Jolicoeur (1963) offered a solution to this problem: perform an analysis of the main components (PCA) of the covariance matrix of transformed log measurements and use the first main component resulting as an estimate of the allometric line. As the first main component (PC1) is this direction in the multidimensional space, which represents the largest possible share of the overall variance, the sum of variances in all directions is perpendicular to it. This use of the main components follows the initial proposal of the method (not yet under this name) as a means of obtaining a most appropriate line for multivariate data according to a smaller square criterion (Pearson 1901). PCA is one of the most fundamental and widespread methods in multivariate statistics (Jolliffe 2002) and is often used for the study of multivariate allometry (see Klingenberg 1996b). The use of PC1 to estimate a most appropriate line is very well suited to multivariate allometrics, as all variables are treated in the same way. Figure 2 shows how the four metrics analyzed for the generated datasets are cut based on the additive and multiplier distortion imposed and the initial correlation between X and Y. A first note about the plots in columns a) and b) in Figure 2 is that there is an intersection point of the iso lines for all metrics. It is assumed that metrics represent a correct decrease in compliance when there is an increased systematic disturbance for all types of correlations. Crossing these lines of iso-smene interference means that this hypothesis is violated. In the case of a slight b shift or a resizing with m, abnormal behavior can also be observed at moderate correlation values (p.B. r between 0.5 and 0.7). For them, all lines intersect only at . This may be considered less uncomfortable, as negative index values could be used to assess how many records match in size, although they do not coincide in the warning signs.
However, this adds ambiguity to the interpretation of the index, which is not desirable. A second point is that Ji-Gallos AC is not negatively limited by zero, as designed. This occurs even if no systematic disturbance is added to the data (i.e. even for a fairly high positive correlation). These are easy conditions to expect for most dataset comparisons, which indicates how Ji-Gallos`s KT should be avoided. A completely different approach was proposed by Mosimann (1970), who proposed explicit geometric definitions of size and shape and developed an analytical framework based on these definitions. Size indicates the total size or scale of an object. Size is a scalar property that can be quantified by a single number (but there may be different ways to calculate the size of a particular object, which leads to different values). The shape is conceptually different from the size: the shapes of two objects are identical when they are geometrically similar, regardless of the size of the objects. For data consisting of length measurements, this means that all measurements in two objects of identical shapes differ only by a constant factor, which relates to the relative size of the objects. The analysis of measurement ranges in relation to the total size is therefore useful for quantifying the shape.
In other words, the shape revolves around the proportions of objects.
