Least-squares property

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

One part of that equation, [AtA] , appears so commonly in chemometric equations that it has been given a special name, it is called the pseudoinverse of the matrix A. The uninverted term ATA is itself fairly commonly found, as well. The pseudoinverse appears as a common component of chemometric equations because it confers the Least Squares property on the results of the computations that is, for whatever is being modeled, the computations defined by equation 69-1 produce a set of coefficients that give the smallest possible sum of the squares of the errors, compared to any other possible linear model. 

The ability of partial least squares to cope with data sets containing very many x values is considered by its proponents to make it particularly suited to modern-day problems, where it is very easy to compute an extremely large number of descriptors for each compound (as in CoMFA). This contrasts with the traditional situation in QSAR, where it could be time-consuming to measure the required properties or where the analysis was restricted to traditional substituent constants. 

The properties of the least squares (LS) method (5 = 0, the non-robust procedure) and the least modules (LM) one (5 = 100%, the robust procedure) are comprehensively compared with the use of several examples of data treatment in the QSAR problems. 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

Substitution of the usual relation between concentration and property, Eq. (2-23), yields one form for linearized graphical analysis and another for least-squares fitting ... 

Partial Least Squares (PLS) regression (Section 35.7) is one of the more recent advances in QSAR which has led to the now widely accepted method of Comparative Molecular Field Analysis (CoMFA). This method makes use of local physicochemical properties such as charge, potential and steric fields that can be determined on a three-dimensional grid that is laid over the chemical stmctures. The determination of steric conformation, by means of X-ray crystallography or NMR spectroscopy, and the quantum mechanical calculation of charge and potential fields are now performed routinely on medium-sized molecules [10]. Modem optimization and prediction techniques such as neural networks (Chapter 44) also have found their way into QSAR. 

The least squares estimator has several desirable properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

Since the HF model already gives good charge densities, and reliably predicts many other diverse properties, it seems reasonable to expect that the charge densities produced from this model will be better than those from conventional least squares fitting. In other words, quantum knowledge is built into the model. 

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