From 9397827eb0a126db94b509efaeca30361a84ddce Mon Sep 17 00:00:00 2001 From: Andrew Date: Wed, 6 Nov 2024 23:29:35 +0000 Subject: [PATCH] [CT4101]: Week 9 lecture notes --- .../notes/CT4101-Notes.pdf | Bin 3917590 -> 3923462 bytes .../notes/CT4101-Notes.tex | 39 ++++++++++++++++++ 2 files changed, 39 insertions(+) diff --git a/year4/semester1/CT4101: Machine Learning/notes/CT4101-Notes.pdf b/year4/semester1/CT4101: Machine Learning/notes/CT4101-Notes.pdf index a8652f5a1056d85e119dd8b5093be40d7eb28d39..263bec75bc35f87499265bead0e4a10ba4049ac1 100644 GIT binary patch delta 89996 zcmY&-JnQ!cZwJNdB5+uf2?Qr zS$X!F{jlEiawa|g3_85)47yU4!wo+&hg&0vk_EB!{l0lJe0&p zdF+Q=7GHFx7@g+SC|(O}G<6~Z;#$ieQ{Cn>*0yZgIflc$H#j-(LxZF^t7#3#Ydm47 zP`|bG1r9Z#vOT)7M?z8Z%P@k5=otD#H1I!K7^8goxJikM1^59?l?q?j{6Qn!H6mWh zijq~pj?xBSo!3cAO`VgQuCSF(HqwQb32{JChB~s5MhZt<6muw;lhBMC0B3fTw6EZR zisbykLD5>dUQyiq7;y|*P&uj(v^zEZ{*6`z^iQ5q?Jn;IHgLEQHOvD8dtjnLc16hi#VJ* zH<%gfvJSJF8cv9PsvVyF;p6XAI?Ffj45Y#lbm3zZeKg8Rsry2+aL`361S_eTm`xRs 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b/year4/semester1/CT4101: Machine Learning/notes/CT4101-Notes.tex index 59840f0f..e952116b 100644 --- a/year4/semester1/CT4101: Machine Learning/notes/CT4101-Notes.tex +++ b/year4/semester1/CT4101: Machine Learning/notes/CT4101-Notes.tex @@ -1575,10 +1575,49 @@ To make such judgements without deep domain knowledge, a normalise \textbf{domai \\\\ The \textbf{$R^2$ coefficient} is a domain-independent measure that compares the performance of a model on a test set with the performance of an imaginary model that always predicts the average values from the test set. $R^2$ values may be interpreted as the amount of variation in the target feature that is explained by the descriptive features in the model. +\begin{align*} + \text{sum of squared errors} =& \frac{1}{2} \sum^n_{i=1} \left( t_i - \mathbb{M} \left( d_i \right) \right)^2 \\ + \text{total sum of squares} =& \frac{1}{2} \sum^n_{i=1} \left( t_i - \overline{t} \right)^2 \\ + R^2 =& \frac{\text{sum of squared errors}}{\text{total sum of squares}} +\end{align*} +where $\overline{t}$ is the average value of the target variable. +\\\\ +$R^2$ values are usually in the range $[0,1]$, with larger values indicating better performance. +However, $R^2$ values can be $< 0$ in certain rare cases (although 1 is always the maximum $R^2$ value). +Negative $R^2$ values indicate a very poor model performance, i.e. that the model performs worse than the horizontal straight-line hypothesis that always predicts the average value of the target feature. +For example, a negative $R^2$ on the test set with a positive $R^2$ value on the training set likely indicates that the model is overfit to the training data. +\subsection{Applying $k$-NN to Regression Tasks} +Previously, we have seen that the $k$-nearest neighbours algorithm bases its prediction on several ($k$) nearest neighbours by computing the distance from the query case to all stored cases, and picking the $k$ nearest neighbours. +When $k$-NN is used for classification tasks, the neighbours vote on the classification of the test case. +In \textbf{regression} tasks, the average value of the neighbours is taken as the label for the query case. +\subsubsection{Uniform Weighting} +Assuming that each neighbour is given an equal weighting: +\begin{align*} + \text{prediction}(q) = \frac{1}{k} \sum^k_{i=1} t_i +\end{align*} + +where $q$ is a vector containing the attribute values for the query instance, $k$ is the number of neighbours, $t_i$ is the target value of neighbour $i. + +\subsubsection{Distance Weighting} +Assuming that each neighbour is given a weight based on the inverse square of its distance from the query instance: + +\begin{align*} + \text{prediction}(q) = \frac{ \sum^k_{i=1} \left( \frac{1}{ \text{dist}(q, d_i)^2 \times t_i } \right) }{ \sum^k_{i=1} \left( \frac{1}{\text{dist}(q, d_i)^2 } \right) } +\end{align*} + +where $q$ is a vector containing the attribute values for the query instance and $\text{dist}(q,d_i)$ returns the distance between the query and the neighbour $i$. + +\subsection{Applying Decision Trees to Regression} +\textbf{Regression trees} are constructed similarly to those for classification; +the main change is that the function used to measure the quality of a split is changed so that it is a measure relevant to regression, e.g. variance, MSE, MAE, etc. +This adapation is easily made to the ID3/C4.5 algorithm. +\\\\ +The aim in regression trees is to group similar target values together at a leaf node. +Typically, a regression tree returns the mean target value at a leaf node.