions, etc. two domains usually do not contain exactly the same features, thus introducing large numbers of when considered over the union of features from both domains. In other words, its marginal distributions are at most overlapping. In the same time, these problems are usually high dimensional, such as, several thousands of features. Thus, the combination of high dimensionality and missing values make the relationship in conditional probabilities between two domains hard to measure and model. To address these challenges, we propose a framework that first brings the marginal distributions of two domains closer by filling up those missing values of disjoint features. Afterwards, it looks for those comparable sub-structures in the latent-space as mapped from the expanded feature vector, where both marginal and conditional distribution are similar. With these sub-structures in latent space, the proposed approach then find common concepts that are transferable across domains with high probability. During prediction, unlabeled instances are treated as queries, the mostly related labeled instances from out-domain are retrieved, and the classification is made by weighted voting using retrieved out-domain examples. We formally show that importing feature values across domains and latent semantic index can jointly make the distributions of two related domains easier to measure than in original feature space, the nearest neighbor method employed to retrieve related out domain examples is bounded in error when predicting in-domain examples. Software and datasets are available for download.

This paper explores how to exploit shape information to perform object class recognition. We use a sparse part-based model to describe object categories defined by shape. The sparseness allows the relative spatial relationship between parts to be described simply. It is possible, with this model, to highlight potential locations of the object and its parts in novel images. Subsequently these areas are examined by a more flexible shape model that measures if the image data provides evidence of the existence of boundary/connecting curves between connected hypothesized parts. From these measurements it is possible to construct a very simple cost function which indicates the presence or absence of the object class. The part-based model is designed to decouple variations due to affine warps and other forms of shape deformations. The latter are modeled probabilistically using conditional probability distributions which describe the linear dependencies between the location of a part and a subset of the other parts. These conditional distributions can then be exploited to search efficiently for the instances of the part model in novel images. Results are reported on experiments performed on the ETHZ shape classes database that features heavily cluttered images and large variations in scale.

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S3. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.

Three-dimensional modeling of a mineral deposit was conducted based on the samples collected from the surficial and deep parts of the deposit. One of the factors that affect the quality of data is the recovery of cores acquired during exploration drilling. As the core recovery reduces, the grade in core or drilling mud increases. Core loss can introduce unpredictable errors and negative or positive bias into the mineral resources estimation. Investigating the grade–recovery relationship in a bivariate space helps to detect the grade bias. In an ideal state, there is no correlation between grade and recovery. In case the grade–recovery relationship possesses a negative or positive correlation, the grade bias is likely to be due to core loss. In practice, however, because there are many data in boreholes sample database, the grade–recovery relationship is not properly determined on a simple scatter plot of grade versus core recovery. Application of conditional distributions in a bivariate space provides the necessary tool for investigating the simultaneous variations of grade and core recovery. Therefore, having converted the grade and core recovery data into the normal space, the conditional grade distribution was determined for various recoveries, and then, the conditional grades expectations were determined and plotted against the recovery to investigate the grade–recovery relationship. Application of the proposed method in three different deposits showed that in low recoveries, due to waste loss, the grade values were overestimated, whereas in high recoveries, due to ore loss, they were underestimated.

Adversarial domain adaptation is a powerful approach to transfer the knowledge of the label-rich source domain to the label-scarce target domain by mitigating domain shifts across distributions. Existing domain adaptation methods align either the marginal distribution with a single-domain discriminator or conditional distributions with multiple-domain discriminators. However, aligning both marginal (global) and conditional (local) distributions should be considered for domain adaptation. This paper proposes a novel adversarial distribution adaptation network (ADAN) to jointly reduce both the global and local distribution discrepancies between different domains for learning domain-invariant representations. ADAN utilizes a single-domain discriminator to adapt the global distribution between two domains, and source decision boundaries to align the local distributions between sub-domains. Furthermore, we extend our ADAN as improved ADAN (iADAN), in which we utilize a feature norm term to regularize the task-specific features to improve model generalization. Extensive experimental results show that our method outperforms other state-of-the-art domain adaptation methods on Office-Home and ImageCLEF-DA datasets.

Modality for scenario analysis and maximum likelihood allocation

A more flexible type of mixture autoregressive model, namely the Burr mixture autoregressive, BMAR model is studied in this article for modeling non linear time series. The model consists of a mixture of K autoregressive components with each conditional distribution of the component following a Burr distribution. The BMAR model enjoys some nice statistical properties which allow it to capture time series with: (1) unimodal or multimodal; (2) asymmetry or symmetry conditional distribution; (3) conditional heteroscedasticity; (4) cyclical or seasonal; and (5) conditional leptokurtic distribution. Sufficient and less restrictive conditions for the ergodicity of the BMAR model are derived and discussed. A more robust constrained optimization algorithm (EM – sequential quadratic programming method) is proposed for the non linear optimization problem. From the simulation studies carried out, the parameters estimation method showed satisfying results. The variance of the estimated parameters is also addressed with the missing information principle. Real datasets from two different fields of study are used to assess the performance of the BMAR model compared to other competing models. The comparison done in the empirical examples reveals the supremacy of the BMAR model in capturing the data behavior.

Modeling Hong Kong’s stock index with the Student t-mixture autoregressive model

A random variable Y 1 is said to be smaller than Y 2 in the increasing concave stochastic order if for all increasing concave functions for which the expected values exist, and smaller than Y 2 in the increasing convex order if for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case and for continuously distributed X.

ABSTRACTThis paper considers the computation of the conditional stationary distribution in Markov chains of level-dependent M/G/1-type, given that the level is not greater than a predefined threshold. This problem has been studied recently and a computational algorithm is proposed under the assumption that matrices representing downward jumps are nonsingular. We first show that this assumption can be eliminated in a general setting of Markov chains of level-dependent G/G/1-type. Next we develop a computational algorithm for the conditional stationary distribution in Markov chains of level-dependent M/G/1-type, by modifying the above-mentioned algorithm slightly. In principle, our algorithm is applicable to any Markov chain of level-dependent M/G/1-type, if the Markov chain is irreducible and positive-recurrent. Furthermore, as an input to the algorithm, we can set an error bound for the computed conditional distribution, which is a notable feature of our algorithm. Some numerical examples are also provided.

Bayesian modeling of joint and conditional distributions