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Journal articles on the topic 'Minimum residual sum of squares'

Author: Grafiati

Published: 8 September 2023

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1

Zhou, Zhen, L. N. Zhang, Y. Qin, D. Z. Ma, and B. Niu. "Statistical Inference of LZL-Type Mass Flowmeter Life Distribution Model." Key Engineering Materials 458 (December 2010): 173–78. http://dx.doi.org/10.4028/www.scientific.net/kem.458.173.

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Characteristics of field failure data are analyzed in this paper. The failure data and sales record of LZL-type mass flowmeter are used to infer life distribution of this conduct. The lines can be fitted in coordinates of six distribution using least square and the residual sum of squares are compared, the minimum correspond is the best distribution type. The results show that the life distribution style of this conduct is the two parameter exponential distribution, which is the base to analyze and predict failure development, research failure mechanism and draw up maintenance policy.

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2

DEVITA, HANY, I. KOMANG GDE SUKARSA, and I. PUTU EKA N. KENCANA. "KINERJA JACKKNIFE RIDGE REGRESSION DALAM MENGATASI MULTIKOLINEARITAS." E-Jurnal Matematika 3, no. 4 (November 28, 2014): 146. http://dx.doi.org/10.24843/mtk.2014.v03.i04.p077.

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Ordinary least square is a parameter estimations for minimizing residual sum of squares. If the multicollinearity was found in the data, unbias estimator with minimum variance could not be reached. Multicollinearity is a linear correlation between independent variabels in model. Jackknife Ridge Regression(JRR) as an extension of Generalized Ridge Regression (GRR) for solving multicollinearity. Generalized Ridge Regression is used to overcome the bias of estimators caused of presents multicollinearity by adding different bias parameter for each independent variabel in least square equation after transforming the data into an orthoghonal form. Beside that, JRR can reduce the bias of the ridge estimator. The result showed that JRR model out performs GRR model.

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Araveeporn, Autcha. "Comparing Parameter Estimation of Random Coefficient Autoregressive Model by Frequentist Method." Mathematics 8, no. 1 (January 2, 2020): 62. http://dx.doi.org/10.3390/math8010062.

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This paper compares the frequentist method that consisted of the least-squares method and the maximum likelihood method for estimating an unknown parameter on the Random Coefficient Autoregressive (RCA) model. The frequentist methods depend on the likelihood function that draws a conclusion from observed data by emphasizing the frequency or proportion of the data namely least squares and maximum likelihood methods. The method of least squares is often used to estimate the parameter of the frequentist method. The minimum of the sum of squared residuals is found by setting the gradient to zero. The maximum likelihood method carries out the observed data to estimate the parameter of a probability distribution by maximizing a likelihood function under the statistical model, while this estimator is obtained by a differential parameter of the likelihood function. The efficiency of two methods is considered by average mean square error for simulation data, and mean square error for actual data. For simulation data, the data are generated at only the first-order models of the RCA model. The results have shown that the least-squares method performs better than the maximum likelihood. The average mean square error of the least-squares method shows the minimum values in all cases that indicated their performance. Finally, these methods are applied to the actual data. The series of monthly averages of the Stock Exchange of Thailand (SET) index and daily volume of the exchange rate of Baht/Dollar are considered to estimate and forecast based on the RCA model. The result shows that the least-squares method outperforms the maximum likelihood method.

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Liu, Kexin, Weimin Bao, Yufeng Hu, Yiqun Sun, Dongjing Li, Kuang Li, and Lili Liang. "Improvement in Ridge Coefficient Optimization Criterion for Ridge Estimation-Based Dynamic System Response Curve Method in Flood Forecasting." Water 13, no. 24 (December 7, 2021): 3483. http://dx.doi.org/10.3390/w13243483.

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The ridge estimation-based dynamic system response curve (DSRC-R) method, which is an improvement of the dynamic system response curve (DSRC) method via the ridge estimation method, has illustrated its good robustness. However, the optimization criterion for the ridge coefficient in the DSRC-R method still needs further study. In view of this, a new optimization criterion called the balance and random degree criterion considering the sum of squares of flow errors (BSR) is proposed in this paper according to the properties of model-simulated residuals. In this criterion, two indexes, namely, the random degree of simulated residuals and the balance degree of simulated residuals, are introduced to describe the independence and the zero mean property of simulated residuals, respectively. Therefore, the BSR criterion is constructed by combining the sum of squares of flow errors with the two indexes. The BSR criterion, L-curve criterion and the minimum sum of squares of flow errors (MSSFE) criterion are tested on both synthetic cases and real-data cases. The results show that the BSR criterion is better than the L-curve criterion in minimizing the sum of squares of flow residuals and increasing the ridge coefficient optimization speed. Moreover, the BSR criterion has an advantage over the MSSFE criterion in making the estimated rainfall error more stable.

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Marjetič, Aleš. "Least-squares adjustment taking into account the errors in variables." Geodetski vestnik 65, no. 02 (2021): 205–18. http://dx.doi.org/10.15292/geodetski-vestnik.2021.02.205-218.

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In this article, we discuss the procedure for computing the values of the unknowns under the condition of the minimum sum of squares of the observation residuals (least-squares method), taking into account the errors in the unknowns. Many authors have already presented the problem, especially in the field of regression analysis and computations of transformation parameters. We present an overview of the theoretical foundations of the least-squares method and extensions of this method by considering the errors in unknowns in the model matrix. The method, which can be called ‘the total least-squares method’, is presented in the paper for the case of fitting the regression line to a set of points and for the case of calculating transformation parameters for the transition between the old and the new Slovenian national coordinate systems. With the results based on relevant statistics, we confirm the suitability of the considered method for solving such tasks.

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Pizarro Inostroza, María Gabriela, Francisco Javier Navas González, Vincenzo Landi, Jose Manuel León Jurado, Juan Vicente Delgado Bermejo, Javier Fernández Álvarez, and María del Amparo Martínez Martínez. "Goat Milk Nutritional Quality Software-Automatized Individual Curve Model Fitting, Shape Parameters Calculation and Bayesian Flexibility Criteria Comparison." Animals 10, no. 9 (September 18, 2020): 1693. http://dx.doi.org/10.3390/ani10091693.

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SPSS syntax was described to evaluate the individual performance of 49 linear and non-linear models to fit the milk component evolution curve of 159 Murciano-Granadina does selected for genotyping analyses. Peak and persistence for protein, fat, dry matter, lactose, and somatic cell counts were evaluated using 3107 controls (3.91 ± 2.01 average lactations/goat). Best-fit (adjusted R2) values (0.548, 0.374, 0.429, and 0.624 for protein, fat, dry matter, and lactose content, respectively) were reached by the five-parameter logarithmic model of Ali and Schaeffer (ALISCH), and for the three-parameter model of parabolic yield-density (PARYLDENS) for somatic cell counts (0.481). Cross-validation was performed using the Minimum Mean-Square Error (MMSE). Model comparison was performed using Residual Sum of Squares (RSS), Mean-Squared Prediction Error (MSPE), adjusted R2 and its standard deviation (SD), Akaike (AIC), corrected Akaike (AICc), and Bayesian information criteria (BIC). The adjusted R2 SD across individuals was around 0.2 for all models. Thirty-nine models successfully fitted the individual lactation curve for all components. Parametric and computational complexity promote variability-capturing properties, while model flexibility does not significantly (p > 0.05) improve the predictive and explanatory potential. Conclusively, ALISCH and PARYLDENS can be used to study goat milk composition genetic variability as trustable evaluation models to face future challenges of the goat dairy industry.

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UTAMI, NI KETUT TRI, and I. KOMANG GDE SUKARSA. "PENERAPAN METODE GENERALIZED RIDGE REGRESSION DALAM MENGATASI MASALAH MULTIKOLINEARITAS." E-Jurnal Matematika 2, no. 1 (January 30, 2013): 54. http://dx.doi.org/10.24843/mtk.2013.v02.i01.p029.

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Ordinary least square is parameter estimation method for linier regression analysis by minimizing residual sum of square. In the presence of multicollinearity, estimators which are unbiased and have a minimum variance can not be generated. Multicollinearity refers to a situation where regressor variables are highly correlated. Generalized Ridge Regression is an alternative method to deal with multicollinearity problem. In Generalized Ridge Regression, different biasing parameters for each regressor variables were added to the least square equation after transform the data to the space of orthogonal regressors. The analysis showed that Generalized Ridge Regression was satisfactory to overcome multicollinearity.

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Annan, Richard Fiifi, Yao Yevenyo Ziggah, John Ayer, and Christian Amans Odutola. "ACCURACY ASSESSMENT OF HEIGHTS OBTAINED FROM TOTAL STATION AND LEVEL INSTRUMENT USING TOTAL LEAST SQUARES AND ORDINARY LEAST SQUARES METHODS." Geoplanning: Journal of Geomatics and Planning 3, no. 2 (October 25, 2016): 87. http://dx.doi.org/10.14710/geoplanning.3.2.87-92.

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Spirit levelling has been the traditional means of determining Reduced Levels (RL’s) of points by most surveyors. The assertion that the level instrument is the best instrument for determining elevations of points needs to be reviewed; this is because technological advancement is making the total station a very reliable tool for determining reduced levels of points. In order to achieve the objective of this research, reduced levels of stations were determined by a spirit level and a total station instrument. Ordinary Least Squares (OLS) and Total Least Squares (TLS) techniques were then applied to adjust the level network. Unlike OLS which considers errors only in the observation matrix, and adjusts observations in order to make the sum of its residuals minimum, TLS considers errors in both the observation matrix and the data matrix, thereby minimising the errors in both matrices. This was evident from the results obtained in this study such that OLS approximated the adjusted reduced levels, which compromises accuracy, whereas the opposite happened in the TLS adjustment results. Therefore, TLS was preferred to OLS and Analysis of Variance (ANOVA) was performed on the preferred TLS solution and the RL’s from the total station in order to ascertain how accurate the total station can be relative to the spirit level.

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Zverev, S. V., V. I. Karpov, and M. A. Nikitina. "Optimization of food compositions according to the ideal protein profile." Food systems 4, no. 1 (April 28, 2021): 4–11. http://dx.doi.org/10.21323/2618-9771-2021-4-1-4-11.

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The paper emphasizes the importance of not only the quantitative but also qualitative composition of protein in nutrition. The authors propose protein classification into three main groups according to the concept of reference (ideal) protein. A mathematical model is examined to solve the task of rational mixture production upon the given profile of reference protein. Two variants of the criterion for formation of optimal composition are described. One of them presents the classical sum of squares of the residual for essential amino acid scores and 1. The second also presents the sum of squares of the residual for essential amino acid scores and 1 but with regard to only those amino acids, which scores are less than 1. The minima of these criteria at the set of variants for the content of ingredients are taken as targeted functions. The algorithm and the program of calculation were realized in the program environment Builder C++ 6.0. The macro flowchart of the algorithm is presented and detailed description of each block is given. The program interface before and after the start of the calculation module is shown. The main windows and interpretation of the presented data are described. An example of realization of the proposed mathematical apparatus when calculating a food model composition is given. Plant components (white kidney beans, flax, peanut, grit “Poltavskaya», dry red carrot) were used as an object of the research. Most plant proteins were incomplete. It is possible to regulate the chemical composition including correction of a protein profile by combination of plant raw materials. Analysis of alternative variants demonstrated that minimum essential amino acid score in the first composition was 0.79 (by the first criterion), in the second 1.0 (by the second criterion); the reference protein proportion in the mixture was 10.8 and 13.5, respectively, according to the first and second criterion. The comparative results by other quality indicators for protein in the mixture are also presented: the coefficient of amino acid score difference (CAASD), biological value (BV), coefficient of utility, essential amino acids index (IEAA).

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Chiang Hsieh, Lin-Han. "How Does the Effect Fade over Distance? An Inquiry into the Decay Pattern of Distance Effect on Property Values in the Case of Taipei, Taiwan." Land 10, no. 11 (November 12, 2021): 1238. http://dx.doi.org/10.3390/land10111238.

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It is generally accepted that the perception of homeowners towards certain potential risks or amenities fades as distance from the risk or amenity increases. This study aims to illustrate the distance–decay pattern with an appropriate mathematical function. Distance–decay functions and parameters that yield the minimum residual sum of squares (RSS) for a given regression model are considered to be the optimal approximation for the pattern of decay. The effect of flood risk and mass rapid transit (MRT) accessibility on residential housing prices in Taipei, Taiwan, are used as examples to test the optimization process. The results indicate that the type of distance function affects both the significance and the magnitude of the regression coefficients. In the case of Taipei, concave functions provide better fits for both the flood risk and MRT accessibility. RSS reduction is up to 10% compared to the blank. Surprisingly, the impact range for the flood risk is found to be larger than that for MRT accessibility, which suggested that the impact range of perception for uncertain risks is larger than expected.

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Liu, Xiang-Yu, Yu-Ling He, and Jian Yao. "A New Hybrid PSS Optimization Method Based on Improved Active Set Algorithm." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/4348257.

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This paper proposes a new hybrid optimization method for the phase-frequency characteristics of the double input power system stabilizer (PSS) based on the improved active set algorithm. This method takes the effect of the filtering section optimization on the parameter improvement into account, and the optimized model focuses on the minimum residual sum of squares between the actual and the target phase-frequency characteristics. The result shows that the improved parameters obtained from the proposed method provide much better phase-frequency characteristics than the widely used engineering parameters. The comparison between the proposed method and the typical commercial software indicates the universal superiority of the proposed method. And the studies on the impact of considering the filter section optimization on the phase-frequency improvement show that taking the filter section optimization into account will be beneficial for the phase-frequency improvement, though in application to the PSS2A model and the PSS2B model there are some differences. The achievements obtained in this paper provide a significant reference for the practical PSS parameter modification and improvement.

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Yang, Zhiwei, Jing Ren, Zichi Zhang, Yuqing Sun, Chunying Zhang, Mengyao Wang, and Liya Wang. "A New Three-Way Incremental Naive Bayes Classifier." Electronics 12, no. 7 (April 5, 2023): 1730. http://dx.doi.org/10.3390/electronics12071730.

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Aiming at the problems of the dynamic increase in data in real life and that the naive Bayes (NB) classifier only accepts or rejects the sample processing results, resulting in a high error rate when dealing with uncertain data, this paper combines three-way decision and incremental learning, and a new three-way incremental naive Bayes classifier (3WD-INB) is proposed. First, the NB classifier is established, and the distribution fitting is carried out according to the minimum residual sum of squares (RSS) for continuous data, so that 3WD-INB can process both discrete data and continuous data, then carry out an incremental learning operation, select the samples with higher data quality according to the confidence of the samples in the incremental training set for incremental learning, solve the problem of data dynamics and filter the poor samples. Then we construct the 3WD-INB classifier and determine the classification rules of the positive, negative and boundary domains of the 3WD-INB classifier, so that the three-way classification of samples can be realized and better decisions can be made when dealing with uncertain data. Finally, five discrete data and five continuous data are selected for comparative experimental analysis with traditional classification methods. The results show that 3WD-INB has high accuracy and recall rate on different types of datasets, and the classification performance is also relatively stable.

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Zhou, Zilong, Riyan Lan, Yichao Rui, Longjun Dong, and Xin Cai. "A New Algebraic Solution for Acoustic Emission Source Localization without Premeasuring Wave Velocity." Sensors 21, no. 2 (January 11, 2021): 459. http://dx.doi.org/10.3390/s21020459.

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The technique of acoustic emission (AE) source localization is critical for studying material failure mechanism and predicting the position of potential hazards. Most existing positioning methods heavily depend on the premeasured wave velocity and are not suitable for complex engineering practices where the wave velocity changes dynamically. To reduce the influence of measurement error of wave velocity on location accuracy, this paper proposes a new algebraic solution for AE source localization without premeasuring wave velocity. In this method, the nonlinear TDOA equations are established and linearized by introducing two intermediate variables. Then, by minimizing the sum of squared residuals of the linear TDOA equations with respect to the AE source coordinate and two intermediate variables separately, the optimal algebraic solution of the AE source coordinate in the least squares sense is obtained. A pencil-lead breaks experiment is performed to validate the positioning effectiveness of the proposed method. The results show that the new method improves the positioning accuracy by more than 40% compared with two pre-existing methods, and the minimum positioning accuracy of the proposed method can reach 1.12 mm. Moreover, simulation tests are conducted to further verify the location performance of the proposed method under different TDOA errors and the number of sensors.

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Zhou, Zilong, Riyan Lan, Yichao Rui, Longjun Dong, and Xin Cai. "A New Algebraic Solution for Acoustic Emission Source Localization without Premeasuring Wave Velocity." Sensors 21, no. 2 (January 11, 2021): 459. http://dx.doi.org/10.3390/s21020459.

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The technique of acoustic emission (AE) source localization is critical for studying material failure mechanism and predicting the position of potential hazards. Most existing positioning methods heavily depend on the premeasured wave velocity and are not suitable for complex engineering practices where the wave velocity changes dynamically. To reduce the influence of measurement error of wave velocity on location accuracy, this paper proposes a new algebraic solution for AE source localization without premeasuring wave velocity. In this method, the nonlinear TDOA equations are established and linearized by introducing two intermediate variables. Then, by minimizing the sum of squared residuals of the linear TDOA equations with respect to the AE source coordinate and two intermediate variables separately, the optimal algebraic solution of the AE source coordinate in the least squares sense is obtained. A pencil-lead breaks experiment is performed to validate the positioning effectiveness of the proposed method. The results show that the new method improves the positioning accuracy by more than 40% compared with two pre-existing methods, and the minimum positioning accuracy of the proposed method can reach 1.12 mm. Moreover, simulation tests are conducted to further verify the location performance of the proposed method under different TDOA errors and the number of sensors.

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Гиззатова, Э. Р., А. Р. Шагиахметов, Г. К. Хисаметдинова, and С. Л. Подвальный. "CALCULATION OF THE RESIDUAL FUNCTION IN THE BASIS SPACE TO SEARCH FOR AREAS OF UNCERTAINTY OF THE RATE CONSTANTS OF THE POLYMERIZATION PROCESS." ВЕСТНИК ВОРОНЕЖСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА, no. 4(-) (August 30, 2022): 37–43. http://dx.doi.org/10.36622/vstu.2022.18.4.005.

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Предлагаeтся к рассмотрению математическая модель процесса полимеризации на катализаторах Циглера-Натта, в предположении существования только трех элементарных стадий: роста полимерной цепи, передачи цепи на мономер и передачи цепи на алюминийорганическое соединение при условии отсутствия стадии инициирования активных центров. Приведены постановки прямой и обратной кинетической задач, последняя из которых заключается в поиске областей неопределенных для констант скоростей стадий роста и передач цепи. Показано, что для рассматриваемого процесса значения констант могут быть определены в базисном пространстве, построенном на векторах констант. В отличие от исходного трехмерного пространства базисное пространство позволяет рассчитывать и визуализировать сеточные поверхности по функции невязки для среднечисленной молекулярной массы. При условии расчета невязки как максимума отклонений или суммы квадратов отклонений получают разные поверхности. Показано, как наложение поверхностей друг на друга локализует области минимума, которые впоследствии можно характеризовать как решение обратных кинетических задач. При этом такой подход позволяет оценить форму и вид «овражных» минимумов и, в целом, определять оптимальные наборы констант, идентифицирующие точки минимума The paper proposes a mathematical model of the polymerization process on Ziegler-Natt catalysts, assuming the existence of only three elementary stages: the growth of the polymer chain, the transfer of the chain to the monomer and the transfer of the chain to the organoaluminium compound, provided there is no stage of initiation of active centers. We give the statements of direct and inverse kinetic problems, the latter of which consists in finding the regions of the chain growth stages and gears that are indeterminate for the rate constants. We show that for the process under consideration, the values of the constants can be determined in the basis space constructed on the vectors of the constants. Unlike the original three-dimensional space, the basis space allows calculating and visualizing grid surfaces by the residual function for the average calculated molecular weight. If the discrepancy is calculated as the maximum of deviations or the sum of the squares of deviations, different surfaces are obtained. We show the way the superposition of surfaces on each other localizes the minimum regions, which can subsequently be characterized as the solution of inverse kinetic problems. At the same time, this approach allows us to evaluate the shape and appearance of "gully" minima and, in general, to determine the optimal set of constants identifying the minimum points

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Dhakal, Madhav, Gurbir Singh, Rachel L. Cook, and Taylor Sievers. "Modeling Hairy Vetch and Cereal Rye Cover Crop Decomposition and Nitrogen Release." Agronomy 10, no. 5 (May 14, 2020): 701. http://dx.doi.org/10.3390/agronomy10050701.

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Empirical models could help us to understand the process of plant residue decomposition and nutrient release into the soil. The objective of this study was to determine an appropriate model to describe the decomposition of hairy vetch (Vicia villosa Roth) and cereal rye (Secale cereale L.) cover crop (CC) residue and nitrogen (N) release. Data pertaining to above and belowground CC residue mass loss and N release for up to 2633 cumulative decomposition degree days (112 d) after litterbag installation were obtained from two cropping system experiments, a 1-yr study conducted in 2015 and a 2-yr study during 2017 to 2018 in the humid subtropical environment of southern IL, USA. Six exponential and two hyperbolic models were fit to percent mass and N remaining data to find the one with minimum Akaike information criterion (AIC) and residual sum of squares. Modified three-parameter single exponential and two- or three-parameter hyperbolic models best met the assumed criteria of selection for above and belowground CC residue, respectively. Fitting a double exponential model to combined data for percent mass and N remaining identified two mass and N pools, a fast and a slow pool with different rate constants. A five-parameter double exponential with an asymptote met the preset criteria and passed all tests for normally distributed population, constant variance, and independence of residuals at α = 0.05 when fit to combined data of hairy vetch shoot mass and N remaining. However, a two-parameter hyperbolic and three-parameter asymptotic hyperbolic model provided the best fit to a combined data of cereal rye shoot mass and N remaining, respectively. Both hyperbolic decay models showed a good fit for belowground mass decomposition and N release for both CCs. Cereal rye had a poorer fit than hairy vetch for mass and N remaining of both above and belowground mass. The best-selected decay models can be used to estimate the decomposition and N release rates of hairy vetch and cereal rye above and belowground residue in a similar environment.

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Khalaf, Akram Abbas. "Estimating Soil Hydraulic Parameters of Different Textured Soils in Semiarid Duhok Conditions - Iraqi-Kurdistan Region by using RETC Program." Kufa Journal for Agricultural Sciences 13, no. 2 (December 1, 2021): 6–17. http://dx.doi.org/10.36077/kjas/2021/v13i2.3651.

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RETC is a computer program that can be used to estimate soil hydraulic functions by easily determining soil properties. Using van Genuchten model in the RETC program gave high accuracy for estimating the hydraulic functions e.g., θ (h), k (θ) and D (h), also the mentioned program is considered a good tool for potential optimization of water management especially though calculating irrigation water after consuming 60% of the available water in root zone restoring technique. In the current study surface, soil samples (0 – 30 cm) from eight locations with a wide range in texture were collected RETC code is used to refine the water retention curve through minimizing the residual error according to van Genuchten – Mualem approach (1980). the obtained results indicated that the sum square SSQ (residual error) values becomes smaller after refining for eight studied soils which were (Avkendal, Berderash, Chemrash, Kanisark, Keperto, Khanik, Qerwola and Terbsbia), which their SSQ values before RETC refining were ( 0.0470264, 0.0680298, 0.0240080, 0.0190478, 0.0396200, 0.0166960, 0.094400 and 0.0199100) whereas these values decrease as fallowing (0.000333400, 0.000942618, 0.0000997008, 0.000167315, 0.002448800, 0.000243300, 0.00080000, and 0.000099917 ) with ha high coefficient of determination (R2) greater than (0.964) and the maximum refining was (0.000099917) at Chemrash silty clay loam soil with 0.9986 (R2), whereas the minimum value was(0.002448800) at Keperto loamy soil with 0.9720(R2) and this means higher precision in fitting using the retention model. RETC code was used to analyze soil water retention curve and hydraulic conductivity functions which are the key parameters in any quantitative description of water flow into the unsaturated zone of soils, van Genuchten et. al. (1991)

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Wang, Jianmei, Ke Ning, Junliang Xu, and Zhixiong Li. "Reliability-based robust design of wind turbine’s shrink disk." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 15 (July 31, 2017): 2685–96. http://dx.doi.org/10.1177/0954406217723364.

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Wind turbine’s shrink disk (WTSD), as a component of transmission system, plays a crucial role in the wind turbine work. In this study, the matrix expression of combined pressure and interference magnitude has been derived by analyzing the geometry relationship of interference fit of multi-layer cylinder. The mechanic model is built by simplifying WTSD as interference fit of multi-layer cylinder. The reliability model is established on the basis of pressure-strength model, and the formula regarding reliability sensitivity is derived. In terms of the minimum principle on the square sum of reliability sensitivity and the minimum reliability principle, the reliability-based robust design (RBRD) model is built under the multi-objective conditions, and the dynamic reliability model is established on the basis of reliability model, load model, and residual strength model. Dynamic reliability sensitivity formula is deduced, and the dynamic reliability-based robust design (DRBRD) model is established under the multi-objective conditions. The effects of the coefficient of friction, assembly clearance, and yield strength on the reliability of WTSD are analyzed by RBRD and DRBRD methods. Taking a type of WTSD as an example, the calculating results validate that the reliability changes along with interference magnitude. The comparison of results shows that the above methods and Monte Carlo method have the same tendency and approximately agree well. Moreover, the traditional design method and RBRD method are compared. It turned out that RBRD can improve the reliability of WTSD by 47.6%, the reliability-based sensitivity is less than that from the traditional design method, and the robustness is stronger. The experiment verified that the design methods of this study can better meet the operation requirements.

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Downey, Peter J. "(Probably) the minimum sum of squares." Communications in Statistics. Stochastic Models 3, no. 3 (January 1987): 457–65. http://dx.doi.org/10.1080/15326348708807066.

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Bochev, Pavel, Leszek Demkowicz, Jay Gopalakrishnan, and Max Gunzburger. "Minimum Residual and Least Squares Finite Element Methods." Computers & Mathematics with Applications 68, no. 11 (December 2014): 1479. http://dx.doi.org/10.1016/j.camwa.2014.11.005.

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Pyatkin, Artem, Daniel Aloise, and Nenad Mladenović. "NP-Hardness of balanced minimum sum-of-squares clustering." Pattern Recognition Letters 97 (October 2017): 44–45. http://dx.doi.org/10.1016/j.patrec.2017.05.033.

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Scott, J. A. "102.52 A sum of squares for the Lester minimum." Mathematical Gazette 102, no. 555 (October 17, 2018): 529–31. http://dx.doi.org/10.1017/mag.2018.133.

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Pal, Jayanta Kumar. "Spiking problem in monotone regression: Penalized residual sum of squares." Statistics & Probability Letters 78, no. 12 (September 2008): 1548–56. http://dx.doi.org/10.1016/j.spl.2008.01.015.

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24

Wu, Bang Ye, and Li-Hsuan Chen. "Parameterized Algorithms for the 2-Clustering Problem with Minimum Sum and Minimum Sum of Squares Objective Functions." Algorithmica 72, no. 3 (February 7, 2014): 818–35. http://dx.doi.org/10.1007/s00453-014-9874-8.

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Gerritsma, Marc, Carsten Carstensen, Leszek Demkowicz, and Jay Gopalakrishnan. "Minimum Residual and Least Squares Finite Element Methods II." Computers & Mathematics with Applications 74, no. 8 (October 2017): 1922. http://dx.doi.org/10.1016/j.camwa.2017.07.021.

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26

Cuong, Tran Hung, Jen-Chih Yao, and Nguyen Dong Yen. "Qualitative properties of the minimum sum-of-squares clustering problem." Optimization 69, no. 9 (June 17, 2020): 2131–54. http://dx.doi.org/10.1080/02331934.2020.1778685.

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27

Hansen, Pierre. "Minimum Sum of Squares Clustering in a Low Dimensional Space." Journal of Classification 15, no. 1 (January 1, 1998): 37–55. http://dx.doi.org/10.1007/s003579900019.

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du Merle, O., P. Hansen, B. Jaumard, and N. Mladenovic. "An Interior Point Algorithm for Minimum Sum-of-Squares Clustering." SIAM Journal on Scientific Computing 21, no. 4 (January 1999): 1485–505. http://dx.doi.org/10.1137/s1064827597328327.

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29

Et.al, Abdul Hadi Bhatti. "Least Square Methods Based on Control Points of Said Ball Curves for Solving Ordinary Differential Equations." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 3 (April 11, 2021): 2597–607. http://dx.doi.org/10.17762/turcomat.v12i3.1261.

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This paper presents the use of Said Ball curve’s control points to approximate the solutions of linear ordinary differential equations (ODEs). Least squares methods (LSM) is proposed to find the control points of Said Ball curves by minimizing the error of residual function.The residual error is measured by taking the sum of squares of the Said Ball curve’s control points of the residual function. Then the approximate solution of ODEs is obtained by minimizing residual error.Two numerical examples are given in term of error and compared with the exact solution to demonstrate the efficiency of the proposed method.

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30

Caimmi, R., C. Marmo, and T. Valentinuzzi. "A numerical fit of analytical to simulated density profiles in dark matter haloes." Serbian Astronomical Journal, no. 170 (2005): 13–32. http://dx.doi.org/10.2298/saj0570013c.

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Analytical and geometrical properties of generalized power-law (GPL) density profiles are investigated in detail. In particular, a one-to-one correspondence is found between mathematical parameters (a scaling radius, r0, a scaling density, ?0, and three exponents, ?, ?, ?), and geometrical parameters (the coordinates of the intersection of the asymptotes, xC, yC, and three vertical intercepts, b, b?, b?, related to the curve and the asymptotes, respectively): (r0,?0,?,?,?) ? (xC,yC,b,b?,b?). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and nonlinear least-absolute values and least-squares fits involving the above mentioned five parameters (RFSM5 method) are prescribed. More specifically, the sum of absolute values or squares of absolute logarithmic residuals, Ri=log?SDH(ri) ? log?GPL(ri), is evaluated on 10 points making a 5dimension hypergrid, through a few iterations. The size is progressively reduced around a fiducial minimum, and superpositions on nodes of earlier hypergrids are avoided. An application is made to a sample of 17 SDHs on the scale of cluster of galaxies, within a flat ?CDM cosmological model (Rasia et al. 2004). In dealing with the mean SDH density profile, a virial radius, Rvir, averaged over the whole sample, is assigned, which allows the calculation of the remaining parameters. Using a RFSM5 method provides a better fit with respect to other methods. The geometrical parameters, averaged over the whole sample of best fitting GPL density profiles, yield (?, ?, ?) ? (0.6,3.1,1.0), to be compared with (?, ?, ?) = (1,3,1), i.e. the NFW density profile (Navarro et al. 1995, 1996, 1997), (?, ?, ?) = (1.5,3, 1.5) (Moore et al. 1998, 1999), (?, ?, ?) = (1,2.5,1) (Rasia et al. 2004); and, in addition, ? ? 1.5 (Hiotelis 2003), deduced from the application of a RFSM5 method, but using a different definition of scaled radius, or concentration; and ? ? 1.21.3 deduced from more recent high-resolution simulations (Diemand et al. 2004, Reed et al. 2005). No evident correlation is found between SDH dynamical state (relaxed or merging) and asymptotic inner slope of the fitting logarithmic density profile or (for SDH comparable virial masses) scaled radius. Mean values and standard deviations of some parameters are calculated, and in particular the decimal logarithm of the scaled radius, ?vir, reads < log?vir >= 0.74 and ?slog?vir = 0.150.17, consistent with previous results related to NFW density profiles. It provides additional support to the idea, that NFW density profiles may be considered as a convenient way to parametrize SDH density profiles, without implying that it necessarily produces the best possible fit (Bullock et al. 2001). A certain degree of degeneracy is found in fitting GPL to SDH density profiles. If it is intrinsic to the RFSM5 method or it could be reduced by the next generation of high-resolution simulations, still remains an open question. .

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31

Nogueira, Maria Cristina Stolf. "Specific residue: application of orthogonal contrasts when heteroscedasticity is present." Scientia Agricola 67, no. 1 (February 2010): 117–25. http://dx.doi.org/10.1590/s0103-90162010000100016.

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When experimental data are submitted to analysis of variance, the assumption of data homoscedasticity (variance homogeneity among treatments), associated to the adopted mathematical model must be satisfied. This verification is necessary to ensure the correct test for the analysis. In some cases, when data homoscedascity is not observed, errors may invalidate the analysis. An alternative to overcome this difficulty is the application of the specific residue analysis, which consists of the decomposition of the residual sum of squares in its components, in order to adequately test the correspondent orthogonal contrasts of interest between treatment means. Although the decomposition of the residual sum of squares is a seldom used procedure, it is useful for a better understanding of the residual mean square nature and to validate the tests to be applied. The objective of this review is to illustrate the specific residue application as a valid and adequate alternative to analyze data from experiments following completely randomized and randomized complete block designs in the presence of heteroscedasticity.

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Martín-Santamaría, R., J. Sánchez-Oro, S. Pérez-Peló, and A. Duarte. "Strategic oscillation for the balanced minimum sum-of-squares clustering problem." Information Sciences 585 (March 2022): 529–42. http://dx.doi.org/10.1016/j.ins.2021.11.048.

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33

Pacheco, Joaquı́n, and Olga Valencia. "Design of hybrids for the minimum sum-of-squares clustering problem." Computational Statistics & Data Analysis 43, no. 2 (June 2003): 235–48. http://dx.doi.org/10.1016/s0167-9473(02)00224-4.

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Bagirov, Adil M. "Modified global -means algorithm for minimum sum-of-squares clustering problems." Pattern Recognition 41, no. 10 (October 2008): 3192–99. http://dx.doi.org/10.1016/j.patcog.2008.04.004.

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35

Aloise, Daniel, Pierre Hansen, and Leo Liberti. "An improved column generation algorithm for minimum sum-of-squares clustering." Mathematical Programming 131, no. 1-2 (April 20, 2010): 195–220. http://dx.doi.org/10.1007/s10107-010-0349-7.

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36

Carrizosa, Emilio, Nenad Mladenović, and Raca Todosijević. "Variable neighborhood search for minimum sum-of-squares clustering on networks." European Journal of Operational Research 230, no. 2 (October 2013): 356–63. http://dx.doi.org/10.1016/j.ejor.2013.04.027.

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37

Demidenko, Eugene. "Criteria for global minimum of sum of squares in nonlinear regression." Computational Statistics & Data Analysis 51, no. 3 (December 2006): 1739–53. http://dx.doi.org/10.1016/j.csda.2006.06.015.

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38

Batra, Luckshay, and Harish Chander Taneja. "Comparison between Information Theoretic Measures to Assess Financial Markets." FinTech 1, no. 2 (May 19, 2022): 137–54. http://dx.doi.org/10.3390/fintech1020011.

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Information theoretic measures were applied to the study of the randomness associations of different financial time series. We studied the level of similarities between information theoretic measures and the various tools of regression analysis, i.e., between Shannon entropy and the total sum of squares of the dependent variable, relative mutual information and coefficients of correlation, conditional entropy and residual sum of squares, etc. We observed that mutual information and its dynamical extensions provide an alternative approach with some advantages to study the association between several international stock indices. Furthermore, mutual information and conditional entropy are relatively efficient compared to the measures of statistical dependence.

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39

Aloise, Daniel, and Pierre Hansen. "A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering." Pesquisa Operacional 29, no. 3 (December 2009): 503–16. http://dx.doi.org/10.1590/s0101-74382009000300002.

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Minimum sum-of-squares clustering (MSSC) consists in partitioning a given set of n points into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. Recently, Peng & Xia (2005) established the equivalence between 0-1 semidefinite programming (SDP) and MSSC. In this paper, we propose a branch-and-cut algorithm for the underlying 0-1 SDP model. The algorithm obtains exact solutions for fairly large data sets with computing times comparable with those of the best exact method found in the literature.

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40

Cho, Seong-Yun. "Simultaneous LTE Signal Propagation Modelling and Base Station Positioning Based on Multiple Virtual Locations." Sensors 22, no. 15 (August 8, 2022): 5917. http://dx.doi.org/10.3390/s22155917.

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In the Long Term Evolution (LTE) system, the Signal Propagation Model (SPM) and the location information of the base stations are required for positioning a smartphone. To this end, this paper proposes a technique for estimating the SPM and the location of the base station at the same time using location-based Reference Signal Received Power (RSRP) information acquired in a limited area. In the proposed technique, multiple Virtual Locations (VLs) for a base station are set within the service area. Signal propagation modelling is performed based on the assumptions that a base station is in each VL and the RSRP measurements are obtained from the corresponding base station. The residuals between the outputs of the estimated SPM and the RSRP measurements are then calculated. The VL with the minimum sum of the squared residuals is determined as the location of the base station. At the same time, the SPM estimated based on the corresponding VL is selected as the SPM of the base station. As a result of the experiment in Seoul, it was confirmed that the positions of seven base stations were estimated with an average accuracy of 40.2 m.

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41

Toral, Fábio Luiz Buranelo, Roberto Augusto de Almeida Torres Júnior, Paulo Sávio Lopes, Luiz Otávio Campos da Silva, and João Cruz Reis Filho. "Modeling the effect of the age of dam at calving on the weaning weight of Charolais-Zebu crossbred calves." Revista Brasileira de Zootecnia 38, no. 7 (July 2009): 1229–37. http://dx.doi.org/10.1590/s1516-35982009000700011.

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The objective of this study was to evaluate alternatives for modeling the age of dam at calving (AOD) effect on the weaning weight of Charolais-Zebu crossbred calves. Data from 56,965 calves were analyzed, using statistical models considering the fixed effects of the contemporary groups, sire and dam genetic groups, and AOD. The AOD effect was fitted to models using annual age classes, and ordinary quadratic to quintic-ordered polynomials (OP) or segmented polynomials (SP) with two, three, four, six and twelve evenly spaced intervals. In the case of segmented polynomials, general linear and quadratic effects and only one quadratic additional term from each knot were considered. The AOD effects were nested within sex of calf in all cases. According to the fitting criteria, the F-test for the reduction of residual sum of squares, coefficient of determination, residual sum of squares and mean of squared residuals, the three interval segmented polynomial (two knots) fitted to the data as well as the more complex polynomials.

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42

DEMETRIOU, I. C., and M. J. D. POWELL. "The Minimum Sum of Squares Change to Univariate Data that gives Convexity." IMA Journal of Numerical Analysis 11, no. 3 (1991): 433–48. http://dx.doi.org/10.1093/imanum/11.3.433.

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43

Hoai An, Le Thi, Le Hoai Minh, and Pham Dinh Tao. "New and efficient DCA based algorithms for minimum sum-of-squares clustering." Pattern Recognition 47, no. 1 (January 2014): 388–401. http://dx.doi.org/10.1016/j.patcog.2013.07.012.

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44

Bagirov, Adil M., Sona Taheri, and Julien Ugon. "Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems." Pattern Recognition 53 (May 2016): 12–24. http://dx.doi.org/10.1016/j.patcog.2015.11.011.

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45

Bagirov, Adil M., and John Yearwood. "A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems." European Journal of Operational Research 170, no. 2 (April 2006): 578–96. http://dx.doi.org/10.1016/j.ejor.2004.06.014.

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46

Liu, Yongguo, Zhang Yi, Hong Wu, Mao Ye, and Kefei Chen. "A tabu search approach for the minimum sum-of-squares clustering problem." Information Sciences 178, no. 12 (June 2008): 2680–704. http://dx.doi.org/10.1016/j.ins.2008.01.022.

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47

Nie, Jiawang, and James W. Demmel. "Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares." Journal of Global Optimization 33, no. 4 (December 2005): 511–25. http://dx.doi.org/10.1007/s10898-005-2099-2.

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48

Ordin, Burak, and Adil M. Bagirov. "A heuristic algorithm for solving the minimum sum-of-squares clustering problems." Journal of Global Optimization 61, no. 2 (March 19, 2014): 341–61. http://dx.doi.org/10.1007/s10898-014-0171-5.

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49

Pacheco, Joaquı́n A. "A scatter search approach for the minimum sum-of-squares clustering problem." Computers & Operations Research 32, no. 5 (May 2005): 1325–35. http://dx.doi.org/10.1016/j.cor.2003.11.006.

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50

Slabospitsky, Alexander. "RECURRENT REPRESENTATION FOR NON-STATIONARY PARAMETER ESTIMATE OF LEAST SQUARES METHOD WITH LEAST DEVIATIONS FROM "ATTRACTION" POINTS FOR BILINEAR DYNAMIC SYSTEMS." Journal of Numerical and Applied Mathematics, no. 2 (131) (2019): 32–38. http://dx.doi.org/10.17721/2706-9699.2019.2.04.

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The estimation problem of non-stationary parameter matrices is considered for bilinear discrete dynamic system in the case when for these unknown parameter matrices their ‘attraction’ points are known at any moment. Explicit and recurrent forms of representation are obtained for these parameter estimates of the least squares method with variable forgetting factor and least deviation norm from ‘attraction’ points under non-classical assumptions. The recurrent algorithm is also proposed for corresponding weighted residual sum of squares.

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