---
title: 'Normalized L1 Distance: Scale-Invariant Metric'
url: https://www.emergentmind.com/topics/normalized-l1-distance
type: topic
---

# Normalized L1 Distance: Scale-Invariant Metric

The normalized $L_1$-distance, denoted $D_{\rm norm}(X, Y)$, is a probabilistic metric between real-valued integrable random variables $X$ and $Y$, widely studied for its applications in theoretical and applied fields, such as economics and physics. This distance is defined as the expected absolute difference between $X$ and $Y$, normalized by the sum of their absolute first moments. Structured to always lie between 0 and 1, it refines the traditional $L_1$-distance by providing a scale-invariant measure, particularly significant when comparing distributions of differing magnitudes. The normalized $L_1$-distance encapsulates and unifies several well-established concepts, including the Gini index, the Lukaszyk–Karmovsky metric, and emerges as a special instance within the framework of 1-Wasserstein optimal transport [2110.04787].

## 1. Formal Definition and Properties

Let $(\Omega, \mathcal A, P)$ be a probability space with $X, Y \in \mathcal L_1(\Omega)$, i.e., both are integrable real-valued random variables. The (compound) $L_1$-distance is
\[
D(X, Y) = E[|X - Y|] = \int_\Omega |X(\omega) - Y(\omega)|\, dP(\omega).
\]
The normalized $L_1$-distance, defined when $E|X| + E|Y| > 0$, is
\[
D_{\rm norm}(X, Y) = \frac{E[|X - Y|]}{E[|X|] + E[|Y|]},
\]
and $D_{\rm norm}(X, Y) = 0$ when both expectations vanish. This yields $0 \leq D_{\rm norm}(X,Y) \leq 1$ for all such $X, Y$.

Analyzing $D_{\rm norm}$ through the axioms of metric spaces:
- **Non-negativity:** $D_{\rm norm}(X, Y) \geq 0$.
- **Symmetry:** $D_{\rm norm}(X, Y) = D_{\rm norm}(Y, X)$.
- **Reflexivity:** $D_{\rm norm}(X, X) = 0$.
- **Identity of indiscernibles:** $D_{\rm norm}(X, Y) = 0$ if and only if $X = Y$ almost surely, considering the standard identification of random variables up to almost sure equality.

In general, $D_{\rm norm}$ does not always satisfy the triangle inequality. However, under the condition that $X$, $Y$, $Z$ are mutually independent, integrable, and nonnegative (with at most one of them concentrated at zero), Rolle proves that $D_{\rm norm}$ satisfies the triangle inequality:
\[
D_{\rm norm}(X, Z) \leq D_{\rm norm}(X, Y) + D_{\rm norm}(Y, Z).
\]
This is achieved via a specific algebraic inequality involving the individual $L_1$-distances and first moments, leveraging what is termed a "Canberra-inequality" for all real $x, y, z$ [2110.04787].

## 2. Closed-form Expressions for Standard Distributions

Explicit evaluation of $D_{\rm norm}$ is important in statistics and applied modeling. In the case of two independent Gaussians
\[
X \sim N(\mu_1, \sigma_1^2), \quad Y \sim N(\mu_2, \sigma_2^2),
\]
the expected absolute difference reads
\[
E|X-Y| = |\mu_1 - \mu_2|\, \left[ 2 \Phi\!\left(\frac{|\mu_1-\mu_2|}{\sqrt{\sigma_1^2 + \sigma_2^2}}\right) - 1 \right] + 2\sqrt{\sigma_1^2 + \sigma_2^2}\, \phi\!\left(\frac{|\mu_1-\mu_2|}{\sqrt{\sigma_1^2 + \sigma_2^2}}\right),
\]
where $\Phi$ and $\phi$ denote the cdf and pdf of the standard normal, respectively. The one-marginal expectation is
\[
E|X| = |\mu| \left[2 \Phi \left(\frac{|\mu|}{\sigma}\right) - 1 \right] + 2\sigma \phi\left(\frac{|\mu|}{\sigma}\right).
\]
$D_{\rm norm}(X, Y)$ is then computed by substituting these closed forms.

For independent uniform variables $X \sim U([a, b])$, $Y \sim U([c, d])$, the mean absolute difference is determined through an explicit double integration:
\[
E|X-Y| = \frac{1}{(b-a)(d-c)} \int_{a}^{b} \int_{c}^{d} |x-y|\, dy\, dx,
\]
with polynomials in endpoints providing concrete values in the cases of interval separation, inclusion, or general overlap. For pure separation ($b < c$), $E|X-Y| = |m_X - m_Y|$ where $m_X$ and $m_Y$ are midpoints of the respective intervals. Table summaries of case enumeration and formulas are presented in [2110.04787].

## 3. Domains of Application and Illustrative Behavior

Normalized $L_1$-distance is prevalent in fields where scale invariance and robust discrepancy measures are essential. In economics, it appears as the Gini index (see §5). In physics, especially error analysis, $D(X, Y)$ is known as the Lukaszyk–Karmovsky metric.

Figures in [2110.04787] exemplify behavior in the bivariate normal setup: as the correlation $\rho$ approaches 1, joint distributions concentrate on the diagonal, and $D_{\rm norm}(X, Y) \to 0$ (total dependence implies null normalized distance). In uniform distributions, $D_{\rm norm}$ interpolates from 0 (total overlap) to 1 (one variable identically zero and the other nondegenerate), with critical dependence on support overlap.

## 4. Connections to Classical Indices and Distances

The normalized $L_1$-distance not only unifies disparate applications but also recovers several established quantities:
- **Gini index:** For a distribution $\mu$, the Gini mean difference is $E_{X, Y \sim \mu} |X-Y|$. The Gini index is its normalized analogue:
  \[
  G(\mu) = \frac{ \mathrm{GMD}(\mu) }{ 2E(X) } = D_{\rm norm}(X, Y ) \text{ for } X, Y \text{ i.i.d.\ } \mu.
  \]
  Thus, $D_{\rm norm}$ is the Gini index viewed as the “autodistance” of a distribution.
- **Lukaszyk–Karmovsky metric:** $D(X, Y)=E|X-Y|$, introduced in physics for uncertainty quantification, possesses reflexivity contrary to early misconceptions.
- **Optimal transport (1-Wasserstein):** If $\mu, \nu$ are probability laws, the Monge–Kantorovich problem with $|x-y|$ cost leads to the 1-Wasserstein distance
  \[
  W_1(\mu, \nu) = \inf_{\pi \in \Pi(\mu, \nu)} E_\pi[|X-Y|] = \int_0^1 |F^{-1}(t) - G^{-1}(t)| \, dt,
  \]
  where $F, G$ are the cdfs of $\mu, \nu$. For independent $X \sim \mu, Y \sim \nu$, $D(X, Y)$ is the cost under the trivial product coupling.

## 5. Mathematical and Probabilistic Structure

The normalized $L_1$-distance defines a semimetric on the space of integrable random variables, becoming a full metric when restricted to independent variables, as established through the generalized triangle inequality. The proof involves verifying a nontrivial algebraic condition, ultimately relying on the positivity of the "Canberra-inequality" for all real $x, y, z$:
\[
|y-z||x| - |x-z||y| + |x-y||z| \geq 0.
\]
This semimetric structure allows for flexible deployment across disparate random variable pairs and distributions, provided integrability conditions are met.

## 6. Illustrative Regimes and Range

$D_{\rm norm}(X, Y)$ assumes values in $[0, 1]$, with limiting cases as follows:
- **$D_{\rm norm}(X, Y) = 0$**: holds if $X = Y$ almost surely or, for instance, in the degenerate case where both random variables vanish.
- **$D_{\rm norm}(X, Y) \to 0$**: as joint law of $(X, Y)$ is concentrated on the diagonal (e.g., perfect dependence, high correlation).
- **$D_{\rm norm}(X, Y) = 1$**: occurs when one variable is almost surely zero while the other is integrable and nondegenerate [2110.04787].

This range captures scenarios of perfect equality, maximal disparity, and interpolation governed by the probabilistic and algebraic relations between the random variables’ distributions.

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Normalized $L_1$-distance thus provides a robust, interpretable, and mathematically grounded similarity measure unifying concepts from diverse fields, with rigorous theoretical guarantees and tractable formulae in common applied cases [2110.04787].

Source: https://www.emergentmind.com/topics/normalized-l1-distance