Sum of squares total

A fourth hierarchical method that is quite popular is Ward s method [Ward 1963]. This method merges those two clusters whose fusion minimises the information toss due to the fusion. Information loss is defined in terms of a function rvhich fdr each cluster i corresponds to the total sum of squared deviations from the mean of the cluster ... 

The total sum of squares of deviation of logZ from their mean. 

Hence is the fraction of the total sum of squares (or inertia) c of the data X that is accounted for by v,. The sum of squares (or inertia) of the projections upon a certain axis is also proportional to the variance of these projections, when the mean value (or sum) of these projections is zero. In data analysis we can assign different masses (or weights) to individual points. This is the case in correspondence factor analysis which is explained in Chapter 32, but for the moment we assume that all masses are identical and equal to one. 

The first term on the right-hand side represents the total sum of squares of Y, that obviously does not depend on R. Likewise, the last term represents the total sum of squares of the transformed X-configuration, viz. XR. Since the rotation/reflection given by R does not affect the distance of an object from the origin, the total sum of squares is invariant under the orthogonal transformation R. (This also follows from tr(R" X XR) = tr(X rXRR T) = tr(X XI) = tr(X" X).) The only term then in eq. (35.2) that depends on R is tr(Y XR), which we must seek to maximize. 

PRESS is the prediction sum of squares SSTotal is the total sum of squares... 

Associated with each data point is a certain degree of freedom, which will be used to attribute more information to, say, 500 data points than to 5 data points. In particular, if N data points are used, the total sum of squares is said to possess N degrees of freedom. The predicted rates estimated from a model containing p parameters have p degrees of freedom, and the remaining N — p degrees of freedom are possessed by the residual sum of squares. 

The total sum of squares, SSj, is defined as the sum of squares of the measured responses. It may be calculated easily using matrix techniques. 

The total sum of squares has n degrees of freedom associated with it, where n is the total number of experiments in a set. 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSf (sometimes called the sum of squares due to regression, SS ) the sum of squares of residuals, SS, and the sum of squares corrected for the mean, (or the total sum of squares, SSj, if there is no Pq term in the model). 

Calculate the total sum of squares, SS, for the nine responses in Section 3.1 (see Equation 9.2). How many degrees of freedom are associated with this sum of squares ... 

Although the partitioning of the total sum of squares into a sum of squares due to the mean and a sum of squares corrected for the mean may be carried out for any data set, it is meaningful only for the treatment of models containing a / 0 term. In effect, the /30 term provides the degree of freedom necessary for offsetting the responses so the mean of the corrected responses can be equal to zero. 

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