D3.0 Specific Example of Inner Most Adaptive Loop

      The purpose of this section is to present a specific example of how the innermost adaptive loop in Figure D-2 (i.e., LOOP III), which corresponds to the 6th Transformation and Part "g" of Figure D-1, could be handled. For illustrative purposes, the situations in the 6th Transformation and Part "g" of Figure D-1, and the situations in the 4th and 5th Transformations and Parts "e" and "f" of Figure D-2 will be simplified in this discussion. Note that the 3rd, 4th, and 5th Transformations, and Parts "e" and "f" of Figure D- 1 all correspond to Loop II in Figure D-2.

      We will assume that the required classification is only twofold instead of threefold. So, assume that we have manifestations of the PHENOMENON that are either High or Low. Furthermore, assume that the classification can be accomplished using only the amplitude portion of the Fourier transform of f(). Hence, we need concern ourselves only with A() in Part "e" and in Part "f" of Figure D-1. (For the simplified case = see below.)

      A brief discussion of the simplified situation in Part "e" and Part "f" will now be given. It will be followed by a discussion of the simplified situation in the 6th Transformation and Part "g", which is the real objective of this section.

      Figure D-3 and Figure D-4 present the simplified situation for Part "e." Therefore, the objective of the adaptive process of LOOP II of Figure D-2 can be stated as follows:

• Find a dual set of parameter vectors , such that the resulting feature vector will permit better classification results in LOOP III than the previous permitted. (Note that for the simplified situation, every iteration of Loop II produces 15 feature vectors of the High PHENOMENON type and 27 feature vectors of the Low PHENOMENON type.)

      The initial values for parameter vectors and may be such that an ordered and uniform partition of the axis is executed, or the parameters of and may initially be set with a random factor involved. The initial values of and will depend upon the form of the search procedure that is established to control the operation of LOOP II. In any case, however, the axis will be partitioned into a set of windows or passbands, each with a center point, c_i, and a passband, d_i, symmetrically positioned around c_i.

      The features upon which the discriminator in LOOP III will operate are simply the integrals of A() over each passband. (It may be the case that the familiar power spectral energy P() = A²() would be a better function to integrate over the various passbands.)

      Figure D-3 presents one possible initial state for LOOP II. Figure D-4 presents a hypothetical final state for LOOP II when the entire adaptive mechanism of Figure D-2 has been stopped in LOOP III because classification has been deemed adequate. In this hypothetical case, the 3 adaptive processes in Figure D-2 have collectively selected four frequency features of some particular form of f() (that was also selected), upon which a particular discriminant operator (that was also selected) in LOOP III can execute the desired classification.

      It is within the context of the above discussion that a particular method for LOOP III will now be given.

      Figure D-5 presents the initial data set that LOOP II provides to LOOP III. Note that there are M frequency feature vector samples for High PHENOMENON and P samples for Low PHENOMENON. (For the 60-Item Perdurabo/TAARP Data Base, M = 15 and P = 27.) This means that there are M High PHENOMENON manifestations, and for each one there is a corresponding geocentric planetary configuration and a corresponding f() and a corresponding G() and a corresponding A() and a corresponding frequency amplitude feature vector. Note that N, the initial number of frequency amplitude samples in LOOP II, is 10 in this example. (Be careful not to get confused by the dual use of the word "sample" in this paragraph. In the first instance, "sample" means an item of the 60-Item Perdurabo/TAARP Data Base. In the second instance, "sample" refers to the fact that A() is being sampled by integrating over various passbands. In other words, in the second instance, "sample" means frequency amplitude feature.)

      The objective of LOOP III, therefore, is to discover a discriminant function of some kind that maps each frequency amplitude feature vector of the High PHENOMENON class into a final product that is easily distinguished from the final product of a similar mapping operating upon each frequency amplitude feature vector of the Low PHENOMENON class.

      For illustrative purposes, assume that N = 2, and thus there are only two frequency amplitude features for LOOP III to work with. One possible way to implement a discriminant function would be to simply see if the feature vectors provided by LOOP II clustered into discrete groups. Figure D-6 presents such a situation. With these assumptions, the objective of LOOP III would be to find a parametric equation for a line in f₁, f₂ space that would serve as a decision boundary between the two classes. In Figure D-6a, after many iterations of LOOP III, it was impossible to find a decision boundary that gave good discrimination. Thus, the performance measure passed from LOOP III to LOOP II for Loop II's k + 1 iteration would be small. This value of the performance measure would be used in LOOP II to help determine the next set of features to send to LOOP III. LOOP III did much better with the features provided by LOOP II on LOOP II's k + 1 input to LOOP III.

      Finally, for LOOP II's k + 2 input of features into LOOP III, LOOP III was able to find an adequate decision boundary. For this success, the many iterations internal to LOOP III involved adaptively adjusting parameters that controlled the decision boundary. For each iteration of LOOP III, a decision boundary was generated and then tested in order to ascertain how well it could be used to separate the two classes. The results of the test were the performance measure values internal to LOOP III that were used to drive the search process involved in manipulating the decision boundary. (Perhaps the best value of this LOOP III performance measure was passed to LOOP II to serve as its performance measure involved in the adjustment of vectors and . Whether or not this occurred depended upon the specific way that LOOP II was designed to operate with LOOP III.)

      In contrast to the decision boundary method of Figure D-6, a mathematically simple method for implementing a LOOP III discriminant function is to adaptively adjust a set of weights = w₁, w₂, ... , w₁₀ such that for some constant T₀:

1. • is greater than or equal to T₀, if belongs to the High PHENOMENON class.
2. • is less than T₀, if belongs to Low PHENOMENON class.

      In this case "•" means dot product so,

1. f₁w₁ + f₂w₂ + ... + f₁₀ w₁₀ is greater than or equal to T₀, if belongs to the High PHENOMENON class.
2. f₁w₁ + f₂w₂ + ... + f₁₀ w₁₀ is less than T₀, if belongs to the Low PHENOMENON class.

      A more complex procedure would involve "•" representing a polynomial operator of some high order. Then, for example, the problem could be stated as:

f₁w₁ + ... + f₁₀w₁₀ + f₁f₂w₁₁ + f₁f₃w₁₂ + ... + f₁²w₂₁ + f₁³w₂₂ + ...
is greater than or equal to T₀, if belongs to the High PHENOMENON class and is less than T₀, if belongs to the Low PHENOMENON class.

      In any case, we have the problem of searching over a multiple dimensional space, the space, in order to try to find a point in the space for which • is acceptably large for all 's belonging to the High PHENOMENON class and acceptably small for all 's belonging to the Low PHENOMENON class. The adaptive search process will require a performance measure. The design of the performance measure must be given careful consideration. For example, it may be that • is greater than or equal to T₀ for only 10% of the High PHENOMENON class 's and • is less than T₀ for only 5% of the Low PHENOMENON 's. This by itself would represent a fairly poor measure of performance. If, however, • is only slightly less than T₀ for the other 90% of the High PHENOMENON 's, and • is only slightly larger than T₀ for the other 95% of the Low PHENOMENON 's, then the measure of performance may be judged to be right on the verge of being very good.

      There are many standard classifier techniques in disciplines such as Decision Theory, Communications Theory, Adaptive Control Theory, and Pattern Recognition Theory which involve such things as Bayes' Rule and other Likelihood Functions and various "Loss" Functions. The discipline of pattern recognition is replete with many different statistical approaches to classification. (See Reference 55 and Reference 56. Also, see Addendum No. 3, which is not included on the website.) All of these disciplines will be taken into account in the performance of Task 2. At the present time TAARP is of the opinion that the Abductive Information Machine (AIM) technology developed and marketed by AbTech Corporation is the most advanced methodology in existence for adaptive processes. Unfortunately, the standard AIM software can not be incorporated into a TAARP custom designed pattern recognition algorithm, and TAARP can not currently afford to pay AbTech to modify an AIM software package. Therefore, TAARP will have to develop all of the adaptive processes implied in Figure D-1 and Figure D-2. However, we do plan to purchase the AbTech AIM StatNet Software, and use it to help train us on adaptive processes.