C++ Neural Networks and Fuzzy Logic
Matrices and Some Arithmetic Operations on Matrices
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C neural networks and fuzzy logic
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- Lyapunov Function
- Kronecker Delta Function
- Gaussian Density Distribution
- C++ Neural Networks and Fuzzy Logic by Valluru B. Rao MTBooks, IDG Books Worldwide, Inc. ISBN
- Adaptive Resonance Theory
- Associative memory
- Brain−State−in−a−Box
- Connection
- Crisp
- Expert system
- Fit vector
- Fuzzy Associative Memory
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Matrices and Some Arithmetic Operations on Matrices A real matrix is a rectangular array of real numbers. A matrix with m rows and n columns is referred to as an
matrix and is denoted by a ij . The transpose of a matrix M is denoted by M T . The element in the ith row and jth column of M T is the same as the element of M in its jth row and ith column. M
is obtained from M by interchanging the rows and columns of M. For example, if 2 7 −3 2 4 M = , then M T = 7 0
4 0 9 −3 9 If X is a vector with m components, x 1 , …, x m , then it can be written as a column vector with components listed one below another. It can be written as a row vector, X = (x 1 , …, x m ). The transpose of a row vector is the column vector with the same components, and the transpose of a column vector is the corresponding row vector.
The addition of matrices is possible if they have the same size, that is, the same number of rows and same number of columns. Then you just add the ij elements of the two matrices to get the ij elements of the sum matrix. For example, 3 −4 5 5 2 −3 8 −2 2 + = 2 3 7 6 0 4 8 3 11 Multiplication is defined for a given pair of matrices, only if a condition on their respective sizes is satisfied. Then too, it is not a commutative operation. This means that if you exchange the matrix on the left with the matrix on the right, the multiplication operation may not be defined, and even if it is, the result may not be the same as before such an exchange. The condition to be satisfied for multiplying the matrices A, B as AB is, that the number of columns in A is equal to the number of rows in B. Then to get the ij element of the product matrix AB, you take the ith row of C++ Neural Networks and Fuzzy Logic:Preface Appendix B Mathematical Background 416
A as one vector and the jth column of B as a second vector and do a dot product of the two. For example, the two matrices given previously to illustrate the addition of two matrices are not compatible for multiplication in whichever order you take them. It is because there are three columns in each, which is different from the number of rows, which is 2 in each. Another example is given as follows. 3 −4 5 5 6 Let A = and B = 2 0 2 3 7 −3 4 Then AB and BA are both defined, AB is a 2x2 matrix, whereas BA is 3x3. −8 38 27 −2 67 Also AB = and BA = 6 −8 10 −5 40 −1 24 13
A Lyapunov function is a function that decreases with time, taking on non−negative values. It is used to correspond between the state variables of a system and real numbers. The state of the system changes as time changes, and the function decreases. Thus, the Lyapunov function decreases with each change of state of the system. We can construct a simple example of a function with the property of decreasing with each change of state as follows. Suppose a real number, x, represents the state of a dynamic system at time t. Also suppose that x is bounded for any t by a positive real number M. That means x is less than M for every value of t. Then the function, f(x,t) = exp(−|x|/(M+|x|+t)) is non−negative and decreases with increasing t.
A function f(x) is defined to have a local minimum at y, with a value z, if f(y) = z, and f(x) e z, for each x, such that there exists a positive real number h such that y – h d x d y + h. In other words, there is no other value of x in a neighborhood of y, where the value of the function is smaller than z. There can be more than one local minimum for a function in its domain. A Step function (with a graph resembling a staircase) is a simple example of a function with an infinite number of points in its domain with local minima.
A function f(x) is defined to have a global minimum at y, with a value z, if f(y) = z, and f(x) e z, for each x in the domain of the function f. C++ Neural Networks and Fuzzy Logic:Preface Global Minimum 417
In other words, there is no other value of x in the domain of the function f, where the value of the function is smaller than z. Clearly, a global minimum is also a local minimum, but a local minimum may not be a global minimum. There can be more than one global minimum for a function in its domain. The trigonometric function f(x) = sinx is a simple example of a function with an infinite number of points with global minima. You may recall that sin(3À/ 2), sin (7À/ 2), and so on are all –1, the smallest value for the sine function. Kronecker Delta Function The Kronecker delta function is a function of two variables. It has a value of 1 if the two arguments are equal, and 0 if they are not. Formally, 1 if x = y ´(x,y)= 0 if x `y Gaussian Density Distribution The Gaussian Density distribution, also called the Normal distribution, has a density function of the following form. There is a constant parameter c, which can have any positive value. Table of Contents Copyright © IDG Books Worldwide, Inc. C++ Neural Networks and Fuzzy Logic:Preface Global Minimum 418
C++ Neural Networks and Fuzzy Logic by Valluru B. Rao MTBooks, IDG Books Worldwide, Inc. ISBN: 1558515526 Pub Date: 06/01/95 Table of Contents References Ahmadian, Mohamad, and Pimmel, Russell, “Recognizing Geometrical Features of Simulated Targets with Neural Networks,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 409–411. Aleksander, Igor, and Morton, Helen, An Introduction to Neural Computing, Chapman and Hall, London, 1990. Aiken, Milam, “Forecasting T−Bill Rates with a Neural Net,” Technical Analysis of Stocks and Commodities, May 1995, Technical Analysis Inc., Seattle. Anderson, James, and Rosenfeld, Edward, eds., Neurocomputing: Foundations of Research, MIT Press, Cambridge, MA, 1988. Anzai, Yuichiro, Pattern Recognition and Machine Learning, Academic Press, Englewood Cliffs, NJ, 1992.
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Carpenter, Gail A., and Ross, William D., “ART−EMAP: A Neural Network Architecture for Object Recognition by Evidence Accumulation,” IEEE Transactions on Neural Networks, Vol. 6., No. 4, July 1995, pp. 805–818. Colby, Robert W., and Meyers, Thomas A., The Encyclopedia of Technical Market Indicators, Business One Irwin, Homewood, IL, 1988. Collard, J. E., “Commodity Trading with a Three Year Old,” Neural Networks in Finance and Investing, pp. 411–420, Probus Publishing, Chicago, 1993. Cox, Earl, The Fuzzy Systems Handbook, Academic Press, Boston, 1994. Dagli, Cihan, et al., eds., Intelligent Engineering Systems through Artificial Neural Networks, Volume 2, ASME Press, New York, 1992. Davalo, Eric, and Naim, Patrick, Neural Networks, MacMillan, New York, 1991. Ercal, F., et al., “Diagnosing Malignant Melanoma Using a Neural Network,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 553–555. Frank, Deborah, and Pletta, J. Bryan, “Neural Network Sensor Fusion for Security Applications,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 745–748.
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Glover, Fred, ORSA CSTS Newsletter Vol 15, No 2, Fall 1994. Goldberg, David E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison−Wesley, Reading, MA, 1989. Grossberg, Stephen, et al., Introduction and Foundations, Lecture Notes, Neural Network Courses and Conference, Boston University, May 1992. Hammerstrom, Dan, “Neural Networks at Work,” IEEE Spectrum, New York, June 1993. Hertz, John, Krogh, Anders, and Palmer, Richard, Introduction to the Theory of Neural Computation, Addison−Wesley, Reading, MA, 1991. Jagota, Arun, “Word Recognition with a Hopfield−Style Net,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 445–448. Johnson, R. Colin, “Accuracy Moves OCR into the Mainstream,” Electronic Engineering Times, CMP Publications, Manhasset, NY, January 16, 1995. Johnson, R. Colin, “Making the Neural−Fuzzy Connection,” Electronic Engineering Times, CMP Publications, Manhasset, NY, September 27, 1993. Johnson, R. Colin, “Neural Immune System Nabs Viruses,” Electronic Engineering Times, CMP Publications, Manhasset, NY, May 8, 1995. Jurik, Mark, “The Care and Feeding of a Neural Network,” Futures Magazine, Oster Communications, Cedar Falls, IA, October 1992. Kimoto, Takashi, et al., “Stock Market Prediction System with Modular Neural Networks,” Neural
Kline, J., and Folger, T.A., Fuzzy Sets, Uncertainty and Information, Prentice Hall, New York, 1988. Konstenius, Jeremy G., “Trading the S&P with a Neural Network,” Technical Analysis of Stocks and Commodities, October 1994, Technical Analysis Inc., Seattle. Kosaka, M., et al., “Applications of Fuzzy Logic/Neural Network to Securities Trading Decision Support System,” Conference Proceedings of the 1991 IEEE International Conference on Systems, Man and Cybernetics, V.3, pp. 1913–1918. Kosko, Bart, and Isaka, Satoru, Fuzzy Logic, Scientific American, New York, July 1993. Kosko, Bart, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine
Laing, Jonathan, “New Brains: How Smart Computers are Beating the Stock Market,” Barron’s, February 27, 1995. Lederman, Jess, and Klein, Robert, eds., Virtual Trading, Probus Publishing, Chicago, 1995. Lin, C.T. and Lee, C.S.G, “A Multi−Valued Boltzman Machine”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 25, No. 4, April 1995 pp. 660−668. MacGregor, Ronald J., Neural and Brain Modeling, Academic Press, Englewood Cliffs, NJ, 1987. Mandelman, Avner, “The Computer’s Bullish! A Money Manager’s Love Affair with Neural Network Programs,” Barron’s, December 14, 1992. Maren, Alianna, Harston, Craig, and Pap, Robert, Handbook of Neural Computing Applications, Academic Press, Englewood Cliffs, NJ, 1990. Marquez, Leorey, et al., “Neural Network Models as an Alternative to Regression,” Neural Networks in Finance and Investing, pp. 435–449, Probus Publishing, Chicago, 1993. Mason, Anthony et al., “Diagnosing Faults in Circuit Boards—A Neural Net Approach,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 839–843. McNeill, Daniel, and Freiberger, Paul, Fuzzy Logic, Simon & Schuster, New York, 1993. McNeill, F. Martin, and Thro, Ellen, Fuzzy Logic: A Practical Approach, Academic Press, Boston, 1994.
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Peters, Edgar E., Fractal Market Analysis: Applying Chaos Theory to Investment and Economics, John Wiley & Sons, New York, 1994. Ressler, James L., and Augusteijn, Marijke F., “Weapon Target Assignment Accessibility Using Neural Networks,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 397–399. Rao, Satyanaryana S,, and Sethuraman, Sriram, “A Modified Recurrent Learning Algorithm for Nonlinear Time Series Prediction,” Conference Proceedings of the 1992 Artificial Neural Networks in Engineering Conference, V.3, pp. 725–730. Rao, Valluru, and Rao, Hayagriva, Power Programming Turbo C++, MIS:Press, New York, 1992. Ruggiero, Murray, “How to Build an Artificial Trader,” Futures Magazine, Oster Communications, Cedar Falls, IA, September 1994. Ruggiero, Murray, “How to Build a System Framework,” Futures Magazine, Oster Communications, Cedar Falls, IA, November 1994. Ruggiero, Murray, “Nothing Like Net for Intermarket Analysis,” Futures Magazine, Oster Communications, Cedar Falls, IA, May 1995. Ruggiero, Murray, “Putting the Components before the System,” Futures Magazine, Oster Communications, Cedar Falls, IA, October 1994. Ruggiero, Murray, “Training Neural Networks for Intermarket Analysis,” Futures Magazine, Oster Communications, Cedar Falls, IA, August 1994. Rumbaugh, James, et al, Object−Oriented Modeling and Design, Prentice−Hall Inc, Englewood Cliffs, New Jersey, 1991. Sharda, Ramesh, and Patil, Rajendra, “A Connectionist Approach to Time Series Prediction: An Empirical Test,” Neural Networks in Finance and Investing, pp. 451–463, Probus Publishing, Chicago, 1993. Sherry, Clifford, “Are Your Inputs Correlated?” Technical Analysis of Stocks and Commodities, February 1995, Technical Analysis Inc., Seattle. Simpson, Patrick K., Artificial Neural Systems: Foundations, Paradigms, Applications and
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C++ Neural Networks and Fuzzy Logic:Preface References 421
Table of Contents Copyright © IDG Books Worldwide, Inc. C++ Neural Networks and Fuzzy Logic:Preface References 422
C++ Neural Networks and Fuzzy Logic by Valluru B. Rao MTBooks, IDG Books Worldwide, Inc. ISBN: 1558515526 Pub Date: 06/01/95 Table of Contents Glossary A Activation The weighted sum of the inputs to a neuron in a neural network. Adaline Adaptive linear element machine. Adaptive Resonance Theory Theory developed by Grossberg and Carpenter for categorization of patterns, and to address the stability–plasticity dilemma.
A step−by−step procedure to solve a problem. Annealing A process for preventing a network from being drawn into a local minimum. ART (Adaptive Resonance Theory) ART1 is the result of the initial development of this theory for binary inputs. Further developments led to ART2 for analog inputs. ART3 is the latest.
The primary object in an artificial neural network to mimic the neuron activity of the brain. The artificial neuron is a processing element of a neural network.
Activity of associating one pattern or object with itself or another. Autoassociative Making a correspondence of one pattern or object with itself. B Backpropagation A neural network training algorithm for feedforward networks where the errors at the output layer are propagated back to the layer before in learning. If the previous layer is not the input layer, then the errors at this hidden layer are propagated back to the layer before. BAM Bidirectional Associative Memory network model. Bias A value added to the activation of a neuron. Binary digit A value of 0 or 1. Bipolar value A value of –1 or +1. Boltzmann machine C++ Neural Networks and Fuzzy Logic:Preface Glossary 423
A neural network in which the outputs are determined with probability distributions. Trained and operated using simulated annealing. Brain−State−in−a−Box Anderson’s single−layer, laterally connected neural network model. It can work with inputs that have noise in them or are incomplete.
Similar to the Boltzmann machine, except that a Cauchy distribution is used for probabilities. Cognitron The forerunner to the Neocognitron. A network developed to recognize characters. Competition A process in which a winner is selected from a layer of neurons by some criterion. Competition suggests inhibition reflected in some connection weights being assigned a negative value.
A means of passing inputs from one neuron to another. Connection weight A numerical label associated with a connection and used in a weighted sum of inputs. Constraint A condition expressed as an equation or inequality, which has to be satisfied by the variables. Convergence Termination of a process with a final result. Crisp The opposite of fuzzy—usually a specific numerical quantity or value for an entity. D Delta rule A rule for modification of connection weights, using both the output and the error obtained. It is also called the LMS rule.
A function of outputs and weights in a neural network to determine the state of the system, e.g., Lyapunov function.
Providing positive weights on connections to enable outputs that cause a neuron to fire. Exemplar An example of a pattern or object used in training a neural network. Expert system A set of formalized rules that enable a system to perform like an expert. F FAM Fuzzy Associative Memory network. Makes associations between fuzzy sets. Feedback The process of relaying information in the opposite direction to the original. C++ Neural Networks and Fuzzy Logic:Preface Glossary
424 Fit vector A vector of values of degree of membership of elements of a fuzzy set. Fully connected network A neural network in which every neuron has connections to all other neurons. Fuzzy As related to a variable, the opposite of crisp. A fuzzy quantity represents a range of value as opposed to a single numeric value, e.g., “hot” vs. 89.4°.
Different concepts having an overlap to some extent. For example, descriptions of fair and cool temperatures may have an overlap of a small interval of temperatures.
A neural network model to make association between fuzzy sets. Fuzzy equivalence relation A fuzzy relation (relationship between fuzzy variables) that is reflexive, symmetric, and transitive. Fuzzy partial order A fuzzy relation (relationship between fuzzy variables) that is reflexive, antisymmetric, and transitive. Download 1.14 Mb. Do'stlaringiz bilan baham: 
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