A Comparison of Simple Methods for Estimating the Mass Flow of Fluoride Discharged into Rivers

1987 ◽  
Vol 19 (5-6) ◽  
pp. 729-740 ◽  
Author(s):  
W. Van Craenenbroeck ◽  
J. Marivoet
Keyword(s):  
Mass Flow ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Fluoride Concentration ◽  
Weighted Least Squares Method ◽  
Use Of Data ◽  
Natural Concentration ◽  
River Meuse ◽  
The Mean ◽  
Pollutant Discharge

In order to follow the long term evolution of the pollutant discharge pattern, and to detect changes in the situation, information about the discharge mass flow is necessary. First, the paper deals with the behaviour of the fluoride concentration in the river Meuse. Data are presented for the stretch Remilly (France) to Keizersveer (Netherlands), and the evolution over the period 1970–1984 is described. Data from waste waters and tributaries are discussed, and a dominating fluoride source is identified. In the second part, a linear first order equation is proposed for fitting the C-Q data at Liège (Belgium). Five methods for estimating the mean fluoride mass flow of the discharge are compared and discussed. The weighted least squares method is proposed for further use. It makes use of data from a routine measuring station, and gives the possibility to estimate a natural concentration which is constant over the studied period 1970–1984. The thus obtained value of 0.19 mg/l is equal to the mean fluoride concentration upstream of the discharge point. The estimated discharge mass flow is about 6 t/d, and this value remained constant since 1979.

Calculation of radioimmunoassay standard curves.

1977 ◽  
Vol 23 (9) ◽  
pp. 1624-1627 ◽  
Author(s):  
A J Naus ◽  
P S Kuppens ◽  
A Borst
Keyword(s):  
Least Squares ◽  
Calibration Curve ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Mean Deviation ◽  
Weighted Least Squares Method ◽  
The Mean ◽  
Good Standard

Abstract A method for calculating radioimmunoassay standard curves, based on the theory of Ekins et al., is described. Because a four-parameter model is used, nonlinear standard curves are the result. The calibration curve is fitted to the measured standard points by means of a weighted least-squares method. The program based on this model can be easily processed on a desk-top calculator. For all 250 runs of six different assays, very good standard curves could be obtained. The mean deviation between the concentrations of the standard points and the corresponding calculated values was about 6%. In 26% of the cases it could be shown that the model we describe gave significantly better results than did two simpler ones.

scholarly journals Novel Optimization-Based Parameter Estimation Method for the Bass Diffusion Model

SAGE Open ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 215824402110269
Author(s):  
Lang Liang
Keyword(s):  
Parameter Estimation ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Estimation Method ◽  
Reference Value ◽  
Diffusion Problem ◽  
Bass Model ◽  
Weighted Least Squares Method ◽  
Estimated Parameters ◽  
Bass Diffusion Model

The Bass model is the most popular model for forecasting the diffusion process of a new product. However, the controlling parameters in it are unknown in practice and need to be determined in advance. Currently, the estimation of the controlling parameters has been approached by various techniques. In this case, a novel optimization-based parameter estimation (OPE) method for the Bass model is proposed in the theoretical framework of system dynamics ( SD). To do this, the SD model of the Bass differential equation is first established and then the corresponding optimization mathematical model is formulated by introducing the controlling parameters as design variable and the discrepancy of the adopter function to the reference value as objective function. Using the VENSIM software, the present SD optimization model is solved, and its effectiveness and accuracy are demonstrated by two examples: one involves the exact solution and another is related to the actual user diffusion problem from Chinese Mobile. The results show that the present OPE method can produce higher predicting accuracy of the controlling parameters than the nonlinear weighted least squares method and the genetic algorithms. Moreover, the reliability interval of the estimated parameters and the goodness of fitting of the optimal results are given as well to further demonstrate the accuracy of the present OPE method.

scholarly journals Spatial interpolation of daily rainfall at catchment scale: a case study of the Ourthe and Ambleve catchments, Belgium

2010 ◽  
Vol 7 (5) ◽  
pp. 7383-7416 ◽  
Author(s):  
S. Ly ◽  
C. Charles ◽  
A. Degré
Keyword(s):  
Spatial Interpolation ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Daily Rainfall ◽  
Daily Basis ◽  
Catchment Scale ◽  
Geostatistical Methods ◽  
Weighted Least Squares Method ◽  
Thiessen Polygon ◽  
Single Moment

Abstract. Spatial interpolation of precipitation data is of great importance for hydrological modelling. Geostatistical methods (krigings) are widely used in spatial interpolation from point measurement to continuous surfaces. However, the majority of existing geostatistical algorithms are available only for single-moment data. The first step in kriging computation is the semi-variogram modelling which usually uses only one variogram model for all-moment data. The objective of this paper was to develop different algorithms of spatial interpolation for daily rainfall on 1 km2 regular grids in the catchment area and to compare the results of geostatistical and deterministic approaches. In this study, we used daily rainfall data from 70 raingages in the hilly landscape of the Ourthe and Ambleve catchments in Belgium (2908 km2). This area lies between 35 and 693 m in elevation and consists of river networks, which are tributaries of the Meuse River. For geostatistical algorithms, Cressie's Approximate Weighted Least Squares method was used to fit seven semi-variogram models (logarithmic, power, exponential, Gaussian, rational quadratic, spherical and penta-spherical) to daily sample semi-variogram on a daily basis. Seven selected raingages were used to compare the interpolation performance of these algorithms applied to many degenerated-raingage cases. Spatial interpolation with the geostatistical and Inverse Distance Weighting (IDW) algorithms outperformed considerably interpolation with the Thiessen polygon that is commonly used in various hydrological models. Kriging with an External Drift (KED) and Ordinary Cokriging (OCK) presented the highest Root Mean Square Error (RMSE) between the geostatistical and IDW methods. Ordinary Kriging (ORK) and IDW were considered to be the best methods, as they provided smallest RMSE value for nearly all cases.

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