Real-valued Parameters Research Articles

Approximation schemes for the parametric knapsack problem

We consider the (linear) parametric 0–1 knapsack problem in which the profits of the items are affine-linear functions of a real-valued parameter and the task is to compute a solution for all values of the parameter. For this problem, it is known that the piecewise linear convex function mapping the parameter to the optimal objective value of the corresponding instance (called the optimal value function) can have exponentially many breakpoints (points of slope change), which implies that every optimal algorithm for the problem must output a number of solutions that is exponential in the number of items.We provide the first (parametric) polynomial time approximation scheme (PTAS) for the parametric 0–1 knapsack problem. Moreover, we exploit the connection between the parametric problem and the bicriteria problem in order to show that the parametric 0–1 knapsack problem admits a parametric FPTAS when the parameter is restricted to the positive real line and the slopes and intercepts of the affine-linear profit functions of the items are nonnegative. The method used to obtain this result applies to many linear parametric optimization problems and provides a general connection between bicriteria and linear parametric optimization problems.

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction.

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here, we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal gain cascades (i.e. when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on the rates and length of the cascade, as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. In particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction.

Revenue-Maximizing Rankings for Online Platforms with Quality-Sensitive Consumers

When a keyword-based search query is received by a search engine, a classified ads website, or an online retailer site, the platform has exponentially many choices in how to sort the search results. Two extreme rules are (a) to use a ranking based on estimated relevance only, which improves customer experience in the long run because of perceived quality and (b) to use a ranking based only on the expected revenue to be generated immediately, which maximizes short-term revenue. Typically, these two objectives and the corresponding rankings differ. A key question then is what middle ground between them should be chosen. We introduce stochastic models that yield elegant solutions for this situation, and we propose effective solution methods to compute a ranking strategy that optimizes long-term revenues. This strategy has a very simple form and is easy to implement if the necessary data is available. It consists of ordering the output items by decreasing order of a score attributed to each, similarly to value models used in practice by e-commerce platforms. This score results from evaluating a simple function of the estimated relevance, the expected revenue of the link, and a real-valued parameter. We find the latter via simulation-based optimization, and its optimal value is related to the endogeneity of user activity in the platform as a function of the relevance offered to them.

A Hybridized Vector Optimal Algorithm for Multi-Objective Optimal Designs of Electromagnetic Devices

Multiple-objective designs exist in most real-world engineering problems in different disciplines. A multi-objective evolutionary algorithm will face a challenge to obtain a series of compromises of different objectives, called Pareto optimal solutions, and to distribute them uniformly. In this regard, it is essential to keep the balance of local and global search abilities of such algorithms. Quantum-behaved particle swarm optimization (QPSO) is a population-based swarm intelligence algorithm, and differential evolutionary (DE) is another simple population-based stochastic search one for global optimization with real-valued parameters. Although the two optimizers have been successfully employed to solve a wide range of design problems, they also suffer from premature convergence and insufficient diversity in the later searching stages. This is probably due to the insufficient dimensional searching strength, especially for problems with many decision parameters. In this paper, a new multi-objective non-dominated optimal methodology combining QPSO, DE, and tabu search algorithm (QPSO-DET) is proposed to guarantee the balance between the local and global searches. The performances of the proposed QPSO-DET are compared with those of other two widely recognized vector optimizers using different case studies.

Comparison of two non-convex mixed-integer nonlinear programming algorithms applied to autoregressive moving average model structure and parameter estimation

In this article, the stochastic modelling approach proposed by Box and Jenkins is treated as a mixed-integer nonlinear programming (MINLP) problem solved with a mesh adaptive direct search and a real-coded genetic class of algorithms. The aim is to estimate the real-valued parameters and non-negative integer, correlated structure of stationary autoregressive moving average (ARMA) processes. The maximum likelihood function of the stationary ARMA process is embedded in Akaike's information criterion and the Bayesian information criterion, whereas the estimation procedure is based on Kalman filter recursions. The constraints imposed on the objective function enforce stability and invertibility. The best ARMA model is regarded as the global minimum of the non-convex MINLP problem. The robustness and computational performance of the MINLP solvers are compared with brute-force enumeration. Numerical experiments are done for existing time series and one new data set.

Exploring stochasticity and imprecise knowledge based on linear inequality constraints.

This paper explores the stochastic dynamics of a simple foodweb system using a network model that mimics interacting species in a biosystem. It is shown that the system can be described by a set of ordinary differential equations with real-valued uncertain parameters, which satisfy a set of linear inequality constraints. The constraints restrict the solution space to a bounded convex polytope. We present results from numerical experiments to show how the stochasticity and uncertainty characterizing the system can be captured by sampling the interior of the polytope with a prescribed probability rule, using the Hit-and-Run algorithm. The examples illustrate a parsimonious approach to modeling complex biosystems under vague knowledge.

Asymmetric Volterra Models Based on Ladder-Structured Generalized Orthonormal Basis Functions

In this paper, an improved method to construct and estimate Volterra models using Generalized Orthonormal Basis Functions (GOBF) is presented. The proposed method extends results obtained in previous works, where an exact technique for optimizing the GOBF parameters (poles) for symmetric Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the new formulation is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles responsible for representing the dominant dynamic of the kernel along a particular dimension; and (ii) the new formulation is based on a ladder-structured GOBF architecture that is characterized by having only real-valued parameters to be estimated, regardless of whether the GOBF poles encoded by these parameters are real- or complex-valued. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically and provide exact search directions for an optimization process that uses only input-output data measured from the dynamic system to be modeled. Computational experiments are presented to illustrate the benefits of the proposed approach when modeling nonlinear systems.

Chaotic sliding mode controllers for uncertain time-delay chaotic systems with input nonlinearity

This paper presents a novel robust chaotic controller for stabilizing uncertain time-delay chaotic systems with input nonlinearity. Based on dynamic output sliding mode control (SMC), Lyapunov theory and linear matrix inequality (LMI) technique, the stability of overall uncertain time-delay chaotic system with input nonlinearity under the proposed control scheme is guaranteed without the state predictor. The selection of sliding surface and the design of control law are two important issues, which have been addressed. A priori knowledge of upper bounds of uncertainties is not required. Furthermore, the proposed scheme ensures robustness against input nonlinearity, time-delays, nonlinear real-valued functions, parameter uncertainties and disturbances simultaneously. Finally, simulation results demonstrate the validity of the proposed control methodology.

Quickest Change Detection and Kullback-Leibler Divergence for Two-State Hidden Markov Models

In this paper, the quickest change detection problem is studied in two-state hidden Markov models (HMM), where the vector parameter $\theta$ of the HMM changes from $\theta_{0}$ to $\theta_{1}$ at some unknown time, and one wants to detect the true change as quickly as possible while controlling the false alarm rate. It turns out that the generalized likelihood ratio (GLR) scheme, while theoretically straightforward, is generally computationally infeasible for the HMM. To develop efficient but computationally simple schemes for the HMM, we first discuss a subtlety in the recursive form of the generalized likelihood ratio (GLR) scheme for the HMM. Then we show that the recursive CUSUM scheme proposed in Fuh (Ann. Statist., 2003) can be regarded as a quasi-GLR scheme for pseudo post-change hypotheses with certain dependence structure between pre- and postchange observations. Next, we extend the quasi-GLR idea to propose recursive score schemes in the scenario when the postchange parameter $\theta_{1}$ of the HMM involves a real-valued nuisance parameter. Finally, the Kullback-Leibler (KL) divergence plays an essential role in the quickest change detection problem and many other fields, however it is rather challenging to numerically compute it in HMMs. Here we develop a non-Monte Carlo method that computes the KL divergence of two-state HMMs via the underlying invariant probability measure, which is characterized by the Fredholm integral equation. Numerical study demonstrates an unusual property of the KL divergence for HMM that implies the severe effects of misspecifying the postchange parameter for the HMM.

Symbolic discrete-time planning with continuous numeric action parameters for agent-controlled processes

In industrial domains such as manufacturing control, a trend away from centralized planning and scheduling towards more flexible distributed agent-based approaches could be observed over recent years. To be of practical relevance, the local control mechanisms of the autonomous agents must be able to dependably adhere and dynamically adjust to complex numeric goal systems like business key performance indicators in an economically beneficial way. However, planning with numeric state variables and objectives still poses a challenging task within the field of artificial intelligence (AI).In this article, a new general-purpose AI planning approach is presented that operates in two stages and extends existing domain-independent modeling formalisms like PDDL with continuous (i.e., infinite-domain) numeric action parameters, which are currently still unsupported by state-of-the-art AI planners. In doing so, it enables the solution of mathematical optimization problems at the action level of the planning tasks, which are inherent to many real-world control problems. To deal with certain difficulties concerning reliable and fast detection of action applicability that arise when planning with real-valued action parameters, the implemented planner allows resorting to an adjustable “satisficing” strategy by means of partial execution and subsequent repair of infeasible plans over the course of time. The functioning of the system is evaluated in a multi-agent simulation of a shop floor control scenario with focus on the effects the possible problem cases and different degrees of satisficing have on attained plan quality and total planning time. As the results demonstrate the basic practicability of the approach for the given setting, this contribution constitutes an important step towards the effective and dependable integration of complex numeric goal systems and non-linear multi-criteria optimization tasks into autonomous agent-controlled industrial processes in a reusable, domain-independent way.

AN INTEGRABLE AND EXACTLY SOLVABLE NON-HERMITIAN BCS HAMILTONIAN AND GENERALISED EXCLUSION STATISTICS

It is more than three decades since the quantum inverse scattering method and the algebraic Bethe ansatz have been presented and applied in the area of many body quantum systems. These methods are playing crucial roles in the development of novel integrable and exactly solvable models. The purpose of the present dissertation is to apply quantum inverse scattering method to new contexts. The first system under consideration describes the interactions of spin half impurities in an isotropic Heisenberg spin chain with periodic boundary condition. By employing quantum inverse scattering method and Bethe ansatz techniques, the Hamiltonian, Bethe ansatz and associated eigenvalues of the model are given. The properties of the system are also discussed. The main results of this thesis are obtained from an integrable and exactly solvable non-Hermitian Bardeen-Cooper-Schrieffer pairing Hamiltonian with two independent, real-valued coupling parameters. Using numerical methods, we investigated the behaviour of the spectrum of the non-Hermitian Hamiltonian as the coupling parameters are varied. This allows for an identification of boundary lines in the coupling parameter space for regions of strictly real spectra. According to their dependence on the coupling parameters, the Bethe ansatz solutions may be categorized into two classes on the boundary lines. It will be discussed how these solutions could be associated to exotic quasi-particles obeying generalised exclusion statistics, in the sense proposed by Haldane in 1991.

Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy between prior and posterior. Therefore, more elaborate methods, such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be used. However, the elaboration of a proposal density for MCMC is a sensitive issue and very difficult in the ABC setting, where the likelihood is intractable. We discuss an automatic proposal distribution useful for ABC-MCMC algorithms. This proposal is inspired by the theory of quasi-likelihood (QL) functions and is obtained by modelling the distribution of the summary statistics as a function of the parameters. Essentially, given a real-valued vector of summary statistics, we reparametrize the model by means of a regression function of the statistics on parameters, obtained by sampling from the original model in a pilot-run simulation study. The QL theory is well established for a scalar parameter, and it is shown that when the conditional variance of the summary statistic is assumed constant, the QL has a closed-form normal density. This idea of constructing proposal distributions is extended to non constant variance and to real-valued parameter vectors. The method is illustrated by several examples and by an application to a real problem in population genetics.

Reverse engineering of logic-based differential equation models using a mixed-integer dynamic optimization approach.

Motivation: Systems biology models can be used to test new hypotheses formulated on the basis of previous knowledge or new experimental data, contradictory with a previously existing model. New hypotheses often come in the shape of a set of possible regulatory mechanisms. This search is usually not limited to finding a single regulation link, but rather a combination of links subject to great uncertainty or no information about the kinetic parameters.Results: In this work, we combine a logic-based formalism, to describe all the possible regulatory structures for a given dynamic model of a pathway, with mixed-integer dynamic optimization (MIDO). This framework aims to simultaneously identify the regulatory structure (represented by binary parameters) and the real-valued parameters that are consistent with the available experimental data, resulting in a logic-based differential equation model. The alternative to this would be to perform real-valued parameter estimation for each possible model structure, which is not tractable for models of the size presented in this work. The performance of the method presented here is illustrated with several case studies: a synthetic pathway problem of signaling regulation, a two-component signal transduction pathway in bacterial homeostasis, and a signaling network in liver cancer cells.Supplementary information: Supplementary data are available at Bioinformatics online.Contact: julio@iim.csic.es or saezrodriguez@ebi.ac.uk

Information-theoretic optimality of observation-driven time series models for continuous responses

We investigate information-theoretic optimality properties of the score function of the predictive likelihood as a device for updating a real-valued time-varying parameter in a univariate observation-driven model with continuous responses.We restrict our attention to models with updates of one lag order. The results provide theoretical justification for a class of score-driven models which includes the generalized autoregressive conditional heteroskedasticity model as a special case. Ourmain contribution is to show that only parameter updates based on the score will always reduce the local Kullback-Leibler divergence between the true conditional density and themodel-implied conditional density. This result holds irrespective of the severity of modelmisspecification. We also show that use of the score leads to a considerably smaller global Kullback- Leibler divergence in empirically relevant settings. We illustrate the theory with an application to time-varying volatility models. We show that the reduction in Kullback-Leibler divergence across a range of different settings can be substantial compared to updates based on, for example, squared lagged observations.

Survey and some new results on performance analysis of complex-valued parameter estimators

Recently, there has been an increased awareness that simplistic adaptation of performance analysis developed for random real-valued signals and parameters to the complex case may be inadequate or may lead to intractable calculations. Unfortunately, many fundamental statistical tools for handling complex-valued parameter estimators are missing or scattered in the open literature. In this paper, we survey some known results and provide a rigorous and unified framework to study the statistical performance of complex-valued parameter estimators with a particular attention paid to properness (i.e., second order circularity), specifically referring to the second-order statistical properties. In particular, some new properties relative to the properness of the estimates, asymptotically minimum variance bound and Whittle formulas are presented. A new look at the role of nuisance parameters is given, proving and illustrating that the noncircular Gaussian distributions do not necessarily improve the Cramer–Rao bound (CRB) with respect to the circular case. Efficiency of subspace-based complex-valued parameter estimators that are presented with a special emphasis is put on noisy linear mixture.

The Fibonacci recurrence with harmonic inhomogeneity

In this paper, a non-homogeneous version of the classical Fibonacci recurrence relation is investigated. The inhomogeneity is defined by a harmonic sequence , with real-valued parameters . It turns out that the golden ratio rule carries over to the solution of the non-homogeneous recurrence for almost all parameter values. The exceptional set is a one-dimensional submanifold in –-space, which is related to the graph of the Lerch transcendent.

Surrogate Modeling of Computer Experiments With Different Mesh Densities

This article considers deterministic computer experiments with real-valued tuning parameters which determine the accuracy of the numerical algorithm. A prominent example is finite-element analysis with its mesh density as the tuning parameter. The aim of this work is to integrate computer outputs with different tuning parameters. Novel nonstationary Gaussian process models are proposed to establish a framework consistent with the results in numerical analysis. Numerical studies show the advantages of the proposed method over existing methods. The methodology is illustrated with a problem in casting simulation. Supplementary material for this article is available online.

CRLB for likelihood functions with parameter-dependent support

In this paper we discuss the regularity conditions required for the classical Craḿer-Rao lower bound (CRLB) for real-valued (nonrandom unknown) parameters to hold. It is shown that the commonly assumed requirement that the support of the likelihood function (LF) should be independent of the parameter to be estimated can be replaced by the much weaker requirement that the LF is continuous at the end points of its support. Parameter-dependent support of the LF arises when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise probability density function (pdf) has a finite support. It is also pointed out that the commonly cited requirements of absolute integrability of the derivatives of the LF should be replaced by requirements on the log-LF (LLF). Some practical examples of finite-support measurement noises, which lead to parameter-dependent LF support, are discussed in light of the above. For the case where the LF is not continuous at the end points of its support, a new modified CRLB -designated as the Craḿer-Rao-Leibniz lower bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide valid bounds for a number of long-standing problems for which the CRLB was shown in the literature as not valid, in particular, for a uniformly distributed measurement noise.

Grammar-based generation of stochastic local search heuristics through automatic algorithm configuration tools

Several grammar-based genetic programming algorithms have been proposed in the literature to automatically generate heuristics for hard optimization problems. These approaches specify the algorithmic building blocks and the way in which they can be combined in a grammar; the best heuristic for the problem being tackled is found by an evolutionary algorithm that searches in the algorithm design space defined by the grammar.In this work, we propose a novel representation of the grammar by a sequence of categorical, integer, and real-valued parameters. We then use a tool for automatic algorithm configuration to search for the best algorithm for the problem at hand. Our experimental evaluation on the one-dimensional bin packing problem and the permutation flowshop problem with weighted tardiness objective shows that the proposed approach produces better algorithms than grammatical evolution, a well-established variant of grammar-based genetic programming. The reasons behind such improvement lie both in the representation proposed and in the method used to search the algorithm design space.

Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images.

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (xi , yi ) pairs, are Euclidean is well studied. The setting where yi is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, xi ∈ R, against a manifold-valued variable, yi ∈ . We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -a manifold-valued dependent variable against multiple independent variables, i.e., f : Rn → . Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL(n)/O(n) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.