Piecewise Monotonicity Research Articles

Piecewise nonlinear materials and Monotonicity Principle

This paper deals with the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. The results obtained are relevant because they enable the use of a fast imaging method based on MP, applied to a wide class of problems with two or more materials, at least one of which is nonlinear. The treatment is very general and makes it possible to model a wide range of practical configurations such as superconducting (SC), perfect electrical conducting (PEC) or perfect electrical insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in Corbo Esposito et al (2021 Inverse Problems 37 045012), where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.

Piecewise monotonicity for unary functions in $o$-stable groups

Piecewise monotonicity for unary functions in $o$-stable groups

Piecewise Monotonicity for Unary Functions in o-Stable Groups

Piecewise Monotonicity for Unary Functions in o-Stable Groups

Incorporating monotonic domain knowledge in support vector learning for data mining regression problems

A common problem of data-driven data mining methods is that they might lack considering domain knowledge, despite possibly having high accuracy with respect to the data. As such, prior knowledge plays an important role in many data mining applications. Incorporating prior knowledge into data mining techniques is not trivial and remains a partially open issue drawing much attention. In this paper, we propose a new support vector regression (SVR) model that takes into account the prior knowledge of domain experts in the form of inequalities, which reflect the monotonic relationship between the output and some of the attributes of the input. A dual quadratic programming problem corresponding to the SVR model is derived, along with algorithms for solving it and creating constraints, respectively. The experiment results, which were conducted on two artificial and two practical datasets, show that the proposed model, which considers the monotonicity defined by domain experts, performs better than the original SVR. Moreover, the proposed method is also suitable for prior domain knowledge of piecewise monotonicity.

Power Systems Stability through Piecewise Monotonic Data Approximations – Part 1: Comparative Benchmarking of L1PMA, L2WPMA and L2CXCV in Overhead Medium-Voltage Broadband over Power Lines Networks

This first paper assesses the performance of three well-known piecewise monotonic data approximations ( i.e. , L1PMA, L2WPMA, and L2CXCV) during the mitigation of measurement differences in the overhead medium-voltage broadband over power lines (OV MV BPL) transfer functions. The contribution of this paper is triple. First, based on the inherent piecewise monotonicity of OV MV BPL transfer functions, L2WPMA and L2CXCV are outlined and applied during the determination of theoretical and measured OV MVBPL transfer functions. Second, L1PMA, L2WPMA, and L2CXCV are comparatively benchmarked by using the performance metrics of the percent error sum (PES) and fault PES. PES and fault PES assess the efficiency and accuracy of the three piecewise monotonic data approximations during the determination of transmission BPL transfer functions. Third, the performance of L1PMA, L2WPMA, and L2CXCV is assessed with respect to the nature of faults — i.e. faults that follow either continuous uniform distribution (CUD) or normal distribution (ND) of different magnitudes—. The goal of this set of two papers is the establishment of a more effective identification and restoration of the measurement differences during the OV MV BPL coupling transfer function determination that may significantly help towards a more stable and self-healing power system. Citation:  Lazaropoulos, A. G. (2017). Power Systems Stability through Piecewise Monotonic Data Approximations – Part 1: Comparative Benchmarking of L1PMA, L2WPMA and L2CXCV in Overhead Medium-Voltage Broadband over Power Lines Networks. Trends in Renewable Energy , 3(1), 2-32. DOI: 10.17737/tre.2017.3.1.0029

Best L1 Piecewise Monotonic Data Approximation in Overhead and Underground Medium-Voltage and Low-Voltage Broadband over Power Lines Networks: Theoretical and Practical Transfer Function Determination

This paper investigates the efficiency and accuracy of the best L1 piecewise monotonic data approximation (best L1PMA) in order either to approximate the transfer functions of distribution BPL networks or to reveal the aforementioned transfer functions when various faults occur during their determination. The contribution of this paper is quadruple. First, based on the inherent piecewise monotonicity of distribution BPL transfer functions, a piecewise monotonic data approximation is first applied in BPL networks; best L1PMA is outlined and applied during the determination of distribution BPL transfer functions. Second, suitable performance metrics such as the percent error sum (PES) and fault PES are reported and applied so as to assess the efficiency and accuracy of the best L1PMA during the determination of distribution BPL transfer functions. Third, the factors of distribution BPL networks that influence the performance of best L1PMA are identified. Fourth, the accuracy of the best L1PMA is assessed with respect to its inherent properties, namely, the assumed number of monotonic sections and the nature of faults, that is, faults that follow either continuous uniform distribution (CUD) or normal distribution (ND), during the determination of distribution BPL transfer functions. Finally, best L1PMA may operate as the necessary intermediate antifault method for the theoretical and practical transfer function determination of distribution BPL networks.

Discrete principal‐monotonicity inference for hydro‐system analysis under irregular nonlinearities, data uncertainties, and multivariate dependencies. Part I: methodology development

AbstractHydrological system analyses are challenged by complexities of irregular nonlinearities, data uncertainties, and multivariate dependencies. Among them, the irregular nonlinearities mainly represent inexistence of regular functions for robustly simulating highly complicated relationships between variables. Few existing studies can enable reliable simulation of hydrological processes under these complexities. This may lead to decreased robustness of the constructed models, unfeasibility of suggestions for human activities, and damages to socio‐economy and eco‐environment. In the first of two companion papers, a discrete principal‐monotonicity inference (DPMI) method is proposed for hydrological systems analysis under these complexities. Normalization of non‐normally distributed samples and invertible restoration of modelling results are enabled through a discrete distribution transformation approach. To mitigate data uncertainties, statistical inference is employed to assess the significance of differences among samples. The irregular nonlinearity between the influencing factors (i.e. predictors) and the hydrological variable of interest (i.e. the predictand) is interpreted as piecewise monotonicity. Monotonicity is further represented as principal monotonicity under multivariate dependencies. Based on stepwise classification and cluster analyses, all paired samples representing the responsive relationship between the predictors and the predictand are discretized as a series of end nodes. A prediction approach is advanced for estimating the predictand value given any combination of predictors. The DPMI method can reveal evolvement rules of hydrological systems under these complexities. Reliance of existing hydro‐system analysis methods on predefined functional forms is removed, avoiding artificial disturbances, e.g. empiricism in selecting model functions under irregular nonlinearities, on the modelling process. Both local and global significances of predictors in driving the evolution of hydrological variables are identified. An analysis of interactions among these complexities is also achieved. The understanding obtained from the DPMI process and associated results can facilitate hydrological prediction, guide water resources management, improve hydro‐system analysis methods, or support hydrological systems analysis in other cases. The effectiveness and advantages of DPMI will be demonstrated through a case study of streamflow simulation in Xingshan Watershed, China, in another paper. Copyright © 2016 John Wiley & Sons, Ltd.

Measurement Differences, Faults and Instabilities in Intelligent Energy Systems – Part 1: Identification of Overhead High-Voltage Broadband over Power Lines Network Topologies by Applying Topology Identification Methodology (TIM)

This first paper considers the identification of the structure of overhead high-voltage broadband over power lines (OV HV BPL) network topologies by applying the best L1 Piecewise Monotonic data Approximation (best L1PMA) to measured OV HV BPL transfer functions. Even if measurement differences occur during the determination of an OV HV BPL transfer function, the corresponding OV HV BPL network topology may be revealed through the curve similarity of the best L1PMA result compared with the available records of the proposed OV HV BPL transfer function database. The contribution of this paper is triple. First, based on the inherent piecewise monotonicity of OV HV BPL transfer functions, best L1PMA is first applied during the determination of theoretical and measured OV HV BPL transfer functions. Second, the creation procedure of the OV HV BPL network topology database is demonstrated as well as the curve similarity performance metric (CSPM). Third, the accuracy of the proposed Topology Identification Methodology (TIM) is examined in comparison with the traditional TIM with respect to the nature of the measurement differences during the determination of OV HV BPL transfer functions. Citation: Lazaropoulos, A. G. (2016). Measurement Differences, Faults and Instabilities in Intelligent Energy Systems – Part 1: Identification of Overhead High-Voltage Broadband over Power Lines Network Topologies by Applying Topology Identification Methodology (TIM). Trends in Renewable Energy, 2(3), 85-112. DOI: 10.17737/tre.2016.2.3.0026

Explicit modified mild-slope equation for wave scattering by piecewise monotonic and piecewise smooth bathymetries

The mild-slope equation and modified mild-slope equation (MMSE) have played an important role in modeling the interaction between waves and bathymetries or structures. However, they are implicit equations whose coefficients are defined by an implicit dispersion relation. In this paper, for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness, we are able to transform the implicit MMSE into an explicit equation by introducing a new independent variable. Besides, an alternative form of the implicit MMSE is also transformed into an explicit equation under the same assumptions. These explicit equations make it much easier to obtain analytic solutions to the MMSE. As practical examples, two analytic solutions to the present explicit MMSE for wave reflection by a single linear slope and for wave scattering by a submerged circular truncated shoal are constructed.

Availability of a Periodically Inspected System Maintained through Several Minimal Repairs before a Replacement or a Perfect Repair

The instantaneous availability of a periodically inspected system with several minimal repairs before a replacement or a perfect repair is studied. First, we investigate two concrete minimal repair models with periodic inspection and constant repair time and give their instantaneous availability formulas. Then the instantaneous availability of the proposed model with constant repair times is presented by a set of recursive formulas, and its piecewise monotonicity is also shown. An example is presented to illustrate its application.

Sufficient conditions for the comonotone interpolation of cubic C 2-splines

We consider the problem of interpolation of a function under the condition of the preservation of the nature of its piecewise monotonicity. We give sufficient conditions for the comonotone interpolation by a classical cubic C2-spline in the representation based on the expansion of its first derivative in a basis consisting of B-splines. These conditions allow to determine whether the soobtained spline is comonotone without solving the interpolation problem.

Strict inequalities for the entropy of transitive piecewise monotone maps

Let $T:[0,1]\to [0,1]$ be a piecewise differentiable piecewise monotone map, and let $r>1$. It is well known that if $|T'|\le r$ (respectively $|T'|\ge r$) then $h_{t o p}(T)\le$ log $r$ (respectively $h_{t o p}(T)\ge$ log $r$). We show that if additionally $|T'| < r $ (respectively $ |T'| > r $) on some subinterval and $T$ is topologically transitive then the inequalities for the entropy are strict. We also give examples that the assumption of piecewise monotonicity is essential, even if $T$ is continuous. In one class of examples the dynamical dimension of the whole interval can be made arbitrarily small.

An adaptive algorithm for least squares piecewise monotonic data fitting

The number of peaks and troughs of measurements of smooth function values can be unacceptably larger than the number of turning points of the function, when the measurements are too rough. It is proposed to make the least sum of squares change to the data subject to a limit on the number of sign changes of their first divided differences, but usually a suitable value of this limit is not known in advance. It is shown how to obtain automatically an adequate value for it. A test is included that attempts to distinguish between genuine trends and data errors. Specifically, if there are trends, then the monotonic sections of a tentative approximation are increased by one, otherwise this approximation seems to meet the trends and the calculation terminates. The numerical work required per iteration, beyond the second one, is quadratic in the number of data. Details for establishing the underlying algorithm are specified, numerical results from a simulation are included and the test is compared to some well-known residual tests. An application of the algorithm on identifying turning points and trends of data from the Dow Jones stock exchange index is presented. A Fortran implementation of our algorithm provides shorter computation times in practice than the complexity indicates in theory. Further, the single monotonicity problem has found many applications in statistical data analyses within various contexts. More generally, piecewise monotonicity is a property that occurs in a wide range of underlying functions and some important applications of it may be found in detrending data for identifying periodicities (eg. business cycles), or in estimating turning points of a function that is known only by some measurements of its values.

L1PMA: A Fortran 77 package for best L1 piecewise monotonic data smoothing

Fortran 77 software is presented for the calculation of a best L 1 approximation to n measurements that include random errors by requiring k−1 sign changes in the first divided differences of the approximation or equivalently k monotonic sections, alternately increasing and decreasing. A dynamic programming algorithm separates the measurements into optimal disjoint sections of adjacent data and applies to each section a single L 1 monotonic calculation. The most distinctive feature of the algorithm is that it terminates at a global minimum in at most n 3+O( kn 2) computer operations, although this calculation can exhibit O( n k ) local minima, because the optimal positions of the turning points are also unknowns of the optimization process. The arithmetic operations involved in this calculation are comparisons mainly spent in finding the medians of subranges of data during the monotonic calculations. The package employs techniques for median and for best L 1 monotonic approximation, while full details of these techniques are specified. The package has been applied and tested on a variety of data that have substantial differences and showed quadratic behaviour in n. Some numerical results demonstrate the performance of the method. Further, there is a commentary on the division of the code into subroutines. Driver programs and numerical examples with output are provided to help new users of the method. Besides that piecewise monotonicity is a property of a wide range of functions, an important application of the method is in estimating turning points of a function from some noisy measurements of its values.

Monotonicity Enhancing Nonlinear Diffusion

We consider the task of filtering the noise from images and other types of inputs which are assumed to be piecewise continuous and piecewise monotone. We show that nonlinear diffusion of the data, a powerful filtering method, is too restrictive for such a case, leading to piecewise constant functions. We claim that the piecewise monotonicity can be enhanced by nonlinear diffusion of first partial derivatives of the input data. The method is developed in this paper; we introduce the algorithms and present experimental results.

Signs of divided differences yield least squares data fitting with constrained monotonicity or convexity

Methods are presented for least squares data smoothing by using the signs of divided differences of the smoothed values. Professor M.J.D. Powell initiated the subject in the early 1980s and since then, theory, algorithms and FORTRAN software make it applicable to several disciplines in various ways. Let us consider n data measurements of a univariate function which have been altered by random errors. Then it is usual for the divided differences of the measurements to show sign alterations, which are probably due to data errors. We make the least sum of squares change to the measurements, by requiring the sequence of divided differences of order m to have at most q sign changes for some prescribed integer q. The positions of the sign changes are integer variables of the optimization calculation, which implies a combinatorial problem whose solution can require about O( n q ) quadratic programming calculations in n variables and n− m constraints. Suitable methods have been developed for the following cases. It has been found that a dynamic programming procedure can calculate the global minimum for the important cases of piecewise monotonicity m=1, q⩾1 and piecewise convexity/concavity m=2, q⩾1 of the smoothed values. The complexity of the procedure in the case of m=1 is O(n 2+qn log 2 n) computer operations, while it is reduced to only O( n) when q=0 (monotonicity) and q=1 (increasing/decreasing monotonicity). The case m=2, q⩾1 requires O( qn 2) computer operations and n 2 quadratic programming calculations, which is reduced to one and n−2 quadratic programming calculations when m=2, q=0, i.e. convexity, and m=2, q=1, i.e. convexity/concavity, respectively. Unfortunately, the technique that receives this efficiency cannot generalize for the highly nonlinear case m⩾3, q⩾2. However, the case m⩾3, q=0 is solved by a special strictly convex quadratic programming calculation, and the case m⩾3, q=1 can be solved by at most 2( n− m) applications of the previous algorithm. Also, as m gets larger, large sets of active constraints usually occur at the optimal approximation, which makes the calculation for higher values of q less expensive than what it seems to. Further, attention is given to the sensitivity of the solution with respect to changes in the constraints and the data. The smoothing technique is an active research topic and there seems to be room for further developments. One strong reason for studying methods that make use of divided differences for data smoothing is that, whereas only the data are provided, the achieved accuracy goes much beyond the accuracy of the data at an order determined automatically by the chosen divided differences.

A No-Trade Theorem under Knightian Uncertainty with General Preferences

This paper derives a no-trade theorem under Knightian uncertainty, which generalizes the theorem of Milgrom and Stokey (1982, Journal of Economic Theory 26, 17) by allowing general preference relations. It is shown that the no-trade theorem holds true as long as agents' preferences are dynamically consistent in the sense of Machina and Schmeidler (1991, Econometrica 60, 745), and satisfies the so-called piece-wise monotonicity axiom. A preference satisfying the piece-wise monotonicity axiom does not necessarily imply the additive utility representation, nor is necessarily based on beliefs.

Monotonicity preserving interpolatory subdivision schemes

A class of local nonlinear stationary subdivision schemes that interpolate equidistant data and that preserve monotonicity in the data is examined. The limit function obtained after repeated application of these schemes exists and is monotone for arbitrary monotone initial data. Next a class of rational subdivision schemes is investigated. These schemes generate limit functions that are continuously differentiable for any strictly monotone data. The approximation order of the schemes is four. Some generalisations, such as preservation of piecewise monotonicity and application to homogeneous grid refinement, are briefly discussed.

The Poincaré–Bendixson Theorem for Monotone Cyclic Feedback Systems with Delay

We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincaré–Bendixson theorem holds for such systems. We also obtain results on piecewise monotonicity and stability of periodic solutions of such systems.

Almost sure uniform convergence rates for M-smoothers with non-monotone score functions

Almost sure uniform convergence rates for M-smoothers have been obtained by Härdle, Janssen and Serfling (1988) in the case of bounded, continuous and monotone score functions. In this paper, we show that if there is an initial estimator whose error is a.s. uniformly bounded, then the same rate result holds without the boundedness assumption on the score function ψ( y, t) and with the assumptions of continuity and monotonicity in t replaced by a.s. continuity at each t and a.s. piecewise monotonicity. Since such initial estimators are easy to construct in most situations, this considerably enlarges the scope of the rate result.