Total least-squares method

Much attention has also been devoted to modal identification without measuring the input time history. In particular, a lot of effort has been dedicated to the case of free vibration (or impulse response) and to the case of ambient vibration. In the former case, often time-domain methods based on auto-regressive moving average (ARMA) models are employed, using least squares as the core ingredient in their formulations. However, it was found that the least-squares method yields biased estimates [76], A number of methods have been developed to eliminate this bias, including the instrumental matrix with delayed observations method [76], the correlation fit method [275], the double least-squares method [114,202] and the total least-squares method [92]. A detailed comparison of these methods can be found in Cooper [61],... 

Toraya s Method. The WPPD as implemented by Toraya et al.11 decomposes the peak profiles and background functions to obtain the best fit to the experimental powder pattern of the individual pure phase data by least-squares refinement. The integrated intensities of the pure phases are then stored with the other refined parameters, such as the profile parameters and the unit-cell parameters of the phases to be quantified. During quantification step, the integrated intensity of the phase being quantified is scaled, as defined in Eq. (12.4), such that the total of the scale factors for the component phases sum to unity. The scale factors of the individual components are then refined by least-squares methods until a best fit is observed with respect to the pattern of unknown composition. 

Total least-squares has been applied to the HSVD method (HTLS),117118129 and, as in LPSVD, has been found to improve its performance slightly. Van Huffel117 found that the TLS-based LP method had better resolution and was more robust than LPSVD. 

Fig. 22. Linear expansion coefficient as of cellulose acetate (CA)-solvent systems plotted as a function of Mw 7). The lines are determined by the least-square method. Numbers on the lines denote the total degree of substitution F of CA. O CA(0.49)-DMAc CA(i.75)-DMAc A CA(2.46)-DMAc A CA(2.46)-acetone jL CA(2.46)-THF CA(2.92)-DMAc...

The first serious attempt at interpreting the protein hydration by crystallographic methods was nude 1978 on rubvedoxin, a molecule consisting of only 54 amino adds [835]. The refinement by conventional (unconstrained) least squares methods included a total of 127 water sites and converged at an R-factor of 12.7% with 1.2 A resolution data. The water oxygen atoms were added to the model only if their temperature factors were <50 A2 and their electron density was >0.3 e/A3, i.e., corresponding to the electron density expected for liquid water, 0.34 e/A3. 

Knowing Ax, (r) and mx, (r), we can find Act (r) from equation (10.92), as in the QL inversion scheme. Note that equation (10.92) should hold for any frequency, because the electrical reflectivity and the material property parameters are functions of frequency as well Ax, = Ax, (r,uj), mx, = mx, (r,cj). In reality, of course, it holds only approximately. Therefore the conductivity Act (r) can be found by using the least squares method to solve equation (10.92) with respect to the logarithm of the total conductivity a (r), similar to equation (10.90) ... 

Determine the parameter values bj andZ>2 by using the data given in Example 9.1 and the nonlinear least squares method. Recall that in Example 9.1 we needed the elements of the Jacobian matrix 7 (see equation (9.142)). In this case, integrate simultaneously the time dependent sensitivity coefficients (i.e., the Jacobi matrix elements dyfdb and dyjdb2 ) and the differential equations. The needed three differential equations can be developed by taking the total derivative (as shown below) of the right hand side of equation (9.149) which we call h ... 

All of the structures tackled have been refined by least squares methods and, with few exceptions, by full-matrix least squares methods. The total computing investment over the past three years amounts to approximately 300 hours of central-processor time on a CDC-6400. When the necessary calculations have exceeded the capacity of the CDC-6400 (about 250 variables), we have turned to a remote hookup with the CDC-7600 at Lawrence Berkeley Laboratory. 

R was determined from the area under the breakthrough curves using a planimeter. Mass eluted compared well with mass injected, indicating that mass balance was achieved. Dispersion (D) for a conservative tracer was determined by fitting the KCl breakthrough curve to the equilibrium model the fitted parameters were R and P. A nonlinear least squares method was used for parameter estimation 7. The sum of the squares of the deviation between model and data (ssq) was used as a measure of total error in the model fit. 

The techniques discussed under this heading often have very little in common and this explains the absence of specific literature on this topic. Note the following two papers which discuss the theoretical aspects of linear predictions and a variant used in some linear prediction methods called total least square . 

The magnitude of the 24-min activity relative to the various monitors was computed from the counter data by a least squares method. The decay period was assumed known as 24.2 min and the slope of the growth curve of UX was frequently measured and known to about 1 percent thus, the net strength of the U activity at a certain time could be calculated as the difference between the total observed activity at that time and the extrapolated UX activity. 

Each compound was described by a set of about 100 descriptors. The descriptors were generated from a fragmentation code or a coded structure (connection table). The set of descriptors is used as a pattern vector that characterizes a compound. Learning machine C261, 3593 and an iterative least-squares method C2603 have been used to train binary classifiers that predict mass spectral peak presence or absence at certain mass numbers. 60 classifiers for 60 mass numbers predict whether the peak is greater than 0.5 % of the total ion current. (The total ion current is the sum of all peak heights in a spectrum.) For 11 of these mass numbers... 

The nsnal total mass balance equations can be written but are not given here. Applying the generalized least squares method like in equilibrium analysis, a set of unknown parameters fPp and KJ can be found searching for the minimum of... 

A qualitatively new treatment was proposed by Dixon, who fitted the electronic energies to polynomial functions of displacement coordinates. Pelikdn and cowork-ers2 i-26,53) independently elaborated the method of evaluation of Jahn-Teller coupling and force constants using a non-linear least-squares fitting of total energy of the system. This treatment consists of two steps. First, the numerical map of APS values should be calculated by any quantum-chemical method. Then, the least-squares method is applied in order to fit the numerical map, E (Qj), to the analytic form. Actually, the set of approximate values, E (Q ), is produced so that the functional... 

The effect of experiinental uncertainties in the amount adsorbed on uncertainties in as-aneilysis and pore volumes was studied for microporous FMl/250 activated carbon cloth (ACC) during low pressure adsorption experiments at 77 K. Non-porous materials with different surface chemistry, such as carbon black and Aerosil 200, were used as standards for the development of -plots. A weighted mean least squares method was applied to a, -data for the evaluation of primary and total micropore volumes and their uncertainties. Primary micropore volumes for FMl/250 ACC calculated using a -data via these non-porous materials only agree within 14.12% with relative combined standard uncertainties (RCSU) equal to 4.15 and 7.46%, respectively. In contrast, total micropore volumes agree within... 

The remaining subsections will examine the results obtained using the FSF approach, FIR models obtained using the least squares method, and models obtained using DMI, a commercially available process identification software package. With this process, there are a total of 15 input-output relationships to be estimated. For brevity, we will only examine a subset of the results to highlight the features of the FSF approach. One key difference between the various approaches that needs to be mentioned at the outset is that the initial value of each step response model (go) has been estimated with the FSF approach but has been set equal to zero with the FIR and DMI approaches. 

Regression Also known as regression analysis, it is the process of estimating the relationship between two or more variables by assuming that one variable, the dependent variable, is a function of the other variables, the independent variables. This analysis uses a measure of total residual error to calculate the function s constants to give the best estimate of the dependent variable over a set of observations or measurements. The least squares method is one of the most common methods for determining the function s constants. 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

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