scratch.txt

which exactly matched the probability of randomly selecting the answer given by 







the sklearn decision tree uses the gini impurity to evaluate the success. 
The algorithm does this by splitting the data into two branches by choosing the split that minimise the gini index, defined as:

This is mathematically formulated as follows.\begin{quote}Given training vectors $x_i \in R^n, i=1,...$, l and a label vector $y \in R^l$, a decision tree recursively partitions the space such that the samples with the same labels are grouped together. Let the data at node m be represented by Q. For each candidate split $\theta = (j, t_m)$ consisting of a feature j and threshold $t_{m}$, partition the data into $Q_{left}(\theta)$ and $Q_{right}(\theta)$ subsets.
\fi

Sklearn can use the Gini impurity or the entropy as the determining how to split the tree. The default method of Gini impurity was used here. 
\begin{equation}
Q_{left}(\theta) = {(x, y) | x_j <= t_m}\end{equation}
\begin{equation}
Q_{right}(\theta) = Q \setminus Q_{left}(\theta)
\end{equation}
The impurity at m is computed using an impurity function H(), the choice of which depends on the task being solved (classification or regression)
\begin{equation}
G(Q, \theta) = \frac{n_{left}}{N_m} H(Q_{left}(\theta)) + \frac{n_{right}}{N_m} H(Q_{right}(\theta))
\end{equation}
Select the parameters that minimises the impurity
\begin{equation}
\theta^{*} = \operatorname{argmin}_\theta G(Q, \theta)
\end{equation}
Recurse for subsets $Q_{left}(\theta^*)$ and $Q_{right}(\theta^*)$ until the maximum allowable depth is reached, $N_m < \min_{samples}$ or $N_m = 1$.
\end{quote}
\cite{sklearn}
The impurity measure used is the gini impurity:
\begin{equation}
H(X_m) = \sum_k p_{mk} (1 - p_{mk})
\end{equation}

\iffalse