Mixing efficiency as estimated by nonlinear least squares

1983 ◽  
Vol 10 (4) ◽  
pp. 703-712
Author(s):  
David T. Chapman
Keyword(s):  
Least Squares ◽  
Dispersion Model ◽  
Nonlinear Least Squares ◽  
Statistical Technique ◽  
Sum Of Squares ◽  
Axial Dispersion ◽  
Mixing Efficiency ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Washout Curves

The suitability of a statistical technique known as nonlinear least squares for use in estimating mixing coefficients was evaluated by fitting models to residence time distribution curves. The washout curves were generated by adding slug inputs of tracers to three different reactors. Each of the reactors, used to treat wastewaters, was a different size and represented a different degree of mixing.Three models, described in the paper, were examined for use in conjuction with the nonlinear least squares technique. They included the axial dispersion, N-tanks-in-series, and Cholette–Cloutier models. The form of the equation for the axial dispersion model depends on the boundary conditions for the reactor being studied. For reactors which cannot be classified as "open" vessels, the required analytical solutions either do not exist or are not suitable for use with the nonlinear least squares technique.Mixing coefficients for the N-tanks and Chollette–Cloutier models were obtained from the tracer washout curves for the three reactors. The residual sum of squares based on nonlinear least squares estimates for the model parameters was compared with the sum of squares obtained using more conventional methods for estimating the parameters. The existence of trailing tails on the tracer curves resulted in misleading parameter estimates for the two models using conventional techniques. Keywords: mixing, least squares, tracer, dispersion, short-circuiting, deadspace.

Integrating Parameter Estimation Solutions From the Time and Time-Scale Domains

2009 ◽  
Author(s):  
James R. McCusker ◽  
Kourosh Danai
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Time Scale ◽  
Prediction Error ◽  
Nonlinear Least Squares ◽  
Integrated Approach ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Single Model ◽  
Iterative Estimation

A method of parameter estimation was recently introduced that separately estimates each parameter of the dynamic model [1]. In this method, regions coined as parameter signatures, are identified in the time-scale domain wherein the prediction error can be attributed to the error of a single model parameter. Based on these single-parameter associations, individual model parameters can then be estimated for iterative estimation. Relative to nonlinear least squares, the proposed Parameter Signature Isolation Method (PARSIM) has two distinct attributes. One attribute of PARSIM is to leave the estimation of a parameter dormant when a parameter signature cannot be extracted for it. Another attribute is independence from the contour of the prediction error. The first attribute could cause erroneous parameter estimates, when the parameters are not adapted continually. The second attribute, on the other hand, can provide a safeguard against local minima entrapments. These attributes motivate integrating PARSIM with a method, like nonlinear least-squares, that is less prone to dormancy of parameter estimates. The paper demonstrates the merit of the proposed integrated approach in application to a difficult estimation problem.

Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

2011 ◽  
Vol 64 (S1) ◽  
pp. S3-S18 ◽  
Author(s):  
Yuanxi Yang ◽  
Jinlong Li ◽  
Junyi Xu ◽  
Jing Tang
Keyword(s):  
Least Squares ◽  
Stochastic Models ◽  
Influence Function ◽  
A Priori ◽  
Functional Model ◽  
Global Navigation Satellite Systems ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Integrated Navigation ◽  
Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

scholarly journals A Monograph on Nonlinear Regression Models

2018 ◽  
Vol 7 (4.10) ◽  
pp. 543
Author(s):  
B. Mahaboob ◽  
B. Venkateswarlu ◽  
C. Narayana ◽  
J. Ravi sankar ◽  
P. Balasiddamuni
Keyword(s):  
Regression Model ◽  
Least Squares ◽  
Nonlinear Regression ◽  
Nonlinear Models ◽  
Nonlinear Least Squares ◽  
Error Variance ◽  
Ordinary Least Squares ◽  
Parameter Estimates ◽  
Nonlinear Regression Model ◽  
Nonlinear Regression Models

This research article uses Matrix Calculus techniques to study least squares application of nonlinear regression model, sampling distributions of nonlinear least squares estimators of regression parametric vector and error variance and testing of general nonlinear hypothesis on parameters of nonlinear regression model. Arthipova Irina et.al [1], in this paper, discussed some examples of different nonlinear models and the application of OLS (Ordinary Least Squares). MA Tabati et.al (2), proposed a robust alternative technique to OLS nonlinear regression method which provide accurate parameter estimates when outliers and/or influential observations are present. Xu Zheng et.al [3] presented new parametric tests for heteroscedasticity in nonlinear and nonparametric models.  

scholarly journals Quantifying soil hydraulic properties and their uncertainties by modified GLUE method

2017 ◽  
Vol 31 (3) ◽  
pp. 433-445
Author(s):  
Yifan Yan ◽  
Jianli Liu ◽  
Jiabao Zhang ◽  
Xiaopeng Li ◽  
Yongchao Zhao
Keyword(s):  
Confidence Interval ◽  
Least Squares ◽  
Water Content ◽  
Soil Water ◽  
Nonlinear Least Squares ◽  
Uncertainty Estimation ◽  
Model Parameters ◽  
Retention Curve ◽  
Least Squares Algorithm ◽  
Soil Hydraulic

AbstractNonlinear least squares algorithm is commonly used to fit the evaporation experiment data and to obtain the ‘optimal’ soil hydraulic model parameters. But the major defects of nonlinear least squares algorithm include non-uniqueness of the solution to inverse problems and its inability to quantify uncertainties associated with the simulation model. In this study, it is clarified by applying retention curve and a modified generalised likelihood uncertainty estimation method to model calibration. Results show that nonlinear least squares gives good fits to soil water retention curve and unsaturated water conductivity based on data observed by Wind method. And meanwhile, the application of generalised likelihood uncertainty estimation clearly demonstrates that a much wider range of parameters can fit the observations well. Using the ‘optimal’ solution to predict soil water content and conductivity is very risky. Whereas, 95% confidence interval generated by generalised likelihood uncertainty estimation quantifies well the uncertainty of the observed data. With a decrease of water content, the maximum of nash and sutcliffe value generated by generalised likelihood uncertainty estimation performs better and better than the counterpart of nonlinear least squares. 95% confidence interval quantifies well the uncertainties and provides preliminary sensitivities of parameters.

Solids Flow Models for Gas - Flowing Solids - Fixed Bed Contactors

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Nikola M Nikacevic ◽  
Milorad P. Dudukovic
Keyword(s):  
Dispersion Model ◽  
Fixed Bed ◽  
Axial Dispersion ◽  
Step Response ◽  
Model Parameters ◽  
Response Curves ◽  
Flow Models ◽  
Solids Holdup ◽  
Axial Dispersion Model ◽  
Static Zone

Three solids flow models for gas – flowing solids – fixed bed contactors are analyzed. They all presume axial dispersion in the dynamic, freely flowing zone, but they differ in the interpretation of the stagnant zone. The models have been examined and the model parameters have been optimized on the basis of two types of tracer experiments. One provides step response curves for flowing solids at the exit and the other presents the response curves of the static flowing solids holdup. The model which assumes axial dispersion and exchange between dynamic and two active static zones, most accurately describes the solids flow pattern. A simpler model which presumes exchange between dynamic and one static zone can be used if there is no need for a precise description of the behavior of stagnant particles. The most simple axial dispersion model is not realistic, as it does not explain stagnancy at all, which was experimentally observed for the gas – flowing solids – fixed bed contactors.

Migratory Catch-Age Analysis

1990 ◽  
Vol 47 (12) ◽  
pp. 2315-2327 ◽  
Author(s):  
Terrance J. Quinn II ◽  
Richard B. Deriso ◽  
Philip R. Neal
Keyword(s):  
Auxiliary Information ◽  
Nonlinear Least Squares ◽  
Sensitivity Study ◽  
Error Model ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Migration Rates ◽  
Process Error ◽  
A New Technique ◽  
And Migration

We review techniques for estimating the abundance of migratory populations and develop a new technique based on catch-age data from geographic regions and our earlier technique, catch-age analysis with auxiliary information (Deriso et al. 1985, 1989). Data requirements are catch-age data over several years, some auxiliary information, and migration rates among regions. The model, containing parameters for year-class abundance, age selectivity, full-recruitment fishing mortality, and catchability, is fitted to data with a nonlinear least squares algorithm. We present a measurement error model and a process error model and favor the process error model because all model parameters can be jointly estimated. By application to data on Pacific halibut, the process error model converges readily and produces estimates with no significant bias. These estimates have relatively high precision compared to those from analyses which did not incorporate migration information. The error structure used in a model has a more significant impact on parameter estimates than migration rates. A sensitivity study of migration rates shows sensitivity of the order of the rates themselves.

scholarly journals Uncertainty Quantification in Machine Learning and Nonlinear Least Squares Regression Models

2021 ◽  
Author(s):  
Ni Zhan ◽  
John Kitchin
Keyword(s):  
Machine Learning ◽  
Least Squares ◽  
Uncertainty Quantification ◽  
Automatic Differentiation ◽  
Nonlinear Least Squares ◽  
Delta Method ◽  
Training Data ◽  
Model Parameters ◽  
Uncertainty Measure ◽  
Least Squares Regression

Machine learning (ML) models are valuable research tools for making accurate predictions. However, ML models often unreliably extrapolate outside their training data. We propose an uncertainty quantification method for ML models (and generally for other nonlinear models) with parameters trained by least squares regression. The uncertainty measure is based on the multiparameter delta method from statistics, which gives the standard error of the prediction. The uncertainty measure requires the gradient of the model prediction and the Hessian of the loss function, both with respect to model parameters. Both the gradient and Hessian can be readily obtained from most ML software frameworks by automatic differentiation. We show that the uncertainty measure is larger for input space regions that are not part of the training data. Therefore this method can be used to identify extrapolation and to aid in selecting training data or assessing model reliability.

scholarly journals Voxel-Based Estimation of Kinetic Model Parameters of the l-[1-11C]Leucine PET Method for Determination of Regional Rates of Cerebral Protein Synthesis: Validation and Comparison with Region-of-Interest-Based Methods

2009 ◽  
Vol 29 (7) ◽  
pp. 1317-1331 ◽  
Author(s):  
Giampaolo Tomasi ◽  
Alessandra Bertoldo ◽  
Shrinivas Bishu ◽  
Aaron Unterman ◽  
Carolyn Beebe Smith ◽  
...  
Keyword(s):  
Protein Synthesis ◽  
Kinetic Model ◽  
Region Of Interest ◽  
Nonlinear Least Squares ◽  
Measured Data ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Time Activity ◽  
Positron Emission ◽  
Cerebral Protein Synthesis

We adapted and validated a basis function method (BFM) to estimate at the voxel level parameters of the kinetic model of the l-[1-11C]leucine positron emission tomography (PET) method and regional rates of cerebral protein synthesis (rCPS). In simulation at noise levels typical of voxel data, BFM yielded low-bias estimates of rCPS; in measured data, BFM and nonlinear least-squares parameter estimates were in good agreement. We also examined whether there are advantages to using voxel-level estimates averaged over regions of interest (ROIs) in place of estimates obtained by directly fitting ROI time-activity curves (TACs). In both simulated and measured data, fits of ROI TACs were poor, likely because of tissue heterogeneity not taken into account in the kinetic model. In simulation, rCPS determined from fitting ROI TACs was substantially overestimated and BFM-estimated rCPS averaged over all voxels in an ROI was slightly underestimated. In measured data, rCPS determined by regional averaging of voxel estimates was lower than rCPS determined from ROI TACs, consistent with simulation. In both simulated and measured data, intersubject variability of BFM-estimated rCPS averaged over all voxels in a ROI was low. We conclude that voxelwise estimation is preferable to fitting ROI TACs using a homogeneous tissue model.

scholarly journals Categorical Omega With Small Sample Sizes via Bayesian Estimation: An Alternative to Frequentist Estimators

2018 ◽  
Vol 79 (1) ◽  
pp. 19-39 ◽  
Author(s):  
Yanyun Yang ◽  
Yan Xia
Keyword(s):  
Least Squares ◽  
Bayesian Estimation ◽  
Weighted Least Squares ◽  
Small Sample ◽  
Estimation Methods ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Analysis Model ◽  
Factor Analysis Model ◽  
Diagonally Weighted Least Squares

When item scores are ordered categorical, categorical omega can be computed based on the parameter estimates from a factor analysis model using frequentist estimators such as diagonally weighted least squares. When the sample size is relatively small and thresholds are different across items, using diagonally weighted least squares can yield a substantially biased estimate of categorical omega. In this study, we applied Bayesian estimation methods for computing categorical omega. The simulation study investigated the performance of categorical omega under a variety of conditions through manipulating the scale length, number of response categories, distributions of the categorical variable, heterogeneities of thresholds across items, and prior distributions for model parameters. The Bayes estimator appears to be a promising method for estimating categorical omega. M plus and SAS codes for computing categorical omega were provided.

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