Least squaresThe method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting.
Linear least squaresLinear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. The three main linear least squares formulations are: Ordinary least squares (OLS) is the most common estimator.
Distributed generationDistributed generation, also distributed energy, on-site generation (OSG), or district/decentralized energy, is electrical generation and storage performed by a variety of small, grid-connected or distribution system-connected devices referred to as distributed energy resources (DER). Conventional power stations, such as coal-fired, gas, and nuclear powered plants, as well as hydroelectric dams and large-scale solar power stations, are centralized and often require electric energy to be transmitted over long distances.
Generalized least squaresIn statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in the regression model. Least squares and weighted least squares may need to be more statistically efficient and prevent misleading inferences. GLS was first described by Alexander Aitken in 1935. In standard linear regression models one observes data on n statistical units.
Ordinary least squaresIn statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable.
Total least squaresIn applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix.
Non-linear least squaresNon-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences.
Numerical methods for linear least squaresNumerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).
Electric power transmissionElectric power transmission is the bulk movement of electrical energy from a generating site, such as a power plant, to an electrical substation. The interconnected lines that facilitate this movement form a transmission network. This is distinct from the local wiring between high-voltage substations and customers, which is typically referred to as electric power distribution. The combined transmission and distribution network is part of electricity delivery, known as the electrical grid.
Electric powerElectric power is the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second. Standard prefixes apply to watts as with other SI units: thousands, millions and billions of watts are called kilowatts, megawatts and gigawatts respectively. A common misconception is that electric power is bought and sold, but actually electrical energy is bought and sold.
