Interval Representation Research Articles

Perceptual fusion of musical notes by native Amazonians suggests universal representations of musical intervals

Music perception is plausibly constrained by universal perceptual mechanisms adapted to natural sounds. Such constraints could arise from our dependence on harmonic frequency spectra for segregating concurrent sounds, but evidence has been circumstantial. We measured the extent to which concurrent musical notes are misperceived as a single sound, testing Westerners as well as native Amazonians with limited exposure to Western music. Both groups were more likely to mistake note combinations related by simple integer ratios as single sounds (‘fusion’). Thus, even with little exposure to Western harmony, acoustic constraints on sound segregation appear to induce perceptual structure on note combinations. However, fusion did not predict aesthetic judgments of intervals in Westerners, or in Amazonians, who were indifferent to consonance/dissonance. The results suggest universal perceptual mechanisms that could help explain cross-cultural regularities in musical systems, but indicate that these mechanisms interact with culture-specific influences to produce musical phenomena such as consonance.

What was visualized? A method for describing content of performance summary displays in feedback interventions

BackgroundVisual displays such as charts and tables may significantly moderate the effects of audit and feedback interventions, but the systematic study of these intervention components will likely remain limited without a method for isolating the information content of a visual display from its form elements. The objective of this study is to introduce such a method based on an application of visualization frameworks to enable a systematic approach to answer the question, “What was visualized?” in studies of audit and feedback.MethodsThe proposed method uses 3 steps to systematically identify and describe the content of visual displays in feedback interventions: 1) identify displays, 2) classify content, and 3) identify elements. The use of a visualization framework led us to identify information content types as representations of measures (metrics or indicators), ascribees (feedback recipients and comparators), performance levels, and time intervals. We illustrate the proposed method in a series of 3 content analyses, one for each step, to identify visual displays and their information content in published example performance summaries.ResultsWe analyzed a convenience sample of 44 published studies of audit and feedback. Through each step, two coders had good agreement. We identified 42 visual displays of performance, containing 6 unique combinations of content types. What was visualized most commonly in the sample was performance levels across a recipient and comparators (i.e. ascribees) for a single measure and single time interval (n = 16). Content types varied in their inclusion of measures, ascribees, and time intervals.ConclusionsThe proposed method appears to be feasible to use as a systematic approach to describing visual displays of performance. The key implication of the method is that it offers more granular and consistent description for empirical, theoretical, and design studies about the information content of feedback interventions.

On Type-2 interval with interval mathematics and order relations: its applications in inventory control

Mathematical decision-making is an important criterion to take the appropriate decision regarding the representation of an imprecise parameter under uncertain situations. In the existing literature, it is observed that one such representation is interval representation. The goal of this paper is to introduce a new concept of Type-2 interval which might be used to represent the impreciseness appropriately. The definitions of arithmatical operations, interval functions and algebraic properties of Type-2 interval are proposed here. Then, the order relations between two Type-2 interval numbers are introduced and all the definitions are illustrated using some numerical examples. Finally, an application of Type-2 interval mathematics and order relation is made in the area of inventory control.

Verified Interval Enclosure Techniques for Robust Gain Scheduling Controllers

In real-life applications, dynamic systems are often subject to uncertainty due to model simplifications, measurement inaccuracy or approximation errors which can be mapped to specific parameters. Uncertainty in dynamic systems can come either in stochastic forms or as interval representations, when they are considered as bounded as it will be done in this paper. The main idea, here, is to find a joint approach for an interval-based gain scheduling controller while simultaneously reducing overestimation by enclosing state intervals with the least amount of conservativity. The robust and/ or optimal control design is realized using linear matrix inequalities (LMIs) to find an efficient solution and aims at a guaranteed stabilization of the system dynamics over a predefined time horizon. Since the resulting system is assumed to be asymptotically stable, a temporal reduction of the widths of intervals representing worst-case bounds of the system states at a specific point of time should occur. However, for commonly used approaches in the computation of interval enclosures those interval widths seemingly blow up due to the wrapping effect in many cases. To avoid this, we provide two interval enclosure techniques --- an exploitation of cooperativity and an exponential approach --- and discuss their applicability taking into account two real-life applications, a high-bay rack feeder and an inverse pendulum.

Assessing the Accuracy of Approximate Confidence Intervals Proposed for the Mean of Poisson Distribution

The Poisson distribution is applied as an appropriate standard model to analyze count data. Because this distribution is known as a discrete distribution, representation of accurate confidence intervals for its distribution mean is extremely difficult. Approximate confidence intervals were presented for the Poisson distribution mean. The purpose of this study is to simultaneously compare several confidence intervals presented, according to the average coverage probability and accurate confidence coefficient and the average confidence interval length criteria.

Utilizing Statistical Information for Interval Analysis: A Method for Analyzing the Interval Uncertainty of Line-of-Sight Measurement Error of Space-Borne Observation Platforms

Analyzing the space-borne observation platform's line-of-sight measurement error (SOPLME) is of great significance for the improvement of the space-borne optical tracking system. In this regard, we consider the space-borne observation platform (SOP) as a multi-body system and build both the target imaging model from the inertial space to the optical camera image plane and the line-of-sight measurement model. The relationship between the SOPLME and the satellite's orbit error, attitude error, platform angular vibration error, inner and outer frame rotation error, and position quantization error of target image point is deduced theoretically. However, since there are many error sources and the modeling process is highly nonlinear, the amplification of the output range caused by the “wrapping effect” may occur when using the interval calculation method to analyze the output range of the SOPLME. To solve this problem, we propose a novel method of interval analysis by using statistical information. This method combines the convenience of interval representation with the high accuracy of probability calculation, which can effectively avoid the overestimation problem caused by the inherent “wrapping effect” of the interval calculation method. Firstly, we describe the parameters of each error source with the interval constructed by upper and lower bounds. Secondly, the optimal Latin hypercube sampling (LHS) matrix of error source parameters is designed by the quasi-optimization method. Then we use the Gaussian mixture model (GMM) to approximate the distribution function of the SOPLME. Finally, the output range of the SOPLME is estimated by Taylor expansion. The simulation test verifies the reliability of the model and interval calculation method in this paper. The simulation example shows that the proposed method can get a more reliable error output interval with less sampling amount, which is suitable for analyzing the dynamic process of the output interval of the SOPLME.

Adaptive flux variability analysis of HEK cell cultures

Measured external fluxes impose constraints to under-determined metabolic networks that narrow the internal flux intervals obtained using Flux Variability Analysis. Nevertheless, these constraints often lead to systems that do not admit a feasible solution. Measurement noise and data smoothing are among the sources of uncertainties that can cause system infeasibility. These constraints are classically released using interval representation of fluxes. This study investigates the use of Adaptive Flux Variability Analysis (AFVA), which allows determining a minimal coefficient of variation of the external fluxes along the time course of the experiment. Especially, AFVA is applied to a medium-size metabolic network and a rich dataset relative to HEK-293 cells cultured in batch, encompassing all 20 amino acids and less commonly measured metabolites, such as urea and pyruvate. AFVA appears as an effective tool for metabolic flux analysis. The impact of data-smoothing and the information provided by the cell growth are thoroughly analyzed.

Estrategias y errores de conversión entre representaciones de intervalos de la recta real

En este artículo se presentan los resultados de un estudio empírico llevado a cabo a través de diferentes ciclos de investigación con alumnos de 4.º de la ESO y 1.º de Bachillerato, centrado en el concepto de intervalo de la recta real. Los objetivos son identificar las estrategias que utilizan los alumnos para realizar conversiones entre diferentes representaciones de intervalos no acotados e interpretar los posibles errores y dificultades derivados de aplicar estas estrategias. Se han detectado diferentes estrategias focalizadas en aspectos particulares, lo que ha corroborado que muchas dificultades están asociadas a la utilización de conversiones por congruencia. A partir de los resultados, se proponen varias orientaciones didácticas para facilitar el aprendizaje de este concepto a través de sus representaciones.

Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity

We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity).

Interval orders with two interval lengths

A poset P=(X,≺) has an interval representation if each x∈X can be assigned a real interval Ix so that x≺y in P if and only if Ix lies completely to the left of Iy. Such orders are called interval orders. In this paper we give a surprisingly simple forbidden poset characterization of those posets that have an interval representation in which each interval length is either 0 or 1. In addition, for posets (X,≺) with a weight of 1 or 2 assigned to each point, we characterize those that have an interval representation in which for each x∈X the length of the interval assigned to x equals the weight assigned to x. For both problems we can determine in polynomial time whether the desired interval representation is possible and in the affirmative case, produce such a representation.

Effect of Presentation Format on Judgment of Long-Range Time Intervals.

Investigations in the temporal estimation domain are quite vast in the range of milliseconds, seconds, and minutes. This study aimed to determine the psychophysical function that best describes long-range time interval estimation and evaluate the effect of numerals in duration presentation on the form of this function. Participants indicated on a line the magnitude of time intervals presented either as a number + time-unit (e.g., “9 months”; Group I), unitless numerals (e.g., “9”; Group II), or tagged future personal events (e.g., “Wedding”; Group III). The horizontal line was labeled rightward (“Very short” = >“Very long”) or leftward (“Very long” = >“Very short”) for Group I and II, but only rightward for Group III. None of the linear, power, logistic or logarithmic functions provided the best fit to the individual participant data in more than 50% of participants for any group. Individual power exponents were different only between the tagged personal events (Group III) and the other two groups. When the same analysis was repeated for the aggregated data, power functions provided a better fit than other tested functions in all groups with a difference in the power function parameters again between the tagged personal events and the other groups. A non-linear mixed effects analysis indicated a difference in the power function exponent between Group III and the other groups, but not between Group I and II. No effect of scale directionality was found in neither of the experiments in which scale direction was included as independent variable. These results suggest that the judgment of intervals in a number + time-unit presentation invoke, at least in part, processing mechanisms other than those used for time-domain. Consequently, we propose the use of event-tagged assessment for characterizing long-range interval representation. We also recommend that analyses in this field should not be restricted to aggregated data given the qualitative variation between participants.

Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning.

One of the most relevant features of musical pieces is the selection and utilization of musical elements by composers. For connecting the musical properties of a melodic line as a whole with those of its constituent elements, we propose a representation for musical intervals based on physical quantities and a statistical model based on the minimization of relative entropy. The representation contains information about the size, location in the register, and level of tonal consonance of musical intervals. The statistical model involves expected values of relevant physical quantities that can be adopted as macroscopic constraints with musical meaning. We studied the occurrences of musical intervals in 20 melodic lines from seven masterpieces of Western tonal music. We found that all melodic lines are strictly ordered in terms of the physical quantities of the representation and that the formalism is suitable for approximately reproducing the final selection of musical intervals made by the composers, as well as for describing musical features as the asymmetry in the use of ascending and descending intervals, transposition processes, and the mean dissonance of a melodic line.

Pose Characterization and Analysis of Soft Continuum Robots With Modeling Uncertainties Based on Interval Arithmetic

This paper introduces a systematical interval-based framework of inherent uncertainties and pose evaluation for a class of soft continuum robots driven by flexible shafts. A more general model of continuum robots driven by shaft tendons is extended from prior kinematic models. On top of the proposed model, the interval-based analysis is presented to analyze and characterize the pose of continuum robots considering uncertainties in kinematic parameters and joint inputs. A 3-D printed bending actuator driven by a flexible shaft is evaluated for case study based on the proposed interval-valued framework. This paper investigates and compares a couple of refinement methods and proposes a new way of sensitivity analysis of model parameters based on interval arithmetic. The kinematic and mechanics parameters are measured and identified experimentally with a representation of intervals. The in-plane motion experiment validates that the computed bounds can enclose all the measured tip positions with consideration of the measurement uncertainty. The method is also validated when external loading is exerted. Note to Practitioners —With the fast development of soft robotics, the increasing number of soft robotic manipulators show great potentials of application in industries such as agriculture, biomedicine, home automation, manufacturing, logistics, and domestic service. Thanks to the mechanical compliance, soft manipulators demonstrate environmental adaptability at the cost of very precise positioning. However, the guaranteed bounds of the reaching range are valuable in order to provide users a good knowledge of the product performance, which is the motivation of this paper. This paper provides not only a method and framework based on interval analysis to evaluate the pose of soft actuators, but also a case study of uncertainty interval characterization procedure which is domain-specific for soft actuators. The presented method framework can be implemented for continuum manipulators and soft actuators as well as any other robotic devices featured by such flexible components.

Equivalence analysis to support environmental safety assessment: Using nontarget organism count data from field trials with cisgenically modified potato.

This paper considers the statistical analysis of entomological count data from field experiments with genetically modified (GM) plants. Such trials are carried out to assess environmental safety. Potential effects on nontarget organisms (NTOs), as indicators of biodiversity, are investigated. The European Food Safety Authority (EFSA) gives broad guidance on the environmental risk assessment (ERA) of GM plants. Field experiments must contain suitable comparator crops as a benchmark for the assessment of designated endpoints. In this paper, a detailed protocol is proposed to perform data analysis for the purpose of assessing environmental safety. The protocol includes the specification of a list of endpoints and their hierarchical relations, the specification of intended levels of data analysis, and the specification of provisional limits of concern to decide on the need for further investigation. The protocol emphasizes a graphical representation of estimates and confidence intervals for the ratio of mean abundances for the GM plant and its comparator crop. Interpretation relies mainly on equivalence testing in which confidence intervals are compared with the limits of concern. The proposed methodology is illustrated with entomological count data resulting from multiyear, multilocation field trials. A cisgenically modified potato line (with enhanced resistance to late blight disease) was compared to the original conventional potato variety in the Netherlands and Ireland in two successive years (2013, 2014). It is shown that the protocol encompasses alternative schemes for safety assessment resulting from different research questions and/or expert choices. Graphical displays of equivalence testing at several hierarchical levels and their interpretation are presented for one of these schemes. The proposed approaches should be of help in the ERA of GM or other novel plants.

Possibilistic compositions and state functions: application to the order promising process for perishables

In this paper, we propose the concepts of the composition of possibilistic variables and state functions. While in conventional compositional data analysis, the interdependent components of a deterministic vector must add up to a specific quantity, we consider such components as possibilistic variables. The concept of state function is intended to describe the state of a dynamic variable over time. If a state function is used to model decay in time, it is called the ageing function. We present a practical implementation of our concepts through the development of a model for a supply chain planning problem, specifically the order promising process for perishables. We use the composition of possibilistic variables to model the existence of different non-homogeneous products in a lot (sub-lots with lack of homogeneity in the product), and the ageing function to establish a shelf life-based pricing policy. To maintain a reasonable complexity and computational efficiency, we propose the procedure to obtain an equivalent interval representation based on α-cuts, allowing to include both concepts by means of linear mathematical programming. Practical experiments were conducted based on data of a Spanish supply chain dedicated to pack and distribute oranges and tangerines. The results validated the functionality of both, the compositions of possibilistic variables and ageing functions, showing also a very good performance in terms of the interpretation of a real problem with a good computational performance.

A Semantic Web Approach to Enable a Smart Route to Historical Archives

In this paper we show that an ontology-based approach can be beneficial for enhancing the access to cultural resources, and in particular historical documents. The paper starts with an overview of our approach, aimed at providing online archival systems with a semantic layer based on Semantic Web standards (OWL 2 and RDF). Two projects are introduced, namely Harlock900 and PRiSMHA, carried out in collaboration with local cultural institutions owning rich historical archives. In particular, the paper describes the computational ontologies supporting the approach, and then focuses on two case studies showing that our framework provides better results if compared with standard access systems. The case studies show the enhancement provided by a semantically rich representation of time intervals and a detailed formal description of events and their participants.

Adaptive Flux Variability Analysis: A Tool To Deal With Uncertainties

Abstract Underdetermined metabolic networks are usually investigated using Flux Variability Analysis (FVA) under the pseudo-steady state assumption of the internal metabolites. When a dynamic overview of the flux map is sought, the time variation of the cell uptake and excretion rates is deduced from extracellular dynamic mass balances and smoothing of the measurement data of the time evolution of the extracellular concentrations. Nevertheless, the resulting system of equations does not always admit a feasible solution under these constraints. Indeed, measurement data is affected by noise, whose processing (smoothing, etc) always entails some subjective user choices, and the network itself might not be perfectly suited to explain the several culture phases. To alleviate these adverse effects, the constraints can be relaxed by introducing coefficients of variation of the external fluxes, and by considering an interval representation of the fluxes. This work presents a systematic method to determine these coefficients of variation, along the time course of the culture, leading to an adaptive scheme where the coefficients are set to the tightest bounds. The methodology is applied to experimental data of cultures of hybridoma in batch and perfusion modes and compared to previously published results.

Improved distributed algorithms for coloring interval graphs with application to multicoloring trees

We give a distributed (1+ϵ)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O(1ϵlog⁎⁡n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CONGEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log⁎⁡n) rounds were known [1]. Linial's ring coloring lower bound implies that the dependency on log⁎⁡n cannot be improved. We further prove that the dependency on 1ϵ is also optimal.To obtain our CONGEST model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a (1+ϵ)-approximate multicoloring of directed trees in O(1ϵlog⁎⁡n) rounds.

Min and Max Operators for Gradual Intervals

This paper presents a new and a more general formulation of the min and max operations for gradual intervals, initially proposed for the special case of triangular fuzzy intervals. Gradual intervals generalize the conventional interval representation by increasing its specificity, thereby making it possible to represent the imprecision and the uncertainty via the notion of gradualness. In this context, a new interpretation of fuzzy intervals through the notion of gradual intervals is developed. The originality of the proposed methodology lies in the extension of inter-interval relations to the gradual case according to midpoint-radius representation.

Derivation and numerical profile analysis of a hierarchically formulated microscopic model of hemoglobin oxygen binding

To address complex thermodynamic systems with multiple interacting events, we have developed the concept of hierarchical thermodynamic interactions. In this study, this concept is extended to protein-ligand systems with similar but not identical protein subunits, and applied to the analysis of previously published NMR and UV–vis monitored hemoglobin oxygen binding data. Non–linear regression provided estimated errors for statistically significant parameters, but not for null (zero) valued parameters. A numerical/graphical profiling approach was therefore used to assess confidence intervals and correlations for both the statistically significant and nulled valued parameters in this model. Individual parameters were set to fixed values around their best–fit value, and the subset of statistically significant parameters re–minimized against hemoglobin oxygen binding data. Plots provide a graphical representation of parameter confidence intervals and correlations, and demonstrate how the two different data types – UV–vis and NMR – constrain the range of values for each parameter. This analysis further illustrates the value of hierarchically formulated models for the analysis of complex state systems, and illuminates the complexity of parameter space around the derived minimum microscopic model of hemoglobin oxygen binding.