From 8fa1147b9f32da5d50dbbd15fc5a9a98b3323766 Mon Sep 17 00:00:00 2001 From: manuel helmerichs Date: Thu, 7 Nov 2024 17:25:12 +0100 Subject: [PATCH 1/5] nb visualized --- .../figure/nb-db.png | Bin 0 -> 948177 bytes .../figure_man/nb-categorial_1.png | Bin 0 -> 47872 bytes .../figure_man/nb-categorial_2.png | Bin 0 -> 54693 bytes .../figure_man/nb-categorial_3.png | Bin 0 -> 60940 bytes .../rsrc/fig-nb-db.R | 106 ++++++++++++++++++ .../slides-classification-naivebayes.tex | 65 +++-------- 6 files changed, 124 insertions(+), 47 deletions(-) create mode 100644 slides/supervised-classification/figure/nb-db.png create mode 100644 slides/supervised-classification/figure_man/nb-categorial_1.png create mode 100644 slides/supervised-classification/figure_man/nb-categorial_2.png create mode 100644 slides/supervised-classification/figure_man/nb-categorial_3.png create mode 100644 slides/supervised-classification/rsrc/fig-nb-db.R diff --git a/slides/supervised-classification/figure/nb-db.png 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z;l6#IYxTsb3;nil{?7B8B-OFz{kHL`XD%Fdj%_@rp*uV2OA85Kc64tq6p(eaI=XbW zf<QZldi92_~wA*>uer5YYXToGG)f29h1^I=}!JLbd5@*e= z%C#L&+l4vcIahLFHh$iYcQ*KqtFy(;&3a}-GhpS2XLn()Ez-@al2Jo+wr$9ejYs~u$D1kBVYoAOL=g6GC1D%7BF@TVC4Zaw2Bd@=^ z*V~yVsI{CYdcla54tLJDl{mTWtblI$iX(*1hFg;s z8kly(gd`|VWkJpgBnYVfJN1IKB+clARDaJTuu?^X?n;bc{AYf#?m-}~Q|t&Y%0CYR zd;^og|Mx1ypS|Akk+4hu^$@}PfFj1l z@Is>h4u}Z_<+#@J|5`QS@A&dk*8cOakziz`U8n&#{~dV-3W^h@srP>ebQ>AQV-fq` zk@JCdNNdRT-vPabLU&;^LjN5(4KhqqeDt54@MjEye2JJ>Q~7)LuBP&@1#`73|B4S+ z8~iF(A_G$WDpvk;N?paut5|uJ4E+@quky;jHpEq|{405QcNHuDS{he*@j1^xa7M_)m|e{GX16#X9% z2e1p_SA@j>mc|vg^FO}niYEKtTwK8!eME)+_S%uVnm+2mgDU{C@{$oT9p| W=QSx-jm;y0FL7aMp row["b"], "a", "b"))) +grid_preds$max_prob = apply(grid_preds[, c("a", "b")], 1, max) +grid = cbind(grid, grid_preds) + +# Recompute density for visualizing distributions +grid_dens1 = grid +grid_dens1$dens = dmvnorm(grid_dens1[, c("x1", "x2")], mean = mu1, sigma = S1) +grid_dens2 = grid +grid_dens2$dens = dmvnorm(grid_dens2[, c("x1", "x2")], mean = mu2, sigma = S2) + +# PLOT ------------------------------------------------------------------------- + +# Generate the plot +orig_data = as.data.frame(task$data()) +pl = ggplot() + + geom_tile(data = grid, aes(x = x1, y = x2, fill = pred_class, alpha = max_prob)) + + geom_contour(data = grid_dens1, aes(x = x1, y = x2, z = dens), color = "#E69F00", alpha = 0.9, lwd = 1.5, bins = 10) + + geom_contour(data = grid_dens2, aes(x = x1, y = x2, z = dens), color = "#56B4E9", alpha = 0.9, lwd = 1.5, bins = 10) + + geom_point(data = orig_data, aes(x = x1, y = x2, color = y), size = point_size) + + guides(shape = FALSE, alpha = FALSE) + + scale_fill_manual(values = c("a" = "#E69F00", "b" = "#56B4E9")) + + scale_color_manual(values = c("a" = "#E69F00", "b" = "#56B4E9")) + + labs(x = expression(x[1]), y = expression(x[2]), color = "class", fill = "class") + + theme_minimal() + + scale_alpha(range = c(0.1, 0.5), guide = 'none') + + theme( + plot.title = element_text(hjust = 0.5, size = base_size, face = "bold"), + axis.title = element_text(size = base_size, face = "bold"), + axis.text = element_text(size = base_size * 0.75, face = "bold"), + legend.title = element_text(size = base_size, face = "bold"), + legend.text = element_text(size = base_size * 0.75, face = "bold") + ) + +# Save directly to PNG +ggsave(filename = "../figure/nb-db.png", plot = pl, width = plot_width, height = plot_height, dpi = plot_dpi) diff --git a/slides/supervised-classification/slides-classification-naivebayes.tex b/slides/supervised-classification/slides-classification-naivebayes.tex index a8237d185..f70e03210 100644 --- a/slides/supervised-classification/slides-classification-naivebayes.tex +++ b/slides/supervised-classification/slides-classification-naivebayes.tex @@ -12,7 +12,7 @@ }{% Lecture title Naive Bayes }{% Relative path to title page image: Can be empty but must not start with slides/ - figure/reg_class_nb_1 + figure/nb-db }{% Learning goals, wrapped inside itemize environment \item Understand the idea of Naive Bayes \item Understand in which sense Naive Bayes is a special QDA model @@ -37,52 +37,33 @@ \begin{vbframe}{NB: Numerical Features} -We use a univariate Gaussian for $p(x_j | y=k)$, and estimate $(\mu_{kj}, \sigma^2_{kj})$ in the standard manner. Because of $\pdfxyk = \prodjp p(x_j|y = k)$, the joint conditional density is Gaussian with diagonal but non-isotropic covariance structure, and potentially different across classes. Hence, NB is a (specific) QDA model, with quadratic decision boundary. +We use a univariate Gaussian for $p(x_j | y=k)$, and estimate $(\mu_{kj}, \sigma^2_{kj})$ in the standard manner. Because of $\pdfxyk = \prodjp p(x_j|y = k)$, the joint conditional density is Gaussian with diagonal but non-isotropic covariance structure, and potentially different across classes. -\begin{knitrout}\scriptsize -\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor} +\begin{center} +\includegraphics[width=0.79\textwidth, clip = true, trim = {0 20 0 20}]{figure/nb-db.png} +\end{center} -{\centering \includegraphics[width=0.95\textwidth]{figure/reg_class_nb_1} +$\Rightarrow$ \textbf{NB is a} (specific) \textbf{QDA model}, with a quadratic decision boundary. -} - - - -\end{knitrout} \end{vbframe} +\begin{frame}{NB: Categorical Features} -\begin{vbframe}{NB: Categorical Features} - - We use a categorical distribution for $p(x_j | y = k)$ and estimate the probabilities $p_{kjm}$ that, in class $k$, our $j$-th feature has value $m$, $x_j = m$, simply by counting the frequencies. +We use a categorical distribution for $p(x_j | y = k)$ and estimate the probabilities $p_{kjm}$ that, in class $k$, our $j$-th feature has value $m$, $x_j = m$, simply by counting the frequencies. $$ -% p(x_j | y = k) = \frac{(\sum x_i)!}{\prod_i x_i!} \prod_m p_{kjm}^{[x_j = m]} p(x_j | y = k) = \prod_m p_{kjm}^{[x_j = m]} $$ -% -% and for the completely observed data this becomes a multinomial distribution -% -% $$ -% \frac{(\sum_i x_i)!}{\prod_i x_i!} \prod_j p_{kj}^{v_{kj}}, -% $$ - -% with ${v_{kj}} = \sum_{i = 1}^n [x_j^{(i)} = 1]$ the number of times $(j, k)$ occurs. Because of the simple conditional independence structure it is also very easy to deal with mixed numerical / categorical feature spaces. -\end{vbframe} - - -% \begin{vbframe}{Categorical NB is linear in frequencies} -% We can now prove that the decision boundaries between klasses k and l are linear: +\begin{flushright} +% SOURCE: https://docs.google.com/presentation/d/1X2FxetT6fewXhoGLZmgJyEHY_pWbKjQ6AHZiZQHvwzA/edit?usp=sharing +\only<1>{\includegraphics[width=\textwidth, clip = true, trim = {50 400 120 350}]{figure_man/nb-categorial_1.png}} +\only<2>{\includegraphics[width=\textwidth, clip = true, trim = {50 410 120 350}]{figure_man/nb-categorial_2.png}} +\only<3>{\includegraphics[width=\textwidth, clip = true, trim = {50 410 120 350}]{figure_man/nb-categorial_3.png}} +\end{flushright} -% $$ -% \log \frac{\pi_k(x)}{\pi_l(x)} \propto \log\frac{\pi_k}{\pi_l} + \sum_j v_{kj} \ln p_{kj} - \sum_j v_{lj} \ln p_{lj} -% $$ - -% This is a linear function in the parameter vector $v = (v_{11}, \ldots, v_{1p}, \ldots, v_{g1} \ldots v_{gp})$. - -% \end{vbframe} +\end{frame} \begin{vbframe}{Laplace Smoothing} If a given class and feature value never occur together in the training data, then the frequency-based probability estimate will be zero. @@ -98,19 +79,6 @@ \end{vbframe} -% \begin{vbframe}{Naive Bayes as a linear classifier} - -% In general, the \emph{Naive Bayes classifier} is \textbf{not} a \emph{linear} classifier. - -% Furthermore, one can show that the Naive Bayes is a linear classifier in a particular feature space if the features are from exponential families (e. g. binomial, multinomial, normal distribution). - -% \lz - -% However, it can be shown that the \emph{Naive Bayes classifier} is a linear classifier in a particular feature space if the features are from exponential families (e. g. binomial, multinomial, normal distribution) . - -% \end{vbframe} - - \begin{vbframe}{Naive Bayes: application as spam filter} \begin{itemize} \item In the late 90s, Naive Bayes became popular for e-mail spam filter programs @@ -120,6 +88,9 @@ \end{itemize} \end{vbframe} +\begin{vbframe}{Benchmarking SPAM} + +\end{vbframe} \endlecture From 96c36a864c4881880947dbe42a4ffeb957a6d761 Mon Sep 17 00:00:00 2001 From: manuel helmerichs Date: Thu, 7 Nov 2024 17:53:39 +0100 Subject: [PATCH 2/5] benchmark --- .../figure/nb-bench.png | Bin 0 -> 94834 bytes .../supervised-classification/rsrc/nb-bench.R | 35 ++++++++++++++++++ 2 files changed, 35 insertions(+) create mode 100644 slides/supervised-classification/figure/nb-bench.png create mode 100644 slides/supervised-classification/rsrc/nb-bench.R diff --git a/slides/supervised-classification/figure/nb-bench.png b/slides/supervised-classification/figure/nb-bench.png new file mode 100644 index 0000000000000000000000000000000000000000..a80c5b2c1ac6c4e740453b270a1555f1d5f2e52a GIT binary patch literal 94834 zcmeFa2UJtp*C-xj6demj6i}*R6p<#PK&Uzfa6n<`y(mMGj&uS9$3jE_XQYS(7(t|j zju1MiAS#eh44p)3B$R*CIm*+(@ZwN$6 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zvXwyh2(^p!LX=Jx#l=kFj3K5u>n<=n<-ANXJhBR6U0R+SKBRMZ+LjrsNU>!=HhXdR zl#KbOUp2X+CR@D=^On^Ug`5vaGn`1yp+Q5I8HwakMKD60SXQASzPyZ>j$(i6=L97V zHyIW9L}KsS6c4|nPG{n=QN_lrafOe#^qHcN%Tiw2zEdUC+HBlt-=z-nmCun+R^t7+ zGTDJ8iTId(n_ifAMLe{O^81nc(x85Bnl}C<#bUv|{#L`KEzj1 z1SNlneY_T=JMtnaOD1Ww0B~fo`eX><*eHrRJ=*`3{e{Dx+vD5NP0O;P`9aE@W1Xhz z@0G?=-S||b)~d3QL}<9w!P$nE=!AePxge1_ev% z7tBoiUq5<-#Q{{5$sz3FqTEBP#elqdX<>rCwp0@b>E$J;z#MB-kFQpQH-m3uxB{6A z{yG^K7^`@f%AV33cAD!OzZtP*aP&1O*KI_?NrDkxLi#oPi3?7J^QfN6+BXEPjakh~z3ZuFeoIZw6F=cak zL_t>P_$ER9{#qBJHX~-7G70&A=NC|aSP!a3oRvx2>|4Mvjh-cqcX&v*9lyEBne-ue zTg;#W=#WgmJ%>IRYyow?7v%>%S}|Wn4he|0-H`<463q@Vl0Y+4)*~Rntn?S6Qx_3I z(O4YRszil&V_y8Hj@KH>Qk(vF)xc`%;Tns emlD|dc~vM-QEGEikIEO`qu>d%>% lrn("classif.qda")) +lrn_nb = GraphLearner$new(pca %>>% lrn("classif.naive_bayes")) +lrn_lda = GraphLearner$new(pca %>>% lrn("classif.lda")) + +learners = list(lrn_nb, lrn_lda, lrn_qda) +bmr = benchmark(benchmark_grid(task, learners, rsmp("cv", folds = 5))) + +a <- autoplot(bmr, type = "boxplot") + + ylab("CE for 5-fold CV") + + xlab("Learners") + + scale_x_discrete(labels = c("QDA", "NB", "LDA")) + + theme_minimal() + + theme( + axis.title = element_text(size = 22, face = "bold"), + axis.text = element_text(size = 20, face = "bold"), + legend.title = element_text(size = 22, face = "bold"), + legend.text = element_text(size = 20, face = "bold"), + axis.text.x = element_text(angle = 45, hjust = 1) + ) +ggsave("../figure/nb-bench.png", plot = a, width = 12, height = 8, dpi = 300) From c74cc4b27a4f472ab8b757b4f0dccbd37e3b8b50 Mon Sep 17 00:00:00 2001 From: manuel helmerichs Date: Thu, 7 Nov 2024 18:50:52 +0100 Subject: [PATCH 3/5] laplace --- .../slides-classification-naivebayes.tex | 24 +++++++++++++++---- 1 file changed, 19 insertions(+), 5 deletions(-) diff --git a/slides/supervised-classification/slides-classification-naivebayes.tex b/slides/supervised-classification/slides-classification-naivebayes.tex index f70e03210..861311904 100644 --- a/slides/supervised-classification/slides-classification-naivebayes.tex +++ b/slides/supervised-classification/slides-classification-naivebayes.tex @@ -66,16 +66,24 @@ \end{frame} \begin{vbframe}{Laplace Smoothing} -If a given class and feature value never occur together in the training data, then the frequency-based probability estimate will be zero. +\begin{small} +If a given class and feature value never occur together in the training data, then the frequency-based probability estimate will be zero, e.g., $p_{\text{no, class, 1st}} = 0$. \lz -This is problematic because it will wipe out all information in the other probabilities when they are multiplied. +When computing $\postk$, any zero probability will cause the entire product to be zero, negating information from other features: + +$$ +\P(\text{no} | \text{class = 1st, sex = male}) = \frac{p_{\text{no, class, 1st}} \cdot p_{\text{no, sex, male}} \cdot \pik[no]}{\sumjg p(\text{class = 1st, sex = male} | y = j)\pi_j} = 0 +$$ + +This is problematic because it will wipe out all information in the other probabilities when they are multiplied! \lz -% -A simple numerical correction is to set these zero probabilities to a small value to regularize against this case. +$\Rightarrow$ A simple numerical correction is to set these zero probabilities to a small value to regularize against this case. + +\end{small} \end{vbframe} @@ -84,12 +92,18 @@ \item In the late 90s, Naive Bayes became popular for e-mail spam filter programs \item Word counts were used as features to detect spam mails (e.g., "Viagra" often occurs in spam mail) \item Independence assumption implies: occurrence of two words in mail is not correlated - \item Seems naive ("Viagra" more likely to occur in context with "Buy now" than "flower"), but leads to less required parameters and therefore better generalization, and often works well in practice. + \item Seems naive ("Viagra" more likely to occur in context with "Buy now" than "flower"), but leads to less required parameters and therefore better generalization. \end{itemize} \end{vbframe} \begin{vbframe}{Benchmarking SPAM} +Benchmarking QDA, Naive Bayes and LDA on $\texttt{spam}$: + +\begin{center} +\includegraphics[clip=true, trim={0 0 0 17}, width=0.80\linewidth]{figure/nb-bench.png} +\end{center} +$\Rightarrow$ In practice, NB often performs well even when the independence assumption is violated! \end{vbframe} \endlecture From 97281299b50a45af4e4947844075911ba2db9eaa Mon Sep 17 00:00:00 2001 From: manuel helmerichs Date: Thu, 7 Nov 2024 19:15:38 +0100 Subject: [PATCH 4/5] finalised discanalysis histogram & latex-math --- .../figure/disc_univariate-1.png | Bin 40013 -> 222175 bytes .../figure/disc_univariate-2.png | Bin 45674 -> 238835 bytes .../figure/disc_univariate-3.png | Bin 50256 -> 231002 bytes .../figure/disc_univariate-4.png | Bin 39157 -> 214384 bytes .../rsrc/fig-disc_univariate.R | 16 ++++++++++++---- .../slides-classification-discranalysis.tex | 12 ++++++------ 6 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zZv@Ecxv%TklUO2BEvo3txq4wK zONa$c#sfAOJ=NELkT!0Ps Date: Thu, 7 Nov 2024 19:19:52 +0100 Subject: [PATCH 5/5] hats --- .../slides-classification-naivebayes.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/slides/supervised-classification/slides-classification-naivebayes.tex b/slides/supervised-classification/slides-classification-naivebayes.tex index 861311904..af560fca6 100644 --- a/slides/supervised-classification/slides-classification-naivebayes.tex +++ b/slides/supervised-classification/slides-classification-naivebayes.tex @@ -74,7 +74,7 @@ When computing $\postk$, any zero probability will cause the entire product to be zero, negating information from other features: $$ -\P(\text{no} | \text{class = 1st, sex = male}) = \frac{p_{\text{no, class, 1st}} \cdot p_{\text{no, sex, male}} \cdot \pik[no]}{\sumjg p(\text{class = 1st, sex = male} | y = j)\pi_j} = 0 +\P(\text{no} | \text{class = 1st, sex = male}) = \frac{p_{\text{no, class, 1st}} \cdot p_{\text{no, sex, male}} \cdot \pikh[no]}{\sumjg p(\text{class = 1st, sex = male} | y = j)\pih_j} = 0 $$ This is problematic because it will wipe out all information in the other probabilities when they are multiplied!