Chapter Two

DESCRIPTIVE CROSS-CORRELATIONAL ANALYSIS

Descriptive analysis of a correlational matrix involves calculation of a number of descriptive dimensions and parameters intrinsic to the correlational table and that allow us to make certain inferences in relation to the matrix--in essence the correlations of the matrix can be treated like a complex set of sample variates. The description is not about the topical items represented by the statistics, but about the structure of the relationships between these things.

In general, different descriptors may be utilized in the depiction and evaluation of such a matrix--for instance, the absolute mean that averages the absolute values of the individual correlations might yield a sense of the overall strength of the correlations of the table, and may be a good quick indicator of the general effectiveness of the particular table in revealing any structural patterning in the data represented.

The descriptive characteristics of a correlation matrix are useful as gross estimators of certain qualities of the matrix that can then be used in simple comparison with the estimators of other matrices. Because these characteristics are "absolute" relative to the matrix, they allow the gross comparison of matrices that can be otherwise of different sizes and dimensions.

Descriptive analysis makes available for other kinds of analysis and representation a number of discrete values which are associated with a given correlational matrix. The relative average gives the average of scores which are not adjusted for sign. While the absolute average is useful for determining the overall significance of the matrix, the relative average is useful for determining the relative "balance" of the matrix--whether it's correlational curve is skewed positively or negatively. This skewing would have important implications for understanding the overall meaning of the matrix--sign assymmetry can be evaluated and even analyzed statistically using chi-square.

The correlation matrix can be flipped along its diagonal axis and the rows or columns totalled and averaged, using either the absolute or relative forms of the average. The rows and columns can then be rearranged in descending rank order according to the relative or absolute average score for each column. Other statistical characteristics of each column can then be deduced, such as standard deviation, median, mode, skewness, probability, etc.

The relative average score of each row or column can be used to "balance" the matrix to its true proportions--the absolute mean score can be used to order the entire matrix from most to least significant correlations.

Frequency histograms can be used to determine the percentage of given values that are significant at a given level of analysis--all negative values, all zero values, all postive values, all absolute values below or above a certain cut-off, etc. Each correlational matrix comprises a set of incomplete but possible correlational sub-matrices. Each possible subset yields a different topography of relative values and meaning.

A correlational matrix may be relatively or even perfectly isomorphic in regard to sign, either positively or negative--a table with near uniform positive values suggests a reflexive identity of the relationships with the reality embodied by the matrix. A table that is uniformly negative in value may be fundamentally different in quality than one which is uniformly positive in value. The uniformity of the table can be analyzed as the function of the degree of variance of values either in sign or in strength--relative uniformity suggests isomorphism of sign, absolute uniformity suggests isomorphism of magnitude or strength of correlation. A correlation table may have relative uniformity of sign but lack uniformity of strength, and vice versa.

Correlational matrices, especially large ones based upon well organized and sorted data tables, can reveal natural clustering of correlational scores or other patterning of correlation within the larger matrix that yield important insight into the pattern and functional organization of the data and their dimensions. High correlations may cluster in small islands in a sea of zeros or low correlations, or disproportionately high correlations may cluster in certain vertices. The patterning of natural clustering of a correlational matrix can be easily checked by "squaring" the table and then creating a grid score table.

It is important at the offset to define the procedural basis for correlational analysis. A correlational matrix of the first order is derived directly from a primary data table of RxC size. The result of the correlation of such a table yields a first order correlational matrix that is of X size. It is this table of correlational values which is then subsequently reanalyzed in different ways, either by systematic comparison with other similar correlational matrices derived from parallel sets of data, or from its internal analysis and possibly even subsequent order recorrelation. As a result of this kind of comparative and descriptive analysis, second and even third or fourth order matrices can be subsequently derived from a range of basic data.

The dimensions of the Matrix is given by the number of rows or columns, which is always the same, and is designated as X.

Matrix Size

Because a correlation table is always squared (R_x X C_x), the size of a first order correlation matrix M_x is given by the formula ((X²)) - (X))/ 2. This is the number of significant cells contained within the matrix without duplication and without a dimension of the matrix being correlated with itself. This number is important in the averaging of the correlation to find the gross descriptive characteristics of the table.

The size of a matrix is also given by the value of (X - 1) d?

The relative value of a first order correlation matrix (R_x), is simply the total sum of all the significant cells of a correlation matrix, not adjusted for sign.

The relative average correlation of a correlation matrix is the total average of the sum of the correlations of the individual cells, with the sign attached, and minus the middle row in which the same dimensions are correlated with themselves, divided by the matrix size, M_x. or:

R_x/M_x

The relative average correlation of the 12 color matrix is -.0095, indicating a very slight negative skewing of the matrix.

The relative average correlation of the 12 rank matrix is .01857, indicating a slight positive correlation.

Neither relative average correlation is very high, indicating that both matrices are fairly well balanced in terms of sign.

The absolute cumulative value (A_x) is simply the sum of the absolute value of all the significant correlational values of the first-order correlational matrix. The absolute average correlation of a correlation matrix is the total average of the sum of the absolute values of the correlations of the individual cells--disregarding the sign--and not including the diagonal middle row of perfect self-correlations, or:

A_x/M_x

The absolute average correlation of the color matrix is .34, indicating the overall strength of the matrix as a whole.

The absolute average correlation of the 12 rank matrix is also .34, indicating that both matrices are equivalent in strength.

A uniform matrix is one which has either all positive or all negative values. Uniformity in a matrix suggests a non-random patterning of relationship, especially if the strength of the matrix is high. In an absolutely uniform matrix, the absolute value of the relative correlational average is equal to the absolute correlational average. While matrices with uniform positive values are not infrequent, and suggest a close relationship of co-occurrence between all the dimensions, a matrix which is negatively uniform is probably very rare and unique.

Uniformity as a relative value of the matrix can be given in a straight forward manner as the sum of the number of positive values (p_x) and the number of negative values (n_x), divided by the total number of values (M_x). In the case of the color correlation matrix, the uniformity value is ((30 + -36)/66) = .09, where a value of 1 is indicative of perfect uniformity.

An equivalent matrix is one in which the posi-valence is equal to the nega-valence. This is probably a rare occurrence, and exists only as an ideal by which we can measure the significance of correlational skewness.

A measure of equivalence is given as the total num?

The nega-valence is the average of the negative correlations of a matrix-- sum total of the negative correlations of a matrix divided by the number of negative values. For the color matrix it is -.37. The number of negative values is 36.

The posi-valence is the average of the positive correlations of a matrix--the sum total of the positive correlations of a matrix divided by the number of negative values. For the color matrix this value is .286, with the number of positive values being 30.

The measure of equivalence of a correlation matrix can be determined by comparing the posi-valence with the nega-valence. Though the relative correlational average will hint at the general skewness of a matrix, the actual ratio or strength of skewness to the positive or negative of a correlation matrix can only be determined by actually summing the values and dividing by the absolute correlational average.

The equivalence value is found by the formula (posi-valence + nega-valence)/ absolute average valence = ((.286 + -.37)/.34)= -.247

The question arises as to whether the skewness of a matrix according to its equivalence value might be significant at some level of confidence. Since the matrix is negatively skewed, we will ask whether negative skewing is significant according to the Z test. We must assume that the normal matrix would be "equivalent" with an equal likelihood of negative and positive values occurring--the equivalence value of such a "normal distribution" being 0. Abs(-.37) - (.34)/(square root of 2.14 X 36/66 X 30/66) = 0.168. The p value of this z score is significant past the .4364 value.

The squared correlational table represents the rotated inversion of the standard correlation matrix upon its diagonal axis. The squared correlational table for the 12 color matrix is given below:

The squared correlational table for the 12 rank matrix is as follows:

note that the dimensional correlation squared with itself down the diagonal axis of the table is always equal to 1.

The relative dimensional average correlations are the average of the sum of each of the dimensional rows or columns of the correlation table, inclusive of sign, divided by the absolute size of the Dimension.

The relative dimensional average correlations of the 12 color table is as follows:

While the relative dimensional average correlations of the 12 rank table is:

The absolute dimensional average correlations are the average of the sum of the absolute value each of the dimensional rows or columns of the correlation table, regardless of sign, divided by the Dimension.

The relative dimensional average correlations of the 12 color table is as follows:

The absolute dimensional average correlations of the 12 rank table are:

While the relative dimensional averages are gross indicators of the relative significance or strength of the correlations within each dimension, the absolute average are gross indicators of the degree of average variance of the dimension about the origin.

The relative and absolute dimensional values can be ranked and sorted by ascending or descending values, such a ranking shedding insight into the pattern of correlational strength among the different dimensions.

The following table presents the rank order of the color-correlations of descending values according to relative and absolute average values:

It is apparent that there is almost no rank order correlation between these two orders--suggesting that relative and absolute average values have very different uses and implications. The absolute average color correlation represents the strength or degree to which that color resembles or may be used to represent all the other colors of the matrix, while the relative average color correlation represents the relative balance or skewing of that color in correlation to all the other colors of the matrix. Thus, while purple is the strongest absolute value of the entire matrix, yellow is the value that is most negatively skewed in relation to all the other colors of the matrix.

Not every value within a correlational matrix need be significant or highly correlated with any other dimension. Only a certain percentage of the entire correlational matrix can be considered to be significant, but even relatively low correlations may yet have significance as part of a relationship with other dimensions. With so few dimensions of freedom (X -2), the insignificance of the p-values of the t-test of values is more a function of so few values than it is of the strength of the correlations involved.

Nevertheless, the significance relationships that do emerge have a relative value compared to all the other values within the matrix, and this topography of significance contains important structural information about the data. When looked at from this standpoint, the topography of significance of a correlation matrix is altered in profile--the critical focus is away from the table as a whole and onto those significant peaks within the matrix and the relations between these peaks.

The gross average p-value is the average of the sum of the p-values of the matrix divided by the size of the matrix. The gross average p-value of both the color and 12 rank matrix is .06 by a one-tailed t-test.

It can be seen by such descriptive analysis that a correlation matrix is inherently complex in terms of the kind of information which it comprises--this complexity renders a straight-foward interpretation of the meaning of the matrix difficult to say the least, but at the same time it makes possible more extensive analysis.

We may introduce the term of the relative complexity of a matrix, as the composite measure of a number of descriptive variables relating to that matrix--in general the greater the uniformity and the symmetry, the less complex the matrix, such that the discrete relative value of simplicity is 1/ uniformity + equivalence and the corresponding derivative value of complexity is 1- simplicity.