Parameters Of Nonlinear Function Research Articles

Design of Quantum Gates Using Quantum Scattering Theory

Given two Hamiltonians \(H_0,H_1=H_0+V\) corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time \(t=-\infty\), entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time \(t\rightarrow\infty\) to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.

A GENERALIZED FRACTAL-BASED APPROACH FOR STRESS-DEPENDENT PERMEABILITY OF POROUS ROCKS

The pore geometry of porous rocks is fundamental for accurate description of stress dependence of effective permeability, which is an important parameter of mass transfer in porous rocks. An important physical assumption that porous rocks contain numerous elliptical or spherical pores has been shown to be successfully applied to many aspects of hydromechanical coupling properties of porous rocks. To investigate the detailed description of pore structures on the degree of effect on the coupled hydromechanical process, in this work, a generalized stress-dependent model for permeability of porous rocks has been proposed based on fractal geometry theory and mechanics of porous rock. The proposed model is expressed as a nonlinear function of pore structure parameters, such as aspect ratio ([Formula: see text]), the fractal dimensions ([Formula: see text] and [Formula: see text]) for tortuosity, the initial fractal dimension ([Formula: see text]), and initial porosity ([Formula: see text]) as well as matrix elastic constants ([Formula: see text] and [Formula: see text]) of porous rocks without any empirical parameter. The validity of the proposed models is verified by the good agreements between available experimental data and theoretical predictions of stress-dependent permeabilities of porous rocks. Detailed discussions of the essential effects of pore structures parameters and material elastic constants of porous rocks on the dimensionless stress-dependent permeabilities are performed. It is found that the stress parameters ([Formula: see text] and [Formula: see text]) have remarkable effects on the dimensionless stress-dependent permeabilities compared with other parameters ([Formula: see text] and [Formula: see text]). The proposed model may contribute to a better quantitative understanding of the coupled hydromechanical properties of porous rocks.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

Adaptive NN control for nominal backstepping form with periodically time‐varying and nonlinearly parameterized switching functions

AbstractIn this paper, the prescribed tracking performance control problem is addressed for uncertain nonlinear systems with unknown periodically time‐varying parameters and arbitrary switching signal. By utilizing radial basis function neural network and fourier series expansion, an approximator is developed to overcome the difficulty of identifying unknown periodically time‐varying and nonlinearly parameterized functions. To achieve the ideal tracking control performance and eliminate the influence of filtering error, a novel command filter‐based adaptive neural network prescribed tracking performance controller is designed by introducing a filtering compensation mechanism. Differently from the standard Backstepping technique, the proposed control scheme eliminates the “explosion of complexity” problem and relaxes the constraint condition on the reference signal. And then, it is warranted that the closed‐loop system is semi‐globally ultimately uniformly bounded and the tracking error is always limited to the specified region bounded by the performance functions. Three simulation examples are used to demonstrate the feasibility of the developed technique in this paper.

Evaluation of frequentist test statistics using constrained statistical inference in the context of the generalized linear model

ABSTRACT When faced with a binary or count outcome, informative hypotheses can be tested in the generalized linear model using the distance statistic as well as modified versions of the Wald, the Score and the likelihood-ratio test (LRT). In contrast to classical null hypothesis testing, informative hypotheses allow to directly examine the direction or the order of the regression coefficients. Since knowledge about the practical performance of informative test statistics is missing in the theoretically oriented literature, we aim at closing this gap using simulation studies in the context of logistic and Poisson regression. We examine the effect of the number of constraints as well as the sample size on type I error rates when the hypothesis of interest can be expressed as a linear function of the regression parameters. The LRT shows the best performance in general, followed by the Score test. Furthermore, both the sample size and especially the number of constraints impact the type I error rates considerably more in logistic compared to Poisson regression. We provide an empirical data example together with R code that can be easily adapted by applied researchers. Moreover, we discuss informative hypothesis testing about effects of interest, which are a non-linear function of the regression parameters. We demonstrate this by means of a second empirical data example.

TI-fuzzy neighborhood measures and generalized Choquet integrals for granular structure reduction and decision making

Granular structures are mathematical representations of knowledge used in granular computing. Any fuzzy information table can be seen as a fuzzy granular structure, which is called a fuzzy β-covering approximation space. Hence, corresponding granular computing methods are widely used in data analysis, granular reduction and decision making. Currently, these methods mainly use rough approximate operators to process data where attributes are independent of each other in rough approximate operators. However, attributes are often associated with each other in real-life problems. As a parametric nonlinear aggregation function, the Choquet integral (CI) with respect to a fuzzy measure (FM) solves the problem of attribute association well. In this paper, we present TI-fuzzy β-neighborhood measures, which are FMs and generalized CIs, to deal with granularity reduction and decision making in a fuzzy β-covering approximation space. Firstly, four pairs of TI-fuzzy β-neighborhood measures under the t-norm “T” and its residual implication “IT”, as FMs, are presented for use in granular computing instead of rough approximation operators. Then, a novel method with TI-fuzzy β-neighborhood measures is presented to deal with granularity reduction in the fuzzy β-covering approximation space. Thirdly, four pairs of generalized CIs based on the TI-fuzzy β-neighborhood measures are constructed. By combining the presented CIs with the fuzzy PROMETHEE method, we propose a new method to solve the problem of decision making. Finally, several numerical examples and UCI data sets are used to illustrate the feasibility and effectiveness of our proposed methods.

A Simple Model of Sea-Surface Cooling under a Tropical Cyclone

A major ocean response to tropical cyclone (TC) wind is the mixing of warm sea-surface water with cool subsurface water, which decreases the sea-surface temperature (SST). The decreased SST (δT) under the TC (rather than the cooled water in the wake after the storm has passed) modifies the storm’s intensity and is of interest to TC intensity studies. Here, the author shows that δT (non-dimensionalized by some reference temperature) is linearly related to Ψ, a dimensionless (nonlinear) function of TC and ocean parameters: the TC maximum wind, radius, and translation speed, as well as the ocean’s 26 °C and 20 °C isothermal depths (Z26 and Z20). The Ψ can be estimated from observations. The modelled δT is validated against sea-surface cooling observed by satellites, δTo, for typhoons during the May–December 2015 period in the western North Pacific. The result yields a best-fit, linear relation between δTo and Ψ that explains ~60% of the observed variance: r2 ≈ 0.6 (99% confidence). Tests show that the cube of the TC maximum wind and the ocean’s Z26 account for 46% and 7%, respectively, of the observed variance, indicating their predominant influence on TC-induced cooling. Contributions from other parameters are less but not negligible.

Non-linear models of species' responses to environmental and spatial gradients.

Species' responses to broad-scale environmental or spatial gradients are typically unimodal. Current models of species' responses along gradients tend to be overly simplistic (e.g., linear, quadratic or Gaussian GLMs), or are suitably flexible (e.g., splines, GAMs) but lack direct ecologically interpretable parameters. We describe a parametric framework for species-environment non-linear modelling ('senlm'). The framework has two components: (i) a non-linear parametric mathematical function to model the mean species response along a gradient that allows asymmetry, flattening/peakedness or bimodality; and (ii) a statistical error distribution tailored for ecological data types, allowing intrinsic mean-variance relationships and zero-inflation. We demonstrate the utility of this model framework, highlighting the flexibility of a range of possible mean functions and a broad range of potential error distributions, in analyses of fish species' abundances along a depth gradient, and how they change over time and at different latitudes.

Modeling and inference for multivariate time series of counts based on the INGARCH scheme

Modeling multivariate time series of counts using the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) scheme is proposed. The key idea is to model each component of the time series with a univariate INGARCH model, where the conditional distribution is modeled with a one-parameter exponential family distribution, and to use a (nonlinear) parametric function of all components to recursively produce the conditional means. It is shown that the proposed multivariate INGARCH (MINGARCH) model is strictly stationary and ergodic. For inference, the quasi-maximum likelihood estimator (QMLE) and the minimum density power divergence estimator (MDPDE) for robust estimation are adopted, and their consistency and asymptotic normality are verified. As an application, the change point test based on the QMLE and MDPDE is illustrated. The Monte Carlo simulation study and real data analysis using the number of weekly syphilis cases in the United States are conducted to confirm the validity of the proposed method.

Adaptive tracking control for a class of stochastic nonlinearly parameterized systems with time-varying input delay using fuzzy logic systems

This paper addresses the problem of adaptive tracking control for a class of stochastic nonlinear systems with time-varying input delays, and the nonlinear functions of the systems are with not only the unknown parameters but also the unknown state time-varying delays, which is different from the previous work. In this paper, through a state transformation, the system can be easily transformed into a system without the time-varying input delay; the appropriate Lyapunov–Krasovskii functionals are used to compensate the unknown time-varying delay terms, and the quadratic functions instead of the quartic functions often utilized in the existing results are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. Fuzzy logic systems (FLSs) in Mamdani type are used to approximate the unknown nonlinear functions. Then, based on the backstepping technique, the adaptive fuzzy controller is designed. The three main advantages of the developed scheme are that (i) unlike the existing results which deal with the nonlinearly parameterized functions by using the separation principle, the nonlinearly parameterized functions are lumped into the continuous functions which can be approximated by using the FLS; (ii) the number of the adjusted parameters only depend on the order of the investigated systems, which can reduce the computational burden greatly; and (iii) the existence of the time-varying input delay such that the controller design becomes much more difficult, and in this paper, it can be dealt with by using an appropriate state transformation. It is proven that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded in probability, whereas the tracking error converges to a small neighborhood of the origin. Finally, simulation results are provided to show the effectiveness of the proposed approach.

Single‐fit bootstrapping: A simple alternative to the delta method

AbstractIn many environmental and ecological settings, the analysis of data involves calculation of a confidence interval for a nonlinear function of the model parameters. In the frequentist approach, this is often achieved by using a Wald interval in which the standard error of the estimate of the function is obtained using the delta method.The purpose of this paper is to promote a simple alternative to the delta method that will provide better coverage properties when the estimates of the model parameters have a distribution that is approximately multivariate normal, as will be the case asymptotically for maximum likelihood estimates in regular models. We propose a simple diagnostic tool that can be used to assess the reliability of this assumption.The method we promote can be thought of as a computationally efficient version of parametric bootstrapping, as it mimics such bootstrapping without a need to refit the model.We provide two examples to illustrate use of the new method and present simulation results which show that its coverage properties are superior to those for a Wald interval based on the delta method when the assumption of normality for the estimates of the model parameters is reliable. We also compare its coverage properties and computational effort with those of parametric bootstrapping.

An innovative method of fault detection in power transformers

The Bayesian classifier is a priori the optimal solution for minimizing the total error in problems of statistical pattern recognition. The article suggests using the classifier as a regular tool to increase the reliability of defect recognition in power oil-filled transformers based on the results of the analysis of gases dissolved in oil. The wide application of the Bayesian method for solving tasks of technical diagnostics of electrical equipment is limited by the problem of the multidimensional distribution of random parameters (features) and the nonlinearity of classification. The application of a generalized feature of a defect in the form of a nonlinear function of the transformer state parameters is proposed. This simultaneously reduces the dimension of the initial space of the controlled parameters and significantly improves the stochastic properties of the random distribution of the generalized feature. A special algorithm has been developed to perform statistical calculations and the procedure for recognizing the current technical condition of the transformer using the generated decision rule. The presented research results illustrate the possibility of the practical application of the developed method in the conditions of real operation of power transformers.

Flexible Bayesian Nonlinear Model Configuration

Regression models are used in a wide range of applications providing a powerful scientific tool for researchers from different fields. Linear, or simple parametric, models are often not sufficient to describe complex relationships between input variables and a response. Such relationships can be better described through flexible approaches such as neural networks, but this results in less interpretable models and potential overfitting. Alternatively, specific parametric nonlinear functions can be used, but the specification of such functions is in general complicated. In this paper, we introduce a flexible approach for the construction and selection of highly flexible nonlinear parametric regression models. Nonlinear features are generated hierarchically, similarly to deep learning, but have additional flexibility on the possible types of features to be considered. This flexibility, combined with variable selection, allows us to find a small set of important features and thereby more interpretable models. Within the space of possible functions, a Bayesian approach, introducing priors for functions based on their complexity, is considered. A genetically modi ed mode jumping Markov chain Monte Carlo algorithm is adopted to perform Bayesian inference and estimate posterior probabilities for model averaging. In various applications, we illustrate how our approach is used to obtain meaningful nonlinear models. Additionally, we compare its predictive performance with several machine learning algorithms.

PSV-14 Using Bayesian Hamiltonian Monte Carlo and a nonlinear model for the estimation of genetic parameters for lactation curves in goats

Abstract Hamiltonian Monte Carlo (HMC) is an algorithm of the Markov Chain Monte Carlo (MCMC) method that uses dynamics to propose samples that follow a target distribution. This algorithm enables more effective and consistent exploration of the probability interval and is more sensitive to correlated parameters. Therefore, Bayesian-HMC is a promising alternative to estimate individual parameters of complex functions such as nonlinear models, especially when using small datasets. Our objective was to estimate genetic parameters for milk traits defined based on nonlinear model parameters predicted using the Bayesian-HMC algorithm. A total of 64,680 milk yield test-day records from 2,624 first, second, and third lactations of Saanen and Alpine goats were used. First, the Wood model was fitted to the data. Second, lactation persistency (LP), peak time (PT), peak yield (PY), and total milk yield [estimated from zero to 50 (TMY50), 100(TMY100), 150(TMY150), 200(TMY200), 250(TMY250), and 300(TMY300) days-in-milk] were predicted for each animal and parity based on the output of the first step (the individual phenotypic parameters of the Wood model). Thereafter, these predicted phenotypes were used for estimating genetic parameters for each trait. In general, the heritability estimates across lactations ranged from 0.10 to 0.20 for LP, 0.04 to 0.07 for PT, 0.26 to 0.27 for PY, and 0.21 to 0.28 for TMY (considering the different intervals). Lower heritabilities were obtained for the nonlinear function parameters (A, b and l) compared to its predicted traits (except PT), especially for the first and second lactations (range: 0.09 to 0.18). Higher heritability estimates were obtained for the third lactation traits. To our best knowledge, this study is the first attempt to use the HMC algorithm to fit a nonlinear model in animal breeding. The two-step method proposed here allowed us to estimate genetic parameters for all traits evaluated.

Estimation of Cole Parameters in ex-vivo tissues as a function of degradation time

Bioimpedance spectroscopy has been used to evaluate and characterize the integrity of different tissues and organs, as well as to detect tissue structural alterations. The tissue electrical model alterations as an influence of the anisotropy of the tissue structure and its intrinsic metabolism on the degradation dynamic have not been completely understood. In this work, the dynamic of Cole parameters was estimated as a function of degradation time in ex vivo skeletal muscle, skin, and white adipose tissues. The results indicate a non-linear behavior function of the Cole parameters through degradation progress in the three evaluated tissues, and such non-linearities might be associated with different capacities of adaptation to anaerobic metabolism, energy production, and water content under stress conditions.

Designing for Parameter Subsets in Gaussian Nonlinear Regression Models

This article presents and illustrates several important subset design approaches for Gaussian nonlinear regression models and for linear models where interest lies in a nonlinear function of the model parameters. These design strategies are particularly useful in situations where currentlyused subset design procedures fail to provide designs which can be used to fit the model function. Our original design technique is illustrated in conjuction with D-optimality, Bayesian D-optimality and Kiefer’s Φk-optimality, and is extended to yield subset designs which take account of curvature.

Qualitative properties of solutions to set optimization problems

In this paper, we consider set optimization problems involving set relations. Using the well-known KKM-Fan Lemma and relaxed convexity assumptions, we study existence conditions for these problems. Moreover, we introduce parametric nonlinear scalarization functions for sets and study their properties. By utilizing the concerning functions, we investigate relationships between set optimization problems and equilibrium problems. Finally, sufficient conditions for the Holder continuity of solution maps to such problems via equilibrium problems are established.

Compositional grading of a 316L-Cu multi-material part using machine learning for the determination of selective laser melting process parameters

The aim of this work is to propose a methodology for rapidly predicting suitable process parameters for additive manufacturing of a 316L-Cu multi-material part with a compositional gradient by using machine learning. Specifically, an algorithm based on a multivariate Gaussian process is developed to predict part density and surface roughness for a given set of laser power, velocity, and hatch spacing values. The training data for the algorithm is collected using a high-throughput experimentation method that allows for rapid measurement of part density, and surface roughness. After the model is validated using leave-one-out cross validation method, process parameter maps are generated for 316L-Cu parts manufactured using selective laser melting with premixed powder at mass fractions of 0.25, 0.50, and 0.75. A set of suitable process parameters are predicted using the process maps. It is shown that process parameters are a nonlinear function of gradient composition and neither process parameters of 316L or Cu are suitable for the graded region of a 316L-Cu multi-material part. Generated process maps provide a firsthand knowledge of process-property relationships for regions of compositional grading in 316L-Cu parts.

Quasi‐LPV PI control of TRMS subject to actuator saturation

This study is concerned on the design of a quasi-linear parameter varying (qLPV) proportional–integral (PI) controller for twin rotor multiple-input multiple-output systems (TRMS). The non-linear model is represented as a qLPV polytopic plant with an affine dependence on a non-linear parametric function of the pitch angle. This representation retains the exact model as opposed to the conventional linearisation around an operating point. Due to the availability of the pitch angle measurement, the non-linear parameter can be obtained in real-time and the controller is designed using qLPV technique. To deal with limited control input for such systems, the proposed controller design also considers the actuator saturation that yields controller with practical gains without any additional gain bound criterion. Further, the transient tracking performance is also considered in the design by using closed-loop eigenvalues assignment in desired damping regions. The control synthesis problem is formulated in the form of linear matrix inequalities for L 2 gain based performance criterion. The designed controller is validated on a two-degree of freedom helicopter experimental setup. Finally, to demonstrate the effectiveness of the proposed design, a comparative analysis is done with the existing algorithms. Also, the efficacy of the decentralised controller vis-a-vis the centralised one is presented.

A multi-objective optimization approach for climate-adaptive building envelope design using parametric behavior maps

This research presents a multi-objective optimization (MOO) framework to support the climate-adaptive building envelope (CABE) design decisionmaking process using a parametric behavior map (PBM). Unlike static shading, CABE systems include dynamic operations that significantly affect their performance; thus, well-informed strategies for scheduling dynamic operations should be integrated to analyze CABE performance. In this study, two conflicting objectives were pursued: minimizing cooling load and maximizing daylighting performance during the summer season in a hot and humid climate (Houston, Texas). Variables in the CABE performance optimization process were defined as dynamic operation schedules having either parametric linear or non-linear relationships between the degree of openness of the CABE model and certain weather stimuli (i.e., solar radiation). Two CABE models were tested with the PBM by integrating a parametric non-linear function that efficiently conducted the optimization process in a large search space. The outcomes of this optimization study included Pareto-front solutions such as optimal CABE performance and their dynamic operation scenarios. These optimal operation scenarios were determined based on the CABE design options available and user's desired objectives; in some cases, static scenarios were found to be superior. Ultimately, combining PBM with a MOO framework will contribute to the field of performance-based CABE design by supporting architects and engineers and facilitating better decisions through well-informed dynamic operation scenarios.