While time series classification and forecasting problems have been
extensively studied, the cases of noisy time series data with arbitrary time
sequence lengths have remained challenging. Each time series instance can be
thought of as a sample realization of a noisy dynamical model, which is
characterized by a continuous stochastic process. For many applications, the
data are mixed and consist of several types of noisy time series sequences
modeled by multiple stochastic processes, making the forecasting and
classification tasks even more challenging. Instead of regressing data naively
and individually to each time series type, we take a latent variable model
approach using a mixtured Gaussian processes with learned spectral kernels.
More specifically, we auto-assign each type of noisy time series data a
signature vector called its motion code. Then, conditioned on each assigned
motion code, we infer a sparse approximation of the corresponding time series
using the concept of the most informative timestamps. Our unmixing
classification approach involves maximizing the likelihood across all the mixed
noisy time series sequences of varying lengths. This stochastic approach allows
us to learn not only within a single type of noisy time series data but also
across many underlying stochastic processes, giving us a way to learn multiple
dynamical models in an integrated and robust manner. The different learned
latent stochastic models allow us to generate specific sub-type forecasting. We
provide several quantitative comparisons demonstrating the performance of our
approach.

使用混合高斯过程和学习的谱核函数的潜变量模型方法来处理具有任意时间序列长度的噪声时间序列数据，使得在多个潜在随机过程上学习多个动力模型成为可能。

使用具有运动编码的随机过程模型稳健地学习噪声时间序列集合

Robust Learning of Noisy Time Series Collections Using Stochastic  Process Models with Motion Codes

In this paper we propose a family of tractable kernels that is dense in the
family of bounded positive semi-definite functions (i.e. can approximate any
bounded kernel with arbitrary precision). We start by discussing the case of
stationary kernels, and propose a family of spectral kernels that extends
existing approaches such as spectral mixture kernels and sparse spectrum
kernels. Our extension has two primary advantages. Firstly, unlike existing
spectral approaches that yield infinite differentiability, the kernels we
introduce allow learning the degree of differentiability of the latent function
in Gaussian process (GP) models and functions in the reproducing kernel Hilbert
space (RKHS) in other kernel methods. Secondly, we show that some of the
kernels we propose require fewer parameters than existing spectral kernels for
the same accuracy, thereby leading to faster and more robust inference.
Finally, we generalize our approach and propose a flexible and tractable family
of spectral kernels that we prove can approximate any continuous bounded
nonstationary kernel.

本文提出了一族可行核函数，这些函数在有界正半定函数族中密集，通过该族函数可以准确地近似任何有界核函数，我们首先讨论了平稳核函数的情况，并提出了一族谱核函数，扩展了现有方法，如谱混合核和稀疏频谱核，进而我们推广了方法并提出了一族灵活、可行的谱核函数，证明其能够近似任何连续有界非平稳核。