Early approaches to multiple-output Gaussian processes (MOGPs) relied on
linear combinations of independent, latent, single-output Gaussian processes
(GPs). This resulted in cross-covariance functions with limited parametric
interpretation, thus conflicting with the ability of single-output GPs to
understand lengthscales, frequencies and magnitudes to name a few. On the
contrary, current approaches to MOGP are able to better interpret the
relationship between different channels by directly modelling the
cross-covariances as a spectral mixture kernel with a phase shift. We extend
this rationale and propose a parametric family of complex-valued cross-spectral
densities and then build on Cram\'er's Theorem (the multivariate version of
Bochner's Theorem) to provide a principled approach to design multivariate
covariance functions. The so-constructed kernels are able to model delays among
channels in addition to phase differences and are thus more expressive than
previous methods, while also providing full parametric interpretation of the
relationship across channels. The proposed method is first validated on
synthetic data and then compared to existing MOGP methods on two real-world
examples.

提出了基于复值交叉频谱密度的参数化多元协方差函数，实现对通道之间时间延迟和相位差异的建模，进而提高了模型的表达能力。该方法在合成数据和真实数据集上进行了验证和比较。

多输出高斯过程的谱混合核函数

Spectral Mixture Kernels for Multi-Output Gaussian Processes

Treatment effects can be estimated from observational data as the difference
in potential outcomes. In this paper, we address the challenge of estimating
the potential outcome when treatment-dose levels can vary continuously over
time. Further, the outcome variable may not be measured at a regular frequency.
Our proposed solution represents the treatment response curves using linear
time-invariant dynamical systems---this provides a flexible means for modeling
response over time to highly variable dose curves. Moreover, for multivariate
data, the proposed method: uncovers shared structure in treatment response and
the baseline across multiple markers; and, flexibly models challenging
correlation structure both across and within signals over time. For this, we
build upon the framework of multiple-output Gaussian Processes. On simulated
and a challenging clinical dataset, we show significant gains in accuracy over
state-of-the-art models.

本研究提出了一种利用线性时不变动力系统和高斯过程，能够弥补观察数据中剂量级别变化以及不定期观测的不足的方法，能够更加准确地进行多变量数据的建模和预测。