## What is the \(\textsf{L}_1\) Metric?

It is a case of the general Minkowski metric:

\[
d_k(\mathbf{x},\mathbf{y}) = \left( \sum_i^m| x_i - y_i |^k \right)^\frac{1}{k}
\] For \(k=1\) - sometimes referred to as the *city block metric* or the *Manhatten metric*, particularly if \(\mathbf{x}\) and \(\mathbf{y}\) are 2D vectors (that is, \(m=2\)), since it approximates road distances when the roads form a regular mesh, such as those in Manhatten. Another common case of the metric is where \(k=2\), where the metric is simply Euclidean distance. A notable property of Euclidean distance is that it is *rotation invariant* - it may easily be verified that that when \(m=2\), if \(\mathbf{R}_\theta\) is a rotation matrix through angle \(\theta\), so we may write

\[
\mathbf{R}_\theta = \left( \begin{array}{rr} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right)
\]

then \[
d_2(\mathbf{x},\mathbf{y}) = d_2(\mathbf{R}_\theta \mathbf{x},\mathbf{R}_\theta \mathbf{y} )
\]

However this is not generally the case:

\[
d_k(\mathbf{x},\mathbf{y}) \ne d_k(\mathbf{R}_\theta \mathbf{x},\mathbf{R}_\theta \mathbf{y} ) \textsf{ if } k \ne 2
\]

In particular \(d_1(.,.)\) is not rotation invariant.

## Kernels based on distances other than \(d_2(.,.)\)

Suppose we wished to impliment a 2D kernel function (for example to use in kernel density estimation or geographically weighted regression) based on the \(\textsf{L}_1\) metric. A symmetrical 2D kernel generally takes the form

\[
\frac{1}{h^2}K\left(\frac{d(\mathbf{x},\mathbf{y})}{h}\right)
\]

Where typically \(K\) takes an argument in the range \([0,\infty)\) and usually

- \(K'(d) < 0 \ \forall \ d\)
- \(\textsf{lim}_{d \rightarrow \infty} \ K(d) = 0\)

Common examples are

- \(K(d) = \exp(-d)\)
- \(K(d) = \exp(-\frac{d^2}{2})\)
- \(K(d) = I_{d<1}(1 - d^2)^2\)

where \(I_{s}\) is 1 if \(s\) is true, and 0 otherwise. Typically, distances are based on \(k=2\) but could be defined otherwise.

Below is an interactive, rotatable and zoomable 3D image of the kernel derived when \(K(d) = \exp(-d)\) and \(d(.,.) = d_1(.,.)\).