Accurate uncertainty measurement is a key step to building robust and reliable machine learning systems. Conformal prediction is a distribution-free uncertainty quantification algorithm popular for its ease of implementation, statistical coverage guarantees, and versatility for underlying forecasters. However, existing conformal prediction algorithms for time series are limited to single-step prediction without considering the temporal dependency. In this paper we propose a Copula Conformal Prediction algorithm for multivariate, multi-step Time Series forecasting, CopulaCPTS. On several synthetic and real-world multivariate time series datasets, we show that CopulaCPTS produces more calibrated and sharp confidence intervals for multi-step prediction tasks than existing techniques.

机器学习方法越来越广泛地用于医疗保健，运输和金融等高危环境中。在这些环境中，重要的是，模型要产生校准的不确定性以反映其自信并避免失败。在本文中，我们调查了有关深度学习的不确定性定量（UQ）的最新著作，特别是针对其数学属性和广泛适用性的无分配保形方法。我们将涵盖共形方法的理论保证，引入在时空数据的背景下提高UQ的校准和效率的技术，并讨论UQ在安全决策中的作用。

Learning multi-agent dynamics is a core AI problem with broad applications in robotics and autonomous driving. While most existing works focus on deterministic prediction, producing probabilistic forecasts to quantify uncertainty and assess risks is critical for downstream decision-making tasks such as motion planning and collision avoidance. Multi-agent dynamics often contains internal symmetry. By leveraging symmetry, specifically rotation equivariance, we can improve not only the prediction accuracy but also uncertainty calibration. We introduce Energy Score, a proper scoring rule, to evaluate probabilistic predictions. We propose a novel deep dynamics model, Probabilistic Equivariant Continuous COnvolution (PECCO) for probabilistic prediction of multi-agent trajectories. PECCO extends equivariant continuous convolution to model the joint velocity distribution of multiple agents. It uses dynamics integration to propagate the uncertainty from velocity to position. On both synthetic and real-world datasets, PECCO shows significant improvements in accuracy and calibration compared to non-equivariant baselines.

机器学习（ML）的指数增长引起了极大的兴趣，以量化用户定义的信心水平的每个预测的不确定性。可靠的不确定性定量至关重要，是迈向增加对AI结果的信任的一步。在高风险决策中，它变得尤为重要，在这种决策中，真正的输出必须在置信度范围内具有很高的可能性。共形预测（CP）是一个无分布的不确定性定量框架，可适用于任何黑框模型，并产生预测间隔（PI），这些预测间隔（PIS）在轻度的交换性假设下有效。 CP型方法由于易于实施和计算便宜而变得越来越流行；但是，交换性假设立即排除时间序列预测。尽管最近的论文解决了协变量的转变，但对于一般时间序列预测生产H-Step提前有效PI的问题还不足。为了实现这样的目标，我们提出了一种称为AENBMIMOCQR的新方法（自适应集合批量多输入多输出保形的分数回归），该方法会产生渐近有效的PIS，适合异质驱动时间序列。我们将提出的方法与NN5预测竞争数据集中的最新竞争方法进行比较。所有用于复制实验的代码和数据都可以使用

Methods for reasoning under uncertainty are a key building block of accurate and reliable machine learning systems. Bayesian methods provide a general framework to quantify uncertainty. However, because of model misspecification and the use of approximate inference, Bayesian uncertainty estimates are often inaccurate -for example, a 90% credible interval may not contain the true outcome 90% of the time. Here, we propose a simple procedure for calibrating any regression algorithm; when applied to Bayesian and probabilistic models, it is guaranteed to produce calibrated uncertainty estimates given enough data. Our procedure is inspired by Platt scaling and extends previous work on classification. We evaluate this approach on Bayesian linear regression, feedforward, and recurrent neural networks, and find that it consistently outputs well-calibrated credible intervals while improving performance on time series forecasting and model-based reinforcement learning tasks.

在过去几十年中，已经提出了各种方法，用于估计回归设置中的预测间隔，包括贝叶斯方法，集合方法，直接间隔估计方法和保形预测方法。重要问题是这些方法的校准：生成的预测间隔应该具有预定义的覆盖水平，而不会过于保守。在这项工作中，我们从概念和实验的角度审查上述四类方法。结果来自各个域的基准数据集突出显示从一个数据集中的性能的大波动。这些观察可能归因于违反某些类别的某些方法所固有的某些假设。我们说明了如何将共形预测用作提供不具有校准步骤的方法的方法的一般校准程序。

我们开发了一个框架，用于在线环境中使用有效的覆盖范围保证构建不确定性集，其中基础数据分布可以急剧（甚至对手）随着时间的推移而发生巨大变化。我们提出的技术非常灵活，因为它可以与任何在线学习算法集成，需要最低限度的实施工作和计算成本。我们方法比现有替代方案的关键优势（也基于共形推断）是我们不需要将数据分为培训和保持校准集。这使我们能够以完全在线的方式拟合预测模型，并利用最新的观察结果来构建校准的不确定性集。因此，与现有技术相反，（i）我们构建的集合可以迅速适应分布的新变化； （ii）我们的过程不需要在每个时间步骤进行改装。使用合成和现实世界的基准数据集，我们证明了理论的有效性以及提案对现有技术的提高绩效。为了证明所提出的方法的更大灵活性，我们展示了如何为多出输出回归问题构造有效的间隔，而以前的顺序校准方法由于不切实际的计算和内存需求而无法处理。

We present a new distribution-free conformal prediction algorithm for sequential data (e.g., time series), called the \textit{sequential predictive conformal inference} (\texttt{SPCI}). We specifically account for the nature that the time series data are non-exchangeable, and thus many existing conformal prediction algorithms based on temporal residuals are not applicable. The main idea is to exploit the temporal dependence of conformity scores; thus, the past conformity scores contain information about future ones. Then we cast the problem of conformal prediction interval as predicting the quantile of a future residual, given a prediction algorithm. Theoretically, we establish asymptotic valid conditional coverage upon extending consistency analyses in quantile regression. Using simulation and real-data experiments, we demonstrate a significant reduction in interval width of \texttt{SPCI} compared to other existing methods under the desired empirical coverage.

可预测的不确定性可以通过两个性能 - 校准和清晰度来表征。本文争辩说明这些属性的不确定性，并提出了在深度学习中强制执行它们的简单算法。我们的方法专注于校准 - 分布校准的最强概念 - 并通过用神经估计器拟合低维密度或定量函数来实施它。由此产生的方法比以前的分类和回归方式更简单，更广泛适用。凭经验，我们发现我们的方法改善了几个任务的预测性不确定性，具有最小的计算和实现开销。我们的见解表明，培训深度学习模式的简单和改进方式，导致应准确的不确定性，应利用，以改善下游应用程序的性能。

This paper presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time series data. EnCQR can be applied on top of a generic forecasting model, including deep learning architectures. EnCQR exploits a bootstrap ensemble estimator, which enables the use of conformal predictors for time series by removing the requirement of data exchangeability. The ensemble learners are implemented as generic machine learning algorithms performing quantile regression, which allow the length of the PIs to adapt to local variability in the data. In the experiments, we predict time series characterized by a different amount of heteroscedasticity. The results demonstrate that EnCQR outperforms models based only on quantile regression or conformal prediction, and it provides sharper, more informative, and valid PIs.

共形预测（CP）是一种多功能的非参数框架，用于量化预测问题中的不确定性。在这项工作中，我们通过首次提出可以应用于时间不断发展的表面，将这种方法扩展到在双变量域上定义的时间序列函数的情况。为了获得有意义有效的预测区域，CP必须与准确的预测算法结合使用，因此，我们扩展了希尔伯特空间中自回旋过程的理论理论，以允许具有双变量域的功能。考虑到该主题的新颖性，我们提出了功能自回旋模型（FAR）的估计技术。实施了仿真研究，以研究不同的点预测因子如何影响所得的预测频段。最后，我们探索了真正数据集中拟议方法的利益和限制，在过去的二十年中，每天都会观察到黑海的海平面异常。

时间变化数量的估计是医疗保健和金融等领域决策的基本组成部分。但是，此类估计值的实际实用性受到它们量化预测不确定性的准确程度的限制。在这项工作中，我们解决了估计高维多元时间序列的联合预测分布的问题。我们提出了一种基于变压器体系结构的多功能方法，该方法使用基于注意力的解码器估算关节分布，该解码器可被学会模仿非参数Copulas的性质。最终的模型具有多种理想的属性：它可以扩展到数百个时间序列，支持预测和插值，可以处理不规则和不均匀的采样数据，并且可以在训练过程中无缝地适应丢失的数据。我们从经验上证明了这些属性，并表明我们的模型在多个现实世界数据集上产生了最新的预测。

We propose Multivariate Quantile Function Forecaster (MQF$^2$), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF$^2$ combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using input-convex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF$^2$: with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-of-the-art methods in terms of single time step metrics while capturing the time dependency structure.

The main objective of Prognostics and Health Management is to estimate the Remaining Useful Lifetime (RUL), namely, the time that a system or a piece of equipment is still in working order before starting to function incorrectly. In recent years, numerous machine learning algorithms have been proposed for RUL estimation, mainly focusing on providing more accurate RUL predictions. However, there are many sources of uncertainty in the problem, such as inherent randomness of systems failure, lack of knowledge regarding their future states, and inaccuracy of the underlying predictive models, making it infeasible to predict the RULs precisely. Hence, it is of utmost importance to quantify the uncertainty alongside the RUL predictions. In this work, we investigate the conformal prediction (CP) framework that represents uncertainty by predicting sets of possible values for the target variable (intervals in the case of RUL) instead of making point predictions. Under very mild technical assumptions, CP formally guarantees that the actual value (true RUL) is covered by the predicted set with a degree of certainty that can be prespecified. We study three CP algorithms to conformalize any single-point RUL predictor and turn it into a valid interval predictor. Finally, we conformalize two single-point RUL predictors, deep convolutional neural networks and gradient boosting, and illustrate their performance on the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) data sets.

准确可靠的流行病预测是对公共卫生规划和疾病缓解影响的重要问题。大多数现有的疫情预测模型无视不确定性量化，导致错误校准的预测。近期神经模型的作品，用于不确定感知的时序预测也有几个限制;例如很难在贝叶斯NNS中指定有意义的前瞻，而Deep Leaseming的方法在实践中是计算昂贵的。在本文中，我们填补了这个重要的差距。我们将预测任务模拟为概率生成过程，并提出了一种名为EPIFNP的功能神经过程模型，其直接模拟预测值的概率密度。 EPIFNP利用动态随机相关图来模拟非参数方式之间序列之间的相关性，并设计不同的随机潜变量以捕获不同视角的功能不确定性。我们在实时流感预测环境中的广泛实验表明，EPIFNP在准确性和校准度量中显着优于先前的最先进模型，精度高达2.5倍，校准2.4倍。此外，由于其生成过程的性质，EPIFNP了解当前季节与历史季节类似模式之间的关系，从而实现可解释的预测。超越疫情预测，EPIFNP可以是独立的利益，以便在深度顺序模型中推进预测性分析的深度顺序模型

最近实现了更准确的短期预测的数据驱动的空气质量预测。尽管取得了成功，但大多数目前的数据驱动解决方案都缺乏适当的模型不确定性的量化，以传达信任预测的程度。最近，在概率深度学习中已经制定了几种估计不确定性的实用工具。但是，在空气质量预测领域的域中没有经验应用和广泛的比较这些工具。因此，这项工作在空气质量预测的真实环境中应用了最先进的不确定性量化。通过广泛的实验，我们描述了培训概率模型，并根据经验性能，信心可靠性，置信度估计和实际适用性评估其预测性不确定性。我们还使用空气质量数据中固有的“自由”对抗培训和利用时间和空间相关性提出改善这些模型。我们的实验表明，所提出的模型比以前的工作更好地在量化数据驱动空气质量预测中的不确定性方面表现出。总体而言，贝叶斯神经网络提供了更可靠的不确定性估计，但可能挑战实施和规模。其他可扩展方法，如深合奏，蒙特卡罗（MC）辍学和随机重量平均-Gaussian（SWAG）可以执行良好，如果正确应用，但具有不同的权衡和性能度量的轻微变化。最后，我们的结果表明了不确定性估计的实际影响，并证明了，实际上，概率模型更适合提出知情决策。代码和数据集可用于\ url {https:/github.com/abdulmajid-murad/deep_probabilistic_forecast}

在回归设置中量化不确定性的许多方法中，指定完整量子函数具有吸引力，随着量级可用于解释和评估。预测每个输入的真实条件定量的模型，在所有量化水平上都具有潜在的不确定性的正确和有效的表示。为实现这一目标，许多基于当前的分位式的方法侧重于优化所谓的弹球损失。然而，这种损失限制了适用的回归模型的范围，限制了靶向许多所需特性的能力（例如校准，清晰度，中心间隔），并且可能产生差的条件量数。在这项工作中，我们开发了满足这些缺点的新分位式方法。特别是，我们提出了可以适用于任何类别的回归模型的方法，允许在校准和清晰度之间选择权衡，优化校准中心间隔，并产生更准确的条件定位。我们对我们的方法提供了彻底的实验评估，其中包括核融合中的高维不确定性量化任务。

机器学习的许多应用涉及预测模型输出的灵活概率分布。我们提出了自动评级分位式流动，这是一种灵活的概率模型，高维变量，可用于准确地捕获预测的炼膜不确定性。这些模型是根据适当评分规则使用新颖目标培训的自回归流动的情况，这简化了培训期间雅各比亚的计算昂贵的决定因素，并支持新型的神经结构。我们证明这些模型可用于参数化预测条件分布，提高时间序列预测和对象检测的概率预测质量。

贝叶斯优化是一种过程，允许获得黑盒功能的全局最佳功能，并且在超参数优化等应用中有用。在目标函数的形状上估计的不确定性估计是引导优化过程的工具。但是，如果客观函数违反基础模型（例如，高斯）的假设，这些估计可能是不准确的。我们提出了一种简单的算法，可以通过目标函数校准后部分布的不确定性作为贝叶斯型优化过程的一部分。我们表明，通过提高校准后分布的不确定性估计，贝叶斯优化使得更好的决策并以较少的步骤到达全球最佳。我们表明，该技术提高了贝叶斯优化对标准基准函数和超参数优化任务的性能。

Probabilistic forecasting, i.e. estimating the probability distribution of a time series' future given its past, is a key enabler for optimizing business processes. In retail businesses, for example, forecasting demand is crucial for having the right inventory available at the right time at the right place. In this paper we propose DeepAR, a methodology for producing accurate probabilistic forecasts, based on training an auto-regressive recurrent network model on a large number of related time series. We demonstrate how by applying deep learning techniques to forecasting, one can overcome many of the challenges faced by widely-used classical approaches to the problem. We show through extensive empirical evaluation on several real-world forecasting data sets accuracy improvements of around 15% compared to state-of-the-art methods.