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📗 [4 points] In a problem where each example has real-valued attributes (i.e. features), where each attribute can be split at possible thresholds (i.e. binary splits), to select the best attribute for a decision tree node at depth , where the root is at depth 0, how many conditional entropies must be calculated (at most)?
📗 Answer: .
📗 [3 points] Consider a training set with 8 items. The first dimension of their feature vectors are: . However, this dimension is continuous (i.e. it is a real number). To build a decision tree, one may ask questions in the form "Is \(x_{1} \geq \theta\)"? where \(\theta\) is a threshold value. Ideally, what is the maximum number of different \(\theta\) values we should consider for the first dimension \(x_{1}\)? Count the values of \(\theta\) such that all instances belong to one class.
📗 Answer: .
📗 [3 points] A decision tree has depth \(d\) = (a decision tree where the root is a leaf node has \(d\) = 0). All its internal node have \(b\) = children. The tree is also complete, meaning all leaf nodes are at depth \(d\). If we require each leaf node to contain at least training examples, what is the minimum size of the training set?
📗 Answer: .
📗 [3 points] Consider a -dimensional feature space where each feature takes integer value from 0 to (including 0 and ). What is the smallest and largest distance between the two distinct (non-overlapping) points in the feature space?
📗 Answer (comma separated vector): .
📗 [3 points] Suppose there is a single integer input \(x\) = {\(0\), \(1\), ..., }, and the label is binary \(y\) = {\(0\), \(1\)}. Let \(\mathcal{H}\) be a hypothesis space containing all possible linear classifiers. How many unique classifiers are there in \(\mathcal{H}\)? For example, the three linear classifiers \(1_{\left\{x < 0.4\right\}}\), \(1_{\left\{x \leq 0.4\right\}}\) and \(1_{\left\{x < 0.6\right\}}\) are considered the same classifier since they classify all possible data sets the same way.
📗 Answer: .
📗 [3 points] Suppose there are \(2\) discrete features \(x_{1}, x_{2}\) that can take on values and , and a binary decision tree is trained based on these features. What is the maximum number of leafs the decision tree can have?
📗 Answer: .
📗 [3 points] Given the following training set, what is the maximum accuracy of a decision tree with depth 1 trained on this set? Enter a number between 0 and 1.
index
\(x_{1}\)
\(y\)
1
2
3
4
5
6
📗 Answer: .
📗 [3 points] A hospital trains a decision tree to predict if any given patient has technophobia or not. The training set consists of patients. There are features. The labels are binary. The decision tree is not pruned. What are the smallest and largest possible training set accuracy of the decision tree? Enter two numbers between 0 and 1. Hint: patients with the same features may have different labels.
📗 Answer (comma separated vector): .
📗 [3 points] Given five decision stumps (decision trees with depth 1) in a random forest in the following table, what is the predicted label for a new data point \(x\) = \(\begin{bmatrix} x_{1} & x_{2} & ... \end{bmatrix}\) = ? Enter a single number (-1 or 1; and 0 in case of a tie).
Index
Decision stump
-
1
Label 1 if
Label -1 otherwise
2
Label 1 if
Label -1 otherwise
3
Label 1 if
Label -1 otherwise
4
Label 1 if
Label -1 otherwise
5
Label 1 if
Label -1 otherwise
📗 Answer: .
📗 [3 points] Given three decision stumps in a random forest in the following table, what is the predicted label for a new data point \(x\) = \(\begin{bmatrix} x_{1} \\ x_{2} \\ ... \end{bmatrix}\) = ? Enter a single number (-1 or 1; and 0 in case of a tie).
Index
Decision stump
-
1
Label 1 if
Label -1 otherwise
2
Label 1 if
Label -1 otherwise
3
Label 1 if
Label -1 otherwise
📗 Answer: .
📗 [4 points] Given the training set below and find the label of the decision tree that achieves 100 percent accuracy. Enter \(\hat{y}_{1}, \hat{y}_{2}, \hat{y}_{3}, \hat{y}_{4}\) as a vector.
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