FIX: typos

tex/chapters/chapter03.tex

@@ -102,11 +102,11 @@ \section{Tree structured models}
 \begin{equation}
 {\cal X}^\Omega_{c_k} = \{ \mathbf{x}_i | i \in \Omega, Y=c_k \}
 \end{equation}
-may not be disjoints.
+may not be disjoint.
 By contrast, since $\varphi$ defines a function from ${\cal X}$ to ${\cal Y}$,
 any input $\mathbf{x} \in {\cal X}$ is mapped to exactly one output value $y \in
 {\cal Y}$ and the subsets ${\cal X}^\varphi_{c_k}$ are therefore necessarily
-disjoints, which means that no model will ever perfectly predict the true output
+disjoint, which means that no model will ever perfectly predict the true output
 value in all cases. As discussed in Section~\ref{sec:2:bayes-model}, this
 limitation is unavoidable and can in fact be viewed as the cause of the residual error.
 \end{remark}
@@ -119,7 +119,7 @@ \section{Tree structured models}
 first define the following concepts:
 
 \begin{definition}
-A \emph{tree} is a directed graph $G=(V,E)$ in which any two vertices (or \emph{nodes})
+A \emph{tree} is a graph $G=(V,E)$ in which any two vertices (or \emph{nodes})
 are connected by exactly one path.
 \end{definition}
 

tex/chapters/chapter05.tex

@@ -806,7 +806,7 @@ \subsubsection{API for random forests}
 \subsection{Internal data structures}
 
 Among all possible ways of representing a decision tree, one of the simplest and most
-direct representation is to adopt an object-oriented approach. In this
+direct representations is to adopt an object-oriented approach. In this
 paradigm, a decision tree is naturally represented as a hierarchy of high-level
 objects, where each object corresponds to a node of the tree and comprises
 attributes referencing its children or storing its split and value. Such a

tex/chapters/chapter07.tex

@@ -232,7 +232,7 @@ \section{Redundant variables}
 the case where $N_c$ redundant variables $X_j^c$ (for $c=1,\dots,N_c$) are
 added to the input variables instead of one, leading to the general
 Proposition~\ref{prop:red:general}. From this result, the same qualitative
-conclusions can drawn, except that the decrease or increase effects discussed
+conclusions can be drawn, except that the decrease or increase effects discussed
 above are even stronger as more redundant variables are added.
 
 \begin{proposition}\label{prop:red:general}
@@ -279,7 +279,7 @@ \section{Redundant variables}
 also with respect to all the other input variables that define the problem}. In
 particular, a variable which is not important is not necessarily uninformative,
 as the example illustrates. A variable may be considered as less important
-because the  information it conveys is also redundantly conveyed and dilluted in other
+because the  information it conveys is also redundantly conveyed and diluted in other
 variables, and not necessarily because it has no information about the output.
 
 \begin{figure}

tex/chapters/chapter08.tex

@@ -338,7 +338,7 @@ \subsection{Small datasets}
 to Figure \ref{fig:8:small-cmp}, only ET is ranked significantly higher than all
 DT-based method but looking at Table \ref{table:rp:small-ttest}, the worse ET-based
 variant (P-ET) is still 9 times significantly better (w.r.t. the 50
-runs over each set) and only 1 times significantly worse than the best DT-based
+runs over each set) and only 1 time significantly worse than the best DT-based
 variant (RP-DT). The separation between these two families of algorithm
 thus appears quite significant. This observation clearly suggests that using
 random split thresholds, instead of optimized ones like in decision trees, pays
@@ -688,7 +688,7 @@ \subsection{Sensitivity to $\alpha_s$ and $\alpha_f$}
 difference is the decrease of the accuracy of RP-DT when $\alpha_s$ and
 $\alpha_f$ grow towards $1.0$. Indeed, since the only source of randomization in RP-DT
 is patch selection, it yields in this case ensembles of identical
-trees and therefore amounts to build a single tree on the whole
+trees and therefore amounts to building a single tree on the whole
 dataset. By contrast, because of the extra-randomization of the split
 thresholds in ET, there is typically no drop of accuracy for RP-ET when $\alpha_s$
 and $\alpha_f$ grow to $1.0$.