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.

当构建回归的预测间隔（具有实值响应）或分类的预测集（具有分类响应）时，不确定性量化对于研究复杂的机器学习方法至关重要。在本文中，我们基于[Xu and Xie，2021]的先前工作，开发了集合正规化的自适应预测集（ERAP），以构建时间序列（具有分类响应）的预测集（具有分类响应）。特别是，我们允许未知的依赖性存在于顺序到达的功能和响应中。在方法论方面，ERAPS是一种不含分发和合奏的框架，适用于任意分类器。从理论上讲，我们在不假设数据交换性的情况下绑定了覆盖差距并显示渐近集收敛。从经验上讲，我们通过ERAP证明了有效的边际和条件覆盖范围，而与竞争方法相比，这也倾向于产生更小的预测集。

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。

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.

在这项工作中，我们对基本思想和新颖的发展进行了综述的综述，这是基于最小的假设的一种无创新的，无分配的，非参数预测的方法 - 能够以非常简单的方式预测集屈服在有限样本案例中，在统计意义上也有效。论文中提供的深入讨论涵盖了共形预测的理论基础，然后继续列出原始想法的更高级的发展和改编。

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。

We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called rank-one-out conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, in order to adapt to heteroskedascity in the data. Finally, we propose a model-free notion of variable importance, called leave-one-covariate-out or LOCO inference. Accompanying this paper is an R package conformalInference that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.

Many scientific and engineering challenges-ranging from personalized medicine to customized marketing recommendations-require an understanding of treatment effect heterogeneity. In this paper, we develop a non-parametric causal forest for estimating heterogeneous treatment effects that extends Breiman's widely used random forest algorithm. In the potential outcomes framework with unconfoundedness, we show that causal forests are pointwise consistent for the true treatment effect, and have an asymptotically Gaussian and centered sampling distribution. We also discuss a practical method for constructing asymptotic confidence intervals for the true treatment effect that are centered at the causal forest estimates. Our theoretical results rely on a generic Gaussian theory for a large family of random forest algorithms. To our knowledge, this is the first set of results that allows any type of random forest, including classification and regression forests, to be used for provably valid statistical inference. In experiments, we find causal forests to be substantially more powerful than classical methods based on nearest-neighbor matching, especially in the presence of irrelevant covariates.

Testing the significance of a variable or group of variables $X$ for predicting a response $Y$, given additional covariates $Z$, is a ubiquitous task in statistics. A simple but common approach is to specify a linear model, and then test whether the regression coefficient for $X$ is non-zero. However, when the model is misspecified, the test may have poor power, for example when $X$ is involved in complex interactions, or lead to many false rejections. In this work we study the problem of testing the model-free null of conditional mean independence, i.e. that the conditional mean of $Y$ given $X$ and $Z$ does not depend on $X$. We propose a simple and general framework that can leverage flexible nonparametric or machine learning methods, such as additive models or random forests, to yield both robust error control and high power. The procedure involves using these methods to perform regressions, first to estimate a form of projection of $Y$ on $X$ and $Z$ using one half of the data, and then to estimate the expected conditional covariance between this projection and $Y$ on the remaining half of the data. While the approach is general, we show that a version of our procedure using spline regression achieves what we show is the minimax optimal rate in this nonparametric testing problem. Numerical experiments demonstrate the effectiveness of our approach both in terms of maintaining Type I error control, and power, compared to several existing approaches.

我们提出\ textbf {jaws}，这是一系列用于无分配的不确定性量化任务的包装方法，以协变量偏移为中心，以我们的核心方法\ textbf {jaw}为中心，\ textbf {ja} ckknife+ \ textbf {w}八 - 重量。下巴还包括使用高阶影响函数的JAW的计算有效\ TextBf {a} pproximations：\ textbf {jawa}。从理论上讲，我们表明JAW放宽了Jackknife+对数据交换性的假设，即使在协变量转移下，也可以实现相同的有限样本覆盖范围保证。 Jawa在轻度假设下进一步以样本量或影响函数顺序的限制接近JAW保证。此外，我们提出了一种通用方法，以重新利用任何无分配不确定性量化方法及其对风险评估的任务的保证：该任务产生了真正标签在用户指定间隔内的估计概率。然后，我们将\ textbf {Jaw-r}和\ textbf {Jawa-r}作为\ textbf {r} ISK评估的建议方法的重新定义版本。实际上，在各种有偏见的现实世界数据集中，下颌的最先进的预测推理基准都超出了间隔生成和风险评估审计任务的偏差。

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.

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。

A flexible method is developed to construct a confidence interval for the frequency of a queried object in a very large data set, based on a much smaller sketch of the data. The approach requires no knowledge of the data distribution or of the details of the sketching algorithm; instead, it constructs provably valid frequentist confidence intervals for random queries using a conformal inference approach. After achieving marginal coverage for random queries under the assumption of data exchangeability, the proposed method is extended to provide stronger inferences accounting for possibly heterogeneous frequencies of different random queries, redundant queries, and distribution shifts. While the presented methods are broadly applicable, this paper focuses on use cases involving the count-min sketch algorithm and a non-linear variation thereof, to facilitate comparison to prior work. In particular, the developed methods are compared empirically to frequentist and Bayesian alternatives, through simulations and experiments with data sets of SARS-CoV-2 DNA sequences and classic English literature.

我们考虑在估计涉及依赖参数的高维滋扰的估计方程中估计一个低维参数。一个中心示例是因果推理中（局部）分位数处理效应（（L）QTE）的有效估计方程，涉及在分位数以估计的分位数评估的协方差累积分布函数。借记机学习（DML）是一种使用灵活的机器学习方法估算高维滋扰的数据分解方法，但是将其应用于参数依赖性滋扰的问题是不切实际的。对于（L）QTE，DML要求我们学习整个协变量累积分布函数。相反，我们提出了局部偏见的机器学习（LDML），该学习避免了这一繁重的步骤，并且只需要对参数进行一次初始粗糙猜测而估算烦恼。对于（L）QTE，LDML仅涉及学习两个回归功能，这是机器学习方法的标准任务。我们证明，在松弛速率条件下，我们的估计量与使用未知的真实滋扰的不可行的估计器具有相同的有利渐近行为。因此，LDML值得注意的是，当我们必须控制许多协变量和/或灵活的关系时，如（l）QTES在（（l）QTES）中，实际上可以有效地估算重要数量，例如（l）QTES。

We discuss an application of Generalized Random Forests (GRF) proposed by Athey et al.(2019) to quantile regression for time series data. We extracted the theoretical results of the GRF consistency for i.i.d. data to time series data. In particular, in the main theorem, based only on the general assumptions for time series data in Davis and Nielsen (2020), and trees in Athey et al.(2019), we show that the tsQRF (time series Quantile Regression Forests) estimator is consistent. Davis and Nielsen (2020) also discussed the estimation problem using Random Forests (RF) for time series data, but the construction procedure of the RF treated by the GRF is essentially different, and different ideas are used throughout the theoretical proof. In addition, a simulation and real data analysis were conducted.In the simulation, the accuracy of the conditional quantile estimation was evaluated under time series models. In the real data using the Nikkei Stock Average, our estimator is demonstrated to be more sensitive than the others in terms of volatility, thus preventing underestimation of risk.

了解特定待遇或政策与许多感兴趣领域有关的影响，从政治经济学，营销到医疗保健。在本文中，我们开发了一种非参数算法，用于在合成控制的背景下检测随着时间的流逝的治疗作用。该方法基于许多算法的反事实预测，而不必假设该算法正确捕获模型。我们介绍了一种推论程序来检测治疗效果，并表明测试程序对于固定，β混合过程渐近有效，而无需对所考虑的一组基础算法施加任何限制。我们讨论了平均治疗效果估计的一致性保证，并为提出的方法提供了遗憾的界限。算法类别可能包括随机森林，套索或任何其他机器学习估计器。数值研究和应用说明了该方法的优势。

Conformal prediction constructs a confidence set for an unobserved response of a feature vector based on previous identically distributed and exchangeable observations of responses and features. It has a coverage guarantee at any nominal level without additional assumptions on their distribution. Its computation deplorably requires a refitting procedure for all replacement candidates of the target response. In regression settings, this corresponds to an infinite number of model fits. Apart from relatively simple estimators that can be written as pieces of linear function of the response, efficiently computing such sets is difficult, and is still considered as an open problem. We exploit the fact that, \emph{often}, conformal prediction sets are intervals whose boundaries can be efficiently approximated by classical root-finding algorithms. We investigate how this approach can overcome many limitations of formerly used strategies; we discuss its complexity and drawbacks.

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。

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

现在通常用于高风险设置，如医疗诊断，如医疗诊断，那么需要不确定量化，以避免后续模型失败。无分发的不确定性量化（无分布UQ）是用户友好的范式，用于为这种预测创建统计上严格的置信区间/集合。批判性地，间隔/集合有效而不进行分布假设或模型假设，即使具有最多许多DataPoints也具有显式保证。此外，它们适应输入的难度;当输入示例很困难时，不确定性间隔/集很大，信号传达模型可能是错误的。在没有多大的工作和没有再培训的情况下，可以在任何潜在的算法（例如神经网络）上使用无分​​发方法，以产生置信度集，以便包含用户指定概率，例如90％。实际上，这些方法易于理解和一般，应用于计算机视觉，自然语言处理，深度加强学习等领域出现的许多现代预测问题。这种实践介绍是针对对无需统计学家的免费UQ的实际实施感兴趣的读者。我们通过实际的理论和无分发UQ的应用领导读者，从保形预测开始，并使无关的任何风险的分布控制，如虚假发现率，假阳性分布检测，等等。我们将包括Python中的许多解释性插图，示例和代码样本，具有Pytorch语法。目标是提供读者对无分配UQ的工作理解，使它们能够将置信间隔放在算法上，其中包含一个自包含的文档。