Monotonicity Constraints Research Articles

Fast and flexible methods for monotone polynomial fitting

ABSTRACTWe investigate an isotonic parameterization for monotone polynomials previously unconsidered in the statistical literature. We show that this parameterization is more flexible than its alternatives through enabling polynomials to be constrained to be monotone over either a compact interval or a semi-compact interval of the form , in addition to over the whole real line. Furthermore, algorithms based on our new parameterization estimate the fitted monotone polynomials much faster than algorithms based on previous isotonic parameterizations which in turn makes the use of standard bootstrap methodology feasible. We investigate the use of the bootstrap under monotonicity constraints to obtain confidence bands for the fitted curves and show that an adjustment by using either the ‘m out of n’ bootstrap or a post hoc symmetrization of the confidence bands is necessary to achieve more uniform coverage probabilities. We illustrate our new methodology with two real world examples which demonstrate not only the need for such techniques, but how restricting the monotonicity constraints to be over either a compact or semi-compact interval allows the fitting of even degree monotone polynomials. We also describe methods for using the ‘m out of n’ bootstrap to select the degree of the fitted monotone polynomial. All algorithms discussed in this paper are available in the R package MonoPoly (version 0.3-6 or later).

Hyperrectangles Selection for Monotonic Classification by Using Evolutionary Algorithms

AbstractIn supervised learning, some real problems require the response attribute to represent ordinal values that should increase with some of the explaining attributes. They are called classification problems with monotonicity constraints. Hyperrectangles can be viewed as storing objects in Rn which can be used to learn concepts combining instance-based classification with the axis-parallel rectangle mainly used in rule induction systems. This hybrid paradigm is known as nested generalized exemplar learning. In this paper, we propose the selection of the most effective hyperrectangles by means of evolutionary algorithms to tackle monotonic classification. The model proposed is compared through an exhaustive experimental analysis involving a large number of data sets coming from real classification and regression problems. The results reported show that our evolutionary proposal outperforms other instance-based and rule learning models, such as OLM, OSDL, k-NN and MID; in accuracy and mean absolute error...

On Ill-Posedness of Nonparametric Instrumental Variable Regression With Convexity Constraints

This note shows that adding monotonicity or convexity constraints on the regression function does not restore well-posedness in nonparametric instrumental variable regression. The minimum distance problem without regularisation is still locally ill-posed.

Stochastic Modeling and Regularity of the Nonlinear Elliptic curl--curl Equation

This paper addresses the nonlinear elliptic curl-curl equation with uncertainties in the material law. It is frequently employed in the numerical evaluation of magnetostatic fields, where the uncertainty is ascribed to the so-called B-H curve. A truncated Karhunen-Lo\`eve approximation of the stochastic B-H curve is presented and analyzed with regard to monotonicity constraints. A stochastic non-linear curl-curl formulation is introduced and numerically approximated by a finite element and collocation method in the deterministic and stochastic variable, respectively. The stochastic regularity is analyzed by a higher order sensitivity analysis. It is shown that, unlike to linear and several nonlinear elliptic problems, the solution is not analytic with respect to the random variables and an algebraic decay of the stochastic error is obtained. Numerical results for both the Karhunen-Lo\`eve expansion and the stochastic curl-curl equation are given for illustration.

What the Partitive Reading of Floating Quantifiers in Korean Tells Us about the Korean Extended DP Structure

This paper aims to present a DP structure of Korean, based on Korean Floating Quantifiers (FQ). As pointed out by Park (2008), among many others, there are various types of FQ constructions in Korean, which have already received much attention. Based on previous studies, such as Schwarzschild (2006) and Park (2008), this paper tries to analyze the following two points, which we believe have not been discussed in detail. First, some measure phrases (MPs) can also appear in FQ constructions, but other MPs cannot, and second, only some types of FQ constructions can receive the partitive interpretation. This paper shows that the distribution of MPs is closely related to Schwarzschild’s (2006) monotonicity constraint on MPs, and analyzes the distribution as well as the interpretation of MPs and classifier phrases in FQ constructions in parallel with Schwarzschild (2006) and Park (2008).

Estimating probabilistic fatigue of Nitinol with scarce samples

Current work estimates probabilistic fatigue life efficiently with scarce samples. The underlying idea of the estimation is to approximate the cumulative distribution function of the fatigue life in a transformed space using a third order polynomial subject to monotonicity constraint. The variations associated with the estimated quantiles are quantified using bootstrap. The proposed approach is validated on a data obtained from literature. It is observed that the life quantiles with reasonable accuracy can be estimated even with 10 samples. Finally, the probabilistic fatigue of Nitinol in austenitic condition is obtained with limited experiments.

Curiosities of the Monotone-Preserving Cubic Spline

In this paper we describe some new features of the monotone-preserving cubic spline and the Hyman monotonicity constraint that is implemented in various spline interpolation methods to ensure monotonicity. We find that, while the Hyman constraint is in general useful to enforce monotonicity, it can safely be omitted when the monotone-preserving cubic spline is considered. We also find that, when computing sensitivities, consistency requires making some specific assumptions about how to deal with non-differentiable locations that become relevant for special values of the parameter space.

Robust time-series retrieval using probabilistic adaptive segmental alignment

Traditional pairwise sequence alignment is based on matching individual samples from two sequences, under time monotonicity constraints. However, in many application settings matching subsequences (segments) instead of individual samples may bring in additional robustness to noise or local non-causal perturbations. This paper presents an approach to segmental sequence alignment that jointly segments and aligns two sequences, generalizing the traditional per-sample alignment. To accomplish this task, we introduce a distance metric between segments based on average pairwise distances and then present a modified pair-HMM (PHMM) that incorporates the proposed distance metric to solve the joint segmentation and alignment task. We also propose a relaxation to our model that improves the computational efficiency of the generic segmental PHMM. Our results demonstrate that this new measure of sequence similarity can lead to improved classification performance, while being resilient to noise, on a variety of sequence retrieval problems, from EEG to motion sequence classification.

Fusing Monotonic Decision Trees

Ordinal classification with a monotonicity constraint is a kind of classification tasks, in which the objects with better attribute values should not be assigned to a worse decision class. Several learning algorithms have been proposed to handle this kind of tasks in recent years. The rank entropy-based monotonic decision tree is very representative thanks to its better robustness and generalization. Ensemble learning is an effective strategy to significantly improve the generalization ability of machine learning systems. The objective of this work is to develop a method of fusing monotonic decision trees. In order to achieve this goal, we take two factors into account: attribute reduction and fusing principle. Through introducing variable dominance rough sets, we firstly propose an attribute reduction approach with rank-preservation for learning base classifiers, which can effectively avoid overfitting and improve classification performance. Then, we establish a fusing principe based on maximal probability through combining the base classifiers, which is used to further improve generalization ability of the learning system. The experimental analysis shows that the proposed fusing method can significantly improve classification performance of the learning system constructed by monotonic decision trees.

A Regularized Monotonic Fuzzy Support Vector Machine Model for Data Mining With Prior Knowledge

Incorporating prior knowledge into data mining is an interesting but challenging problem, and this study proposes a novel fuzzy support vector machine (SVM) model to explore this issue. It considers the fact that in many applications, each input point may not be exactly labeled as one particular class, and thus, it applies a fuzzy membership to each input point. It also utilizes expert knowledge concerning the monotonic relations between the response and predictor variables, which is represented in the form of monotonicity constraints. We formulate the classification problem of a monotonically constrained fuzzy SVM, called a monotonic FSVM, derive its dual optimization problem, and theoretically analyze its monotonic property. The Tikhonov regularization method is further applied to ensure that the solution is unique and bounded. A new measure, i.e., the frequency monotonicity rate, is proposed to evaluate the ability of the model to retain the monotonicity. The results of the experiments on real-world and synthetic datasets show that this method, which considers different contributions of each data and the prior knowledge of the monotonicity, has a number of advantages with regard to predictive ability and retaining monotonicity over the original FSVM and SVM models when applied to classification problems.

Вычислительная сложность отбора объектов и признаков для задач классификации с ограничениями монотонности

Decision rules with monotonicity constraints are often used in biomedical diagnostics. Simultaneous feature selection and prototype selection can significantly affect the degree of monotonicity of the data set and, as a consequence, the classification quality. In this paper we propose a systematization of discrete optimization problems of simultaneous feature selection and prototype selection and estimate their computational complexity.

On the monotonic and conservative transport on overset/Yin–Yang grids

In this paper, we outline a simple and a general methodology to achieve positivity, monotonicity and mass conservation with transport schemes on general overset grids. The main feature of the approach is its reduced complexity, which simplifies the use of higher-order schemes and higher dimensions on general grids and in particular for overset grids. The method also does not degrade substantially the order of the overall scheme despite the extra constraints of monotonicity and conservation. The approach is applied to achieve mass conservation with semi-Lagrangian schemes and its performance is analyzed using simple one-dimensional overlapping grids and a two-dimensional spherical Yin–Yang grid. The Yin–Yang grid is a special overset grid for the sphere and it is of a special interest in the atmospheric modeling community, as it is one of the grids that may resolve the scaling issue of existing longitude–latitude-grid based atmospheric models on massively parallel machines.

MoNGEL: monotonic nested generalized exemplar learning

In supervised prediction problems, the response attribute depends on certain explanatory attributes. Some real problems require the response attribute to represent ordinal values that should increase with some of the explaining attributes. They are called classification problems with monotonicity constraints. In this paper, we aim at formalizing the approach to nested generalized exemplar learning with monotonicity constraints, proposing the monotonic nested generalized exemplar learning (MoNGEL) method. It accomplishes learning by storing objects in $${\mathbb {R}}^n$$Rn, hybridizing instance-based learning and rule learning into a combined model. An experimental analysis is carried out over a wide range of monotonic data sets. The results obtained have been verified by non-parametric statistical tests and show that MoNGEL outperforms well-known techniques for monotonic classification, such as ordinal learning model, ordinal stochastic dominance learner and k-nearest neighbor, considering accuracy, mean absolute error and simplicity of constructed models.

Optimal Nonlinear Taxation Revisited

A method for handling monotonicity constraints in optimal control problems is applied in order to correct a technical weakness of optimal nonlinear taxation models with many commodities, caused by the neglect of a second order constraint for utility maximization. We conclude that all the optimal tax-rules previously obtained remain in force under the extended approach, even without assuming that demand functions are piecewise smooth. Furthermore, in a large class of cases most of such tax-rules hold as well on intervals where the second order constraint binds. The last does not apply, however, for the celebrated zero tax-rate result at both endpoints of the tax-schedule, derived by Seade (1977) and others under the first order approach. The reason lies in the different characterization that arises with a binding monotonicity constraint, independently of the existence or not of bunching, when there are more than two goods.

Monotonic Random Forest with an Ensemble Pruning Mechanism based on the Degree of Monotonicity

In classification problems, it is very common to have ordinal data in the variables, in both explanatory and class variables. When the class variable should increase according to a subset of explanatory variables, the problem must satisfy the monotonicity constraint. It is well known that standard classification tree algorithms, such as CART or C4.5, are not guaranteed to produce monotonic trees, even if the data set is completely monotone. Recently, some classifiers have been designed to handle these kinds of problems. In decision trees, growing and pruning mechanisms have been updated to improve the monotonicity of the trees. In this paper, we study the suitability of using these mechanisms in the generation of Random Forests. For this, we propose a simple ensemble pruning mechanism based on the degree of monotonicity of the resulting trees. The performance of several decision trees are evaluated through experimental studies on monotonic data sets. We deduce that the trees produced by the Random Forest also hold the monotonicity restriction but achieve a slightly better predictive performance than standard algorithms.

Bargaining under monotonicity constraints

We study unanimity bargaining on the division of a surplus in the presence of monotonicity constraints. The monotonicity constraints specify a complete order on the players, which has to be respected by the shares in the surplus the players obtain in any bargaining outcome. A player higher in the order should not receive a lower share of the surplus. We analyze the resulting subgame perfect equilibria in stationary strategies and show that they are characterized by the simpler notion of bargaining equilibrium. Bargaining equilibria are shown to be unique and to have the property that players ranked strictly higher obtain strictly higher shares in the surplus. The key question is whether the bargaining advantage of a higher-ranked player persists when the probability of breakdown of bargaining tends to zero. We argue that such is not the case by showing that bargaining equilibria have a unique limit equal to an equal division of the surplus. It then follows that the limit also coincides with the Nash bargaining solution for this problem.

Parameter estimation of partial linear model under monotonicity constraints with censored data

We propose a weighted least square method for estimation in the partial linear model with monotonicity constraints and right-censored data. This method uses the Kaplan–Meier weights to account for censoring and monotone B-splines to approximate the unknown monotone function. We show that the proposed estimator of regression coefficients is root-n consistent and asymptotically normal under appropriate assumptions. One advantage is that our method can be easily computed using existing software. A simulation study is conducted to evaluate the finite sample performance of the proposed method.

Monotone B-Spline Smoothing for a Generalized Linear Model Response

Various methods have been proposed for smoothing under the monotonicity constraint. We review the literature and implement an approach of monotone smoothing with B-splines for a generalized linear model response. The approach is expressed as a quadratic programming problem and is easily solved using the statistical software R. In a simulation study, we find that the approach performs better than other approaches with much faster computation time. The approach can also be used for smoothing under other shape constraints or mixed constraints. Supplementary materials of the appendices and R code to implement the developed approach is available online.

Existence and Uniqueness of Solutions to the Generalized Riemann Problem for Isentropic Flow

We consider network models for isentropic flow and evaluate the application of three different momentum-related coupling constants: pressure, momentum flux, and the Bernoulli invariant. Subsonic solutions to generalized Riemann problems with subsonic initial conditions are proved to be unique, and the region where such initial conditions yield subsonic solutions is identified. The proof is restricted to monotone coupling constants. We also prove that the three momentum-related coupling constants fulfill the monotonicity constraint. Further, we analyze the existence of entropic solutions using a commonly applied entropy condition. The condition states that nonphysical solutions are characterized by production of mechanical energy at a junction. Both pressure and momentum flux are seen to yield nonphysical solutions in a constructed test case consisting of three pipe sections of equal cross-sectional area that are connected at a junction. The Bernoulli invariant is proved to yield entropic solutions for all subsonic flow conditions in the general case of a junction connecting \(N\) pipe sections of arbitrary cross-sectional area. Hence, among the three considered network models, only the model with the Bernoulli invariant as the momentum-related coupling constant yields physical solutions for all subsonic flow conditions at the pipe-junction interfaces.

Flexible regression models for rate differences, risk differences and relative risks.

Generalized additive models (GAMs) based on the binomial and Poisson distributions can be used to provide flexible semi-parametric modelling of binary and count outcomes. When used with the canonical link function, these GAMs provide semi-parametrically adjusted odds ratios and rate ratios. For adjustment of other effect measures, including rate differences, risk differences and relative risks, non-canonical link functions must be used together with a constrained parameter space. However, the algorithms used to fit these models typically rely on a form of the iteratively reweighted least squares algorithm, which can be numerically unstable when a constrained non-canonical model is used. We describe an application of a combinatorial EM algorithm to fit identity link Poisson, identity link binomial and log link binomial GAMs in order to estimate semi-parametrically adjusted rate differences, risk differences and relative risks. Using smooth regression functions based on B-splines, the method provides stable convergence to the maximum likelihood estimates, and it ensures that the estimates always remain within the parameter space. It is also straightforward to apply a monotonicity constraint to the smooth regression functions. We illustrate the method using data from a clinical trial in heart attack patients.