Performance of a solution of the direct geodetic problem by Taylor series of Cartesian coordinates

2021 ◽  
Vol 11 (1) ◽  
pp. 122-130
Author(s):  
C. Marx
Keyword(s):  
Test Data ◽  
Taylor Series ◽  
Solution Method ◽  
Mathematical Formulation ◽  
Current Method ◽  
Series Expansions ◽  
Triaxial Ellipsoid ◽  
Present Contribution ◽  
Data Set ◽  
Cartesian Coordinates

Abstract The direct geodetic problem is regarded on the biaxial and triaxial ellipsoid. A known solution method suitable for low eccentricities, which uses differential equations in Cartesian coordinates and Taylor series expansions of these coordinates, is advanced in view of its practical application. According to previous works, this approach has the advantages that no singularities occur in the determination of the coordinates, its mathematical formulation is simple and it is not computationally intensive. The formulas of the solution method are simplified in the present contribution. A test of this method using an extensive test data set on a biaxial earth ellipsoid shows its accuracy and practicability for distances of any length. Based on the convergence behavior of the series of the test data set, a truncation criterion for the series expansions is compiled taking into account accuracy requirements of the coordinates. Furthermore, a procedure is shown which controls the truncation of the series expansions by accuracy requirements of the direction to be determined in the direct problem. The conducted tests demonstrate the correct functioning of the methods for the series truncation. However, the considered solution method turns out to be significantly slower than another current method for biaxial ellipsoids, which makes it more relevant for triaxial ellipsoids.

scholarly journals Estimation of Weibull parameters using artificial neural network

2021 ◽  
Author(s):  
Md. Sujauddin Mallick
Keyword(s):  
Neural Network ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Test Data ◽  
Mathematical Formulation ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Time To Failure ◽  
Data Set

Weibull distribution is an important distribution in the field of reliability. In this distribution usually there are two parameters. The usual parameter estimation method is maximum likelihood estimation. Maximum likelihood estimation requires mathematical formulation and prior assumption. Non parametric method such as neural network does not require prior assumption and mathematical formulation. They need data to formulate the model. In this report feed forward neural network with back propagation is used to estimate the parameters of a two-parameter Weibull distribution based on four Scenarios. The Scenario consists of training and test data set. Training and test data set generated through simulated time to failure events using wblrnd function in MATLAB. The input to the network is time to failure, and the output is shape and scale parameters. The network is trained and tested using trainbr algorithm in MATLAB. The network performed better on Scenario 2 which has the larger number of training examples of shape and scale.

The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid

2020 ◽  
Vol 14 (2) ◽  
pp. 205-213
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Method ◽  
Analytical Method ◽  
Taylor Series Expansion ◽  
Oblate Spheroid ◽  
Accurate Solution ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Similar Accuracy ◽  
Geodesic Problem

AbstractIn this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.

scholarly journals Estimation of Weibull parameters using artificial neural network

2021 ◽  
Author(s):  
Md. Sujauddin Mallick
Keyword(s):  
Neural Network ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Test Data ◽  
Mathematical Formulation ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Time To Failure ◽  
Data Set

Weibull distribution is an important distribution in the field of reliability. In this distribution usually there are two parameters. The usual parameter estimation method is maximum likelihood estimation. Maximum likelihood estimation requires mathematical formulation and prior assumption. Non parametric method such as neural network does not require prior assumption and mathematical formulation. They need data to formulate the model. In this report feed forward neural network with back propagation is used to estimate the parameters of a two-parameter Weibull distribution based on four Scenarios. The Scenario consists of training and test data set. Training and test data set generated through simulated time to failure events using wblrnd function in MATLAB. The input to the network is time to failure, and the output is shape and scale parameters. The network is trained and tested using trainbr algorithm in MATLAB. The network performed better on Scenario 2 which has the larger number of training examples of shape and scale.

scholarly journals Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

2019 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Oblate Spheroid ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Ellipsoidal Coordinates ◽  
Extensive Test ◽  
Geodesic Problem

Abstract In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

How to Assess Prognostic Models for Survival Data: A Case Study in Oncology

2003 ◽  
Vol 42 (05) ◽  
pp. 564-571 ◽  
Author(s):  
M. Schumacher ◽  
E. Graf ◽  
T. Gerds
Keyword(s):  
Test Data ◽  
Survival Data ◽  
Prediction Error ◽  
Classification Scheme ◽  
Neural Nets ◽  
Brier Score ◽  
Data Set ◽  
Independent Test ◽  
Artificial Neural

Summary Objectives: A lack of generally applicable tools for the assessment of predictions for survival data has to be recognized. Prediction error curves based on the Brier score that have been suggested as a sensible approach are illustrated by means of a case study. Methods: The concept of predictions made in terms of conditional survival probabilities given the patient’s covariates is introduced. Such predictions are derived from various statistical models for survival data including artificial neural networks. The idea of how the prediction error of a prognostic classification scheme can be followed over time is illustrated with the data of two studies on the prognosis of node positive breast cancer patients, one of them serving as an independent test data set. Results and Conclusions: The Brier score as a function of time is shown to be a valuable tool for assessing the predictive performance of prognostic classification schemes for survival data incorporating censored observations. Comparison with the prediction based on the pooled Kaplan Meier estimator yields a benchmark value for any classification scheme incorporating patient’s covariate measurements. The problem of an overoptimistic assessment of prediction error caused by data-driven modelling as it is, for example, done with artificial neural nets can be circumvented by an assessment in an independent test data set.

Demographic Characterization of Anonymous Trace Travel Data

2015 ◽  
Vol 2526 (1) ◽  
pp. 19-28 ◽  
Author(s):  
Joshua Auld ◽  
Abolfazl (Kouros) Mohammadian ◽  
Marcelo Simas Oliveira ◽  
Jean Wolf ◽  
William Bachman
Keyword(s):  
Test Data ◽  
Large Scale ◽  
Travel Demand ◽  
Null Model ◽  
Demographic Characteristics ◽  
Travel Pattern ◽  
Data Set ◽  
Trace Data ◽  
Demographic Models ◽  
Pattern Information

Research was undertaken to determine whether demographic characteristics of individual travelers could be derived from travel pattern information when no information about the individual was available. This question is relevant in the context of anonymously collected travel information, such as cell phone traces, when used for travel demand modeling. Determining the demographics of a traveler from such data could partially obviate the need for large-scale collection of travel survey data, depending on the purpose for which the data were to be used. This research complements methodologies used to identify activity stops, purposes, and mode types from raw trace data and presumes that such methods exist and are available. The paper documents the development of procedures for taking raw activity streams estimated from GPS trace data and converting these into activity travel pattern characteristics that are then combined with basic land use information and used to estimate various models of demographic characteristics. The work status, education level, age, and license possession of individuals and the presence of children in their households were all estimated successfully with substantial increases in performance versus null model expectations for both training and test data sets. The gender, household size, and number of vehicles proved more difficult to estimate, and performance was lower on the test data set; these aspects indicate overfitting in these models. Overall, the demographic models appear to have potential for characterizing anonymous data streams, which could extend the usability and applicability of such data sources to the travel demand context.

scholarly journals Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Oblate Spheroid ◽  
Initial Value ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Research Article ◽  
Fast Solution ◽  
Geodesic Problem

AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

scholarly journals A Shortcut to LAD Estimator Asymptotics

1991 ◽  
Vol 7 (4) ◽  
pp. 450-463 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Asymptotic Expansion ◽  
Regression Model ◽  
Taylor Series ◽  
Asymptotic Theory ◽  
Random Variables ◽  
Generalized Functions ◽  
Infinite Variance ◽  
Series Expansions ◽  
Linear Functionals ◽  
Taylor Series Expansions

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

Improving SAR Altimeter processing over the coastal zone and inland waters - the ESA HYDROCOASTAL project

2021 ◽  
Author(s):  
David Cotton ◽  
Keyword(s):  
Coastal Zone ◽  
Test Data ◽  
River Discharge ◽  
Altimeter Data ◽  
Inland Waters ◽  
Data Sets ◽  
Data Set ◽  
Discharge Data ◽  
Processing Algorithms ◽  
The Impact

<p><strong>Introduction</strong></p><p>HYDROCOASTAL is a two year project funded by ESA, with the objective to maximise exploitation of SAR and SARin altimeter measurements in the coastal zone and inland waters, by evaluating and implementing new approaches to process SAR and SARin data from CryoSat-2, and SAR altimeter data from Sentinel-3A and Sentinel-3B. Optical data from Sentinel-2 MSI and Sentinel-3 OLCI instruments will also be used in generating River Discharge products.</p><p>New SAR and SARin processing algorithms for the coastal zone and inland waters will be developed and implemented and evaluated through an initial Test Data Set for selected regions. From the results of this evaluation a processing scheme will be implemented to generate global coastal zone and river discharge data sets.</p><p>A series of case studies will assess these products in terms of their scientific impacts.</p><p>All the produced data sets will be available on request to external researchers, and full descriptions of the processing algorithms will be provided</p><p> </p><p><strong>Objectives</strong></p><p>The scientific objectives of HYDROCOASTAL are to enhance our understanding  of interactions between the inland water and coastal zone, between the coastal zone and the open ocean, and the small scale processes that govern these interactions. Also the project aims to improve our capability to characterize the variation at different time scales of inland water storage, exchanges with the ocean and the impact on regional sea-level changes</p><p>The technical objectives are to develop and evaluate  new SAR  and SARin altimetry processing techniques in support of the scientific objectives, including stack processing, and filtering, and retracking. Also an improved Wet Troposphere Correction will be developed and evaluated.</p><p><strong>Project  Outline</strong></p><p>There are four tasks to the project</p><ul><li>Scientific Review and Requirements Consolidation: Review the current state of the art in SAR and SARin altimeter data processing as applied to the coastal zone and to inland waters</li> <li>Implementation and Validation: New processing algorithms with be implemented to generate a Test Data sets, which will be validated against models, in-situ data, and other satellite data sets. Selected algorithms will then be used to generate global coastal zone and river discharge data sets</li> <li>Impacts Assessment: The impact of these global products will be assess in a series of Case Studies</li> <li>Outreach and Roadmap: Outreach material will be prepared and distributed to engage with the wider scientific community and provide recommendations for development of future missions and future research.</li> </ul><p> </p><p><strong>Presentation</strong></p><p>The presentation will provide an overview to the project, present the different SAR altimeter processing algorithms that are being evaluated in the first phase of the project, and early results from the evaluation of the initial test data set.</p><p> </p>

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