Least squares minimisation

A matrix for each F value was set up and diagonalised with the aid of a least squares minimisation programme, and the spectra were fitted with seven constants, whose values (in MHz) were found to be ... 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

An experiment for a simultaneous measurement of D(H, C ) and Z)(H ,C ) was proposed by Chow and Bax. The sequence is based on a 3D CB(CA)CONH experiment with the use of a quantitative approach to evaluate the coupling. Three spectra with different positions of H 180° pulse in the constant-time period are recorded in an interleaved fashion and the couplings are derived with a least-square minimisation routine. For methylene groups only the sum of /(C,H) couplings is available from the experiment. The... 

This is a study of the complexation of zirconium by fluoride employing the potenti-ometric technique, using a Fe VFe electrode. Experiments were conducted at (25.0 0.2)"C and in 0.5 M NH4CIO4. Results were analysed using least squares minimisation. The stability constants estimated are given in the table, according to the reaction... 

The parameter estimation presented in the previous section is based on a least squares minimisation of the errors between measured system outputs and outputs of a system model evaluated by using estimated parameter values. If the real system is replaced by a model in preferred integral causality, measured outputs can be obtained by solving the model equations for given initial conditions and can be used for offline parameter estimation in order to isolate multiple faults deliberately introduced into the system model. In real-time FDI, initial conditions are either not known or difficult to obtain. Therefore, in online parameter estimation, they have to be considered as additional unknowns that are to be estimated. 

Alternatively, multiple faults may be isolated by least squares minimisation of ARR residuals. The latter are indicators for the errors between measurements from a faulty system and outputs of a model computed by using estimated parameters. ARRs obtained from a DBG do not depend on initial conditions but use derivatives of measurements with respect to time which entails the drawback that differentiation carried out in discrete time amplifies noise if not properly filtered. 

For least squares minimisation of ARR residuals only the m parameters contributing to the unstructured part of a FSM need to be considered. Residuals in that part structurally depend on more than one parametric fault and the unstructured part of a FSM can be further subdivided into subspaces (rows of the FSM) with common component fault signatures. Which parameters in which subspaces can be identified... 

Once the form of the correlation is selected, the values of the constants in the equation must be determined so that the differences between calculated and observed values are within the range of assumed experimental error for the original data. However, when there is some scatter in a plot of the data, the best line that can be drawn representing the data must be determined. If it is assumed that all experimental errors (s) are in thejy values and the X values are known exacdy, the least-squares technique may be appHed. In this method the constants of the best line are those that minimise the sum of the squares of the residuals, ie, the difference, a, between the observed values,jy, and the calculated values, Y. In general, this sum of the squares of the residuals, R, is represented by... 

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 procedure of Lifson and Warshel leads to so-called consistent force fields (OFF) and operates as follows First a set of reliable experimental data, as many as possible (or feasible), is collected from a large set of molecules which belong to a family of molecules of interest. These data comprise, for instance, vibrational properties (Section 3.3.), structural quantities, thermochemical measurements, and crystal properties (heats of sublimation, lattice constants, lattice vibrations). We restrict our discussion to the first three kinds of experimental observation. All data used for the optimisation process are calculated and the differences between observed and calculated quantities evaluated. Subsequently the sum of the squares of these differences is minimised in an iterative process under variation of the potential constants. The ultimately resulting values for the potential constants are the best possible within the data set and analytical form of the chosen force field. Starting values of the potential constants for the least-squares process can be derived from the same sources as mentioned in connection with trial-and-error procedures. 

Later, in Chapter 4.4, General Optimisation, we discuss non-linear least-squares methods where the sum of squares is minimised directly. What is meant with that statement is, that ssq is calculated for different sets of parameters p and the changes of ssq as a function of the changes in p are used to direct the parameter vector towards the minimum. 

The task is to compute the best parameter shift vector 8p that minimises the new residuals r(p+8p) in the least-squares sense. This is a linear regression equation with the explicit solution. 

The alternating computation of the matrices A and C in linear least-squares fits, each followed by setting negative values to zero, is simple but very crude. This fact is reflected in the slow process of the sum of squares minimisation. 

The most common calibration model or function in use in analytical laboratories assumes that the analytical response is a linear function of the analyte concentration. Most chromatographic and spectrophotometric methods use this approach. Indeed, many instruments and software packages have linear calibration (regression) functions built into them. The main type of calculation adopted is the method of least squares whereby the sums of the squares of the deviations from the predicted line are minimised. It is assumed that all the errors are contained in the response variable, T, and the concentration variable, X, is error free. Commonly the models available are Y = bX and Y = bX + a, where b is the slope of the calibration line and a is the intercept. These values are the least squares estimates of the true values. The following discussions are only... 

The presentation of Eq. 1 also applies well where y is implicit that is, stated as a solution to a set of conditions and not attainable by direct calculation (for example, in least squares fitting to a circular shape, the result for the radius is that value which minimises a sum of squared residuals, and cannot be calculated directly). In these cases, while the function/may not be known explicitly, we can write, for example,... 

In relation (5.3), where Pq p. . etc. are the unknown parameters, we can observe that among the different methods to identify these parameters, the method of least-squares can be used without any restriction. So, the identification of Po Pj . . etc. coefficients has been reduced to the functional minimisation shown in relation (5.5) ... 

This deviation is now minimised by variation of the parameters. The combination of parameter values that best fit the experimental data using this deviation as the criterion of best fit is the desired solution of the analysis. This process of finding a solution is termed iteration because the solution is located by trying out many possible combinations of parameters since the equations being fitted are in general non-linear, the process is more specifically one of iterative non-linear least-squares fitting. 

There is peak overlap. In this case, it is necessary to decompose the peak by modelling using software. A least squares error minimisation procedure can be employed to adjust the positions, intensities and full width half maxima of the components and provides an indication of the quality of the model in relation to the actual shape of the peak. Various peak shapes are available for modelling purposes in particular the use of experimentally obtained shapes becomes extremely useful in the case of asymmetric peaks of transition metals (Fig. 5.7). 

The decay law G(t) was extracted from the experimental decay curve using nonlinear least-squares iterative reconvolution. The parameters were varied until the xl values were minimised and initially the trial function for G(t) was chosen to be a sum of 10 exponential terms viz... 

Both E and n were calculated using the Marquart-Levenberg least square algorithm. The sum = w CRroic p) was explicitly used to minimise effects of non-uniform errors. By applying the weight function w = all data points are considered... 

The modelled output signal is then calculated by performing the inverse FFT on the modelled output spectrum. The model parameters of the model of the combustion camber arc found by fitting the modelled output signal with the output signal measured. The fitting is effected by minimising the least square error. 

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