We have a vector \(x\) and we define the function tetha that corresponds to the coefficient of variation (i.e. the statistics we want to estimate) x i) jack <- x The previous equation is the bias-corrected jackknife estimate. We then turn this i-th partial estimate into the i-th pseudo value \(\hat\) is the estimate using the full dataset. More formally, you have to generate a jackknife sample that has the value \(x_i\) removed and then compute the i-th partial estimate of the statistic using this sample, The general idea behind Jackknife is that a parameter estimate can be found by estimating the parameter for each subsample omitting the i-th observation. Jackknife was proposed by John Tukey in 1956. The wager approach uses the Jackknife resampling technique for estimating the CI. But the one I used for the above example is based on the article Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife (Wager, Hastie and Efron 2014) So, how is CI calculated for random forests?, there are several approaches (see the list of papers at the end of the post). The article states that for each sample, the ensemble will tend toward a particular mean prediction value for that sample as the number of trees tends towards infinity. An intuition about this is given in this article. If you use more trees, you are reducing the variance of the of de model and the RF will be more confident with its own predictions. Notice the results in Figure 1 are consistent with the way RF works for binary classification problems. Predictions against estimates of standard error ($se$) for all four RF models. On the other hand, the model using 1000 trees and all \(se\) values under 0.25, it is certainly more confident about its predictions. Moreover, It is possible that the model is overfitting the data. So it basically means that the model is not sure about anything and the results from Table 1 should not be taken so seriously. The model using 50 trees has a very large \(se\) values, in some cases very close to 1. Now, you can see some observable differences between the four RF models. The solid curves are smoothing splines that fit through the data. Orange dots refer to instances incorrectly classified while blue to correctly classified. shows a plot of test-set predictions against estimates of standard error ( \(se\)) for all four RF models. However, if you look at the CI distribution for each of the output probabilities, things could be different.įigure 1. Then, RF with the number of trees = 50 will be the choice model. If you follow a traditional policy of selecting the best value. And even for specificity you have all but the model with 100 trees showing the same values. Accuracy and Sensitivity are practically the same. The problem here (notice it is oversimplified) is which model you will choose. In the end, you get the following table: Table 1: Prediction results for RF using different numbers of trees. You do the usual 70/30 split in train and test sets and then you train four different models using Random Forest varying the number of trees (aka estimators). Let’s try an example using a dataset containing information about malicious and normal network traffic connections. Q: Given a prediction on a particular example, how sure is Random Forest about it? For the latter, there are several approaches, but clearly, the term Confidence interval (CI) arises as a possible solution. For dealing with the former, you will use approaches such as K-fold cross-validation or the classic Monte Carlo split. We are not interested in the overall performance, but in a particular example in our test dataset. Despite their similarities, the key here is the word overall. (read about average generalization error here). The traditional question is more focused on how different will be the overall performance of a given model if we use a different dataset. You should notice this question is a little different from the usual Machine learning issue regarding model future performance estimation. Or if you put it in different words: if we change the training dataset just a little bit, will Random Forest give you the same result for that particular example? You train the model and test it on unseen data and surprisingly you got some very decent results.īut now, you asked yourself: given a prediction on a particular example, how sure is Random Forest about it. After cleaning and doing the usual Machine Learning procedure, you decide to try the good ol’ Random Forest from our friend Leo. The problem is quite simple: you have collected a good tabular dataset.
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