Least-squares criterion

Tarantola, A., 1987, Inverse Problem Theory, Elsevier Tarantola, A., Valette, B., 1982, Generalized nonlinear inverse problems solved using the least squares criterion. Reviews of Geophysics and Space Physics 20, 219... 

Firstly, the krlglng estimator is optimal only for the least square criterion. Other criteria are known which yield no more complicated estimators such as the minimization of the mean absolute deviation (mAD), E P(2c)-P (3c), yielding median-type regression estimates. 

If L(e) e, l.e. the loss Is proportional to the squared error, the least square criterion Is apparent, and the best estimator P(x) Is the conditional expectation defined In (3). Note that this estimator Is usually different from that provided by ordinary krlglng for the simple fact that expression (3) Is usually non-llnear In the N data values. If L(e) e, l.e. the loss Is proportional to the absolute value of the error, the best estimator Is the conditional median, l.e. the value ... 

In multiple linear regression (MLR) we are given an nxp matrix X and an n vector y. The problem is to find an unknown p vector b such that the product y of X with b is as close as possible to the original y using a least squares criterion ... 

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. 

Principal covariates regression (PCovR) is a technique that recently has been put forward as a more flexible alternative to PLS regression [17]. Like CCA, RRR, PCR and PLS it extracts factors t from X that are used to estimate Y. These factors are chosen by a weighted least-squares criterion, viz. to fit both Y and X. By requiring the factors to be predictive not only for Y but also to represent X adequately, one introduces a preference towards the directions of the stable principal components of X. 

A selected subset of the reported densities is fit to functions of temperature using the least squares criterion. Up to a boundary temperature Tb (approximately 0.87 c), the calculated density px is represented by a polynomial in temperature with coefficients ak of order p,... 

A special adaptation of this analysis was applied by Donachie (D8) to estimate the pipe resistances using field measurements of pressures. In that application the u s correspond to pipe resistances and the x s correspond to the nodal pressures. Using linearized sensitivity information adjustments of u s are made, one at a time, to reduce the discrepancies between calculated and measured pressures according to the least-squares criterion. The Am which causes the largest change in the sum of squares of discrepancies is given precedence over others. 

Depending on the fulfilment of the conditions mentioned above, namely (i) to (iii) and following, the least squares criterion has to be modified as follows ... 

Therefore, let us start by proposing a definition, one that can at least serve as a basis for our own discussions. Let us define linearity as The property of data comparing test results to actual concentrations, such that a straight line provides as good a fit (using the least-squares criterion) as any other mathematical function. ... 

The concept of metric tensor becomes central whenever distances and projections are considered, particularly when least-square criterion are used, a point that will be discussed in Chapter 5. Let us ask the frequently raised question of how to find an expression in terms of old coordinates (e.g., oxide proportions) for a projection made in the non-Euclidian space. This could be the case for finding oxide abundances of a basalt composition projected in the Yoder and Tilley tetrahedron, or the oxide abundance of a metamorphic rock composition projected into an ACF diagram assuming that quartz is present. 

None of the population parameters x and y can be found since their determination would require the whole range of attainable values to be measured. The least-square criterion provides estimates x and of x and y, respectively, which also satisfy the model, i.e.,... 

Given a geochemical variable y, m measurements at times tu t2,..., tm produce the unevenly spaced time series yl5 y2,..., ym, which we lump together as the vector y. In order to find out eventual periodicities, Lomb (1976) suggests fitting the data by a sine wave using a least-square criterion. For any arbitrary frequency /, the fitting function is written... 

The least-square criterion suggests to minimize the modulus eTs of this error vector subject to the condition that the modulus iTi of the estimate is unity. The problem is therefore to minimize the sum c2 such as... 

The dimensionality of the model, a, is estimated so as to give the model as good predictive properties as possible. Geometrically, this corresponds to the fitting of an a-dimensional hyperplane to the object points in the measurement space. The fitting is made using the least squares criterion, i.e. the sum of squared residuals is minimized for the class data set. 

In most biological cases/(C) is nonlinear and analytical integration is difficult or impossible. Numerical integration again allows calculation of concentration at the end of each experiment. Differences between simulated and experimental data are then minimized using, e.g., the least squares criterion. Both experimental approaches are compared in Table 1, especially with respect to their suitability for kinetic screening. 

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

Knowing which factor and interaction have an influence on the response (y), we can try to fit an empirical model by the common least-squares criterion ... 

Figure 1. Intensity of second harmonic light (I ) produced by LB films of compound 6 as a function of layer number (N). Both the fundamental and second harmonic beams are p-polarized. The line (with slope 2) is the best fit according to the least square criterion.

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