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Counterfactual explanations (CEs) play an important role in detecting bias and improving the explainability of data-driven classification models. A CE is a minimal perturbed data point for which the decision of the model changes. Most of the existing methods can only provide one CE, which may not be achievable for the user. In this work, we derive an iterative method to calculate robust CEs (i.e., CEs that remain valid even after the features are slightly perturbed). To this end, our method provides a whole region of CEs, allowing the user to choose a suitable recourse to obtain a desired outcome. We use algorithmic ideas from robust optimization and prove convergence results for the most common machine learning methods, including decision trees, tree ensembles, and neural networks. Our experiments show that our method can efficiently generate globally optimal robust CEs for a variety of common data sets and classification models. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015, Optimization for and with Machine Learning (OPTIMAL)]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0153. [ABSTRACT FROM AUTHOR]

In statistical process monitoring, the CUSUM and EWMA control charts have received considerable attention because of their remarkable ability to detect small sustained shifts. In practice, small process variation and shifts are anticipated beforehand in many processes, so the focus should be on detecting a moderate to a large shift. The aforementioned charts identify minor changes in population parameters as out-of-control scenarios; thus, "small" and potentially practically insignificant shifts are producing signals. To counteract this, both charts are amended to accommodate an indifference region by optimizing the detection of a shift at the outer boundaries of the indifference region. The results show that the adapted CUSUM and EWMA monitoring schemes yield comparable results. On nearly all occasions, the CUSUM chart outperforms the EWMA chart, yet the EWMA chart seems more robust and is easier to interpret. Furthermore, we provide two practical examples to illustrate the use-case of optimized charts to mitigate small (unimportant) variations, such as seasonality and modest temporary shifts. Overall, this work provides a general approach tailored to practice in quality control, e.g., as prescribed by ISO standards. It also answers a recent call in statistical process monitoring literature to reconsider the design of control charts. [ABSTRACT FROM AUTHOR]

Predictive monitoring techniques produce signals in case of a high predicted probability of an undesirable event, such as mortality, heart attacks, or machine failure. When using these predicted probabilities to classify the unknown outcome, a decision threshold needs to be chosen in statistical and machine learning models. In many cases, this is set to 0.5 by default. However, this may not lead to an acceptable model performance. To mitigate this issue, different threshold optimization approaches have been proposed in the literature. In this paper, we compare existing thresholding techniques to achieve a desired false alarm rate, and also evaluate the corresponding precision and recall performance metrics. A simulation study is conducted and a real‐world example on a medical dataset is provided. [ABSTRACT FROM AUTHOR]

We propose a method for monitoring proportions when the in-control proportion and the sample sizes vary over time. Our approach is able to overcome some of the performance issues of other commonly used methods, as we demonstrate in this paper using analytical and numerical methods. The derivations and results are shown mainly for monitoring proportions, but we show how the method can be extended to the monitoring of count data. [ABSTRACT FROM AUTHOR]

In this paper we derive correction factors for Shewhart control charts that monitor individual observations as well as subgroup averages. In practice, the distribution parameters of the process characteristic of interest are unknown and, therefore, have to be estimated. A well-known performance measure within Statistical Process Monitoring is the expectation of the average run length (ARL), defined as the unconditional ARL. A practitioner may want to design a control chart such that, in the in-control situation, it has a certain expected ARL. However, accurate correction factors that lead to such an unconditional ARL are not yet available. We derive correction factors that guarantee a certain unconditional in-control ARL. We use approximations to derive the factors and show their accuracy and the performance of the control charts – based on the new factors – in out-of-control situations. We also evaluate the variation between the ARLs of the individually estimated control charts. [ABSTRACT FROM AUTHOR]

Because the in-control distribution and parameters are generally unknown, control limits have to be estimated using a Phase I reference sample. Because different practitioners obtain different samples, their control limit estimates will vary and, consequently, also their control chart performance. We propose the use of nonparametric tolerance intervals in statistical process monitoring to guarantee a minimum control chart performance with a prespecified probability. We evaluate the performance of the proposed limits for various distributions and sample sizes. Note that this nonparametric set-up includes control charts for location and dispersion. Moreover, we compare the performance with other existing methods involving data transformations and a bootstrap procedure. It turns out that the use of nonparametric tolerance intervals performs very well in statistical process monitoring, especially when moderately large sample sizes are available in Phase I. [ABSTRACT FROM AUTHOR]

When in-control process parameters are estimated, this can have a substantial effect on the control chart performance. In particular, it may lead to high false alarm rates for a large number of practitioners. In recent literature, control limit adjustments have been proposed for Shewhart, CUSUM and EWMA control charts in order to provide a specified in-control performance with a specified high probability. In this paper, we compare the out-of-control performance of these adjusted Shewhart, CUSUM and EWMA control charts for sustained shifts in the process mean. We find that the CUSUM control chart has faster detection of sustained shifts compared to both the EWMA and Shewhart control charts. This finding generalizes to almost all shift sizes and estimation errors considered in this paper. The performance of the EWMA is not far worse than that of the CUSUM, but the Shewhart control chart is much slower in detecting sustained shifts in the mean compared to these other two charts. [ABSTRACT FROM AUTHOR]

Abstract: Because of digitalization, many organizations possess large datasets. Furthermore, measurement data are often not normally distributed. However, when samples are sufficiently large, the central limit theorem may be used for the sample means. In this article, we evaluate the use of the central limit theorem for various distributions and sample sizes, as well as its effects on the performance of a Shewhart control chart for these large non‐normally distributed datasets. To this end, we use the sample means as individual observations and a Shewhart control chart for individual observations to monitor processes. We study the unconditional performance, expressed as the expectation of the in‐control average run length (ARL), as well as the conditional performance, expressed as the probability that the control chart based on estimated parameters will have a lower in‐control ARL than a specified desired in‐control ARL. We use recently developed factors to correct the control limits to obtain a specified conditional or unconditional in‐control performance. The results in this paper indicate that the X ¯ control chart should be applied with caution, even with large sample sizes. [ABSTRACT FROM AUTHOR]

When in-control parameters are unknown, they have to be estimated using a reference sample. Due to the use of different reference samples in phase I, the control chart performance in phase II will vary across practitioners. This variation is especially large for small sample sizes. To prevent low in-control average run lengths, new corrections for Shewhart control charts are proposed that guarantee a minimum in-control performance with a specified probability. However, a minimum in-control performance guarantee generally lowers the out-of-control performance. To balance the tradeoff between in-control and out-of-control performance, the minimum performance threshold and specified probability can be adjusted as desired. The corrections are given in a closed form so that the bootstrap method, which has recently been suggested, is no longer required. The performance of our proposed correction is illustrated by simulating some practical situations. Furthermore, a comparison is made with tolerance intervals and self-starting control charts. [ABSTRACT FROM AUTHOR]

The article discusses the theory of Professor William H. Woodall on statistical process monitoring (SPM), which points out the necessity of narrowing the gap between theories and practice owing to Phase I data analysis. It states the use of strong estimation methods in phase I, control charts, bootstrap methods and analytical approximation methods will narrow the gap in the estimation errors.

Recently, two methods have been published in this journal to determine adjusted control limits for the Shewhart control chart in order to guarantee a pre-specified in-control performance. One is based on the bootstrap approach (Saleh et al. (2015)), and the other is an analytical approach (Goedhart, Schoonhoven, and Does (2017)). Although both methods lead to the desired control chart performance, they are still difficult to implement by the practitioner. The bootstrap is rather computationally intensive, while the analytical approach consists of multiple integrals and derivatives. In this letter to the editorwe simplify the analytical expressions provided in Goedhart, Schoonhoven, and Does (2017) by using the theory on tolerance intervals for individual observations as given in Krishnamoorthy and Mathew (2009). [ABSTRACT FROM AUTHOR]