An abductive model of population change

Logical abduction is a relatively unknown technique that may be used to classify or fore­ cast. This paper describes abduction and applies it to a classification problem involving popu­ lation change in the United States to test its accuracy. Results show that for this problem, it is as accurate as logistic regression, used in a prior study. Guidelines for using abduction are also presented. INTRODUCTION Although relatively unknown, logical abduction has been used to forecast precipitation (Saburo, 1984), seasonal climate (Leobow, Mehra, & Toldalagi, 1984), fish populations (Brooks & Probert, 1984), shrimp catches (Prager & Saila, 1984), interest rates (Ohashi, 1984; Scott & Hutchinson, 1984), and many other variables (Barron, Mucciardi, Cook, Craig, & Barron, 1984). Few studies have compared abduction with other techniques, however. One study found abduc­ tion to be more accurate than a neural network for residential property assessment, bank loan approval, and the "exclusive or" problem and was faster to develop (Aiken, Paolillo, & Vanjani, 1999). A second study found abduction to be superior to a neural network forecasting housing prices in the Boston area (Aiken & Vanjani, 1999), and a third study found it to be superior to logistic regression for the "exclusive or" problem and bank loan approval (Aiken & Alonzo, in press). The purpose of this paper is to describe abduction and to illustrate its use with a population change classification problem. This problem has been used at least three times before to study different classification techniques including discriminant analysis, logistic regression, and artifi­ cial neural networks. Results show that abduction is just as accurate as logistic regression for this problem, but slightly less accurate than a neural network. The paper concludes with guidelines for using abduction. 45 1 Aiken: An abductive model of population change Published by CSUSB ScholarWorks, 1999 Journal Of International inforrnation Management Volume 8, Number 2 ABDUCTION Abduction was first defined as a form of logical inference by Charles Peirce in the 1860s (Fann, 1970), but a mathematical formulation of this theory known as the Group Method of Data Handling (GMDH) was developed about 100 years later (Ivakhnenko, Krotov, & Visotsky, 1979) and is perhaps the most commonly-used implementation of the method (Farlow, 1984). Using numeric functions to describe complex relationships, abduction differs from deduc­ tion and induction in that abduction may be used for problems with a high degree of uncertainty. Like induction, however, abduction learns from examples. By iteratively evaluating a large num­ ber of potential models, abduction determines the functional element coefficients, number of network elements, types of network elements, and the connectivity among the elements. Abduc­ tion utilizes a network of functions so that only the relationships among small subsets of vari­ ables need to be discovered at a time. Using logical abduction, a mathematical model is con­ structed in a manner similar to that of evolution. Starting with a few basic equations, a new generation of more complex equations and a survival-of-the-fittest principle is used to determine which equations live and which equations die. Like neltral networks, abduction automatically learns the relationships among variables in a model and does not require the user to make assumptions about the underlying distribution or that the input variables are independent (Wasserman, 1989). Multi-linear and logistic regression and discriminant analysis have certain assumptions that must be met; otherwise, their results may be unreliable. Again, like neural network learning, abduction automatically evaluates a large number of potential models and develops a network with nodes containing mathematical func­ tions, more complex than the nodes in a neural network. Unlike a neural network using back propagation, abduction automatically selects the network architecture. Thus, it is similar to that used in neural network applications using the genetic algorithm (Dorsey, Johnson, & Mayer, 1994). Thus, the primary differences between abduction and neural networks are that the former: (1) typically results in a network with fewer, more powerful nodes; (2) often results in faster network development; and (3) automatically determines the network architecture. The Abductive Information Modeler {AIM) from Abteeh Corporation is an example of how abduction may be implemented. Using this software, the synthesized abductive network may consist of seven types of elements described below and shown in Figure 1 (AIM User Manual, 1994): 1. Singles Wg+(w,x,)+(w2X,^)-l-(w3X,') 2. Doubles Wg+(w,x,)-i-(w2X2)+(w3X|^)+(w^X2^)-i-(w,X|X2)-f-(w^x,^)+(w.yX2') 3. Triples w„+(w,x,)+(w2X2)+(w3X3)-i-(w^x,2)+(w5X2-)+(WgX32)+(w,x,X2) +(WgX,X3)-|-(W^X3X3)-H(W|„X,X3X3)+(Wj|X,')+(w,3X/)+(Wj3X35) 4. White Elements A linear weighted sum of all the outputs of the previous layer. W,X,+W.,X.+W,X-t-. . . -tW X 1 1 2 2 3 3 n n 5. Normalizers Normalizers transform all of the original input variables into a rela­ tively common region with a mean of 0 and a variance of 1 using mean46 2 Journal of International Information Management, Vol. 8 [1999], Iss. 2, Art. 4 http://scholarworks.lib.csusb.edu/jiim/vol8/iss2/4 An AbductiveModel Journal of Internationallnformation Management sigma normalization. Wo + (w,x,) 6. Unitizers A unitizer converts the range of the network outputs to a range with the mean and variance of the output values used to train the network. W o + ( W , X , ) 7. Wire Elements Wire elements are used for a network that consists only of a normalizer and a unitizer. Figure 1. Four Input, Three-Layer Abductive Network Normalizers First Layer Second Layer Third Layer Unitizer T Input A •M Single V Double