The Pareto front of a set of vectors is the subset which is comprised solely
of all of the best trade-off points. By interpolating this subset, we obtain
the optimal trade-off surface. In this work, we prove a very useful result
which states that all Pareto front surfaces can be explicitly parametrised
using polar coordinates. In particular, our polar parametrisation result tells
us that we can fully characterise any Pareto front surface using the length
function, which is a scalar-valued function that returns the projected length
along any positive radial direction. Consequently, by exploiting this
representation, we show how it is possible to generalise many useful concepts
from linear algebra, probability and statistics, and decision theory to
function over the space of Pareto front surfaces. Notably, we focus our
attention on the stochastic setting where the Pareto front surface itself is a
stochastic process. Among other things, we showcase how it is possible to
define and estimate many statistical quantities of interest such as the
expectation, covariance and quantile of any Pareto front surface distribution.
As a motivating example, we investigate how these statistics can be used within
a design of experiments setting, where the goal is to both infer and use the
Pareto front surface distribution in order to make effective decisions. Besides
this, we also illustrate how these Pareto front ideas can be used within the
context of extreme value theory. Finally, as a numerical example, we applied
some of our new methodology on a real-world air pollution data set.

通过极坐标，我们证明了帕累托前沿可以显式地参数化，从而基于此表示可以将线性代数、概率统计和决策理论推广至帕累托前沿空间中的函数。我们以随机设置为重点，研究了帕累托前沿作为随机过程的情况，并展示了如何定义和估计期望值、协方差和分位数等统计量。同时，我们还通过设计实验探讨了如何利用帕累托前沿表面分布来做出有效决策，并在极值理论的背景下展示了帕累托前沿思想的应用。最后，我们通过对实际空气污染数据集的数值示例，应用了我们的新方法。