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Tracer Kinetic Analysis for PET and SPECT
Published in Troy Farncombe, Krzysztof Iniewski, Medical Imaging, 2017
In weighted least squares methods, the structure of the error variance can be concerned by weighting the measurement error with the relative accuracy of the measurement.28 However, direct estimation of the error variance of every pixel is hard to achieve practically. An approximate weighting formula is, therefore, commonly used.29,30 Generalized least squares (GLS) and gen-eralized weighted least squares (GWLS) are the generalized formula of LLS and LWLS, respectively. Since the dependency between the measurement error at each sampling point in the LLS and LWLS methods leads to bias in the estimated parameters, GLS and GWLS were suggested as alternatives to eliminate this bias.29,30
Generalized least squares and the analysis of heteroscedasticity
Published in Raymond J. Carroll, David Ruppert, Transfor mation and Weighting in Regression, 2017
Raymond J. Carroll, David Ruppert
The interpretation of (2.3) as weighted least squares is natural: the larger the values of the weight wi, the larger the contribution of that squared deviation. When we note that the weights are the inverses of the variances, we see that weighted least squares gives most weight to data points with low variability, which seems desirable.
Evaluating the impacts of vehicle-mounted Variable Message Signs on passing vehicles: implications for protecting roadside incident and service personnel
Published in Journal of Intelligent Transportation Systems, 2023
Jun Liu, Xing Fu, Alexander Hainen, Chenxuan Yang, Leon Villavicencio, William J. Horrey
Due to variations in video quality, such as lighting conditions, the number of frames in which the same vehicle was detected differed across observations. Recognizing that observations with different frame numbers may have an uneven influence on the model parameter estimates, this study employed the Weighted Regression method to estimate the model parameters. Weighted least squares take into account the behavior of random errors in the model by incorporating weights associated with each data point into the fitting criterion. The weight size indicates the amount of information contained in the corresponding observation. By optimizing the weighted fitting criterion to determine the parameter estimates, the weights determine the contribution of each observation to the final parameter estimates. In this study, the number of frames detected for each observation serves as the weighting factor integrated into the model estimation.
Efficient computational method for the dynamic responses of a floating wind turbine
Published in Ships and Offshore Structures, 2020
A floating wind turbine is an offshore wind turbine mounted on a floating structure that allows the turbine to generate electricity in water depths where fixed-foundation turbines are not feasible. In recent years a lot of research studies (see e. g., Skaare 2017; Tomasicchio et al. 2017; Tomasicchio et al. 2018; Ahn and Shin 2019; Chow et al. 2019; Hegseth and Bachynski 2019; Utsunomiya et al. 2019, etc.) have been carried out regarding the experimental modelling, dynamic modelling and design development of floating wind turbines. This paper investigates the computational methods for calculating the dynamic and motion responses of a floating wind turbine. In the worldwide offshore wind energy community, the dynamic and motion response analysis of a floating wind turbine is typically carried out by solving the turbine motion equation with a complicated convolution integral term representing the hydrodynamic memory effects (see e. g., the publications: Jonkman 2007, 2009, 2010; Jonkman and Buhl 2007; Jonkman and Matha 2011; Robertson and Jonkman 2011a, 2011b; Wang et al. 2013; Xia and Wang 2013; Xia 2014, etc.). Calculating the convolution integral term is difficult, time consuming and requiring a significant amount of computer memory. To the best knowledge of the author of this article, in the present literature, there is only one work (Duarte et al. 2013) that has tried to fit a parametric model (state space model) to approximate the convolution integral term when performing hydrodynamic responses predictions of floating wind turbines. The computational efficiency can indeed be improved by solving the turbine motion equation with a state space model representing the hydrodynamic memory effects. However, when performing the frequency–domain identification of the state space model and solving an optimisation problem regarding the transfer function, Duarte et al. (2013) used a weighted least squares method. The biggest disadvantage of weighted least squares is the fact that the theory behind this method is based on the assumption that the weights are known exactly. This is almost never the case in real applications, of course, so estimated weights must be used instead. The effect of using estimated weights is difficult to assess. Meanwhile, weighted least squares regression is also sensitive to the effects of outliers. If potential outliers are not investigated and dealt with appropriately, they will likely have a negative impact on the parameter estimation and other aspects of a weighted least squares analysis. If a weighted least squares regression actually increases the influence of an outlier, the results of the analysis may be far inferior to an un-weighted least squares analysis.