Model moving least squares

H. Noguchi and G. Gompper, Meshless membrane model based on the moving least-squares method, Phys. Rev. E 73, 021903 (2006). 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

Fig. 8, Examples of a least squares fit of a shrinking core model to measurements of moving ion exchange fronts which develop within the acidic form of poly(acrylamide-co-sodium methacrylate) gel cylinders of different radii when immersed in pH 12 NaOH solution. Legend All gels were made from a solution containing 1 mol % of solution methacrylate A = 1.8 mm radius B = 2.8 mm C = 4.2 mm D = 6.4 mm. Reprinted with permission from [127]. Copyright [1992] American Chemical Society...

For this case of study, it is supposed that the four states are directly measurable. The estimation problem is posed as a least squares objective function subject to the model nonlinear differential equations as constraints, restricting the mathematical program to the size of the moving window, and therefore ignoring the data outside such window. 

PCA is a least square method and therefore its results depend on data scaling. The initial variance of a column variable partly determines its importance in the model. In order to avoid the problem of over- or under-representation of variables, column variables are scaled to unit variance before analysis. The column average is then subtracted from each variable, which, from a statistical point of view, corresponds to moving the multivariate system to the center of the data, which becomes the starting point of the mathematical analysis. The same auto-scaUng and centering procedures are applied in PLS discriminant analysis. 

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],... 

Modern devices use photodiodes or CCD arrays (Sect. 4.5.2) instead of photoplates. With a diode width of 25 p.m, the peak of a symmetric line profile extending over 3—5 diodes can be determined by a least-squares fit to a model profile within 1—5 xm, depending on the S/N ratio. When the array is placed behind a spectrometer with a dispersion of 1 mm/nm, the center of the line can be determined within 10 nm. Since the signals are read electronically, there are no moving parts in the device and any mechanical error source (backlash) is eliminated. 

S. Kasemsumran, Y. P. Du, K. Maruo, and Y. Ozaki, Improvement of Partial Least-Squares Models for In tro and In vo Glucose Quantifications by Using Near-Infrared Spectroscopy and Searching Combination Moving Window Partial Least-Squares, Chemometrics Intell. Lab. Syst., 82,97 (2006). 

Davies and Seaton (1998) found that pore-size distributions depend strongly on the shape of pore assumed in the modeling. Pores, with square and rectangular shapes, have the same role in the analysis based on slit-shaped pores. This causes pore-size distributions based on these models to be flatter and to move to larger pore sizes compared with those determined using slit-shaped pores. These authors found that it was possible to resolve square and rectangular pores into an equivalent distribution of slit-shaped pores, this suggesting that slit-shaped pores represent the clearest, least complicated description of the internal structure of activated carbon. 

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