From 6eea1b4d8768c183e936ed5145aeebeb0cc43baa Mon Sep 17 00:00:00 2001 From: Andrew Date: Thu, 31 Oct 2024 10:46:52 +0000 Subject: [PATCH] [CT4101]: Week 8 lecture materials & slides --- year4/semester1/CT4101: Machine Learning/exam | 3 + .../topic5/CT4101 - 05 - Data Processing.pdf | Bin 0 -> 2555913 bytes .../notes/CT4101-Notes.pdf | Bin 2619055 -> 3158624 bytes .../notes/CT4101-Notes.tex | 196 ++++++++++++++++++ .../notes/images/equalbins.png | Bin 0 -> 191572 bytes .../notes/images/freqbins.png | Bin 0 -> 183415 bytes .../notes/images/numbins.png | Bin 0 -> 154543 bytes 7 files changed, 199 insertions(+) create mode 100644 year4/semester1/CT4101: Machine Learning/exam create mode 100644 year4/semester1/CT4101: Machine Learning/materials/topic5/CT4101 - 05 - Data Processing.pdf create mode 100644 year4/semester1/CT4101: Machine Learning/notes/images/equalbins.png create mode 100644 year4/semester1/CT4101: Machine Learning/notes/images/freqbins.png create mode 100644 year4/semester1/CT4101: Machine Learning/notes/images/numbins.png diff --git a/year4/semester1/CT4101: Machine Learning/exam b/year4/semester1/CT4101: Machine Learning/exam new file mode 100644 index 00000000..09e2bdc7 --- /dev/null +++ b/year4/semester1/CT4101: Machine Learning/exam @@ -0,0 +1,3 @@ +equal frequency and equal width binning very well may come up on exam + +need to know entropy formnulae as often asked on exam diff --git a/year4/semester1/CT4101: Machine Learning/materials/topic5/CT4101 - 05 - Data Processing.pdf b/year4/semester1/CT4101: Machine Learning/materials/topic5/CT4101 - 05 - Data Processing.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ebb79ea48c51fd792b2a93b382358bfcc8df25b3 GIT binary patch literal 2555913 zcma%i1#n!umaUm7W@bB%soTuV?3kIEnb|Qj#LN(5%*@Q}n3zw* z3;=0jMiv$p4hANG9Pvjr4uA+T6Oe(Cl>s12%)$Ur`tz8XgBb`AC)OlpVPYg^WnkCl z>Z$4Q-8#<@Ft`9Y1;$vbA!yws9n80mwTT8<`tA**Xw2 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provides various classes to generate ROC curves: \url{https://scikit-learn.org/stable/auto_examples/model_selection/plot_roc.html}. The link also gives some details on how ROC curves can be computed for mutli-class classification problems. +\section{Data Processing \& Normalisation} +\subsection{Data Normalisation} +One major problem in data normalisation is \textbf{scaling}. +For example, if Attribute 1 has a range of 0-10 and Attribute 2 has a range from 0-1000, then Attribute 2 will dominate calculations. +The solution for this problem is to rescale all dimensions independently: +\begin{itemize} + \item Z-normalisation: calculated by subtracting the population mean from an individual raw score and then dividing it by the population standard deviation. + \[ + z = \frac{x - \mu}{\sigma} + \] + where $z$ is the z-score, $x$ is the raw score, $\mu$ is the population mean, \& $\sigma$ is the standard deviation of the population. + \\\\ + This can be achieved in scikit-learn using the \mintinline{python}{StandardScaler} utility class. + \item Min-Max data scaling: also called 0-1 normalisation or range normalisation. + \[ + X' = \frac{X - X_{\text{min}}}{X_{\text{max}} - X_{\text{min}}} + \] + This can be achieved in scikit-learn using the \mintinline{python}{MinMaxScaler} utility class. +\end{itemize} +It is generally good practice to normalise continuous variables before developing an ML model. +Some algorithms (e.g., $k$-NN) are much more susceptible to the effects of the relative scale of attributes than others (e.g., decision trees are more robust to the effects of scale). +\subsection{Binning} +\textbf{Binning} involves converting a continuous feature into a categorical feature. +To perform binning, we define a series of ranges called \textbf{bins} for the continuous feature that corresponds to the levels of the new categorical feature we are creating. +Two of the more popular ways of defining bins include equal-width binning \& equal-frequency binning. +\\\\ +Deciding on the number of bins can be complex: +in general, if we set the number of bins to a very low number we may lose a lot of information, but if we set the number of bins to a very high number then we might have very few instances in each bin or even end up with empty bins. + +\begin{figure}[H] + \centering + \includegraphics[width=0.7\textwidth]{images/numbins.png} + \caption{The effect of different numbers of bins} +\end{figure} + +\subsubsection{Equal-Width Binning} +The \textbf{equal-width binning} approach splits the range of the feature values into $b$ bins, each of size $\frac{\text{range}}{b}$. + +\begin{figure}[H] + \centering + \includegraphics[width=0.7\textwidth]{images/equalbins.png} + \caption{Equal-width binning} +\end{figure} + +\subsubsection{Equal-Frequency Binning} +\textbf{Equal-frequency binning} first sorts the continuous feature values into ascending order, and then places an equal number of instances into each bin, starting with bin 1. +The number of instances placed in each bin is simply the total number of instances divided by the number of bins $b$. + +\begin{figure}[H] + \centering + \includegraphics[width=0.7\textwidth]{images/freqbins.png} + \caption{Equal-frequency binning} +\end{figure} + +\subsection{Sampling} +Sometimes, the dataset that we have is so large that we do not use all the data available to us and instead take a smaller percentage from the larger dataset. +For example, we may wish to only use part of the data because training will take a long time with very many examples for some algorithms. +In the case of $k$-NN, a very large training set may lead to long prediction times. +However, we need to be careful when sampling to ensure that the resulting datasets are still representative of the original data and that no unintended bias is introduced during this process. +Common forms of sampling include: top sampling, random sampling, stratified sampling, under-sampling, \& over-sampling. + +\subsubsection{Top Sampling} +\textbf{Top sampling} simply selects the top $s\%$ of instances from a dataset to create a sample. +It runs a serious risk of introducing bias as the sample will be affected by any ordering of the original dataset; therefore, top sampling should be avoided. + +\subsubsection{Random Sampling} +\textbf{Random sampling} is a good default sampling strategy as it randomly selects a proportion ($s\%$) of the instances from a large dataset to create a smaller set. +It is a good choice in most cases as the random nature of the selection of instances should avoid introducing bias. + +\subsubsection{Stratified Sampling} +\textbf{Stratified sampling} is a sampling method that ensures that the relative frequencies of the levels of a specific stratification feature are maintained in the sampled dataset. +To perform stratified sampling, the instances in a dataset are divided into groups (or \textit{strata}), where each group contains only instances that have a particular level for the stratification feature. +$s\%$ of the instances in each stratum are randomly selected and these selections are combined to give an overall sample of $s\%$ of the original dataset. +\\\\ +In contrast to stratified sampling, sometimes we would like a sample to contain different relative frequencies of the levels of a particular discrete feature to the distribution in the original dataset. +To do this, we can use under-sampling or over-sampling. + +\subsubsection{Under-Sampling} +\textbf{Under-sampling} begins by dividing a dataset into groups, where each group contains only instances that have a particular level for the feature to be under-sampled. +The number of instances in the smallest group is the under-sampling target size. +Each group containing more instances than the smallest one is then randomly sampled by the appropriate percentage to create a subset that is the under-sampling target size. +These under-sampled groups are then combined to create the overall under-sampled dataset. + +\subsubsection{Over-Sampling} +\textbf{Over-sampling} addresses the same issue as under-sampling but in the opposite way: +after dividing the dataset into groups, the number of instances in the largest group becomes the over-sampling target size. +From each smaller group, we then create a sample containing that number of instances using random sampling without replacement. +These larger samples are combined to form the overall over-sampled dataset. + +\subsection{Feature Selection} +\subsubsection{The Curse of Dimensionality} +Some attributes are much more significant than others are considered equally in distance metric, possibly leading to bad predictions. +$k$-NN uses all attributes when making a prediction, whereas other algorithms (e.g., decision trees) use only the most useful features and so are not as badly affected by the curse of dimensionality. +Any algorithm that considers all attributes in a high-dimensional space equally has this problem, not just $k$-NN + Euclidean distance. +Two solutions to the curse of dimensionality are: +\begin{itemize} + \item Assign weighting to each dimension (not the same as distance-weighted $k$-NN). + Optimise weighting to minimise error. + \item Give some dimensions 0 weight: feature subset selection. +\end{itemize} + +Consider $~$ cases with $d$ dimensions, in a hypercube of \textit{unit} volume. +Assume that neighbourhoods are hypercubes with length $b$; volume is $b^d$. +To contain $k$ points, the average neighbourhood must occupy $\frac{k}{N}$ of the entire volume. +\begin{align*} + \Rightarrow& b^d = \frac{k}{N} \\ + \Rightarrow& b = \left( \frac{k}{N}\right)^\frac{1}{d} +\end{align*} + +In high dimensions, e.g. $k=10$, $N=1,000,000$, $d=100$, then $b = 0.89$, i.e., a neighbourhood must occupy 90\% of each dimension of space. +In low dimensions, with $k=10$, $N=1,000,000$, $d=2$, then $b= 0.003$ which is acceptable. +High dimensional spaces are generally very sparse, and each neighbour is very far away. + +\subsubsection{Feature Selection} +Fortunately, some algorithms partially mitigate the effects of the curse of dimensionality (e.g., decision tree learning). +However, this is not true for all algorithms, and heuristics for search can sometimes be misleading. +$k$-NN and many other algorithms use all attributes when making a prediction. +Acquiring more data is not always a realistic option; the best way to avoid the curse of dimensionality is to use only the most useful features during learning: \textbf{feature selection}. +\\\\ +We may wish to distinguish between different types of descriptive features: +\begin{itemize} + \item \textbf{Predictive:} provides information that is useful when estimating the correct target value. + \item \textbf{Interacting:} provides useful information only when considered in conjunction with other features. + \item \textbf{Redundant:} features that have a strong correlation with another feature. + \item \textbf{Irrelevant:} doesn't provide any useful information for estimating the target value. +\end{itemize} + +Ideally, a good feature selection approach should identify the smallest subset of features that maintain prediction performance. + +\subsubsection{Feature Selection Approaches} +\begin{itemize} + \item \textbf{Rank \& Prune:} rank features according to their predictive power and keep only the top $X\%$. + A \textbf{filter} is a measure of the predictive power used during ranking, e.g., information gain. + A drawback of rank \& prune is that features are evaluated in isolation, so we will miss useful \textit{interacting features}. + \item \textbf{Search for useful feature subsets:} + we can pick out useful interacting features by evaluating feature subsets. + We could generate, evaluate, \& rank all possible feature subsets then pick the best (essentially a brute force approach, computationally expensive). + A better approach is a \textbf{greedy local search}, which builds the feature subset iteratively by starting out with an empty selection, then trying to add additional features incrementally. + This requires evaluation experiments along the way. + We stop trying to add more features to the selection once termination conditions are met. +\end{itemize} + +\subsection{Covariance \& Correlation} +As well as visually inspecting scatter plots, we can calculate formal measures of the relationship between two continuous features using \textbf{covariance} \& \textbf{correlation}. + +\subsubsection{Measuring Covariance} +For two features $a$ \& $b$ in a dataset of $n$ instances, the \textbf{sample covariance} between $a$ \& $b$ is: +\[ + \text{cov}(a,b) = \frac{1}{n-1} \sum^n_{i=1} \left( \left( a_i - \overline{a} \right) \times \left( b_i - \overline{b} \right) \right) +\] + +where $a_i$ \& $b_i$ are the $i^\text{th}$ instances of features $a$ \& $b$ in a dataset, and $\overline{a}$ \& $\overline{b}$ are the sample means of features $a$ \& $b$. +\\\\ +Covariance values fall into the range $[- \infty, \infty]$, where negative values indicate a negative relationship, positive values indicate a positive relationship, \& values near to zero indicate that there is little to no relationship between the features. +\\\\ +Covariance is measured in the same units as the features that it measures, so comparing something like weight of a basketball player to the height of a basketball player doesn't really make sense as the features are not in the same units. +To solve this problem, we use the \textbf{correlation coefficient}, also known as the Pearson product-moment correlation coefficient or Pearson's $r$. + +\subsubsection{Measuring Correlation} +\textbf{Correlation} is a normalised form of covariance with range $[-1,1]$. +The correlation between two features $a$ \& $b$ can be calculated as +\[ + \text{corr}(a,b) = \frac{\text{cov}(a,b)}{\text{sd}(a) \times \text{sd}(b)} +\] +where $\text{cov}(a,b)$ is the covariance between features $a$ \& $b$, and $\text{sd}(a)$ \& $\text{sd}(b)$ are the standard deviations of $a$ \& $b$ respectively. +\\\\ +Correlation values fall into the range $[-1,1]$ where values close to $-1$ indicate a very strong negative correlation (or covariance), values close to $1$ indicate a very strong positive correlation, \& values around $0$ indicate no correlation. +Features that have no correlation are said to be \textbf{independent}. +\\\\ +The \textbf{covariance matrix}, usually denoted as $\sum$, between a set of continuous features $\{a,b, \dots, z\}$, is given as +\[ + \sum_{\{ a, b, \dots, z \}} = + \begin{bmatrix} + \text{var}(a) & \text{cov}(a,b) & \cdots & \text{cov}(a,z) \\ + \text{cov}(a,b) & \text{var}(b) & \cdots & \text{cov}(b,z) \\ + \vdots & \vdots & \vdots & \vdots \\ + \text{cov}(z,a) & \text{cov}(z,b) & \cdots & \text{var}(z) + \end{bmatrix} +\] + +Similarly, the \textbf{correlation matrix} is just a normalised version of the covariance matrix and shows the correlation between each pair of features: +\[ + \underset{\{ a,b,\dots,z \}}{\text{correlation matrix}} = + \begin{bmatrix} + \text{corr}(a,a) & \text{corr}(a,b) & \cdots & \text{corr}(a,z) \\ + \text{corr}(b,a) & \text{corr}(b,b) & \cdots & \text{corr}(b,z) \\ + \vdots & \vdots & \cdots & \vdots \\ + \text{corr}(z,a) & \text{corr}(z,b) & \cdots & \text{corr}(z,z) + \end{bmatrix} +\] + +Correlation is a good measure of the relationship between two continuous features, but is not perfect. +Firstly, the correlation measure given earlier responds only to linear relationships between features. +In a linear relationship between two features, as one feature increases or decreases, the other feature increases or decreases by a corresponding amount. +Frequently, features will have a very strong non-linear relationship that correlation does not respond to. +Some limitations of measuring correlation are illustrated very clearly in the famous example of Anscombe's Quartet, published by the famous statistician Francis Anscombe in 1973. diff --git a/year4/semester1/CT4101: Machine Learning/notes/images/equalbins.png b/year4/semester1/CT4101: Machine Learning/notes/images/equalbins.png new file mode 100644 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