Interval Representation Research Articles

Efficient representations of upper limits and confidence intervals for experimental data

Upper limits and confidence intervals are a convenient way to present experimental results. With modern experiments producing more and more data, it is often necessary to reduce the volume of the results for convenient distribution. A common approach is to take a maximum over a set of upper limits, which yields an upper limit valid for the entire set. This, however, can result in significant sensitivity loss, because we discard more constraining data. In this paper, we introduce upper limits and confidence intervals that allow one to summarize results with a small relaxation of constraints. A toy example and an application to upper limits in all-sky continuous gravitational wave searches are worked out. Published by the American Physical Society 2024

Intuitionistic Fuzzy Sets for Spatial and Temporal Data Intervals

Spatial and temporal uncertainties are found in data for many critical applications. This paper describes the use of interval-based representations of some spatial and temporal information. Uncertainties in the information can arise from multiple sources in which degrees of support and non-support occur in evaluations. This motivates the use of intuitionistic fuzzy sets to permit the use of the positive and negative memberships to capture these uncertainties. The interval representations will include both simple and complex or nested intervals. The relationships between intervals such as overlapping, containing, etc. are then developed for both the simple and complex intervals. Such relationships are required to support the aggregation approaches of the interval information. Both averaging and merging approaches to interval aggregation are then developed. Furthermore, potential techniques for the associated aggregation of the interval intuitionistic fuzzy memberships are provided. A motivating example of maritime depth data required for safe navigation is used to illustrate the approach. Finally, some potential future developments are discussed.

A formalization of constraint interval: A precussor to fuzzy interval analysis

This presentation outlines some aspects of interval analysis from the constraint interval point of view, focusing on some issues associated with mathematical analysis, especially those that are fundamental to fuzzy mathematical analysis. We begin by reviewing interval representations from the standard interval point of view and the study of intervals as functions of parameters varying in hypercubes, called constraint interval representation (CI). These concepts are used to define arithmetic operations between intervals, as well, interval-valued and interval functions for rational and transcedental functions. Issues of semantics as they apply to interval analysis and fuzzy interval analysis are discussed. The definition of constrained interval function extension is used to prove that each CI arithmetic operation is an isotonic inclusion.

A General Framework for Providing Interval Representations of Pareto Optimal Outcomes for Large-Scale Bi- and Tri-Criteria MIP Problems

The Multi-Objective Mixed-Integer Programming (MOMIP) problem is one of the most challenging. To derive its Pareto optimal solutions one can use the well-known Chebyshev scalarization and Mixed-Integer Programming (MIP) solvers. However, for a large-scale instance of the MOMIP problem, its scalarization may not be solved to optimality, even by state-of-the-art optimization packages, within the time limit imposed on optimization. If a MIP solver cannot derive the optimal solution within the assumed time limit, it provides the optimality gap, which gauges the quality of the approximate solution. However, for the MOMIP case, no information is provided on the lower and upper bounds of the components of the Pareto optimal outcome. For the MOMIP problem with two and three objective functions, an algorithm is proposed to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization. Such interval representations can be used to navigate on the Pareto front. The results of several numerical experiments on selected large-scale instances of the multi-objective multidimensional 0–1 knapsack problem illustrate the proposed approach. The limitations and possible enhancements of the proposed method are also discussed.

Developing Constrained Interval Operators for Fuzzy Logic with Interval Values

A well-known problem in the interval analysis literature is the overestimation and loss of information. In this article, we define new interval operators, called constrained interval operators, that preserve information and mitigate overestimation. These operators are investigated in terms of correction, algebraic properties, and orders. It is shown that a large part of the properties studied is preserved by this operator, while others remain preserved with the condition of continuity, as is the case of the exchange principle. In addition, a comparative study is carried out between this operator g¨ and the best interval representation: g^. Although g¨⊆g^ and g¨ do not preserve the Moore correction, we do not have a loss of relevant information since everything that is lost is irrelevant, mitigating the overestimation.

Is There a Role of Inferior Frontal Cortex in Motor Timing? A Study of Paced Finger Tapping in Patients with Non-Fluent Aphasia.

The aim of the present study was to investigate the deficits in timing reproduction in individuals with non-fluent aphasia after a left hemisphere lesion including the inferior frontal gyrus, in which Broca's region is traditionally localised. Eighteen stroke patients with non-fluent aphasia and twenty-two healthy controls were recruited. We used a finger-tapping Test, which consisted of the synchronisation and the continuation phase with three fixed intervals (450 ms, 650 ms and 850 ms). Participants firstly had to tap simultaneously with the device's auditory stimuli (clips) (synchronisation phase) and then continue their tapping in the same pace when the stimuli were absent (continuation phase). Patients with aphasia demonstrated less accuracy and greater variability during reproduction in both phases, compared to healthy participants. More specifically, in the continuation phase, individuals with aphasia reproduced longer intervals than the targets, whereas healthy participants displayed accelerated responses. Moreover, patients' timing variability was greater in the absence of the auditory stimuli. This could possibly be attributed to deficient mental representation of intervals and not experiencing motor difficulties (due to left hemisphere stroke), as the two groups did not differ in tapping reproduction with either hand. Given that previous findings suggest a potential link between the IFG, timing and working memory, we argue that patients' extra-linguistic cognitive impairments should be accounted for, as possible contributing factors to timing disturbances.

Improved lower bound on the on-line chain partitioning of semi-orders with representation

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains in the partition. The on-line chain partitioning problem involves finding the minimal number of chains needed by an on-line algorithm. Chrobak and Ślusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most 3w−2 chains, where w is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most 2w−1 chains, over 30 years later, it remains unknown whether this is an optimal algorithm. In this paper, we improve upon previously known bounds and show that any on-line algorithm can be forced to use ⌈32w⌉ chains to partition an order presented in the form of its unit interval representation. As a consequence, we completely solve the problem for w=3. Lastly, we show that loosening the restriction from unit intervals to proper intervals in the bandwidth variant allows us to improve the lower bound by w/3.

Graph embedding-based Anomaly localization for HVAC system

As a major energy consumption system in buildings, anomaly detection on multivariate time series monitored by sensors in HVAC systems has been a significant challenge. However, existing data-driven anomaly detection methods are dedicated to improving the performance of abnormal time interval detection that fails to localize abnormal time series within abnormal time intervals. In this study, we design a novel anomaly localization method based on graph embedding and apply it to the HVAC system. The proposed method aims to learn feature representations of time intervals and time series by graph embedding and to detect abnormal time intervals and time series by identifying outlier degrees of embedding vectors. It constructs dynamic graphs representing time intervals by extracting correlations between multivariate time series and proposes an entropy-based graph embedding model to embed the dynamic graph. In the embedding space, the local outlier factor is used to measure abnormal embedding vectors to detect abnormal time intervals. After detecting abnormal time intervals, large-scale information network embedding is applied to embed each time series within the abnormal time interval for detecting abnormal time series. Our method is evaluated on three different abnormal patterns of the HVAC system. The experimental results reveal that the F1 scores of our method beat the existing methods in terms of abnormal time interval detection and abnormal time series detection of the HVAC system.

Approximation by interval-decomposables and interval resolutions of persistence modules

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

Prediction of acute hypertensive episodes in critically ill patients

Prevention and treatment of complications are the backbone of medical care, particularly in critical care settings. Early detection and prompt intervention can potentially prevent complications from occurring and improve outcomes. In this study, we use four longitudinal vital signs variables of intensive care unit patients, focusing on predicting acute hypertensive episodes (AHEs). These episodes represent elevations in blood pressure and may result in clinical damage or indicate a change in a patient’s clinical situation, such as an elevation in intracranial pressure or kidney failure. Prediction of AHEs may allow clinicians to anticipate changes in the patient’s condition and respond early on to prevent these from occurring. Temporal abstraction was employed to transform the multivariate temporal data into a uniform representation of symbolic time intervals, from which frequent time-intervals-related patterns (TIRPs) are mined and used as features for AHE prediction. A novel TIRP metric for classification, called coverage, is introduced that measures the coverage of a TIRP’s instances in a time window. For comparison, several baseline models were applied on the raw time series data, including logistic regression and sequential deep learning models, are used. Our results show that using frequent TIRPs as features outperforms the baseline models, and the use of the coverage, metric outperforms other TIRP metrics. Two approaches to predicting AHEs in real-life application conditions are evaluated: using a sliding window to continuously predict whether a patient would experience an AHE within a specific prediction time period ahead, our models produced an AUC-ROC of 82%, but with low AUPRC. Alternatively, predicting whether an AHE would generally occur during the entire admission resulted in an AUC-ROC of 74%.

Modeling of logistics of war reserve stockpiling for successful combat operations

This paper formulates and solves a multivariate problem related to modeling the logistics of war reserve stockpiling for successful combat operations in the armed conflict area. The relevance of the study is related to a comprehensive solution to the problem of war reserve stockpiling to fulfill the objectives of a military operation, considering the capabilities of materiel suppliers, complex logistics of war materiel transportation to armed conflict areas, and military threats arising from the martial law in the country. The study creates a set of models that allow: prioritizing war materiel types for reserve stockpiling; formulating requirements for the size of reserves; selection of a rational structure of logistics supply chain, considering the risks of military threats; ensuring the success of combat operations in the armed conflict area. The article analyzes the existing problems of war reserve stockpiling under martial law, which arose due to: the variety of war materiel; small supply batches; different materiel manufacturers and suppliers located at a great distance from the armed conflict area; heterogeneous transport environment of supply; risks associated with military threats during war materiel reserve stockpiling. A method of presenting the importance of certain war materiel types for supply to the armed conflict area is proposed, considering the combat capability of individual weapons and assessments of military experts. The requirements for the amount of war materiel reserve in the interval representation, from the minimum to the maximum value of the inventory, have been formed. The minimum war materiel reserve to be used in the armed conflict area provides the necessary parity of military forces. Simultaneously, the nature of hostilities does not change, but the risks associated with the enemy's actions to destroy the war materiel reserve (war of reserves) may manifest themselves. The maximum war materiel reserve provides confidence in achieving the objectives of a military operation in the armed conflict area, but is difficult to implement due to the limited capabilities of weapons and materiel suppliers and existing military threats. An optimization model for the war reserve stockpiling was created, considering the limited capabilities of suppliers, permissible time for reserve stockpiling, and the risks that may arise from military threats and their impact on the logistics of war materiel supply (war of logistics). An optimization model for the war reserve stockpiling was developed, considering the combat capability of certain types of weapons, which makes it possible to create asymmetry in the military parity of forces due to the increased combat characteristics of modern weapons (quality over quantity). Due to the multitude of materiel manufacturers (possible suppliers) and various variants for the formation of supply chain, the task of enumeration of possible variants for the composition and structures of the logistics supply chain for war reserve stockpiling was formed and solved. The number of warehouse variants and logistics supply chain structures is estimated using the methods of enumeration theory, and a set of variants is formed for subsequent comparison and selection of a rational option. An optimization model for the formation of logistics links in the supply chain was created to form the necessary war reserve stocks in the armed conflict area. The example of HIMARS MLRS stockpiling illustrates the effectiveness of the proposed approach for successful combat operations in the armed conflict area, using both quantitative and qualitative assessments when comparing possible stockpiling variants. The scientific novelty of the study is related to the development of a set of original optimization models, models of the variant enumeration for the structures of the logistics supply chain, which allows scientifically sound formulation of requirements for the size of the war reserve stock for successful fulfillment of the objectives of a military operation in the armed conflict area, considering the capabilities of materiel suppliers, long supply chains in a heterogeneous transport environment, short delivery time, and the risks of military threats. It is advisable to use the results of the study for planning war reserve stock, for their effective use in the combat zone, justifying the composition of materiel manufacturers and suppliers, forming a logistics supply chain, which will ensure the success of combat operations in the armed conflict area.

Gauging energy poverty in developing countries with a composite metric of electricity access

Energy poverty is a significant development issue that is not univocally interpreted. In many parts of the world, people do not have access to modern and reliable energy, which can be a big problem, particularly for vulnerable and developing countries. When vulnerable people do not have access to energy, they will often not be able to get the power they need to improve their lives. In addition, they may not be able to get proper food, education, health, sanitation, or basic needs for daily living. We use interval-based composite indicators and triplex representations of intervals to measure and assess access to electricity in 54 developing countries as a concept approximating relevant aspects of energy poverty. The proposed composite metric is innovative because it accurately quantifies how much electricity is available to people and how resilient and vulnerable people in developing countries are when they lose access to energy. After comparing the different representations, we found that one group of developing countries is more vulnerable to national and international events than the other. However, other countries are more resilient to electricity access problems. Conflicts and wars can impact the index used to measure energy poverty. However, as measured by the index, these factors contribute to poor performance.

Local Characterizations for Decomposability of 2-Parameter Persistence Modules

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.

A new graph parameter to measure linearity

AbstractConsider a sequence of Lexicographic Breadth‐First‐Search vertex orderings where each ordering is used to break ties for . Since the total number of vertex orderings of a finite graph is finite, this sequence must end in a cycle of vertex orderings. The possible length of this cycle is the main subject of this work. Intuitively, we prove for graphs with a known notion of linearity (e.g., interval graphs with their interval representation on the real line), this cycle cannot be too big, no matter which vertex ordering we start with. More precisely, it was conjectured by Dusart and Habib that for cocomparability graphs, the size of this cycle is always 2, independent of the starting order. Furthermore Stacho asked whether for arbitrary graphs, the size of such a cycle is always bounded by the asteroidal number of the graph. In this work, while we answer this latter question negatively, we provide support for the conjecture on cocomparability graphs by proving it for the subclass of domino‐free cocomparability graphs. This subclass contains cographs, proper interval, interval and cobipartite graphs. We also provide simpler independent proofs for each of these cases which lead to stronger results on this subclasses.

Automatic Generation Control with Virtual Synchronous Renewables

As synchronous generators (SGs) are gradually displaced by renewable energy sources (RESs), the frequency stability of power systems deteriorates because RESs, represented by utility-scale solar and wind power sources, do not provide the inertial response, primary frequency response, secondary frequency response, and tertiary frequency regulation. As a result, the remaining SGs may not be sufficient to maintain the power balance and frequency stability. The concept and control strategies of virtual synchronous generators (VSGs) enable the inverter-based wind and solar power sources to emulate the outer characteristics of traditional SGs and participate in the active power and frequency control of power systems. This paper focuses on the automatic generation control (AGC) with virtual synchronous renewables (VSRs). First, the VSR strategy that enables the RESs to participate in AGC is introduced. Second, based on the interval representation of uncertainty, the output of RES is transformed into two portions, i.e., the dispatchable portion and the stochastic portion. In the dispatchable portion, the RESs can participate in AGC jointly with SGs. Accordingly, a security-constrained economic dispatch (SCED) model is built considering the RESs operating in VSR mode. Third, the solution strategy that employs the slack variables to acquire deterministic constraints is introduced. Finally, the proposed SCED model is solved based on the 6-bus and 39-bus systems. The results show that, compared with the maximum power point tracking (MPPT) mode, VSRs can participate in the active power and frequency control jointly with SGs, increase the maximum penetration level of RESs, and decrease the operating cost.

Temporal patterns selection for All-Cause Mortality prediction in T2D with ANNs

Mortality prevention in T2D elderly population having Chronic Kidney Disease (CKD) may be possible thorough risk assessment and predictive modeling. In this study we investigate the ability to predict mortality using heterogeneous Electronic Health Records data. Temporal abstraction is employed to transform the heterogeneous multivariate temporal data into a uniform representation of symbolic time intervals, from which then frequent Time Intervals Related Patterns (TIRPs) are discovered. However, in this study a novel representation of the TIRPs is introduced, which enables to incorporate them in Deep Learning Networks. We describe here the use of iTirps and bTirps, in which the TIRPs are represented by a integer and binary vector representing the time respectively. While bTirp represents whether a TIRP’s instance was present, iTirp represents whether multiple instances were present. While the framework showed encouraging results, a major challenge is often the large number of TIRPs, which may cause the models to under-perform. We introduce a novel method for TIRPs’ selection method, called TIRP Ranking Criteria (TRC), which is consists on the TIRP’s metrics, such as the differences in its recurrences, its frequencies, and the average duration difference between the classes. Additionally, we introduce an advanced version, called TRC Redundant TIRP Removal (TRC-RTR), TIRPs that highly correlate are candidates for removal. Then the selected subset of iTirp/bTirps is fed into a Deep Learning architecture like a Recurrent Neural Network or a Convolutional Neural Network. Furthermore, a predictive committee is utilized in which raw data and iTirp data are both used as input. Our results show that iTirps-based models that use a subset of iTirps based on the TRC-RTR method outperform models that use raw data or models that use full set of discovered iTirps.

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

On interval decomposability of 2D persistence modules

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and thin indecomposable representations. We show that over the “equioriented” commutative 2D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an nD persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For 2D persistence modules, we provide an algorithm for determining interval-decomposability, together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative 2D grid. We also propose several heuristics to speed up the computation.

Level-Crossing Sampling with Multiple Temporal Resolutions for Speech Signals

Level-crossing sampling (LCS) is an alternative to the uniform sampling method and is suitable for sparse and bursty signals. LCS, which is adaptive to amplitude variations of the signals, mainly focuses on reducing the total number of samples. Presently, speech signals are sampled with LCS methods that have a constant temporal resolution, and they require time-to-digital converters (TDC) with a wide dynamic range. A long sequence of overflow counts of the TDC counter may be generated during silence regions if the dynamic range is reduced. This paper proposes level-crossing samplers with multiple temporal resolutions for speech signals. Frequency scaling is applied to standard LCS, adaptive LCS and peak sampling. During the silence region, the frequency of the TDC clock is scaled down, which enables the representation of longer intervals with smaller words. The number of bits per sample and hence the total number of samples generated are decreased due to the reduction in the dynamic range of the TDC. Simulation results indicate that the proposed method outperforms the conventional level-crossing samplers, which do not employ frequency scaling. In addition to a significant reduction in data size, this method can also be used for real-time automatic detection of silence regions in a speech signal.

On fuzzy numerical model dealing with the control of glucose in insulin therapies for diabetes via nonsingular kernel in the fuzzy sense

<abstract><p>Very recently, several novel conceptions of fractional derivatives have been proposed and employed to develop numerical simulations for a wide range of real-world configurations with memory, background, or non-local effects via an uncertainty parameter $ [0, 1] $ as a confidence degree of belief. Under the complexities of the uncertainty parameter, the major goal of this paper is to develop and examine the Atangana-Baleanu derivative in the Caputo sense for a convoluted glucose-insulin regulating mechanism that possesses a memory and enables one to recall all foreknowledge. However, as compared to other existing derivatives, this is a vitally important point, and the convenience of employing this derivative lessens the intricacy of numerical findings. The Atangana-Baleanu derivative in the Caputo sense of fuzzy valued functions (FVF) in parameterized interval representation is established initially in this study. Then, it is leveraged to demonstrate that the existence and uniqueness of solutions were verified using the theorem suggesting the Banach fixed point and Lipschitz conditions under generalized Hukuhara differentiability. In order to explore the regulation of plasma glucose in diabetic patients with impulsive insulin injections and by monitoring the glucose level that returns to normal in a finite amount of time, we propose an impulsive differential equation model. It is a deterministic mathematical framework that is connected to diabetes mellitus and fractional derivatives. The framework for this research and simulations was numerically solved using a numerical approach based on the Adams-Bashforth-Moulton technique. The findings of this case study indicate that the fractional-order model's plasma glucose management is a suitable choice.</p></abstract>