# misclassification rate decision tree example

Understand the resubstitution error rate and the cost-complexity measure, their differences, and why the cost-complexity measure is introduced.

The following three figures are three classification trees constructed from the same data, but each using a different bootstrap sample. Assign each observation to a final category by a majority vote over the set of trees. Understand the fact that the best-pruned subtrees are nested and can be obtained recursively. Algorithm: Consider the following steps in a fitting algorithm with a dataset having N observations and a binary response variable. Bagging exploits that idea to address the overfitting issue in a more fundamental manner. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. Bagging introduces a new concept, "margins." It draws a random sample of predictors to define each split. Boosting, like bagging, is another general approach for improving prediction results for various statistical learning methods. 2 classes; with or without signs of diabetes. Random forests are in principle an improvement over bagging. Thus, either branch may have a higher proportion of 0 values for the response value than the other. The CP column lists the values of the complexity parameter, the number of splits is listed undernsplit, and the column xerror contains cross-validated classification error rates; the standard deviation of the cross-validation error rates are in the xstd column. The misclassification rate on a separate test dataset of size 5000 is 0.28. stream For example, if over all trees an observation is correctly classified 75\% of the time, the margin is 0.75 - 0.25 = 0.50. When a logistic regression was applied to the data, not a single incident of serious domestic violence was identified. Operationally, the "margin" is the difference between the proportion of times a case is correctly classified and the proportion of times it is incorrectly classified. The pruning is performed by function prune, which takes the full tree as the first argument and the chosen complexity parameter as the second. A regression equation, with one set of regression coefficients or smoothing parameters. To get a better evaluation of the model, the prediction error is estimated only based on the "out-of-bag'' observations. By contrast, the functions in the tree library draw the tree in such a way that all the decision statements contain "less than" (<) symbols. Ideally, there should be large margins for all of the observations. This average value for each observation is the bagged fitted value. It is also particularly well suited to decision trees. Bagging can help with the variance. Drop the out-of-bag data down the tree. Number of Predictors Sampled: the number of predictors sampled at each split would seem to be a key tuning parameter that should affect how well random forests perform. Therefore, local feature predictors will have the opportunity to define a split. But it is clear that the variance across trees is large. Much of the time a dominant predictor will not be included. For each observation in the dataset, count the number of trees that it is classified in one category over the number of trees. Thus, if 51\% of the time over a large number of trees a given observation is classified as a "1'', that becomes its classification. F1 Score vs. Thus, if 51% of the time over a large number of trees a given observation is classified as a "1", that becomes its classification. The pool of candidate splits that we might select from involves a set, The candidate split is evaluated using a goodness of split criterion $$\Phi(s, t)$$ that can be evaluated for any split. These "out-of-bag'' observations can be treated as a test dataset and dropped down the tree. Your email address will not be published. This means the model incorrectly predicted the outcome for 27.5% of the players. The conceptual advantage of bagging is to aggregate fitted values from a large number of bootstrap samples. This bodes well for generalization to new data. The core of bagging's potential is found in the averaging over results from a substantial number of bootstrap samples. Repeat this process for each node until the tree is large enough. The procedure is much in the same spirit as disproportional stratified sampling used for data collection (Thompson, 2002). This is pretty close to the cross-validation estimate! - if another tree achieves the minimum at the same , then the other tree is guaranteed to be bigger than the smallest minimized tree, $$R(t)$$ , the resubstitution error rate for node, $$R(T_t)$$ , the resubstitution error rate for the branch coming out of node. $$\Phi$$ achieves minimum only at the points (1, 0, , 0), (0, 1, 0, , 0), , (0, 0, , 0, 1), i.e., when the probability of being in a certain class is 1 and 0 for all the other classes. Although there remain some important variations and details to consider, these are the key steps to producing "bagged'' classification trees. The Bayes error rate is 0.26. Section 8.2.3 in the textbook provides details. The Bayes classification rule can be derived because we know the underlying distribution of the three classes. Keep going until all terminal nodes are pure (contain only one class). control gives control details of the rpart algorithm. xZ[ Fm'Om6vt3qng calculate $$p( j | t ) = N_j (t) / N (t)$$ - for all the points that land in node.

Large margins are desirable because a more stable classification is implied. The search of the optimal subtree should be computationally tractable. Normally, we select a tree size that minimizes the cross-validated error, which is shown in the xerror column printed by ()\$cptable. Denote the collection of all the nodes in the tree by. F1 Score vs. Understand the definition of the impurity function and several example functions. All of the concerns about overfitting apply, especially given the potential impact that outliers can have on the fitting process when the response variable is quantitative. The goal is to grow trees with as little bias as possible. The error rate estimated by using an independent test dataset of size 5000 is 0.30. Take a random sample without replacement of the predictors. A stop-splitting rule, i.e., we have to know when it is appropriate to stop splitting. <> Accuracy: Which Should You Use? They differ by whether costs are imposed on the data before each tree is built, or at the end when classes are assigned. Number of Trees: in practice, 500 trees is often a good choice. Q/0YD;e3dzX0[ n8 0\|STt8k@'PLj0_7? It is not clear how much bias exists in the three trees. How do we assign these class labels? Suppose we have obtained the maximum tree grown on the original data set. The regression coefficients estimated for particular predictors may be very unstable, but it does not necessarily follow that the fitted values will be unstable as well. There is little room for improvement over the tree classifier. Remember, we know the exact distribution for generating the simulated data. The opposite of misclassification rate would be accuracy, which is calculated as: This means the model correctly predicted the outcome for 72.5% of the players. In other words, the averaging for a given observation is done only using the trees for which that observation was not used in the fitting process. In R, the bagging procedure (i.e., bagging() in the ipred library) can be applied to classification, regression, and survival trees. Interpretations from the results of a single tree can be quite risky when a classification tree performs in this manner. After completing the reading for this lesson, please finish the Quiz and R Lab on Canvas (check the course schedule for due dates). If we know how to make splits or 'grow' the tree, how do we decide when to declare a node terminal and stop splitting? $$R_{\alpha}(T(\alpha)) = min_{T \leq Tmax} R_{\alpha}(T)$$, If $$R_{\alpha}(T) = R_{\alpha}(T(\alpha))$$ , then $$T(\alpha) \leq T$$. Further information on the pruned tree can be accessed using the summary() function. Construct a split by using predictors selected in Step 2. The following tutorials provide additional information about common machine learning concepts: Introduction to Logistic Regression Take a random sample of size N with replacement from the data (bootstrap sample). There is a need to consider the relative costs of false negatives (fail to predict a felony incident) and false positives (predict a case to be a felony incident when it is not). The idea of classifying by averaging over the results from a large number of bootstrap samples generalizes easily to a wide variety of classifiers beyond classification trees. Applying this rule to the test set yields a misclassification rate of 0.14. The following confusion matrix summarizes the predictions made by the model: Here is how to calculate the misclassification rate for the model: The misclassification rate for this model is 0.275 or 27.5%. Upon successful completion of this lesson, you should be able to: 11.3 - Estimate the Posterior Probabilities of Classes in Each Node, 11.5 - Advantages of the Tree-Structured Approach, 11.8.4 - Related Methods for Decision Trees. The error rate estimated by cross-validation using the training dataset which only contains 200 data points is also 0.30. Understand the three elements in the construction of a classification tree. The random forests algorithm is very much like the bagging algorithm. The ways in which bagging aggregates the fitted values are the basis for many other statistical learning developments. For each tree, observations not included in the bootstrap sample are called "out-of-bag'' observations. The functions in the rpart library draw the tree in such a way that left branches are constrained to have a higher proportion of 0 values for the response variable than right branches. $$\Phi$$ achieves maximum only for the uniform distribution, that is all the. It constructs a large number of trees with bootstrap samples from a dataset. Then, the average of these assigned means over trees is computed for each observation. However, when a classification tree is used solely as a classification tool, the classes assigned may be relatively stable even if the tree structure is not. Your email address will not be published. Understand the purpose of model averaging. An obvious gain with random forests is that more information may be brought to reduce bias of fitted values and estimated splits. :Z|w(8n~%dB 01_*XP;q8oQT@PtF?V]>d5mmhGhq1fyylsq&lr|} #>%!tQ 63=s2 A classification regression tree with one set of leaf nodes. In the example of domestic violence, the following predictors were collected from 500+ households: Household size and number of children; Male / female age (years); Marital duration; Male / female education (years); Employment status and income; The number of times the police had been called to that household before; Alcohol or drug abuse, etc. And they would be extremely difficult to forecast. Understand the advantages of tree-structured classification methods. Ideally, many sets of fitted values, each with low bias but high variance, may be averaged in a manner that can effectively reduce the bite of the bias-variance tradeoff. We can also use the random forest procedure in the "randomForest" package since bagging is a special case of random forests. Finally, we need a rule for assigning every terminal node to a class. The selection of the splits, i.e., how do we decide which node (region) to split and how to split it? One can 'grow' the tree very big. There are often a few predictors that dominate the decision tree fitting process because on the average they consistently perform just a bit better than their competitors. Each tree is produced from a random sample of cases and at each split a random sample of predictors. The following example show how to calculate misclassification rate for a logistic regression model in practice. There are 29 felony incidents which are very small as a fraction of all domestic violence calls for service (4%). It is apparent that random forests are a form of bagging, and the averaging over trees can substantially reduce instability that might otherwise result. Selection of the optimal subtree can also be done automatically using the following code: opt stores the optimal complexity parameter. We have to assign each terminal node to a class. % So, some of the decision statements contain "less than" (<) symbols and some contain "greater than or equal to" (>=) symbols (whatever is needed to satisfy this constraint). Our goal is not to forecast new domestic violence, but only those cases in which there is evidence that serious domestic violence has actually occurred. As long as the tree is sufficiently large, the size of the initial tree is not critical. The cross-validation estimate of misclassification rate is 0.29. It was invented by Leo Breiman, who called it "bootstrap aggregating" or simply "bagging" (see the reference: "Bagging Predictors,"Machine Learning, 24:123-140, 1996, cited by 7466). E%1OnX_Kj$ )5i3(/^n^ Nq4}B)!_g?"ziRiHm72v{s3[ctlwN.})&mQO&k91;6!|hgnQi6EiLA9{vSN.I>m,n&>q\j&b34_Eq5.f1-G+3uIgXq>*u\3q%.$\nqhxU2S; Consequently, many other predictors, which could be useful for very local features of the data, are rarely selected as splitting variables. In random forests, there are two common approaches. At each node, we move down the left branch if the decision statement is true and down the right branch if it is false. Bagging constructs a large number of trees with bootstrap samples from a dataset. Take a random sample of size N with replacement from the data (a bootstrap sample). Variance reduction: the trees are more independent because of the combination of bootstrap samples and random draws of predictors. Understand the basic idea of decision trees. Repeat Steps 1-5 a large number of times (e.g., 500). For a sample of households to which sheriff's deputies were dispatched for domestic violence incidents, the deputies collected information on a series of possible predictors of future domestic violence, for example, whether police officers had been called to that household in the recent past. The cptable provides a brief summary of the overall fit of the model. #% N {&P! wzj[(AU >7!}L0pi]6T$A=QavA Pon9m\$]4#)M&e'whxP 9>@M@.DJ%9QSk|Ofd.l BWroM]:1c0'gn(%'MNg:b(HqDv)cZm?w6Mz,q;OpP/e)g%|"\R2?%Q^?pCrZ1 9oPEKef?rw}eo;~}M. In practice, we often don't go that far. Otherwise, the best prediction would be assuming no serious domestic violence with an error rate of 4%. Why Bagging Works? Data were collected to help forecast incidents of domestic violence within households. Indeed, random forests are among the very best classifiers invented to date (Breiman, 2001a). $$\Phi$$ is a symmetric function of $$p_1, \cdots , p_K$$, i.e., if we permute $$p_j$$ , $$\Phi$$ remains constant.