Pooled variance-covariance matrix

The use of a pooled variance-covariance matrix implies that the variance-covariance matrices for both populations are assumed to be the same. The consequences of this are discussed in Section 33.2.3. 

A simple two-dimensional example concerns the data from Table 33.1 and Fig. 33.9. The pooled variance-covariance matrix is obtained as [K K -1- L L]/(n, + 3 - 2), i.e. by first computing for each class the centred sum of squares (for the diagonal elements) and the cross-products between variables (for the other... 

Computation of the cross-product term in the pooled variance-covariance matrix for the data of Table 33. 

When all are considered equal, this means that they can be replaced by S, the pooled variance-covariance matrix, which is the case for linear discriminant analysis. The discrimination boundaries then are linear and is given by... 

Calculate the variance-covariance matrix for each group (minus the outliers) separately and hence the pooled variance-covariance matrix CAB. 

If the data arise from a single parent population, then a pooled variance-covariance matrix may be calculated from... 

Linear discriminant analysis (LDA) is a classification method that uses the distance between the incoming sample and the class centroid to classify the sample. For LDA using Mahalanobis distances, the classification metric uses the pooled variance-covariance matrix to weight the Mahalanobis distance ) between the incoming... 

If overhtting occurs, then the prediction ability will be much worse than the classihcation ability. To avoid it, it is very important that the sample size is adequate to the problem and to the technique. A general rule is that the number of objects should be more than hve times (at least, no less than three times) the number of parameters to be estimated. LDA works on a pooled variance-covariance matrix this means that the total number of objects should be at least hve times the number of variables. QDA computes a variance-covariance matrix for each category, which makes it a more powerful method than LDA, but this also means that each category should have a number of objects at least hve times higher than the number of variables. This is a good example of how the more complex, and therefore better methods, sometimes cannot be used in a safe way because their requirements do not correspond to the characterishcs of the data set. 

Values of the Group Mean Matrix, Inverse Pooled Variance-Covariance Matrix, and RMS Group Size for the Data of Table 15.1 ... 

The most obvious and straightforward use of these concepts is the application to qualitative analysis. Having generated a set of group means and an inverse pooled variance-covariance matrix for a given set of materials, it then becomes possible to determine whether unknown materials can be classified as one or another of the knowns, by calculating the distance of the unknown to each of the known... 

In particular, the pooled variance/covariance matrix S, common to aU the categories, is defined as the weighted average of the individual matrices S ... 

In eq. (33.3) and (33.4) x, and Xj are the sample mean vectors, that describe the location of the centroids in m-dimensional space and S is the pooled sample variance-covariance matrix of the training sets of the two classes. 

As stated earlier, LDA requires that the variance-covariance matrices of the classes being considered can be pooled. This is only so when these matrices can be considered to be equal, in the same way that variances can only be pooled, when they are considered equal (see Section 2.1.4.4). Equal variance-covariance means that the 95% confidence ellipsoids have an equal volume (variance) and orientation in space (covariance). Figure 33.10 illustrates situations of unequal variance or covariance. Clearly, Fig. 33.1 displays unequal variance-covariance, so that one must expect that QDA gives better classification, as is indeed the case (Fig. 33.2). When the number of objects is smaller than the number of variables m, the variance-covariance matrix is singular. Clearly, this problem is more severe for QDA (which requires m < n ) than for LDA, where the variance-covariance matrix is pooled and therefore the number of objects N is the sum of all objects... 

With the STS approach estimates of individual parameters are combined as if the set of estimates were a true Wsample from a multivariate distribution. It has been recommended as a very simple and valuable approach for pooling individual estimates of PK parameters derived from experimental PK studies (29). The advantage of the STS approach is its simplicity, but the validity of its results should not be overemphasized. However, it has been shown from simulation studies that the STS approach tends to overestimate parameter dispersion (the variance-covariance matrix) (20, 30). 

Typical values for a group mean matrix and for a pooled inverse variance-covariance matrix are illustrated in Table 15.3, in which the values have been computed from the wheat and soy data of Figure 15.1 and Figure 15.2, and which is partially listed in Table 15.1. 

This represents the ratio of the separation of the means of the two groups to the within-group variance for the groups as given by the pooled covariance matrix, S. 

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