In this step, the numerical values of the elements of the
transformation-matrix 
of the intermediate solution will be changed
continuously, until specific criteria are reached. However, not all
numerical values of a given precision will be entered as elements of the
transformation-matrix (this would need too much time, even if only values
of three digits are entered). The iterative change or adjustment of the
elements of the transformation-matrix is restricted to the strictly
neccessary. Positive and negative increments will be added alternately to
all non-diagonal elements of the transformation-matrix. Therefore, the
procedure is called
Transformation-matrix-Search and
Identification
(Trasid). After every iterative change of the transformation-matrix, the
solution will be checked with regard to the maximisation of the hyperplane
count. The orientation on the main loadings will not be lost, because the
iterative change will be performed only within a certain range around the
elements of the transformation-matrix.
Two ways of computing the hyperplane count are performed: the count for
absolute loadings 
0.10 (or another value, if intended), which are
divided by the square root of the communality (h) and the count for
absolute loadings 0.10 (without division by h). The priority of
maximisation is given to the hyperplane count based on the absolute
loadings divided by h. A priority for the absolute loadings without
division by h may produce solutions, which depend strongly on unreliable
variables, because low reliability may lead to low communality. Low
communality means that the loadings of the variables will be low, so that
they will more likely fall in the hyperplane and thereby may influence the
rotation. In addition to the argument that unreliable variables should not
have a strong influence on the rotation, it is not clear why a variable
with low communality, i.e. which is not really represented by one factor
of a solution, may influence the rotation of the factors. So, the simple
way of computing the hyperplane count, as proposed by Cattell (1952), must
be criticised: when a solution has many variables with low communalities,
the simple sum of the absolute loadings 0.10 is not very informative.
With very similar arguments Bargmann (1955) used the absolute loadings
divided by h for his test of significance of simple structure.
However, it is possible to maximise the hyperplane count without division by h within the range of solutions, which is determined by the hyperplane count with division by h. In this case, the hyperplane count with division by h has the priority of maximisation. When a maximisation of the hyperplane count without division by h is not in conflict with the maximisation of hyperplane count with division by h, there is no reason to avoid this maximisation, because in this case the solution which is only based on the variables with high communalities is not very different from the solution which includes also variables with low communalities in the maximisation.
The combined criterion, which realises the abovementioned priority, and
which will be maximised in the Trasid procedure, can be formalised as
follows: For n variables and m factors a matrix B is defined, with the
elements 
, according to the first condition for the hyperplane count
(
are the factor loadings):
Depending on sample size, other values for the hyperplane count can be
chosen (see above), but in order to be compatible with most of the previous
studies a value of 0.10 will be chosen for the cut-off here. A second
matrix C is defined, with the elements 
, according to the second,
simple condition for the hyperplane count:
Then, the following criterion will be maximised:
Dividing the simple hyperplane count by the total number of loadings in
the matrix (
) ensures that the increase in the latter can never be
greater than the increase of one loading falling in the hyperplane
according to the first hyperplane count. So, when one loading falls
into the hyperplane according to the first criterion, no increase in
the second, simple hyperplane count can be greater or equal (remembering
that never all absolute loadings of a solution will be 0.10
or even 0.20).
Thus, the present combined criterion realises the priority of the hyperplane
count with division of the absolute loadings by h over the simple hyperplane
count.
The criterion in formula 6 is maximised according to the iterative change
of elements of the intermediate transformation-matrix . Because the
matrix
is changed only in a range or width (W, see below) around the values of
, this criterion is not maximised over the whole matrix.
Besides the fact, that in a first step, a rotation for the orientation on the main loadings is performed, the present combined criterion (formula 6), which takes into account division by h, is the main difference between Trasid and other methods which maximise the hyperplane count as Maxplane-, PPFP-, and Hyball-rotation. The above-mentioned problems, which may occur, when only the hyperplane count for the absolute loadings without division by h is maximised, are demonstrated with an example in a further section (see below).
Because the hyperplane count is a discrete criterion, and because there is no gradient for maximising it, micro-local optima may easily occur (Katz & Rohlf, 1974). In PPFP-rotation Katz and Rohlf (1974, 1975) circumvent the problem using an exponential function of the loadings, which is continuous. The disadvantage of this method is that it is not clear how far the hyperplane count is still directly maximised.
That is why, as in Cattell and Muerle (1966), a trial-and-error maximisation of the hyperplane count will be performed with Trasid, however with the difference that in the cases, in which the iterative procedure cannot maximise the hyperplane count (i.e. the combined criterion), a maximisation of a continuous parameter will be performed. In this case, the parameter is called continuous, because with every change in the transformation-matrix and the factor pattern a change of the parameter will occur. The hyperplane count will only change, when absolute loadings fall into the cut-off value, whereas the continuous parameter will be sensitive to all absolute loadings above the cut-off value.
The parameter used here is the mean difference between the absolute loadings (divided by h) and the cut-off value for the hyperplanes (here 0.10). When the mean difference becomes smaller, the loadings are ``closer'' to the hyperplanes and the probability increases for the variables to fall into the hyperplanes in the iteration process. The difference between the absolute loadings and the cut-off value makes sense only when the absolute loadings are greater than the cut-off value, because a zero loading can and should stay at zero and should not be maximised in order to come closer to the cut-off value. So, the mean difference is only computed for the absolute loadings greater then the cut-off value.
Instead of minimising the mean difference between the absolute loadings
divided by h and the cut-off value, the mean of the reciprocal value of
the differences is maximised. The hyperbolic function yields extreme values
for very low differences, so that the mean value is influenced mostly by
variables with low differences. Because the parameter corresponds to the
mean proximity of the loadings to the cut-off value, it will be called P.
The parameter will be computed over the whole factor matrix as for the
hyperplane count. We define a matrix D of differences with the following
elements 
, according to the following conditions:
If the absolute loadings divided by h are greater than the cut-off value,
the matrix contains 
,
otherwise it contains zero-elements.
Then the sum of is divided by the number of non-zero elements,
i.e. the number n of loadings in the matrix (
) minus the
number of zero elements, i.e. the sum of (see formula 4).
Maximising P will tend to increase the hyperplane count, also if this is not guarenteed. Of course, other functions which maximise the hyperplane count are possible.
Maximising of the proximity P has the lowest priority, i.e. when the combined hyperplane count can only be maximised at the cost of minimising P, P will be minimised and the combined hyperplane count maximised. Maximising of P supports the maximisation of the hyperplane count, because the lower loadings are pushed in the direction of the hyperplanes (e.g. a loading of .20 will become .15 when P is maximised, thus it will be nearer to the cut-off value), so that the trial-and-error algorithm will have a greater chance to ``catch'' this loading into the hyperplane. The maximisation of P thus increases the propability for maximising the combined hyperplane count by means of the trial-and-error-algorithm.
The elements of the transformation-matrix will be changed (i.e. increments will be added or subtracted to all non-diagonal elements of the matrix, see below), until the above-mentioned parameters are maximal. The main problem is to avoid local optima. Of course, local optima can be considered local, only if an even better solution has been found. Because local optima cannot be excluded theoretically or mathematically, a complex algorithm for the iterative change of the transformation-matrix is proposed. Several different algorithms for the maximisation would be possible. The present algorithm is a first attempt to reach a very high hyperplane count (see below) within a reasonable amount of time. Different algorithms and different parameters within the present algorithm have been checked, and did not produce higher hyperplane counts.
Two parameters of the algorithm are varied systematically: (1) the amount of
change C of the increments (see formula 9, below), which are successively
added to the elements of the transformation-matrix and (2) the maximal width
W of the increments, i.e. the maximal value, which is added to the elements.
The increments are added alternately to the transformation-matrix of
the intermediate solution, which is the result of the rotation for
orientation of the factors on the main loadings and to the optimised
transformation-matrix 
, which is the transformation-matrix of the until
then best solution. The fact that the increments are added to and
To ensures that the algorithm starts from two different, but sensible
matrices (for more on the influence of starting positions on the rotation,
see Rozeboom, 1992).
The increments, which are added to the elements of the transformation
matrices, are computed according to formula 9. For C, values of 0.002,
0.003, 0.004, 0.005 and 0.006 are entered successively in formula 9. Even
smaller values could be entered for C, but the algorithm would run much
more slowly. In the empirical tests of the algorithm smaller values for C
did not improve the simple structure of the solutions. The widths W of the
increments are 0.005, 0.010, 0.015, 0.020...0.150, i.e. the iterative change
of the transformation-matrix is performed for all widths between 0.005 and
0.015 in steps of 0.005. Of course, even greater values can be chosen for
the maximal width, e.g. 0.20 may sometimes be necessary in order to get
the optimal solution. But the values should not be too big (e.g. > 0.30)
in order to avoid collapsing of the factors and in order to maintain the
previously established orientation on the main loadings. For an empirical
illustration of the maximisation of the hyperplane count depending on W,
see below. The increments 
are computed on the
basis of the previous
increments i. The starting value is an increment of 0.0001.
Figure 2 contains the recursive sequence of i, which is computed according
to formula 9 with a change C of 0.002 and 0.006 and a width of W of 0.10.
Note that the increments are positive and negative, so that positive and
negative increments will be added to the elements of the intermediate
transformation-matrix and the until then optimised matrix .
Figure 2: Sequence of the increment i according to formula 9
When the absolute value of the increment i becomes equal to or greater than W, the iteration process stops (as for the 17th iteration with C = 0.006 in Figure 2). For a given W, there are many more iterations when C is small. Because the transformation-matrix is normalised, the exact values of the increment will be slightly smaller.
An example for the meaning of W: when W is 0.02, (positive and negative)
increments will be added to every non-diagonal element of the
transformation-matrices and , until the absolute size of the
increments becomes equal or greater than W. The change C of the increments
will be 0.002, until the increment i becomes equal to or greater than the
W of 0.02. Then, the same iteration process starts with a value of 0.003
for C until the increment reaches W. This process continues until C
becomes 0.006.
The iterative change of the transformation-matrix can be described as
follows: the first change value C will be 0.002. According to formula 9
the first increment will be -0.0021 for the first non-diagonal element
in the first column and second row (
) of the transformation-matrix of
the intermediate solution (). The new matrix will be called 
.
i will be a bit smaller, because will be normalised.
The width W will be 0.005 in this first step.
is the matrix to compute the factor structure from the orthogonal factor
matrix. But the simple structure should be maximised in the factor pattern.
So, the factor pattern will be computed according to formula 10.
is the factor pattern, A is the orthogonal starting solution
and 
is the inverse of the transposed
(Formula 7 for computation of the factor
pattern is according to Harman, 1976, p. 268, formula 12.21). When the
factor pattern is computed, it will be checked, if the criterion for the
hyperplane count (formula 6) has been maximised. If the criterion has been
maximised, the new transformation-matrix will be saved as optimised
transformation-matrix , if not, two cases can be distinguished: (1)
if the hyperplane count according to formula 6 has been reduced, will
not be saved. (2) If the hyperplane count according to formula 6 has not
changed, it will be checked, if the proximity parameter P (formula 8) has
increased. If P has increased, again will be saved as optimised matrix
, if it has not increased, will not be saved.
After has been computed for the first time, all subsequent iterations
will be performed alternately starting from and from the intermediate
solution . The next change of the element will be performed
with an
increment of 0.0041. The next increment will be -0.0061, so that the
absolute value of i is more than the W of 0.005. Only this increment will
be added to the element of and .
Then, the iterative procedure
will be performed with a C of 0.002 for the next non-diagonal elements
(t31) of and . When the procedure has been performed for all
non-diagonal elements of and a new cycle begins with a change C
of 0.003. After the last cycle with C=0.006, a new iteration of the whole
procedure starts with a width W=0.01. The next starts with W=0.015, up
to the maximal value of W (here 0.15).
The optimised factor pattern (according to the above-mentioned criteria)
and transformation-matrix will be saved.