An important mathematical tool in the analysis of dynamical systems is the
approximation of the reach set, i.e., the set of states reachable after a given
time from a given initial state. This set is difficult to compute for complex
systems even if the system dynamics are known and given by a system of ordinary
differential equations with known coefficients. In practice, parameters are
often unknown and mathematical models difficult to obtain. Data-based
approaches are promised to avoid these difficulties by estimating the reach set
based on a sample of states. If a model is available, this training set can be
obtained through numerical simulation. In the absence of a model, real-life
observations can be used instead. A recently proposed approach for data-based
reach set approximation uses Christoffel functions to approximate the reach
set. Under certain assumptions, the approximation is guaranteed to converge to
the true solution. In this paper, we improve upon these results by notably
improving the sample efficiency and relaxing some of the assumptions by
exploiting statistical guarantees from conformal prediction with training and
calibration sets. In addition, we exploit an incremental way to compute the
Christoffel function to avoid the calibration set while maintaining the
statistical convergence guarantees. Furthermore, our approach is robust to
outliers in the training and calibration set.

通过利用基于数据的方法中的克里斯托费尔函数进行接近集的近似计算，我们改进了现有结果，提高了样本效率，并通过利用训练和校准集的统计保证，放宽了一些假设。此外，我们采用了增量计算克里斯托费尔函数的方法，避免使用校准集，同时保持统计收敛保证，并且我们的方法对训练集和校准集中的异常值具有鲁棒性。

基于数据驱动的可达性使用 Christoffel 函数和共形预测

Data-driven Reachability using Christoffel Functions and Conformal  Prediction

The increase in the use of photovoltaic (PV) energy in the world has shown
that the useful life and maintenance of a PV plant directly depend on
theability to quickly detect severe faults on a PV plant. To solve this problem
of detection, data based approaches have been proposed in the
literature.However, these previous solutions consider only specific behavior of
one or few faults. Most of these approaches can be qualified as supervised,
requiring an enormous labelling effort (fault types clearly identified in each
technology). In addition, most of them are validated in PV cells or one PV
module. That is hardly applicable in large-scale PV plants considering their
complexity. Alternatively, some unsupervised well-known approaches based on
data try to detect anomalies but are not able to identify precisely the type of
fault. The most performant of these methods do manage to efficiently group
healthy panels and separate them from faulty panels. In that way, this article
presents an unsupervised approach called DTW K-means. This approach takes
advantages of both the dynamic time warping (DWT) metric and the Kmeans
clustering algorithm as a data-driven approach. The results of this mixed
method in a PV string are compared to diagnostic labels established by visual
inspection of the panels.

文章提出了一种基于 DTW K-means 的无监督方法，旨在对光伏电池组的严重故障进行检测。