Least squares models

The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows ... 

As noted earlier, the x -test for goodness-of-fit gives a more balanced view of the concept of fit than does the pure least-squares model however, there is no direct comparison between x and the reproducibility of an analytical method. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

S. Wold, Non-linear partial least squares modelling. II. Spline inner relation. Chemom. Intell. Lab. Syst., 14(1992)71-84. 

Restori, R. and Schwarzenbach, D. (1995) Maximum-entropy versus least-squares modelling of the electron-density in K2PtCl6, Acta Cryst., B51, 261-263. 

We originally proposed NNM to be present in metallic beryllium [30] based on analysis of the X-ray diffraction data measured by Larsen and Hansen [24], Based on Fourier maps and elaborate multipole least-squares modeling, indisputable evidence... 

M. Sjostrom, S. Wold, W. Lindberg, J.A. Persson and H. Martens, A multivariate calibration problem in analytical chemistry solved by partial least squares models in latent variables. Anal. Chim. Acta, 150, 61-70 (1983). 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

Factor The result of a transformation of a data matrix where the goal is to reduce the dimensionality of the data set. Estimating factors is necessary to construct principal component regression and partial least-squares models, as discussed in Section 5.3.2. (See also Principal Component.)... 

D. C. Baxter and J. Ohman, Multi-component standard additions and partial least squares modelling, a multivariate calibration approach to the resolution of spectral interferences in graphite furnace atomic absorption spectrometry, Spectrochim. Acta, Part B, 45(4 5), 1990, 481 491. 

D. C. Baxter, W. Freeh and 1. Berglund, Use of partial least squares modelling to compesate for spectral interferences in electrothermal atomic... 

Sjoestroem, M., Wold, S., Lindberg, W., Persson, J.A. and Martens, H., A Multivariate Calibration Problem in Analytical Chemistry Solved by Partial Least Squares Models in Latent Variables Anal. Chim. Acta 1983, 150, 61-70. 

Factorial methods - factor analysis (FA) - principal components analysis ( PCA) - partial least squares modeling (PLS) - canonical correlation analysis Finding factors (causal complexes)... 

Usually, linear models are preferable (linear ordinary, i.e., unweighted, least squares regression model is not appropriate in many cases, in which weighted least squares model should be applied), but, if necessary, nonlinear (e.g., second order) models can be used [60],... 

Hasegawa, K., Kimura, T., Miyashita, Y. and Funatsu, K. (1996a). Nonlinear Partial Least Squares Modeling of Phenyl Alkylamines with the Monoamine Qxidase Inhibitory Activities. J.Chem.Inf.Comput.Sci.,36,1025-1029. 

Hasegawa, K., Arakawa, M. and Funatsu, K. (1999a). 3D-QSAR Study of Insecticidal Neonicoti-noid Compounds Based on 3-Way Partial Least Squares Model. Chemom.Intel Lab.Syst., 47, 33-AO. 

Nord, L.I., Fransson, D. and Jacobsson, S.P. (1998). Prediction of Liquid Chromatographic Retention Times of Steroids by Three-Dimensional Structiure Descriptors and Partial Least Squares Modeling. Chemom.InteliLab.Syst, 44,257-269. 

Oberrauch, E. and Mazzanti, V. (1990). Partial-Least-Squares Models for the Octane Number of Alkanes Based on Subgraph Descriptors. Anal.Chim.Acta, 235,177-188. 

M. Kasper and W. H. Ray, 1993, Partial Least Squares Modeling as Successive Singular Value Decomposition, Comput. Chem. Engng., Vol. 17, Issue 10, 985... 

Netzeva TI, Schultz TW, Aptula AO, Cronin MTD. Partial least squares modelling of the acute toxicity of aliphatic compounds to Tetrahymena pyriformis. SAR QSAR Environ Res 2003 14(4) 265-83. 

If a second term, say the absorbance at 2i> is added to the model equation, the predictive ability is improved considerably. Thus by including A21, the least-squares model is... 

Least squares models, 39, 158 Linear combination, normalized, 65 Linear combination of variables, 64 Linear discriminant analysis, 134 Linear discriminant function, 132 Linear interpolation, 47 Linear regression, 156 Loadings, factor, 74 Lorentzian distribution, 14... 

Some commercial RTO systems either back-calculate model parameters from available process measurements or use some form of least-squares parameter estimation.P The back-calculation approach has been shown to be less robust to errors in the data and model than the least-squares approach and, as a result, is less desirable in RTO applications. The least-squares model updating problem can be solved by a number of nonlinear programming solvers, and the estimated parameters used for model-based optimization. 

The model matrix X in least squares modelling describes the variation of the variables included in the model. The matrix X X is symmetric, and hence also the dispersion matrix, (X X). The eigenvalues of the dispersion matrix are related to the precision of the estimated model parameters. The determinant of the dispersion matrix is the product of its eigenvalues. The "Volume" of the joint confidence region of the estimated model parameters is proportional to the square root of the determinant of the dispersion matrix. 

The development of a model for the entire cartilage spectrum, without the use of pure-component spectra, holds much promise because it does not require user intervention. In addition, the incorporation of many (15-i-) factors into a least-squares model may be necessary to describe the interactions between collagen and proteoglycan molecules. Moreover, the incorporation of a parsimony measure to reduce noise contributions in model development may provide a better classification of osteoarthritic-related damage. 

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