Weighting coefficient

Each column of S represents a row-principal component of X and can be interpreted as a linear combination of the columns of X using the elements of V as weighting coefficients ... 

When all weight coefficients w and are constant then we have the special case 1. 1... 

In this respect, the weight coefficients are proportional to the column-sums. Distances of Chi-square form the basis of correspondence factor analysis (CFA) which is discussed in Chapter 32. 

Table of double-closed data Z, with weighted means, weighted sums of squares and weight coefficients added in the margins, from Table 32.4... 

Weight coefficients can be used to define weighted measures of location and spread. In particular, we define the weighted mean m of a vector x by means of ... 

The usual expressions for the mean and the variance result when the weight coefficients are constant and defined as ... 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

In this analysis, weight coefficients for rows and for columns have been defined as constants. They could have been made proportional to the marginal sums of Table 32.10, but this would weight down the influence of the earlier years, which we wished to avoid in this application. As with CFA, this analysis yields three latent vectors which contribute respectively 89, 10 and 1% to the interaction in the data. The numerical results of this analysis are very similar to those in Table 32.11 and, therefore, are not reproduced here. The only notable discrepancies are in the precision of the representation of the early years up to 1972, which is less than in the previous application, and in the precision of the representation of the category of women chemists which is better than in the previous analysis by CFA (0.960 vs 0.770). 

In this equation, f is the molecular wave function, is an atomic wave function, and a is a weighting coefficient that gives the relative weight in the "mix" of the atomic wave functions. The summation is... 

Although the combination has been written as a sum, the difference is also an acceptable linear combination. The weighting coefficients are variables that must be determined. 

These equations are known as the secular equations, and in them the weighting coefficients a1 and a2 are the unknowns. These equations constitute a pair of linear equations that can be written in the form... 

When these relationships between the weighting coefficients are used, it is found that... 

Having shown that the weighting coefficient (A) of the term giving the contribution of an ionic structure to the molecular wave function is related to the dipole moment of the molecule, it is logical to expect that equations could be developed that relate the ionic character of a bond to the electronegativities of the atoms. Two such equations that give the percent ionic character of the bond in terms of the electronegativities of the atoms are... 

By making use of the weighting coefficients and the populations of the orbitals, the electron density at each atom can be calculated as before. For the allyl radical,... 

Implicit in this notation, the wavefunction is written as a function of the electronic coordinates, xi,X2,..., xn, and the bracket indicates integration over these coordinates. In this equation, the weighting coefficients, represent the probability of observing the system in the state associated with and thus must satisfy the constraints... 

A g-density, pg, is N-representable if and only if there exists some set of antisymmetric wavefunctions,, , and weighting coefficients, p,, for which Eq. (2) holds. 

According to spectral-kinetic parameters, the optimal conditions of luminescence excitation and detection, so called selection window (SW) parameters, were calculated in the following way. At optimal for the useful component excitation, the liuninescence spectra, decay time and intensity were determined for this mineral and for the host rock. After that, on the personal computer was calculated the proportion between useful and background signals for the full spectral region for each 50 ns after laser impulse. For calculation the spectral band was simulated by the normal distribution and the decay curve by the mono-exponential function. The useful intensity was multiplied by the weight coefficient, which corresponds to the concentration at which this component must be detected. 

The first layer transmits the value of the predictors to the second— hidden—layer. All the neurons of the input layer are connected to the / neurons of the second layer by means of weight coefficients, meaning that the / elements of the hidden layer receive, as information, a weighted sum S of the values from the input layer. They transform the information received (S) by means of a suitable transfer function, frequently a sigmoid. 

In this example the sum of the b weights is 1. Therefore, we may say that b is normalized. We see that each element of row a is a linear combination of a corresponding subset of the g elements. Sequential values of a are computed by sliding the set of b factors along row g. It is usually convenient to write each a value opposite the largest weighting coefficient in row b ... 

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