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<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html"><i class="fa fa-check"></i>Preface</a><ul>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#what-this-book-is-not-about"><i class="fa fa-check"></i>What this book is not about</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#the-targeted-audience"><i class="fa fa-check"></i>The targeted audience</a></li>
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<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#companion-website"><i class="fa fa-check"></i>Companion website</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#why-r"><i class="fa fa-check"></i>Why R?</a></li>
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</ul></li>
<li class="part"><span><b>I Introduction</b></span></li>
<li class="chapter" data-level="1" data-path="notdata.html"><a href="notdata.html"><i class="fa fa-check"></i><b>1</b> Notations and data</a><ul>
<li class="chapter" data-level="1.1" data-path="notdata.html"><a href="notdata.html#notations"><i class="fa fa-check"></i><b>1.1</b> Notations</a></li>
<li class="chapter" data-level="1.2" data-path="notdata.html"><a href="notdata.html#dataset"><i class="fa fa-check"></i><b>1.2</b> Dataset</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="intro.html"><a href="intro.html"><i class="fa fa-check"></i><b>2</b> Introduction</a><ul>
<li class="chapter" data-level="2.1" data-path="intro.html"><a href="intro.html#context"><i class="fa fa-check"></i><b>2.1</b> Context</a></li>
<li class="chapter" data-level="2.2" data-path="intro.html"><a href="intro.html#portfolio-construction-the-workflow"><i class="fa fa-check"></i><b>2.2</b> Portfolio construction: the workflow</a></li>
<li class="chapter" data-level="2.3" data-path="intro.html"><a href="intro.html#machine-learning-is-no-magic-wand"><i class="fa fa-check"></i><b>2.3</b> Machine learning is no magic wand</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="factor.html"><a href="factor.html"><i class="fa fa-check"></i><b>3</b> Factor investing and asset pricing anomalies</a><ul>
<li class="chapter" data-level="3.1" data-path="factor.html"><a href="factor.html#introduction"><i class="fa fa-check"></i><b>3.1</b> Introduction</a></li>
<li class="chapter" data-level="3.2" data-path="factor.html"><a href="factor.html#detecting-anomalies"><i class="fa fa-check"></i><b>3.2</b> Detecting anomalies</a><ul>
<li class="chapter" data-level="3.2.1" data-path="factor.html"><a href="factor.html#challenges"><i class="fa fa-check"></i><b>3.2.1</b> Challenges</a></li>
<li class="chapter" data-level="3.2.2" data-path="factor.html"><a href="factor.html#simple-portfolio-sorts"><i class="fa fa-check"></i><b>3.2.2</b> Simple portfolio sorts </a></li>
<li class="chapter" data-level="3.2.3" data-path="factor.html"><a href="factor.html#factors"><i class="fa fa-check"></i><b>3.2.3</b> Factors</a></li>
<li class="chapter" data-level="3.2.4" data-path="factor.html"><a href="factor.html#predictive-regressions-sorts-and-p-value-issues"><i class="fa fa-check"></i><b>3.2.4</b> Predictive regressions, sorts, and p-value issues</a></li>
<li class="chapter" data-level="3.2.5" data-path="factor.html"><a href="factor.html#fama-macbeth-regressions"><i class="fa fa-check"></i><b>3.2.5</b> Fama-Macbeth regressions</a></li>
<li class="chapter" data-level="3.2.6" data-path="factor.html"><a href="factor.html#factor-competition"><i class="fa fa-check"></i><b>3.2.6</b> Factor competition</a></li>
<li class="chapter" data-level="3.2.7" data-path="factor.html"><a href="factor.html#advanced-techniques"><i class="fa fa-check"></i><b>3.2.7</b> Advanced techniques</a></li>
</ul></li>
<li class="chapter" data-level="3.3" data-path="factor.html"><a href="factor.html#factors-or-characteristics"><i class="fa fa-check"></i><b>3.3</b> Factors or characteristics?</a></li>
<li class="chapter" data-level="3.4" data-path="factor.html"><a href="factor.html#hot-topics-momentum-timing-and-esg"><i class="fa fa-check"></i><b>3.4</b> Hot topics: momentum, timing and ESG</a><ul>
<li class="chapter" data-level="3.4.1" data-path="factor.html"><a href="factor.html#factor-momentum"><i class="fa fa-check"></i><b>3.4.1</b> Factor momentum</a></li>
<li class="chapter" data-level="3.4.2" data-path="factor.html"><a href="factor.html#factor-timing"><i class="fa fa-check"></i><b>3.4.2</b> Factor timing</a></li>
<li class="chapter" data-level="3.4.3" data-path="factor.html"><a href="factor.html#the-green-factors"><i class="fa fa-check"></i><b>3.4.3</b> The green factors</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="factor.html"><a href="factor.html#the-links-with-machine-learning"><i class="fa fa-check"></i><b>3.5</b> The links with machine learning</a><ul>
<li class="chapter" data-level="3.5.1" data-path="factor.html"><a href="factor.html#a-short-list-of-recent-references"><i class="fa fa-check"></i><b>3.5.1</b> A short list of recent references</a></li>
<li class="chapter" data-level="3.5.2" data-path="factor.html"><a href="factor.html#explicit-connections-with-asset-pricing-models"><i class="fa fa-check"></i><b>3.5.2</b> Explicit connections with asset pricing models</a></li>
</ul></li>
<li class="chapter" data-level="3.6" data-path="factor.html"><a href="factor.html#coding-exercises"><i class="fa fa-check"></i><b>3.6</b> Coding exercises</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Data.html"><a href="Data.html"><i class="fa fa-check"></i><b>4</b> Data preprocessing</a><ul>
<li class="chapter" data-level="4.1" data-path="Data.html"><a href="Data.html#know-your-data"><i class="fa fa-check"></i><b>4.1</b> Know your data</a></li>
<li class="chapter" data-level="4.2" data-path="Data.html"><a href="Data.html#missing-data"><i class="fa fa-check"></i><b>4.2</b> Missing data</a></li>
<li class="chapter" data-level="4.3" data-path="Data.html"><a href="Data.html#outlier-detection"><i class="fa fa-check"></i><b>4.3</b> Outlier detection</a></li>
<li class="chapter" data-level="4.4" data-path="Data.html"><a href="Data.html#feateng"><i class="fa fa-check"></i><b>4.4</b> Feature engineering</a><ul>
<li class="chapter" data-level="4.4.1" data-path="Data.html"><a href="Data.html#feature-selection"><i class="fa fa-check"></i><b>4.4.1</b> Feature selection</a></li>
<li class="chapter" data-level="4.4.2" data-path="Data.html"><a href="Data.html#scaling"><i class="fa fa-check"></i><b>4.4.2</b> Scaling the predictors</a></li>
</ul></li>
<li class="chapter" data-level="4.5" data-path="Data.html"><a href="Data.html#labelling"><i class="fa fa-check"></i><b>4.5</b> Labelling</a><ul>
<li class="chapter" data-level="4.5.1" data-path="Data.html"><a href="Data.html#simple-labels"><i class="fa fa-check"></i><b>4.5.1</b> Simple labels</a></li>
<li class="chapter" data-level="4.5.2" data-path="Data.html"><a href="Data.html#categorical-labels"><i class="fa fa-check"></i><b>4.5.2</b> Categorical labels</a></li>
<li class="chapter" data-level="4.5.3" data-path="Data.html"><a href="Data.html#the-triple-barrier-method"><i class="fa fa-check"></i><b>4.5.3</b> The triple barrier method</a></li>
<li class="chapter" data-level="4.5.4" data-path="Data.html"><a href="Data.html#filtering-the-sample"><i class="fa fa-check"></i><b>4.5.4</b> Filtering the sample</a></li>
<li class="chapter" data-level="4.5.5" data-path="Data.html"><a href="Data.html#horizons"><i class="fa fa-check"></i><b>4.5.5</b> Return horizons</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="Data.html"><a href="Data.html#pers"><i class="fa fa-check"></i><b>4.6</b> Handling persistence</a></li>
<li class="chapter" data-level="4.7" data-path="Data.html"><a href="Data.html#extensions"><i class="fa fa-check"></i><b>4.7</b> Extensions</a><ul>
<li class="chapter" data-level="4.7.1" data-path="Data.html"><a href="Data.html#transforming-features"><i class="fa fa-check"></i><b>4.7.1</b> Transforming features</a></li>
<li class="chapter" data-level="4.7.2" data-path="Data.html"><a href="Data.html#macrovar"><i class="fa fa-check"></i><b>4.7.2</b> Macro-economic variables</a></li>
<li class="chapter" data-level="4.7.3" data-path="Data.html"><a href="Data.html#active-learning"><i class="fa fa-check"></i><b>4.7.3</b> Active learning</a></li>
</ul></li>
<li class="chapter" data-level="4.8" data-path="Data.html"><a href="Data.html#additional-code-and-results"><i class="fa fa-check"></i><b>4.8</b> Additional code and results</a><ul>
<li class="chapter" data-level="4.8.1" data-path="Data.html"><a href="Data.html#impact-of-rescaling-graphical-representation"><i class="fa fa-check"></i><b>4.8.1</b> Impact of rescaling: graphical representation</a></li>
<li class="chapter" data-level="4.8.2" data-path="Data.html"><a href="Data.html#impact-of-rescaling-toy-example"><i class="fa fa-check"></i><b>4.8.2</b> Impact of rescaling: toy example</a></li>
</ul></li>
<li class="chapter" data-level="4.9" data-path="Data.html"><a href="Data.html#coding-exercises-1"><i class="fa fa-check"></i><b>4.9</b> Coding exercises</a></li>
</ul></li>
<li class="part"><span><b>II Common supervised algorithms</b></span></li>
<li class="chapter" data-level="5" data-path="lasso.html"><a href="lasso.html"><i class="fa fa-check"></i><b>5</b> Penalized regressions and sparse hedging for minimum variance portfolios</a><ul>
<li class="chapter" data-level="5.1" data-path="lasso.html"><a href="lasso.html#penalized-regressions"><i class="fa fa-check"></i><b>5.1</b> Penalized regressions</a><ul>
<li class="chapter" data-level="5.1.1" data-path="lasso.html"><a href="lasso.html#penreg"><i class="fa fa-check"></i><b>5.1.1</b> Simple regressions</a></li>
<li class="chapter" data-level="5.1.2" data-path="lasso.html"><a href="lasso.html#forms-of-penalizations"><i class="fa fa-check"></i><b>5.1.2</b> Forms of penalizations</a></li>
<li class="chapter" data-level="5.1.3" data-path="lasso.html"><a href="lasso.html#illustrations"><i class="fa fa-check"></i><b>5.1.3</b> Illustrations</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="lasso.html"><a href="lasso.html#sparse-hedging-for-minimum-variance-portfolios"><i class="fa fa-check"></i><b>5.2</b> Sparse hedging for minimum variance portfolios</a><ul>
<li class="chapter" data-level="5.2.1" data-path="lasso.html"><a href="lasso.html#presentation-and-derivations"><i class="fa fa-check"></i><b>5.2.1</b> Presentation and derivations</a></li>
<li class="chapter" data-level="5.2.2" data-path="lasso.html"><a href="lasso.html#sparseex"><i class="fa fa-check"></i><b>5.2.2</b> Example</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="lasso.html"><a href="lasso.html#predictive-regressions"><i class="fa fa-check"></i><b>5.3</b> Predictive regressions</a><ul>
<li class="chapter" data-level="5.3.1" data-path="lasso.html"><a href="lasso.html#literature-review-and-principle"><i class="fa fa-check"></i><b>5.3.1</b> Literature review and principle</a></li>
<li class="chapter" data-level="5.3.2" data-path="lasso.html"><a href="lasso.html#code-and-results"><i class="fa fa-check"></i><b>5.3.2</b> Code and results</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="lasso.html"><a href="lasso.html#coding-exercise"><i class="fa fa-check"></i><b>5.4</b> Coding exercise</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="trees.html"><a href="trees.html"><i class="fa fa-check"></i><b>6</b> Tree-based methods</a><ul>
<li class="chapter" data-level="6.1" data-path="trees.html"><a href="trees.html#simple-trees"><i class="fa fa-check"></i><b>6.1</b> Simple trees</a><ul>
<li class="chapter" data-level="6.1.1" data-path="trees.html"><a href="trees.html#principle"><i class="fa fa-check"></i><b>6.1.1</b> Principle</a></li>
<li class="chapter" data-level="6.1.2" data-path="trees.html"><a href="trees.html#treeclass"><i class="fa fa-check"></i><b>6.1.2</b> Further details on classification</a></li>
<li class="chapter" data-level="6.1.3" data-path="trees.html"><a href="trees.html#pruning-criteria"><i class="fa fa-check"></i><b>6.1.3</b> Pruning criteria</a></li>
<li class="chapter" data-level="6.1.4" data-path="trees.html"><a href="trees.html#code-and-interpretation"><i class="fa fa-check"></i><b>6.1.4</b> Code and interpretation</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="trees.html"><a href="trees.html#random-forests"><i class="fa fa-check"></i><b>6.2</b> Random forests</a><ul>
<li class="chapter" data-level="6.2.1" data-path="trees.html"><a href="trees.html#principle-1"><i class="fa fa-check"></i><b>6.2.1</b> Principle</a></li>
<li class="chapter" data-level="6.2.2" data-path="trees.html"><a href="trees.html#code-and-results-1"><i class="fa fa-check"></i><b>6.2.2</b> Code and results</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="trees.html"><a href="trees.html#adaboost"><i class="fa fa-check"></i><b>6.3</b> Boosted trees: Adaboost</a><ul>
<li class="chapter" data-level="6.3.1" data-path="trees.html"><a href="trees.html#methodology"><i class="fa fa-check"></i><b>6.3.1</b> Methodology</a></li>
<li class="chapter" data-level="6.3.2" data-path="trees.html"><a href="trees.html#illustration"><i class="fa fa-check"></i><b>6.3.2</b> Illustration</a></li>
</ul></li>
<li class="chapter" data-level="6.4" data-path="trees.html"><a href="trees.html#boosted-trees-extreme-gradient-boosting"><i class="fa fa-check"></i><b>6.4</b> Boosted trees: extreme gradient boosting</a><ul>
<li class="chapter" data-level="6.4.1" data-path="trees.html"><a href="trees.html#managing-loss"><i class="fa fa-check"></i><b>6.4.1</b> Managing loss</a></li>
<li class="chapter" data-level="6.4.2" data-path="trees.html"><a href="trees.html#penalization"><i class="fa fa-check"></i><b>6.4.2</b> Penalization</a></li>
<li class="chapter" data-level="6.4.3" data-path="trees.html"><a href="trees.html#aggregation"><i class="fa fa-check"></i><b>6.4.3</b> Aggregation</a></li>
<li class="chapter" data-level="6.4.4" data-path="trees.html"><a href="trees.html#tree-structure"><i class="fa fa-check"></i><b>6.4.4</b> Tree structure</a></li>
<li class="chapter" data-level="6.4.5" data-path="trees.html"><a href="trees.html#boostext"><i class="fa fa-check"></i><b>6.4.5</b> Extensions</a></li>
<li class="chapter" data-level="6.4.6" data-path="trees.html"><a href="trees.html#boostcode"><i class="fa fa-check"></i><b>6.4.6</b> Code and results</a></li>
<li class="chapter" data-level="6.4.7" data-path="trees.html"><a href="trees.html#instweight"><i class="fa fa-check"></i><b>6.4.7</b> Instance weighting</a></li>
</ul></li>
<li class="chapter" data-level="6.5" data-path="trees.html"><a href="trees.html#discussion"><i class="fa fa-check"></i><b>6.5</b> Discussion</a></li>
<li class="chapter" data-level="6.6" data-path="trees.html"><a href="trees.html#coding-exercises-2"><i class="fa fa-check"></i><b>6.6</b> Coding exercises</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="NN.html"><a href="NN.html"><i class="fa fa-check"></i><b>7</b> Neural networks</a><ul>
<li class="chapter" data-level="7.1" data-path="NN.html"><a href="NN.html#the-original-perceptron"><i class="fa fa-check"></i><b>7.1</b> The original perceptron</a></li>
<li class="chapter" data-level="7.2" data-path="NN.html"><a href="NN.html#multilayer-perceptron"><i class="fa fa-check"></i><b>7.2</b> Multilayer perceptron</a><ul>
<li class="chapter" data-level="7.2.1" data-path="NN.html"><a href="NN.html#introduction-and-notations"><i class="fa fa-check"></i><b>7.2.1</b> Introduction and notations</a></li>
<li class="chapter" data-level="7.2.2" data-path="NN.html"><a href="NN.html#universal-approximation"><i class="fa fa-check"></i><b>7.2.2</b> Universal approximation</a></li>
<li class="chapter" data-level="7.2.3" data-path="NN.html"><a href="NN.html#backprop"><i class="fa fa-check"></i><b>7.2.3</b> Learning via back-propagation</a></li>
<li class="chapter" data-level="7.2.4" data-path="NN.html"><a href="NN.html#further-details-on-classification"><i class="fa fa-check"></i><b>7.2.4</b> Further details on classification</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="NN.html"><a href="NN.html#howdeep"><i class="fa fa-check"></i><b>7.3</b> How deep we should go and other practical issues</a><ul>
<li class="chapter" data-level="7.3.1" data-path="NN.html"><a href="NN.html#architectural-choices"><i class="fa fa-check"></i><b>7.3.1</b> Architectural choices</a></li>
<li class="chapter" data-level="7.3.2" data-path="NN.html"><a href="NN.html#frequency-of-weight-updates-and-learning-duration"><i class="fa fa-check"></i><b>7.3.2</b> Frequency of weight updates and learning duration</a></li>
<li class="chapter" data-level="7.3.3" data-path="NN.html"><a href="NN.html#penalizations-and-dropout"><i class="fa fa-check"></i><b>7.3.3</b> Penalizations and dropout</a></li>
</ul></li>
<li class="chapter" data-level="7.4" data-path="NN.html"><a href="NN.html#code-samples-and-comments-for-vanilla-mlp"><i class="fa fa-check"></i><b>7.4</b> Code samples and comments for vanilla MLP</a><ul>
<li class="chapter" data-level="7.4.1" data-path="NN.html"><a href="NN.html#regression-example"><i class="fa fa-check"></i><b>7.4.1</b> Regression example</a></li>
<li class="chapter" data-level="7.4.2" data-path="NN.html"><a href="NN.html#classification-example"><i class="fa fa-check"></i><b>7.4.2</b> Classification example</a></li>
<li class="chapter" data-level="7.4.3" data-path="NN.html"><a href="NN.html#custloss"><i class="fa fa-check"></i><b>7.4.3</b> Custom losses</a></li>
</ul></li>
<li class="chapter" data-level="7.5" data-path="NN.html"><a href="NN.html#recurrent-networks"><i class="fa fa-check"></i><b>7.5</b> Recurrent networks</a><ul>
<li class="chapter" data-level="7.5.1" data-path="NN.html"><a href="NN.html#presentation"><i class="fa fa-check"></i><b>7.5.1</b> Presentation</a></li>
<li class="chapter" data-level="7.5.2" data-path="NN.html"><a href="NN.html#code-and-results-2"><i class="fa fa-check"></i><b>7.5.2</b> Code and results</a></li>
</ul></li>
<li class="chapter" data-level="7.6" data-path="NN.html"><a href="NN.html#other-common-architectures"><i class="fa fa-check"></i><b>7.6</b> Other common architectures</a><ul>
<li class="chapter" data-level="7.6.1" data-path="NN.html"><a href="NN.html#generative-aversarial-networks"><i class="fa fa-check"></i><b>7.6.1</b> Generative adversarial networks</a></li>
<li class="chapter" data-level="7.6.2" data-path="NN.html"><a href="NN.html#autoencoders"><i class="fa fa-check"></i><b>7.6.2</b> Autoencoders</a></li>
<li class="chapter" data-level="7.6.3" data-path="NN.html"><a href="NN.html#a-word-on-convolutional-networks"><i class="fa fa-check"></i><b>7.6.3</b> A word on convolutional networks</a></li>
<li class="chapter" data-level="7.6.4" data-path="NN.html"><a href="NN.html#advanced-architectures"><i class="fa fa-check"></i><b>7.6.4</b> Advanced architectures</a></li>
</ul></li>
<li class="chapter" data-level="7.7" data-path="NN.html"><a href="NN.html#coding-exercise-1"><i class="fa fa-check"></i><b>7.7</b> Coding exercise</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="svm.html"><a href="svm.html"><i class="fa fa-check"></i><b>8</b> Support vector machines</a><ul>
<li class="chapter" data-level="8.1" data-path="svm.html"><a href="svm.html#svm-for-classification"><i class="fa fa-check"></i><b>8.1</b> SVM for classification</a></li>
<li class="chapter" data-level="8.2" data-path="svm.html"><a href="svm.html#svm-for-regression"><i class="fa fa-check"></i><b>8.2</b> SVM for regression</a></li>
<li class="chapter" data-level="8.3" data-path="svm.html"><a href="svm.html#practice"><i class="fa fa-check"></i><b>8.3</b> Practice</a></li>
<li class="chapter" data-level="8.4" data-path="svm.html"><a href="svm.html#coding-exercises-3"><i class="fa fa-check"></i><b>8.4</b> Coding exercises</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="bayes.html"><a href="bayes.html"><i class="fa fa-check"></i><b>9</b> Bayesian methods</a><ul>
<li class="chapter" data-level="9.1" data-path="bayes.html"><a href="bayes.html#the-bayesian-framework"><i class="fa fa-check"></i><b>9.1</b> The Bayesian framework</a></li>
<li class="chapter" data-level="9.2" data-path="bayes.html"><a href="bayes.html#bayesian-sampling"><i class="fa fa-check"></i><b>9.2</b> Bayesian sampling</a><ul>
<li class="chapter" data-level="9.2.1" data-path="bayes.html"><a href="bayes.html#gibbs-sampling"><i class="fa fa-check"></i><b>9.2.1</b> Gibbs sampling</a></li>
<li class="chapter" data-level="9.2.2" data-path="bayes.html"><a href="bayes.html#metropolis-hastings-sampling"><i class="fa fa-check"></i><b>9.2.2</b> Metropolis-Hastings sampling</a></li>
</ul></li>
<li class="chapter" data-level="9.3" data-path="bayes.html"><a href="bayes.html#bayesian-linear-regression"><i class="fa fa-check"></i><b>9.3</b> Bayesian linear regression</a></li>
<li class="chapter" data-level="9.4" data-path="bayes.html"><a href="bayes.html#naive-bayes-classifier"><i class="fa fa-check"></i><b>9.4</b> Naive Bayes classifier</a></li>
<li class="chapter" data-level="9.5" data-path="bayes.html"><a href="bayes.html#BART"><i class="fa fa-check"></i><b>9.5</b> Bayesian additive trees</a><ul>
<li class="chapter" data-level="9.5.1" data-path="bayes.html"><a href="bayes.html#general-formulation"><i class="fa fa-check"></i><b>9.5.1</b> General formulation</a></li>
<li class="chapter" data-level="9.5.2" data-path="bayes.html"><a href="bayes.html#priors"><i class="fa fa-check"></i><b>9.5.2</b> Priors</a></li>
<li class="chapter" data-level="9.5.3" data-path="bayes.html"><a href="bayes.html#sampling-and-predictions"><i class="fa fa-check"></i><b>9.5.3</b> Sampling and predictions</a></li>
<li class="chapter" data-level="9.5.4" data-path="bayes.html"><a href="bayes.html#code"><i class="fa fa-check"></i><b>9.5.4</b> Code</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>III From predictions to portfolios</b></span></li>
<li class="chapter" data-level="10" data-path="valtune.html"><a href="valtune.html"><i class="fa fa-check"></i><b>10</b> Validating and tuning</a><ul>
<li class="chapter" data-level="10.1" data-path="valtune.html"><a href="valtune.html#mlmetrics"><i class="fa fa-check"></i><b>10.1</b> Learning metrics</a><ul>
<li class="chapter" data-level="10.1.1" data-path="valtune.html"><a href="valtune.html#regression-analysis"><i class="fa fa-check"></i><b>10.1.1</b> Regression analysis</a></li>
<li class="chapter" data-level="10.1.2" data-path="valtune.html"><a href="valtune.html#classification-analysis"><i class="fa fa-check"></i><b>10.1.2</b> Classification analysis</a></li>
</ul></li>
<li class="chapter" data-level="10.2" data-path="valtune.html"><a href="valtune.html#validation"><i class="fa fa-check"></i><b>10.2</b> Validation</a><ul>
<li class="chapter" data-level="10.2.1" data-path="valtune.html"><a href="valtune.html#the-variance-bias-tradeoff-theory"><i class="fa fa-check"></i><b>10.2.1</b> The variance-bias tradeoff: theory</a></li>
<li class="chapter" data-level="10.2.2" data-path="valtune.html"><a href="valtune.html#the-variance-bias-tradeoff-illustration"><i class="fa fa-check"></i><b>10.2.2</b> The variance-bias tradeoff: illustration</a></li>
<li class="chapter" data-level="10.2.3" data-path="valtune.html"><a href="valtune.html#the-risk-of-overfitting-principle"><i class="fa fa-check"></i><b>10.2.3</b> The risk of overfitting: principle</a></li>
<li class="chapter" data-level="10.2.4" data-path="valtune.html"><a href="valtune.html#the-risk-of-overfitting-some-solutions"><i class="fa fa-check"></i><b>10.2.4</b> The risk of overfitting: some solutions</a></li>
</ul></li>
<li class="chapter" data-level="10.3" data-path="valtune.html"><a href="valtune.html#the-search-for-good-hyperparameters"><i class="fa fa-check"></i><b>10.3</b> The search for good hyperparameters</a><ul>
<li class="chapter" data-level="10.3.1" data-path="valtune.html"><a href="valtune.html#methods"><i class="fa fa-check"></i><b>10.3.1</b> Methods</a></li>
<li class="chapter" data-level="10.3.2" data-path="valtune.html"><a href="valtune.html#example-grid-search"><i class="fa fa-check"></i><b>10.3.2</b> Example: grid search</a></li>
<li class="chapter" data-level="10.3.3" data-path="valtune.html"><a href="valtune.html#example-bayesian-optimization"><i class="fa fa-check"></i><b>10.3.3</b> Example: Bayesian optimization</a></li>
</ul></li>
<li class="chapter" data-level="10.4" data-path="valtune.html"><a href="valtune.html#short-discussion-on-validation-in-backtests"><i class="fa fa-check"></i><b>10.4</b> Short discussion on validation in backtests</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="ensemble.html"><a href="ensemble.html"><i class="fa fa-check"></i><b>11</b> Ensemble models</a><ul>
<li class="chapter" data-level="11.1" data-path="ensemble.html"><a href="ensemble.html#linear-ensembles"><i class="fa fa-check"></i><b>11.1</b> Linear ensembles</a><ul>
<li class="chapter" data-level="11.1.1" data-path="ensemble.html"><a href="ensemble.html#principles"><i class="fa fa-check"></i><b>11.1.1</b> Principles</a></li>
<li class="chapter" data-level="11.1.2" data-path="ensemble.html"><a href="ensemble.html#example"><i class="fa fa-check"></i><b>11.1.2</b> Example</a></li>
</ul></li>
<li class="chapter" data-level="11.2" data-path="ensemble.html"><a href="ensemble.html#stacked-ensembles"><i class="fa fa-check"></i><b>11.2</b> Stacked ensembles</a><ul>
<li class="chapter" data-level="11.2.1" data-path="ensemble.html"><a href="ensemble.html#two-stage-training"><i class="fa fa-check"></i><b>11.2.1</b> Two-stage training</a></li>
<li class="chapter" data-level="11.2.2" data-path="ensemble.html"><a href="ensemble.html#code-and-results-3"><i class="fa fa-check"></i><b>11.2.2</b> Code and results</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="ensemble.html"><a href="ensemble.html#extensions-1"><i class="fa fa-check"></i><b>11.3</b> Extensions</a><ul>
<li class="chapter" data-level="11.3.1" data-path="ensemble.html"><a href="ensemble.html#exogenous-variables"><i class="fa fa-check"></i><b>11.3.1</b> Exogenous variables</a></li>
<li class="chapter" data-level="11.3.2" data-path="ensemble.html"><a href="ensemble.html#shrinking-inter-model-correlations"><i class="fa fa-check"></i><b>11.3.2</b> Shrinking inter-model correlations</a></li>
</ul></li>
<li class="chapter" data-level="11.4" data-path="ensemble.html"><a href="ensemble.html#exercise"><i class="fa fa-check"></i><b>11.4</b> Exercise</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="backtest.html"><a href="backtest.html"><i class="fa fa-check"></i><b>12</b> Portfolio backtesting</a><ul>
<li class="chapter" data-level="12.1" data-path="backtest.html"><a href="backtest.html#protocol"><i class="fa fa-check"></i><b>12.1</b> Setting the protocol</a></li>
<li class="chapter" data-level="12.2" data-path="backtest.html"><a href="backtest.html#turning-signals-into-portfolio-weights"><i class="fa fa-check"></i><b>12.2</b> Turning signals into portfolio weights</a></li>
<li class="chapter" data-level="12.3" data-path="backtest.html"><a href="backtest.html#perfmet"><i class="fa fa-check"></i><b>12.3</b> Performance metrics</a><ul>
<li class="chapter" data-level="12.3.1" data-path="backtest.html"><a href="backtest.html#discussion-1"><i class="fa fa-check"></i><b>12.3.1</b> Discussion</a></li>
<li class="chapter" data-level="12.3.2" data-path="backtest.html"><a href="backtest.html#pure-performance-and-risk-indicators"><i class="fa fa-check"></i><b>12.3.2</b> Pure performance and risk indicators</a></li>
<li class="chapter" data-level="12.3.3" data-path="backtest.html"><a href="backtest.html#factor-based-evaluation"><i class="fa fa-check"></i><b>12.3.3</b> Factor-based evaluation</a></li>
<li class="chapter" data-level="12.3.4" data-path="backtest.html"><a href="backtest.html#risk-adjusted-measures"><i class="fa fa-check"></i><b>12.3.4</b> Risk-adjusted measures</a></li>
<li class="chapter" data-level="12.3.5" data-path="backtest.html"><a href="backtest.html#transaction-costs-and-turnover"><i class="fa fa-check"></i><b>12.3.5</b> Transaction costs and turnover</a></li>
</ul></li>
<li class="chapter" data-level="12.4" data-path="backtest.html"><a href="backtest.html#common-errors-and-issues"><i class="fa fa-check"></i><b>12.4</b> Common errors and issues</a><ul>
<li class="chapter" data-level="12.4.1" data-path="backtest.html"><a href="backtest.html#forward-looking-data"><i class="fa fa-check"></i><b>12.4.1</b> Forward looking data</a></li>
<li class="chapter" data-level="12.4.2" data-path="backtest.html"><a href="backtest.html#backov"><i class="fa fa-check"></i><b>12.4.2</b> Backtest overfitting</a></li>
<li class="chapter" data-level="12.4.3" data-path="backtest.html"><a href="backtest.html#simple-safeguards"><i class="fa fa-check"></i><b>12.4.3</b> Simple safeguards</a></li>
</ul></li>
<li class="chapter" data-level="12.5" data-path="backtest.html"><a href="backtest.html#implication-of-non-stationarity-forecasting-is-hard"><i class="fa fa-check"></i><b>12.5</b> Implication of non-stationarity: forecasting is hard</a><ul>
<li class="chapter" data-level="12.5.1" data-path="backtest.html"><a href="backtest.html#general-comments"><i class="fa fa-check"></i><b>12.5.1</b> General comments</a></li>
<li class="chapter" data-level="12.5.2" data-path="backtest.html"><a href="backtest.html#the-no-free-lunch-theorem"><i class="fa fa-check"></i><b>12.5.2</b> The no free lunch theorem</a></li>
</ul></li>
<li class="chapter" data-level="12.6" data-path="backtest.html"><a href="backtest.html#first-example-a-complete-backtest"><i class="fa fa-check"></i><b>12.6</b> First example: a complete backtest</a></li>
<li class="chapter" data-level="12.7" data-path="backtest.html"><a href="backtest.html#second-example-backtest-overfitting"><i class="fa fa-check"></i><b>12.7</b> Second example: backtest overfitting</a></li>
<li class="chapter" data-level="12.8" data-path="backtest.html"><a href="backtest.html#coding-exercises-4"><i class="fa fa-check"></i><b>12.8</b> Coding exercises</a></li>
</ul></li>
<li class="part"><span><b>IV Further important topics</b></span></li>
<li class="chapter" data-level="13" data-path="interp.html"><a href="interp.html"><i class="fa fa-check"></i><b>13</b> Interpretability</a><ul>
<li class="chapter" data-level="13.1" data-path="interp.html"><a href="interp.html#global-interpretations"><i class="fa fa-check"></i><b>13.1</b> Global interpretations</a><ul>
<li class="chapter" data-level="13.1.1" data-path="interp.html"><a href="interp.html#surr"><i class="fa fa-check"></i><b>13.1.1</b> Simple models as surrogates</a></li>
<li class="chapter" data-level="13.1.2" data-path="interp.html"><a href="interp.html#variable-importance"><i class="fa fa-check"></i><b>13.1.2</b> Variable importance (tree-based)</a></li>
<li class="chapter" data-level="13.1.3" data-path="interp.html"><a href="interp.html#variable-importance-agnostic"><i class="fa fa-check"></i><b>13.1.3</b> Variable importance (agnostic)</a></li>
<li class="chapter" data-level="13.1.4" data-path="interp.html"><a href="interp.html#partial-dependence-plot"><i class="fa fa-check"></i><b>13.1.4</b> Partial dependence plot</a></li>
</ul></li>
<li class="chapter" data-level="13.2" data-path="interp.html"><a href="interp.html#local-interpretations"><i class="fa fa-check"></i><b>13.2</b> Local interpretations</a><ul>
<li class="chapter" data-level="13.2.1" data-path="interp.html"><a href="interp.html#lime"><i class="fa fa-check"></i><b>13.2.1</b> LIME</a></li>
<li class="chapter" data-level="13.2.2" data-path="interp.html"><a href="interp.html#shapley-values"><i class="fa fa-check"></i><b>13.2.2</b> Shapley values</a></li>
<li class="chapter" data-level="13.2.3" data-path="interp.html"><a href="interp.html#breakdown"><i class="fa fa-check"></i><b>13.2.3</b> Breakdown</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="14" data-path="causality.html"><a href="causality.html"><i class="fa fa-check"></i><b>14</b> Two key concepts: causality and non-stationarity</a><ul>
<li class="chapter" data-level="14.1" data-path="causality.html"><a href="causality.html#causality-1"><i class="fa fa-check"></i><b>14.1</b> Causality</a><ul>
<li class="chapter" data-level="14.1.1" data-path="causality.html"><a href="causality.html#granger"><i class="fa fa-check"></i><b>14.1.1</b> Granger causality</a></li>
<li class="chapter" data-level="14.1.2" data-path="causality.html"><a href="causality.html#causal-additive-models"><i class="fa fa-check"></i><b>14.1.2</b> Causal additive models</a></li>
<li class="chapter" data-level="14.1.3" data-path="causality.html"><a href="causality.html#structural-time-series-models"><i class="fa fa-check"></i><b>14.1.3</b> Structural time series models</a></li>
</ul></li>
<li class="chapter" data-level="14.2" data-path="causality.html"><a href="causality.html#nonstat"><i class="fa fa-check"></i><b>14.2</b> Dealing with changing environments</a><ul>
<li class="chapter" data-level="14.2.1" data-path="causality.html"><a href="causality.html#non-stationarity-yet-another-illustration"><i class="fa fa-check"></i><b>14.2.1</b> Non-stationarity: yet another illustration</a></li>
<li class="chapter" data-level="14.2.2" data-path="causality.html"><a href="causality.html#online-learning"><i class="fa fa-check"></i><b>14.2.2</b> Online learning</a></li>
<li class="chapter" data-level="14.2.3" data-path="causality.html"><a href="causality.html#homogeneous-transfer-learning"><i class="fa fa-check"></i><b>14.2.3</b> Homogeneous transfer learning</a></li>
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</ul></li>
<li class="chapter" data-level="15" data-path="unsup.html"><a href="unsup.html"><i class="fa fa-check"></i><b>15</b> Unsupervised learning</a><ul>
<li class="chapter" data-level="15.1" data-path="unsup.html"><a href="unsup.html#corpred"><i class="fa fa-check"></i><b>15.1</b> The problem with correlated predictors</a></li>
<li class="chapter" data-level="15.2" data-path="unsup.html"><a href="unsup.html#principal-component-analysis-and-autoencoders"><i class="fa fa-check"></i><b>15.2</b> Principal component analysis and autoencoders</a><ul>
<li class="chapter" data-level="15.2.1" data-path="unsup.html"><a href="unsup.html#a-bit-of-algebra"><i class="fa fa-check"></i><b>15.2.1</b> A bit of algebra</a></li>
<li class="chapter" data-level="15.2.2" data-path="unsup.html"><a href="unsup.html#pca"><i class="fa fa-check"></i><b>15.2.2</b> PCA</a></li>
<li class="chapter" data-level="15.2.3" data-path="unsup.html"><a href="unsup.html#ae"><i class="fa fa-check"></i><b>15.2.3</b> Autoencoders</a></li>
<li class="chapter" data-level="15.2.4" data-path="unsup.html"><a href="unsup.html#application"><i class="fa fa-check"></i><b>15.2.4</b> Application</a></li>
</ul></li>
<li class="chapter" data-level="15.3" data-path="unsup.html"><a href="unsup.html#clustering-via-k-means"><i class="fa fa-check"></i><b>15.3</b> Clustering via k-means</a></li>
<li class="chapter" data-level="15.4" data-path="unsup.html"><a href="unsup.html#nearest-neighbors"><i class="fa fa-check"></i><b>15.4</b> Nearest neighbors</a></li>
<li class="chapter" data-level="15.5" data-path="unsup.html"><a href="unsup.html#coding-exercise-2"><i class="fa fa-check"></i><b>15.5</b> Coding exercise</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="RL.html"><a href="RL.html"><i class="fa fa-check"></i><b>16</b> Reinforcement learning</a><ul>
<li class="chapter" data-level="16.1" data-path="RL.html"><a href="RL.html#theoretical-layout"><i class="fa fa-check"></i><b>16.1</b> Theoretical layout</a><ul>
<li class="chapter" data-level="16.1.1" data-path="RL.html"><a href="RL.html#general-framework"><i class="fa fa-check"></i><b>16.1.1</b> General framework</a></li>
<li class="chapter" data-level="16.1.2" data-path="RL.html"><a href="RL.html#q-learning"><i class="fa fa-check"></i><b>16.1.2</b> Q-learning</a></li>
<li class="chapter" data-level="16.1.3" data-path="RL.html"><a href="RL.html#sarsa"><i class="fa fa-check"></i><b>16.1.3</b> SARSA</a></li>
</ul></li>
<li class="chapter" data-level="16.2" data-path="RL.html"><a href="RL.html#the-curse-of-dimensionality"><i class="fa fa-check"></i><b>16.2</b> The curse of dimensionality</a></li>
<li class="chapter" data-level="16.3" data-path="RL.html"><a href="RL.html#policy-gradient"><i class="fa fa-check"></i><b>16.3</b> Policy gradient</a><ul>
<li class="chapter" data-level="16.3.1" data-path="RL.html"><a href="RL.html#principle-2"><i class="fa fa-check"></i><b>16.3.1</b> Principle</a></li>
<li class="chapter" data-level="16.3.2" data-path="RL.html"><a href="RL.html#extensions-2"><i class="fa fa-check"></i><b>16.3.2</b> Extensions</a></li>
</ul></li>
<li class="chapter" data-level="16.4" data-path="RL.html"><a href="RL.html#simple-examples"><i class="fa fa-check"></i><b>16.4</b> Simple examples</a><ul>
<li class="chapter" data-level="16.4.1" data-path="RL.html"><a href="RL.html#q-learning-with-simulations"><i class="fa fa-check"></i><b>16.4.1</b> Q-learning with simulations</a></li>
<li class="chapter" data-level="16.4.2" data-path="RL.html"><a href="RL.html#RLemp2"><i class="fa fa-check"></i><b>16.4.2</b> Q-learning with market data</a></li>
</ul></li>
<li class="chapter" data-level="16.5" data-path="RL.html"><a href="RL.html#concluding-remarks"><i class="fa fa-check"></i><b>16.5</b> Concluding remarks</a></li>
<li class="chapter" data-level="16.6" data-path="RL.html"><a href="RL.html#exercises"><i class="fa fa-check"></i><b>16.6</b> Exercises</a></li>
</ul></li>
<li class="part"><span><b>V Appendix</b></span></li>
<li class="chapter" data-level="17" data-path="data-description.html"><a href="data-description.html"><i class="fa fa-check"></i><b>17</b> Data description</a></li>
<li class="chapter" data-level="18" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html"><i class="fa fa-check"></i><b>18</b> Solutions to exercises</a><ul>
<li class="chapter" data-level="18.1" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-3"><i class="fa fa-check"></i><b>18.1</b> Chapter 3</a></li>
<li class="chapter" data-level="18.2" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-4"><i class="fa fa-check"></i><b>18.2</b> Chapter 4</a></li>
<li class="chapter" data-level="18.3" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-5"><i class="fa fa-check"></i><b>18.3</b> Chapter 5</a></li>
<li class="chapter" data-level="18.4" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-6"><i class="fa fa-check"></i><b>18.4</b> Chapter 6</a></li>
<li class="chapter" data-level="18.5" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-7-the-autoencoder-model"><i class="fa fa-check"></i><b>18.5</b> Chapter 7: the autoencoder model</a></li>
<li class="chapter" data-level="18.6" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-8"><i class="fa fa-check"></i><b>18.6</b> Chapter 8</a></li>
<li class="chapter" data-level="18.7" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-11-ensemble-neural-network"><i class="fa fa-check"></i><b>18.7</b> Chapter 11: ensemble neural network</a></li>
<li class="chapter" data-level="18.8" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-12"><i class="fa fa-check"></i><b>18.8</b> Chapter 12</a><ul>
<li class="chapter" data-level="18.8.1" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#ew-portfolios-with-the-tidyverse"><i class="fa fa-check"></i><b>18.8.1</b> EW portfolios with the tidyverse</a></li>
<li class="chapter" data-level="18.8.2" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#advanced-weighting-function"><i class="fa fa-check"></i><b>18.8.2</b> Advanced weighting function</a></li>
<li class="chapter" data-level="18.8.3" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#functional-programming-in-the-backtest"><i class="fa fa-check"></i><b>18.8.3</b> Functional programming in the backtest</a></li>
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<li class="chapter" data-level="18.9" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-15"><i class="fa fa-check"></i><b>18.9</b> Chapter 15</a></li>
<li class="chapter" data-level="18.10" data-path="solutions-to-exercises.html"><a href="solutions-to-exercises.html#chapter-16"><i class="fa fa-check"></i><b>18.10</b> Chapter 16</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Machine Learning for Factor Investing</a>
</h1>
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<section class="normal" id="section-">
<div id="notdata" class="section level1">
<h1><span class="header-section-number">Chapter 1</span> Notations and data</h1>
<div id="notations" class="section level2">
<h2><span class="header-section-number">1.1</span> Notations</h2>
<p>This section aims at providing the formal mathematical conventions that will be used throughout the book.</p>
<p>Bold notations indicate vectors and matrices. We use capital letters for matrices and lower case letters for vectors.
<span class="math inline">\(\mathbf{v}'\)</span> and <span class="math inline">\(\mathbf{M}'\)</span> denote the transposes of <span class="math inline">\(\mathbf{v}\)</span> and <span class="math inline">\(\mathbf{M}\)</span>. <span class="math inline">\(\mathbf{M}=[m]_{i,j}\)</span>, where <span class="math inline">\(i\)</span> is the row index and <span class="math inline">\(j\)</span> the column index.</p>
<p>We will work with two notations in parallel. The first one is the pure machine learning notation in which the <strong>labels</strong> (also called <strong>output</strong>, <strong>dependent</strong> variables or <strong>predicted</strong> variables) <span class="math inline">\(\mathbf{y}=y_i\)</span> are approximated by functions of features <span class="math inline">\(\mathbf{X}_i=(x_{i,1},\dots,x_{i,K})\)</span>. The dimension of the feature matrix <span class="math inline">\(\mathbf{X}\)</span> is <span class="math inline">\(I\times K\)</span>: there are <span class="math inline">\(I\)</span> <strong>instances</strong>, <strong>records</strong>, or <strong>observations</strong> and each one of them has <span class="math inline">\(K\)</span> <strong>attributes</strong>, <strong>features</strong>, <strong>inputs</strong>, or <strong>predictors</strong> which will serve as <strong>independent</strong> and <strong>explanatory</strong> variables (all these terms will be used interchangeably). Sometimes, to ease notations, we will write <span class="math inline">\(\textbf{x}_i\)</span> for one instance (one row) of <span class="math inline">\(\textbf{X}\)</span> or <span class="math inline">\(\textbf{x}_k\)</span> for one (feature) column vector of <span class="math inline">\(\textbf{X}\)</span>.</p>
<p>The second notation type pertains to finance and will directly relate to the first. We will often work with discrete returns <span class="math inline">\(r_{t,n}=p_{t,n}/p_{t-1,n}-1\)</span> computed from price data. Here <span class="math inline">\(t\)</span> is the time index and <span class="math inline">\(n\)</span> the asset index. Unless specified otherwise, the return is always computed over one period, though this period can sometimes be one month or one year. Whenever confusion might occur, we will specify other notations for returns.</p>
<p>In line with our previous conventions, the number of return dates will be <span class="math inline">\(T\)</span> and the number of assets, <span class="math inline">\(N\)</span>. The features or characteristics of assets will be denoted with <span class="math inline">\(x_{t,n}^{(k)}\)</span>: it is the time-<span class="math inline">\(t\)</span> value of the <span class="math inline">\(k^{th}\)</span> attribute of firm or asset <span class="math inline">\(n\)</span>. In stacked notation, <span class="math inline">\(\mathbf{x}_{t,n}\)</span> will stand for the vector of characteristics of asset <span class="math inline">\(n\)</span> at time <span class="math inline">\(t\)</span>. Moreover, <span class="math inline">\(\mathbf{r}_t\)</span> stands for all returns at time <span class="math inline">\(t\)</span> while <span class="math inline">\(\mathbf{r}_n\)</span> stands for all returns of asset <span class="math inline">\(n\)</span>. Often, returns will play the role of the dependent variable, or label (in ML terms). For the riskless asset, we will use the notation <span class="math inline">\(r_{t,f}\)</span>.</p>
<p>The link between the two notations will most of the time be the following. One <strong>instance</strong> (or <strong>observation</strong>) <span class="math inline">\(i\)</span> will consist of one couple (<span class="math inline">\(t,n\)</span>) of one particular date and one particular firm (if the data is perfectly rectangular with no missing field, <span class="math inline">\(I=T\times N\)</span>). The label will usually be some performance measure of the firm computed over some future period, while the features will consist of the firm attributes at time <span class="math inline">\(t\)</span>. Hence, the purpose of the machine learning engine in factor investing will be to determine the model that maps the time-<span class="math inline">\(t\)</span> characteristics of firms to their future performance.</p>
<p>In terms of canonical matrices: <span class="math inline">\(\mathbf{I}_N\)</span> will denote the <span class="math inline">\((N\times N)\)</span> identity matrix.</p>
<p>From the probabilistic literature, we employ the expectation operator <span class="math inline">\(\mathbb{E}[\cdot]\)</span> and the conditional expectation <span class="math inline">\(\mathbb{E}_t[\cdot]\)</span>, where the corresponding filtration <span class="math inline">\(\mathcal{F}_t\)</span> corresponds to all information available at time <span class="math inline">\(t\)</span>. More precisely, <span class="math inline">\(\mathbb{E}_t[\cdot]=\mathbb{E}[\cdot | \mathcal{F}_t]\)</span>. <span class="math inline">\(\mathbb{V}[\cdot]\)</span> will denote the variance operator. Depending on the context, probabilities will be written simply <span class="math inline">\(P\)</span>, but sometimes we will use the heavier notation <span class="math inline">\(\mathbb{P}\)</span>. Probability density functions (pdfs) will be denoted with lowercase letters (<span class="math inline">\(f\)</span>) and cumulative distribution functions (cdfs) with uppercase letters (<span class="math inline">\(F\)</span>). We will write equality in distribution as <span class="math inline">\(X \overset{d}{=}Y\)</span>, which is equivalent to <span class="math inline">\(F_X(z)=F_Y(z)\)</span> for all <span class="math inline">\(z\)</span> on the support of the variables. For a random process <span class="math inline">\(X_t\)</span>, we say that it is <strong>stationary</strong> if the law of <span class="math inline">\(X_t\)</span> is constant through time, i.e., <span class="math inline">\(X_t\overset{d}{=}X_s\)</span>, where <span class="math inline">\(\overset{d}{=}\)</span> means equality in distribution.</p>
<p>Sometimes, asymptotic behaviors will be characterized with the usual Landau notation <span class="math inline">\(o(\cdot)\)</span> and <span class="math inline">\(O(\cdot)\)</span>. The symbol <span class="math inline">\(\propto\)</span> refers to proportionality: <span class="math inline">\(x\propto y\)</span> means that <span class="math inline">\(x\)</span> is proportional to <span class="math inline">\(y\)</span>. With respect to derivatives, we use the standard notation <span class="math inline">\(\frac{\partial}{\partial x}\)</span> when differentiating with respect to <span class="math inline">\(x\)</span>. We resort to the compact symbol <span class="math inline">\(\nabla\)</span> when all derivatives are computed (gradient vector).</p>
<p>In equations, the left-hand side and right-hand side can be written more compactly: l.h.s. and r.h.s., respectively.</p>
<p>Finally, we turn to functions. We list a few below:<br />
- <span class="math inline">\(1_{\{x \}}\)</span>: the indicator function of the condition <span class="math inline">\(x\)</span>, which is equal to one if <span class="math inline">\(x\)</span> is true and to zero otherwise.<br />
- <span class="math inline">\(\phi(\cdot)\)</span> and <span class="math inline">\(\Phi(\cdot)\)</span> are the standard Gaussian pdf and cdf.<br />
- card<span class="math inline">\((\cdot)=\#(\cdot)\)</span> are two notations for the cardinal function which evaluates the number of elements in a given set (provided as argument of the function).<br />
- <span class="math inline">\(\lfloor \cdot \rfloor\)</span> is the integer part function.<br />
- for a real number <span class="math inline">\(x\)</span>, <span class="math inline">\([x]^+\)</span> is the positive part of <span class="math inline">\(x\)</span>, that is <span class="math inline">\(\max(0,x)\)</span>.<br />
- tanh<span class="math inline">\((\cdot)\)</span> is the hyperbolic tangent: tanh<span class="math inline">\((x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\)</span>.<br />
- ReLu<span class="math inline">\((\cdot)\)</span> is the rectified linear unit: ReLu<span class="math inline">\((x)=\max(0,x)\)</span>.<br />
- s<span class="math inline">\((\cdot)\)</span> will be the softmax function: <span class="math inline">\(s(\textbf{x})_i=\frac{e^{x_i}}{\sum_{j=1}^Je^{x_j}}\)</span>, where the subscript <span class="math inline">\(i\)</span> refers to the <span class="math inline">\(i^{th}\)</span> element of the vector.</p>
</div>
<div id="dataset" class="section level2">
<h2><span class="header-section-number">1.2</span> Dataset</h2>
<p>Throughout the book, and for the sake of <strong>reproducibility</strong>, we will illustrate the concepts we present with examples of implementation based on a single financial dataset available at <a href="https://github.com/shokru/mlfactor.github.io/tree/master/material" class="uri">https://github.com/shokru/mlfactor.github.io/tree/master/material</a>. This dataset comprises information on 1,207 stocks listed in the US (possibly originating from Canada or Mexico). The time range starts in November 1998 and ends in March 2019. For each point in time, 93 <strong>characteristics</strong> describe the firms in the sample. These attributes cover a wide range of topics:</p>
<ul>
<li><strong>valuation</strong> (earning yields, accounting ratios);<br />
</li>
<li><strong>profitability</strong> and quality (return on equity);<br />
</li>
<li><strong>momentum</strong> and technical analysis (past returns, relative strength index);<br />
</li>
<li><strong>risk</strong> (volatilities);<br />
</li>
<li><strong>estimates</strong> (earnings-per-share);<br />
</li>
<li><strong>volume</strong> and <strong>liquidity</strong> (share turnover).</li>
</ul>
<p>The sample is not perfectly rectangular: there are no missing points, but the number of firms and their attributes is not constant through time. This makes the computations in the backtest more tricky, but also more realistic.</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="notdata.html#cb3-1"></a><span class="kw">library</span>(tidyverse) <span class="co"># Activate the data science package</span></span>
<span id="cb3-2"><a href="notdata.html#cb3-2"></a><span class="kw">library</span>(lubridate) <span class="co"># Activate the date management package</span></span>
<span id="cb3-3"><a href="notdata.html#cb3-3"></a><span class="kw">load</span>(<span class="st">"data_ml.RData"</span>) <span class="co"># Load the data</span></span>
<span id="cb3-4"><a href="notdata.html#cb3-4"></a>data_ml <-<span class="st"> </span>data_ml <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb3-5"><a href="notdata.html#cb3-5"></a><span class="st"> </span><span class="kw">filter</span>(date <span class="op">></span><span class="st"> "1999-12-31"</span>, <span class="co"># Keep the date with sufficient data points</span></span>
<span id="cb3-6"><a href="notdata.html#cb3-6"></a> date <span class="op"><</span><span class="st"> "2019-01-01"</span>) <span class="op">%>%</span></span>
<span id="cb3-7"><a href="notdata.html#cb3-7"></a><span class="st"> </span><span class="kw">arrange</span>(stock_id, date) <span class="co"># Order the data</span></span>
<span id="cb3-8"><a href="notdata.html#cb3-8"></a>data_ml[<span class="dv">1</span><span class="op">:</span><span class="dv">6</span>, <span class="dv">1</span><span class="op">:</span><span class="dv">6</span>] <span class="co"># Sample values</span></span></code></pre></div>
<pre><code>## # A tibble: 6 x 6
## stock_id date Advt_12M_Usd Advt_3M_Usd Advt_6M_Usd Asset_Turnover
## <int> <date> <dbl> <dbl> <dbl> <dbl>
## 1 1 2000-01-31 0.41 0.39 0.42 0.19
## 2 1 2000-02-29 0.41 0.39 0.4 0.19
## 3 1 2000-03-31 0.4 0.37 0.37 0.2
## 4 1 2000-04-30 0.39 0.36 0.37 0.2
## 5 1 2000-05-31 0.4 0.42 0.4 0.2
## 6 1 2000-06-30 0.41 0.47 0.42 0.21</code></pre>
<p></p>
<p>The data has 99 columns and 268336 rows. The first two columns indicate the stock identifier and the date. The next 93 columns are the features (see Table <a href="data-description.html#tab:appendix1">17.1</a> in the Appendix for details). The last four columns are the labels. The points are sampled at the monthly frequency. As is always the case in practice, the number of assets changes with time, as is shown in Figure <a href="notdata.html#fig:nbassets">1.1</a>.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="notdata.html#cb5-1"></a>data_ml <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb5-2"><a href="notdata.html#cb5-2"></a><span class="st"> </span><span class="kw">group_by</span>(date) <span class="op">%>%</span><span class="st"> </span><span class="co"># Group by date</span></span>
<span id="cb5-3"><a href="notdata.html#cb5-3"></a><span class="st"> </span><span class="kw">summarize</span>(<span class="dt">nb_assets =</span> stock_id <span class="op">%>%</span><span class="st"> </span><span class="co"># Count nb assets</span></span>
<span id="cb5-4"><a href="notdata.html#cb5-4"></a><span class="st"> </span><span class="kw">as.factor</span>() <span class="op">%>%</span><span class="st"> </span><span class="kw">nlevels</span>()) <span class="op">%>%</span></span>
<span id="cb5-5"><a href="notdata.html#cb5-5"></a><span class="st"> </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> date, <span class="dt">y =</span> nb_assets)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_col</span>() <span class="op">+</span><span class="st"> </span><span class="co"># Plot</span></span>
<span id="cb5-6"><a href="notdata.html#cb5-6"></a><span class="st"> </span><span class="kw">coord_fixed</span>(<span class="dv">3</span>)</span></code></pre></div>
<div class="figure" style="text-align: center"><span id="fig:nbassets"></span>
<img src="ML_factor_files/figure-html/nbassets-1.png" alt="Number of assets through time." width="600" />
<p class="caption">
FIGURE 1.1: Number of assets through time.
</p>
</div>
<p></p>
<p>There are four immediate <strong>labels</strong> in the dataset: R1M_Usd, R3M_Usd, R6M_Usd and R12M_Usd, which correspond to the 1-month, 3-month, 6-month and 12-month future/forward returns of the stocks. The returns are <strong>total returns</strong>, that is, they incorporate potential <strong>dividend</strong> payments over the considered periods. This is a better proxy of financial gain compared to price returns only. We refer to the analysis of <span class="citation">Hartzmark and Solomon (<a href="#ref-hartzmark2019dividend" role="doc-biblioref">2019</a>)</span> for a study on the impact of decoupling price returns and dividends. These labels are located in the last 4 columns of the dataset. We provide their descriptive statistics below.</p>
<pre><code>## # A tibble: 4 x 5
## Label mean sd min max
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 R12M_Usd 0.137 0.738 -0.991 96.0
## 2 R1M_Usd 0.0127 0.176 -0.922 30.2
## 3 R3M_Usd 0.0369 0.328 -0.929 39.4
## 4 R6M_Usd 0.0723 0.527 -0.98 107.</code></pre>
<p></p>
<p>In anticipation for future models, we keep the name of the predictors in memory. In addition, we also keep a much shorter list of predictors.</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="notdata.html#cb7-1"></a>features <-<span class="st"> </span><span class="kw">colnames</span>(data_ml[<span class="dv">3</span><span class="op">:</span><span class="dv">95</span>]) <span class="co"># Keep the feature's column names (hard-coded, beware!)</span></span>
<span id="cb7-2"><a href="notdata.html#cb7-2"></a>features_short <-<span class="st"> </span><span class="kw">c</span>(<span class="st">"Div_Yld"</span>, <span class="st">"Eps"</span>, <span class="st">"Mkt_Cap_12M_Usd"</span>, <span class="st">"Mom_11M_Usd"</span>, </span>
<span id="cb7-3"><a href="notdata.html#cb7-3"></a> <span class="st">"Ocf"</span>, <span class="st">"Pb"</span>, <span class="st">"Vol1Y_Usd"</span>)</span></code></pre></div>
<p></p>
<p>The predictors have been uniformized, that is, for any given feature and time point, the distribution is uniform. Given 1,207 stocks, the graph below cannot display a perfect rectangle.</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="notdata.html#cb8-1"></a>data_ml <span class="op">%>%</span></span>
<span id="cb8-2"><a href="notdata.html#cb8-2"></a><span class="st"> </span><span class="kw">filter</span>(date <span class="op">==</span><span class="st"> "2000-02-29"</span>) <span class="op">%>%</span></span>
<span id="cb8-3"><a href="notdata.html#cb8-3"></a><span class="st"> </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(<span class="dt">x =</span> Div_Yld)) <span class="op">+</span><span class="st"> </span><span class="kw">geom_histogram</span>(<span class="dt">bins =</span> <span class="dv">100</span>) <span class="op">+</span><span class="st"> </span><span class="kw">coord_fixed</span>(<span class="fl">0.03</span>)</span></code></pre></div>
<div class="figure" style="text-align: center"><span id="fig:datarectangle"></span>
<img src="ML_factor_files/figure-html/datarectangle-1.png" alt="Distribution of the dividend yield feature on date 2000-02-29." width="600" />
<p class="caption">
FIGURE 1.2: Distribution of the dividend yield feature on date 2000-02-29.
</p>
</div>
<p></p>
<p>The original labels (future returns) are numerical and will be used for regression exercises, that is, when the objective is to predict a scalar real number. Sometimes, the exercises can be different and the purpose may be to forecast <strong>categories</strong> (also called <strong>classes</strong>), like “buy”, “hold” or “sell”. In order to be able to perform this type of classification analysis, we create additional labels that are categorical.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="notdata.html#cb9-1"></a>data_ml <-<span class="st"> </span>data_ml <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb9-2"><a href="notdata.html#cb9-2"></a><span class="st"> </span><span class="kw">group_by</span>(date) <span class="op">%>%</span><span class="st"> </span><span class="co"># Group by date</span></span>
<span id="cb9-3"><a href="notdata.html#cb9-3"></a><span class="st"> </span><span class="kw">mutate</span>(<span class="dt">R1M_Usd_C =</span> R1M_Usd <span class="op">></span><span class="st"> </span><span class="kw">median</span>(R1M_Usd), <span class="co"># Create the categorical labels</span></span>
<span id="cb9-4"><a href="notdata.html#cb9-4"></a> <span class="dt">R12M_Usd_C =</span> R12M_Usd <span class="op">></span><span class="st"> </span><span class="kw">median</span>(R12M_Usd)) <span class="op">%>%</span></span>
<span id="cb9-5"><a href="notdata.html#cb9-5"></a><span class="st"> </span><span class="kw">ungroup</span>() <span class="op">%>%</span></span>
<span id="cb9-6"><a href="notdata.html#cb9-6"></a><span class="st"> </span><span class="kw">mutate_if</span>(is.logical, as.factor)</span></code></pre></div>
<p></p>
<p>The new labels are binary: they are equal to 1 (true) if the original return is above that of the median return over the considered period and to 0 (false) if not. Hence, at each point in time, half of the sample has a label equal to zero and the other half to one: some stocks overperform and others underperform.</p>
<p>In machine learning, models are estimated on one portion of data (<strong>training set</strong>) and then tested on another portion of the data (<strong>testing set</strong>) to assess their quality. We split our sample accordingly.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="notdata.html#cb10-1"></a>separation_date <-<span class="st"> </span><span class="kw">as.Date</span>(<span class="st">"2014-01-15"</span>)</span>
<span id="cb10-2"><a href="notdata.html#cb10-2"></a>training_sample <-<span class="st"> </span><span class="kw">filter</span>(data_ml, date <span class="op"><</span><span class="st"> </span>separation_date)</span>
<span id="cb10-3"><a href="notdata.html#cb10-3"></a>testing_sample <-<span class="st"> </span><span class="kw">filter</span>(data_ml, date <span class="op">>=</span><span class="st"> </span>separation_date)</span></code></pre></div>
<p></p>
<p>We also keep in memory a few key variables, like the list of asset identifiers and a rectangular version of returns. For simplicity, in the computation of the latter, we shrink the investment universe to keep only the stocks for which we have the maximum number of points.</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="notdata.html#cb11-1"></a>stock_ids <-<span class="st"> </span><span class="kw">levels</span>(<span class="kw">as.factor</span>(data_ml<span class="op">$</span>stock_id)) <span class="co"># A list of all stock_ids</span></span>
<span id="cb11-2"><a href="notdata.html#cb11-2"></a>stock_days <-<span class="st"> </span>data_ml <span class="op">%>%</span><span class="st"> </span><span class="co"># Compute the number of data points per stock</span></span>
<span id="cb11-3"><a href="notdata.html#cb11-3"></a><span class="st"> </span><span class="kw">group_by</span>(stock_id) <span class="op">%>%</span><span class="st"> </span><span class="kw">summarize</span>(<span class="dt">nb =</span> <span class="kw">n</span>()) </span>
<span id="cb11-4"><a href="notdata.html#cb11-4"></a>stock_ids_short <-<span class="st"> </span>stock_ids[<span class="kw">which</span>(stock_days<span class="op">$</span>nb <span class="op">==</span><span class="st"> </span><span class="kw">max</span>(stock_days<span class="op">$</span>nb))] <span class="co"># Stocks with full data</span></span>
<span id="cb11-5"><a href="notdata.html#cb11-5"></a>returns <-<span class="st"> </span>data_ml <span class="op">%>%</span><span class="st"> </span><span class="co"># Compute returns, in matrix format, in 3 steps:</span></span>
<span id="cb11-6"><a href="notdata.html#cb11-6"></a><span class="st"> </span><span class="kw">filter</span>(stock_id <span class="op">%in%</span><span class="st"> </span>stock_ids_short) <span class="op">%>%</span><span class="st"> </span><span class="co"># 1. Filtering the data</span></span>
<span id="cb11-7"><a href="notdata.html#cb11-7"></a><span class="st"> </span>dplyr<span class="op">::</span><span class="kw">select</span>(date, stock_id, R1M_Usd) <span class="op">%>%</span><span class="st"> </span><span class="co"># 2. Keep returns along with dates & firm names</span></span>
<span id="cb11-8"><a href="notdata.html#cb11-8"></a><span class="st"> </span><span class="kw">spread</span>(<span class="dt">key =</span> stock_id, <span class="dt">value =</span> R1M_Usd) <span class="co"># 3. Put in matrix shape </span></span></code></pre></div>
<p></p>
</div>
</div>
<h3>References</h3>
<div id="refs" class="references">
<div id="ref-hartzmark2019dividend">
<p>Hartzmark, Samuel M, and David H Solomon. 2019. “The Dividend Disconnect.” <em>Journal of Finance</em> 74 (5): 2153–99.</p>
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