Shepard diagram

The analyst should check the Shepard diagram that represents a step line so-called D-hat values. If all reproduced distances fall onto the step-line, then the rank ordering of distances (or similarities) would be perfectly reproduced by the dimensional model, while deviations from the step-line mean lack of fit. The interpretation of the dimensions usually represents the final step of this multivariate procedure. As in factor analysis, the final orientation of axes in the plane (or space) is mostly the result of a subjective decision by the researcher since the distances between objects remain invariable regardless of the type of the rotation. However, it must be remembered that MDS and FA are different methods. FA requires that the underlying data be distributed as multivariate normal, whereas MDS does not impose such a restriction. MDS often yields more interpretable solutions than FA because the latter tends to extract more factors. MDS can be applied to any kind of distances or similarities (those described in cluster analysis), whereas FA requires firstly the computation of the correlation matrix. Figure 7.3 shows the results of applying MDS to the samples described in the CA and FA sections (7.3.1 and 7.3.2). 

Such dimensional reduction may be achieved by Multidimensional Scaling (MDS) techniques. MDS projects the n-dimensional distance in a lower-dimensional space (2- or 3-dimensions) under the constraint of maximizing the retention of the stmcture of the inter-molecular distance matrix. The representation of the n-dimensional space is optimized in the lower-dimensional projection by minimizing what is known as stress . The smaller the stress the better the projection up to the (generally unattainable) limit of zero which is a projection that preserves the distance matrix completely. The quality of the projection can be gleaned from what is known as a Shepard diagram . 

The Shepard diagram is a scatter plot in which the dissimilarities between the molecules, measured as the distances in the fuU n-dimensional space, are compared with the corresponding distances in the projected 2-dimensional space. A large spread is an indicator of a poor MDS projection and vice versa up to the unreachable extreme where all points fall on one line which indicates a perfect MDS projection. 

Using the data in Table 3.4 and the MDS algorithm in the software package XLSTAT M the Shepard diagram displayed in Fig. 3.6 is obtained. One can glean... 

A variation of the Shepard Cane incorporates both electrodes on a single rod. A circuit diagram of this type of instrument is shown in Fig. 10.51. The single rod has a calibration constant, determined by the area of the positive electrode. At balance the resistivity is given by the resistance multiplied by the rod constant. This instrument is portable and simple to operate, but it cannot be recommended when accurate results are required. 

Figure 5.2 Sectional diagram of Potter precision laboratory spray tower. (From Potter, C. and Way, M.J., Precision spraying, in Methods of Testing Chemicals on Insects, Shepard, H.H., Ed., Vol. 1, Burgess Publishing Company, Minneapolis, 1958, p. 154.)...

Fig. 1.7 Ternary diagram of sand-silt-clay grain-size distribution showing principal names for siliciclastic, terrigeneous sediments (from Shepard 1954).

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