Commit 55c16ce6 authored by Michael Kohlhase's avatar Michael Kohlhase

Minkowski distance

parent a899309b
\begin{mhmodnl}[creators=miko]{Minkowski-distance}{de}
\begin{definition}
Ist $\realmorethan{p}1$ eine \mtrefii[realnumbers?real-number]{reelle}{Zahl}, so
nennen wir die von der \mtrefi[pnorm?pnorm]{$\ell_p$-norm} auf $\ndim\RealNumbers{n}$
\mtrefii[norm-induced-metric?induced-metric]{induzierte}{Metrik} die
\defi[name=Minkowski-distance]{Minkowskimetrik} $\MinkowskiDistOp{p}$. Es gilt
\[\fundefeq{p,x,y}
{\MinkowskiDist{p}xy}
{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{\realminus{x_i,y_i}}}}{\frac1p}}
\]
f"ur $\defeq{x}{\ntupli{x}1n}$ und $\defeq{y}{\ntupli{y}1n}$.
\end{definition}
\begin{definition}
Die durch die \mtrefi[pnorm?taxicab-norm]{Summennorm}
\mtrefi[norm-induced-metric?induced-metric]{induzierte}
\mtrefi[metric-space?metric]{Metrik} $\MinkowskiDistOp1$ hei"st
\defi[name=Manhattan-distance]{Manhattan-Metrik},
\defi[name=Manhattan-distance]{Manhattan-Distanz},
\defii[name=Manhattan-distance]{Mannheimer}{Metrik},
\defi[name=Manhattan-distance]{Taxi-Metrik}
oder \defi[name=Manhattan-distance]{Cityblock-Metrik}.
\end{definition}
\begin{definition}
Die durch die \mtrefi[pnorm?pnorm]{$\ell_2$-Norm}
\mtrefi[norm-induced-metric?induced-metric]{induzierte}
\mtrefi[metric-space?metric]{Metrik} $\MinkowskiDistOp2$ hei"st
\defii[name=Euclidean-distance]{Euklidsche}{Distanz}.
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=miko]{Minkowski-distance}{en}
\begin{definition}
Let $\realmorethan{p}1$ be a \trefii[realnumbers]{real}{number}, then we call
\atrefii[norm-induced-metric]{metric induced}{induced}{metric} on
$\ndim\RealNumbers{n}$ by the \mtrefi[pnorm?pnorm]{$\ell_p$-norm} the
\defii{Minkowski}{distance} $\MinkowskiDistOp{p}$. We have
\[\fundefeq{p,x,y}
{\MinkowskiDist{p}xy}
{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{\realminus{x_i,y_i}}}}{\frac1p}}
\]
where $\defeq{x}{\ntupli{x}1n}$ and $\defeq{y}{\ntupli{y}1n}$.
\end{definition}
\begin{definition}
The \trefi[metric-space]{metric} $\MinkowskiDistOp1$
\atrefii[norm-induced-metric]{induced}{induced}{metric} by the
\trefii[pnorm]{taxicab}{norm} is called the
\defii{Manhattan}{distance}, the
\defii[name=Manhattan-distance]{rectilinear}{distance},
\defii[name=Manhattan-distance]{snake}{distance}, or
\defiii[name=Manhattan-distance]{city}{block}{distance}.
\end{definition}
\begin{definition}
The \trefi[metric-space]{metric} $\MinkowskiDistOp2$
\atrefii[norm-induced-metric]{induced}{induced}{metric} by the
\mtrefi[pnorm?pnorm]{$\ell_2$-norm} is called the \defii{Euclidean}{distance}.
\end{definition}
\end{mhmodnl}
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\begin{modsig}[creators=miko]{Minkowski-distance}
\gimport{pnorm}
\gimport{norm-induced-metric}
\symdef{MinkowskiDistOp}[1]{d_{#1}}
\symdef{MinkowskiDist}[3]{\prefix{\MinkowskiDistOp{#1}}{#2,#3}}
\symtest{MinkowskiDist}{\MinkowskiDist{p}xy}
\end{modsig}
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