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finish lecture 4
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Lecture 2 - 07-04-2020}
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\subsection{Argomento}
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@ -193,3 +196,5 @@ We want to replace this process with a predictor (so we don’t have to bored a
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person).\\
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y is the ground truth for x $\rightarrow$ mean reality!\\
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If i want to predict stock for tomorrow, i will wait tomorrow to see the ground truth.
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\end{document}
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(lecture3.tex
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@ -210,9 +210,9 @@ $\hat{y} = + \quad or \quad \hat{y} = - $
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\\\
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I can came up with some sort of classifier.
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\\\\
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Given $S$ training set, i can define $h_NN X \rightarrow \{-1,1\}\\
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Given $S$ training set, i can define $\hnn$ $X \rightarrow \{-1,1\}\\
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$
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$h_NN(x) = $ label $y_t$ of the point $x_t$ in $S$ closest to $X$\\
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$\hnn(x) = $ label $y_t$ of the point $x_t$ in $S$ closest to $X$\\
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\textbf{(the breaking rule for ties)}
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\\
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For the closest we mean euclidian distance
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\relax
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\@writefile{toc}{\contentsline {section}{\numberline {1}Lecture 4 - 07-04-2020}{1}\protected@file@percent }
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Lecture 4 - 07-04-2020}
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We spoke about Knn classifier with voronoi diagram
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$$
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\hat{\ell}(\hnn) = 0 \qquad \forall Traning set
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$$
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\\
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$\hnn$ predictor needs to store entire dataset.
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\\
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\subsection{Computing $\hnn$}
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Computing $\hnn(x)$ requires computing distances between x and points in the traning set.
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\\
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$$
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\Theta(d) \quad \textit{time for each distance}
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$$\\
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NN $\rightarrow$ 1-NN\\
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We can generalise NN in K-NN with $k = 1,3,5,7$ so odd $K$ \\
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$\hknn(x)$ = label corresponding to the majority of labels of the k closet point to
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x in the training set.\\\\
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How big could $K$ be if i have $n$ point?\\
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I look at the $k$ closest point\\
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When $k = m$?\\
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The majority, will be a constant classifier
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$\hknn$ is constant and corresponds to the majority of training labels\\
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Training error is always 0 for $\hnn$, while for $\hknn$ will be typically $>0$, with $k >
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1$\\
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Image: one dimensional classifier and training set is repeated.
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Is the plot of 1-NN classifier.\\
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Positive and negative.
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$K = 1$ error is 0.\\
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In the second line we switch to $k =3$. Second point doesn’t switch and third will
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be classify to positive and we have training mistake.\\
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Switches corresponds to border of voronoi partition.
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$$\knn \qquad \textit{For multiclass classification}$$\\
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$$
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(|Y| > 2 ) \qquad \textit{for regression } Y\equiv \barra{R}
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$$
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\\
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Average of labels of $K$ neighbours $\rightarrow$ i will get a number with prediction.
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\\
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I can weight average by distance
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\\
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You can vary this algorithm as you want.\\\\
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Let’s go back to Binary classification.\\
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The $k$ parameter is the effect of making the structure of classifier more
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complex and less complex for small value of $k$.\\\\
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--.. DISEGNO ..--
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\\
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Fix training set and test set\\
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Accury as oppose to the error
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\\\\
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Show a plot. Training error is 0 at $k = 0$.\\
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As i go further training error is higher and test error goes down. At some point
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after which training and set met and then after that training and test error goes
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up (accuracy goes down).\\
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If i run algorithm is going to be overfitting: training error and test error is high and also underfitting since testing and training are close and both high.
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Trade off point is the point in $x = 23$ (more or less).\\
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There are some heuristic to run NN algorithm without value of $k$.
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\\\\
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\textbf{History}
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\begin{itemize}
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\item $\knn$: from 1960 $\rightarrow$ $X \equiv \barra{R}^d$
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\item Tree predictor: from 1980
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\\
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\end{itemize}
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\subsection{Tree Predictor}
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If a give you data not welled defined in a Euclidean space.
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\\
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$X = X_1 \cdot x \cdot ... \cdot X_d \cdot x$ \qquad Medical Record
|
||||
\\
|
||||
$X_1 = \{Male, Female\}$\\
|
||||
$X_2 = \{Yes, No\}$
|
||||
\\
|
||||
so we have different data
|
||||
\\\\
|
||||
I want to avoid comparing $x_i$ with $x_j$, $i\neq j $\\
|
||||
so comparing different feature and we want to compare each feature with
|
||||
each self. I don’t want to mix them up.\\
|
||||
We can use a tree!
|
||||
\\
|
||||
I have 3 features:
|
||||
\begin{itemize}
|
||||
\item outlook $= \{sunny, overcast, rain\}$
|
||||
\item humidity $= \{[0,100]\}$
|
||||
\item windy $ = \{yes,no\}$
|
||||
\end{itemize}
|
||||
... -- DISEGNO -- ...\\\\
|
||||
Tree is a natural way of doing decision and abstraction of decision process of
|
||||
one person. It is a good way to deal with categorical variables.\\
|
||||
What kind of tree we are talking about?\\
|
||||
Tree has inner node and leaves. Leaves are associated with labels $(Y)$ and
|
||||
inner nodes are associated with test.
|
||||
\begin{itemize}
|
||||
\item Inner node $\rightarrow$ test
|
||||
\item Leaves $\rightarrow$ label in Y
|
||||
\end{itemize}
|
||||
%... -- DISEGNO -- ...
|
||||
Test if a function $f$ (NOT A PREDICTOR!) \\
|
||||
Test $ \qquad f_i \, X_i \rightarrow \{1,...,k\}$
|
||||
\\ where $k$ is the number of children (inner node) to which test is assigned
|
||||
\\
|
||||
In a tree predictor we have:
|
||||
\begin{itemize}
|
||||
\item Root node
|
||||
\item Children are ordered(i know the order of each branch that come out from the node)
|
||||
\end{itemize}
|
||||
$$
|
||||
X = \{Sunny, 50\%, No \} \quad \rightarrow \quad \textit{are the parameters for } \{outlook. humidity, windy \}
|
||||
$$
|
||||
\\
|
||||
$
|
||||
f_i =
|
||||
\begin{cases}
|
||||
1, & \mbox{if } x_2 \in [30 \%,60 \% ]
|
||||
\\
|
||||
2, & \mbox{if } otherwise \end{cases}
|
||||
$
|
||||
\\ where the numbers 1 and 2 are the children
|
||||
\\
|
||||
A test is partitioning the range of values of a certain attribute in a number of
|
||||
elements equal to number of children of of the node to which the test is
|
||||
assigned.
|
||||
\\
|
||||
$h_T(x)$ is always the label of a leaf of T\\
|
||||
This leaf is the leaf to which $x$ is \textbf{routed}
|
||||
\\
|
||||
Data space for this problem (outlook,..) is partitioned in the leaves of the tree.
|
||||
It won’t be like voronoi graph.
|
||||
How do I build a tree given a training set?
|
||||
How do i learn a tree predictor given a training set?
|
||||
\begin{itemize}
|
||||
\item Decide tree structure (how • many node, leaves ecc..)
|
||||
\item Decide test on inner nodes
|
||||
\item Decide labels on leaves
|
||||
\end{itemize}
|
||||
We have to do this all together and process will be more dynamic.
|
||||
For simplicity binary classification and fix two children for each inner node.\\\\
|
||||
$ Y = \{-1, +1 \}$
|
||||
\\ $2$ children for each inner node
|
||||
\\\\
|
||||
What's the simplest way?\\
|
||||
Initial tree and correspond to a costant classifier
|
||||
\\\\
|
||||
-- DISEGNO --
|
||||
\\\\
|
||||
\textbf{Majority of all example}
|
||||
\\\\
|
||||
-- DISEGNO --
|
||||
\\\\
|
||||
$(x_1, y_1) ... (x_m, y_m)$ \\
|
||||
$ x_t \in X$ \qquad $ y_t \in \{-1,+1\}$\\
|
||||
Training set $S = \{ (x,y) \in S$, x is routed to $\ell\}$\\
|
||||
$S_{\ell}^+$
|
||||
\\\\
|
||||
-- DISEGNO --
|
||||
\\\\
|
||||
$ S_{\ell}$ and $ S’_{\ell}$ are given by the result of the test, not the labels and $\ell$ and $\ell'$.
|
||||
\end{document}
|
||||
@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.12) 12 APR 2020 15:12
|
||||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.12) 12 APR 2020 15:21
|
||||
entering extended mode
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**./main.tex
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(main.tex
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[9]
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[10]
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[]
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)
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[]
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[11] (lectures/lecture3.tex
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||||
) [11] (lectures/lecture3.tex
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[16]) [17] (lectures/lecture4.tex) [18] (lectures/lecture5.tex) [19]
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[16]
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[]
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) [17] (lectures/lecture4.tex) [18] (lectures/lecture5.tex) [19]
|
||||
(lectures/lecture6.tex) [20] (lectures/lecture7.tex) [21]
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(lectures/lecture8.tex) [22] (lectures/lecture9.tex) [23]
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(lectures/lecture10.tex
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12.pfb><E:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/symbols/msbm10.
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pfb>
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Output written on main.pdf (27 pages, 198551 bytes).
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Output written on main.pdf (27 pages, 198691 bytes).
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PDF statistics:
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\newcommand\barra[1]{\mathbb{#1}}
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\newcommand\hnn{h_{NN}}
|
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|
||||
\newcommand\hknn{h_{k-NN}}
|
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\newcommand\knn{K_{NN}}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user