Sum-square splitting

Sum Square Splitting and Root Mean Square Splitting as Measures of Configurational Splitting... 

Our first example concerns an analysis of the pectrym of gaseous Ni. The experimental value for the sum square splitting, of the characterizing... 

As we know from the section on ANOVA (analysis of variance see Section 2.3) in univariate cases, where only one feature is investigated, the sum of the squares of deviations of all n measuring values from the total mean is split into a part determined by... 

In this case, conversely to the residual variance, we can propose two zero hypotheses the first is HjqI the variance of the response values determined by the change of factor Xj has the same value as the residual variance the second one is H20 the variance of the response values (when X2 factor changes) is similar to the residual variance . With these hypotheses we indirectly start the validation of two others assumptions (i) the equality of the mean values of the lines (related to Hjo), (ii) the equality of the mean values of the columns (related to H2o°)-The splitting of the total variance into parts associated to Table 5.41 follows a procedure similar to that for the analysis of the variances of a monofactor process, as previously explained. In this case, we introduce the sums of the squares Sj, S2,... 

This method is more informative, but it can be quite slow. Its major objective is to produce a weighted sum of squares surface diagram. In addition the minimum calculated WSS is estimated. The three-dimensional plot in Fig. 4 was calculated by this method. The calculation is set up by inputting the upper and lower limits of each parameter of interest. This range is split into a number of... 

This sum of squares of the deviations about the mean is then divided into two parts and SS/ sid- We will see later that each of these may be split up... 

The advantage of this high-temperature expansion is that we need not determine the individual energy levels but can use the theorem of the invariance of the trace of a matrix. For example, to find the value of e2 we square the Hamiltonian and pick out only the diagonal terms. On summing these terms over all possible values, many of the sums cancel. For example, the zero-field splitting Hamiltonian of the form... 

Table 1 Double-ionizations of the benzene molecule to singlet dication states, predicted in the standard enhanced ADC(2) approximation [5] and in the diagonal approximation described in the text. In this and subsequent tables, Term indicates the term symbol for a transition. Character the sum of the squares of the normalised transition eigenvector coefficients associated with the dominant basis configuration, and AE the small symmetry-breaking energy splitting in degenerate irreducible representations introduced by the diagonal approximation (see text)...

An increasingly popular stopping rule for variable selection involves calculating a statistic referred to as PRESS. PRESS stands for predicted residual error sum of squares. To calculate PRESS, the data must be split into at least... 

Although the complete data set consisted of 417 measured IR spectra, it covered only 18 different rockets, i.e. it contained 399 replicates. These replicates were not used in the validation of the models. Instead, leave-one-out cross-validation (Hjorth, 1994) was used to assess the quality of the models, i.e. the set of n (= 18) independent samples was split into n-1 training samples, while the n h point was reserved for model validation. The training-validation split was repeated n times until each data point had been omitted once for validation. A validation set of n predictions on the unseen data was therefore derived from all the available data and a predicted residual estimate sum of squares (PRESS) was calculated on the validation set. 

Experimental results obtained from the tests may demonstrate that manipulations with independent variables in some way affect the dependent variable. The question, however, is whether this is caused by the controlled change in the levels of the independent variables or is purely due to the experimental errors. The idea of ANOVA is to split the total variation observed into its components and then to estimate the significance of each component. This results in a relatively simple procedure whereby individual variances can be estimated using sum of squares of the differences between the measurements or sums of measurements and corresponding averages (Montgomery, 1991). 

The TSS can be split into two terms the first one accounts for the effect of the studied factor and is called Sum of Squares Between Columns, SSBC, whereas, the second one takes the experimental uncertainty into account and is regarded as the Sum of Squares Error, SSE. The related expressions are shown in Eqs. (A2) and (A3). 

The total region of integration—the square area of Fig. 4.7—is spht into two regions above and below the diagonal line. The integral on the Ihs of (4.5.4) is split into the sum... 

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