Integral transforms are invaluable mathematical tools to map functions into
spaces where they are easier to characterize. We introduce the hyperdimensional
transform as a new kind of integral transform. It converts square-integrable
functions into noise-robust, holographic, high-dimensional representations
called hyperdimensional vectors. The central idea is to approximate a function
by a linear combination of random functions. We formally introduce a set of
stochastic, orthogonal basis functions and define the hyperdimensional
transform and its inverse. We discuss general transform-related properties such
as its uniqueness, approximation properties of the inverse transform, and the
representation of integrals and derivatives. The hyperdimensional transform
offers a powerful, flexible framework that connects closely with other integral
transforms, such as the Fourier, Laplace, and fuzzy transforms. Moreover, it
provides theoretical foundations and new insights for the field of
hyperdimensional computing, a computing paradigm that is rapidly gaining
attention for efficient and explainable machine learning algorithms, with
potential applications in statistical modelling and machine learning. In
addition, we provide straightforward and easily understandable code, which can
function as a tutorial and allows for the reproduction of the demonstrated
examples, from computing the transform to solving differential equations.

超维变换是一种新型的积分变换，将方块可积函数转换为噪声鲁棒、全息、高维表示的超维向量。它提供了辅助其他积分变换的理论基础和见解，与傅里叶、拉普拉斯和模糊变换密切相关。此外，它为超维计算领域提供了强大而灵活的框架，该计算范式在高效和可解释的机器学习算法方面引起了广泛关注，具有在统计建模和机器学习中应用的潜力。

超维变换：函数的全息表示

The Hyperdimensional Transform: a Holographic Representation of  Functions

We present "interoperability" as a guiding framework for statistical
modelling to assist policy makers asking multiple questions using diverse
datasets in the face of an evolving pandemic response. Interoperability
provides an important set of principles for future pandemic preparedness,
through the joint design and deployment of adaptable systems of statistical
models for disease surveillance using probabilistic reasoning. We illustrate
this through case studies for inferring spatial-temporal coronavirus disease
2019 (COVID-19) prevalence and reproduction numbers in England.

提出以 "互操作性" 为指导框架的统计建模，以协助政策制定者在面对不断演变的疫情应对中使用多样化数据询问多个问题。通过概率推理，我们设计和部署适应性统计模型系统，从而为未来的疫情准备工作提供重要的原则，具有疾病监测的功能。文中举例说明了在英格兰推断时空 COVID-19 预 valence 和再生数的情况。