On: “A propagating algorithm for determining nth‐order polynomial, least‐squares fits,” by A. F. Gangi and J. N. Shapiro (GEOPHYSICS, October 1977, p. 1265–1276).

Geophysics ◽  
1979 ◽  
Vol 44 (9) ◽  
pp. 1588-1589
Author(s):  
Yoich Ohta ◽  
Masanori Saito
Keyword(s):  
Least Squares ◽  
Recursive Algorithm ◽  
Decomposition Algorithm ◽  
Triangular Decomposition ◽  
Mean Square ◽  
Polynomial Fit ◽  
Order Polynomial ◽  
Optimum Order ◽  
The Mean ◽  
Mean Square Errors

Gangi and Shapiro (1977) proposed a recursive algorithm for determining coefficients of least‐squares polynomials. The algorithm is simpler and more efficient than Trench’s (1965) algorithm or Phillips’ (1971) triangular decomposition algorithm and has an advantage that by monitoring the mean‐square errors at each iteration we can find an optimum order of polynomial fit. We have tried their algorithm and encountered a difficulty. It may be worth recording the source of the difficulty.

scholarly journals Direct Calculation of Thermodynamic Wet-Bulb Temperature as a Function of Pressure and Elevation

2013 ◽  
Vol 30 (8) ◽  
pp. 1757-1765 ◽  
Author(s):  
Sayed-Hossein Sadeghi ◽  
Troy R. Peters ◽  
Douglas R. Cobos ◽  
Henry W. Loescher ◽  
Colin S. Campbell
Keyword(s):  
Air Temperature ◽  
Mean Absolute Error ◽  
Direct Calculation ◽  
Ambient Air ◽  
Absolute Error ◽  
Mean Square ◽  
Air Temperatures ◽  
Order Polynomial ◽  
The Mean ◽  
Very High

Abstract A simple analytical method was developed for directly calculating the thermodynamic wet-bulb temperature from air temperature and the vapor pressure (or relative humidity) at elevations up to 4500 m above MSL was developed. This methodology was based on the fact that the wet-bulb temperature can be closely approximated by a second-order polynomial in both the positive and negative ranges in ambient air temperature. The method in this study builds upon this understanding and provides results for the negative range of air temperatures (−17° to 0°C), so that the maximum observed error in this area is equal to or smaller than −0.17°C. For temperatures ≥0°C, wet-bulb temperature accuracy was ±0.65°C, and larger errors corresponded to very high temperatures (Ta ≥ 39°C) and/or very high or low relative humidities (5% < RH < 10% or RH > 98%). The mean absolute error and the root-mean-square error were 0.15° and 0.2°C, respectively.

scholarly journals Adaptive recursive algorithm for optimal weighted suprathreshold stochastic resonance

2017 ◽  
Vol 4 (9) ◽  
pp. 160889 ◽  
Author(s):  
Liyan Xu ◽  
Fabing Duan ◽  
Xiao Gao ◽  
Derek Abbott ◽  
Mark D. McDonnell
Keyword(s):  
Stochastic Resonance ◽  
Recursive Algorithm ◽  
Signal Strength ◽  
Distinct Form ◽  
Mean Square ◽  
Least Mean Square ◽  
Complete Knowledge ◽  
The Mean ◽  
Suprathreshold Stochastic Resonance ◽  
Over Time

Suprathreshold stochastic resonance (SSR) is a distinct form of stochastic resonance, which occurs in multilevel parallel threshold arrays with no requirements on signal strength. In the generic SSR model, an optimal weighted decoding scheme shows its superiority in minimizing the mean square error (MSE). In this study, we extend the proposed optimal weighted decoding scheme to more general input characteristics by combining a Kalman filter and a least mean square (LMS) recursive algorithm, wherein the weighted coefficients can be adaptively adjusted so as to minimize the MSE without complete knowledge of input statistics. We demonstrate that the optimal weighted decoding scheme based on the Kalman–LMS recursive algorithm is able to robustly decode the outputs from the system in which SSR is observed, even for complex situations where the signal and noise vary over time.

scholarly journals Realization of the Primary Terrestrial Reference Frame

1991 ◽  
Vol 127 ◽  
pp. 108-115
Author(s):  
W. Kosek ◽  
B. Kołaczek
Keyword(s):  
Reference Frame ◽  
Geographic Distribution ◽  
Terrestrial Reference Frame ◽  
Mean Square ◽  
Minimum Value ◽  
Collocation Points ◽  
Station Coordinates ◽  
The Mean ◽  
Transformation Parameters ◽  
Mean Square Errors

AbstractThe PTRF is based on 43 sites with 64 SSC collocation points with the optimum geographic distribution, which were selected from all stations of the ITRF89 according to the criterion of the minimum value of the errors of 7 parameters of transformation. The ITRF89 was computed by the IERS Terrestrial Frame Section in Institut Geographique National - IGN and contains 192 VLBI and SLR stations (points) with 119 collocation ones. The PTRF has been compared with the ITRF89. The errors of the 7 parameters of transformation between the PTRF and 18 individual SSC as well as the mean square errors of station coordinates are of the same order as those for the ITRF89. The transformation parameters between the ITRF89 and the PTRF are negligible and their errors are of the order of 3 mm.

Mathematical-Statistical Methods for the Analysis of Aerial Triangulation Adjustment Errors Using a Computer Program

1975 ◽  
Vol 29 (2) ◽  
pp. 175-188
Author(s):  
M. Mosaad Allam
Keyword(s):  
Computer Program ◽  
Frequency Interval ◽  
Mean Square ◽  
Fisher Criterion ◽  
Statistical Parameters ◽  
Sample Distribution ◽  
Development Section ◽  
The Mean ◽  
Kolmogorov Criterion ◽  
Mean Square Errors

In practice, photogrammetrists use a single statistic reliability interval criterion, based on the mean square errors, to judge the accuracy of adjustment of photogrammetric blocks. Even in some cases, if the practical and theoretical distributions of frequency interval agree, such a test does not make it possible to establish the closeness of their convergence nor the degree of their difference. In other words, to get a complete picture of the character of the distribution of errors in the adjusted photogrammetric blocks, it is insufficient to investigate any single statistic. In the Research and Development Section of the Topographical Survey Directorate, a computer program (SABA) has been designed to analyze the errors of photogrammetric block adjustments, compute various statistical parameters and check the sample distribution using Kolmogorov criterion. Based on the decision taken, the correspondence between the empirical and theoretical distribution series are checked using the criterion χ2. The program divides the adjusted block to make a comparative evaluation of accuracies in the different sub-blocks. In this case, in addition to Kolmogorov and χ2 tests, the program checks the reliability intervals of the means and mean square errors of the samples and uses Fisher criterion ‘F’ to check the hypothesis of the equality of dispersion. SABA is coded in Fortran IV and Compass for the CDC CYBER 74 and requires a central memory of 28K decimal works. SABA is the acronym for Statistical Analysis of Block Adjustment.

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

2017 ◽  
Vol 19 ◽  
pp. 1-23
Author(s):  
Iryna Golichenko ◽  
Oleksand Masyutka ◽  
Mikhail Moklyachuk
Keyword(s):  
Random Field ◽  
Spectral Characteristics ◽  
Mean Square ◽  
Spectral Densities ◽  
Optimal Linear ◽  
Extrapolation Problem ◽  
Time Argument ◽  
The Mean ◽  
Mean Square Errors ◽  
Isotropic Random

The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t< 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

Application of on-line Adaptive Weighted Fusion in Mine Gas Measurement System

2012 ◽  
Vol 239-240 ◽  
pp. 1395-1398
Author(s):  
Yan Ju Wang ◽  
Li Kun Yang ◽  
Yu Tian Wang
Keyword(s):  
Measurement System ◽  
Mean Square ◽  
Least Mean Square ◽  
Fusion Algorithm ◽  
Important Indicator ◽  
Weighted Fusion ◽  
Environmental Monitoring System ◽  
The Mean ◽  
Mine Gas ◽  
Mean Square Errors

In mine environmental monitoring system, the concentration of mine gas is an important indicator. Aiming at the redundant information from multi-gas sensors in the measurement system, adaptive weighted fusion algorithm was presented. Using this algorithm, it was unnecessary to be aware of any pre-defined knowledge about these datas measured by the sensors. That the algorithm could adjust the fused sensor’s weight in time according to the variation in sensors’ variances makes the mean square error minimal. It was also proved theoretically that this fusion algorithm is linear and unbiased, in respect of the least mean square errors. Simulation results showed that this fusion algorithm is effective and the result of fused data is superior to the mean estimate algorithm in respect of accuracy and fault tolerance.

IV.—On Least Squares and Linear Combination of Observations

1936 ◽  
Vol 55 ◽  
pp. 42-48 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Least Squares ◽  
Linear Combination ◽  
Mean Square Error ◽  
Mean Square ◽  
Approximate Representation ◽  
Linear Combinations ◽  
The Mean ◽  
Image Position

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–(i) The data beingu1, u2, …, un, the representation is to be given by linear combinations(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than thekth.(iii) The sum of squared coefficientswhich measures the mean square error ofyi, is to be a minimum for each value ofi.

scholarly journals The VIF and MSE in Raise Regression

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 605 ◽  
Author(s):  
Román Salmerón Gómez ◽  
Ainara Rodríguez Sánchez ◽  
Catalina García García ◽  
José García Pérez
Keyword(s):  
Least Squares ◽  
Mean Square Error ◽  
Ordinary Least Squares ◽  
Least Squares Estimation ◽  
Mean Square ◽  
Variance Inflation Factor ◽  
Inflation Factor ◽  
The Mean

The raise regression has been proposed as an alternative to ordinary least squares estimation when a model presents collinearity. In order to analyze whether the problem has been mitigated, it is necessary to develop measures to detect collinearity after the application of the raise regression. This paper extends the concept of the variance inflation factor to be applied in a raise regression. The relevance of this extension is that it can be applied to determine the raising factor which allows an optimal application of this technique. The mean square error is also calculated since the raise regression provides a biased estimator. The results are illustrated by two empirical examples where the application of the raise estimator is compared to the application of the ridge and Lasso estimators that are commonly applied to estimate models with multicollinearity as an alternative to ordinary least squares.

Huit campagnes d’observations de Saturne a l’Astrolabe Danjon a San Fernando

1978 ◽  
Vol 48 ◽  
pp. 471-478
Author(s):  
M. Sanchez
Keyword(s):  
Theoretical Development ◽  
Mean Square ◽  
Mean Values ◽  
San Fernando ◽  
The Mean ◽  
Mean Square Errors ◽  
Right Ascension

Abstract:This paper contains an analysis of Saturn observations with Danjon astrolabe at San Fernando. These observations were obtained during eight winter campaigns (1970-1978). Table 1 gives the mean values for each of the quantities Δα and Δδ (astrolabe - American Ephemeris) and the mean square errors. Figure 1 to 8 shows the results (right ascension and declination) and the corresponding smoothing curves. The accuracy of these curves is also given in table 1. The analysis of the values Δα and Δδ seem to show that there are differences, between the theoretical development of ephemeris and the observations, of periodical character.

Realization of the Precise Point Positioning (PPP) technique and its accuracy

2017 ◽  
Vol 927 (9) ◽  
pp. 42-49
Author(s):  
A.V. Voytenko
Keyword(s):  
Precise Point Positioning ◽  
Russian Language ◽  
Convergence Time ◽  
Practical Implementation ◽  
Mean Square ◽  
Dual Frequency ◽  
Russian Scientist ◽  
The Mean ◽  
Point Positioning ◽  
Mean Square Errors

The article notes that the replacement of the English name «Precise Point Positioning» (PPP) in Russian-language sources is possible using the term «accurate differential positioning» (TDP) technique. The author proposes to use both terms. This article contains information about the practical implementation of the PPP in the on-line service. The author has analyzed the research on the accuracy of PPP foreign and domestic experts and scholars. The author analyzed the data about the convergence time for PPP solutions. These data belong to another Russian scientist. The results of evaluating the accuracy of the PPP of different scientists led to the next. The author of this article gave the mean square errors topocentric coordinates of the geodetic points. The coordinates of the points must be obtained by dual-frequency GPS-measurements for a period of 24 hours with the help of PPP. The author proposed a formula for the calculation of the mean square error of the spatial position of geodetic point, if its position is obtained in the processing of dual-frequency GPS-observations of less than 24 hours. The article written conclusions about the features, defects and PPP development.

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