Application of Total Least Squares for Spatial Point Process Analysis

Felus, Yaron A.

doi:10.1061/(asce)0733-9453(2004)130:3(126)

Cited by 51 publications

(28 citation statements)

References 5 publications

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“…This method incorporates prior information on all or some of the parameters by using a priori variances (or full variance-covariance matrices) as stochastic constraints or weights, and is considered a generalization-based adjustment with stochastic constraints (SCLESS) (Snow 2002) or Generalized Least Squares (GLS) (Felus 2004). In practical terms, pseudoobservations together with their variance-covariance matrices are combined with the set of measurements (carrier phases and pseudoranges), as shown in the algorithm below.…”

Section: Approach and Methodology: Approachmentioning

confidence: 99%

On accuracy and reliability of instantaneous network RTK as a function of network geometry, station separation, and data processing strategy

2005

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The network-based approach to kinematic GPS positioning significantly increases the distance, over which carrier-phase ambiguity resolution can be performed. This can be achieved either by introducing geometric conditions based on the fixed reference locations, and/or through the use of reference network data to estimate the corrections to GPS observations that can be broadcast to the users. The Multi Purpose GPS Processing Software (MPGPS) developed at The Ohio State University uses the multiple reference station approach for wide area and regional differential kinematic GPS positioning. The primary processing algorithm uses the weighted free-net (WFN) approach with the distance-dependent weighting scheme to derive optimal estimates of the user coordinates and realistic accuracy measures. The WFN approach, combined with the single epoch (instantaneous) ambiguity resolution algorithm is presented here as one approach to realtime kinematic (RTK) GPS. Since for baselines exceeding $100 km, the instantaneous ambiguity resolution may not always be possible due to the increasing observation noise and insufficient number of observations to verify the integer selection, an alternative approach, based on a single-(or multiple-) baseline solution, supported by a double-difference (DD) ionospheric delay propagated from the previous epoch is also presented. In this approach, some data accumulation, supported by the network-derived atmospheric corrections, is required at the beginning of the rover data processing to obtain the integer ambiguities; after this initialization period, the processing switches to the instantaneous RTK positioning mode. This paper presents a discussion on the effects of the network geometry, station separation and the data reduction technique on the final quality and reliability of the rover positioning solution. A 24-h data set of August 31, 2003, collected by the Ohio Continuously Operating Reference Station (CORS) network was processed by both techniques under different network geometry and reference station separation. Various solutions, such as (1) single-baseline solution for varying base-rover separation, (2) multi-baseline solution with medium-range base separation (over 100 km), and (3) multi-baseline solution with long-range base separation (up to 377 km), were obtained and compared for accuracy and consistency. The horizontal positioning accuracy achieved in these tests, expressed as the difference between the estimated coordinates and the known rover coordinates, is at the sub-decimeter level for the first approach, and at the centimeter-level for the second method, for baselines over 100 km. GPS Solut (2005) 9: 212-225

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Section: Approach and Methodology: Approachmentioning

confidence: 99%

On accuracy and reliability of instantaneous network RTK as a function of network geometry, station separation, and data processing strategy

2005

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“…Schaffrin et al 2006) and, most often, coordinate transformation (see e.g. Felus 2004;Schaffrin et al 2006;Akyilmaz 2007;Schaffrin and Wieser 2008). In this section, we will simulate an example of industrial surveying.…”

Section: Examplesmentioning

confidence: 99%

“…The TLS method has recently been extensively studied and applied in geodesy, in particular, for industrial surveying and coordinate transformation (see e.g. Felus 2004;Schaffrin 2006;Akyilmaz 2007;Schaffrin and Wieser 2008;Felus and Burtch 2009;Schaffrin and Felus 2009;Neitzel 2010;Shen et al 2011). TLS solutions have also been extended to the case with equality constraints (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis

Liu

Shi

2012

J Geod

161

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The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight.

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“…A typical EIV model is similar to a GaussMarkov (GM) model but all the variables are subject to random errors. For further reading, see e.g., Van Huffel and Vandewalle (1991), Schaffrin and Wieser (2008), Felus (2004), Schaffrin et al (2012a, b), Mahboub et al (2012), Fang (2013Fang ( , 2014, Snow and Schaffrin (2012), Snow (2012) and Mahboub (2014), etc. Meanwhile, some other researchers investigated this problem traditionally; see e.g., Neitzel (2010) and Shen et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Adjustment of non-typical errors-in-variables models

2015

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In this contribution, non-typical errors-in-variables (EIV) model is introduced as a more general form of the so-called EIV model which is highly relevant for geodetic data processing. Total least-squares (TLS) algorithms within a typical EIV model cannot deal with the non-typical EIV models since the first order moment/mean of the random design matrix in a non-typical EIV model is not linear. In this paper, we propose a new algorithm besides a standard solution to deal with this model. To achieve this goal, first we review a classic algorithm in order to solve this problem using traditional non-linear leastsquares method within a non-linear mixed model. Then by comparison, weighted TLS algorithm within the typical EIV model can be replaced by the proposed approach after some slight modifications which results in an excellent approximate solution. This foundation is important because there is no need for linearization in the TLS algorithms. The proposed way is not sensitive to the approximate initial values of the unknown parameters and it is applicable to curve fitting, surface reconstruction or other non-typical EIV models. Here we employ it to the curve fitting. Also two examples convincingly demonstrate that the standard TLS solution of the non-typical EIV model is not admissible when it is incorrectly considered as a typical EIV model; i.e., the non-linear relationships of the elements of the random design matrix are neglected.

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Application of Total Least Squares for Spatial Point Process Analysis

Cited by 51 publications

References 5 publications

On accuracy and reliability of instantaneous network RTK as a function of network geometry, station separation, and data processing strategy

On accuracy and reliability of instantaneous network RTK as a function of network geometry, station separation, and data processing strategy

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis

Adjustment of non-typical errors-in-variables models

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