Error Distribution Parameters Research Articles

Stochastic analysis and modeling of pharmaceutical screw feeder mass flow rates

Screw feeders, as the initial operation in continuous manufacturing of drug product processes, greatly influence the mass flow rate of pharmaceutical powders downstream. Existing flowsheet models can quickly simulate the average powder mass flow rate while custom Discrete Element Method models require prohibitively long times to simulate a minute of realistic, high-variance particle flow. We propose a hybrid deterministic-stochastic feeder flowsheet model that leverages time series analysis and an Autoregressive Moving Average (ARMA) model to quantify and simulate the observed non-random variation in feeder powder flow. To allow for improved process and controller design, our approach is quick-to-solve, high-variance, and has a low experimental overhead. By examining the deterministic model errors of three different volumetrically fed excipients, we demonstrate that the errors are leptokurtic, heavy-tailed, and display a linear dependence on their prior two seconds of state. These errors are all reasonably modeled by an ARMA(2,1) model and are parametrically distinct from each other. Furthermore, we show that refilling the feeder online significantly alters the error distribution, autocorrelation structure, and ARMA parameters. These findings lay the groundwork necessary to model and predict the realistic feeder dynamics of a much broader range of powders and operating conditions.

Stochastic side-lobe printing in EUV lithography: a simulation study

Background: Extreme ultraviolet (EUV) phase shifting masks (PSMs) may be introduced at the N3 node as alternative to double patterning or high-NA exposures. To be optically advantageous, PSMs may have to come with phase different from 180 deg and high transmission. Aim: The main concern is how such exposures could be affected by stochastic side-lobe printing. Approach: We present a simulation study where we compare stochastic distributions for contact layers with different mask approaches: binary mask, PSM with and without assist features, and TriTone masks. The purpose of this work is to evaluate whether PSMs in EUV lithography may become a valuable option to further extend single-exposure EUV at NA = 0.33. Results: We identify the best exposure conditions for lowest stochastic defectivity for each of the masks considered and perform a large-scale study at these conditions. We report CD and placement error distribution parameters, along with frequencies of observed defects. Conclusions: We find that PSMs are more limited in range of CDs that can be printed without stochastic defects, compared with binary masks. When compared at target CD that is optimal for PSM, proposed phase shift masks did not exhibit a clear advantage, suggesting careful consideration of stochastic defects will be needed during future process development.

Lateral collision risk assessment of parallel routes in ocean area based on space-based ADS-B

In order to ensure the safe operation of parallel routes in the ocean area, it is necessary to assess the risk of lateral collision of routes. The data used in the assessment often come from the real flights, while the traditional land-based ADS-B equipment has limited coverage and cannot be used for research. The space-based ADS-B system can receive and send the flight position of the aircraft over the ocean area. In this paper, the ocean tracks in space-based ADS-B data are first extracted. According to the different route structures, the data of fixed route and flexible route are preprocessed by segmented Kalman filter and cubic polynomial fitting method respectively. Then the processed track position error is calculated, and Q-Q (Quantile-Quantile) median method is used to test the normal distribution of the error, and k-means clustering is used to obtain the clustering center and occurrence frequency of positioning error and flight error. Finally, a collision risk model based on Bayesian network is established, and the error distribution parameters and occurrence frequency are brought into the model to evaluate the collision risk of parallel routes in the ocean area. The results show that the lateral collision risk based on space-based ADS-B ocean flight data is lower than 0.5 × 10−8 times/hour specified by ICAO when the fixed route separation is 14n miles and the flexible route separation is 49n miles. And the actual lateral interval of the Pacific flexible route is 50n miles (Jiang, 2010), which meets the safety target level.

Novel sparseness-inducing dual Kalman filter and its application to tracking time-varying spatially-sparse structural stiffness changes and inputs

While the theory of Bayesian system identification provides a probabilistic means for reliably and robustly inferring models of a dynamic system and their parameters based on measured dynamic response, exploiting sparseness during online tracking of changing model parameters is not well understood. The focus in this study is to implement the dual Kalman filter for real-time Bayesian sequential state and parameter identification based on noisy sensor signals while incorporating sparse Bayesian learning to impose sparse model parameter changes from their initial reference values. We also want out model to be able to capture the evolution of sparse model parameter changes between two successive time instants. To this end, we present a hierarchical Bayesian model for tracking the joint posterior distribution of the state and model parameter vectors for a monitored dynamical system, where the two afore-mentioned sparseness constraints (sparse changes from reference values and with time) are also effectively incorporated for each time. We show our stochastic model of the structural dynamical system can be represented as a coupled conditionally-linear Gaussian state space model for the state and model parameter vectors, leading to some interesting analytical properties of the method that allow quantities of interest to be calculated in real time by using Kalman filtering equations. The parameters for the measurement and state prediction errors are learned solely from the available data up to the current time and so our method resolves the well-known instability problem in Kalman filtering due to arbitrary assignment of the error-distribution parameters. Finally, two illustrative applications are presented, one for identification of stiffness degradation and the other for input time–history identification where both are based on noisy dynamic response measurements from a structure.

Robust design method for optimizing the static accuracy of a vertical machining center

In this paper, a robust design method for optimizing the static accuracy of a vertical machining center is proposed. First, the accuracy prediction model was established using screw theory to determine the output accuracy of the machine tool, which was verified by using a DBB and laser interferometer. Then, combining the machine tool’s output accuracy and accuracy design requirements to identify the performance equation of the machining accuracy, a model to calculate machining accuracy reliability is derived. Because the reliability calculation model is highly nonlinear, this paper uses response surface methodology to obtain a highly approximate solution for reliability, and the worst reliable working position can also be found from this. On this basis, according to the sensitivity analysis of machining accuracy reliability, the key distribution parameters of geometric error elements that have an important impact are identified. According to the principle of accuracy balance, the error distribution parameters are reasonably adjusted to reduce the effect of errors on the reliability to create a robust design for optimizing the machine tool static accuracy. The research results show that the method proposed only needs to optimize the error distribution parameters five times to make the machining accuracy reliability meet design requirements.

Analysis and Prediction of Reliability for Motion Error of Polish Mechanisms for Composite Box Based on Monte-carlo Method

The polish mechanisms for composite material box is produced for polishing box within demanding error range, it is made up of gear system, shaft systems and other transmission parts, the clearance and eccentric in the assembly lead to motion error in the polish mechanism; hence, the error reliability of polish mechanisms should be investigated. The error distributions follow normal distribution, the parameter are defined according to the multi-object software ADAMS simulation experiments results. The position accuracy reliability analysis bases on the Monte-carol method to estimate reliability of the polish equipment. Error distribution parameters are calculated with ADAMS simulation experiments instead of the real experiment, which can spare the experimental cost and time and makes it easy to predict reliability with little consumption. The error analysis focuses on the gear system and shaft system mainly, the other factor from other parts are neglected due to their little influence on the error, the reduced analysis benefits to the rapid obtaining reliability results.

Improving the Robustness of Entangled States by Basis Transformation.

In the practical application of quantum entanglement, entangled particles usually need to be distributed to many distant parties or stored in different quantum memories. In these processes, entangled particles unavoidably interact with their surrounding environments, respectively. We here systematically investigate the entanglement-decay laws of cat-like states under independent Pauli noises with unbalanced probability distribution of three kinds of errors. We show that the robustness of cat-like entangled states is not only related to the overall noise strength and error distribution parameters, but also to the basis of qubits. Moreover, we find that whether a multi-qubit state is more robust in the computational basis or transversal basis depends on the initial entanglement and number of qubits of the state as well as the overall noise strength and error distribution parameters of the environment. However, which qubit basis is conductive to enhancing the robustness of two-qubit states is only dependent on the error distribution parameters. These results imply that one could improve the intrinsic robustness of entangled states by simply transforming the qubit basis at the right moment. This robustness-improving method does not introduce extra particles and works in a deterministic manner.

Accounting for measurement error in analyzing data

Key PointAccounting for measurement error, whether the measurement error distribution parameters are known or not, is easily done using Bayesian inference and thereby improves analyses of data.

Estimating hydrologic model uncertainty in the presence of complex residual error structures

Hydrologic models provide a comprehensive tool to estimate streamflow response to environmental variables. Yet, an incomplete understanding of physical processes and challenges associated with scaling processes to a river basin, introduces model uncertainty. Here, we apply generalized additive models of location, scale and shape (GAMLSS) to characterize this uncertainty in an Atlantic coastal plain watershed system. Specifically, we describe distributions of residual errors in a two-step procedure that includes model calibration of the soil and water assessment tool (SWAT) using a sequential Bayesian uncertainty algorithm, followed by time-series modeling of residual errors of simulated daily streamflow. SWAT identified dominant hydrological processes, performed best during moderately wet years, and exhibited less skill during times of extreme flow. Application of GAMLSS to model residuals efficiently produced a description of the error distribution parameters (mean, variance, skewness, and kurtosis), differentiating between upstream and downstream outlets of the watershed. Residual error distribution is better described by a non-parametric polynomial loess curve with a smooth transition from a Box–Cox t distribution upstream to a skew t type 3 distribution downstream. Overall, the fitted models show that low flow events more strongly influence the residual probability distribution, and error variance increases with streamflow discharge, indicating correlation and heteroscedasticity of residual errors. These results provide useful insights into the complexity of error behavior and highlight the value of using GAMLSS models to conduct Bayesian inference in the context of a regression model with unknown skewness and/or kurtosis.

Trans-dimensional matched-field geoacoustic inversion with hierarchical error models and interacting Markov chains

This paper develops a trans-dimensional approach to matched-field geoacoustic inversion, including interacting Markov chains to improve efficiency and an autoregressive model to account for correlated errors. The trans-dimensional approach and hierarchical seabed model allows inversion without assuming any particular parametrization by relaxing model specification to a range of plausible seabed models (e.g., in this case, the number of sediment layers is an unknown parameter). Data errors are addressed by sampling statistical error-distribution parameters, including correlated errors (covariance), by applying a hierarchical autoregressive error model. The well-known difficulty of low acceptance rates for trans-dimensional jumps is addressed with interacting Markov chains, resulting in a substantial increase in efficiency. The trans-dimensional seabed model and the hierarchical error model relax the degree of prior assumptions required in the inversion, resulting in substantially improved (more realistic) uncertainty estimates and a more automated algorithm. In particular, the approach gives seabed parameter uncertainty estimates that account for uncertainty due to prior model choice (layering and data error statistics). The approach is applied to data measured on a vertical array in the Mediterranean Sea.

Linear and nonlinear regression with stable errors

In this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution. The asymptotic Fisher information matrix for both the regression coefficients and the error distribution parameters are derived, giving large sample confidence intervals for all parameters. Simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable, as well as a real example using financial data. The results are then extended to nonlinear models and to non-homogeneous error terms.

Restraint of tool path ripple based on surface error distribution and process parameters in deterministic finishing

The influence from the regular tool path to micro fabrication errors in deterministic finishing is studied through the simulations and experiments. The random pitch tool path based on the surface error distribution and the process parameters is designed to reduce this residual error when adopting the regular path to achieve the corrective polish. A nucleated glass flat mirror is polished with this method on the experimental installation UPF700-7 developed by ourselves. The surface accuracy is improved from the initial λ/30(RMS, 90%aperture, λ = 632.8 nm) to the final λ/200 in 5 minutes and the medium-high spatial frequency errors induced by the regular path is restricted well at the same time. The accuracy of the simulation and the validity of the random pitch tool path are both proved through the experiments.

An introduction to volatility models with indices

This paper considers a class of volatility models generated by autoregressive (AR) type models with indices. Some results associated with the autocorrelation function (acf) of this class are given and the spectral density is obtained in terms of the kurtosis of the error distribution and model parameters.

A Transform Approach to the Simulation of Transient Gas Flow in Porous Media

Abstract The mathematical description of transient gas flow through porous media leads to a second-order nonlinear partial differential equation for which no analytical solution bas been found. For flow in one linear dimension, this paper outlines a procedure for using finite Fourier transforms to reduce the partial differential equations to a set of first-order, partial differential equations to a set of first-order, nonlinear ordinary differential equations with time as the independent variable. The dependent variables of these equations are the Fourier transform coefficients of the gas pressure in the porous media The procedure is applied to an porous media The procedure is applied to an example problem and calculated results are compared with other published results obtained by a numerical finite-difference technique. The good agreement shown between the two sets of results tends to confirm the validity of both solutions. Introduction The differential equation describing one-dimensional gas flow in porous media is a nonlinear equation with no known analytical solution. Numerical finite-difference methods are normally used to solve the flow equation. This paper presents a method which uses linear analysis to transform a second-order problem into a set of first-order problems. Basically the approach used is to assume an approximation of the solution consisting of known functions and unknown parameters and apply some error distribution parameters and apply some error distribution principles to determine the parameters of the principles to determine the parameters of the approximation. Although there are several error distribution principles, one of the most common is the orthogonal method. Recently special cases of the orthogonal method, the Galerkin or Bubnov-Galerkin have been applied by Price et al., to petroleum reservoir problems with considerable success. A classical orthogonal method is finite Fourier transforms. Recently several authors have demonstrated that this method can be used with considerable success in the simulation of systems described by a limited class of linear partial differential equations. This paper is intended to demonstrate that the finite Fourier transform method can be used to reduce the nonlinear partial differential equation describing gas flow in a linear porous media to a set of ordinary differential equations. THEORY The basic equation being studied is: ............(1) which is derived from the continuity equation ...............(2) the equation of state ..............(3) and Darcy's law ................(4) For the orthogonal method of error distribution, the procedure is to assume an approximate solution Pa, define an error L[pa] and require that this error Pa, define an error L[pa] and require that this error be orthogonal to N independent functions n(x) (n=1,2.... N) ...................(5) SPEJ P. 135