DR. MARCOS LÓPEZ DE PRADO is a principal at AQR Capital Management, and its head of machine learning. Marcos is also a research fellow at Lawrence Berkeley National Laboratory (U.S. Department of Energy, Office of Science). SSRN ranks him as one of the most-read authors in economics, and he has published dozens of scientific articles on machine learning and supercomputing in the leading academic journals. Marcos earned a PhD in financial economics (2003), a second PhD in mathematical finance (2011) from Universidad Complutense de Madrid, and is a recipient of Spain's National Award for Academic Excellence (1999). He completed his post-doctoral research at Harvard University and Cornell University, where he teaches a graduate course in financial machine learning at the School of Engineering. Marcos has an Erdös #2 and an Einstein #4 according to the American Mathematical Society.

About the Author xxi PREAMBLE 1 1 Financial Machine Learning as a Distinct Subject 3 1.1 Motivation, 3 1.2 The Main Reason Financial Machine Learning Projects Usually Fail, 4 1.2.1 The Sisyphus Paradigm, 4 1.2.2 The Meta-Strategy Paradigm, 5 1.3 Book Structure, 6 1.3.1 Structure by Production Chain, 6 1.3.2 Structure by Strategy Component, 9 1.3.3 Structure by Common Pitfall, 12 1.4 Target Audience, 12 1.5 Requisites, 13 1.6 FAQs, 14 1.7 Acknowledgments, 18 Exercises, 19 References, 20 Bibliography, 20 Part 1 Data Analysis 21 2 Financial Data Structures 23 2.1 Motivation, 23 2.2 Essential Types of Financial Data, 23 2.2.1 Fundamental Data, 23 2.2.2 Market Data, 24 2.2.3 Analytics, 25 2.2.4 Alternative Data, 25 2.3 Bars, 25 2.3.1 Standard Bars, 26 2.3.2 Information-Driven Bars, 29 2.4 Dealing with Multi-Product Series, 32 2.4.1 The ETF Trick, 33 2.4.2 PCA Weights, 35 2.4.3 Single Future Roll, 36 2.5 Sampling Features, 38 2.5.1 Sampling for Reduction, 38 2.5.2 Event-Based Sampling, 38 Exercises, 40 References, 41 3 Labeling 43 3.1 Motivation, 43 3.2 The Fixed-Time Horizon Method, 43 3.3 Computing Dynamic Thresholds, 44 3.4 The Triple-Barrier Method, 45 3.5 Learning Side and Size, 48 3.6 Meta-Labeling, 50 3.7 How to Use Meta-Labeling, 51 3.8 The Quantamental Way, 53 3.9 Dropping Unnecessary Labels, 54 Exercises, 55 Bibliography, 56 4 Sample Weights 59 4.1 Motivation, 59 4.2 Overlapping Outcomes, 59 4.3 Number of Concurrent Labels, 60 4.4 Average Uniqueness of a Label, 61 4.5 Bagging Classifiers and Uniqueness, 62 4.5.1 Sequential Bootstrap, 63 4.5.2 Implementation of Sequential Bootstrap, 64 4.5.3 A Numerical Example, 65 4.5.4 Monte Carlo Experiments, 66 4.6 Return Attribution, 68 4.7 Time Decay, 70 4.8 Class Weights, 71 Exercises, 72 References, 73 Bibliography, 73 5 Fractionally Differentiated Features 75 5.1 Motivation, 75 5.2 The Stationarity vs. Memory Dilemma, 75 5.3 Literature Review, 76 5.4 The Method, 77 5.4.1 Long Memory, 77 5.4.2 Iterative Estimation, 78 5.4.3 Convergence, 80 5.5 Implementation, 80 5.5.1 Expanding Window, 80 5.5.2 Fixed-Width Window Fracdiff, 82 5.6 Stationarity with Maximum Memory Preservation, 84 5.7 Conclusion, 88 Exercises, 88 References, 89 Bibliography, 89 Part 2 Modelling 91 6 Ensemble Methods 93 6.1 Motivation, 93 6.2 The Three Sources of Errors, 93 6.3 Bootstrap Aggregation, 94 6.3.1 Variance Reduction, 94 6.3.2 Improved Accuracy, 96 6.3.3 Observation Redundancy, 97 6.4 Random Forest, 98 6.5 Boosting, 99 6.6 Bagging vs. Boosting in Finance, 100 6.7 Bagging for Scalability, 101 Exercises, 101 References, 102 Bibliography, 102 7 Cross-Validation in Finance 103 7.1 Motivation, 103 7.2 The Goal of Cross-Validation, 103 7.3 Why K-Fold CV Fails in Finance, 104 7.4 A Solution: Purged K-Fold CV, 105 7.4.1 Purging the Training Set, 105 7.4.2 Embargo, 107 7.4.3 The Purged K-Fold Class, 108 7.5 Bugs in Sklearn's Cross-Validation, 109 Exercises, 110 Bibliography, 111 8 Feature Importance 113 8.1 Motivation, 113 8.2 The Importance of Feature Importance, 113 8.3 Feature Importance with Substitution Effects, 114 8.3.1 Mean Decrease Impurity, 114 8.3.2 Mean Decrease Accuracy, 116 8.4 Feature Importance without Substitution Effects, 117 8.4.1 Single Feature Importance, 117 8.4.2 Orthogonal Features, 118 8.5 Parallelized vs. Stacked Feature Importance, 121 8.6 Experiments with Synthetic Data, 122 Exercises, 127 References, 127 9 Hyper-Parameter Tuning with Cross-Validation 129 9.1 Motivation, 129 9.2 Grid Search Cross-Validation, 129 9.3 Randomized Search Cross-Validation, 131 9.3.1 Log-Uniform Distribution, 132 9.4 Scoring and Hyper-parameter Tuning, 134 Exercises, 135 References, 136 Bibliography, 137 Part 3 Backtesting 139 10 Bet Sizing 141 10.1 Motivation, 141 10.2 Strategy-Independent Bet Sizing Approaches, 141 10.3 Bet Sizing from Predicted Probabilities, 142 10.4 Averaging Active Bets, 144 10.5 Size Discretization, 144 10.6 Dynamic Bet Sizes and Limit Prices, 145 Exercises, 148 References, 149 Bibliography, 149 11 The Dangers of Backtesting 151 11.1 Motivation, 151 11.2 Mission Impossible: The Flawless Backtest, 151 11.3 Even If Your Backtest Is Flawless, It Is Probably Wrong, 152 11.4 Backtesting Is Not a Research Tool, 153 11.5 A Few General Recommendations, 153 11.6 Strategy Selection, 155 Exercises, 158 References, 158 Bibliography, 159 12 Backtesting through Cross-Validation 161 12.1 Motivation, 161 12.2 The Walk-Forward Method, 161 12.2.1 Pitfalls of the Walk-Forward Method, 162 12.3 The Cross-Validation Method, 162 12.4 The Combinatorial Purged Cross-Validation Method, 163 12.4.1 Combinatorial Splits, 164 12.4.2 The Combinatorial Purged Cross-Validation Backtesting Algorithm, 165 12.4.3 A Few Examples, 165 12.5 How Combinatorial Purged Cross-Validation Addresses Backtest Overfitting, 166 Exercises, 167 References, 168 13 Backtesting on Synthetic Data 169 13.1 Motivation, 169 13.2 Trading Rules, 169 13.3 The Problem, 170 13.4 Our Framework, 172 13.5 Numerical Determination of Optimal Trading Rules, 173 13.5.1 The Algorithm, 173 13.5.2 Implementation, 174 13.6 Experimental Results, 176 13.6.1 Cases with Zero Long-Run Equilibrium, 177 13.6.2 Cases with Positive Long-Run Equilibrium, 180 13.6.3 Cases with Negative Long-Run Equilibrium, 182 13.7 Conclusion, 192 Exercises, 192 References, 193 14 Backtest Statistics 195 14.1 Motivation, 195 14.2 Types of Backtest Statistics, 195 14.3 General Characteristics, 196 14.4 Performance, 198 14.4.1 Time-Weighted Rate of Return, 198 14.5 Runs, 199 14.5.1 Returns Concentration, 199 14.5.2 Drawdown and Time under Water, 201 14.5.3 Runs Statistics for Performance Evaluation, 201 14.6 Implementation Shortfall, 202 14.7 Efficiency, 203 14.7.1 The Sharpe Ratio, 203 14.7.2 The Probabilistic Sharpe Ratio, 203 14.7.3 The Deflated Sharpe Ratio, 204 14.7.4 Efficiency Statistics, 205 14.8 Classification Scores, 206 14.9 Attribution, 207 Exercises, 208 References, 209 Bibliography, 209 15 Understanding Strategy Risk 211 15.1 Motivation, 211 15.2 Symmetric Payouts, 211 15.3 Asymmetric Payouts, 213 15.4 The Probability of Strategy Failure, 216 15.4.1 Algorithm, 217 15.4.2 Implementation, 217 Exercises, 219 References, 220 16 Machine Learning Asset Allocation 221 16.1 Motivation, 221 16.2 The Problem with Convex Portfolio Optimization, 221 16.3 Markowitz's Curse, 222 16.4 From Geometric to Hierarchical Relationships, 223 16.4.1 Tree Clustering, 224 16.4.2 Quasi-Diagonalization, 229 16.4.3 Recursive Bisection, 229 16.5 A Numerical Example, 231 16.6 Out-of-Sample Monte Carlo Simulations, 234 16.7 Further Research, 236 16.8 Conclusion, 238 Appendices, 239 16.A.1 Correlation-based Metric, 239 16.A.2 Inverse Variance Allocation, 239 16.A.3 Reproducing the Numerical Example, 240 16.A.4 Reproducing the Monte Carlo Experiment, 242 Exercises, 244 References, 245 Part 4 Useful Financial Features 247 17 Structural Breaks 249 17.1 Motivation, 249 17.2 Types of Structural Break Tests, 249 17.3 CUSUM Tests, 250 17.3.1 Brown-Durbin-Evans CUSUM Test on Recursive Residuals, 250 17.3.2 Chu-Stinchcombe-White CUSUM Test on Levels, 251 17.4 Explosiveness Tests, 251 17.4.1 Chow-Type Dickey-Fuller Test, 251 17.4.2 Supremum Augmented Dickey-Fuller, 252 17.4.3 Sub- and Super-Martingale Tests, 259 Exercises, 261 References, 261 18 Entropy Features 263 18.1 Motivation, 263 18.2 Shannon's Entropy, 263 18.3 The Plug-in (or Maximum Likelihood) Estimator, 264 18.4 Lempel-Ziv Estimators, 265 18.5 Encoding Schemes, 269 18.5.1 Binary Encoding, 270 18.5.2 Quantile Encoding, 270 18.5.3 Sigma Encoding, 270 18.6 Entropy of a Gaussian Process, 271 18.7 Entropy and the Generalized Mean, 271 18.8 A Few Financial Applications of Entropy, 275 18.8.1 Market Efficiency, 275 18.8.2 Maximum Entropy Generation, 275 18.8.3 Portfolio Concentration, 275 18.8.4 Market Microstructure, 276 Exercises, 277 References, 278 Bibliography, 279 19 Microstructural Features 281 19.1 Motivation, 281 19.2 Review of the Literature, 281 19.3 First Generation: Price Sequences, 282 19.3.1 The Tick Rule, 282 19.3.2 The Roll Model, 282 19.3.3 High-Low Volatility Estimator, 283 19.3.4 Corwin and Schultz, 284 19.4 Second Generation: Strategic Trade Models, 286 19.4.1 Kyle's Lambda, 286 19.4.2 Amihud's Lambda, 288 19.4.3 Hasbrouck's Lambda, 289 19.5 Third Generation: Sequential Trade Models, 290 19.5.1 Probability of Information-based Trading, 290 19.5.2 Volume-Synchronized Probability of Informed Trading, 292 19.6 Additional Features from Microstructural Datasets, 293 19.6.1 Distibution of Order Sizes, 293 19.6.2 Cancellation Rates, Limit Orders, Market Orders, 293 19.6.3 Time-Weighted Average Price Execution Algorithms, 294 19.6.4 Options Markets, 295 19.6.5 Serial Correlation of Signed Order Flow, 295 19.7 What Is Microstructural Information?, 295 Exercises, 296 References, 298 Part 5 High-performance Computing Recipes 301 20 Multiprocessing and Vectorization 303 20.1 Motivation, 303 20.2 Vectorization Example, 303 20.3 Single-Thread vs. Multithreading vs. Multiprocessing, 304 20.4 Atoms and Molecules, 306 20.4.1 Linear Partitions, 306 20.4.2 Two-Nested Loops Partitions, 307 20.5 Multiprocessing Engines, 309 20.5.1 Preparing the Jobs, 309 20.5.2 Asynchronous Calls, 311 20.5.3 Unwrapping the Callback, 312 20.5.4 Pickle/Unpickle Objects, 313 20.5.5 Output Reduction, 313 20.6 Multiprocessing Example, 315 Exercises, 316 Reference, 317 Bibliography, 317 21 Brute Force and Quantum Computers 319 21.1 Motivation, 319 21.2 Combinatorial Optimization, 319 21.3 The Objective Function, 320 21.4 The Problem, 321 21.5 An Integer Optimization Approach, 321 21.5.1 Pigeonhole Partitions, 321 21.5.2 Feasible Static Solutions, 323 21.5.3 Evaluating Trajectories, 323 21.6 A Numerical Example, 325 21.6.1 Random Matrices, 325 21.6.2 Static Solution, 326 21.6.3 Dynamic Solution, 327 Exercises, 327 References, 328 22 High-Performance Computational Intelligence and Forecasting Technologies 329Kesheng Wu and Horst D. Simon 22.1 Motivation, 329 22.2...