scholarly journals Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2210
Author(s):  
Nikolai Krivulin
Keyword(s):  
Linear Regression ◽  
Algebraic Approach ◽  
Chebyshev Approximation ◽  
Direct Solution ◽  
Unknown Parameters ◽  
Absolute Deviation ◽  
Maximum Absolute Deviation ◽  
Regression Problems ◽  
Approximation Problems ◽  
Parameter Elimination

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples.

MONTE CARLO METHODS IN FUZZY NON-LINEAR REGRESSION

2008 ◽  
Vol 04 (02) ◽  
pp. 123-141 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Linear Regression ◽  
Fuzzy Numbers ◽  
Random Number ◽  
Random Number Generator ◽  
Search Space ◽  
Triangular Fuzzy Numbers ◽  
Non Linear ◽  
Regression Problems

We apply our new fuzzy Monte Carlo method to certain fuzzy non-linear regression problems to estimate the best solution. The best solution is a vector of triangular fuzzy numbers, for the fuzzy coefficients in the model, which minimizes an error measure. We use a quasi-random number generator to produce random sequences of these fuzzy vectors which uniformly fill the search space. We consider example problems to show that this Monte Carlo method obtains solutions comparable to those obtained by an evolutionary algorithm.

Density and Viscosity of Athabasca Bitumen Samples at Temperatures Up to 200°C and Pressures Up to 10 MPa

2015 ◽  
Vol 18 (03) ◽  
pp. 375-386 ◽  
Author(s):  
Hossein Nourozieh ◽  
Mohammad Kariznovi ◽  
Jalal Abedi
Keyword(s):  
Oil Recovery ◽  
Viscosity Data ◽  
Vapor Extraction ◽  
Absolute Deviation ◽  
Athabasca Bitumen ◽  
Steam Assisted Gravity Drainage ◽  
Recovery Processes ◽  
Wide Range ◽  
Maximum Absolute Deviation ◽  
Temperature And Pressure

Summary This paper presents the measurements of bitumen thermophysical properties (density and viscosity) over a wide range of temperatures (ambient to 200°C) and pressures (atmospheric to 14 MPa). The measurements have been conducted on three Athabasca bitumen samples taken from different locations. A new method was proposed to correlate the density data as a function of temperature and pressure, with a maximum absolute deviation of 1.7 kg/m3. The viscosity data were also correlated with two correlations available in literature considering the effect of pressure and temperature on viscosity of bitumen, with an average absolute relative deviation of 9.2%. The measured data and correlations are applicable for the prediction and optimization of oil recovery in the solvent- and thermal-based bitumen-recovery processes such as expanding- solvent steam assisted gravity drainage (ES-SAGD) and heated vapor extraction (VAPEX).

scholarly journals QML Estimators in Linear Regression Models with Functional Coefficient Autoregressive Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-30 ◽  
Author(s):  
Hongchang Hu
Keyword(s):  
Maximum Likelihood ◽  
Linear Regression ◽  
Regression Models ◽  
Asymptotic Properties ◽  
Asymptotic Distributions ◽  
Autoregressive Processes ◽  
Linear Regression Models ◽  
Unknown Parameters ◽  
Quasi Maximum Likelihood ◽  
Functional Coefficient

This paper studies a linear regression model, whose errors are functional coefficient autoregressive processes. Firstly, the quasi-maximum likelihood (QML) estimators of some unknown parameters are given. Secondly, under general conditions, the asymptotic properties (existence, consistency, and asymptotic distributions) of the QML estimators are investigated. These results extend those of Maller (2003), White (1959), Brockwell and Davis (1987), and so on. Lastly, the validity and feasibility of the method are illuminated by a simulation example and a real example.

Comparison of Redundancy of French and English through Use of a Regular Finite Markov Process

1972 ◽  
Vol 34 (3) ◽  
pp. 712-714
Author(s):  
J. Gérard Muise ◽  
Renaud S. Leblanc ◽  
Clarence J. Jeffrey
Keyword(s):  
Markov Process ◽  
Absolute Deviation ◽  
Sequential Dependencies ◽  
Language Studies ◽  
French And English ◽  
Finite Markov Process ◽  
Maximum Absolute Deviation ◽  
Equilibrium Matrix ◽  
Transitional Probabilities ◽  
Average Information

Digram transitional probabilities were used to compare the uncertainty of the English and French languages. An equilibrium matrix ( A), an average information ( H) value and the redundancy ( C) were computed for both languages using a regular Markov process. Six exponentiations were required to reach an equilibrium with a maximum absolute deviation of 0.005. The value of H for English and French was 4.11 bits and 3.96 bits respectively. Redundancy C for English was 12.6% and for French 15.8%. Differences between languages were observed warranting caution in the use of sequential dependencies of letters in inter-language studies.

A Strategy for Efficient Sample Selection in Simple Linear Regression Problems with Unequal per unit Sampling Costs

1986 ◽  
Vol 62 (1) ◽  
pp. 16-19 ◽  
Author(s):  
P. L. Marshall ◽  
J. P. Demaerschalk
Keyword(s):  
Linear Regression ◽  
Sample Selection ◽  
Search Procedure ◽  
Trial And Error ◽  
Simple Linear Regression ◽  
Efficient Design ◽  
Iterative Search ◽  
Regression Problems ◽  
Sample Allocation ◽  
Unequal Sampling

Simple linear regression is widely used in forestry, but often only a vaguely defined strategy for selecting sampling units is followed. Trial and error methods exist for aiding efficient sample allocation for simple linear regression purposes. These methods are computationally tedious and often impractical without the aid of a computer. This paper briefly describes a computerized iterative search procedure that can provide an efficient design for sample allocation in simple linear regression problems with equal or unequal sampling costs and balanced or unbalanced prediction intervals. Potential savings gained by employing an efficient design over other more easily derived but less efficient designs are illustrated by an example. Key words: Simple linear regression, optimal sampling design, iterative search procedure.

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