Ordinal Problems Research Articles

Finger counting to relieve working memory in children with developmental coordination disorder: Insights from behavioral and three-dimensional motion analyses

A limited number of studies have attempted to understand how motor deficits affect numerical abilities in children with developmental coordination disorder (DCD). The purpose of this study was to explore the functionality of finger-counting (FC) in children with DCD. The participants, 15 children with DCD and 15 typically developing (TD) children matched on school level and fluid reasoning abilities, were asked to use FC to solve an ordinal task with high working memory (WM) load. Behavioral measures supplemented with biomechanical measures, from three-dimensional motion analysis synchronized to a voice recording were used to assess children’s performance and FC functionality (total duration, inter-finger [IF] transition, IF variance, finger/voice synchronization, and automatization of FC movements). Children with DCD were less accurate than TD children in using FC to solve ordinal problems with high WM load. This group difference could not be accounted for by poor FC skills given that FC movement turned out to be as functional in children with DCD as in their TD peers. When added to the model as a covariate, WM captured a greater proportion of intergroup variability than manual dexterity, further suggesting that their difficulties would be better accounted for by limited WM resources than by fine motor skills.

Generalised triangular distributions for ordinal deep learning: Novel proposal and optimisation

Deep learning techniques for ordinal classification have recently gained significant attention. Predicting an ordinal variable, that is, a variable that demonstrates a natural relationship between categories, is of relevance for a number of real-world problems in various fields of knowledge. For example, a medical diagnosis can occur at different stages of the disease. Applying standard classifiers to ordered labels can lead to errors in distant categories, when errors in an ordinal problem ideally tend to be produced in adjacent classes because of their similarity. To address this issue, we propose a soft labelling approach based on generalised triangular distributions, which are asymmetric and different for each class. The parameters of these distributions are determined using a metaheuristic and are specifically adapted to the given problem. Moreover, this approach enables the model to avoid errors in distant classes (e.g. classifying a patient with a severe disease as healthy). A comprehensive comparison was performed using eight datasets and five performance metrics. The main advantage of the proposed soft-labelling approach is that it adapts the distributions to each problem, resulting in greater flexibility and better performance. The results and statistical analysis show that the proposed methodology significantly outperforms all other methods.

Metacognitive prompts and numerical ordinality in solving word problems: An eye-tracking study.

The ability to translate concrete manipulatives into abstract mathematical formulas can aid in the solving of mathematical word problems among students, and metacognitive prompts play a significant role in enhancing this process. Based on the concept of semantic congruence, we explored the effects of metacognitive prompts and numerical ordinality on information searching and cognitive processing, throughout the process of solving mathematical word problems among primary school students in China. Participants included 73 primary school students (38 boys and 35 girls) with normal or corrected visual acuity. This study was based on a 2 (prompt information: no-prompt, metacognitive-prompt) × 2 (number attribute: cardinal number, ordinal number) mixed experimental design. We analysed multiple eye-movement indices, such as fixation duration, saccadic amplitude, and pupil size, since they pertained to the areas of interest. When solving both types of problems, pupil sizes were significantly smaller under the metacognitive-prompt condition compared with the no-prompt condition, and shorter dwell time for specific sentences, conditional on metacognitive prompts, indicated the optimization of the presented algorithm. Additionally, the levels of fixation durations and saccadic amplitudes were significantly higher when solving ordinal number word problems compared with solving ordinal number problems, indicating that primary school students were less efficient in reading and faced increased levels of difficulty when solving ordinal number problems. The results indicate that for Chinese upper-grade primary school students, cognitive load was lower in the metacognitive prompting condition and when solving cardinal problems, and higher when solving ordinal problems.

Overcoming the ordinal imbalanced data problem by combining data processing and stacked generalizations

Ordinal imbalanced datasets are pervasive in real world applications but remain challenging to analyse as they require specific methods to account for the ordering information and imbalanced classes. Failure to account for both those characteristics can substantially impact the model predictive performance. However, existing methods tend to focus either on ordinality or imbalance, rather than addressing both simultaneously. The few approaches that do account for both characteristics are not always easy to implement for non-advanced analysts and simpler approaches are needed to facilitate appropriate data processing. Here, we developed a general approach using some of the most popular machine learning algorithms to ensure appropriate processing of ordinal imbalanced datasets and to optimize the predictions of all classes. After transforming the multi-class ordinal problem into a well-known binary problem, we implemented several different resampling methods in a decision-tree classifier. We then used a stacked generalization algorithm to combine the classifiers to improve model predictive performance. To test our approach, we used two ordinal imbalanced datasets on student performance and wine quality. Individual resampling techniques tended to improve the accuracy of minority classes, while simultaneously increasing the number of false positives in those classes. This resulted in a decrease, sometimes substantial, in accuracy of other classes. The stacking model offered a good compromise between improvement in accuracy of minority classes and mitigation of reduced accuracy in other classes. Our approach provided useful insights into modelling strategies that should be favoured for implementation in production that involve these common datasets, depending on the end-user interests.

Unimodal regularisation based on beta distribution for deep ordinal regression

Currently, the use of deep learning for solving ordinal classification problems, where categories follow a natural order, has not received much attention. In this paper, we propose an unimodal regularisation based on the beta distribution applied to the cross-entropy loss. This regularisation encourages the distribution of the labels to be a soft unimodal distribution, more appropriate for ordinal problems. Given that the beta distribution has two parameters that must be adjusted, a method to automatically determine them is proposed. The regularised loss function is used to train a deep neural network model with an ordinal scheme in the output layer. The results obtained are statistically analysed and show that the combination of these methods increases the performance in ordinal problems. Moreover, the proposed beta distribution performs better than other distributions proposed in previous works, achieving also a reduced computational cost.

Ordinal losses for classification of cervical cancer risk.

Cervical cancer is the fourth leading cause of cancer-related deaths in women, especially in low to middle-income countries. Despite the outburst of recent scientific advances, there is no totally effective treatment, especially when diagnosed in an advanced stage. Screening tests, such as cytology or colposcopy, have been responsible for a substantial decrease in cervical cancer deaths. Cervical cancer automatic screening via Pap smear is a highly valuable cell imaging-based detection tool, where cells must be classified as being within one of a multitude of ordinal classes, ranging from abnormal to normal. Current approaches to ordinal inference for neural networks are found to not sufficiently take advantage of the ordinal problem or to be too uncompromising. A non-parametric ordinal loss for neuronal networks is proposed that promotes the output probabilities to follow a unimodal distribution. This is done by imposing a set of different constraints over all pairs of consecutive labels which allows for a more flexible decision boundary relative to approaches from the literature. Our proposed loss is contrasted against other methods from the literature by using a plethora of deep architectures. A first conclusion is the benefit of using non-parametric ordinal losses against parametric losses in cervical cancer risk prediction. Additionally, the proposed loss is found to be the top-performer in several cases. The best performing model scores an accuracy of 75.6% for seven classes and 81.3% for four classes.

On the spectral Green’s function-constant nodal method for fixed-source SN problems in X,Y-geometry with arbitrary L’th-order anisotropic scattering

Presented here is an extension of the spectral Green’s function-constant nodal (SGF-CN) method for the numerical solution of energy multigroup, fixed-source, discrete ordinates (SN) problems in X, Y-geometry with arbitrary L’th-order of scattering anisotropy, provided L<N. This analytical coarse-mesh method uses the multigroup SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. The only approximations in the present version of the SGF-CN method occur in these transverse leakage terms, as the energy-group transfer scattering source terms are treated analytically within the offered method. Numerical results to typical model problems are given to illustrate the method’s accuracy and to analyze the efficiency of the offered SGF-CN computer code for neutral particle transport calculations.

The modified spectral deterministic method applied to fixed–source discrete ordinates problems in X,Y–geometry

A new approach for the application of the coarse–mesh Modified Spectral Deterministic method to numerically solve the two–dimensional neutron transport equation in the discrete ordinates (Sn) formulation is presented in this work. The method is based on within node general solution of the conventional one–dimensional Sn transverse integrated equations considering constant approximations for the transverse leakage terms and obtaining the Sn spatial balance equations. The discretized equations are solved by using a modified Source Iteration scheme without additional approximations since the average angular fluxes are computed analytically in each iteration. The numerical algorithm of the method presented here is algebraically simpler than other spectral nodal methods in the literature for the type of problems we have considered. Numerical results to two typical model problems are presented to test the accuracy of the offered method.

Perceptron Ranking Using Interval Labels with Ramp Loss for Online Ordinal Regression

Due to its wide applications and learning efficiency, online ordinal regression using perceptron algorithms with interval labels (PRIL) has been increasingly applied to solve ordinal ranking problems. However, it is still a challenge for the PRIL method to handle noise labels, in which case the ranking results may change dramatically. To tackle this problem, in this paper, we propose noise-resilient online learning algorithms using ramp loss function, called PRIL-RAMP, and its nonlinear variant K-PRIL-RAMP, to improve the performance of PRIL method for noisy data streams. The proposed algorithms iteratively optimize the decision function under the framework of online gradient descent (OGD), and we justify the algorithms by showing the order preservation of thresholds. It is validated in the experiments that both approaches are more robust and efficient to noise labels than state-of-the-art online ordinal regression algorithms on real-world datasets.

A response matrix spectral nodal method for energy multigroup X,Y-geometry discrete ordinates problems in non-multiplying media

A new spectral nodal method is proposed for coarse-mesh numerical solution of energy multigroup X,Y-geometry discrete ordinates (SN) transport problems in non-multiplying media. This method uses the Response Matrix (RM) technique to solve analytically the multigroup one-dimensional transverse-integrated SN nodal equations with approximations only for the node-edge angular fluxes in the transverse leakage terms. We approximate these leakage terms by constants, hence we refer to the present spectral nodal method as the RM-Constant Nodal (RM-CN) method. We show in numerical calculations that the offered RM-CN method generates as accurate solutions as the companion spectral Green's function-constant nodal (SGF-CN) method, whose mathematical preliminaries are essentially the same, except that the multigroup RM-CN algorithm is both algebraically and computationally simpler than the SGF-CN method.

An Extended Linear Discontinuous Method for One-group Fixed Source Discrete Ordinates Problems with Isotropic Scattering in Slab Geometry

Nowadays, the obtainment of an accurate numerical solution of fixed source discrete ordinates problems is relevant in many areas of engineering and science. In this work, we extend the hybrid Finite Element Spectral Green's Function method (FEM-SGF), originally developed to solve eigenvalue diffusion problems, for fixed source problems using as a mathematical model, the discrete ordinates formulation in one energy group with isotropic scattering in slab geometry. This new method, Extended Linear Discontinuous Discrete Ordinates (ELD-SN), is based on the use of neutron balance equations and the construction of a hybrid auxiliary equation. This auxiliary equation combines a linear discontinuous approximation and spectral parameters to approximate the neutron angular flux inside the cell. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of our methodology. ELD-SN method is free from spatial truncation errors in S2 quadrature, and generate good results in the other quadrature sets. This method is more accurate than the conventional Diamond Difference (DD) and Linear Discontinuous (LD) methods, but surpassed by the Spectral Green's Function (SGF) method, for quadrature order greater than two.

An approximation to solve regression problems with a genetic fuzzy rule ordinal algorithm

Regression problems try estimating a continuous variable from a number of characteristics or predictors. Several proposals have been made for regression models based on the use of fuzzy rules; however, all these proposals make use of rule models in which the irrelevance of the input variables in relation to the variable to be approximated is not taken into account. Regression problems share with the ordinal classification the existence of an explicit relationship of order between the values of the variable to be predicted. In a recent paper, the authors have proposed an ordinal classification algorithm that takes into account the detection of the irrelevance of input variables. This algorithm extracts a set of fuzzy rules from an example set, using as the basic model a sequential covering strategy along with a genetic algorithm. In this paper, a proposal for a regression algorithm based on this ordinal classification algorithm is presented. The proposed model can be interpreted as a multiclassifier and multilevel system that learns at each stage using the knowledge gained in previous stages. Due to similarities between regression and ordinal problems as well as the use of a set of ordinal algorithms, an error interval can be returned with the regression output value. Experimental results show the good behavior of the proposal as well as the results of the error interval.

A multi-group extended linear discontinuous method for fixed-source discrete ordinates problems in slab geometry

At present, neutron density calculation in non-multiplying media is relevant in many areas of engineering and science. In this paper, we propose the Extended Linear Discontinuous (ELD) method in multi-group discrete ordinates formulation, originally formulated for one-energy group fixed-source problems with isotropic scattering source in slab geometry. The proposed auxiliary equations are uncoupled on angular directions and combine the linear discontinuous approximation of the finite element method and the quasi-analytical general solution of the spectral nodal method. Thus, we can implement an efficient and simple algorithm using the conventional source iteration scheme for the sweeping equations. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of the ELD method. The work shows that the main advantages of the proposed method are that the numerical scheme is stable for coarse-meshes, and its numerical results are more accurate than those generated by the Diamond Difference (DD) and Linear Discontinuous (LD) methods.

A multi-group extended linear discontinuous method for fixed-source discrete ordinates problems in slab geometry

At present, neutron density calculation in non-multiplying media is relevant in many areas of engineering and science. In this paper, we propose the Extended Linear Discontinuous (ELD) method in multi-group discrete ordinates formulation, originally formulated for one-energy group fixed-source problems with isotropic scattering source in slab geometry. The proposed auxiliary equations are uncoupled on angular directions and combine the linear discontinuous approximation of the finite element method and the quasi-analytical general solution of the spectral nodal method. Thus, we can implement an efficient and simple algorithm using the conventional source iteration scheme for the sweeping equations. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of the ELD method. The work shows that the main advantages of the proposed method are that the numerical scheme is stable for coarse-meshes, and its numerical results are more accurate than those generated by the Diamond Difference (DD) and Linear Discontinuous (LD) methods.

A New Look at the Models for Ordinal Categorical Data Analysis

The multinomial/categorical responses, whether they are nominal or ordinal, are recorded in counts under all categories/cells involved. The analysis of this type of multinomial data is traditionally done by exploiting the marginal cell probabilities-based likelihood function. As opposed to the nominal setup, the computation of the marginal probabilities is not easy in the ordinal setup. However, as the ordinal responses in practice are interpreted by collapsing multiple categories either to binary data using a single cut-point or to tri-nomial data using two cut-points, most of the studies over the last four decades, first modeled the associated cumulative probabilities using a suitable such as logit, probit, log-log, or complementary log-log link function. Next the marginal cell probabilities were computed by subtraction, in order to construct the desired estimating function such as moment or likelihood function. In this paper we take a new look at this ordinal categorical data analysis problem. As opposed to the existing studies, we first model the ordinal categories using a multinomial logistic marginal approach by pretending that the adjacent categories are nominal, and then construct the cumulative probabilities to develop the final model for ordinal responses. For inferences, we develop the cut points based likelihood or generalized quasi-likelihood (GQL) estimating functions for the purpose of the estimation of the underlying regression parameters. The new GQL estimation approach is developed in details by utilizing both tri-nomial and binary (or binomial) collapsed structures. The likelihood analysis is also discussed. A data example is given to illustrate the proposed models and the estimation methodologies. Furthermore, we also examine the asymptotic properties of the likelihood and GQL estimators for the regression parameters for both tri-nomial and binary types of cumulative response based models.

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

A spectral nodal method for energy multi-group X, Y-geometry, discrete ordinates (SN) problems in non-multiplying medium is developed. This analytical coarse-mesh method is referred to as the multi-group spectral Green's function - constant nodal (SGF-CN) method. The SGF-CN method uses the multi-group SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. As the energy-group transfer scattering source terms are treated analytically in the offered method, the only approximations occur in the group transverse leakage terms. Numerical results are given to illustrate the method's accuracy for coarse-mesh calculations.

Assessment of the “Disrupt-O-Meter” model by ordinal multicriteria methods

The objective of this article is to explore a potential diagnostic model, called “Disrupt-O-Meter”, about the Christensen's disruptive innovation theory. The diagnostic model was analyzed under multi-criteria decision aid (MCDA) methods. This diagnosis presents a typical data structure of multi-criteria ordinal problems. Different alternatives were evaluated under a set of criteria, using a scale of ordinal preferences. The steps of a MCDA problem were followed. The chosen methods were the Borda, the Condorcet and the Probabilistic Composition of Preferences (CPP). This article used a database from other research, about 3D printing technology startups. The results showed the best discrimination power by the CPP method, revealing the business category with the most disruptive potential, among other alternatives.

A Response Matrix Method for One-Speed Slab-Geometry Discrete Ordinates Adjoint Calculations in Source-Detector Problems

ABSTRACTPresented here is an application of the one-speed Response Matrix (RM) method for adjoint discrete ordinates (SN) problems in slab geometry applied to source-detector problems. The adjoint RM method is free from spatial truncation errors, as it generates numerical results for the adjoint angular fluxes in multilayer slabs that agree with the numerical values obtained from the analytical solution of the adjoint SN equations. Numerical results are given for two typical source-detector problems to illustrate the accuracy and the efficiency of the offered RM computer code.

On the median in imprecise ordinal problems

When having to make a prediction under probabilistic uncertainty in ordinal problems, the median offers a number of interesting properties compared to other statistics such as the expected value. In particular, it does not depend on a particular metric defined over the elements, but still takes account of the ordinal nature of the data. It can also be shown to be the minimizer of the $$L_1$$ loss function. In this paper, we show that similar results can be obtained when the uncertainty is described not by a single probability distribution, but by a convex set of those. In particular, we relate the lower and upper medians to the $$L_1$$ loss function via the notion of lower and upper expectations (and extend these results to general quantiles). We also show that, using a different decision rule, the lower and upper median can be retrieved when assuming the cost to be strictly monotonic and symmetric, and nothing more. Finally, we run some tests to show the interest of using Median based predictions with convex sets of probabilities in ordinal regression problems.

Recent Studies on the Asymptotic Convergence of the Spatial Discretization for Two-Dimensional Discrete Ordinates Solutions

ABSTRACTIn this work, four types of quadrature schemes are used to define discrete directions in the solution of a two-dimensional fixed-source discrete ordinates problem in Cartesian geometry. Such schemes enable generating numerical results for averaged scalar fluxes over specified regions of the domain with high number (up to 105) of directions per octant. Two different nodal approaches, the ADO and AHOT-N0 methods, are utilized to obtain the numerical results of interest. The AHOT-N0 solutions on a sequence of refined meshes are then used to develop an asymptotic analysis of the spatial discretization error in order to derive a reference solution. It was more clearly observed that the spatial discretization error converges asymptotically with second order for the source region with all four quadratures employed, while for the other regions refined meshes along with tighter convergence criterion must be applied to evidence the same behavior. However, in that case, some differences among the four quadrature schemes results were found.