Approximate Probability Research Articles

ESPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems

Classification problems in the small data regime (with small data statistic T and relatively large feature space dimension D) impose challenges for the common machine learning (ML) and deep learning (DL) tools. The standard learning methods from these areas tend to show a lack of robustness when applied to data sets with significantly fewer data points than dimensions and quickly reach the overfitting bound, thus leading to poor performance beyond the training set. To tackle this issue, we propose eSPA+, a significant extension of the recently formulated entropy-optimal scalable probabilistic approximation algorithm (eSPA). Specifically, we propose to change the order of the optimization steps and replace the most computationally expensive subproblem of eSPA with its closed-form solution. We prove that with these two enhancements, eSPA+ moves from the polynomial to the linear class of complexity scaling algorithms. On several small data learning benchmarks, we show that the eSPA+ algorithm achieves a many-fold speed-up with respect to eSPA and even better performance results when compared to a wide array of ML and DL tools. In particular, we benchmark eSPA+ against the standard eSPA and the main classes of common learning algorithms in the small data regime: various forms of support vector machines, random forests, and long short-term memory algorithms. In all the considered applications, the common learning methods and eSPA are markedly outperformed by eSPA+, which achieves significantly higher prediction accuracy with an orders-of-magnitude lower computational cost.

Predicting Potential Drug-Drug-Gene Interactions in a Population of Individuals Utilizing a Community-Based Pharmacy

Adverse drug reactions (ADRs) are associated with increased morbidity, mortality, and resource utilization. Drug interactions (DDIs) are among the most common causes of ADRs, and estimates have cited that up to 22% of patients take interacting medications. DDIs are often due to the propensity for agents to induce or inhibit enzymes responsible for the metabolism of concomitantly administered drugs. However, this phenomenon is further complicated by genetic variants of such enzymes. The aim of this study is to quantify and describe potential drug-drug, drug-gene, and drug-drug-gene interactions in a community-based patient population. A regional pharmacy with retail outlets in Arkansas provided deidentified prescription data from March 2020 for 4761 individuals. Drug-drug and drug-drug-gene interactions were assessed utilizing the logic incorporated into GenMedPro, a commercially available digital gene-drug interaction software program that incorporates variants of 9 pharmacokinetic (PK) and 2 pharmacodynamic (PD) genes to evaluate DDIs and drug-gene interactions. The data were first assessed for composite drug-drug interaction risk, and each individual was stratified to a risk category using the logic incorporated in GenMedPro. To calculate the frequency of potential drug-gene interactions, genotypes were imputed and allocated to the cohort according to each gene's frequency in the general population. Potential genotypes were randomly allocated to the population 100 times in a Monte Carlo simulation. Potential drug-drug, gene-drug, or gene-drug-drug interaction risk was characterized as minor, moderate, or major. Based on prescription data only, the probability of a DDI of any impact (mild, moderate, or major) was 26% [95% CI: 0.248-0.272] in the population. This probability increased to 49.6% [95% CI: 0.484-0.507] when simulated genetic polymorphisms were additionally assessed. When assessing only major impact interactions, there was a 7.8% [95% CI: 0.070-0.085] probability of drug-drug interactions and 10.1% [95% CI: 0.095-0.108] probability with the addition of genetic contributions. The probability of drug-drug-gene interactions of any impact was correlated with the number of prescribed medications, with an approximate probability of 77%, 85%, and 94% in patients prescribed 5, 6, or 7+ medications, respectively. When stratified by specific drug class, antidepressants (19.5%), antiemetics (21.4%), analgesics (16%), antipsychotics (15.6%), and antiparasitics (49.7%) had the highest probability of major drug-drug-gene interaction. In a community-based population of outpatients, the probability of drug-drug interaction risk increases when genetic polymorphisms are attributed to the population. These data suggest that pharmacogenetic testing may be useful in predicting drug interactions, drug-gene interactions, and severity of interactions when proactively evaluating patient medication profiles. Genomind, Inc.

Updated Italian cohort data continues to confirm lack of mesothelioma risk in pooled cohort of international cosmetic talc miners and millers

Objectives To assess potential mesothelioma risk following inhalation of cosmetic talc, we updated previous iterations of a pooled cohort analysis, post-study statistical power analysis, and confidence interval function analysis for a pooled cohort of international cosmetic talc miners/millers given new Italian cohort data. Methods Five cohorts of cosmetic talc miners/millers were pooled. Expected numbers of mesotheliomas for each cohort were reported by the original authors. We based our post-study statistical power analysis on an a priori one-sided significance level of 0.05, and exact Poisson and approximate distribution probabilities. To evaluate the confidence interval function for the observed pooled mesothelioma standardized mortality ratios (SMRs), we calculated the probability for the upper 100(1–2α)% confidence limit that equals various SMRs of interest. Results The pooled cohorts generated a total observation time of 135,524.38 person-years. Overall, 4.14 mesotheliomas were expected (mid-value estimate), though only one case of mesothelioma has been confirmed in the pooled cohort to date. We calculated 71% and 87% post-study power to detect a 2.5-fold or greater and a 3.0-fold or greater increase in mesothelioma, respectively. Our complimentary confidence interval function analysis demonstrated that the probability that the true mesothelioma SMR for the pooled cohort was at or above 2.0 or at or above 3.0 was 0.00235 and 0.00005, respectively. Conclusions Based on the updated results of our various analyses, the current epidemiological evidence from cosmetic talc miner/miller cohort studies continues to not support the hypothesis that the inhalation of cosmetic talc is associated with an increased risk of mesothelioma.

Second moments of work and heat for a single-particle stochastic heat engine in a breathing harmonic potential

Abstract We consider a simple model of a stochastic heat engine, which consists of a single Brownian particle moving in a one-dimensional periodically breathing harmonic potential. The overdamped limit is assumed. Expressions for the second moments (variances and covariances) of heat and work are obtained in the form of integrals, whose integrands contain functions satisfying certain differential equations. The results in the quasi-static limit are simple functions of the temperatures of hot and cold thermal baths. The coefficient of variation of the work is suggested to give an approximate probability for the work to exceed a given threshold. During the derivation, we get the expression of the cumulant-generating function.

Multiresolution approximation and consistent estimation of a multivariate density function

In this paper we extend multiresolution analysis structures on to approximate multivariate probability density functions. We propose a consistent estimator for a multivariate multiresolution approximation (MMR) of a multivariate pdf. And we also develop an algorithm to estimate the MMR pdf that behaves well when handling big data. This algorithm performs better, in terms of running time, than traditional optimization algorithms. For large samples, the estimations are as good as those obtained by maximum likelihood. Numerical results are provided to illustrate the method.

Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

An integrated cluster-wise significance measure for fMRI analysis.

Cluster‐wise inference is widely used in fMRI analysis. The cluster‐level statistic is often obtained by counting the number of intra‐cluster voxels which surpass a voxel‐level statistical significance threshold. This measure can be sub‐optimal regarding the power and false‐positive error rate because the suprathreshold voxel count neglects the voxel‐wise significance levels and ignores the dependence between voxels. This article aims to provide a new Integrated Cluster‐wise significance Measure (ICM) for cluster‐level significance determination in cluster‐wise fMRI analysis by integrating cluster extent, voxel‐level significance (e.g., p values), and activation dependence between within‐cluster voxels. We develop a computationally efficient strategy for ICM based on probabilistic approximation theories. Consequently, the computational load for ICM‐based cluster‐wise inference (e.g., permutation tests) is affordable. We validate the proposed method via extensive simulations and then apply it to two fMRI data sets. The results demonstrate that ICM can improve the power with well‐controlled family‐wise error (FWE).

Parameter estimation of MIMO FSO systems using saddlepoint approximation

In this paper, the challenges of parameter estimation for multiple-input multiple-out (MIMO) free-space optical (FSO) communication systems employing equal gain combining diversity techniques over the Gamma–Gamma turbulence channels with pointing errors are addressed. According to the goodness-of-fit statistical tests results previously reported by Kolmogorov-Smirnov, the approximate probability density function obtained by a saddlepoint approximation (SAP) method provide a highly efficient approximation of the actual channel models over a wide range of channel conditions. To reveal the importance of the SAP approximation method, we propose a maximum likelihood estimation algorithm to estimate its parameters. The accuracy of the proposed estimator is investigated by using the mean square error (MSE) and normalized mean square error (NMSE); they exhibit satisfactory performance in both noiseless and noisy environments. The bit error rate (BER) performance for MIMO FSO systems with an SAP estimator is also discussed.

Covert Wireless Communication With Noise Uncertainty in Space-Air-Ground Integrated Vehicular Networks

In this paper, we propose a covert wireless uplink transmission strategy in space-air-ground integrated vehicular networks, where the source vehicle transmits its own message over the channel that being used by the host communication system, to avoid being detected by the warden. It is obvious that the data transmission efficiency of the covert communication system is limited due to the co-channel interference. To improve the data transmission efficiency, we consider that the covert communication system adopts improper Gaussian signaling (IGS). We formulate a joint transmit power and IGS factor optimization problem to minimize the outage probability of the covert communication system. The minimum error detection probability of the warden is first analyzed with noise uncertainty, which is used to measure the system covertness. Under the constraints of the quality of service (QoS) of host communication system and the covertness requirement, the optimal transmit power is first derived with proper Gaussian signaling (PGS) scheme. Then, with the approximate outage probability derived under IGS scheme, the optimization problem is solved by jointly designing the transmit power and IGS factor. Finally, we provide extensive numerical results to validate the proposed covert transmission strategy, and demonstrate that the IGS scheme is beneficial in improving the data transmission efficiency in terms of outage probability compared to PGS scheme.

Global and saturated probabilistic approximations based on generalized maximal consistent blocks

Abstract In this paper incomplete data sets, or data sets with missing attribute values, have three interpretations, lost values, attribute-concept values and ‘do not care’ conditions. Additionally, the process of data mining is based on two types of probabilistic approximations, global and saturated. We present results of experiments on mining incomplete data sets using six approaches, combining three interpretations of missing attribute values with two types of probabilistic approximations. We compare our six approaches, using the error rate computed as a result of ten-fold cross validation as a criterion of quality. We show that for some data sets the error rate is significantly smaller (5% level of significance) for lost values, for some data sets the smaller error rate is associated with attribute-concept values, and sometimes with ‘do not care’ conditions. Again, for some approaches the error rate is significantly smaller for saturated probabilistic approximations than for global probabilistic approximations, while for some approaches it is the other way around. Thus, for an incomplete data set, the best approach to data mining should be chosen by trying all six approaches.

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Three-way decision based on confidence level change in rough set

Three-way decision (3WD) and rough set are two influential theories of study knowledge discovery and uncertain artificial intelligence. A central notion of 3WD is a tri-level thinking paradigm consisting of trisecting, acting, and outcome (i.e., TAO model). As is well known, movement-based on three-way decision (M-3WD) and change-based TAO model, mainly started from the perspective of effectiveness measure, are two outcome evolution studies about the three-way decision, which could lead to some limitations in application. This paper builds a change-based three-way decision (C-3WD) based on confidence level, and an application to rough set is also discussed. Furthermore, the (α,β)-approximate probability regions of rough set are re-decided by the change model, and a medical decision example is introduced to explain how to make C-3WD in the classification of rough set. By comparing the effectiveness of the traditional three-way decision method with ours, it is again verified from the two aspects of cost and earns and concludes that the model in this paper is more suitable for the decision process that includes trisecting and acting. Some experiments on various datasets to demonstrate the effectiveness of our methods.

A suggestion for the quantification of precise and bounded probability to quantify epistemic uncertainty in scientific assessments

An honest communication of uncertainty about quantities of interest enhances transparency in scientific assessments. To support this communication, risk assessors should choose appropriate ways to evaluate and characterize epistemic uncertainty. A full treatment of uncertainty requires methods that distinguish aleatory from epistemic uncertainty. Quantitative expressions for epistemic uncertainty are advantageous in scientific assessments because they are nonambiguous and enable individual uncertainties to be characterized and combined in a systematic way. Since 2019, the European Food Safety Authority (EFSA) recommends assessors to express epistemic uncertainty in conclusions of scientific assessments quantitatively by subjective probability. A subjective probability can be used to represent an expert judgment, which may or may not be updated using Bayes's rule to integrate evidence available for the assessment and could be either precise or approximate. Approximate (or bounded) probabilities may be enough for decision making and allow experts to reach agreement on certainty when they struggle to specify precise subjective probabilities. The difference between the lower and upper bound on a subjective probability can also be used to reflect someone's strength of knowledge. In this article, we demonstrate how to quantify uncertainty by bounded probability, and explicitly distinguish between epistemic and aleatory uncertainty, by means of robust Bayesian analysis, including standard Bayesian analysis through precise probability as a special case. For illustration, the two analyses are applied to an intake assessment.

A New Dynamic Substitution Box for Data Security Using an Innovative Chaotic Map

As the motivations and capabilities of threat actors continue to evolve, providing data security has become more important than ever. For this purpose, different ciphers using various techniques are being developed. Currently, chaotic maps are designed and applied in the development of these ciphers. Modern ciphers utilize a substitution box (S-Box) as a core module to provide data security. In this article, an innovative chaotic map is suggested for the design of new and dynamic S-Box. Criteria like Bijectiveness, Nonlinearity (NL), Strict Avalanche Criterion (SAC), Bit Independence Criterion (BIC), Linear Approximation Probability (LP), and Differential Approximation Probability (DP) are used to critically analyze and evaluate the proposed S-Box performance against various attacks. The cryptanalytic strength of the proposed S-Box is equated with freshly designed S-Boxes for its customization in real-life security applications. The comparative analysis gratifies the true potential of the proposed S-Box for its solicitation in data security domain.

Lp CONVERGENCE OF APPROXIMATE MAPS AND PROBABILITY DENSITIES FOR FORWARD AND INVERSE PROBLEMS IN UNCERTAINTY QUANTIFICATION

Lp CONVERGENCE OF APPROXIMATE MAPS AND PROBABILITY DENSITIES FOR FORWARD AND INVERSE PROBLEMS IN UNCERTAINTY QUANTIFICATION

Unsupervised Bayesian method for colour matching of product packaging

There are some problems in the traditional colour matching methods of product packaging, such as high accuracy of colour matching and long consumption time. This paper proposes a colour matching method of product packaging based on unsupervised Bayes. First, the distance threshold between different colours of product packaging was obtained. Second, the colour features were extracted by sparse representation method, and the colour features were preprocessed by fuzzy c-means clustering. Third, the colour of product outer packaging is sorted according to the proportion to realise the quantification of product outer packaging colour. Then, the colour matching parameters of product packaging are determined by Dirichlet distribution in unsupervised Bayesian. Finally, the optimal colour matching scheme is determined by introducing multiple beta functions. The importance weight of matching scheme is modified by approximate posterior probability to complete the matching. The results show that the minimum error of the proposed method is about 0.5%.

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

We study the performance -- and specifically the rate at which the error probability converges to zero -- of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say $\sim \exp\left(-n\,I + o(n) \right)$, where $n$ is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and $I$ is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and, consequently, the related error rate $I$, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for $n$ large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of $n$. Furthermore, approximate error probability curves $\sim \zeta_n \exp\left(-n\,I \right)$ are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where $\zeta_n$ represents the most representative sub-exponential terms of the error probabilities.

Construction of Key-dependent S-box for Secure Cloud Storage

As cloud storage systems have developed and been applied in complex environments, their data security has become more prevalent in recent years. The issue has been approached through many models. Data is encrypted and stored in these models. One of the most widely used encryption methods is the Advanced Encryption Standard (AES). In AES, the Substitution box(S-box) is playing a significant part in imparting the job of confusion. The security of the entire cryptosystem depends on its nonlinearity. In this work, a robust and secure S-box is constructed using a novel method, i.e., fingerprint features-based permutation function. Two stages are considered to construct a strong S-box. Firstly, random numbers are generated from the fingerprint features such as bifurcation and ridge endings of the user transmitting data. Subsequently, the permutation function is adapted on the random numbers (developed in the first stage) to augment the strength of the S-box. National Institute of Standards and Technology (NIST) STS 800-22 test suite is considered to evaluate the randomness of the enhanced fingerprint-based S-box. Also, the robustness of the constructed S-box is tested using cryptographical properties, namely Strict Avalanche Criterion (SAC), Nonlinearity (NL), Differential Approximation (DA) probability and output Bits Independence Criterion (BIC). Later, the cryptographical properties of the proposed S-box are compared with several existing S-boxes. After analyzing the characteristics of the proposed scheme, it is revealed that the newly constructed S-box is powerful, robust, and safe against linear and differential assaults.

A Strong Hybrid S-Box Scheme Based on Chaos, 2D Cellular Automata and Algebraic Structure

Substitution-boxes are the main deciding components in symmetric-key cryptosystems for resisting many cryptanalytic attacks. It has been a challenging task for the designers to construct strong S-box which satisfies multiple cryptographic properties simultaneously. A number of S-box studies have been investigated in literature; but, the generated S-box found to exhibit one single property with good score. This paper proposes a novel creation of S-boxes which possess excellent scores of multiple cryptographic properties instead of only one property. The suggested hybrid S-box method explores the science of two-dimensional cellular automata theory, discrete chaotic maps, and algebraic group structure. The proposed anticipated 8×8 S-box holds excellent security performance features such as: minimum nonlinearity as high as 110, no fixed points, satisfaction of strict avalanche and bits independence criterions, differential uniformity as low as 6, linear approximation probability as low as 0.0703, and auto-correlation function (absolute indicator) of 40. The performance comparison indicates the proposed S-box has superior features, greater inherent security and robustness strength than many available state of the art S-box methods.

A Cryptographic Scheme for Construction of Substitution Boxes Using Quantic Fractional Transformation

In the new era of cryptography, Substitution Boxes (S-Boxes) are very important to raise confusion in cipher text and the security of encryption directly depends on the algebraic strength of S-box. To avoid a hacker attack, researchers are focusing on creating dynamic S-boxes that are much stronger. The dominating concept for developing S-boxes is linear fractional transformation. In this article, we proposed a novel technique to generate cryptographically strong S-box by using fractional transformation based on finite field. The substitute box is constructed in two phases. Firstly, general form of dynamic fractional transformation designed which work for odd exponents in the range [1-255]. The S-box is then constructed using quantic fractional transformation as an example. Secondly, in order to increase the unpredictability of proposed S-box, we use the symmetric group’s S256 permutation. The usefulness of the constructed S-box was also tested using several criteria such as nonlinearity, differential uniformity, strict avalanche criteria, linear approximation probability and bit independence criteria. To assess the reliability of S-box, its performance outcomes are compared to those of previously developed S-boxes. Furthermore, we utilized the suggested S-Box to the image encryption approach. Then, to determine the effectiveness of the encryption scheme, use several tests such as contrast, correlation, homogeneity, entropy, and energy. We have compared our results with different algorithms which ensured that the proposed strategy for ciphered image is excellent.