Least-squares norm

For each molecule the errors in the a/a- and the a/ S-blocks of the 2-RDM are reported in Table IV. The 2-RDM errors are measured through a least-squares norm, which is defined by... 

A variety of techniques is nowadays available for the solution of inverse problems [26,27], However, one common approach relies on the minimization of an objective function that generally involves the squared difference between measured and estimated variables, like the least-squares norm, as well as some kind of regularization term. Despite the fact that the minimization of the least-squares norm is indiscriminately used, it only yields maximum likelihood estimates if the following statistical hypotheses are valid the errors in the measured variables are additive, uncorrelated, normally distributed, with zero mean and known constant standard-deviation only the measured variables appearing in the objective function contain errors and there is no prior information regarding the values and uncertainties of the unknown parameters. 

Although very popular and useful in many situations, the minimization of the least-squares norm is a non-Bayesian estimator. A Bayesian estimator [28] is basically concerned with the analysis of the posterior probability density, which is the conditional probability of the parameters given the measurements, while the likelihood is the conditional probability of the measurements given the parameters. If we assume the parameters and the measurement errors to be independent Gaussian random variables, with known means and covariance matrices, and that the measurement errors are additive, a closed form expression can be derived for the posterior probability density. In this case, the estimator that maximizes the posterior probability... 

It is seen that this process is essentially a least square fit of atp eg and pifirregby (f>i and (fE, subject to a minimum energy condition which allows to determine a and /3. Note that a and fi are related by the norm of so that there is in fact a single parameter in this minimisation. 

The least squares criterion states that the norm of the error between observed and predicted (dependent) measurements 11 y - yl I must be minimal. Note that the latter condition involves the minimization of a sum of squares, from which the unknown elements of the vector b can be determined, as is explained in Chapter 10. 

Step 2. Multivariate least squares regression of Y on the major A principal components, using either the unit-norm singular vectors or the principal components = XVj j = U[a,S[aj ... 

Other error norms have boon considered in Sections 1.8.2 and 1.8.3. Why the least squares method is the most popular Where does it come from If it is good at least for a well defined class of problems, why to experiment with... 

Without information on the errors any error norm is as good as the others. Thus, to explain the popularity of the least squares method we have to make a number of assumptions. In particular, for model (3.2) we assume that... 

The least squares solution x of an unsolvable linear system Ax = b such as our system is the vector x that minimizes the error Ax — 6 in the euclidean vector norm a defined by x Jx +. .. + x% when the vector x has n real entries x. ... 

In these two equations, the norm of the residuals between the PCA-reproduced data, DpcA, using the selected number of components, andjhe ALS-reproduced data using the least-squares estimatespf C and ST matrices, C and ST, is alternatively minimized by keeping constant C (Equation 11.11) or ST (Equation 11.12). The least-squares solution of Equation 11.11 is ... 

Linear and nonlinear fitting methods, from least-squares and logarithmic to fitting procedures based on the entropy norm... 

This equation shows that an element of the normed rate sensitivity matrix is the ratio of the rate of formation or consumption of species i in reaction j, to the production rate of species i. It is possible to consider the effect of each parameter on several production rates simultaneously using a least-squares objective function. This approach leads to the application of the following overall sensitivity type measure ... 

The main role of the stabilizing functional (a stabilizer) is to select the appropriate class of models for inverse problem solution. The examples listed above show that there are several common choices for a stabilizer. One is based on the least squares criterion, or, in other words, on the Lo norm for functions describing model parameters ... 

The Least-Squares formulation is based on the minimization of a norm-equivalent functional. This method consists in finding the minimizer of the residual in a certain norm. Consider the following linear problem ... 

T T. X, A continuous linear form, used in least squares method outline J i g) norm equivalent functional in least squares method outline ... 

The vector y has been shown deliberately as a function of the constants, a, because the size of the components of this vector will depend on the selection of those parameters. A procedure often adopted is to minimize some norm of the error, e, by a best choice of the parameters, a, for example time-domain least-squares minimization. Application of this process in itself provides a check on the validity of the model, since the optimal choice of parameters, a p,. should lie within ranges expected on physical grounds. The corresponding transient states will be y(a p, f), and thus we may define the minimum error as... 

Figure 2.20 shows a typical result of the problem. The dashed line is the tme relationship and the crosses are the measurements. Even though the samples are quite scattered in the low range of X, the Bayesian approach reflects the correct weighting for different measurements. On the other hand, the traditional least-squares method simply minimizes the 2-norm of the difference... 

The location of the function values >) must be determined in order to provide for the match-point. This is achieved by minimizing a goal function which measures the distances between the grey levels in template and patch. The goal function to be minimized in this approach is the L2-norm of the residuals of least squares estimation. The location is described by shift parameters Ax, Ay, which are counted with respect to an initial position of gix, y), the approximation of the conjugate patch g x, y). 

The determination of output weights between hidden and output layers is to find the least-square solution to the given linear system. The minimum norm least-square solution to hnear system (1) is M Y, where M is the Moore-Penrose generalized inverse of matrix M. The minimum norm least-square solution is unique and has the smallest norm among the least-square solutions. 

A little more expensive [n (m + I7nl3) flops and 2mn space versus n (m—nl3) and mn in the Householder transformation] but completely stable algorithm relies on computing the singular value decomposition (SVD) of A. Unlike Householder s transformation, that algorithm always computes the least-squares solution of the minimum 2-norm. The SVD of an m x n matrix A is the factorization A = ITEV, where U and V are two square orthogonal matrices (of sizes mxm and nxn, respectively), U U = Im, y V = In, and where the m x n matrix S... 

A common approach is the method of least squares (L2 norm) which leads to root mean squared residuals (where the residual is the difference between the observed and calculated travel times). However, the use of least squares procedures requires the assumption that the distribution of the residuals is of Gaussian nature (Mendecki 1997). This is generally not true. 

Workers in this field use several methods to derive the optimized parameters for the pseudopotentials and the pseudo-orbitals. Generally, the parameters can be obtained by a fit procedure taking the shape or the norm of the orbitals as a reference function. The pseudo-orbitals are derived by fitting them to numerical valence orbitals of all-electron calculations. Some methods generate the potentials on a numerical grid by inverting Fock equations for pseudo-orbitals derived from numerical atomic wavefunctions. The numerically tabulated potentials are then least-squares fit with analytical Gaussian... 

In the special case of an underdetermined linear system we may look for the minimum norm least squares solution of this system. With F = 7, o = 0 we obtain from Eq. (2.3.4)... 

This restoration technique involves estimating 6, such that H6 approximates i in the least squares sense [21]. In the absence of constraint, this is achieved by finding the value of 6 that minimizes i - H6 p - n p. Assuming that the norm of the noise is small, this is equivalent to minimizing the function... 

The basic idea of the least-squares method is to minimize the integral of the square of the residual over the computational domain. In order to find the approximation in the sense of the least squares error, the following norm-equivalent least-squares function is defined for the generalized problem formulation (12.402) ... 

It is noticed that a quadrature rule is applied in (12.477) and (12.478). In order to reduce time consuming operations, the quadrature points are normally chosen the same as the collocation points in the approximation of the norm integrals of the least-squares method. The quadrature points and the collocation points are defined at the same locations when both type of points are determined as the roots of the same type of orthogonal polynomial of the same order. In this case [f] = fj coincides with [f] =flQ. 

Considering the strong formulation of the boundary conditions, the following norm-equivalent least-squares function is defined for the generalized problem formulation... 

Norm equivalent functional in least squares method outline... 

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