Simplex Method of Nonlinear Least-Squares - A Logical Complementary Method to Linear Least-Square Analysis of Data

1997 ◽  
Vol 74 (8) ◽  
pp. 1008 ◽  
Author(s):  
Shannon G. Lieb
Keyword(s):  
Least Squares ◽  
Simplex Method ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Complementary Method ◽  
Linear Least Square

scholarly journals Comparison Results on Preconditioned GAOR Methods for Weighted Linear Least Squares Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Guangbin Wang ◽  
Yanli Du ◽  
Fuping Tan
Keyword(s):  
Least Squares ◽  
Least Square ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Numerical Example ◽  
Comparison Results ◽  
Linear Least Squares Problems ◽  
Linear Least Square ◽  
Least Square Problems ◽  
Theoretical Results

We present preconditioned generalized accelerated overrelaxation methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we give a numerical example to confirm our theoretical results.

scholarly journals Prediction of Individual Tree Diameter and Height to Crown Base Using Nonlinear Simultaneous Regression and Airborne LiDAR Data

2020 ◽  
Vol 12 (14) ◽  
pp. 2238
Author(s):  
Zhaohui Yang ◽  
Qingwang Liu ◽  
Peng Luo ◽  
Qiaolin Ye ◽  
Guangshuang Duan ◽  
...  
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Equation System ◽  
Simultaneous Equation ◽  
Forest Growth ◽  
Airborne Lidar ◽  
Least Square ◽  
Projection Area ◽  
Individual Tree ◽  
Simultaneous Equation System

The forest growth and yield models, which are used as important decision-support tools in forest management, are commonly based on the individual tree characteristics, such as diameter at breast height (DBH), crown ratio, and height to crown base (HCB). Taking direct measurements for DBH and HCB through the ground-based methods is cumbersome and costly. The indirect method of getting such information is possible from remote sensing databases, which can be used to build DBH and HCB prediction models. The DBH and HCB of the same trees are significantly correlated, and so their inherent correlations need to be appropriately accounted for in the DBH and HCB models. However, all the existing DBH and HCB models, including models based on light detection and ranging (LiDAR) have ignored such correlations and thus failed to account for the compatibility of DBH and HCB estimates, in addition to disregarding measurement errors. To address these problems, we developed a compatible simultaneous equation system of DBH and HCB error-in-variable (EIV) models using LiDAR-derived data and ground-measurements for 510 Picea crassifolia Kom trees in northwest China. Four versatile algorithms, such as nonlinear seemingly unrelated regression (NSUR), two-stage least square (2SLS) regression, three-stage least square (3SLS) regression, and full information maximum likelihood (FIML) were evaluated for their estimating efficiencies and precisions for a simultaneous equation system of DBH and HCB EIV models. In addition, two other model structures, namely, nonlinear least squares with HCB estimation not based on the DBH (NLS and NBD) and nonlinear least squares with HCB estimation based on the DBH (NLS and BD) were also developed, and their fitting precisions with a simultaneous equation system compared. The leave-one-out cross-validation method was applied to evaluate all estimating algorithms and their resulting models. We found that only the simultaneous equation system could illustrate the effect of errors associated with the regressors on the response variables (DBH and HCB) and guaranteed the compatibility between the DBH and HCB models at an individual level. In addition, such an established system also effectively accounted for the inherent correlations between DBH with HCB. However, both the NLS and BD model and the NLS and NBD model did not show these properties. The precision of a simultaneous equation system developed using NSUR appeared the best among all the evaluated algorithms. Our equation system does not require the stand-level information as input, but it does require the information of tree height, crown width, and crown projection area, all of which can be readily derived from LiDAR imagery using the delineation algorithms and ground-based DBH measurements. Our results indicate that NSUR is a more reliable and quicker algorithm for developing DBH and HCB models using large scale LiDAR-based datasets. The novelty of this study is that the compatibility problem of the DBH model and the HCB EIV model was properly addressed, and the potential algorithms were compared to choose the most suitable one (NSUR). The presented method and algorithm will be useful for establishing similar compatible equation systems of tree DBH and HCB EIV models for other tree species.

Nonlinear Measurement Data Processing Based on Improved Damping Least Squares

2011 ◽  
Vol 105-107 ◽  
pp. 2034-2038
Author(s):  
Gui Ling Li
Keyword(s):  
Data Processing ◽  
Least Squares ◽  
Accuracy Assessment ◽  
Measurement Data ◽  
Nonlinear Least Squares ◽  
Taylor Series Expansion ◽  
Least Square ◽  
High Tech ◽  
Damping Factor ◽  
Observation Data

Datum are the key of “Digital Earth”.In measurement, dealing with nonlinear models of observation datum, we may take their approximate values at observation values by Taylor series expansion, say, taking first-order item as a linear function of classical adjustment. But requirements of observation data, processing and accuracy assessment are higher and higher with today's fast-growing of high-tech mapping and surveying. So study on nonlinear least squares adjustment has been paid more and more attention. Damping least squares, as a modified algorithm of Gauss-Newton’s algorithm, is necessary to add a damping factor to improve the nature of a coefficient matrix. But it is difficult to choose a suitable damping factor, and needs to solve a group of linear equations repeatedly. In this paper, an improved damping least square was utilized for the non-linear processing of measurement datum in order to reduce a lot of computational workload.

GPS-DERIVED SECULAR VELOCITY FIELD AROUND SANGIHE ISLAND AND ITS IMPLICATION TO THE MOLUCCA SEA SEISMICITY

GEOMATIKA ◽  
2020 ◽  
Vol 26 (2) ◽  
pp. 107
Author(s):  
Leni Sophia Heliani ◽  
Cecep Pratama ◽  
Parseno Parseno ◽  
Nurrohmat Widjajanti ◽  
Dwi Lestari
Keyword(s):  
Surface Displacement ◽  
Active Regions ◽  
Least Square ◽  
The Other ◽  
Gps Data ◽  
West Side ◽  
Active Deformation ◽  
Secular Motion ◽  
Small Change ◽  
Linear Least Square

<p><em>Sangihe-Moluccas region is the most active seismicity in Indonesia. Between 2015 to 2018 there is four M6 class earthquake occurred close to the Sangihe-Moluccas region. These seismic active regions representing active deformation which is recorded on installed GPS for both campaign and continuous station. However, the origin of those frequent earthquakes has not been well understood especially related to GPS-derived secular motion. Therefore, we intend to estimate the secular motion inside and around Sangihe island. On the other hand, we also evaluate the effect of seismicity on GPS sites. Since our GPS data were conducted on yearly basis, we used an empirical global model of surface displacement due to coseismic activity. We calculate the offset that may be contained in the GPS site during its period</em><em>. </em><em>We remove the offset and estimate again the secular motion using linear least square. Hence, in comparison with the secular motion without considering the seismicity, we observe small change but systematically shifting the motion. We concluded the seismicity in the Molucca sea from 2015 to 2018 systematically change the secular motion around Sangihe Island at the sub-mm level. Finally, we obtained the secular motion toward each other between the east and west side within 1 to 5.5 cm/year displacement. </em></p>

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