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Multi-Agent Market Modeling Based on Neural Networks

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Multi-Agent Market Modeling Based on Neural Networks MULTI-AGENT MARKET MODELING BASED ON NEURAL NETWORKS by RALPH GROTHMANN University of Bremen, Germany Siemens AG, Corporate Technology, Munich, Germany Thesis presented for the Degree of Doctor of Economics (Dr. rer. pol.) Faculty of Economics, University of Bremen, Germany Contents Acronyms xi List of Figures xiii List of Tables xvii 1 Introduction 1 2 Neural Networks: Introduction & Historical Notes 15 1. From Biological to Artificial Neurons 16 1.1 Biological Neurons 16 1.2 Artificial Neurons 18 1.3 Common Activation Functions 20 1.4 Neurons as Elementary Models of Decision Making 21 2. 3-Layer Feedforward Networks 23 2.1 Architecture of the Standard Model 24 2.2 Universal Functional Approximation Abilities 26 2.3 Representation of Temporal Structures 26 2.4 Input Preprocessing 27 2.5 The Dilemma of Overfitting 28 2.6 Early Stopping 29 2.7 Late Stopping 30 3. Historical Notes on Neural Networks 32 3.1 McCulloch and Pitts Model of a Neuron (1943) 33 3.2 Hebbian Learning Rule (1949) 33 3.3 Perceptron of Rosenblatt (1958) 34 3.4 Adaptive Linear Element (1960) 36 3.5 Minsky and Papert's Critics of Perceptrons (1969) 36 3.6 Self-Organizing Maps (1982) 37 3.7 Hopfield Networks (1982) 38 3.8 Boltzmann Machine (1985) 39 3.9 Standard Error Backpropagation (1974 & 1986) 41 v vi MULTI-AGENT MARKET MODELING BY NEURAL NETWORKS 3.10 Radial Basis Function Networks (1988) 42 3.11 Elman's Recurrent Neural Networks (1990) 42 3.12 Dilemma of Overfitting & Purely Data Driven Modeling 45 4. Preliminary Conclusion 46 3 Modeling Dynamical Systems by Feedforward Neural Networks 49 1. Network Internal Preprocessing 51 2. Merging MLP and RBF Networks 53 3. Forces and Embeddings 56 4. Multi-Forecasts and Statistical Averaging 60 5. The 11-Layer Neural Network 63 6. Empirical Study: Forecasting the German Bond Rate 66 7. Preliminary Conclusion 69 4 Modeling Dynamical Systems by Recurrent Neural Networks 71 1. Basic Time-Delay Recurrent Neural Networks 73 1.1 Representing Dynamic Systems by Recurrent Networks 74 1.2 Finite Unfolding in Time 76 1.3 Overshooting 78 1.4 Embedding of Open Systems 79 1.5 Causality & Feature Selection 80 2. Error Correction Neural Networks (ECNN) 81 2.1 Approaching Error Correction Neural Networks 82 2.2 Unfolding in Time of Error Correction Neural Networks 83 2.3 Combining Overshooting & ECNN 84 2.4 Alternating Errors & ECNN 85 3. Undershooting 86 3.1 Uniform Causality 87 3.2 Approaching the Concept of Undershooting 89 3.3 Combining ECNN and Undershooting 90 4. Variants-Invariants Separation 91 4.1 Variants-Invariants Separation by Neural Networks 93 4.2 Combining Variants-Invariants Separation & ECNN 94 5. Optimal State Space Reconstruction for Forecasting 95 5.1 Finite Unfolding in Space & Time 96 5.2 Smoothness 99 5.3 Combining State Space Reconstruction & ECNN 100 6. Empirical Study: Forecasting the USD / DEM FX-Rate 100 6.1 Outline of the Empirical Study 101 6.1.1 Data Set 101 6.1.2 Input Signals & Preprocessing 101 6.1.3 Network Architectures & Benchmarks 102 6.1.4 Performance measures 105 6.2 Results of the Empirical Study 107 6.2.1 Comparing the time-delay recurrent networks 108 6.2.2 Benchmark Comparison 110 7. Preliminary Conclusion 111 Contents vii 5 Training of Neural Networks by Error Backpropagation 113 1. Standard Error Backpropagation Algorithm 114 2. Robust Estimation of Neural Network Parameters 119 3. Shared Weights Extension of Error Backpropagation 122 4. Estimating the Maximal Inter-Temporal Connectivity 128 5. Pattern-by-Pattern & Vario-Eta Learning Rule 131 5.1 Gradient descent 132 5.2 Pattern-by-pattern learning 133 5.3 Vario-eta learning 134 6. Exploring Invariant Structures in Time 135 6.1 Stochastic pruning 137 6.2 Early brain damage (EBD) 137 6.3 Instability pruning 137 6.4 Partial Learning 139 7. Empirical Study: German Interest Rate Forecasting 141 8. Preliminary Conclusion 145 6 Multi-Agent Market Modeling: A Guide to Literature 147 1. Multi-Agent Models as a New Theory of Financial Markets 149 2. Microeconomic Design Issues of Multi-Agent Models: Agents 161 2.1 Decision Making 163 2.1.1 Rule-based Agents 165 2.1.2 Forecasting Agents 168 2.1.3 Intertemporal Decision Making Schemes 171 2.1.4 Contagious Decision Making 172 2.2 Objective Functions 173 2.2.1 Explicit Objective Functions 175 2.2.2 Implicit Objective Functions 178 2.3 Heterogeneity 180 2.3.1 Information Basis 181 2.3.2 Parameter Settings 181 2.3.3 Agent Types 182 2.3.4 Learning Algorithms 183 2.4 Learning and Evolution 184 2.4.1 Non-learning Agents 185 2.4.2 Type Switching Agents 186 2.4.3 Evolutionary Learning 187 2.4.4 Gradient-based Learning 189 3. Macroeconomic Design Issues of Multi-Agent Models: Markets 190 3.1 Traded Assets 192 3.1.1 Single vs. Multiple Markets 192 3.1.2 Types of Assets 194 3.1.3 Asset Properties 197 3.2 Market Structure and Organization 199 3.2.1 Arrangement of the Agents 199 3.2.2 Trading Synchroneity 202 3.2.3 Additional Market Settings 204 3.3 Price Formation Mechanism 206 3.3.1 Price Response to Excess Demand 207 3.3.2 Temporary Equilibrium Price 209 viii MULTI-AGENT MARKET MODELING BY NEURAL NETWORKS 3.3.3 Real-world Price Mechanism 211 4. Preliminary Conclusion 213 7 Multi-Agent Modeling by Feedforward Neural Networks 217 1. Modeling the Behavior of Agents in FX-Markets 218 1.1 The Behavior of Agents in a Single FX-Market 219 1.2 The Behavior of Agents in Multiple FX-Markets 223 2. The Explicit Price Dynamics of Single & Multiple FX-Markets 225 3. Modeling the Explicit Price Dynamics of FX-Markets 227 3.1 Modeling a Single FX-Market 227 3.2 Modeling Multiple FX-Markets 229 4. The Implicit Price Dynamics of Single & Multiple FX-Markets 231 5. Modeling the Implicit Price Dynamics of FX-Markets 234 5.1 Modeling a Single FX-Market 234 5.2 Modeling Multiple FX-Markets 235 6. Empirical Study: Multi-Agent Modeling of FX-Markets 237 6.1 Outline of the Empirical Study 238 6.1.1 Outline of the Single FX-Market Analysis 239 6.1.2 Outline of the Multiple FX-Market Analysis 240 6.2 Results of the Empirical Study 240 6.2.1 Results of the Single FX-Market Analysis 241 6.2.2 Results of the Multiple FX-Market Analysis 243 7. Preliminary Conclusion 244 8 Multi-Agent FX-Market Modeling based on Cognitive Systems 247 1. An Inductive Approach to Cognitive Agents 248 1.1 Necessary Conditions for Cognitive Systems 249 1.1.1 Perception 250 1.1.2 Internal processing 250 1.1.3 Action 251 1.2 Modeling Cognitive Systems by ECNN 252 1.2.1 Perception 253 1.2.2 Internal processing 254 1.2.3 Action 255 1.3 Agents' Trading Scheme & Market Price Formation 256 1.4 FX-Market Modeling by Inductive Cognitive Agents 258 1.4.1 Arrangement of the agents 258 1.4.2 Market price formation 259 1.5 Empirical Study: Modeling the USD/DEM FX-Market 260 1.5.1 Outline of the empirical study 260 1.5.2 Results of the empirical study 264 2. A Deductive Approach to Cognitive Agents 266 2.1 Approaching Deductive Cognitive Systems 267 2.1.1 Homeostasis 267 2.1.2 Perception 268 2.1.3 Internal processing 268 2.1.4 Action 268 2.2 Modeling Deductive Cognitive Systems 269 Contents ix 2.2.1 A Structural Representation of Deductive Cog- nitive Systems 269 2.2.2 Unfolding in Time of Deductive Cognitive Sys- tems 272 2.3 Identification of Entities: Binding 277 2.3.1 Variants-invariants separation 277 2.3.2 Variants-invariants separation by neural networks 278 2.3.3 Cognitive systems & binding of entities 279 2.4 FX-Market Modeling by Deductive Cognitive Agents 281 2.4.1 Arrangement of the agents 282 2.4.2 Market price formation 282 2.5 Empirical Study 283 2.5.1 Outline of the empirical study 283 2.5.2 Results of the empirical study 284 3. Preliminary Conclusion 286 9 Summary and Conclusion 289 References 303 Acronyms adaline adaptive linear element approx. approximately AR autoregressive ARIMA autoregressive integrated moving average ARMA autoregressive moving average CARA constant absolute risk aversion chp. chapter CRB Commodity Research Bureau CRRA constant relative risk aversion EBD early brain damage ECNN error correction neural network Ed. editor EMH efficient market hypothesis et al. et aliter etc. et cetera eq. equation e. g. exempli gratia fig. figure FTSE 100 Financial Times Stock Exchange 100 stock index FX foreign exchange FX-market foreign exchange market FX-rate foreign exchange rate xi xii MULTI-AGENT MARKET MODELING BY NEURAL NETWORKS GARCH generalized autoregressive conditional heteroskedasticity DAX German stock market index DEM German Mark i. e. id est Ind. index LMS least mean square MIC maximum intertemporal connectivity MLP multi-layer perceptron MSE mean square error NARMA nonlinear autoregressive moving average NARX nonlinear autoregressive with exogenous inputs NYSE New York stock exchange OBD optimal brain damage OLS ordinary least squares Oz. ounce PCA principal component analysis RBF radial basis function REE rational expectations equilibrium resp. respectively RNN recurrent neural network sec. section SR Sharpe ratio SRN simple recurrent network SOM self-organizing map tab. table TAR threshold autoregressive USD US-Dollar VAR vector autoregressive vs. versus YEN Japanese Yen List of Figures 2.1 Schematical illustration of the human nervous system 17 2.2 Biological neuron: The pyramidal cell 18 2.3 A nonlinear artificial neuron 18 2.4 Threshold activation function 21 2.5 Piecewise linear activation function 21 2.6 Logistic activation function 21 2.7 A neuron: an elementary model of decision making 22 2.8 A 3-layer feedforward neural network 24 2.9 The early stopping concept 30 2.10 The late stopping concept 31 2.11 Model of a neuron suggested
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