Dependencies

Let $\mathrm{LZSet}(A)$ be the Lempel–Ziv dictionary for a sequence $A$. We define LZ–Jaccard distance $\mathrm{LZJD}$ by

The Szymkiewicz–-Simpson coefficient is defined by

For each $1\le p\le \mathrm{\infty }$, let ¹

For each $1\le p\le \mathrm{\infty }$, ${\mathcal{V}}_p$ is a metric.

The Tversky dissimilarity $d_{\alpha ,\beta }^T$ is a metric iff $\alpha =\beta \ge 1$.

Let $\mathcal{X}$ be a set. A metric on $\mathcal{X}$ is a function $d:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ such that

1. $d(x,y)\ge 0$,

2. $d(x,y)=0$ if and only if $x=y$,

3. $d(x,y)=d(y,x)$ (symmetry),

4. $d(x,y)\le d(x,z)+d(z,y)$ (the triangle inequality)

for all $x,y,z\in \mathcal{X}$. If $d$ satisfies Enumi 1, Enumi 2, Enumi 3 but not necessarily Enumi 4 then $d$ is called a semimetric.

The Sørensen–Dice coefficient is defined by

Let $\delta :=\alpha \tilde{m}+\bar{\alpha }\tilde{M}$. Let $X=\{0\},Y=\{1\},Z=\{0,1\}$. Then $\delta (X,Y)=1$, $\delta (X,Z)=\delta (Y,Z)=\bar{\alpha }$.

For each $1\le p\le \mathrm{\infty }$, ${\mathrm{\Delta }}_p$ is a metric.

$d_{\alpha ,\beta }^T$ is a metric only if $\alpha =\beta \ge 1$.

Let $f(A,B)=|A\setminus B|+|B\setminus A|$. Then $f$ is a metric.

If $d_1$ and $d_2$ are metrics and $a,b$ are nonnegative constants, not both zero, then $a{d}_1+b{d}_2$ is a metric.

Let $0\le \alpha \le 1$ and $\beta >0$. Then $D_{\alpha ,\beta }$ is a metric if and only if $0\le \alpha \le 1/2$ and $\beta \ge 1/(1-\alpha )$.

Let $d(x,y)$ be a metric and let $a(x,y)$ be a nonnegative symmetric function. If $a(x,z)\le a(x,y)+d(y,z)$ for all $x,y,z$, then $d^{\prime }(x,y)=\frac{d(x,y)}{a(x,y)+d(x,y)}$, with $d^{\prime }(x,y)=0$ if $d(x,y)=0$, is a metric.

Let $f(A,B)$ be a metric such that

for all $A,B$. Then the function $d$ given by

is a metric.

${\delta }_{\alpha }=\alpha \tilde{m}+\bar{\alpha }\tilde{M}$ satisfies the triangle inequality if and only if $0\le \alpha \le 1/2$.

Let $f(A,B)=max\{|A\setminus B|,|B\setminus A|\}$. Then $f$ is a metric.

The optimal constant $\rho$ such that $d_{\alpha ,\beta }^T(X,Y)\le \rho (d_{\alpha ,\beta }^T(X,Y)+d_{\alpha ,\beta }^T(Y,Z))$ for all $X,Y,Z$ is

The function $D_{\alpha ,\beta }$ is a metric on all finite power sets only if $\alpha \le 1/2$.

Let $\mathcal{X}$ be a collection of finite sets. We define $S:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ as follows. The symmetric Tversky ratio model is defined by

The unbiased symmetric TRM ($\mathbf{ustrm}$) is the case where $\mathrm{bias}=0$, which is the case we shall assume we are in for the rest of this paper. The Tversky semimetric $D_{\alpha ,\beta }$ is defined by $D_{\alpha ,\beta }(X,Y)=1-\mathbf{ustrm}(X,Y)$, or more precisely

$J_3$ is not a metric.

Suppose $d$ is a metric on a collection of nonempty sets $\mathcal{X}$, with $d(X,Y)\le 2$ for all $X,Y\in \mathcal{X}$. Let $\hat{\mathcal{X}}=\mathcal{X}\cup \{\mathrm{\varnothing }\}$ and define $\hat{d}:\hat{\mathcal{X}}\times \hat{\mathcal{X}}\to \mathbb{R}$ by stipulating that for $X,Y\in \mathcal{X}$,

Then $\hat{d}$ is a metric on $\hat{\mathcal{X}}$.

Let $f(A,B)=mmin\{|A\setminus B|,|B\setminus A|\}+Mmax\{|A\setminus B|,|B\setminus A|\}$ with $0<m\le M$ and $1\le M$. Then the function $d$ given by

is a metric.

For sets $A,B,C$, we have $|A\setminus B|\le |A\setminus C|+|C\setminus B|$.

For sets $X$ and $Y$ the Tversky index with parameters $\alpha ,\beta \ge 0$ is a number between 0 and 1 given by

We also define the corresponding Tversky dissimilarity $d_{\alpha ,\beta }^T$ by

The function $D_{\alpha ,\beta }$ is a metric only if $\beta \ge 1/(1-\alpha )$.