3
$\begingroup$

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via $\exp:\mathbb R\to\mathbb R^{+}$ I get the norm $\|\cdot\|_{\log}$ on $(\mathbb R^{+}, \cdot)$ which is defined by $$\|a\|_{\log}=|\log(a)|$$ and the distance $$\operatorname{dist}_{\log}(a,b) = \left|\log\left(\frac ab\right)\right| = |\log(a)-\log(b)|$$

My Question: Is there a name for the norm $\|\cdot\|_{\log}$ or for the error/distance $\operatorname{dist}_{\log}(a,b)$? Can you point me to a textbook/paper, where properties of this norm/distance are discussed?

Reason for my Question: I'm interested in interval arithmetic where intervals are defined like the following:

$$\begin{align} [a]_\epsilon &= \left\{ y \in \mathbb R^{+}: \left\|\frac{y}{a}\right\|_{\log} = |\log(y)-\log(a)| < \epsilon \right\} \\ &= \left[ae^{-\epsilon};ae^{\epsilon}\right] \end{align}$$

Unfortunately I do not now how the norm $\|\cdot\|_{\log}$ nor the error $|\log(y)-\log(a)|$ is called. So I do not know for what I shall look for...

$\endgroup$
4
$\begingroup$

The metric $$ \mathrm{dist}(x,y) = |\ln(x) - \ln(y)| = \left| \ln\left( \frac{x}{y} \right) \right| $$ is frequently used in compositional data analysis, where it was introduced by J. Aitchison. It is known as the log-ratio distance.

It is far less well knwon that this metric also appears in the log-normal distribution. Conventionally, the probability density function of the log-normal distribution is written down as $$ f_\mathcal{LN}(x \mid \mu,\sigma^2) = \dfrac{1}{\sqrt{2\pi} \sigma x} \exp\left( -\dfrac{ \left(\ln(x) - \mu \right)^2 }{2\sigma^2} \right). $$ However, certain conceptual relationships, such as the involvement of the squared distance of observations from the mean, are obscured in this expression. In fact, if parameterized with the geometric mean $m =\mathrm{e}^\mu$, the probability density function becomes $$ \begin{array}{rcl} f_\mathcal{LN}(x \mid m,\sigma^2) &=& \dfrac{1}{\sqrt{2\pi\sigma^2 x^2}} \exp\left( -\dfrac{ \left| \ln(x) - \ln(m) \right|^2 }{2\sigma^2} \right) \\ &=& \dfrac{1}{\sqrt{2\pi\sigma^2 x^2}} \exp\left( -\dfrac{ \mathrm{dist}^2(x,m) }{2\sigma^2} \right). \end{array} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.