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A I I E Transactions 2023年第8期55卷

作者： John Alasdair Warwicker Steffen Rebennack Stochastic Optimization Institute of Operations Research Karlsruhe Institute of Technology Karlsruhe Germany

Using piecewise linear (PWL) functions to model discrete data has applications for example in healthcare, engineering and pattern recognition. Recently, mixed-integer linear programming (MILP) approaches have been used to optimally fit continuous PWL functions. We extend these formulations to allow for outliers. The resulting MILP models rely on binary variables and big-Mconstructs to model logical implications. The combinatorial Benders decomposition (CBD) approach removes the dependency on the big-Mconstraints by separating the MILP model into a master problem of the complicating binary variables and a linear sub problem over the continuous variables, which feeds combinatorial solution information into the master problem. We use the CBD approach to decompose the proposed MILP model and solve for optimal PWL functions. Computational results show that vast speedups can be found using this robust approach, with problem-specific improvements including smart initialization, strong cut generation and special branching approaches leading to even faster solve times, up to more than 12,000 times faster than the standard MILP approach.

关键词： piecewise linear function combinatorial Benders decomposition mixed-integer linear programming (MILP) function fitting outlier detection

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Improved Ultra-Rapid UT1-UTC Determination and Its Preliminary Impact on GNSS Satellite Ultra-Rapid Orbit Determination

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REMOTE SENSING 2020年第21期12卷 1-16页

作者： Ye, Fei Yuan, Yunbin Deng, Zhiguo Chinese Acad Sci Innovat Acad Precis Measurement Sci & Technol State Key Lab Geodesy & Earths Dynam Wuhan 430077 Peoples R China Deutsch GeoForschungsZentrum TELEGRAFENBERG 1 D-14473 Potsdam Germany

Errors in ultra-rapid UT1-UTC primarily affect the overall rotation of spatial datum expressed by GNSS (Global Navigation Satellite System) satellite ultra-rapid orbit. In terms of existing errors of traditional strategy, e.g., piecewise linear functions, for ultra-rapid UT1-UTC determination, and the requirement to improve the accuracy and consistency of ultra-rapid UT1-UTC, the potential to improve the performance of ultra-rapid UT1-UTC determination based on an LS (Least Square) + AR (Autoregressive) combination model is explored. In this contribution, based on the LS+AR combination model and by making joint post-processing/rapid UT1-UTC observation data, we propose a new strategy for ultra-rapid UT1-UTC determination. The performance of the new strategy is subsequently evaluated using data provided by IGS (International GNSS Services), iGMAS (international GNSS Monitoring and Assessment System), and IERS (International Earth Rotation and Reference Systems Service). Compared to the traditional strategy, the numerical results over more than 1 month show that the new strategy improved ultra-rapid UT1-UTC determination by 29-43%. The new strategy can provide a reference for GNSS data processing to improve the performance of ultra-rapid products.

关键词： GNSS ultra-rapid UT1-UTC determination LS plus AR piecewise linear function ultra-rapid orbit determination

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Flexible piecewise linear model for investigating dose-response relationship in meta-analysis: Methodology, examples, and comparison

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JOURNAL OF EVIDENCE BASED MEDICINE 2019年第1期12卷 63-68页

作者： Xu, Chang Thabane, Lehana Liu, Tongzu Borhan, Asm Sun, Xin Sichuan Univ West China Hosp Chinese Evidence Based Med Ctr 37 Guoxue Rd Chengdu 610041 Peoples R China Sichuan Univ West China Hosp Chinese Cochrane Ctr Chengdu Peoples R China McMaster Univ Dept Hlth Res Methods Evidence & Impact Hamilton ON Canada St Josephs Healthcare Biostat Unit Father Sean OSullivan Res Ctr Hamilton ON Canada Wuhan Univ Dept Urol Zhongnan Hosp Wuhan Peoples R China

Objectives Dose-response meta-analysis (DRMA) is widely employed in establishing the potential dose-response relationship between continuous exposures and disease outcomes. However, there is no valid DRMA method readily for discrete exposures, especially when the possible dose-response trend not likely to be linear. We proposed a piecewise linear DRMA model as a solution to this issue. Methods We illustrated the methodology of piecewise linear model in both one-stage DRMA approach and two-stage DRMA approach. The method by testing the equality of slopes of each piecewise was employed to judge if there is "piecewise effect" against a simple linear trend. We then used sleep (continuous exposure) and parity (discrete exposure) data as examples to illustrate how to apply the model in DRMA using the Stata code attached. We also empirically compared the slopes of piecewise linear model with simple linear as well as restricted cubic spline model. Results Both one-stage and two-stage piecewise linear DRMA model fitted well in our examples, and the results were similar. Obvious "piecewise effects" were detected in both the two samples by the method we used. In our example, the new model showed a better fitting effect and practical, reliable results compared to the simple linear model, while similar results for to restricted cubic spline model. Conclusion piecewise linear function is a valid and straightforward method for DRMA and can be used for discrete exposures, especially when the simple linear function is under fitted. It represents a superior model to linear model in DRMA and may be an alternative model to the nonlinear model.

关键词： dose-response meta-analysis discrete exposure piecewise linear function

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Construction of flux-controlled memristor and circuit simulation based on smooth cellular neural networks module

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IET CIRCUITS DEVICES & SYSTEMS 2018年第3期12卷 263-270页

作者： Zhang, Xiaohong Jiang, Wei Jiangxi Univ Sci & Technol Sch Informat Engn Ganzhou 341000 Peoples R China

A new-type four-dimensional cellular neural network (CNN) is designed, whose piecewise linear function is replaced by smooth continuous function as CNN output module, theoretical analysis proves the new system has chaotic characteristics. By adding two cells, this study generates flux-controlled memristor and creates a brand new six-dimensional memristive CNNs. With the purpose of verifying the validity of this scheme, universal electronic components are adopted to build a memristive module and apply them to the integrity circuit of this system. Multisim circuit simulation software shows that this memristive module has the characteristic of the hysteresis loop as well, and circuit output curves are approximately in agreement with those of MATLAB numerical calculation results within the range of the permitted errors. Finally, via analyses of phase trajectories, equilibrium points, bifurcation diagrams, Lyapunov exponents, and dimension, this study demonstrates that this new type memristive CNNs possesses more abundant dynamic characteristics.

关键词： circuit simulation cellular neural nets neural chips piecewise linear techniques chaos memristor circuits flux-controlled memristor circuit simulation smooth cellular neural network module four-dimensional cellular neural network piecewise linear function smooth continuous function chaotic characteristics six-dimensional memristive CNNs universal electronic components memristive module integrity circuit Multisim circuit simulation software hysteresis loop circuit output curves Matlab numerical calculation phase trajectories analysis equilibrium points bifurcation diagrams Lyapunov exponents

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Nonlinear model predictive control based on piecewise linear Hammerstein models

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NONlinear DYNAMICS 2018年第3期92卷 1001-1021页

作者： Zhang, Jian Chin, Kwai-Sang Lawrynczuk, Maciej City Univ Hong Kong Dept Syst Engn & Engn Management Kowloon Hong Kong Peoples R China Warsaw Univ Technol Inst Control & Computat Engn Fac Elect & Informat Technol Warsaw Poland

This paper develops a nonlinear model predictive control (MPC) algorithm for dynamic systems represented by piecewise linear (PWL) Hammerstein models. At each sampling instant, the predicted output trajectory is linearized online at an assumed input trajectory such that the control actions can be easily calculated by solving a quadratic programming optimization problem, and such linearization and optimization may be repeated a few times for good linear approximation accuracy. A three-step procedure is developed to linearize a PWL function, where the derivatives of a PWL function are obtained by a computationally efficient look-up table approach. Unlike many existing MPC algorithms for Hammerstein systems, it does not require the inversion of static nonlinearity and can directly cope with input constraints even in multivariable systems. Two benchmark chemical reactors are studied to illustrate the effectiveness of the proposed algorithm.

关键词： Nonlinear model predictive control Hammerstein model piecewise linear function Multistep linearization Chemical reactor

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Rule number and approximation of the hybrid fuzzy system based on binary tree hierarchy

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2018年第6期9卷 979-991页

作者： Wang, Guijun Yang, Yang Li, Xiaoping Tianjin Normal Univ Sch Math Sci Tianjin 300387 Peoples R China Yaohua Middle Sch Tianjin City Math Grp Tianjin 300000 Peoples R China

To effectively avoid internal rule explosion of a fuzzy system or computer memory overflow caused by increased input variables, a hybrid fuzzy system is established by unifying the Takagi-Sugeno and the Mamdani fuzzy systems based on a binary tree hierarchical method. This method can greatly reduce the total number of rules within the system. Firstly, a calculation formula of the total number of rules for the hybrid fuzzy system is given, by comparing with other layered systems, the total number of rules based on the binary tree hierarchy has the largest decline. Secondly, a new K-integral norm is redefined by introducing a K-quasi-subtraction operator. Using the piecewise linear function the approximation capability of the hybrid fuzzy system after hierarchy to a kind of integrable functions is studied. Finally, the binary tree hierarchical structure expressions of the hybrid fuzzy system are given through two simulation examples.

关键词： Binary tree hierarchy The total number of rules Hybrid fuzzy system piecewise linear function K-integral norm Approximation

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The computation of conformal map by harmonic diffeomorphisms between surfaces

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APPLIED MATHEMATICS AND COMPUTATION 2018年 333卷 536-546页

作者： Zhang, Jialing Qian, Kun Kunming Univ Sci & Technol Fac Civil Engn & Mech Kunming 650500 Yunnan Peoples R China

Harmonic map is the critical point of the corresponding integral with respect to the square norm of the gradient or energy density. The harmonic energy defined on Riemann surfaces will decrease along its gradient line direction and reduce to the limit conformal map. Harmonic map between surfaces is a diffeomorphism which is associated to a unique Beltrami differential. So a harmonic diffeomorphism sequence corresponds to an Beltrami differential sequence. When boundary map is restricted on a unit circle, the Beltrami differential sequence changes with constant conformal modulus. In this paper, we consider the conformal map between surfaces by the Beltrami differential sequence with constant conformal modulus, which is equivalent to a decreasing harmonic energy sequence with fixed boundary correspondence, and provide the corresponding algorithms for numerical computation. Furthermore, we will discuss the convergence of proposed algorithms, which provides theoretical foundation for numerical experiments. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Harmonic map Conformal map Beltrami differential Gradient line method piecewise linear function Covariant derivative

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Examples of the Best piecewise linear Approximation with Free Nodes

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VESTNIK ST PETERSBURG UNIVERSITY-MATHEMATICS 2018年第4期51卷 380-385页

作者： Malozemov, V. N. Tamasyan, G. Sh. St Petersburg State Univ St Petersburg 199034 Russia

The monograph published by ***. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

关键词： Chebyshev approximations piecewise linear function partition with equal deviations

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Classification of short single-lead electrocardiograms (ECGs for atrial fibrillation detection using piecewise linear spline and XGBoost

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PHYSIOLOGICAL MEASUREMENT 2018年第10期39卷 104006页

作者： Chen, Yao Wang, Xiao Jung, Yonghan Abedi, Vida Zand, Ramin Bikak, Marvi Adibuzzaman, Mohammad Purdue Univ Regenstrief Ctr Healthcare Engn W Lafayette IN 47907 USA Purdue Univ Dept Stat W Lafayette IN 47907 USA Purdue Univ Dept Ind Engn W Lafayette IN 47907 USA Geisinger Med Ctr Danville PA 17822 USA Indiana Univ Sch Med Indianapolis IN 46202 USA

Objective: Detection of atrial fibrillation is important for risk stratification of stroke. We developed a novel methodology to classify electrocardiograms (ECGs) to normal, atrial fibrillation and other cardiac dysrhythmias as defined by the PhysioNet Challenge 2017 . Approach: More specifically, we used piecewise linear splines for the feature selection and a gradient boosting algorithm for the classifier. In the algorithm, the ECG waveform is fitted by a piecewise linear spline, and morphological features relating to the piecewise linear spline coefficients are extracted. XGBoost is used to classify the morphological coefficients and heart rate variability features. Main results: The performance of the algorithm was evaluated by the PhysioNet Challenge database (3658 ECGs classified by experts). Our algorithm achieved an average F-1 score of 81% for a 10-fold cross-validation and also achieved 81% for F-1 score on the independent testing set. This score is similar to the top 9th score (81%) in the official phase of the PhysioNet Challenge 2017. Significance: Our algorithm presents a good performance on multi-label short ECG classification with selected morphological features.

关键词： atrial fibrillation electrocardiography piecewise linear function XGBoost

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Response-Time-Optimized Service Deployment: MILP Formulations of Piece-Wise linear functions Approximating Bivariate Mixed-Integer functions

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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT 2017年第1期14卷 121-135页

作者： Keller, Matthias Karl, Holger Univ Paderborn D-33098 Paderborn Germany

A current trend in networking and cloud computing is to provide compute resources at widely distributed sites;this is exemplified by developments such as network function virtualization. This paves the way for wide-area service deployments with improved service quality: e.g., user-perceived response times can be reduced by offering services at nearby sites. But always assigning users to the nearest site can be a bad decision if this site is already highly utilized. This paper formalizes two related decisions of allocating compute resources at different sites and assigning users to them with the goal of minimizing the response times while the total number of resources to be allocated is limited-a non-linear capacitated facility location problem with integrated queuing systems. To efficiently handle its non-linearity, we introduce five linear problem linearizations and adapt the currently best heuristic for a similar scenario to our scenario. All six approaches are compared in experiments for solution quality and solving time. Surprisingly, our best optimization formulation outperforms the heuristic in both time and quality. Additionally, we evaluate the influence of distributions of available compute resources in the network on the response time: the time was halved for some configurations. The presented formulation techniques for our problem linearizations are applicable to a broader optimization domain.

关键词： Cloud computing computer network management virtual network function network function virtualisation resource management placement facility location queueing model piecewise linear function problem linearisation optimisation mixed integer function derivative erlang delay formula next generation networking

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