Nonparametric Scheme Research Articles

A Non-Parametric Scheme for Identifying Data Characteristic Based on Curve Similarity Matching

A Non-Parametric Scheme for Identifying Data Characteristic Based on Curve Similarity Matching

RAFaRe: Learning Robust and Accurate Non-parametric 3D Face Reconstruction from Pseudo 2D&3D Pairs

We propose a robust and accurate non-parametric method for single-view 3D face reconstruction (SVFR). While tremendous efforts have been devoted to parametric SVFR, a visible gap still lies between the result 3D shape and the ground truth. We believe there are two major obstacles: 1) the representation of the parametric model is limited to a certain face database; 2) 2D images and 3D shapes in the fitted datasets are distinctly misaligned. To resolve these issues, a large-scale pseudo 2D&3D dataset is created by first rendering the detailed 3D faces, then swapping the face in the wild images with the rendered face. These pseudo 2D&3D pairs are created from publicly available datasets which eliminate the gaps between 2D and 3D data while covering diverse appearances, poses, scenes, and illumination. We further propose a non-parametric scheme to learn a well-generalized SVFR model from the created dataset, and the proposed hierarchical signed distance function turns out to be effective in predicting middle-scale and small-scale 3D facial geometry. Our model outperforms previous methods on FaceScape-wild/lab and MICC benchmarks and is well generalized to various appearances, poses, expressions, and in-the-wild environments. The code is released at https://github.com/zhuhao-nju/rafare.

A comparative study on the nonparametric memory-type charts for monitoring process location

Control charts may help keep industrial and manufacturing processes running smoothly and optimize their performance. In situations when it is uncertain how the data will be distributed, nonparametric control charts are more reliable and useful than parametric charts. The sign test statistic and arcsine transformation are rapidly used in designing the nonparametric control charts. This article has designed Exponentially weighted moving average and Cumulative sum control charts using sign test statistic and arcsine transformation under zero-state and steady-state at head-to-head optimal design parameter choices. The proposed study quantifies the effect of both tests for the in-control (IC) and out-of-control (OOC) processes. The run-length (RL) properties have been calculated by using the Monte Carlo simulation method to assess the performance of the designed schemes. The RL profiles show that all the nonparametric schemes have IC robust behavior against the non-normality of the distributions. The results of the study reveal that the sign test statistic reveals almost equal but quite better results at small shifts, if the actual shifts are medium to large then arcsine transformation may respond quite better. Application of the charts under sign and arcsine transformation has been executed using an artificial dataset.

Quasi-interpolation for multivariate density estimation on bounded domain

The paper proposes a new nonparametric scheme for multivariate density estimation under the framework of quasi-interpolation, a classical function approximation scheme in approximation theory. Given samples of a random variable obeyed by an unknown density function with compact support, we first partition the support into several bins and compute frequency of samples falling into each bin. Then, by viewing these frequencies as (approximate) integral functionals of density function over corresponding bins, we construct a quasi-interpolation scheme for approximating the density function. Maximal mean squared errors of the scheme demonstrates that our scheme keeps the same optimal convergence rate as classical nonparametric density estimations. In addition, the scheme includes classical boundary kernel density estimation as a special case when the number of bins equals to the number of samples. Moreover, it can dynamically allocate different amounts of smoothing via selecting the bin widths and shape parameters of kernels with the prior knowledge. Numerical simulations provide evidence that the proposed nonparametric scheme is robust and is capable of producing high-performance estimation of density function.

Design and implementation of distribution-free Phase-II charting schemes based on unconditional run-length percentiles

Traditionally, the mean of the run-length distribution (ARL) of an in-control (IC) process is used to design and implement statistical process charting schemes. When standards are unknown (Case U), the unconditional ARL is considered during Phase-II monitoring—surprisingly, by suppressing the term “unconditional.” The literature has recently highlighted the difference between the unconditional and the conditional ARL in studying the properties of Phase-II charting schemes under the Case U. The effects of bias in the Phase-I sample may lead to remarkably high rates of early false alarms. We explore the idea of restricting the probability of unconditional early false alarms by using lower percentile points of the unconditional run-length distribution to design nonparametric charting schemes. This new approach is named “the lower percentile-based (LPL) design.” We consider the design and implementation of six distribution-free schemes: five precedence-type schemes and the rank-sum scheme. We carry out simulations to compare the six schemes with a prefixed value of some lower percentile point of the IC run-length distribution. The best scheme is the one with the lowest value for a specific higher percentile point of the out-of-control run-length distribution. We illustrate the new design and implementation strategies with real data, and offer a summary and concluding remarks.

Nonparametric multivariate processes monitoring with guaranteed in-control performance for changes in location

Multivariate statistical process control (MSPC) addresses the concurrent monitoring of several measurements. Since multivariate normality is rare in practice, nonparametric schemes become useful alternatives. Inspired by the development of distribution free Shewhart-type control charts to monitor multivariate data streams as a mean to address high-dimensional processes where distance measures are ranked and compared using two-sample test statistics, a novel distribution-free multivariate chart for location that combines Mahalanobis distances with the Mann-Whitney statistic is proposed. By considering, probably for the first time in nonparametric MSPC, practitioner-to-practitioner variation, we account for the uncertainty of the run length distribution, where computational complexity is addressed using the cumulative conditional false alarm probability as a proxy of the conditional average run length. This approach enables quicker determination of control limits that guarantee a minimum performance given a Phase-I sample, a common situation in parametric approaches, but scarcely discussed in nonparametric monitoring. Since no direct alternative exists, results are compared with the traditional approach to show the benefits of a guarantee in-control performance with comparable sensitivity when dealing with medium to large shifts. An implementation is illustrated in a real scenario related to Wine Quality Data.

Nonparametric multivariate control chart for numerical and categorical variables

Multivariate statistical process control (MSPC) was developed for the monitoring of variables that are either all numerical or all categorical. In the present paper, we describe a nonparametric control scheme that can be used to monitor a mixture of numerical and categorical variables simultaneously. It integrates Principal Component Analysis Mix (PCA Mix), a multivariate statistical tool, with the conventional Hotelling T 2 chart. To estimate the control limit for the PCA Mix-based T 2 statistic, two nonparametric approaches – kernel density estimation (KDE) and bootstrap – are employed, because of the unknown nature of the underlying distribution. The simulation results demonstrate that with an appropriate number of principal components, both bootstrap and KDE exhibit convincing performance in terms of generating the same, or nearly the same, number of false alarms (ARL 0) as expected, and being able to detect process shifts efficiently (ARL 1). Compared with bootstrap, KDE is shown to work better with small sample sizes (n < 800) and to be slightly more sensitive to small shifts. However, the results also show the instability of the estimated nonparametric control limit when highly imbalanced categorical variables are included, which indicates the need for further research on this topic.

A comprehensive distribution‐free scheme for tri‐aspect surveillance of complex processes

AbstractDistribution‐free charting schemes for process monitoring are more robust and reliable than their parametric counterparts when the process distributions are unknown and complicated. Most of the existing control charts are uni‐aspect schemes because they can detect a shift in only one aspect of the process distribution, like location or scale. Research on schemes for simultaneous surveillance of location and scale parameters has been very active in recent years, leading to many bi‐aspect schemes. These schemes do not explicitly deal with the shape of the process distribution. However, a shift in the shape parameter with or without a location or scale shift can occur in practice, especially in production or time‐to‐event processes. Note that there are nonparametric schemes for detecting general shifts based on goodness‐of‐fit test statistics or empirical likelihood ratio statistics, which cannot isolate if there is a shift in location, scale, shape, or combination of these parameters. This article introduces a new distribution‐free process monitoring scheme that can detect a shift in the location, scale or shape parameters, or any combinations of them. The new scheme is based on a combined statistic designed via Euclidean distance of the standardized Wilcoxon statistic for location, the standardized Ansari–Bradley statistic for scale, and the standardized Savage‐type statistic for shape. We discuss the implementation design using the average run‐length as the performance metric and investigate the proposed scheme's in‐control performance. It is shown that the new charting scheme is in‐control robust irrespective of the underlying process distribution and therefore applicable to monitor any univariate continuous processes. An out‐of‐control performance comparison study of the new scheme with many existing schemes shows that the new scheme is preferable to the existing schemes. We illustrate the proposed chart with real data to monitor a passenger train's arrival delays in Italy's regional route. The proposed scheme is comprehensive because it integrates the follow‐up procedure via integrated sub‐charts for classifying the signaling component.

Bayesian bootstrapping for symmetric distributions

In this paper, we describe a Bayesian nonparametric approach to make inference for a spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all spherically symmetric distributions and we derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process to a spherically symmetric distribution. In addition, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches the mixtures of Dirichlet processes. Moreover, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support is symmetric with respect to the parallel lines of axes. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.

Two economically optimized nonparametric schemes for monitoring process variability

AbstractControlling and reducing process variability is an essential aspect for maintaining the product or service quality. Even though most practitioners believe that an increasing process variability is often a more severe concern than a shift in location, barely a few research paid attention to the cost‐efficient monitoring of process variability. Some of the existing studies addressed the dispersion aspect, assuming that the quality characteristic is Gaussian. Non‐normal and complex distributions are not uncommon in modern production processes, time to event processes, or processes involving service quality. Unfortunately, we find no literature on economically designed nonparametric (distribution‐free) schemes for monitoring process variability. This article introduces two Shewhart‐type cost‐optimized nonparametric schemes for monitoring the variability of any unknown but continuous processes to fill the research gap. The proposed monitoring schemes are based on two popular two‐sample rank statistics for differences in scale parameters, known as the Ansari–Bradley statistic and the Mood statistic. We assess their actual performance for a set of process scenarios and illustrate the design along with the implementation steps. We discuss a practical problem related to product quality management. It is expected that the proposed schemes will be beneficial in various industrial operations.

An EWMA-type chart based on signed ranks with exact run length properties

ABSTRACT In this work, a new distribution-free Phase II EWMA-type chart based on the Wilcoxon signed rank statistic is introduced and its exact Run Length properties are discussed. A proper discrete Markov-chain approach is used for the determination of the Average Run Length in the in-control and out-of-control process operating conditions and its exact performance is derived without any knowledge of the distribution of sample observations. In addition, an algorithm for the determination of the chart's optimal design has been developed. Moreover, the performance of our proposed chart is compared with several existing nonparametric schemes available from the literature. Finally, two illustrative examples are provided to show the practical implementation of our proposed chart.

An information-theoretic approach to study spatial dependencies in small datasets.

From epidemiology to economics, there is a fundamental need of statistically principled approaches to unveil spatial patterns and identify their underpinning mechanisms. Grounded in network and information theory, we establish a non-parametric scheme to study spatial associations from limited measurements of a spatial process. Through the lens of network theory, we relate spatial patterning in the dataset to the topology of a network on which the process unfolds. From the available observations of the spatial process and a candidate network topology, we compute a mutual information statistic that measures the extent to which the measurement at a node is explained by observations at neighbouring nodes. For a class of networks and linear autoregressive processes, we establish closed-form expressions for the mutual information statistic in terms of network topological features. We demonstrate the feasibility of the approach on synthetic datasets comprising 25-100 measurements, generated by linear or nonlinear autoregressive processes. Upon validation on synthetic processes, we examine datasets of human migration under climate change in Bangladesh and motor vehicle deaths in the United States of America. For both these real datasets, our approach is successful in identifying meaningful spatial patterns, begetting statistically-principled insight into the mechanisms of important socioeconomic problems.

Nonparametric Phase-II control charts for monitoring high-dimensional processes with unknown parameters

Monitoring multivariate and high-dimensional data streams is often an essential requirement for quality management in manufacturing and service sectors in the Industry 4.0 era. Identifying a suitable distribution for a multivariate data set, especially when the number of variables is much larger than the sample size, is often challenging. Consequently, in a high-dimensional set-up, that is, when the number of variables under investigation exceeds sample size, parametric methods are generally not reliable in practice. There are various nonparametric schemes based on data depth for multivariate process monitoring, which are applicable only when the sample size is reasonably larger than the number of variables in the process but not in a high-dimensional set-up. We discuss that most of these charts are not robust when the true process parameters are unknown. There are, however, some nonparametric schemes for a high-dimensional process, when true process parameters are known. Nevertheless, when process parameters are unknown, a highly robust nonparametric scheme for monitoring high-dimensional processes is not yet available. In this paper, we propose some Shewhart-type nonparametric monitoring schemes based on specific distance metrics for surveillance of multivariate as well as high-dimensional processes. Our proposed charts are easy to implement, interpret and also advantageous in simultaneous monitoring of multiple aspects. We discuss the design and implementation issues in details. We carry out a performance study using Monte Carlo simulations and illustrate the proposed methods using a dataset related to industrial production.

One-sided precedence monitoring schemes for unknown shift sizes using generalized 2-of-(h+1) and w-of-w improved runs-rules

Parametric monitoring schemes are expected to perform better than their nonparametric counterparts when the assumption of a specific form of a distribution such as the normal is met. In practice, such assumption is often violated. Consequently, the performance of parametric schemes deteriorates considerably. To solve this problem, more efficient and flexible monitoring schemes based on nonparametric tests are recommended. In this paper, efficient and robust one-sided monitoring schemes are developed using generalized {1-of-1 or 2-of-(h + 1)} and {1-of-1 or w-of-w} improved runs-rules (IRR) without any distributional assumption in the zero- and steady-states modes. Moreover, the zero- and steady-states run-length properties of the resulting one-sided IRR schemes are formulated and empirically evaluated using the Markov chain technique. Comparisons with other well-known one-sided Shewhart-type nonparametric schemes (e.g., basic precedence scheme and precedence scheme with standard runs-rules) indicate that the proposed schemes have a better performance. Real data are used to demonstrate the design and implementation of the one-sided improved runs-rules precedence schemes. Finally, a discussion on the two-sided IRR precedence schemes is also provided.

Nonparametric “anti-Bayesian” quantile-based pattern classification

Parametric and nonparametric pattern recognition have been studied for almost a century based on a Bayesian paradigm, which is, in turn, founded on the principles of Bayes theorem. It is well known that the accuracy of the Bayes classifier cannot be exceeded. Typically, this reduces to comparing the testing sample to mean or median of the respective distributions. Recently, Oommen and his co-authors have presented a pioneering and non-intuitive paradigm, namely that of achieving the classification by comparing the testing sample with another descriptor, which could also be quite distant from the mean. This paradigm has been termed as being “anti-Bayesian,” and it essentially uses the quantiles of the distributions to achieve the pattern recognition. Such classifiers attain the optimal Bayesian accuracy for symmetric distributions even though they operate with a non-intuitive philosophy. While this paradigm has been applied in a number of domains (briefly explained in the body of this paper), its application for nonparametric domains has been limited. This paper explains, in detail, how such quantile-based classification can be extended to the nonparametric world, using both traditional and kernel-based strategies. The paper analyzes the methodology of such nonparametric schemes and their robustness. From a fundamental perspective, the paper utilizes the so-called large sample theory to derive strong asymptotic results that pertain to the equivalence between the parametric and nonparametric schemes for large samples. Apart from the new theoretical results, the paper also presents experimental results demonstrating their power. These results pertain to artificial data sets and also involve a real-life breast cancer data set obtained from the University Hospital Centre of Coimbra. The experimental results clearly confirm the power of the proposed “anti-Bayesian” procedure, especially when approached from a nonparametric perspective.

Distribution-free monitoring schemes based on order statistics: a general approach

In this article, we establish a new class of distribution-free Shewhart-type monitoring schemes based on order statistics. The setup of the proposed family of nonparametric control charts is presented in detail. Specific monitoring schemes, already introduced in the literature, are confirmed to be members of the new class. In addition, a new nonparametric monitoring scheme that belongs to the class is established, while explicit formulae for its basic characteristics are reached. The numerical study carried out reveals that the proposed scheme achieves adversarial in-control and out-of-control performance.

Overlapping community detection for count-value networks

Detecting network overlapping community has become a very hot research topic in the literature. However, overlapping community detection for count-value networks that naturally arise and are pervasive in our modern life, has not yet been thoroughly studied. We propose a generative model for count-value networks with overlapping community structure and use the Indian buffet process to model the community assignment matrix Z; thus, provide a flexible nonparametric Bayesian scheme that can allow the number of communities K to increase as more and more data are encountered instead of to be fixed in advance. Both collapsed and uncollapsed Gibbs sampler for the generative model have been derived. We conduct extensive experiments on simulated network data and real network data, and estimate the inference quality on single variable parameters. We find that the proposed model and inference procedure can bring us the desired experimental results.

Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity

In this article, we propose three nonparametric schemes based on the combination of p-values for simultaneous monitoring of shifts in location and scale parameters of a process. Li and Qiu first introduced the idea of dynamic nonparametric monitoring based on the p-value in the context of location shift of a process. This article extends their idea to a more general problem of joint monitoring of location and scale. In this article, we use three approaches for combining the p-values, namely Fisher, Liptak, and Tippett methods, to design three new schemes. We refer to them as the SL-Fisher (hereafter, SLF), the SL-Liptak (hereafter, SLL), and the SL-Tippett (hereafter, SLT) schemes. These schemes are based on a single plotting statistic attained by combining the p-values of the Wilcoxon-rank-sum (WRS) and Ansari-Bradley (AB) test statistics along with the combining functions to control the multiplicity effect. We use a real application based on water turbidity data to explain the implementation of the proposed schemes. We examine and compare the performance characteristics including the mean, median, and several percentiles of the proposed schemes thoroughly using Monte-Carlo simulation. We observe that, in general, the SLT scheme is effective for the detection of shifts in scale followed by either no shift or small shift in the location and the SLL scheme for other cases of shifts. Unlike existing schemes, the proposed schemes offer easier post-signal follow-up approach. We illustrate the design for one-sided shifts in both location and scale, but it can be easily extended for two sided shifts in both location and scale or two-sided shifts in location along with one-sided shifts in scale.

Flexible semiparametric generalized Pareto modeling of the entire range of rainfall amount

AbstractPrecipitation amounts at daily or hourly scales are skewed to the right, and heavy rainfall is poorly modeled by a simple gamma distribution. An important yet challenging topic in hydrometeorology is to find a probability distribution that is able to model well low, moderate, and heavy rainfall. To address this issue, we present a semiparametric distribution suitable for modeling the entire range of rainfall amount. This model is based on a recent parametric statistical model called the class of extended generalized Pareto distributions (EGPDs). The EGPD family is in compliance with extreme value theory for both small and large values, while it keeps a smooth transition between these tails and bypasses the hurdle of selecting thresholds to define extremes. In particular, return levels beyond the largest observation can be inferred. To add flexibility to this EGPD class, we propose to model the transition function in a nonparametric fashion. A fast and efficient nonparametric scheme based on Bernstein polynomial approximations is investigated. We perform simulation studies to assess the performance of our approach. It is compared to two parametric models: a parametric EGPD and the classical generalized Pareto distribution (GPD), the latter being only fitted to excesses above a high threshold. We also apply our semiparametric version of EGPD to a large network of 180 precipitation time series over France.

Smoothed Conditional Scale Function Estimation in AR(1)-ARCH(1) Processes

The estimation of the Smoothed Conditional Scale Function for time series was taken out under the conditional heteroscedastic innovations by imitating the kernel smoothing in nonparametric QAR-QARCH scheme. The estimation was taken out based on the quantile regression methodology proposed by Koenker and Bassett. And the proof of the asymptotic properties of the Conditional Scale Function estimator for this type of process was given and its consistency was shown.