Weighted centroid

By far the most popular technique is based on simplex methods. Since its development around 1940 by DANTZIG [1951] the simplex method has been widely used and continually modified. BOX and WILSON [1951] introduced the method in experimental optimization. Currently the modified simplex method by NELDER and MEAD [1965], based on the simplex method of SPENDLEY et al. [1962], is recognized as a standard technique. In analytical chemistry other modifications are known, e.g. the super modified simplex [ROUTH et al., 1977], the controlled weighted centroid , the orthogonal jump weighted centroid [RYAN et al., 1980], and the modified super modified simplex [VAN DERWIEL et al., 1983]. CAVE [1986] dealt with boundary conditions which may, in practice, limit optimization procedures. 

As stated in Chap. 2, in a PET scanner, block detectors are cut into small detectors and coupled with four PM tubes, which are arranged in arrays of rings. Each detector is connected in coincidence to as many as N/2 detectors, where N is the number of small detectors in the ring. So which two detectors detected a coincidence event within the time window must be determined. Pulses produced in PM tubes are used to determine the locations of the two detectors (Fig. 3.2). As in scintillation cameras, the position of each detector is estimated by a weighted centroid algorithm. This algorithm estimates... 

The intercept guarantees that the straight line need not pass the origin of the coordinate system, but passes the centroids of the variables, that is, x and y. For estimation of the parameters for a weighted regression, first the weighted centroids are calculated as follows ... 

In equation (5.16) y and x represent the coordinates of the weighted centroid, through which the weighted regression line must pass. These coordinates are given as expected by x = %WiXjn and = %w,yjn. 

The foregoing are volume integrals evaluated over the entire volume of the rigid body and dw is an infinitesimal element of weight. If the body is of uniform density, then the center of gravity is also called the centroid. Centroids of common lines, areas, and volumes are shown in Tables 2-1, 2-2, and 2-3. For a composite body made up of elementary shapes with known centroids and known weights the center of gravity can be found from... 

The vector of column-means nip defines the coordinates of the centroid (or center of mass) of the row-pattern P" that represents the rows in column-space Sf . Similarly, the vector of row-means m defines the coordinates of the center of mass of the column-pattern that represents the columns in row-space S". If the column-means are zero, then the centroid will coincide with the origin of SP and the data are said to be column-centered. If both row- and column-means are zero then the centroids are coincident with the origin of both 5" and S . In this case, the data are double-centered (i.e. centered with respect to both rows and columns). In this chapter we assume that all points possess unit mass (or weight), although one can extend the definitions to variable masses as is explained in Chapter 32. 

These weights depend on several characteristics of the data. To understand which ones, let us first consider the univariate case (Fig. 33.7). Two classes, K and L, have to be distinguished using a single variable, Jt,. It is clear that the discrimination will be better when the distance between and (i.e. the mean values, or centroids, of 3 , for classes K and L) is large and the width of the distributions is small or, in other words, when the ratio of the squared difference between means to the variance of the distributions is large. Analytical chemists would be tempted to say that the resolution should be as large as possible. 

The centroids represent the positions of Gauss kernels and are in this example positioned in the same place as the objects in the input space. The width factors do not change during training. The weights of each kernel function are obtained by training. 

As already stated, the mass spectrum is a two-dimensional graph that reports the m/z ratio of ions (abscissa) and their relative intensity (ordinate). The most abundant ions are assigned as 100%. A mass spectrum can be displayed as peak profiles or as bar graphs corresponding to the peak centroids, i.e. the weighted centre of mass of the peak, or as a table (Figure 2.16). The most abundant ions in a mass spectrum constitute the base peak whose intensity is assumed equal to 100%. 

The simplest inverse model consists in finding liquid and solid phase proportions assuming a melt and source composition. This case is depicted in Figure 9.1 and may be modeled quantitatively with no extra assumption. Equation (9.2.1) expresses the fact that, in the m-dimensional composition space, the source composition must be the centroid of melt and residual mineral compositions, each being weighted by the... 

Autoscaled data have a mean of zero and a variance (or standard deviation) of one, thereby giving all variables an equal statistical weight. Autoscaling shifts the centroid... 

So, the cluster centroids are weighted averages of all observations, with weights based on the membership coefficients of all observations to the corresponding cluster. When using only memberships of 0 and 1, this algorithm reduces to /.--means. 

The unknown cubic and quartic terms, which are small, enter with a small weight and may safely be neglected in this case. Even the second term in this series amounts to less than 1% of the leading term (sometimes referred to as the r-centroid approximation), except at small separations where R is comparable to the collision diameter ( 5.71 bohr) and corrections amount to +2%. 

Figure 5. Experimental s-weighted intensity function (a) compared with calculated si(s) curves for (b) random chain according to conditional probabilities derived from the statistical weights of Abe, Jernigan, and Flory ( i) with near-neighbor dependence of rotation angles, and (c) an assemblage of randomly packed spheres with average segments placed at their centroids (see text)...

The element side surfaces are formed by lines that connect the centroid of the triangular side and the midpoint of the edge. Kim s definition of the control volume fill factors are the same as described in the previous section. Once the velocity field within a partially filled mold has been solved for, the melt front is advanced by updating the nodal fill factors. To test their simulation, Turng and Kim compared it to mold filling experiments done with the optical lenses shown in Fig. 9.34. The outside diameter of each lens was 96.19 mm and the height of the lens at the center was 19.87 mm. The thickest part of the lens was 10.50 mm at the outer rim of the lens. The thickness of the lens at the center was 6 mm. The lens was molded of a PMMA and the weight of each lens was 69.8 g. 

The iterative adjustment of weight vectors is similar to the iterative refinement of k-means clustering to derive cluster centroids. The main difference is that adjustment affects neighboring weight vectors at the same time. Kohonen mapping requires O(Nmn) time and 0(N) space, where m is the number of cycles and n the number of neurons. 

Figure 13, Thermoluminescence-dated bronzes and Hsien, with centroids data normalized by weight to 100%...

It is the centroid of the energies of the states /) of the manifold a, where the weights are the spectroscopic factors. 

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