[PS] Examples

semester4/ps/ps-rb/parts/02_variables.tex

@@ -279,6 +279,12 @@ $$
     Z &\sim \mathcal{N}\Biggl( \underbrace{m_0 + \sum_{i=1}^n \lambda_iX_i}_{m_Z}\quad,\quad \underbrace{\sum_{i=1}^n \lambda_i^2 \sigma_i^2}_{\sigma^2_Z} \Biggr)
 \end{align*}
 
+{\footnotesize
+    \remark $X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu, \sigma^2) \implies \Bigl(\displaystyle\sum_{i=1}^n X_i\Bigr) \sim \mathcal{N}(n\mu,n\sigma^2)$
+
+        \remark $X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu, \sigma^2) \implies  \Bigl(\underbrace{\frac{1}{n}\sum_{i=1}^n X_i}_{\bar{X}_n}\Bigr) \sim \mathcal{N}(\mu,\frac{\sigma^2}{n})$
+}
+
 \definition \textbf{Standardnormalverteilung} 
 $$X \sim \mathcal{N}(0,1) \qquad \Phi(x) := \P[X \leq x]$$
 

semester4/ps/ps-rb/parts/03_expectation.tex

@@ -84,8 +84,8 @@ $$
     \begin{align*}
         \V[X]               &= \E\bigl[X^2\bigr] - \E[X]^2      & (X \sim \text{Poisson}(\lambda))      \\
         \lambda             &= \E\bigl[X^2\bigr] - \lambda^2                                            \\
-        \lambda + \lambda^2 &= \E\bigl[X^2\bigr]                                                        \\
-    \end{align*}
+        \boxed{\lambda + \lambda^2} &= \E\bigl[X^2\bigr]                                                        \\
+    \end{align*}    
 }
 
 \lemma \textbf{Eigenschaften}

semester4/ps/ps-rb/parts/06_limits.tex

@@ -58,4 +58,18 @@ $$
 
 \newpage
 
-div % s.d. Statistik auf einer Seite bleibt
+{\footnotesize
+    \textbf{Beispiel}: Zentraler Grenzwertsatz als Teststatistik
+
+    $X_1,\ldots,X_{72} \overset{\text{i.i.d.}}{\sim} \text{Ber}(\theta)$, $S_{72} = \sum_{i=1}^{72} X_i$. Stichprobe $\bar{s}_{72}=32$.
+    $$
+        H_0: \theta = \frac{1}{3} \qquad H_A: \theta > \frac{1}{3}
+    $$
+    Mit ZGS, wobei wir $\theta = \frac{1}{3}$ ($H_0$) nutzen, erhalten wir $T$:
+    $$
+        T \overset{\text{ZGS}}{=} \frac{S_n - n\mu}{\sigma\sqrt{n}} = \frac{S_n - n\E[X_i]}{\sqrt{\V[X_i]n}} \overset{\text{Ber.}}{=} \frac{S_n - 72\cdot\frac{1}{3}}{\sqrt{\frac{2}{9}\cdot 72}} = \boxed{\frac{S_n-24}{4}}
+    $$
+    Der Vorteil hiervon ist, der ZGS garantiert: $T \sim \mathcal{N}(n\mu, n\sigma^2)$:
+
+    Realisierungen $T(\omega) = T(\bar{s}_n) = T(32) = 2$ bilden mit $z_{1-\alpha}$: $K = (z_{1-\alpha},\infty)$ und können direkt ausgewertet werden, z.B. für $p$-Wert.
+}

semester4/ps/ps-rb/parts/07_stats.tex

@@ -94,6 +94,7 @@ $$
         \item $f(\vartheta) := \ln\Bigl( L(x_1,\ldots,x_n;\vartheta) \Bigr)$
         \item $\vartheta^*$ finden, s.d. $f'(\vartheta^*) = 0$
         \item Argumentieren, dass $\vartheta^*$ das Maximum ist (z.B. via $f''$)
+        \item Folglich: $T_\text{ML} = \theta^*$
     \end{enumerate}
 }
 
@@ -370,7 +371,7 @@ $$
 \remark \textbf{Interpretation}
 \begin{align*}
     \alpha > p      &\implies (T, K_t) \text{ lehnt } H_0 \text{ ab} \\
-    \alpha \leq p   &\implies (T, K_t) \text{ akzeptier } H_0 
+    \alpha \leq p   &\implies (T, K_t) \text{ akzeptiert } H_0 
 \end{align*}
 
 {\footnotesize

semester6/iml/parts/00_intro.tex

@@ -1 +1 @@
-\textit{placeholder}
+\subtext{Wherever $\Vert\cdot\Vert_p$ isn't specified, $p=2$.}

semester6/iml/parts/04_networks.tex

@@ -501,6 +501,35 @@ $$
 $$
 }
 
+\newpage
+
+\subsection{Network sizes}
+
+
+
+{\footnotesize
+    \textbf{Example}: CNN vs FCNN (MLP) Parameter counts.
+
+    We can model an RGB Image as $\mathbf{I}\in\R^{1920\times1080\times3}$ where
+    $(\mathbf{I}_{i,j,1},\mathbf{I}_{i,j,2},\mathbf{I}_{i,j,3})$ are the RGB values of pixel $(i,j)$. We use a CNN \& FCNN to analyze.
+
+    For an FCNN $N$ with $1$ hidden layer with $h$ nodes, and $o$ output nodes:
+    $$
+        \text{size}(N) = \underbrace{(1920 \cdot 1080 \cdot h)}_{\text{All-to-All}} + \underbrace{(h \cdot o)}_\text{Output} + \underbrace{(h + o)}_\text{Bias}
+    $$
+    For a CNN $N'$ with an $n\times n$ Filter, padding $p$ and stride $s$:
+    $$
+        \text{size}(N') = \biggl(\frac{1920 + 2\cdot p - n}{s}+1\biggr)\cdot\biggl(\frac{1080+2\cdot p - n}{s}+1\biggr)
+    $$
+    The advantage of CNNs becomes clear when plugging in values, e.g.
+    $$
+        N:\ h=256, o=10 \qquad N':\ n=4, p=2, s=2 
+    $$
+    $$
+    \text{size}(N) = 1'592'527'626 \qquad \text{size}(N') = 519'901
+    $$
+}
+
 \subsubsection{Multidimensional Convolution}
 \textbf{TODO} add explanation
 

semester6/iml/parts/05_unsupervised.tex

@@ -18,6 +18,8 @@ There are many use-cases:
 The goal here is to group inputs into clusters, based on some definiton of similarity, e.g. $l_2$ distance for $\mathcal{D} \subset \R^2$.\\
 \subtext{This can be seen as the unsupervised analogy to classification}
 
+\subsubsection{Basic Methods}
+
 \method \textbf{Hierarchical Clustering}
 
 A simple method, using the "similarity" measure directly.
@@ -58,11 +60,14 @@ $$
 $$
 \subtext{(minimize the sum of sq. distances between points \& their centers)}
 
-So we are searching:
+{\footnotesize
+    \remark $\Vert\cdot\Vert_2$ corresponds to the \textit{mean}. $\Vert\cdot\Vert_1$ would use the \textit{median}.
+}
+
+So we are searching: (non-convex \& NP-hard)
 $$
     \underset{\mu}{\text{arg min}} \Bigl( \hat{R}(\mu) \Bigr) \qquad {\color{gray}\footnotesize \text{(optimal $k$-means cluster)}}
 $$
-\remark This is non-convex \& NP-hard.
 
 \method \textbf{Lloyd's Heuristic}