On a Unified Theory of Estimation in Linear Models—A Review of Recent Results

1975 ◽  
Vol 12 (S1) ◽  
pp. 89-104
Author(s):  
C. Radhakrishna Rao
Keyword(s):  
Least Squares ◽  
Numerical Evaluation ◽  
Linear Models ◽  
Least Square ◽  
Unified Theory

The paper deals with two approaches to the estimation of the parameters β and σ2 in the General Gauss-Markoff (GGM) model represented by the triplet (Y, Xß, σ2V), where E(Y)=Xβ and D(Y) =σ2V, when no assumptions are made about the ranks of X and V. One is called Inverse Partition Matrix (IPM) method, which depends on the numerical evaluation of the g-inverse of a partitioned matrix. The second is an analogue of least squares theory applicable even when V is singular, unlike Atiken's method which is applicable only for non-singular V, and is called Unified Least Square (ULS) method.

Fuzzy C-Regression Models

2013 ◽  
Vol 278-280 ◽  
pp. 1323-1326
Author(s):  
Yan Hua Yu ◽  
Li Xia Song ◽  
Kun Lun Zhang
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Linear Models ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Estimation Methods ◽  
Statistical Regression ◽  
Fuzzy Linear Regression ◽  
Fuzzy Degree

Fuzzy linear regression has been extensively studied since its inception symbolized by the work of Tanaka et al. in 1982. As one of the main estimation methods, fuzzy least squares approach is appealing because it corresponds, to some extent, to the well known statistical regression analysis. In this article, a restricted least squares method is proposed to fit fuzzy linear models with crisp inputs and symmetric fuzzy output. The paper puts forward a kind of fuzzy linear regression model based on structured element, This model has precise input data and fuzzy output data, Gives the regression coefficient and the fuzzy degree function determination method by using the least square method, studies the imitation degree question between the observed value and the forecast value.

Determination of Pyridostigmine Bromide in Presence of its Related Impurities by Four Modified Classical Least Square Based Models: A Comparative Study

2020 ◽  
Vol 17 (1) ◽  
pp. 87-94
Author(s):  
Ibrahim A. Naguib ◽  
Fatma F. Abdallah ◽  
Aml A. Emam ◽  
Eglal A. Abdelaleem
Keyword(s):  
Comparative Study ◽  
Least Squares ◽  
Orthogonal Projection ◽  
Predictive Ability ◽  
Least Square ◽  
Projection To Latent Structures ◽  
Preprocessing Method ◽  
Spectral Residual ◽  
Latent Structures

: Quantitative determination of pyridostigmine bromide in the presence of its two related substances; impurity A and impurity B was considered as a case study to construct the comparison. Introduction: Novel manipulations of the well-known classical least squares multivariate calibration model were explained in detail as a comparative analytical study in this research work. In addition to the application of plain classical least squares model, two preprocessing steps were tried, where prior to modeling with classical least squares, first derivatization and orthogonal projection to latent structures were applied to produce two novel manipulations of the classical least square-based model. Moreover, spectral residual augmented classical least squares model is included in the present comparative study. Methods: 3 factor 4 level design was implemented constructing a training set of 16 mixtures with different concentrations of the studied components. To investigate the predictive ability of the studied models; a test set consisting of 9 mixtures was constructed. Results: The key performance indicator of this comparative study was the root mean square error of prediction for the independent test set mixtures, where it was found 1.367 when classical least squares applied with no preprocessing method, 1.352 when first derivative data was implemented, 0.2100 when orthogonal projection to latent structures preprocessing method was applied and 0.2747 when spectral residual augmented classical least squares was performed. Conclusion: Coupling of classical least squares model with orthogonal projection to latent structures preprocessing method produced significant improvement of the predictive ability of it.

scholarly journals Association of Serum Levels of Zinc, Copper, and Iron with Risk of Metabolic Syndrome

Nutrients ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 548
Author(s):  
Chia-Wen Lu ◽  
Yi-Chen Lee ◽  
Chia-Sheng Kuo ◽  
Chien-Hsieh Chiang ◽  
Hao-Hsiang Chang ◽  
...  
Keyword(s):  
Insulin Resistance ◽  
Metabolic Syndrome ◽  
Linear Models ◽  
Serum Zinc ◽  
Serum Levels ◽  
Least Square ◽  
Model Assessment ◽  
Serum Concentrations ◽  
Metabolic Factors ◽  
Homeostatic Model Assessment

The association between serum concentrations of zinc, copper, or iron and the risk of metabolic syndrome are inconclusive. Therefore, we conduct a case-control study to explore the relationship between serum levels of zinc, copper, or iron and metabolic syndrome as well as each metabolic factor and insulin resistance. We enrolled 1165 adults, aged ≥ 40 (65.8 ± 10) years in a hospital-based population to compare the serum levels of zinc, copper, and iron between subjects with and without metabolic syndrome by using multivariate logistic regression analyses. The least square means were computed by general linear models to compare serum concentrations of zinc, copper, and iron in relation to the number of metabolic factors. The mean serum concentrations of zinc, copper, and iron were 941.91 ± 333.63 μg/L, 1043.45 ± 306.36 μg/L, and 1246.83 ± 538.13 μg/L, respectively. The odds ratios (ORs) of metabolic syndrome for the highest versus the lowest quartile were 5.83 (95% CI: 3.35–10.12; p for trend < 0.001) for zinc, 2.02 (95% CI: 1.25–3.25; p for trend: 0.013) for copper, and 2.11 (95% CI: 1.24–3.62; p for trend: 0.021) for iron after adjusting for age, sex, personal habits, body mass index, and homeostatic model assessment insulin resistance. Additionally, the serum zinc, copper, and iron concentrations increased as the number of metabolic factors rose (p for trend < 0.001). This was the first study to clearly demonstrate that higher serum levels of zinc, copper, and iron were associated with the risk of metabolic syndrome and the number of metabolic factors independent of BMI and insulin resistance.

The use of linear models and matrix least squares in clinical chemistry.

1979 ◽  
Vol 25 (6) ◽  
pp. 840-855 ◽  
Author(s):  
S N Deming ◽  
S L Morgan
Keyword(s):  
Least Squares ◽  
Precise Measurement ◽  
Linear Models ◽  
Clinical Chemistry ◽  
Experimental Designs ◽  
Unified Approach

Abstract We present a unified approach to the use of linear models and matrix least squares with the intention of providing a better understanding of the techniques themselves and of the statistics that arise from these techniques as they are used in clinical chemistry. Emphasis is placed on the importance of appropriate experimental designs and adequately precise measurement processes for efficiently obtaining the desired information.

Application of Fitting Estimation Method in GPS Height Fitting

2013 ◽  
Vol 694-697 ◽  
pp. 2545-2549 ◽  
Author(s):  
Qian Wen Cheng ◽  
Lu Ben Zhang ◽  
Hong Hua Chen
Keyword(s):  
Least Squares ◽  
Estimation Method ◽  
Least Square ◽  
Least Squares Collocation ◽  
Normal Height ◽  
Geodetic Height ◽  
The Given ◽  
Surveying And Mapping

The key point researched by many scholars in the field of surveying and mapping is how to use the given geodetic height H measured by GPS to obtain the normal height. Although many commonly-used fitting methods have solved many problems, they all value the pending parameters as the nonrandom variables. Figuring out the best valuations, according to the traditional least square principle, only considers its trend or randomness, which is theoretically incomprehensive and have limitations in practice. Therefore, a method is needed not only considers its trend but also takes randomness into account. This method is called the least squares collocation.

Research on the Wire Drawing Die Pass Parameter Calculation Based on the Least Square Algorithm

2012 ◽  
Vol 591-593 ◽  
pp. 850-853
Author(s):  
Huai Xing Wen ◽  
Yong Tao Yang
Keyword(s):  
Pore Structure ◽  
Least Squares ◽  
Least Squares Method ◽  
Piecewise Linear ◽  
Wire Drawing ◽  
Least Square ◽  
Structure Characteristics ◽  
Drawing Die ◽  
The Wire ◽  
Drawing Dies

Drawing Dies meter A / D acquisition module will be collected from the mold hole contour data to draw a curve in Matlab. According to the mold pore structure characteristics of the curve, the initial cut-off point of each part of contour is determined and iteratived optimization to find the best cut-off point, use the least squares method for fitting piecewise linear and fitting optimization to find the function of the various parts of the curve function, finally calculate the pass parameters of drawing mode. Parameters obtained compare with the standard mold, both of errors are relatively small that prove the correctness of the algorithm. Also a complete algorithm flow of pass parameters is designed, it can fast and accurately measure the wire drawing die hole parameters.

scholarly journals PREDICTION OF PASSENGER FLOW ON THE HIGHWAY BASED ON THE LEAST SQUARE SUPPOERT VECTOR MACHINE / MAŽIAUSIŲ KVADRATŲ ATRAMINIŲ VEKTORIŲ METODO TAIKYMAS KELEIVIŲ SRAUTUI GREITKELYJE PROGNOZUOTI / ПРИМЕНЕНИЕ МЕТОДА ОПОРНЫХ ВЕКТОРОВ С КВАДРАТИЧНОЙ ФУНКЦИЕЙ ПОТЕРЬ ДЛЯ ПРОГНОЗИРОВАНИЯ ПАССАЖИРСКИХ ПОТОКОВ НА АВТОМАГИСТРАЛЯХ

Transport ◽  
2011 ◽  
Vol 26 (2) ◽  
pp. 197-203 ◽  
Author(s):  
Yanrong Hu ◽  
Chong Wu ◽  
Hongjiu Liu
Keyword(s):  
Support Vector Machine ◽  
Least Squares ◽  
Least Square ◽  
Small Samples ◽  
Support Vector ◽  
Risk Minimization ◽  
Structural Risk Minimization ◽  
Passenger Flow ◽  
Regressive Model ◽  
Structural Risk

A support vector machine is a machine learning method based on the statistical learning theory and structural risk minimization. The support vector machine is a much better method than ever, because it may solve some actual problems in small samples, high dimension, nonlinear and local minima etc. The article utilizes the theory and method of support vector machine (SVM) regression and establishes the regressive model based on the least square support vector machine (LS-SVM). Through predicting passenger flow on Hangzhou highway in 2000–2008, the paper shows that the regressive model of LS-SVM has much higher accuracy and reliability of prediction, and therefore may effectively predict passenger flow on the highway. Santrauka Atraminių vektorių metodas (Support Vector Machine – SVM) yra skaičiuojamasis metodas, paremtas statistikos teorija, struktūriniu požiūriu mažinant riziką. SVM metodas, palyginti su kitais metodais, yra patikimesnis metodas, nes juo remiantis galima išspręsti realias problemas, esant įvairioms sąlygoms. Tyrimams naudojama SVM metodo regresijos teorija ir sukuriamas regresinis modelis, kuris grindžiamas mažiausių kvadratų atraminių vektorių metodu (Least Squares Support Vector Machine – LS-SVM). Straipsnio autoriai prognozuoja keleivių srautą Hangdžou (Kinija) greitkelyje 2000–2008 m. Gauti rezultatai rodo, kad regresinis LS-SVM modelis yra labai tikslus ir patikimas, todėl gali būti efektyviai taikomas keleivių srautams prognozuoti greitkeliuose. Резюме Метод опорных векторов (Support Vector Machine – SVM) – это набор аналогичных алгоритмов вида «обучение с учителем», использующихся для задач классификации и регрессионного анализа. Метод SVM принадлежит к семейству линейных классификаторов. Основная идея метода SVM заключается в переводе исходных векторов в пространство более высокой размерности и поиске разделяющей гиперплоскости с максимальным зазором в этом пространстве. Алгоритм работает в предположении, что чем больше разница или расстояние между параллельными гиперплоскостями, тем меньше будет средняя ошибка классификатора. В сравнении с другими методами метод SVM более надежен и позволяет решать проблемы с различными условиями. Для исследования был использован метод SVM и регрессионный анализ, затем создана регрессионная модель, основанная на методе опорных векторов с квадратичной функцией потерь (Least Squares Support Vector Machine – LS-SVM). Авторы прогнозировали пассажирский поток на автомагистрали Ханчжоу (Китай) в 2000–2008 гг. Полученные результаты показывают, что регрессионная модель LS-SVM является надежной и может быть применена для прогнозирования пассажирских потоков на других магистралях.

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