Uniqueness in Finite Measurement

Unique finite conjoint measurement

A definition of binaural additivity is given in terms of the theory of simultaneous conjoint measurement. Additivity is then tested and verified by a conjoint measurement procedure. Methods for deriving psychophysical scales from such procedures are discussed, and the experimental scales are compared with the usual ratio scales for loudness, derived from extensive measurement such as magnitude estimation. The functions are in good agreement and it is concluded that binaural additivity of loudness holds for non‐zero stimulation of the ears.

Additive conjoint measurement with ordered categories

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe nonprobabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which μ(A ∪ B) = max(μ(A), μ(B)). In this paper, we show that possibility measures are, in some sense, universal approximators: for every e > 0, every non-additive measure which satisfies a certain reasonable boundedness property is equivalent to a measure which is e-close to a possibility measure. I. ADDITIVE MEASURES AND FUZZY (NON-ADDITIVE) MEASURES: BRIEF REMINDER Formulation of the problem. One of the main motivations behind fuzzy and other non-probabilistic uncertainty is that the traditional probability theory is sometimes not very adequate for describing uncertainty. From the mathematical viewpoint, probability theory is based on probability (additive) measures. To describe nonprobabilistic uncertainty, researchers therefore came up with a general notion of non-additive (fuzzy) measures, in particular, possibility measures. In this paper, we show that every fuzzy measure satisfying several reasonable properties is isomorphic to an “almost”possibility measure. In order to formulate our result, let us first briefly recall definitions and main properties of probability measures and of fuzzy measures. Definition 1. Let X be a set called a universal set. By an algebra of sets (or algebra, for short) A, we mean a non-empty class of subsets A ⊆ X which is closed under complement and union and intersection, i.e.: • if A ∈ A, then its complement −A also belongs to A; • if A ∈ A and B ∈ A, then A ∪B ∈ A; • if A ∈ A and B ∈ A, then A ∩B ∈ A. Definition 2. By an additive measure, we mean a function μ which maps each set A from some algebra of sets to a nonnegative number μ(A) ≥ 0 and for which μ(A ∪B) = μ(A) + μ(B) for every two sets A and B for which A ∩B = ∅. Comment. Example of additive measures include length of sets on a line, area of planar sets, volume of sets in 3-D space, and probability of different events. Let us recall the main properties of additive measures. The first property is that μ(∅) = 0. Indeed, since the class A is non-empty, it contains some set A. Since A is an algebra, with set A, it also contains sets −A and A ∩ −A = ∅. Since A ∩ ∅ = ∅, additivity implies μ(A) = μ(A) + μ(∅) and thus, μ(∅) = 0. Definition 3. A function μ(A) defined on sets is called monotonic if A ⊆ B implies μ(A) ≤ μ(B). Proposition 1. Every additive measure is monotonic. Proof. Indeed, if A,B ∈ A, then B − A = B ∩ (−A) ∈ A. Due to additivity, we have μ(B) = μ(A) + μ(B − A). Since μ(B − A) ≥ 0, this implies μ(A) ≤ μ(B). The statement is proven. Definition 4. A function μ(A) defined on an algebra of sets is called subadditive if for every two sets A and B, we have μ(A ∪B) ≤ μ(A) + μ(B). Proposition 2. Every additive measure is subadditive. Proof. Indeed, we have B − A ∈ A and, due to additivity, μ(A∪B) = μ(A)+μ(B−A). Since B−A ⊆ B, monotonicity implies that μ(B −A) ≤ μ(B) and thus, μ(A ∪B) ≤ μ(A) + μ(B). The statement is proven. Definition 5. Let X be a set called a universal set. By an σ-algebra A, we mean a non-empty class of subsets A ⊆ X which is closed under complement and countable union and intersection, i.e.: • if A ∈ A, then its complement −A also belongs to A; • if A1 ∈ A, . . . , An ∈ A, . . . , then ∪ i Ai ∈ A; • if A1 ∈ A, . . . , An ∈ A, . . . , then ∩

This paper considers a variety of combinatorial and number-theoretic questions that arise from considerations of uniqueness in the theory of measurement. Given $n + 1$ objects linearly ordered from worst (smallest) to best (largest), let $d_i $ stand for the difference between objects i and $i + 1$. Three types of assumptions that allow different inequality and equality comparisons between certain subsets of differences are considered. The three types of assumptions arise from problems concerning the uniqueness of finite algebraic difference measurement and finite subjective probability measurement. If the number of equality comparisons is sufficient to imply that the $d_i $ in $d = ( d_1 ,d_2 , \cdots ,d_n )$ are unique up to multiplication by a positive constant, then d is said to be unique. Results on the number of unique d’s and relationships among the components of such d’s are obtained for each of the three types.

“Parallel Training Considered Harmful?”: Comparing series-parallel and parallel feedforward network training

Conjoint measurement in utility theory with incomplete product sets

The axioms of additive conjoint measurement provide a means of testing the hypothesis that testing data can be placed onto a scale with equal-interval properties. However, the axioms are difficult to verify given that item responses may be subject to measurement error. A Bayesian method exists for imposing order restrictions from additive conjoint measurement while estimating the probability of a correct response. In this study an improved version of that methodology is evaluated via simulation. The approach is then applied to data from a reading assessment intentionally designed to support an equal-interval scaling.

Necessary and sufficient conditions are specified for a general theory of additive measurement that presumes very little set-theoretic structure. The theory is illustrated for numerical representations in extensive, conjoint, difference, threshold, expected utility, probability, ambiguity, and subset measurement. © 1992 John Wiley & Sons, Inc.

Scientific findings involve description, and description requires measurements on the dimensions descriptive of the phenomena described. The richer a description the more multidimensional is the abstract space in which that description in effect locates what is described. Many of the dimensions of human psychological phenomena, including those of psychotherapy, are naturally gradated only ordinally. So descriptions of these phenomena locate them in merely ordinal hyperspaces, which imposes severe constraints on data analysis for inducing or testing explanatory theory involving them. Therefore, it is important to be clear about what these constraints are and so what properly can be concluded on the basis of ordinal-scale multivariate data, which also provides a test for methods that are proposed to transform ordinal-scale data into ratio-scale data (e.g., classical test theory, item response theory, additive conjoint measurement), because such transformations must not violate these constraints and so distort descriptions of studied phenomena.

Measurement foundations for multiattribute psychophysical theories based on first order polynomials

A foundational area of mathematics, group theory has broad applications in many scientific domains. An integrated review of its use in cryptography and chemistry is presented in this work. Group theory simplifies structural predictions in chemistry by making it easier to analyze molecular symmetries, molecular orbitals, and spectroscopy. Group theory in cryptography provides the security underpinnings of contemporary encryption techniques, especially through discrete logarithm issues, homomorphisms, and permutation groups. We examine group-theoretical isomorphisms, homomorphism stability, and representations using both recent and classic literature, emphasizing their dual applicability in the information and biological sciences. Opportunities for interdisciplinary advances, such as improved chemical modeling and post-quantum cryptography protocols, are suggested by the synergy between these domains.

One of the persistent challenges in computer vision has been tracking objects under varying lighting conditions. In this paper we present a method for estimation of 3D motion of a rigid object from a monocular video sequence under arbitrary changes in the illumination conditions under which the video was captured. This is achieved by alternately estimating motion and illumination parameters using a generative model for integrating the effects of motion, illumination and structure within a unified mathematical framework. The motion is represented in terms of translation and rotation of the object centroid, and the illumination is represented using a spherical harmonics linear basis. The method does not assume any model for the variation of the illumination conditions - lighting can change slowly or drastically. For the multi-camera tracking scenario, we propose a new photometric constraint that is valid over the overlapping field of view between two cameras. This is similar in nature to the well-known epipolar constraint, except that it relates the photometric parameters, and can provide an additional constraint for illumination invariant multi-camera tracking. We demonstrate the effectiveness of our tracking algorithm on single and multi-camera video sequences under severe changes of lighting conditions.

This paper considers the scenario of the (1+1) evolutionary algorithm (EA) and randomized local search (RLS) with memory. Previously explored solutions are stored in memory until an improvement in fitness is obtained; then the stored information is discarded. This results in two new algorithms: (1+1) EA-m (with a raw list and hash table option) and RLS-m+ (and RLS-m if the function is a priori known to be unimodal). These two algorithms can be regarded as very simple forms of tabu search. Rigorous theoretical analysis of the expected time to find the globally optimal solutions for these algorithms is conducted for both unimodal and multimodal functions. A unified mathematical framework, involving the new concept of spatially invariant neighborhood, is proposed. Under this framework, both (1+1) EA with standard uniform mutation and RLS can be considered as particular instances and in the most general cases, all functions can be considered to be unimodal. Under this framework, it is found that for unimodal functions, the improvement by memory assistance is always positive but at most by one half. For multimodal functions, the improvement is significant; for functions with gaps and another hard function, the order of growth is reduced; for at least one example function, the order can change from exponential to polynomial. Empirical results, with a reasonable fitness evaluation time assumption, verify that (1+1) EA-m and RLS-m+ are superior to their conventional counterparts. Both new algorithms are promising for use in a memetic algorithm. In particular, RLS-m+ makes the previously impractical RLS practical, and surprisingly, does not require any extra memory in actual implementation.

Credit scoring models serve as pivotal instruments for lenders and financial institutions, facilitating the assessment of creditworthiness. Traditional models, while instrumental, grapple with challenges related to efficiency and subjectivity. The advent of machine learning heralds a transformative era, offering data-driven solutions that transcend these limitations. This research delves into a comprehensive analysis of various machine learning algorithms, emphasizing their mathematical underpinnings and their applicability in credit score classification. A comprehensive evaluation is conducted on a range of algorithms, including logistic regression, decision trees, support vector machines, and neural networks, using publicly available credit datasets. Within the research, a unified mathematical framework is introduced, which encompasses preprocessing techniques and critical algorithms such as Particle Swarm Optimization (PSO), the Light Gradient Boosting Model, and Extreme Gradient Boosting (XGB), among others. The focal point of the investigation is the LIME (Local Interpretable Model-agnostic Explanations) explainer. This study offers a comprehensive mathematical model using the LIME explainer, shedding light on its pivotal role in elucidating the intricacies of complex machine learning models. This study’s empirical findings offer compelling evidence of the efficacy of these methodologies in credit scoring, with notable accuracies of 88.84%, 78.30%, and 77.80% for the Australian, German, and South German datasets, respectively. In summation, this research not only amplifies the significance of machine learning in credit scoring but also accentuates the importance of mathematical modeling and the LIME explainer, providing a roadmap for practitioners to navigate the evolving landscape of credit assessment.