class: center, middle, inverse, title-slide # Machine Learning for Social Scientists ## Loss functions and decision trees ### Jorge Cimentada ### 2022-02-18 --- layout: true <!-- background-image: url(./figs/upf.png) --> background-position: 100% 0%, 100% 0%, 50% 100% background-size: 10%, 10%, 10% --- # What are loss functions? * Social Scientists use metrics such as the `\(R^2\)`, `\(AIC\)`, `\(Log\text{ }likelihood\)` or `\(BIC\)`. * We almost always use these metrics and their purpose is to inform some of our modeling choices. * In machine learning, metrics such as the `\(R^2\)` and the `\(AIC\)` are called 'loss functions' * There are two types of loss functions: continuous and binary --- # Root Mean Square Error (RMSE) Subtract the actual `\(Y_{i}\)` score of each respondent from the predicted `\(\hat{Y_{i}}\)` for each respondent: <img src="03_loss_trees_files/figure-html/unnamed-chunk-3-1.png" width="70%" style="display: block; margin: auto;" /> `$$RMSE = \sqrt{\sum_{i = 1}^n{\frac{(\hat{y_{i}} - y_{i})^2}{N}}}$$` --- # Mean Absolute Error (MAE) * This approach doesn't penalize any values and just takes the absolute error of the predictions. * Fundamentally simpler to interpret than the `\(RMSE\)` since it's just the average absolute error. <img src="03_loss_trees_files/figure-html/unnamed-chunk-4-1.png" width="70%" style="display: block; margin: auto;" /> `$$MAE = \sum_{i = 1}^n{\frac{|\hat{y_{i}} - y_{i}|}{N}}$$` --- # Confusion Matrices * The city of Berlin is working on developing an 'early warning' system that is aimed at predicting whether a family is in need of childcare support. * Families which received childcare support are flagged with a 1 and families which didn't received childcare support are flagged with a 0: <img src="../../../img/base_df_lossfunction.svg" width="15%" style="display: block; margin: auto;" /> --- # Confusion Matrices * Suppose we fit a logistic regression that returns a predicted probability for each family: <img src="../../../img/df_lossfunction_prob.svg" width="35%" style="display: block; margin: auto;" /> --- # Confusion Matrices * We could assign a 1 to every respondent who has a probability above `0.5` and a 0 to every respondent with a probability below `0.5`: <img src="../../../img/df_lossfunction_class.svg" width="45%" style="display: block; margin: auto;" /> --- # Confusion Matrices The accuracy is the sum of all correctly predicted rows divided by the total number of predictions: <img src="../../../img/confusion_matrix_50_accuracy.svg" width="55%" style="display: block; margin: auto;" /> * Accuracy: `\((3 + 1) / (3 + 1 + 1 + 2) = 50\%\)` --- # Confusion Matrices * **Sensitivity** of a model is a fancy name for the **true positive rate**. * Sensitivity measures those that were correctly predicted only for the `1`: <img src="../../../img/confusion_matrix_50_sensitivity.svg" width="55%" style="display: block; margin: auto;" /> * Sensitivity: `\(3 / (3 + 1) = 75\%\)` --- # Confusion Matrices * The **specificity** of a model measures the true false rate. * Specificity measures those that were correctly predicted only for the `0`: <img src="../../../img/confusion_matrix_50_specificity.svg" width="55%" style="display: block; margin: auto;" /> * Specificity: `\(1 / (1 + 2) = 33\%\)` --- # ROC Curves and Area Under the Curve * The ROC curve is just another fancy name for something that is just a representation of sensitivity and specificity. <br> <br> <br> * In our previous example, we calculated the sensitivity and specificity assuming that the threshold for being 1 in the probability of each respondent is `0.5`. <br> <br> > What if we tried different cutoff points? --- # ROC Curves and Area Under the Curve <div id="htmlwidget-eed7aff265bbbec827fa" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-eed7aff265bbbec827fa">{"x":{"filter":"none","vertical":false,"data":[["1","2","3"],[0.3,0.5,0.7],[0.74,0.87,0.96],[0.87,0.8,0.53]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>cutoff<\/th>\n <th>sensitivity<\/th>\n <th>specificity<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"paging":true,"pageLength":5,"bLengthChange":false,"columnDefs":[{"className":"dt-right","targets":[1,2,3]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> * Assigning a 1 if the probability was above `0.3` is associated with a true positive rate (sensitivity) of `0.74`. * Switching the cutoff to `0.7`, increases the true positive rate to `0.95`, quite an impressive benchmark. * At the expense of increasing sensitivity, the true false rate decreases from `0.87` to `0.53`. --- # ROC Curves and Area Under the Curve * We want a cutoff that maximizes both the true positive rate and true false rate. * Try all possible combinations: <div id="htmlwidget-a1bfc39547a0f4afe7b8" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-a1bfc39547a0f4afe7b8">{"x":{"filter":"none","vertical":false,"data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99"],[0.01,0.02,0.03,0.04,0.05,0.06,0.07,0.08,0.09,0.1,0.11,0.12,0.13,0.14,0.15,0.16,0.17,0.18,0.19,0.2,0.21,0.22,0.23,0.24,0.25,0.26,0.27,0.28,0.29,0.3,0.31,0.32,0.33,0.34,0.35,0.36,0.37,0.38,0.39,0.4,0.41,0.42,0.43,0.44,0.45,0.46,0.47,0.48,0.49,0.5,0.51,0.52,0.53,0.54,0.55,0.56,0.57,0.58,0.59,0.6,0.61,0.62,0.63,0.64,0.65,0.66,0.67,0.68,0.69,0.7,0.71,0.72,0.73,0.74,0.75,0.76,0.77,0.78,0.79,0.8,0.81,0.82,0.83,0.84,0.85,0.86,0.87,0.88,0.89,0.9,0.91,0.92,0.93,0.94,0.95,0.96,0.97,0.98,0.99],[0.26,0.35,0.39,0.39,0.39,0.39,0.39,0.39,0.43,0.52,0.57,0.57,0.61,0.61,0.61,0.61,0.61,0.61,0.65,0.65,0.65,0.65,0.7,0.7,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.74,0.78,0.78,0.78,0.83,0.83,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.91,0.91,0.91,0.91,0.91,0.91,0.91,0.91,0.91,0.91,0.91,0.96,0.96,0.96,0.96,0.96,0.96,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,0.93,0.93,0.93,0.93,0.93,0.93,0.93,0.93,0.93,0.93,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.87,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.73,0.6,0.6,0.6,0.6,0.6,0.6,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.53,0.47,0.47,0.47,0.4,0.4,0.4,0.4,0.33,0.27,0.2,0.13,0.13,0.07,0.07]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>cutoff<\/th>\n <th>sensitivity<\/th>\n <th>specificity<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"paging":true,"pageLength":5,"bLengthChange":false,"columnDefs":[{"className":"dt-right","targets":[1,2,3]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- # ROC Curves and Area Under the Curve * This result contains the sensitivity and specificity for many different cutoff points. 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For example: <img src="03_loss_trees_files/figure-html/unnamed-chunk-16-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> * The more points are located in the top left quadrant, the higher the overall accuracy of our model * 90% of the space of the plot is under the curve. --- # Precision and recall <img src="../../../img/precision_recall.png" width="55%" style="display: block; margin: auto;" /> --- # Precision and recall <img src="../../../img/precision_recall_plot.png" width="55%" style="display: block; margin: auto;" /> --- # Precision and recall - When to use ROC curves and precision-recall: <br> 1. ROC curves should be used when there are roughly equal numbers of observations for each class <br> 2. Precision-Recall curves should be used when there is a moderate to large class imbalance --- <br> <br> <br> .center[ # Decision trees ] --- # Decision trees * Decision trees are tree-like diagrams. * They work by defining `yes-or-no` rules based on the data and assign the most common value for each respondent within their final branch. <img src="03_loss_trees_files/figure-html/unnamed-chunk-19-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-20-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-21-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-22-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-23-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-24-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # How do decision trees work <img src="../../../img/decision_trees_adv1.png" width="55%" style="display: block; margin: auto;" /> --- # How do decision trees work <img src="../../../img/decision_trees_adv2.png" width="55%" style="display: block; margin: auto;" /> --- # How do decision trees work <img src="../../../img/decision_trees_adv3.png" width="55%" style="display: block; margin: auto;" /> --- # How do decision trees work <img src="../../../img/decision_trees_adv4.png" width="55%" style="display: block; margin: auto;" /> --- # How do decision trees work <img src="../../../img/decision_trees_adv5.png" width="55%" style="display: block; margin: auto;" /> --- # Bad things about Decision trees * They overfit a lot <img src="03_loss_trees_files/figure-html/unnamed-chunk-31-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Bad things about Decision trees How can you address this? * Not straight forward * `min_n` and `tree_depth` are sometimes useful * You need to tune these <img src="03_loss_trees_files/figure-html/unnamed-chunk-32-1.png" width="70%" height="70%" style="display: block; margin: auto;" /> --- # Tuning decision trees * Model tuning can help select the best `min_n` and `tree_depth` ``` # A tibble: 5 x 7 tree_depth min_n .metric .estimator mean n std_err <dbl> <dbl> <chr> <chr> <dbl> <int> <dbl> 1 9 50 rmse standard 0.459 5 0.0126 2 9 100 rmse standard 0.459 5 0.0126 3 3 50 rmse standard 0.518 5 0.0116 4 3 100 rmse standard 0.518 5 0.0116 5 1 50 rmse standard 0.649 5 0.0102 ``` --- # Tuning decision trees <img src="03_loss_trees_files/figure-html/unnamed-chunk-34-1.png" width="95%" height="950%" style="display: block; margin: auto;" /> --- # Best tuned decision tree * As usual, once we have out model, we predict on our test set and compare: .center[ ``` Testing error Training error 0.4644939 0.4512248 ``` ] --- .center[ # Break ]