\Section[
    \eng{Machine-Learning}
    \ger{Maschinelles Lernen}
]{ml}
    \Subsection[
        \eng{Performance metrics}
        \ger{Metriken zur Leistungsmessung}
    ]{performance}
        \eng[cp]{correct predictions}
        \ger[cp]{richtige Vorhersagen}
        \eng[fp]{false predictions}
        \ger[fp]{falsche Vorhersagen}

        \eng[y]{ground truth}
        \eng[yhat]{prediction}
        \ger[y]{Wahrheit}
        \ger[yhat]{Vorhersage}

        \begin{formula}{accuracy}
            \desc{Accuracy}{}{}
            \desc[german]{Genauigkeit}{}{}
            \eq{a = \frac{\tgt{cp}}{\tgt{fp} + \tgt{cp}}}
        \end{formula}
        \TODO{is $n$ the nuber of predictions or the number of output features?}
        \begin{formula}{mean_abs_error}
            \desc{Mean absolute error (MAE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ ?}
            \desc[german]{Mittlerer absoluter Fehler (MAE)}{}{}
            \eq{\text{MAE} = \frac{1}{n} \sum_{i=1}^n \abs{y_i - \hat{y}_i}}
        \end{formula}
        \begin{formula}{root_mean_square_error}
            \desc{Root mean squared error (RMSE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ ?}
            \desc[german]{Standardfehler der Regression}{Quadratwurzel des mittleren quadratischen Fehlers (RSME)}{}
            \eq{\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}}
        \end{formula}

    \Subsection[
        \eng{Regression}
        \ger{Regression}
    ]{reg}
        \Subsubsection[
            \eng{Linear Regression}
            \ger{Lineare Regression}
        ]{linear}
            \begin{formula}{eq}
                \desc{Linear regression}{Fits the data under the assumption of \hyperref[f:math:pt:distributions:cont:normal]{normally distributed errors}}{$\mat{x}\in\R^{N\times M}$ input data, $\mat{y}\in\R^{N\times L}$ output data, $\mat{b}$ bias, $\vec{W}$ weights, $N$ samples, $M$ features, $L$ output variables}
                \desc[german]{Lineare Regression}{Fitted Daten unter der Annahme \hyperref[f:math:pt:distributions:cont:normal]{normalverteilter Fehler}}{}
                \eq{\mat{y} = \mat{b} + \mat{x} \cdot \vec{W}}
            \end{formula}
            \begin{formula}{design_matrix}
                \desc{Design matrix}{Stack column of ones to the feature vector\\Useful when $b$ is scalar}{$x_{ij}$ feature $j$ of sample $i$}
                \desc[german]{Designmatrix Ansatz}{}{}
                \eq{
                    \mat{X} = \begin{pmatrix} 1 & x_{11} & \ldots & x_{1M} \\  \vdots & \vdots & \vdots & \vdots \\ 1 & x_{N1} & \ldots & x_{NM} \end{pmatrix} 
                }
            \end{formula}       
            \begin{formula}{scalar_bias}
                \desc{Linear regression with scalar bias}{Using the design matrix, the scalar is absorbed into the weight vector}{$\mat{y}$ output data, $\mat{X}$ \fqEqRef{comp:ml:reg:design_matrix}, $\vec{W}$ weights}
                \desc[german]{Lineare Regression mit skalarem Bias}{Durch die Designmatrix wird der Bias in den Gewichtsvektor absorbiert}{}
                \eq{\mat{y} = \mat{X} \cdot \vec{W}}
            \end{formula}
            \begin{formula}{normal_equation}
                \desc{Normal equation}{Solves \fqEqRef{comp:ml:reg:linear:scalar_bias}}{$\mat{y}$ output data, $\mat{X}$ \fqEqRef{comp:ml:reg:linear:design_matrix}, $\vec{W}$ weights}
                \desc[german]{Normalengleichung}{Löst \fqEqRef{comp:ml:reg:linear:scalar_bias}}{}
                \eq{\vec{W} = \left(\mat{X}^\T \mat{X}\right)^{-1} \mat{X}^T \mat{y}}
            \end{formula}

        \Subsubsection[
            \eng{Ridge regression}
            \ger{Ridge Regression}
        ]{ridge}
            \TODO{ridge reg, Kernel ridge reg, gaussian process reg}
    % \Subsection[
    %     \eng{Bayesian probability theory}
    %     % \ger{}
    % ]{bayesian}
        


    \Subsection[
        \eng{Gradient descent}
        \ger{Gradientenverfahren}
    ]{gd}
        \TODO{in lecture 30 CMP}