Piecewise Linear Research Articles

Interpretable (not just posthoc-explainable) medical claims modeling for discharge placement to reduce preventable all-cause readmissions or death

We developed an inherently interpretable multilevel Bayesian framework for representing variation in regression coefficients that mimics the piecewise linearity of ReLU-activated deep neural networks. We used the framework to formulate a survival model for using medical claims to predict hospital readmission and death that focuses on discharge placement, adjusting for confounding in estimating causal local average treatment effects. We trained the model on a 5% sample of Medicare beneficiaries from 2008 and 2011, based on their 2009–2011 inpatient episodes (approximately 1.2 million), and then tested the model on 2012 episodes (approximately 400 thousand). The model scored an out-of-sample AUROC of approximately 0.75 on predicting all-cause readmissions—defined using official Centers for Medicare and Medicaid Services (CMS) methodology—or death within 30-days of discharge, being competitive against XGBoost and a Bayesian deep neural network, demonstrating that one need-not sacrifice interpretability for accuracy. Crucially, as a regression model, it provides what blackboxes cannot—its exact gold-standard global interpretation, explicitly defining how the model performs its internal “reasoning” for mapping the input data features to predictions. In doing so, we identify relative risk factors and quantify the effect of discharge placement. We also show that the posthoc explainer SHAP provides explanations that are inconsistent with the ground truth model reasoning that our model readily admits.

Predicting fundamental gaps accurately from density functional theory with non-empirical local range separation.

We present an exchange-correlation approximation in which the Coulomb interaction is split into long- and short-range components and the range separation is determined by a non-empirical density functional. The functional respects important constraints, such as the homogeneous and slowly varying density limits, leads to the correct long-range potential, and eliminates one-electron self-interaction. Our approach is designed for spectroscopic purposes and closely approximates the piecewise linearity of the energy as a function of the particle number. The functional's accuracy for predicting the fundamental gap in generalized Kohn-Sham theory is demonstrated for a large number of systems, including organic semiconductors with a notoriously difficult electronic structure.

Optimal dispatch of integrated electricity and heating systems considering the quality-quantity regulation of heating systems to promote renewable energy consumption

The integrated electricity and heating system (IEHS) can improve energy efficiency and promote renewable energy consumption. When quality-quantity regulation (QQR) is adopted, namely adjusting hydraulic and thermal conditions of heating system, the increase in operation flexibility is more remarkable. However, present IEHS dispatch models considering QQR overlook hydraulic characteristics of valves and variable frequency pumps resulting in inexecutable dispatch results easily and hindering IEHS’s flexibility potential. Furthermore, loop networks and the access of multiple heat sources in heating systems may lead to flow reversal, but current research adopts a fixed flow direction which suppresses the potential of flow reversal to improve flexibility. In this paper, we propose an IEHS dispatch model considering QQR to promote renewable energy consumption, which considers hydraulic characteristics of valves and variable frequency pumps, and flow reversal. Specifically, we introduce two binary flow direction labels for every branch in heating system model and then the IEHS dispatch model which can deal with flow reversal is established. A sequential solution process combining linear and nonlinear optimization is formulated to overcome the non-convex feature of IEHS dispatch model. Specifically, piecewise linearization and piecewise McCormick relaxation are combined to handle complex nonlinear terms in the dispatch model. Therefore, a mixed integer linear programming model is obtained and solved, of which results are used as initial values for nonlinear optimization. Results in the case study show that operation cost is decreased by 0.97% and renewable power consumption rate is increased from 83.31% to 84.42% after considering valve adjustment. Operation cost is further decreased by 6.06% and renewable power consumption rate is increased to 94.04% after considering flow reversal.

A ReLU-based linearization approach for maximizing oil production in subsea platforms: An application to flow splitting

Oil and gas extraction from subsea reservoirs is a complex process that is highly reliant on accurate mathematical modeling for optimization. A key challenge is identifying control settings to maximize oil production while accounting for the nonlinear behavior and constraints of the underlying physical systems. This paper introduces a new method using Rectified Linear Units (ReLU) neural networks to model these complexities. We focus on a subsea system connecting multiple oil wells to a manifold with multiple headers, explicitly modeling flow splitting to optimize oil yield. Our approach employs mixed-integer reformulations of ReLU neural networks to approximate pressure-drop functions in risers, allowing efficient optimization of production platforms considering routing possibility and flow-splitting operations. Computational analysis employing a branch-and-cut approach and p-split partitioning (for the ReLU model) shows that our ReLU-based methodology converges faster than standard piecewise-linear (PWL) approaches while delivering comparably effective results. Notably, ReLU found solutions in scenarios where PWL failed to find an incumbent solution within the time limit. These results underscore a significant advancement in the optimization of offshore production platforms by using ReLU-based models.

Bidding Strategy for Wind and Thermal Power Joint Participation in the Electricity Spot Market Considering Uncertainty

As the proportion of new energy sources, such as wind power, in the electricity system rapidly increases, their participation in spot market competition has become an inevitable trend. However, the uncertainty of clearing price and wind power output will lead to bidding deviation and bring revenue risks. In response to this, a bidding strategy is proposed for wind farms to participate in the spot market jointly with carbon capture power plants (CCPP) that have flexible regulation capabilities. First, a two-stage decision model is constructed in the day-ahead market and real-time balancing market. Under the joint bidding mode, CCPP can help alleviate wind power output deviations, thereby reducing real-time imbalanced power settlement. On this basis, a tiered carbon trading mechanism is introduced to optimize day-ahead bidding, aiming at maximizing revenue in both the electricity spot market and carbon trading market. Secondly, conditional value at risk (CVaR) is introduced to quantitatively assess the risks posed by uncertainties in the two-stage decision model, and the risk aversion coefficient is used to represent the decision-maker’s risk preference, providing corresponding strategies. The model is transformed into a mixed-integer linear programming model using piecewise linearization and McCormick enveloping. Finally, the effectiveness of the proposed model and methods is verified through numerical examples.

Floating-Point Approximation Enabling Cost-Effective and High-Precision Digital Implementation of FitzHugh-Nagumo Neural Networks.

The study of neuron interactions and hardware implementations are crucial research directions in neuroscience, particularly in developing large-scale biological neural networks. The FitzHugh-Nagumo (FHN) model is a popular neuron model with highly biological plausibility, but its complexity makes it difficult to apply at scale. This paper presents a cost-saving and improved precision approximation algorithm for the digital implementation of the FHN model. By converting the computational data into floating-point numbers, the original multiplication calculations are replaced by adding the floating-point exponent part and fitting the mantissa part with piecewise linear. In the hardware implementation, shifters and adders are used, greatly reducing resource overhead. Implementing FHN neurons by this approximation calculations on FPGA reduces the normalized root mean square error (RMSE) to 3.5% of the state-of-the-art (SOTA) while maintaining a performance overhead ratio improvement of 1.09 times. Compared to implementations based on approximate multipliers, the proposed method achieves a 20% reduction in error at the cost of a 2.8% increase in overhead.This model gained additional biological properties compared to LIF while reducing the deployment scale by only 9%. Furthermore, the hardware implementation of nine coupled circular networks with eight nodes and directional diffusion was carried out to demonstrate the algorithm's effectiveness on neural networks. The error decreased to 60% compared to the single neuron of the SOTA. This hardware-friendly algorithm allows for the low-cost implementation of high-precision hardware simulation, providing a novel perspective for studying large-scale, biologically plausible neural networks.

Longitudinal effects of data-based instructional changes for students with intensive learning needs: A piecewise linear–linear mixed-effects modeling approach.

Longitudinal effects of data-based instructional changes for students with intensive learning needs: A piecewise linear–linear mixed-effects modeling approach.

Geometric stress functions, continuous and discontinuous

In his work on stress functions Maxwell noted that given a planar truss the internal force distribution may be described by a piecewise linear, C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} continuous version of the Airy stress function. Later Williams and McRobie proposed that one can consider planar moment-bearing frames, where the stress function need not be even C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} continuous. The two authors also proposed a discontinuous stress function for the analysis of space-frames, which however suffers from incompleteness. This paper provides a discontinuous stress function for n-dimensional space frames that is complete and minimal, along with its derivation from an n-dimensional continuous stress function. The continuous stress function generalizes both Günther’s and Maxwell’s stress functions.

DC‐bias and temperature included CSWPL model for RF power transistors

AbstractA novel frequency domain behavioral modeling method for gallium‐nitride (GaN) devices is presented in this article. By utilizing a multi‐dimensional polynomial function, the proposed technique interpolates DC‐bias voltage and temperature based on the Canonical section‐wise piecewise linear (CSWPL) model framework. A detailed description of the model's theory is provided. With data from 10‐W GaN devices, the model was implemented in commercial software and validated through both DC and radio frequency (RF) tests. A robust predictive ability is demonstrated by the obtained results, thus proving the accuracy of the developed modeling method. This model is superior to the standard CSWPL model in that it is capable of predicting transistor behavior at different bias voltages and temperatures using a single set of parameters, thereby greatly reducing the complexity of the model and the time required for extraction.

Approaches to simplify industrial energy models for operational optimisation

Minimising energy consumption in today's industrial sector is a crucial objective for achieving established climate goals. One effective strategy to enhance efficiency is optimising energy system operations within industries. The initial step in establishing operational optimisation involves developing a comprehensive model of the energy system. This model necessitates a specific structure to meet optimisation requirements. However, creating a model from scratch incurs substantial effort. While numerous companies possess energy models, they often lack the requisite structure for optimisation. Consequently, simplifying existing models can significantly reduce the effort needed to implement operational optimisation.This paper investigates the simplification of intricate industrial energy system models for optimisation purposes. The subsequent sections analyse two distinct approaches. One approach involves linearisation, while the other utilises neural networks. To facilitate a comparative analysis of these approaches, a reference model is developed. The assessment of these methodologies includes an investigation into optimisation robustness, computation time, accuracy concerning the reference model, and the effort required for developing and maintaining the simplified models. It proved that both approaches are suitable for operational optimisation. Linearisation exhibits superior computational efficiency compared to the neural network approach. The linearisation modelling approach together with the optimisation only required a few milliseconds for the calculation. The neural network approach needed 3 h for the calculation of the optimum with the genetic algorithm. The simulation of the neural network itself only required a few milliseconds. Hence, an improvement of the genetic algorithm is needed. However, the accuracy of linearisation falls short of that achieved by neural networks. The linearisation achieves a mean average percentage error from only 13%. In comparison the neural network's mean average error is 2.3%. Therefore, the linearisation must be improved. The impact using a piecewise linearisation on the results will be analysed in further research.

Moving Sum Procedure for Change Point Detection under Piecewise Linearity

We propose a computationally and statistically efficient procedure for segmenting univariate data under piecewise linearity. The proposed moving sum (MOSUM) methodology detects multiple change points where the underlying signal undergoes discontinuous jumps and/or slope changes. It controls the family-wise error rate at a given significance level and achieves consistency in multiple change point detection, with a minimax optimal estimation rate when the signal is piecewise linear and continuous, all under weak assumptions permitting serial dependence and heavy-tailedness. Computationally, the complexity of the MOSUM procedure is O(n), which, combined with its good performance on simulated datasets, makes it highly attractive compared to the existing methods. We further demonstrate its good performance on a real data example on rolling element-bearing prognostics.

Theoretical investigation of the free fall of a spherical particle in a viscous fluid using continuous piecewise linearization method

This article presents a theoretical investigation of the problem of free fall of a spherical particle in a viscous fluid. The classic Boussinesq-Basset-Oseen (BBO) model for particle motion in laminar flow was modified for generalized flow by using a drag law that is applicable for 0<Re≤2.0×105. By assuming that the acceleration in the Basset force integral is constant, the Basset force effect was approximated to form an integrated added mass coefficient. Consequently, the integro-differential equation of the BBO model was transformed to a first-order nonlinear ordinary differential equation that accounts for the Basset force effect and was solved using the continuous piecewise linearization method (CPLM). The CPLM algorithm was developed based on the jerk-velocity relationship and is applicable to zero and non-zero initial conditions, steady motion, increasing or decreasing velocities and the corresponding acceleration and jerk responses. The CPLM algorithm was shown to predict published experimental results accurately and compared very well with numerical solutions and existing analytical solutions. Examination of the fall response under varying parameters showed that the fall distance, fall time and terminal velocity depend strongly on the sphere diameter, sphere density, and the density and viscosity of the fluid medium. Also, an analytical solution for the power dissipated in the fluid medium as the sphere falls to reach its terminal velocity was derived. The power dissipated was found to increase exponentially as the initial velocity deviates positively from the terminal velocity.

Real-time clothing insulation level classification based on model transfer learning and computer vision for PMV-based heating system optimization through piecewise linearization

Achieving a balance between energy efficiency and thermal comfort is a key aspect of sustainable building design. Traditional control methods typically maintain indoor air temperature within predetermined limits, disregarding variable factors such as occupancy activity and clothing levels, which influence thermal comfort. Conversely, comfort-based control strategies present an opportunity to automate heating and cooling systems, enabling them to dynamically respond to variations in thermal comfort. To achieve this, real-time information on clothing insulation (and its adjustment) is indispensable for accurately estimating thermal comfort. In this study, we explore the potential of a novel detection approach capable of classifying clothing insulation in real-time and utilizing this information to optimize the operation of building energy systems. By doing so, the proposed method facilitates the delivery of indoor conditions tailored to user requirements and potentially reduces energy wastage. The development of a two-stage, computer vision-based framework for occupancy detection and clothing insulation classification forms the core of this approach. Leveraging deep learning algorithms, this framework performs detection and recognition tasks, even with limited training data effectively, enabling real-time classification of light, medium and heavy clothing. To address the nonlinearity of traditional predicted mean vote (PMV) models, we applied a piecewise linearization approach to our PMV-based optimal control strategy. We evaluated the performance of this detection method through initial experimental field tests conducted in a case study university building. The results demonstrate the proposed method's ability to classify clothing insulation levels and generate real-time profiles. We further analyze the impact of our proposed approach on thermal comfort and energy performance through scenario-based modelling and simulations. The initial results showed the potential of integrating our method with PMV-based controls to enhance thermal comfort and overcome the limitations of predefined values or fixed schedules. However, while our study confirms the feasibility of classifying clothing insulation levels for multiple occupants engaged in diverse indoor activities, it also highlights the need for further refinement to enhance detection accuracy and ensure seamless integration with building energy systems.

Algorithms and complexity for the almost equal maximum flow problem

AbstractIn the equal maximum flow problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set, so called homologous arc sets. In this article, we study the closely related almost equal maximum flow problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called deviation function . We prove that the integer AEMFP is in general ‐complete, and show that even the problem of finding a fractional maximum flow in the case of convex deviation functions is also ‐complete. This is in contrast to the EMFP, which is polynomial time solvable in the fractional case. Additionally, we provide inapproximability results for the integral AEMFP. For the (fractional) concave AEMFP we state a strongly polynomial algorithm for the linear and concave piecewise polynomial deviation function case for a fixed number of homologous sets using a parametric search approach.

Spatial-temporal heterogeneity of spring phenology in boreal forests as estimated by satellite solar-induced chlorophyll fluorescence and vegetation index

Changes in spring phenology in boreal forests around the subarctic region have a profound impact on the global carbon and water cycles. To date, photosynthesis and greenness dynamics in evergreen coniferous forests have been found to be decoupled on a large scale, but whether the spatial heterogeneity of seasonal dynamics in boreal forests is coupled in various datasets remains unclear. Therefore, based on the solar-induced chlorophyll fluorescence (SIF) retrieved by satellite and the modelled SIF (GOME-2 SIF and GOSIF) during 2007–2018, we calculated the start of the growing season (SOS) of boreal forests (45–70°N) and compared the results with 2-band Enhanced Vegetation Index (EVI2) phenological products (MCD12Q2 C6) and flux data (Gross primary productivity, GPP) phenology. The spatial-temporal variations in the SOS were analyzed by linear regression and piecewise linear regression. The results exhibited that GOME-2 SIF showed an SOS of 11 days earlier than GOSIF, 8 days earlier than EVI2 in North America, and 5 days earlier than GOSIF, 3 days earlier than EVI2 in Eurasia. However, all relationships between SOSs of the three remote sensing datasets were significant in North America and Eurasia, and the significance was higher in Eurasia. With increasing latitude and longitude, almost all SOSs estimated from the three datasets significantly increased. Also, both in North America and Eurasia, the SOSs exhibited a trend of being advanced to being delayed from the western to the eastern parts of the continents over 2007–2018. This indicated the spatial consistency of photosynthesis and greenness dynamics in the spring phenology of boreal forests, especially in Eurasia. In this study, the validity of the spatial-temporal variation in spring phenology of boreal forests based on GOME-2 SIF retrieved from satellites and MCD12Q2 C6 was confirmed, and further exploration is needed to determine their potential advantages.

A joint solution framework for the optimal operation and evaluation of integrated energy systems based on exergy

Exergy analysis provides a unified scale for evaluating the quality and quantity of energy, and is suitable and essential for scientifically and rationally guiding the optimization and evaluation of integrated energy systems with intricate multi-component configurations. The present work proposes an exergy analysis model and joint solution framework for integrated energy systems based on exergy, and presents an efficient and accurate solution method. The exergy analysis model is analogous to the energy-based model and possesses a succinct form. This model is then applied to establish an input exergy model defining the exergy of inputs, a destroyed exergy model defining the exergy lost due to the conversions, transmissions, and storages, and a benefit exergy model defining the remaining useful exergy for the loads. The joint solution framework optimizes the system from the basis of exergy conservation, and evaluates the destroyed exergy of each link and the prices of the various exergy products from an economic perspective based on the established exergy analysis model. The complexity of the framework is then reduced by establishing a sequential optimization and evaluation model. The optimization model is solved accurately and efficiently using a stringent cutting plane method to address the problem of relaxation inaccuracy, and piecewise linearization to accommodate the non-convex constraints. Numerical computations on a 97-node system demonstrate the accuracy of the proposed models in exergy-based optimization, destruction, and price evaluation, preventing substantial deviations (e.g., a 23% discrepancy in total destroyed exergy). The adopted cost allocation method based on energy quality reflects fair pricing, rectifying conventional pricing disparities (e.g., a 47.56% underestimation of the electric load using energy-based pricing). Therefore, the accuracy and efficiency of the developed exergy analysis model, solution framework, and solution method are validated.

Support vector machines within a bivariate mixed-integer linear programming framework

Support vector machines (SVMs) are a powerful machine learning paradigm, performing supervised learning for classification and regression analysis. A number of SVM models in the literature have made use of advances in mixed-integer linear programming (MILP) techniques in order to perform this task efficiently. In this work, we present three new models for SVMs that make use of piecewise linear (PWL) functions. This allows effective separation of data points where a simple linear SVM model may not be sufficient. The models we present make use of binary variables to assign data points to SVM segments, and hence fit within a recently presented framework for machine learning MILP models. Alongside presenting an inbuilt feature selection operator, we show that the models can benefit from robust inbuilt outlier detection. Experimental results show when each of the presented models is effective, and we present guidelines on which of the models are preferable in different scenarios.

Density functional theory of material design: fundamentals and applications—II

Abstract This is the second and the final part of the review on density functional theory (DFT), referred to as DFT-II. In the first review, DFT-I, we have discussed wavefunction-based methods, their complexity, and basics of density functional theory. In DFT-II, we focus on fundamentals of DFT and their implications for the betterment of the theory. We start our presentation with the exact DFT results followed by the concept of exchange-correlation (xc) or Fermi-Coulomb hole and its relationship with xc energy functional. We also provide the exact conditions for the xc-hole, xc-energy and xc-potential along with their physical interpretation. Next, we describe the extension of DFT for non-integer number of electrons, the piecewise linearity of total energy and discontinuity of chemical potential at integer particle numbers, and derivative discontinuity of the xc potential, which has consequences on fundamental gap of solids. After that, we present how one obtains more accurate xc energy functionals by going beyond the LDA. We discuss the gradient expansion approximation (GEA), generalized gradient approximation (GGA), and hybrid functional approaches to designing better xc energy functionals that give accurate total energies. However, these functionals fail to predict properties like the ionization potential and the band gap. Thus, we next describe different methods of modelling these potentials and results of their application for calculation of the band gaps of different solids to highlight accuracy of different xc potentials. Finally, we conclude with a glimpse on orbital-free density functional theory and the machine learning approach.

Efficient continuous piecewise linear regression for linearising univariate non-linear functions

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

The magnitude and direction of the relationship between risk factor and cognition depends on age: a pooled analysis of 5 community-based studies

The mixed evidence of the association between high levels of cardiovascular risk factors (CVRF) and the risk for cognitive impairment may be due to confounding of age across studies. We pooled and harmonized individual-level data (30,967 persons, age range 42–96 years) from five prospective cohorts to investigate by 1 year age increments to investigate whether or not there is change in slope describing the association of CVRF to a cognitive outcome (Digit Symbol Substitution Test; DSST). The CVRF included: systolic and diastolic blood pressure, total cholesterol, fasting glucose and body mass index. Linear and quadratic piecewise regression models were fit to the trajectory patterns of these slopes (betas). The pattern of yearly slope changes showed higher CVRF were associated with lower DSST, but associations attenuated toward zero as age increased for all but DBP where 1 year slopes for DBP changed direction from negative to positive from mid- to late-age. Age is not only a driver of cognitive decline—age also modifies the direction and strength of the association of cognitive function to CVRF and cohort age may be one reason why the evidence for CVRF-CD association is mixed.