Statistical Decision Theory Research Articles

Statistical learning modeling based health indicator construction for machine condition monitoring

Machine condition monitoring aims to evaluate machine health conditions by analyzing machine vibration signals, which is helpful to make timely maintenance decisions and prevent unexpected accidents. Currently, constructions of virtual and physical health indicators (HIs) are commonly used methods for machine condition monitoring. However, most classic physical and virtual HIs lack inherent thresholds, robustness, monotonicity, and interpretability for machine condition monitoring. In this paper, a statistical learning modeling based HI construction method for machine condition monitoring is proposed to solve these problems. Firstly, a statistical decision theory is suggested to clearly describe a machine condition monitoring objective, and subsequently shapes of square envelope spectra are robustly modeled by using a parametric statistical model called a penalized B-spline approximation. Further, an interpretable HI named B-spline weight HI (BSWHI) as well as an inherent statistical threshold is accordingly constructed based on the Mahalanobis distance between B-spline weights of testing samples and a healthy sample. Experiments on bearing and gear run-to-failure datasets are studied to show that the proposed BSWHI and its inherent statistical threshold can effectively detect early machine faults and simultaneously provide monotonic degradation assessment trends. The proposed interpretable BSWHI has achieved a substantial improvement over existing classic HIs.

Targeted Double Scoring of Performance Tasks Using a Decision-Theoretic Approach.

Targeted double scoring, or, double scoring of only some (but not all) responses, is used to reduce the burden of scoring performance tasks for several mastery tests (Finkelman, Darby, & Nering, 2008). An approach based on statistical decision theory (e.g., Berger, 1989; Ferguson, 1967; Rudner, 2009) is suggested to evaluate and potentially improve upon the existing strategies in targeted double scoring for mastery tests. An application of the approach to data from an operational mastery test shows that a refinement of the currently used strategy would lead to substantial cost savings.

Probabilistic prediction for binary treatment choice: With focus on personalized medicine

This paper extends my research applying statistical decision theory to treatment choice with sample data, using maximum regret to evaluate the performance of statistical treatment rules. The specific new contribution is to study as-if optimization using estimates of illness probabilities in a class of medical decisions. Beyond its specifics, the paper sends a broad message. Statisticians and computer scientists have addressed conditional prediction for decision making in indirect ways, the former applying classical statistical theory and the latter measuring prediction accuracy in test samples. Neither approach is satisfactory. Statistical decision theory provides a coherent, generally applicable methodology.

Causation and decision: On Dawid’s “Decision theoretic foundation of statistical causality”

Abstract In a recent issue of this journal, Philip Dawid (2021) proposes a framework for causal inference that is based on statistical decision theory and that is, in many aspects, compatible with the familiar framework of causal graphs (e.g., Directed Acyclic Graphs (DAGs)). This editorial compares the methodological features of the two frameworks as well as their epistemological basis.

Experimental Cost of Information

We relate two main representations of the cost of acquiring information: a cost that depends on the experiment performed, as in statistical decision theory, and a cost that depends on the distribution of posterior beliefs, as in applications of rational inattention. We show that in many cases of interest, posterior-based costs are inconsistent with a primitive model of costly experimentation. The inconsistency is at the core of known limits to the application of rational inattention in games and, more broadly, in equilibrium analyses where beliefs are endogenous; we show that an experiment-based approach helps to understand and overcome these difficulties. (JEL D82, D83)

A response function that maps associative strengths to probabilities.

Bridging associative and normative theories of animal learning, I show that an associative system can behave as if performing probabilistic inference by using the function f(V) = 1 - e-cV to transform associative strengths (V) into response probabilities. For example, using this function, an associative system can respond normatively to a compound stimulus AB, given previous separate experiences with the components A and B. The CR probability formulae that result from the proposed function have a normative interpretation in terms of statistical decision theory. The formulae also suggest a normative interpretation of stimulus generalization as a heuristic to infer whether different stimuli are likely to convey redundant or independent information about reinforcement. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Tillyer Award Lecture: Visual search in noise and natural backgrounds

I will describe evidence for a theory of covert visual search developed within the framework of natural scene statistics and Bayesian statistical decision theory. The theory is unique in several ways: (1) it directly takes into account the statistical properties of natural images, (2) it takes into account the variation in neural processing with retinal location, as well as other known properties of the visual system, and hence contains almost no free parameters, and (3) it includes a principled attentional mechanism that efficiently allocates sensitivity gain across the visual field. This latter mechanism was discovered in experiments measuring covert search in white-noise backgrounds, where the target could appear anywhere within a large search area. In a separate experiment, target detectability (d') was measured across the visual field when the target location was cued/known. The shape of this "d' map" was consistent with the theory. The overall performance in the covert search task was also predicted quite well from this d' map, with no free parameters, assuming parallel unlimited-capacity processing. However, paradoxically, detection accuracy was low in the foveal region, even though it was predicted to be very high. We show that this "foveal neglect" is the expected consequence of efficiently allocating a fixed total attentional sensitivity gain across neurons in visual cortex, rather than across locations in visual space (the traditional assumption). Furthermore, the theory predicts the detailed pattern of covert search performance in the white-noise backgrounds. Finally, I will describe predictions of the theory for search in natural images.

Josephson-Based Scheme for the Detection of Microwave Photons

We propose a scheme for the detection of microwave-induced photons through a current-biased Josephson junction, from the point of view of the statistical decision theory. Our analysis is based on the numerical study of the zero-voltage lifetime distribution in response to a periodic train of pulses, that mimics the absorption of photons. The statistical properties of the detection are retrieved comparing the thermally induced transitions with the distribution of the switchings to the finite voltage state due to the joint action of thermal noise and of the incident pulses. The capability to discriminate the photon arrival can be quantified through the Kumar-Caroll index, which is a good indicator of the signal-to-noise ratio. The index can be exploited to identify the system parameters best suited for the detection of weak microwave photons.

Measure Transportation and Statistical Decision Theory

Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantiles, signs, and ranks is anything but obvious. Tentative definitions have been proposed in the literature but do not enjoy the basic properties (e.g., distribution-freeness of ranks, their independence with respect to the order statistic, their independence with respect to signs) they are expected to satisfy. Based on measure transportation ideas, new concepts of distribution and quantile functions, ranks, and signs have been proposed recently that, unlike previous attempts, do satisfy these properties. These ranks, signs, and quantiles have been used, quite successfully, in several inference problems and have triggered, in a short span of time, a number of applications: fully distribution-free testing for multiple-output regression, MANOVA, and VAR models; R-estimation for VARMA parameters; distribution-free testing for vector independence; multiple-output quantile regression; nonlinear independent component analysis; and so on.

Mathematical modelling of professional risk at Ukrainian metallurgical industry enterprises

Purpose: To develop a more advanced methodology, the application of which will provide an informational and computational and analytical basis for planning and implementing effective preventive measures aimed at minimizing occupational risks with limited resources, as well as in the absence of organizational and technical capabilities to create absolutely safe working conditions Design/methodology/approach: For the study, statistical data were used that obtained from enterprises of the metallurgical industry of Ukraine. Research methods: analysis and generalization of known scientific results, methods of statistical analysis, mathematical modelling, expert assessments and decision theory. Findings: The results of experimental studies have confirmed the possibility of an objective assessment of various options for the OSH management strategy, which allows justifying the allocation of funds for OSH in the required amounts. It is shown that professional risk management strategies are characterized by different efficiency in the use of available financial resources, and the most effective strategy is one that allows you to minimize the level of risk (in comparison with other strategies) with the same amount of funding. Research limitations/implications: The study focuses on enterprises of the metallurgical industry in Ukraine. Practical implications: The application of the developed mathematical models demonstrates the effectiveness of financing certain preventive and protective measures, and stimulates the head to ensure industrial safety. Originality/value: The developed mathematical models allow justifying the allocation of funds for OSH in the required amounts.

Charles Stein and invariance: Beginning with the Hunt–Stein theorem

When statistical decision theory was emerging as a promising new paradigm, Charles Stein was to play a major role in the development of minimax theory for invariant statistical problems. In some of his earliest work with Gil Hunt, he set out to prove that, in problems where invariant procedures have constant risk, any best invariant test would be minimax among all tests. Although finding it not quite true in general, this led to the legendary Hunt–Stein theorem, which established the result under restrictive conditions on the underlying group of transformations. In decision problems invariant under such suitable groups, an overall minimax test was guaranteed to reside within the class of invariant procedures where it would typically be much easier to find. But when it did not seem possible to establish this result for invariance under the full linear group, he instead turned to prove its impossibility with counterexamples such as the nonminimaxity of the usual sample covariance estimator where the full linear group was just too big for the Hunt–Stein theorem to apply. Further explorations of invariance such as the sometimes problematic inference under a fiducial distribution, or the characterization of a best invariant procedure as a formal Bayes procedure under a right Haar prior, are further examples of the far reaching influence of Stein’s contributions to invariance theory.

The integrals and integral transformations connected with the joint vector Gaussian distribution

In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.

Detectability assessment of an x-ray imaging system using the nodes in a wavelet packet decomposition of a star-bar object.

Using linear transformations of the acquired data can expand the study of detectability in an imaging system. From one image, an appropriate transformation will produce a set of signals with different contrast and different frequency contents. In this work this strategy is explored to present a task-based test for the detectability of an x-ray imaging system. Images of a new star-bar phantom are acquired with different entrance air KERMA and with different beam qualities. Then, after a wavelet packet is applied to both input and output of the system, statistical decision theory is applied to determine detectability of the different images or nodes resulting from the transformation. An ideal Bayesian observer (IBO) is applied to the data in the spatial domain to perform ROC analysis and to determine a detectability index for each of the nodes. In addition, image quality is characterized in terms of noise equivalent quanta (NEQ) and a 5mm nodule detection task is performed. AUC maps resulting from the analysis show the area under the ROC curve over the whole 2D frequency space for the different doses and beam qualities. Also, AUC curves, obtained by radially averaging the AUC maps, allow comparing detectability of the different techniques as a function of the frequency in a single figure. The results obtained show differences between images acquired with different doses for each of the beam qualities analyzed. Classifying image quality by means of detectability indexes agrees with that of the AUC curves and the nodule detection task but differs from the NEQ for the low air KERMA images. Combining a star-bar as test object, a wavelet packet as linear transformation and ROC analysis provides an appropriate task-based test for detectability performance of an imaging system. The test presented in this work produces maps and curves quantifying system detectability as a function of the frequency characterizing the signal to detect and allows calculating detectability differences between different acquisition techniques.

Тotal probability formula for vector Gaussian distributions

The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.

Statistical Decision Properties of Imprecise Trials Assessing Coronavirus Disease 2019 (COVID-19) Drugs

ObjectivesResearchers studying treatment of coronavirus disease 2019 (COVID-19) have reported findings of randomized trials comparing standard care with care augmented by experimental drugs. Many trials have small sample sizes, so estimates of treatment effects are imprecise. Hence, clinicians may find it difficult to decide when to treat patients with experimental drugs. A conventional practice when comparing standard care and an innovation is to choose the innovation only if the estimated treatment effect is positive and statistically significant. This practice defers to standard care as the status quo. We study treatment choice from the perspective of statistical decision theory, which considers treatment options symmetrically when assessing trial findings. MethodsWe use the concept of near-optimality to evaluate criteria for treatment choice. This concept jointly considers the probability and magnitude of decision errors. An appealing criterion from this perspective is the empirical success rule, which chooses the treatment with the highest observed average patient outcome in the trial. ResultsConsidering the design of some COVID-19 trials, we show that the empirical success rule yields treatment choices that are much closer to optimal than those generated by prevailing decision criteria based on hypothesis tests. ConclusionUsing trial findings to make near-optimal treatment choices rather than perform hypothesis tests should improve clinical decision making.

Minimax Efficient Random Experimental Design Strategies With Application to Model-Robust Design for Prediction

In game theory and statistical decision theory, a random (i.e., mixed) decision strategy often outperforms a deterministic strategy in minimax expected loss. As experimental design can be viewed as a game pitting the Statistician against Nature, the use of a random strategy to choose a design will often be beneficial. However, the topic of minimax-efficient random strategies for design selection is mostly unexplored, with consideration limited to Fisherian randomization of the allocation of a predetermined set of treatments to experimental units. Here, for the first time, novel and more flexible random design strategies are shown to have better properties than their deterministic counterparts in linear model estimation and prediction, including stronger bounds on both the expectation and survivor function of the loss distribution. Design strategies are considered for three important statistical problems: (i) parameter estimation in linear potential outcomes models, (ii) point prediction from a correct linear model, and (iii) global prediction from a linear model taking into account an L 2-class of possible model discrepancy functions. The new random design strategies proposed for (iii) give a finite bound on the expected loss, a dramatic improvement compared to existing deterministic exact designs for which the expected loss is unbounded. Supplementary materials for this article are available online.

Econometrics for Decision Making: Building Foundations Sketched by Haavelmo and Wald

Haavelmo (1944) proposed a probabilistic structure for econometric modeling, aiming to make econometrics useful for decision making. His fundamental contribution has become thoroughly embedded in econometric research, yet it could not answer all the deep issues that the author raised. Notably, Haavelmo struggled to formalize the implications for decision making of the fact that models can at most approximate actuality. In the same period, Wald (1939, 1945) initiated his own seminal development of statistical decision theory. Haavelmo favorably cited Wald, but econometrics did not embrace statistical decision theory. Instead, it focused on study of identification, estimation, and statistical inference. This paper proposes use of statistical decision theory to evaluate the performance of models in decision making. I consider the common practice ofas‐if optimization: specification of a model, point estimation of its parameters, and use of the point estimate to make a decision that would be optimal if the estimate were accurate. A central theme is that one should evaluate as‐if optimization or any other model‐based decision rule by its performance across the state space, listing all states of nature that one believes feasible, not across the model space. I apply the theme to prediction and treatment choice. Statistical decision theory is conceptually simple, but application is often challenging. Advancing computation is the primary task to complete the foundations sketched by Haavelmo and Wald.

ЗАСТОСУВАННЯ ЗМІШАНИХ СТРАТЕГІЙ ПРИ УПРАВЛІННІ МІЖНАРОДНИМИ ВАНТАЖНИМИ ПЕРЕВЕЗЕННЯМИ

This article considers the use of the mathematical apparatus of the theory of games with nature to model such a socio-economic phenomenon as the process of international freight transportation when performing delivery by own and borrowed vehicles on Kyiv-Warsaw route. Calculations are made with the use of existing costs and tariffs for the selected direction of transportation. The object of research is the process of international freight transportation management. The purpose of the work is to study the peculiarities of decision-making with the use of mixed strategies. The research method is the theory of statistical decisions, namely the theory of decision-making in the conditions of playing with nature. The player, i.e. the person who will make the decision, is the company that performs the carriage of goods by road in international traffic and is fully responsible for organizing the process of delivery of goods by road on the route. The situation is modeled using variants of various demand for transportation. This demand acts as the uncertainty of different variations of nature's strategies. Recommendations are given for the application of four decision making criteria: Wald, Savage, Laplace and Hurwitz criteria. Appropriate calculations were performed and the obtained results were commented on, because each criterion offers a new solution to a given situation. Calculations of application of mixed strategies are presented in the Excel spreadsheet. KEY WORDS: GAME THEORY, DECISION MAKING CRITERIA, UNCERTAINTY CONDITIONS, RISK, STRATEGY, INTERNATIONAL FREIGHT TRANSPORTATION

A Novel Unified Computational Approach to Constructing Shortest-Length or Equal Tails Confidence Intervals in Terms of Pivotal Quantities and Quantile Functions

A confidence interval is a range of values that gives the user a sense of how precisely a statistic estimates a parameter. In the present paper, a new simple computational method is proposed for simultaneous constructing and comparing confidence intervals of shortest length and equal tails in order to make efficient decisions under parametric uncertainty. This method represents a simple and computationally attractive numerical technique for finding the shortest-length and equal tails confidence intervals using pivotal quantities, which are developed from either maximum likelihood estimates or sufficient statistics. In statistics, a pivotal quantity (or pivot) is a function of observations and unobservable parameters such that the function’s probability distribution does not depend on the unknown parameters (including nuisance parameters). Finding a pivotal quantity is not discussed, but the choice a “good” pivotal quantity is essential for the resulting confidence interval to be useful. The unified computational technique yields intervals in several situations which have previously required separate analyses using more advanced techniques and tables for numerical solutions. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate nuisance parameters from the problem via the technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE&APQ). It should be noted that the well-known classical approach to constructing confidence intervals of the shortest length considers at least three versions of possible solutions and is in need of information about the form of the probability distribution of pivotal quantity in order to determine an adequate version of the correct solution. The proposed method does not need such information. It receives this information through the quantiles of the probability distribution of pivotal quantity. Therefore, the proposed method automatically recognizes an adequate version of the correct solution. To illustrate this method, numerical examples are given. In detail, the Pareto distribution is discussed.

Comprehensive evaluation of combining CFD simulation and entropy weight to predict natural ventilation strategy in a sliding cover solar greenhouse

This study proposed a comprehensive evaluation system to incorporate the contribution of both numerical simulation and statistical decision theory in ventilation performance assessment. A high-resolution model based on the finite volume approach was established to analyze the influence of rotation angles (i.e., side vent flip angle and roof vent flip angle) of the rack-and-pinion ventilated structure on the greenhouse microclimate. The water circulating system and tomato seeding canopies were considered. Heat removal efficiency and mean age of air were employed as quantitative attributes to reflect the internal thermal environment and the airflow organization in the sliding cover solar greenhouse. The simulation model was verified with the temperature profile measured and the average relative error was 1.74%. The results demonstrate that the rotating angles of ventilation schemes have a substantial impact on the microclimate and inhomogeneity of the tomato seeding canopies. The results suggest the average velocity and its inhomogeneity are the crucial predictors, and their entropy weight values are 0.231 and 0.218, respectively. The relative degree of membership of the side vent flip angle of 45° is 36% and 97% higher than that of the side vent flip angle of 35° and the side vent flip angle of 25°, respectively. This study can provide a reference to evaluate the ventilated strategies of the sliding cover solar greenhouse for the regional and central government. Keywords: sliding cover solar greenhouse, natural ventilation, CFD simulation, entropy weight, comprehensive evaluation DOI: 10.25165/j.ijabe.20211406.6793 Citation: Li H, Ji D, Hu X, Xie T, Song W T, Tian S B. Comprehensive evaluation of combining CFD simulation and entropy weight to predict natural ventilation strategy in a sliding cover solar greenhouse. Int J Agric & Biol Eng, 2021; 14(6): 213–221.