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Topological Optimisation of Artificial Neural Networks for Financial Asset View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by LSE Theses Online The London School of Economics and Political Science Topological Optimisation of Artificial Neural Networks for Financial Asset Forecasting Shiye (Shane) He A thesis submitted to the Department of Management of the London School of Economics for the degree of Doctor of Philosophy. April 2015, London 1 Declaration I certify that the thesis I have presented for examination for the MPhil/PhD degree of the London School of Economics and Political Science is solely my own work other than where I have clearly indicated that it is the work of others (in which case the extent of any work carried out jointly by me and any other person is clearly identified in it). The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without the prior written consent of the author. I warrant that this authorization does not, to the best of my belief, infringe the rights of any third party. 2 Abstract The classical Artificial Neural Network (ANN) has a complete feed-forward topology, which is useful in some contexts but is not suited to applications where both the inputs and targets have very low signal-to-noise ratios, e.g. financial forecasting problems. This is because this topology implies a very large number of parameters (i.e. the model contains too many degrees of freedom) that leads to over fitting of both signals and noise. This results in the ANN having very good in-sample performance on the data used for its training but poor performance out- of-sample for forecasting. The main contribution of my research is to develop a new heuristic method called “ANN reduction” for optimising the topological structure of a feed-forward ANN in order to improve its out-of-sample performance (using an RMS measure). The research concentrated on the topological optimization of the graph representing an ANN, which reduces the effective degrees of freedom of the ANN whilst still maintaining its feed-forward (but incomplete) topology. Such reductions in the number of parameters have been attempted before in the literature, but our procedure is of a different (graph theoretic) nature and (in extremis) optimal for small-size ANNs. Two applications of the ANN reduction are also implemented and programmed for empir- ical simulations. For this purpose, two datasets generated from deterministic functions and three datasets derived from foreign exchange market prices are used for evaluating the ANN reduction applications. These applications generate new ANN topologies with some clear per- formance advantages over those obtained by the best complete ANNs, improving the general- ization (out-of-sample) performance by up to 27.6% compared to the complete ANN on the function generated datasets and up to 14.1% on the financial forecasting problem for the FX data. 3 Acknowledgements Foremost, I would like to thank my advisors, Prof. Nicos Christofides, for guiding me through the whole PhD life. His continuous encouragement, invaluable guidance and constant support helped me greatly in pursuing my goals in life. Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Nigel Meade and Dr. Cormac Lucas for their encouragement, insightful comments, and hard questions. I owe special thanks to my mom, dad and my fiancé, for their continuous encouragement, care and support during my study. This dissertation is simply impossible without them. 4 Contents 1 Introduction9 1.1 The Financial Forecasting Problem (FFP).....................9 1.2 Traditional forecasting methodologies........................ 11 1.2.1 Time series methods............................. 12 1.2.2 Causal methods................................ 14 1.2.3 Practical discussion.............................. 15 1.3 Machine learning methods.............................. 16 1.3.1 Machine learning classes........................... 16 1.3.2 Machine learning models........................... 18 1.3.3 Machine learning in the FFP........................ 19 1.4 ANN and applications in the FFP......................... 20 2 Background 22 2.1 Topology of ANN................................... 22 2.1.1 Feed-forward ANN.............................. 25 2.1.2 Approximation using ANN......................... 31 2.2 Training of ANN................................... 38 5 2.2.1 Sampled input-target data pairs....................... 39 2.2.2 Objective function.............................. 40 2.2.3 Back-propagation............................... 41 2.2.4 Conjugate gradient methods......................... 46 2.2.5 Local minima and global minima...................... 55 2.3 ANN for forecasting................................. 56 2.3.1 Over-fitting problem............................. 56 2.4 Inputs to ANN.................................... 58 2.5 Survey of ANN topological optimisation...................... 60 2.5.1 Growing methods............................... 61 2.5.2 Pruning methods............................... 63 3 Theoretical framework 67 3.1 Representing ANN topological structures as graphs................ 67 3.1.1 Graph and ANN network architecture................... 69 3.1.2 ANN topological optimisation problem................... 70 3.2 Heuristic implementation of static vertex optimisation.............. 74 3.2.1 λ-optimal on ANN static vertex optimisation............... 76 3.2.2 Enhanced ANN-Reduction Procedure (ERP)............... 80 3.2.3 Cascaded Enhanced ANN-Reduction Procedure (CERP)........ 89 3.3 Conclusion of the chapter.............................. 90 4 Empirical simulation on function approximation 93 4.1 Introduction...................................... 93 6 4.2 Simulation with deterministic function one..................... 95 4.2.1 Optimisation problem settings........................ 96 4.2.2 Result from complete ANN......................... 97 4.2.3 Result of topologically-optimising complete ANN for function one using ERP...................................... 101 4.2.4 Result of optimising complete ANN for function one using CERP.... 106 4.3 Simulate with sample data from deterministic function two........... 110 4.3.1 Problem settings............................... 110 4.3.2 Result from complete ANN......................... 111 4.3.3 Results of ERP................................ 114 4.3.4 Results of CERP............................... 119 4.4 Summary of result.................................. 123 5 Empirical simulation on foreign exchange price forecasting 125 5.1 Introduction...................................... 125 5.2 Data construction for the ANN forecasting problem................ 126 5.3 Design and setting of the ANN forecasting System................ 135 5.4 Forecasting performance of complete ANN..................... 138 5.4.1 Forecasting Problem 1............................ 139 5.4.2 Forecasting Problem 2............................ 140 5.4.3 Forecasting Problem 3............................ 141 5.5 ERP results...................................... 142 5.5.1 Forecasting Problem 1............................ 142 5.5.2 Forecasting Problem 2............................ 144 7 5.5.3 Forecasting Problem 3............................ 146 5.6 CERP results..................................... 148 5.6.1 Forecasting Problem 1............................ 148 5.6.2 Forecasting Problem 2............................ 150 5.6.3 Forecasting Problem 3............................ 152 5.7 Robustness test.................................... 154 5.8 Comparative study of ANN and AR model..................... 155 5.9 Summary and comparison of simulation results.................. 156 6 Conclusion 159 6.1 Introduction...................................... 159 6.2 Theoretical framework and implementation.................... 160 6.3 Empirical evidence.................................. 163 6.4 Limitation and future research........................... 164 A 25 Possible topology of 5-3-1 Basic ANN 176 8 Chapter 1 Introduction 1.1 The Financial Forecasting Problem (FFP) Forecasting is a process of making statements about events, in which actual outcomes have not yet been observed (output) often by employing time series analysis on historical data. The basic idea of forecasting assumes that future occurrences are partially based on present and past observations, e.g. a forecast of variable Xt is E(Xt+hjWt) where Wt is the information set used and available at time t, and the forecast is for h periods ahead. The success of a forecasting model E relies on the accuracy of finding the relationships between particular input and output sets. Mathematical modeling and trend forecasting for stochastic financial time series has long been a popular research topic. Most modern methodologies have involved statistical models such as time series analysis. These models usually use real market data as inputs such as unexpected growth in GDP, changing industry structure or more specifically corporate fundamentals i.e. price earnings ratios, etc. The FFP involves making statements about a financial asset’s future value based on the observed information and assumptions that reflect existing conditions in the financial market. The term "forecasting" is reserved for estimating quantitative values at specific times in the future, thus the FFP is associated with quantitative methods and quantified information. 9 A commonplace application of the FFP is to estimate the holding period return of a portfolio in the next period. The
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