diff --git a/Readme.md b/Readme.md index 0abac8c..b0202ef 100644 --- a/Readme.md +++ b/Readme.md @@ -21,8 +21,8 @@ Dieses Repo beinhaltet die $\LaTeX$ Informationen für die Zusammenfassung im Fa - [x] Äquivalenztabelle aus Übung 6 (Prädikatenlogik) - [x] Skolemisierungsverfahren hinzufügen - [ ] Unifikationsverfahren hinzufügen -- [ ] Kosinusähnlichkeit hinzufügen (Übung 8) -- [ ] Mahalanobis Abstand hinzufügen (Übung 8) +- [x] Kosinusähnlichkeit hinzufügen (Übung 8) +- [x] Mahalanobis Abstand hinzufügen (Übung 8) ## Hinweise Requires you to enable [--shell escape](https://tex.stackexchange.com/questions/516604/how-to-enable-shell-escape-or-write18-visual-studio-code-latex-workshop) diff --git a/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex b/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex index ff8c880..ab3a8ef 100644 --- a/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex +++ b/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex @@ -5,6 +5,13 @@ \section{Ähnlichkeitsmaß} \label{aehnlichkeitsmass} + \begin{wrapfigure}{h}{.4\textwidth} + \vspace{-30mm} + \includegraphics[width = .4\textwidth]{manhattan_euclidean_1.pdf}\\ + \includegraphics[width = .4\textwidth]{cosine1.pdf} + \vspace{-20mm} + \end{wrapfigure} + \paragraph{Euklidischer Abstand} \large $$dist_e(a,b))\sqrt{\sum^n_{i=1}\left(a[i]-b[i]\right)^2}$$ @@ -15,6 +22,23 @@ $$dist_m(a,b)=\sum^n_{i=1}|a[i]-b[i]|$$ \normalsize + \paragraph{Kosinus-Ähnlichkeit} + \large + $$dist_c(a,b)=\cos(\theta)$$ + $$=\frac{a\cdot b}{||a||\cdot||b||} = \frac{\sum^n_{i=1}a_i\cdot b_i}{\sqrt{\sum^n_{i=1}a_i^2}\cdot\sqrt{\sum^n_{i=1}b_i^2}}$$ + \normalsize + + \paragraph{Mahalanobis-Abstand} + berücksichtigt die Richtung, in die die Daten verteilt sind.\\ + ($\sum^{-1}$ ist die inverse der Kovarianzmatrix) + \large + $$dist_{\text{Mahalanobis}}(a,b) = \sqrt{[a_1-b_1,\dots,a_n-b_n]\sum^{-1}\begin{bmatrix} + a_1-b_1\\ + \vdots\\ + a_n-b_n + \end{bmatrix}}$$ + \normalsize + \section{K-Nächste-Nachbarn} \label{k-nearest-neighbour} Beim \say{K-Nächste-Nachbarn}-Verfahren wird dem System eine Reihe von gelabelten Trainingsdaten übergeben. diff --git a/images/cosine1.pdf b/images/cosine1.pdf new file mode 100644 index 0000000..c0e6508 --- /dev/null +++ b/images/cosine1.pdf @@ -0,0 +1,471 @@ +%PDF-1.5 % +2 0 obj <> endobj 13 0 obj <>stream + + + + + 2014-07-25T15:32:11Z + R + 2015-02-09T19:48:48Z + 2015-02-09T19:48:48Z + + + + 256 + 236 + JPEG + /9j/4AAQSkZJRgABAgEASABIAAD/7QAsUGhvdG9zaG9wIDMuMAA4QklNA+0AAAAAABAASAAAAAEA AQBIAAAAAQAB/+4ADkFkb2JlAGTAAAAAAf/bAIQABgQEBAUEBgUFBgkGBQYJCwgGBggLDAoKCwoK DBAMDAwMDAwQDA4PEA8ODBMTFBQTExwbGxscHx8fHx8fHx8fHwEHBwcNDA0YEBAYGhURFRofHx8f Hx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8f/8AAEQgA7AEAAwER AAIRAQMRAf/EAaIAAAAHAQEBAQEAAAAAAAAAAAQFAwIGAQAHCAkKCwEAAgIDAQEBAQEAAAAAAAAA AQACAwQFBgcICQoLEAACAQMDAgQCBgcDBAIGAnMBAgMRBAAFIRIxQVEGE2EicYEUMpGhBxWxQiPB 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