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Forecasting with Linear Programming and Machine Learning
Published in William P. Fox, Robert E. Burks, Modeling Change and Uncertainty, 2022
William P. Fox, Robert E. Burks
The method of least-squares curve fitting, also known as ordinary least squares and linear regression, is simply the solution to a model that minimizes the sum of the squares of the deviations between the observations and predictions. Least squares will find the parameters of the function, f(x) that will minimize the sum of squared differences between the real data and the proposed model, shown in Equation (6.2). Minimize SSE=∑i=1n(yi−f(xi))2
Supervised Statistical Techniques
Published in Richard J. Roiger, Just Enough R!, 2020
A common statistical measure used to compute a and b is the least-squares criterion. The least-squares criterion minimizes the sum of squared differences between actual and predicted output values. Deriving a and b via the least-squares method requires a knowledge of differential calculus. Therefore, we simply state the formulas for computing a and b. For a total of n instances, we have: b=ΣxyΣx2a=ΣyΣn−bΣyn
Curve Fitting
Published in Karan S. Surana, Numerical Methods and Methods of Approximation, 2018
When the data points (xi, fi) are close together and when there is large variation in fi values, the interpolation technique may produce wildly oscillating behavior that may not be a reasonable mathematical representation of this data set. In such cases linear least squares fit is meritorious. As we have seen the least squares fit requires that we know what function and their combinations are a reasonable mathematical description of the data. Weighted linear least squares fit provides means to assign weight factors greater than one to data points that are more accurate so that the least squares fit becomes biased towards these data points. The non-linear least squares fit for special forms suitable for taking log or natural log described in section 7.4 is a convenient way to treat special classes of non-linearities in the coefficients by modifying the linear process provided it is possible to take log natural log of both sides and obtain linear least squares fit in log or natural log. The general non-linear least squares fit presented in section 7.5 is the most general linearized approach to non-linear least squares fit with or without weight factors that is applicable to any non-linear least squares fit. This formulation automatically degenerates to linear least squares fit. Thus, the least squares fit formulation in section 7.5 is meritorious for linear as well as non-linear least squares fit, both weighted as well as without weight factors.
An ensemble machine learning-based solar power prediction of meteorological variability conditions to improve accuracy in forecasting
Published in Journal of the Chinese Institute of Engineers, 2023
Priyadharshini Ramu, Sivasankar Gangatharan
The least-squares regression approach calculates the model’s regression coefficients to minimize the total square errors between the predicted and the observed value. The main goal of utilizing LR is to find the line that best matches the data, which signifies the difference between the projected value and the actual value should be as near to zero as feasible. The line with the slightest error is the one with the best fit. Using regression analysis, finding a functional connection (model or equation) between the predictor, explanatory, or independent or dependent variables is possible. By using a linear equation on the data, the LR method demonstrates the relationship between the variables (Fumo and Rafe Biswas 2015). For the hourly analysis, the simple quadratic model performs better than the simple linear model compared to the daily study. The results show a considerable improvement in addressing the unstable prediction issues in other models when the suggested strategy was applied to a case study in France. A significant association between two variables rather than a causal link indicates that the predictor and responder variables are related (Pino-Mejías et al. 2017).
Remarks on the experimental behaviour of curved surface sliders of the new Polcevera Bridge in Genoa
Published in Structure and Infrastructure Engineering, 2023
Andrea Miano, Marcello Cademartori, Antimo Fiorillo, Angelo Figundio, Marco Di Ludovico, Andrea Prota
In order to maximize the available information about the static and dynamic frictional coefficients, all the results (from both certification and acceptance tests performed according to EN 15129) have been used and discussed in this section. Regressions laws have been derived from the data, by also comparing them with the outcomes of previous literature studies. Linear least squares fitting is a mathematical procedure for finding the best-fitting line to a given set of points by minimizing the sum of the squares of the residuals (offsets) of the data from the line. Herein, linear regression is used to find the relationship between the (logarithm of) frictional coefficient, lnμ, and the (logarithm of) contact pressure, lnσ. Logarithmic linear regression in structural reliability is a widely used approach in literature (see e.g. Cornell, Jalayer, Hamburger, & Foutch, 2002; Jalayer, Ebrahimian, & Miano, 2020; Jalayer et al. 2021; Miano, Sezen, Jalayer, & Prota, 2017). This is equivalent to fitting a power-law curve to the μ−σ response in the arithmetic scale that predicts the conditional median of μ for a given value of the contact pressure σ, denote as ημ |σ, as follows: where ln a and b are the logarithmic linear regression parameters; βμ|σ is the logarithmic standard deviation of regression (i.e. the standard error of regression); {σi, μi}, i = 1: n, are the n outcomes of the tests.
Evaluating the impacts of vehicle-mounted Variable Message Signs on passing vehicles: implications for protecting roadside incident and service personnel
Published in Journal of Intelligent Transportation Systems, 2023
Jun Liu, Xing Fu, Alexander Hainen, Chenxuan Yang, Leon Villavicencio, William J. Horrey
Due to variations in video quality, such as lighting conditions, the number of frames in which the same vehicle was detected differed across observations. Recognizing that observations with different frame numbers may have an uneven influence on the model parameter estimates, this study employed the Weighted Regression method to estimate the model parameters. Weighted least squares take into account the behavior of random errors in the model by incorporating weights associated with each data point into the fitting criterion. The weight size indicates the amount of information contained in the corresponding observation. By optimizing the weighted fitting criterion to determine the parameter estimates, the weights determine the contribution of each observation to the final parameter estimates. In this study, the number of frames detected for each observation serves as the weighting factor integrated into the model estimation.