Joint plots

It is useful to go back to two-way visualization in principal component analysis to find what really is seen in a plot. A score plot for a two-way PCA model has an orthonormal basis, because the loadings are orthonormal. This can be compared to projecting all points in multidimensional space on a movie screen using a strong light source at a large distance. What is seen in this projection is true Euclidean distance in the reduced space, if both 

Multi-way Analysis With Applications in the Chemical Sciences 

When making biplots (overlayed scatterplots), a number of matters have to be considered (ignoring scaling of the variables before the analysis)  

The relative scaling of the different components, e.g when making a scatterplot of loadings with a large and a small eigenvalue. 

The relative scaling of the scores and loadings. Do variables and objects use the same scale or does each axis have two scales  

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Figure 8.35. The Tucker3 decomposition. For the joint plot, one of the components for the B-loadings is selected and the corresponding plane Gq of the core is extracted (see dashed lines).

This equation is also given by Gemperline et ol. [1992], This article also contains a good application of Tucker3 analysis on trace element data and joint plots. 

Figure 8.37. The joint plot of the blue crab disease data for the first plane of the core array. The plot shows individuals in three groups sitel (asterisk), site2 healthy (triangle) and site2 diseased (hexagon) together with some selected elements (asterisk with an element symbol) that are related to the separation.

Interpretation of the Tucker3 core is very important and sometimes core rotations are needed. Some authors rotate in order to simplify the core, while others see the need for a varimax rotation of the loadings. These rotations are very subjective and require extensive background knowledge. The loadings are used in line or scatter plots and allow the detection of outliers, groupings and trends. Joint plots are used frequently. 

Figure 40 is a joint plot of ApAiR+ and ApATa (using data from Refs. 278, 284-287) against AjG°(40) and AfG°(41). Squares denote ApAia/ArG°(41) and open circles denote ApAiR+/AfG°(40). They define a straight line of very good statistical quality... 

In Fig. 20, the fatigue curves for both riveted and Redux-bonded joints — plotting cyclic load against number of cycles to failure — are shown. It is clear that all of the bonded specimens failed outside the bonded area. Further, it can be seen that the bonded joint can support about 6-7 times the fatigue load for any given cycles to failure value or, to look at it in another way, at any given cyclic load, the bonded joint can be seen to withstand many orders of magnitude more cycles. 

Fio. 10. Original and corrected solutions of Goland and Reissner for the maximum transverse tensile stresses in the adhesive layer of a single-lap joint plotted against Young s modulus of the adhesive (from Adams and Peppiatt,... 

In Fig. 13.10, the fatigue curves for both riveted and Redux-bonded joints—plotting cychc load... 

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