Distance least squares method

Distance Least Squares Methods.- Perhaps the earliest computational method in this field was that due to Meier and Villinger, who devised the method of distance least squares (DLS) to provide a suitable method for the refinement of zeolite crystal structures. 

Kroll, H., Maurer, H., Stockelmann, D., Becker, W., Fulst, J., Kriisemann, R., et al. (1992). Simulation of crystal structures by a combined distance-least-squares valence-rule method. Zeit. Kristallogr. 199, 49 66. 

Because the ions in electrolyte solutions are often more or less associated, Eq. (7.5) is useful in analyzing conductivity data. The experimental data for A and c are subjected to computer analysis, by applying the least-squares method, and optimum values of such parameters as A°°, KA and a are obtained. Sometimes the ion parameter a (i.e. the distance of closest approach) is replaced by the Bjerrum s distance q in Section 2.6. In this case, the parameters obtained from Eq. (7.5) are of two kinds, A°° and KA. 

A method for investigating all four types of problems (of which 2a is the most complicated and lb is the least complicated) has been derived by Meier and Villiger (12). The distance least squares (DLS) procedure uses the well-known distances D for Si-O, Al-O, 0-0 (and eventually others) of the framework and refines atomic parameters by a least-squares procedure minimizing... 

Distance least squares (DLS), a method developed by Meier and Vill-iger (1) for generating model structures (DLS models) of prescribed symmetry and optimum interatomic distances, can supply atomic coordinates which closely approach the values obtained by extensive structure refinement. DLS makes use of the available information on interatomic distances, bond angles, and other geometric features. It is primarily based on the fact that the number of crystallographically non-equivalent interatomic distances exceeds the number of coordinates in framework-type structures. A general DLS program is available (8) which allows any combination of prescribed parameters (interatomic distances, ratios of distances, unit cell constants etc). In addition, subsidiary conditions (as discussed in Refs. 1 and 8) can also be prescribed. 

In the simple least-squares method in two dimensions, the aim is to find a function y =f(x) that fits a series of observations (x y, (x2,y2),...(xi-,yJ.), where each observation is a data point, a measured value of the independent variable x at some selected value y. (For example, y might be the temperature of a gas, and x might be its measured pressure.) The solution to the problem is a function fix) for which the sum of the squares of distances between the data points and the function itself is as small as possible. In other words,/(x) is the function that minimizes D, the sum of the squared differences between observed (yf.) and calculated [f(x )] values, as follows... 

The structure of the closely related molecule, 1,2-cyclopentenophen-anthrene, has been determined and refined with partial three-dimensional data by least-squares methods by Entwhistle and Iball (1961). Independent confirmation of the correctness of this structure has been provided by Basak and Basak (1959) who did not, however, carry out any refinement of the structure. Entwhistle and Iball s results show that the molecule is not planar the deviations of the carbon atoms from the mean molecular plane are shown in Fig. 9 (the standard deviations of the atomic coordinates lie between 0-009 and 0-015 A). The three aromatic rings appear to be linked in a slightly twisted arrangement. Atoms H and K, which are bonded to the overcrowded hydrogen atoms, are displaced almost the same distance on opposite sides of the mean plane. In the five-membered ring, atoms C and E are below the molecular plane by about 0-10 A while atom D lies 0-18 A... 

According to a complete X-ray diffraction analysis, Se6 consists of ring molecules with the molecular symmetry of Dzd the crystal and molecular parameters are listed in Table II (17) and the crystal structure is shown in Fig. 2. Refinement by the least squares method resulted in the following atomic parameters of the single atom in the asymmetric unit x = 0.1602 0.00048, y = 0.20227 0.00047, z = 0.12045 0.00120 calculated density, 4.71 g/cm3. An earlier investigation of selenium vapor by electron diffraction led to an internuclear distance of 234 1 pm and an average bond angle of 102 0.5° for the chairlike cyclic Se6 molecule (23). 

The structure of the monoclinic form has been refined by the least squares method. The Mo-O distances are therefore known with an accuracy that allows a detailed discussion of the distortions from regular coordination around the molybdenum atoms. The tetrahedral coordination is very regular, while the octahedral coordination is rather irregular, especially in octahedra joined to tetrahedra. 

The CjjY are parameters fit to each pair of atoms X and Y on molecules A and B, and is the distance separating these atoms. Szczesniak et al. ° fit these parameters by a least-squares method so as to reproduce their calculated dispersion as closely as possible. 

It may be tempted to believe that the NFM reduces to a simple least-squares method, where only the relative weights (are used to rank the entire Pareto domain, when the three thresholds Qu, Pk, and 14) are either made all equal to zero or to very high values. This is not the case and, in fact, the three threshold values play an important role in the ranking of the Pareto domain over the whole range of threshold values. The role of thresholds is to use the distance between two values of a given criterion to create a zone of preference around each solution of the Pareto domain and to identify the solutions that are systematically better than the other solutions. 

The molecular term jIm, or sometimes the ratio j(Im/Ib) = sM(s), is analyzed by a least-squares method and the bond distances rg, bond angles and some of the amplitudes of vibration are determined (see more about this in the next section). The rg distance, sometimes called an operational parameter, is an ill-defined parameter since it refers to the maximum position of any peak of the P(r)/r function. The better parameter, the Vg distance. 

It is desirable to express the calibration data in functional form with a curve fit for real-time processing. Polynomial fits of various orders by the least-square method may be used in various regions of the data to represent the intensity-to-distance relationship with the following form ... 

In principle, only three detectors are needed to determine the tracer position. The availability of measured distances from 16 detectors resulted in data redundancy for location determination. To take advantage of this planned redundancy, a weighted least-square method based on an linearization scheme was used to determine the optimum tracer position. 

Numerous methods exist for estimating A and B. The most common approach is the least squares method. The least sguares method is based on minimizing the square of the distance between the observed value yobs the "fitted" value yfit = a+bx. 

This distance is represented by the interval d in Figure 1. Thus, the least squares method is based on minimizing XI d =X(Yobs yfit) ... 

The least squares method is a mathematical way of doing what was just described. Vertical distances of the line to the data point ()>/ - y ) are used. (Here, y, represents the ith data value, and y represents the estimated value of y that lies on the best-fit Une.) But because distances may be either positive or negative, and so may cancel, the distances are squared, (y - y ), to make the numbers all positive. Next, all data points are used (y - y where X denotes... 

It is a calibration method in conventional photogrammetry to solve Eq. (7) with the theory of the least squares method. On the other hand, in the case where distances are used as geodetic data, the distance Dij between the points Pi and Pj is written... 

Vertical neutron distributions were studied also ,typical results.-will be found in Fig. A2.F.. -It will be recalled that no reflector was present at top or bottom of the experimental pile, and the solid lines are cosine curves fitted to. points near the center by a. least-squares method. The slight lateral shift of. the data from center is apparently due to the presence of a portion of a control rod in the top of the pile.- It is clear that the extrapolation distances for thermal and epithermal neutrons are the -same, within the experimental limits. 

The minimization process use ICP algorithm (Iterative Closest Point) described by Greespan [8] and Besl et al. [9]. Defining D the set of data points of the surface Sj and M the set of points of the model or surface S2, this method establish a matching of D and Mpoints. Thus for each point of D there is a point (the nearest) of the model M. By the correspondence established above, the transformation that minimizes the distance criterion is calculated and applied to the points of the set D and the overall error is calculated using least squares method. 

The structural parameters (r, u, sometimes k, and possibly other parameters) are then obtained by least-squares fitting, usually on the intensity curves, but occasionally on the RD curves (c/. p. 45). If the intensity data are used, the data from all the nozzle-to-plate distances are not necessarily combined into one composite curve an individual scale factor may be refined for each set of data. Since the observed data are considerably correlated, i.e. each point cannot be regarded as an independent observation, a number of problems are encountered in the application of the least-squares method. This problem is most important for the estimation of the standard deviations, since the parameters may be obtained fairly satis-factorally by conventional least-squares refinement. (A more detailed discussion is set out on p. 45.)... 

---