Chapter Three

INTRA-CORRELATIONAL ANALYSIS

A correlational matrix forms an uneven landscape of values which can be topographically mapped. It is the pattern of this landscape which yields insight into the functional organization of the data. This patterning is carried over into second and subsequent order cross-correlational matrices.

Intracorrelational analysis proceeds on the basis that a correlational matrix represents a complex, meaningful distribution of data that to some unknown extent are interlinked in inter-dependent relationships. When correlational matrices are organized, they may then be analyzed as sub clusters of points, and even subdivided into smaller matrices which represent these clusterings. The sub-matrices can then be analytically compared to one another, and the derivative intracorrelations yield a measure of similarity/difference. At the same time, a correlational table allows us to intracorrelate specific dimensions with one another, and even to generate an infinite number of subsequent correlational matrices which are based upon this retabulation of the previous matrix.

The balance of a matrix consists of analyzing the distribution of values across the dimensions as these are grouped. Before the matrix can be analyzed in these ways, it is necessary to group and organize the data of the original tables in a meaningful order that is congruent with the nature of the data.

Whatever the organization of the original data table, the resulting matrix can be tested for apparent balance and symmetry by "splitting" the table in half and creating three general areas of the table--resulting in two tails and a central region. A matrix is split in half by dividing the number of dimensions (X) by 2. If the number of dimensions is odd numbered, then the dimension is split with the middle value overlapping.

This frequency distribution table yields two equal sized correlation matrices, the first one below based upon the correlation by colors (columns) and the second by correlation of rank order (rows) each of which have been subdivided into four quadrants:

These correlation matrices will be used in all examples for the first sections of this work, until another independent sample of another 12 color task is introduced for comparative purposes.

If we take the correlation table of the first generation matrix, we can generate subsequent generation matrices by correlating the dimensions of the correlation table, a process which systematically measures the degree to which any two dimensions of the table are related to one another, and the degree to which these same two dimensions are similarly related to all the other dimensions. This process has the effect of "stretching" out the values of the matrix such that high values (either positive or negative) increase, while low values tend to drift in one direction or the other. It appears that this drift of values follows a regular curvilinear regression which cannot be predicted by the first generation values--positive values may become negative, and then positive again. The structure of the final matrix is somewhat different from that of the first matrix. It is my opinion that with each successive generation, the range of dispersion of the original values is lost to a more linear form of relationship between successive generations.

This process was repeated four times, for a total of five generations of matrix. What is evident in this process is that at each successive generation with the descriptive values of each subsequent generation matrix compared, as well as the intercorrelation of the matrices themselves.

Second Generation 12 color matrix

Third Generation 12 Color Matrix

Fourth Generation 12 Color Matrix

Second Generation 12 Rank matrix

Third Generation 12 Rank Matrix

Fourth Generation 12-Rank Matrix

Besides generational recorrelation, there are successive correlational levels of analysis. In general, the original data table represents the first level. At each successive level, the dimensions relevant at the previous level become replaced by a more restrictive set of dimensions at the higher level. At each successive level, therefore, it appears that the resulting correlational values do not speak directly about the original frequency data, but increasingly more about the internal relationships of the dimensions of the matrix itself. Generational recorrelation at each successive level intensifies these sets of relationships, and creates the foundation for the next higher level.

The table below represents the successive generations of intracorrelations at the third level, one remove from the color matrices above.

Five generations of intracorrelational matrices were run upon this table, producing the following table at the fourth level of intracorrelation:

Note, besides locating these tables, which are presumably in my spreadsheets, it is not clear what is the difference between "level" and generational order in a matrix.

What appears to occur by the fifth generation is a basic transformation of the successive points, such that the structure of the matrix which was apparently strongly monovalent in the first generation, eventually becomes bivalent by the fifth generation, with strong and apparently stable values.

In short, it can be seen that when a correlation is inverted along its diagonal axis and then recorrelated, and this operation is repeated successively, then a basic pattern emerges of the data, such that the "tails" of the correlation matrix tend to remain stable and monovalent, while the region of the matrix which constitutes the interrelationships of the tail tends to drift to a negative pole opposite of the tails. This is an expected outcome of such recorrelation. The second expected outcome is that with each successive recorrelation, the absolute average value will increase, and eventually approach one, while the relative average value will tend to remain the same as in the first correlation.

Successive recorrelation results in a stabilization of the matrix in terms of sign. It is evident that any first generation matrix may contain relatively low correlational values that are ambivalent in terms of sign. Attempts to analyze and represent these matrixes without adjusting the sign of the matrix, will result in difficulty in determining some points or relationships.

The preceding matrix represents the third recorrelation of the relationships across the five dimensions above, by which trial the matrix reached unity of value. The assymmetry of sign of the matrix is readily apparent.

Certain patterns are apparently consistent in recorrelation of matrices which suggests that within a significantly patterned matrix, there may be one or more significant clusters representing distinct relations between dimensions, and which do not radically change during recorrelation. These clusters of values serve as "anchor" points about which the curvilinear movement or drift of other, less significant values, can be mapped in the manner of an orthogonal rotation. What this seems to signify is that the anchor point represents a linear dimension of largely unknown magnitude or directionality, about which the other less determinate points will vary and may be mapped in their movement over successive trials. The less determinate points will therefore move or drift in a less precise manner, and can even switch direction. What recorrelation appears to accomplish is the partial mapping of the space or area about the anchor point about which this movement is most likely to occur until it reaches some plateau--it is not the actual movement of points or variation, merely the suggested possible patterning or structure for such movement to occur.

This patterning can be imagined as if it were a sidewise projection in time--the shape of an unknown curvilinear surface about one or more central axis of rotation. A bipolar or multipolar projection would suggest a much more complex pattern of oscillation within and about the central axis, determined in part by the more or less fixed relationship of one axis to the others. The axii themselves may drift in relationship about one another.

The effect of performing multiple recorrelation appears to be the "polarization" of values about a negative or positive axis--this polarization has significance in the sense that highly correlated values bearing an actual linear relationship should shift in one or another direction--values which bear no actual relationship are likely to drift in one direction or the other, or both, but should not "polarize" as effectively.

A measure of the degree of "polarization" of subsequent recorrelated matrices might be given by the average relative difference of values between each subsequent value, tracing the overall movement. The following matrix introduces a new measure, that of the average recorrelational stability of a given set of relationships, given by the formula:

In this matrix, the high values represent the more stable set of relationships, and low correlations represent those which are most prone towards movement or drift. In this graph there can be seen to be one main cluster of stable points, comprising the relationship between blue and (violet, pink, purple, orange) and between red and (violet, pink and purple, orange) There appear to be another cluster, that comprising the interrelationships between Grey, Brown, White, and Black, excluding the Black/Brown relation, as well as two possibly "negative" clusters (Red (Green, Yellow, Grey, Brown, White, Black)) and (Blue(Green, Yellow, Grey, Brown, White, Black)).

This interpretation of the patterning of the matrix offers insight into the cluster analysis of the distribution of values within a matrix. For instance, recognizing the relationships that designate the anchor points in the original matrix or in the original data-table might allow us to reorganize or regroup the values of the orginal tables in a more consistent manner.