From fd70cb5f8cd5185d9aab8d8351a04e231dc27e85 Mon Sep 17 00:00:00 2001 From: Andrew Date: Thu, 10 Oct 2024 08:14:51 +0100 Subject: [PATCH] [CT4101]: Week 5 lecture notes --- .../notes/CT4101-Notes.pdf | Bin 708784 -> 2050123 bytes .../notes/CT4101-Notes.tex | 175 +++++++++++++++++- .../notes/images/1nnboundary.png | Bin 0 -> 201531 bytes .../notes/images/anyonefortennis.png | Bin 0 -> 106667 bytes .../notes/images/distanceweightingknn.png | Bin 0 -> 142659 bytes .../notes/images/increasingk.png | Bin 0 -> 121848 bytes .../notes/images/increasingk2.png | Bin 0 -> 125796 bytes .../notes/images/knn.png | Bin 0 -> 87813 bytes .../notes/images/morevoronoi.png | Bin 0 -> 132633 bytes .../notes/images/smoothingeffectk.png | Bin 0 -> 663137 bytes .../notes/images/voronoi.png | Bin 0 -> 185784 bytes 11 files changed, 173 insertions(+), 2 deletions(-) create mode 100644 year4/semester1/CT4101: Machine Learning/notes/images/1nnboundary.png create mode 100644 year4/semester1/CT4101: Machine 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z&7$sJUmXRyrTRd>6*q5X%h>!j!-sF^ zG2KmrV91IORlx1et|@#qQa%&ul*Lo`v&3{E!wWAddjVVoGZx=| zlqK$rW_aggW%-h%vlhSmmcluUKONKPip9UYt 3$ and often in the range $5$ -- $21$. +Increasing $K$ has a \textbf{smoothing effect}: +\begin{itemize} + \item If $k$ is too low, it tends to overfit if the data is noisy. + \item If $k$ is too high, it tends to underfit. +\end{itemize} +In imbalanced datasets, the majority target class tends to dominate for large $k$ values. +It's important to note that $k$ does not affect computational cost much: most of the computation is in calculating the distances from the query to all stored instances. +\begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{images/increasingk.png} + \caption{Effect of Increasing $k$ (1)} +\end{figure} +\begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{images/increasingk2.png} + \caption{Effect of Increasing $k$ (2)} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{images/smoothingeffectk.png} + \caption{Smoothing Effect of $k$} +\end{figure} + +\subsubsection{Distance-Weighted $k$-NN} +In \textbf{distance-weighted $k$-NN}, we give each neighbour a weight equal to the inverse of its distance from the target. +We then take the weighted vote or weighted average to classify the target case. +It's reasonable to use $k = [\text{all training cases}]$. + +\begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{images/distanceweightingknn.png} + \caption{Effect of Distance Weighting} +\end{figure} + +\section{Decision Trees} +\textbf{Decision trees} are a fundamental structure used in information-based machine learning. +The idea is to use a decision tree as a predictive model to decide what category/label/class an item belongs to based on the values of its features. +Decision trees consist of \textbf{nodes} (where two branches intersect) which are decision points which partition the data. +Observations about an item (values of features) are represented using branches. +The terminal nodes are called \textbf{leaves} and specify the target label for an item. +The inductive learning of a decision tree is as follows: +\begin{enumerate} + \item For all attributes that have not yet been used in the tree, calculate their impurity (\textbf{entropy} or \textbf{Gini index}) and \textbf{information/Gini gain} values for the training samples. + \item Select the attribute that has the \textbf{highest} information gain. + \item Make a tree node containing that attribute. + \item This node \textbf{partitions} the data: apply the algorithm recursively to each partition. +\end{enumerate} + +The main class used in scikit-learn to implement decision tree learning for classification tasks is \mintinline{python}{DecisionTreeClassifier}. +The default measure of impurity is the Gini index, but entropy is also an option. + +\subsection{Computing Entropy} +We already saw how some descriptive features can more effectively discriminate between (or predict) classes which are present in the dataset. +Decision trees partition the data at each node, so it makes sense to use features which have higher discriminatory power ``higher up'' in a decision tree. +Therefore, we need to develop a formal measure of the discriminatory power of a given attribute. +\\\\ +Claude Shannon (often referred to as ``the father of information theory'') proposed a measure of the impurity of the elements in the set called \textbf{entropy}. +Entropy may be used to measure the uncertainty of a random variable. +The term ``entropy'' generally refers to disorder or uncertainty, so the use of this term in the context of information theory is analogous to other well-known uses of the term such as in statistical themodynamics. +The acquisition of information (\textbf{information gain}) corresponds to a \textbf{reduction in entropy}. +\\\\ +The \textbf{entropy} of a dataset $S$ with $n$ different classes may be calculated as: +\[ + \text{Ent}(S) = \sum^n_{i=1} -p_i \log_2 p_i +\] +where $p_i$ is the proportion of the class $i$ in the dataset. +This is an example of a probability mass function. +Entropy is typically measured in \textbf{bits} (note the $\log_2$ in the equation above): +the lowest possible entropy output from this function is 0 ($\log_2 1 = 0$), while the highest possible entropy is $\log_2n$ (which is equal to 1 when there are only two classes). +\\\\ +We use the binary logarithm because a useful measure of uncertainty should assign high uncertainty to outcomes with a low probability and assign low uncertainty values to outcomes with a high probability. +$\log_2$ returns large negative values when $P$ is close to 0 and small negative values when $P$ is close to 1. +We use $-\log_2$ for convenience, as it returns positive entropy values with 0 as the lowest entropy. + +\begin{tcolorbox}[colback=gray!10, colframe=black, title=\textbf{Worked Entropy Example}] + \begin{figure}[H] + \centering + \includegraphics[width=0.6\textwidth]{images/anyonefortennis.png} + \caption{Example Data} + \end{figure} + + Workings; + \begin{align*} + \text{Ent}(S) =& \text{Ent}([9+, 5-]) \\ + =& \frac{-9}{14} \log_2 \left( \frac{9}{14} \right) - \frac{5}{14} \log_2 \left( \frac{5}{14} \right) \\ + =& 0.9403 + \end{align*} + + Note that if you are calculating entropy using a spreadsheet application such as Excel, make sure that you are using $\log_2$, e.g. \verb|LOG(9/14,2)|. +\end{tcolorbox} + +\subsection{Computing Information Gain} +The \textbf{information gain} of an attribute is the reduction of entropy from partitioning the data according to that attribute: +\[ + \text{Gain}(S,A) = \text{Ent}(S) - \sum_{v \in \text{Values}(A)} \frac{\left| S_v \right|}{\left| S \right|} \text{Ent}(S_v) +\] + +Here $S$ is the entire set of data being considered and $S_v$ refers to each partition of the data according to each possible value $v$ for the attribute. +$\left| S \right|$ \& $\left| S_v \right|$ refer to the cardinality or size of the overall dataset, and the cardinality or size of a partition respectively. +When selecting an attribute for a node in a decision tree, we use whichever attribute $A$ that gives the greatest information gain. + +\begin{tcolorbox}[colback=gray!10, colframe=black, title=\textbf{Worked Information Gain Example}] + Given $\left| S \right| = 14$, $\left| S_{\text{windy} = \text{true}} \right| = 14$, \& $\left| S_{\text{windy} = \text{false}} \right| = 14$, calculate the information gain of the attribute ``windy''. + + \begin{align*} + \text{Gain}(S, \text{windy}) =& \text{Ent}(S) - \frac{\left| S_{\text{windy} = \text{true}} \right|}{\left| S \right|} \text{Ent}(S_\text{windy} = \text{true}) + - \frac{\left| S_{\text{windy} = \text{false}} \right|}{\left| S \right|} \text{Ent}(S_\text{windy} = \text{false}) \\ + =& \text{Ent}(S) - \left( \frac{6}{14} \right) \text{Ent}(\left[3+,3-\right]) - \left( \frac{8}{14} \right) \text{Ent}(\left[ 6+,2- \right]) \\ + =& 0.940 - \left( \frac{6}{14} \right) 1.00 - \left( \frac{8}{14} \right) 0.811\\ + =& 0.048 + \end{align*} +\end{tcolorbox} \end{document} diff --git a/year4/semester1/CT4101: Machine Learning/notes/images/1nnboundary.png b/year4/semester1/CT4101: Machine Learning/notes/images/1nnboundary.png new file mode 100644 index 0000000000000000000000000000000000000000..9a07158a8a9960ead60219e130f0cec3fcbe8604 GIT binary patch literal 201531 zcmcG$hdbAA8$PZjD_bZdZ(A}VJF;h1cJ?MaWJSX!d+!h-SqWJQSs^QA@4ff_Uay|# zIlkXN;P-PJeLjaWUiW=p_jR4ud7jrRNLf+(2G%VsG&HmuvN95?XlUp`XlPeRFwo&| z3dShe(a`Ro$x4W-xuvX*y>L@AJN~(~kr+3Q@!B`Zmm!0jnMe6i@J9@-2sSq3OD-hX znU{lu3x-_vBuzyL2(Il1yb1h-aT%47dB-2c5}1It^sujMBm9wurraw!#<_D{O}4gQ zi3g+6j#2yz{2M#Ba=I;0|2|YHhio($@d=QRy!*~q>HqWB|M|Boc`|hWpMK7NzM1;? z0qwuv_MiVmCOuN8`F}sZB_oOY|Ml}$BbQgDv#!;{753C4Uu|cS?dy!-|Jm8ulLseD z6ZInD5BIh5esVsJc3JsT;xxShb2@g$o}C3a(eaot!ir_qlkP=Cv&wM!~PD*e&*df22hAsd_=8gzeui zQ{{vDS@Q90+|T}m{`#fB_H7Zhzc!+o-*Ry{6E1Wx`n+zi^W@=`pQlN2f 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