Deterministic Quantities Research Articles

Discussion: Deterioration of performance of the lasso with many predictors

Oracle inequalities provide probability loss bounds for the lasso estimator at a deterministic choice of the regularization parameter and are commonly cited as theoretical justification for the lasso and its ability to handle high-dimensional settings. Unfortunately, in practice, the regularization parameter is not selected to be a deterministic quantity, but is instead chosen using a random, data-dependent procedure, often making these inequalities misleading in their implications. We discuss general results and demonstrate empirically for data using categorical predictors that the amount of deterioration in performance of the lasso as the number of unnecessary predictors increases can be far worse than the oracle inequalities suggest, but imposing structure on the form of the estimates can reduce this deterioration substantially.

Modelling stochastic correlation

This work deals with the stochastic modelling of correlation in finance. It is well known that the correlation between financial products, financial institutions, e.g., plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that the correlation is hardly a deterministic quantity. For example, we illustrate this issue with the analysis of correlation between daily returns time series of S&P Index and Euro/USD exchange rates. The approach of modelling the correlation as a hyperbolic function of a stochastic process has been recently proposed. Here, we review this novel concept and generalize this approach to derive stochastic correlation processes (SCP) from a hyperbolic transformation of the modified Ornstein-Uhlenbeck process. We determine a transition density function of this SCP in closed form which could be used easily to calibrate SCP models to historical data. As an illustrating example of our new approach, we compute the price of a quantity adjusting option (Quanto) and discuss concisely the effect of considering stochastic correlation on pricing the Quanto.

Hybrid manufacturing/remanufacturing lot-sizing problem with returns supplier’s selection under, carbon emissions constraint

In this study, we discuss a lot sizing problem in manufacturing systems incorporating returns remanufacturing options under a carbon emission constraint. The system studied is a single production line where both regular production and remanufacturing process are carried out, with different set-up costs for each operation. We consider also the returns collection phase from a certain number of customers/distributors with a deterministic returns quantity in each period of the planning horizon. The paper proposes a mathematical model for the problem and two versions of a hybrid method (HM1 & HM2) providing an approximate solution. The mathematical model was tested on CPLEX (Software optimizer), and the obtained results were compared with the ones provided by the hybrid methods. The analysis of the results showed that hybrid methods provide good quality solutions in a moderate computational time.

Improved PPP Ambiguity Resolution Considering the Stochastic Characteristics of Atmospheric Corrections from Regional Networks.

With the increased availability of regional reference networks, Precise Point Positioning (PPP) can achieve fast ambiguity resolution (AR) and precise positioning by assimilating the satellite fractional cycle biases (FCBs) and atmospheric corrections derived from these networks. In such processing, the atmospheric corrections are usually treated as deterministic quantities. This is however unrealistic since the estimated atmospheric corrections obtained from the network data are random and furthermore the interpolated corrections diverge from the realistic corrections. This paper is dedicated to the stochastic modelling of atmospheric corrections and analyzing their effects on the PPP AR efficiency. The random errors of the interpolated corrections are processed as two components: one is from the random errors of estimated corrections at reference stations, while the other arises from the atmospheric delay discrepancies between reference stations and users. The interpolated atmospheric corrections are then applied by users as pseudo-observations with the estimated stochastic model. Two data sets are processed to assess the performance of interpolated corrections with the estimated stochastic models. The results show that when the stochastic characteristics of interpolated corrections are properly taken into account, the successful fix rate reaches 93.3% within 5 min for a medium inter-station distance network and 80.6% within 10 min for a long inter-station distance network.

Exact enclosures of roots of interval quadratic equations by Sridhara's and Fagnano's or modified Fagnano's formulas

In this study we investigate the question of accurate determination of the root enclosures of quadratic equations whose coefficients constitute interval variables. We treat several special cases where either Sridhara's classical formula or Fagnano's alternative expression provide exact interval enclosures for the roots of the quadratic equation. In the case of a single coefficient serving as an interval, it is shown that the classical interval analysis by either Sridhara's, Fagnano's or specifically modified Fagnano's formulas provide lower and upper bounds of roots. Then we follow with the case of two coefficients being intervals whereas the third coefficient is a deterministic quantity. Numerous examples are provided in which the solutions are compared with the direct numerical evaluation of the roots' enclosures. In three arising cases either Sridhara's or Fagnano's expressions suffice to obtain exact enclosures.

Security of Supply and Generation Reserve Management Delegation under Extremely High Load Curtailment Cost

Power failures in a number of electric systems worldwide emphasize the importance of security of electric supply. Security of supply involves long term resource adequacy and medium to short term generation reserve management. Generation reserve requirements are usually determined using empirical rules considering the demand as a deterministic quantity. Ignoring randomness could lead to suboptimal decisions. The research presented here differs from that prevailing in the literature by considering the demand as a random variable following a doubly truncated normal distribution this has the advantage of allowing considering day to day and seasonal variations. In this paper two new mathematical models are presented to determine generation reserve requirements for a market in which the regulator imposes an extremely high load curtailment cost in any unmet demand and delegates the role of buying generation reserves on the generators in a secondary market. These models are helpful tools to study analytically or by numerical simulation the interactions between the two markets. A potential scope for application of the proposed models is presented for the Colombian electric market.

Direct shrinkage estimation of large dimensional precision matrix

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. At the end, a simulation is provided where the suggested estimator is compared with the estimators for the precision matrix proposed in the literature. The optimal shrinkage estimator shows significant improvement and robustness even for non-normally distributed data.

On the Mathematical Theories and Cognitive Foundations of Information

A recent discovery in computer and software sciences is that information in general is a deterministic abstract quantity rather than a probability-based property of the nature. Information is a general form of abstract objects represented by symbolical, mathematical, communication, computing, and cognitive systems. Therefore, information science is one of the contemporary scientific disciplines collectively known as abstract sciences such as system, information, cybernetics, cognition, knowledge, and intelligence sciences. This paper presents the cognitive foundations, mathematical models, and formal properties of information towards an extended theory of information science. From this point of view, information is classified into the categories of classic, computational, and cognitive information in the contexts of communication, computation, and cognition, respectively. Based on the three generations of information theories, a coherent framework of contemporary information is introduced, which reveals the nature of information and the fundamental principles of information science and engineering.

A mixed approach for determination of initial cable forces in cable-stayed bridges and the parameters variability

The determination of initial cable forces in cable-stayed bridges is an important first step in design and analysis of the structure under external loads. Adjustments of stay forces are often required during construction in order to assure the requested behaviour of the bridge in terms of final geometrical configuration and internal force distribution. An accurate assessment of the stay tensioning system allows designers to obtain a good result at the end of construction, by considering the parameters involved as deterministic quantities, assuring the observance of the execution tolerances during works. Actual loads and their variations need instead a stochastic approach which can give useful indications about the effects of parameter variations. Particularly, the measurement on site of the actual values of stay elongations contains errors and the actual values of prestressing forces and working site loads are subjected to unknown variations. In this paper a procedure is implemented which takes into account the stochastic variation of stay elongations and the related uncertainties. The presented method does not require large computational efforts or computer memory. By approximating the probability density distribution of the variables involved by the Gaussian curve, a closed form expression of the imposed elongations given to stays and their variations during erection is provided. The main aim is to obtain an accurate prediction of the differences between the deterministic quantities found by the design analyses and the actual values of deck deformations and stresses in the erection of steel cable-stayed bridges.

Noncontextuality with marginal selectivity in reconstructing mental architectures.

We present a general theory of series-parallel mental architectures with selectively influenced stochastically non-independent components. A mental architecture is a hypothetical network of processes aimed at performing a task, of which we only observe the overall time it takes under variable parameters of the task. It is usually assumed that the network contains several processes selectively influenced by different experimental factors, and then the question is asked as to how these processes are arranged within the network, e.g., whether they are concurrent or sequential. One way of doing this is to consider the distribution functions for the overall processing time and compute certain linear combinations thereof (interaction contrasts). The theory of selective influences in psychology can be viewed as a special application of the interdisciplinary theory of (non)contextuality having its origins and main applications in quantum theory. In particular, lack of contextuality is equivalent to the existence of a “hidden” random entity of which all the random variables in play are functions. Consequently, for any given value of this common random entity, the processing times and their compositions (minima, maxima, or sums) become deterministic quantities. These quantities, in turn, can be treated as random variables with (shifted) Heaviside distribution functions, for which one can easily compute various linear combinations across different treatments, including interaction contrasts. This mathematical fact leads to a simple method, more general than the previously used ones, to investigate and characterize the interaction contrast for different types of series-parallel architectures.

Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises

The mean exit time and escape probability are deterministic quantities that can quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. Both deterministic quantities are characterized by differential–integral equations (i.e., differential equations with nonlocal terms) but with different exterior conditions. A convergent numerical scheme is developed and validated for computing the mean exit time and escape probability for two-dimensional stochastic systems with rotationally symmetric α-stable type Lévy motions. The effects of drift, Gaussian noises, intensity of jump measure and domain sizes on the mean exit time are discussed. The difference between the one-dimensional and two-dimensional cases is also presented.

Isoperimetry in Two‐Dimensional Percolation

We study isoperimetric sets, i.e., sets with minimal boundary for a prescribed volume, on the unique infinite connected component of supercritical bond percolation on the square lattice. In the limit of the volume tending to infinity, properly scaled isoperimetric sets are shown to converge (in the Hausdorff metric) to the solution of an isoperimetric problem in ℝ2 with respect to a particular norm. As part of the proof we also show that the anchored isoperimetric profile as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the square lattice. © 2015 Wiley Periodicals, Inc.

Matched-Field Processing Performance Under the Stochastic and Deterministic Signal Models

Matched-field processing (MFP) is commonly used in underwater acoustics to estimate source position and/or oceanic environmental parameters. Performance prediction of the multisnapshot and multifrequency MFP problem is of critical importance. To this end, two signal models are usually considered: the stochastic model, which assumes that the source signal is a stochastic process, and the deterministic model, which assumes that the source signal is a deterministic quantity. The Ziv-Zakai bound (ZZB) and the method of interval errors (MIE), which both rely on the computation of a so-called pairwise error probability, proved to be useful tools for MFP performance prediction. However, only the stochastic model has been considered so far. This paper provides a method that allows to compute the pairwise error probability, hence to use the ZZB and MIE, under both the stochastic and deterministic signal models. The proposed approach, based on recent results on quadratic forms in Gaussian variables, unifies the two models under the same formalism. The results are illustrated through the computation of the ZZB and MIE performance analysis. The Bayesian and the hybrid Cramer-Rao bounds are also given for comparison.

Modeling disease transmission near eradication: An equation free approach

Although disease transmission in the near eradication regime is inherently stochastic, deterministic quantities such as the probability of eradication are of interest to policy makers and researchers. Rather than running large ensembles of discrete stochastic simulations over long intervals in time to compute these deterministic quantities, we create a data-driven and deterministic “coarse” model for them using the Equation Free (EF) framework. In lieu of deriving an explicit coarse model, the EF framework approximates any needed information, such as coarse time derivatives, by running short computational experiments. However, the choice of the coarse variables (i.e., the state of the coarse system) is critical if the resulting model is to be accurate. In this manuscript, we propose a set of coarse variables that result in an accurate model in the endemic and near eradication regimes, and demonstrate this on a compartmental model representing the spread of Poliomyelitis. When combined with adaptive time-stepping coarse projective integrators, this approach can yield over a factor of two speedup compared to direct simulation, and due to its lower dimensionality, could be beneficial when conducting systems level tasks such as designing eradication or monitoring campaigns.

On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix

In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free estimators obey almost surely the smallest Frobenius loss over all linear shrinkage estimators for the covariance matrix. The case we consider includes the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. Additionally, we prove that the Frobenius norm of the sample covariance matrix tends almost surely to a deterministic quantity which can be consistently estimated.

An economic production quantity model for a deteriorating system integrated with predictive maintenance strategy

While there has been considerable work over the years on the basic deterministic economic production quantity (EPQ) and its derivative models, there have been few extensions of these models that recognize the potential effects of machine degradation. As maintenance activities can keep machines in good operation, it should be integrated into EPQ models to meet real situations. Due to machine degradation, this paper integrates predictive maintenance into EPQ model in which autoregressive integrated moving average model is adopted to predict system's healthy indicator. Moreover, two kinds of system out-of-control states are considered in this proposed EPQ model: in State I, the system produces non-conforming items; and in State II, the system fails. Aiming at minimizing the expected average total cost and optimizing the EPQ, suitable maintenance intervals and frequency are determined prior to any predicted failure. Finally, a case study is presented and the computational results are discussed to show the efficiency of this integrated EPQ model.

Bayesian calibration of power plant models for accurate performance prediction

Abstract Gas turbine combined cycles are expected to play an increasingly important role in the balancing of supply and demand in future energy markets. Thermodynamic modeling of these energy systems is frequently applied to assist in decision making processes related to the management of plant operation and maintenance. In most cases, model inputs, parameters and outputs are treated as deterministic quantities and plant operators make decisions with limited or no regard of uncertainties. As the steady integration of wind and solar energy into the energy market induces extra uncertainties, part load operation and reliability are becoming increasingly important. In the current study, methods are proposed to not only quantify various types of uncertainties in measurements and plant model parameters using measured data, but to also assess their effect on various aspects of performance prediction. The authors aim to account for model parameter and measurement uncertainty, and for systematic discrepancy of models with respect to reality. For this purpose, the Bayesian calibration framework of Kennedy and O’Hagan is used, which is especially suitable for high-dimensional industrial problems. The article derives a calibrated model of the plant efficiency as a function of ambient conditions and operational parameters, which is also accurate in part load. The article shows that complete statistical modeling of power plants not only enhances process models, but can also increases confidence in operational decisions.

Survey and Review

The Survey and Review article in this issue is “The Narrow Escape Problem,” by D. Holcman and Z. Schuss. We are all familiar with the jagged plots formed by share prices and interest rates as they fluctuate over time. It is natural to ask questions like -when will my portfolio of Facebook shares reach $1 Million? -when will my bank's interest rate leave the 2% -- 3% range? These questions concern hitting times. Modeling the evolution as a stochastic process (a random variable that changes over time), we want to know the first occasion when the value reaches a specified level. More generally, for processes in a higher dimension, such as Robert Brown's pollen particles suspended in a fluid, we could ask when a tagged particle will first meet a specified region on the boundary of a given domain. The hitting time is itself a random variable, and we can reduce to a simpler deterministic quantity by asking for its expected value. It will then be no surprise to many readers that this deterministic value can be obtained by solving an appropriate partial differential equation. The topic of this article is a commonly arising special case which has wide practical significance and offers the added attraction of allowing mathematicians to introduce a small parameter. The narrow escape time (NET) relates to the first time that Brownian motion meets a small absorbing window on the boundary of a domain that is otherwise reflecting. Figure 1.3 illustrates the setting very clearly. A closely related problem is to determine when a particle traverses a narrow passage between almost disconnected regions, as illustrated in Figure 1.1. The article addresses the development of asymptotic methods that deal with various geometries, motivated by applications in cell biology. The underlying mathematical challenge can be viewed as solving a mixed Neumann--Dirichlet boundary value problem for the Poisson equation in a bounded domain, and the authors survey a range of developments in nonstandard asymptotics that have been brought to bear in recent years. This article will be a valuable resource for readers with interests at the interface between probability theory and partial differential equations, and those who study asymptotic analysis of singular perturbation problems and applications to physics, biochemistry, and neuroscience.

A Model-Based Approach for Predicting the Remaining Driving Range in Electric Vehicles


 
 
 The limited driving range has been pointed out as one of the main technical factors affecting the acceptance of electric vehicles. Offering the driver accurate information about the remaining driving range (RDR) reduces the range anxiety and increases the acceptance of the driver. The integration of electric vehicles into future transportation systems demands advanced driving assistance systems that offer reliable information regarding the RDR. Unfortunately, the RDR is, due to many sources of uncertainty, difficult to predict. The driving style, the road conditions or the traffic situation are some of these uncertain factors. A model-based approach for predicting the RDR by combining unscented filtering and Markov chains is introduced in this paper. Detailed models are implemented for representing the electric vehicle and its energy storage system. The RDR prediction is validated by a set of simulation based experiments for different driving scenarios. Whereas traditional approaches consider the RDR as a deterministic quantity, to our knowledge, this approach is the first to represent the RDR by a probability density function. We aim to provide initial steps towards a solution for generating reliable information regarding the RDR which can be used by driving assistance systems in electric vehicles.
 
 

SU-D-137-05: Impact of Dosimetric Uncertainties On the Computed Relative Biological Effectiveness (RBE) of High Linear Energy Transfer (LET) Particles

Purpose: Quantify uncertainties in cellular and sub‐cellular dosimetry required for the analysis of biological experiments in which cell cultures are irradiated by charged particles. Assess the impact of dosimetric uncertainties on the computed RBE of high LET charged particles. Methods: Estimates of the frequency‐mean specific energy, lineal energy and absorbed dose per unit fluence (D/Φ) are computed using analytical, deterministic (MCDS) and Monte Carlo (MCNPX and TOPAS/Geant4) methods for representative cell culture geometries. Uncertainties in dosimetric quantities associated with thickness of the culture dish (0.85 mg/cm2 of Melinex) and the location of the nucleus (0–8 urn from the Melinex surface) within a 2 mm layer of culture medium (water) are quantified for a variety of high LET particles (e−, 1H+, 2H+, 3He2 + 4He2 +, and 12C6 +). Results: Uncertainties in D/Φ and related quantities may be as large as 75% for a 4 MeV 4He2 + (range about 27 μm). Substantial differences are seen in dosimetric quantities for all of the considered charged particles with a CSDA range less than about 150 μm. For particles with a CSDA range larger than about 30 μm, dosimetric quantities computed using the deterministic MCDS algorithm are typically within 3% of the estimates obtained with Monte Carlo. The default TOPAS/Geant4 physics parameters sometimes produce estimates of fluence and absorbed dose that differ from the MCDS and MCNPX estimates by as much as 15%. Conclusion: The size and location of the cell nucleus during mitotic cell cycle has a significant impact on the dose delivered to cellular and subcellular targets for short range (high LET) particles. Path and energy straggling have a small impact on deterministic dose quantities for particles with a CSDA range larger than about 30 μm (Z < 6). Dosimetric uncertainties may have a significant impact on the inferred RBE of high LET particles.