Proportion Of Outliers Research Articles

Matchability and Uncertainty-Aware Iterative Disparity Refinement for Stereo Matching

After significant progress in stereo matching, the pursuit of robust and efficient ill-posed-region disparity refinement methods remains challenging. To further improve the performance of disparity refinement, in this paper, we propose the matchability and uncertainty-aware iterative disparity refinement neural network. Firstly, a new matchability and uncertainty decoder (MUD) is proposed to decode the matchability mask and disparity uncertainties, which are used to evaluate the reliability of feature matching and estimated disparity, thereby reducing the susceptibility to mismatched pixels. Then, based on the proposed MUD, we present two modules: the uncertainty-preferred disparity field initialization (UFI) and the masked hidden state global aggregation (MGA) modules. In the UFI, a multi-disparity window scan-and-select method is employed to provide a further initialized disparity field and more accurate initial disparity. In the MGA, the adaptive masked disparity field hidden state is globally aggregated to extend the propagation range per iteration, improving the refinement efficiency. Finally, the experimental results on public datasets show that the proposed model achieves a reduction up to 17.9% in disparity average error and 16.9% in occluded outlier proportion, respectively, demonstrating its more practical handling of ill-posed regions.

Stability Detection of Canopy RGB Images for Different Underlying Surfaces Based on SVM

This study aims to investigate the impact of different environmental conditions on the stability of RGB images in ecological sites and the anti-interference properties of vegetation images on different underlying surfaces. Three vegetation types including sugarcane, forest, and karst (mainly shrub and grass) are used to segment green vegetation using machine learning, and the RGB vegetation indices are calculated using color channel data. Then, The effect of weather, season, and time period on different types of vegetation indices are studied, which provide technical references for quantitative application of RGB image data. The results indicate the following: ① For the vegetation with high canopy density, the SVM machine learning segmentation algorithm used in this study is more applicable, as the RGB image segmentation accuracy of sugarcane and forest is significantly higher than karst. For different weather conditions, the segmentation accuracy of sugarcane and forest RGB images on sunny or cloudy days is higher than that on rainy or foggy days, but the effect on sparse vegetation in karst is not obvious. Additionally, the segmentation accuracy of different vegetation types has a small increase with NLM filter processing. ② Change in weather, season, and time can affect the stability of the image index. For different weather conditions, the vegetation indices of sugarcane and forest images on sunny or cloudy days are the more stable (ExGR, outlier proportion between 3.25% and 5.63%), while on rainy and foggy days they are less stable (ExR, outlier proportion between 17.60% and 21.59%). For different seasons, the stability of the sugarcane and karst image index obtained in spring is better (ExG, outlier proportion between 3.32% and 6.88%), while the stability of the forest image index obtained in summer is better (ExR, outlier proportion = 10.32%). For different times within a day, the sugarcane image index obtained in the morning is more stable (ExGR, outlier proportion = 4.62%), while the stability of the forest image index obtained in the afternoon is better (ExGR, outlier proportion = 9.24%). ③ The stability of the sugarcane image index is more affected by weather and season. For forest, the influence of the weather is more obvious than the season. But, for karst, the effect of season on the vegetation index is greater than that of weather.

Estimating the proportions and latencies of reaction time outliers: A pooling method and case study of lexical decision tasks

A methodological problem in most reaction time (RT) studies is that some measured RTs may be outliers—that is, they may be very fast or very slow for reasons unconnected to the task-related processing of interest. Numerous ad hoc methods have been suggested to discriminate between such outliers and the valid RTs of interest, but it is extremely difficult to determine how well these methods work in practice because virtually nothing is known about the actual characteristics of outliers in real RT datasets. This article proposes a new method of pooling cumulative distribution function values for examining empirical RT distributions to assess both the proportions of outliers and their latencies relative to those of the valid RTs. As the method is developed, its strengths and weaknesses are examined using simulations based on previously suggested ad hoc models for RT outliers with particular assumed proportions and distributions of valid RTs and outliers. The method is then applied to several large RT datasets from lexical decision tasks, and the results provide the first empirically based description of outlier RTs. For these datasets, fewer than 1% of the RTs seem to be outliers, and the median outlier latency appears to be approximately 4–6 standard deviations of RT above the mean of the valid RT distribution.

PMA-Net: Progressive multi-stage adaptive feature learning for two-view correspondence

Establishing high-quality correspondences is a fundamental step for many computer vision tasks. Leveraging the local consistency of correct correspondences (i.e., inliers) has been a prevalent approach to distinguish them from incorrect correspondences (i.e., outliers). However, the random distribution of a significant proportion of outliers complicates the local neighborhood construction of inliers, making it inevitably contain many outliers, which compromises the reliability of the consistency information. In this paper, we propose a Progressive Multi-Stage Adaptive Feature Learning Network (PMA-Net) to solve this problem, which progressively and adaptively learns over multiple stages to remove random outliers embedded in the initial consistency. Specifically, we propose an Adaptive Dynamic Graph Construction (ADGC) module to construct a global topological graph with a high inlier ratio by calculating the affinity between the corresponding neighbors. Additionally, we also design a Feature Mapping Processing (FMP) block and Multi-Stage Prediction (MSP) block to improve the accuracy of subsequent local neighborhood information aggregation, as well as information loss due to channel dimension compression in the prediction phase. Experimental results in camera pose estimation, remote sensing image registration, and homography estimation demonstrate that the proposed PMA-Net performs better than other state-of-the-art methods on various public datasets. Code: https://github.com/XiaojieLi11/PMA-Net

Robust and efficient estimation in ordinal response models using the density power divergence

In real life, we frequently encounter ordinal variables depending upon independent covariates. The latent linear regression model is useful for modelling such data. One can find the model's parameters' maximum likelihood estimate (MLE). Though noted for its optimum properties, a small proportion of outliers may destabilize the MLE. This paper uses the minimum density power divergence estimate (MDPDE) as a robust alternative. The roles of different link functions are analysed in this context. We discuss their asymptotic properties in this setup. Unlike the MLE, the MDPDEs are robust for– lower values of the gross error sensitivity, and very high breakdown point. Also, the slope’s MDPDEs never implode. In simulation studies for pure data, MDPDEs perform almost as good as the MLE. However, the MDPDEs outperform the MLE in data contamination. Moreover, MDPDEs are very competitive with the other robust alternatives. Finally, this article is wrapped up with a real-data example.

Development of a simplified new method of bone age estimation using three bones of the hand and wrist.

1. Devising a simplified method utilizing three bones of the hand and wrist for bone age (BA) assessment. 2. Testing whether the 3 bone method gives comparable results to standard methods (GP,TW2,TW3) in Indian children. Developmental stages and corresponding BA for radius, hamate, terminal phalanx (left middle finger) epiphyses combining stages from GP,TW3 atlases were described; BA were rated by two blinded observers. 3 bone method ratings were compared with the same dataset analyzed earlier using GP,TW2,TW3 (4 raters). Radiographs analysed:493 (Girls=226). Mean chronological age:9.4 ± 4.6 yrs, mean BA 3 bone:9.8 ± 4.8 yrs, GP:9.6 ± 4.8 yrs, TW3:9.3 ± 4.5 yrs, TW2:9.9 ± 5.0 yrs. The 3 bone method demonstrated no significant inter-observer variability (p = 0.3, mean difference = 0.02 ± 0.6 yrs); a strong positive correlation (p < 0.0001) with GP (r = 0.985), TW3 (r = 0.983) and TW2 (r = 0.982) was noted. Bland-Altman plots demonstrated good agreement; the root mean square errors between 3 bone and GP,TW3,TW2 ratings were 0.6,0.7,0.6 years; mean differences were 0.19,0.49,-0.14 years respectively. Greatest proportion of outliers (beyond ±1.96 SD of mean difference) was between 6 and 8 years age for difference in 3 bone and GP, and between 4-6 years for difference in 3 bone and TW3,TW2. The 3 bone method has multiple advantages; it is easier, tackles differential maturation of wrist and hand bones, has good reproducibility, without compromising on accuracy rendering it suitable for office practice.

Alzheimer's disease heterogeneity revealed by neuroanatomical normative modeling.

Overlooking the heterogeneity in Alzheimer's disease (AD) may lead to diagnostic delays and failures. Neuroanatomical normative modeling captures individual brain variation and may inform our understanding of individual differences in AD-related atrophy. We applied neuroanatomical normative modeling to magnetic resonance imaging from a real-world clinical cohort with confirmed AD (n=86). Regional cortical thickness was compared to a healthy reference cohort (n=33,072) and the number of outlying regions was summed (total outlier count) and mapped at individual- and group-levels. The superior temporal sulcus contained the highest proportion of outliers (60%). Elsewhere, overlap between patient atrophy patterns was low. Mean total outlier count was higher in patients who were non-amnestic, at more advanced disease stages, and without depressive symptoms. Amyloid burden was negatively associated with outlier count. Brain atrophy in AD is highly heterogeneous and neuroanatomical normative modeling can be used to explore anatomo-clinical correlations in individual patients.

Multi-Stage Network with Geometric Semantic Attention for Two-View Correspondence Learning.

The removal of outliers is crucial for establishing correspondence between two images. However, when the proportion of outliers reaches nearly 90%, the task becomes highly challenging. Existing methods face limitations in effectively utilizing geometric transformation consistency (GTC) information and incorporating geometric semantic neighboring information. To address these challenges, we propose a Multi-Stage Geometric Semantic Attention (MSGSA) network. The MSGSA network consists of three key modules: the multi-branch (MB) module, the GTC module, and the geometric semantic attention (GSA) module. The MB module, structured with a multi-branch design, facilitates diverse and robust spatial transformations. The GTC module captures transformation consistency information from the preceding stage. The GSA module categorizes input based on the prior stage's output, enabling efficient extraction of geometric semantic information through a graph-based representation and inter-category information interaction using Transformer. Extensive experiments on the YFCC100M and SUN3D datasets demonstrate that MSGSA outperforms current state-of-the-art methods in outlier removal and camera pose estimation, particularly in scenarios with a high prevalence of outliers. Source code is available at https://github.com/shuyuanlin.

Depression, anxiety, and burnout in academia: topic modeling of PubMed abstracts

The problem of mental health in academia is increasingly discussed in literature, and to extract meaningful insights from the growing amount of scientific publications, text mining approaches are used. In this study, BERTopic, an advanced method of topic modeling, was applied to abstracts of 2,846 PubMed articles on depression, anxiety, and burnout in academia published in years 1975–2023. BERTopic is a modular technique comprising a text embedding method, a dimensionality reduction procedure, a clustering algorithm, and a weighing scheme for topic representation. A model was selected based on the proportion of outliers, the topic interpretability considerations, topic coherence and topic diversity metrics, and the inevitable subjectivity of the criteria was discussed. The selected model with 27 topics was explored and visualized. The topics evolved differently with time: research papers on students' pandemic-related anxiety and medical residents' burnout peaked in recent years, while publications on psychometric research or internet-related problems are yet to be presented more amply. The study demonstrates the use of BERTopic for analyzing literature on mental health in academia and sheds light on areas in the field to be addressed by further research.

Fast Robust Matrix Completion via Entry-Wise l0 -Norm Minimization.

Matrix completion (MC) aims at recovering missing entries, given an incomplete matrix. Existing algorithms for MC are mainly designed for noiseless or Gaussian noise scenarios and, thus, they are not robust to impulsive noise. For outlier resistance, entry-wise lp -norm with and M-estimation are two popular approaches. Yet the optimum selection of p for the entrywise lp -norm-based methods is still an open problem. Besides, M-estimation is limited by a breakdown point, that is, the largest proportion of outliers. In this article, we adopt entrywise l0 -norm, namely, the number of nonzero entries in a matrix, to separate anomalies from the observed matrix. Prior to separation, the Laplacian kernel is exploited for outlier detection, which provides a strategy to automatically update the entrywise l0 -norm penalty parameter. The resultant multivariable optimization problem is addressed by block coordinate descent (BCD), yielding l0 -BCD and l0 -BCD-F. The former detects and separates outliers, as well as its convergence is guaranteed. In contrast, the latter attempts to treat outlier-contaminated elements as missing entries, which leads to higher computational efficiency. Making use of majorization-minimization (MM), we further propose l0 -BCD-MM and l0 -BCD-MM-F for robust non-negative MC where the nonnegativity constraint is handled by a closed-form update. Experimental results of image inpainting and hyperspectral image recovery demonstrate that the suggested algorithms outperform several state-of-the-art methods in terms of recovery accuracy and computational efficiency.

Outlier detection using iterative adaptive mini-minimum spanning tree generation with applications on medical data.

As an important technique for data pre-processing, outlier detection plays a crucial role in various real applications and has gained substantial attention, especially in medical fields. Despite the importance of outlier detection, many existing methods are vulnerable to the distribution of outliers and require prior knowledge, such as the outlier proportion. To address this problem to some extent, this article proposes an adaptive mini-minimum spanning tree-based outlier detection (MMOD) method, which utilizes a novel distance measure by scaling the Euclidean distance. For datasets containing different densities and taking on different shapes, our method can identify outliers without prior knowledge of outlier percentages. The results on both real-world medical data corpora and intuitive synthetic datasets demonstrate the effectiveness of the proposed method compared to state-of-the-art methods.

Detection of Outliers in Univariate Circular Data Using New Cut-off Points for the Circular Distance

The aim of this study is to propose two new cut-off points for outlier detection in univariate circular data using the concept of circular distance. The first cut-off point involves using a quantile of the gamma distribution based on adjusted circular distances, whereas the second cut-off point employs the upper fence of a modified boxplot for skewed data. Simulation studies are conducted using both uncontaminated and contaminated data, and the performance of the proposed cut-off points is evaluated in the proportion of outliers, probability of all outliers being successfully detected, probability of outliers being falsely detected as inliers (masking effect), and probability of inliers detected as outliers (swamping effect). Real data examples are also used to demonstrate the efficacy of the proposed cut-off points. The results of the simulation and real data experiments show that the proposed cut-off point involves using a quantile of the gamma distribution based on adjusted circular distances and is successful in outlier detection compared to the existing cut-off points.

Clustering asymmetrical data with outliers: Parsimonious mixtures of contaminated mean-mixture of normal distributions

Mixture modeling has emerged as a statistical tool to perform unsupervised model-based clustering for heterogeneous data. A framework of using contaminated mean-mixture of normal distributions as the components of the mixture model is designed to accommodate asymmetric data with outliers. Fourteen parsimonious variants of the postulated model are introduced by employing an eigenvalue decomposition of the component scale matrices. Simultaneously clustering and outliers detection is an outstanding advantage of the proposed model in analyzing non-normally distributed data. A computationally feasible and flexible EM-type algorithm is outlined for obtaining maximum likelihood parameter estimates. Moreover, the score vector and empirical information matrix for calculating asymptotic standard errors of the parameter estimates are derived by offering an information-based approach. The applicability of the proposed method is demonstrated through the analysis of simulated and real datasets with varying proportions of outliers.

Consistency between Treatment Effects on Clinical and Brain Atrophy Outcomes in Alzheimer’s Disease Trials

BackgroundLongitudinal changes in volumetric MRI outcome measures have been shown to correlate well with longitudinal changes in clinical instruments and have been widely used as biomarker outcomes in clinical trials for Alzheimer’s disease (AD). While instances of discordant findings have been noted in some trials, especially the recent amyloid-removing therapies, the overall relationship between treatment effects on brain atrophy and clinical outcomes, and how it might depend on treatment target or mechanism, clinical instrument or imaging variable is not yet clear.ObjectiveTo systematically assess the consistency and therapeutic class-dependence of treatment effects on clinical outcomes and on brain atrophy in published reports of clinical trials conducted in mild cognitive impairment (MCI) and/or AD.DesignQuantitative review of the published literature. The consistency of treatment effects on clinical and brain atrophy outcomes was assessed in terms of statistical agreement with hypothesized equal magnitude effects (e.g., 30% slowing of both) and nominal directional concordance, as a function of therapeutic class.SettingInterventional randomized clinical trials.ParticipantsMCI or AD trial participants.InterventionTreatments included were those that involved ingestion or injection of a putatively active substance into the body, encompassing both pharmacological and controlled dietary interventions.MeasurementsEach trial included in the analysis reported at least one of the required clinical outcomes (ADAS-Cog, CDR-SB or MMSE) and at least one of the required imaging outcomes (whole brain, ventricular or hippocampal volume).ResultsData from 35 trials, comprising 185 pairwise comparisons, were included. Overall, the 95% confidence bounds overlapped with the line of identity for 150/185 (81%) of the imaging-clinical variable pairs. The greatest proportion of outliers was found in trials of anti-amyloid antibodies that have been shown to dramatically reduce the level of PET-detectable amyloid plaques, for which only 13/33 (39%) of observations overlapped the identity line. A Deming regression calculated using all data points yielded a slope of 0.54, whereas if data points from the amyloid remover class were excluded, the Deming regression line had a slope of 0.92. Directional discordance of treatment effects was also most pronounced for the amyloid-removing class, and for comparisons involving ventricular volume.ConclusionOur results provide a frame of reference for the interpretation of clinical and brain atrophy results from future clinical trials and highlight the importance of mechanism of action in the interpretation of imaging results.

Robust solution of coordinate transformation parameters with a high breakdown point

Because the M estimation method may lead to poor robustness or even failure owing to excessive outliers, a robust algorithm with a high breakdown point was proposed and applied to the estimation of coordinate transformation parameters. Firstly, the sampling method was used to calculate multiple sets of model parameters, and some sampling results were sifted according to posterior information. Then, the samples were sorted according to their number in the sampling results, and the F-test was adopted to screen and reserve valid information. Finally, the initial values of the reliable parameters were computed using the valid information, and the final parameters were obtained by the Institute of Geodesy and Geophysics III scheme. Monte Carlo method was adopted for the simulation test, and a case analysis was chosen for verification. The results show that the proposed method can identify and process outliers more accurately than those of Rousseeuw and Hubert (2011 Wires Data Min. Knowl. 1 73–79) and Tao et al (2016 Acta Geod. Cartogr. Sin. 45 297–301). When the proportion of outliers exceeded 50%, the proposed algorithm maintained a strong robustness and had a high breakdown point.

Revealing Individual Neuroanatomical Heterogeneity in Alzheimer Disease Using Neuroanatomical Normative Modeling.

Alzheimer disease (AD) is highly heterogeneous, with marked individual differences in clinical presentation and neurobiology. To explore this, we used neuroanatomical normative modeling to index regional patterns of variability in cortical thickness. We aimed to characterize individual differences and outliers in cortical thickness in patients with AD, people with mild cognitive impairment (MCI), and controls. Furthermore, we assessed the relationships between cortical thickness heterogeneity and cognitive function, β-amyloid, phosphorylated-tau, and ApoE genotype. Finally, we examined whether cortical thickness heterogeneity was predictive of conversion from MCI to AD. Cortical thickness measurements across 148 brain regions were obtained from T1-weighted MRI scans from 62 sites of the Alzheimer's Disease Neuroimaging Initiative. AD was determined by clinical and neuropsychological examination with no comorbidities present. Participants with MCI had reported memory complaints, and controls were cognitively normal. A neuroanatomical normative model indexed cortical thickness distributions using a separate healthy reference data set (n = 33,072), which used hierarchical Bayesian regression to predict cortical thickness per region using age and sex, while adjusting for site noise. Z-scores per region were calculated, resulting in a Z-score brain map per participant. Regions with Z-scores <-1.96 were classified as outliers. Patients with AD (n = 206) had a median of 12 outlier regions (out of a possible 148), with the highest proportion of outliers (47%) in the parahippocampal gyrus. For 62 regions, over 90% of these patients had cortical thicknesses within the normal range. Patients with AD had more outlier regions than people with MCI (n = 662) or controls (n = 159) (F(2, 1,022) = 95.39, p = 2.0 × 10-16). They were also more dissimilar to each other than people with MCI or controls (F(2, 1,024) = 209.42, p = 2.2 × 10-16). A greater number of outlier regions were associated with worse cognitive function, CSF protein concentrations, and an increased risk of converting from MCI to AD within 3 years (hazard ratio 1.028, 95% CI 1.016-1.039, p = 1.8 × 10-16). Individualized normative maps of cortical thickness highlight the heterogeneous effect of AD on the brain. Regional outlier estimates have the potential to be a marker of disease and could be used to track an individual's disease progression or treatment response in clinical trials.

Automatic quality control of the numerical accuracy of EEG lead fields

Precise individualized EEG source localization is predicated on having accurate subject-specific Lead Fields (LFs) obtained from their Magnetic Resonance Images (MRI). LF calculation is a complex process involving several error-prone steps that start with obtaining a realistic head model from the MRI and finalizing with computationally expensive solvers such as the Boundary Element Method (BEM) or Finite Element Method (FEM). Current Big-Data applications require the calculation of batches of hundreds or thousands of LFs. LF. Quality Control is conventionally checked subjectively by experts, a procedure not feasible in practice for larger batches. To facilitate this step, we introduce the Lead Field Automatic-Quality Control Index (LF-AQI) that flags LF with potential errors. We base our LF-AQI on the assumption that LFs obtained from simpler head models, i.e., the homogeneous head model LF (HHM-LF) or spherical head model LF (SHM-LF), deviate only moderately from a "good" realistic test LF. Since these simpler LFs are easier to compute and check for errors, they may serve as "reference LF" to detect anomalous realistic test LF. We investigated this assumption by comparing correlation-based channel ρmin(ref,test)and source τmin(ref,test) similarity indices (SI) between "gold standards," i.e., very accurate FEM and BEM LFs, and the proposed references (HHM-LF and SHM-LF). Surprisingly we found that the most uncomplicated possible reference, HHM-LF had high SI values with the gold standards—leading us to explore further use of the channel ρmin(HHM−LF,test)and source τmin(HHM−LF,test)SI as a basis for our LF-AQI. Indeed, these SI successfully detected five simulated scenarios of LFs artifacts. This result encouraged us to evaluate the SI on a large dataset and thus define our LF-AQI. We thus computed the SI of 1251 LFs obtained from the Child Mind Institute (CMI) MRI dataset. When ρmin(HHM−LF,test)and source τmin(HHM−LF,test) were plotted for all test subjects on a 2D space, most were tightly clustered around the median of a high similarity centroid (HSC), except for a smaller proportion of outliers. We define the LF-AQI for a given LF as the log Euclidean distance between its SI and the HSC median. To automatically detect outliers, the threshold is at the 90th percentile of the CMI LF-AQIs (-0.9755). LF-AQI greater than this threshold flag individual LF to be checked. The robustness of this LF-AQI screening was checked by repeated out-of-sample validation. Strikingly, minor corrections in re-processing the flagged cases eliminated their status as outliers. Furthermore, the "doubtful" labels assigned by LF-AQI were validated by neuroscience students using a Likert scale questionnaire designed to manually check the LF's quality. Item Response Theory (IRT) analysis was applied to the questionnaire results to compute an optimized model and a latent variable θ for that model. A linear mixed model (LMM) between the θ and LF-AQI resulted in an effect with a Cohen's d value of 1.3 and a p-value <0.001, thus validating the correspondence of LF-AQI with the visual quality control. We provide an open-source pipeline to implement both LF calculation and its quality control to allow further evaluation of our index.

KNN robustification equivariant nonparametric regression estimators for functional ergodic data

We discuss in this paper the robust equivariant nonparametric regression estimators for ergodic data with the k Nearst Neighbour (kNN) method. We consider a new robust regression estimator when the scale parameter is unknown. The principal aim is to prove the almost complete convergence (with rate) for the proposed estimator. Furthermore, a comparison study based on simulated data is also provided to illustrate the finite sample performances and the usefulness of the kNN approach and to prove the highly sensitive of the kNN approach to the presence of even a small proportion of outliers in the data.

Postoperative limb alignment in total knee replacement. Conventional versus navigated versus robotic techniques.

Our objective was to compare the coronal mechanical axis after total knee replacement (TKR) obtained in three groups of patients subjected to conventional, navigated, and robotic surgery. Retrospective analysis. 124 knees were included (36 conventional, 41 navigated, 47 robotic). No statistically significant differences were found between the postOp tibiofemoral angle of the conventional, navigated and robotic groups (p=0.396). A repeated-measure analysis of preOp-to-postOp also found no significant differences (p=0.387). There were no differences in the proportion of outliers (3-degree) found (p=0.211). Nevertheless, a higher proportion of patients in the robotic group improved their mechanical alignment, as compared with conventional surgery (p=0.023), although no differences were found when comparing with navigation (p=0.121). No statistically significant differences were found with respect to the postOp alignment achieved. However, statistically significant differences were detected between robotic and conventional surgery when considering the percentage of patients with improved limb alignment.

Phase I Analysis of Nonlinear Profiles Using Anomaly Detection Techniques

In various industries, the process or product quality is evaluated by a functional relationship between a dependent variable y and one or a few input variables x, expressed as y=fx. This relationship is called a profile in the literature. Recently, profile monitoring has received a lot of research attention. In this study, we formulated profile monitoring as an anomaly-detection problem and proposed an outlier-detection procedure for phase I nonlinear profile analysis. The developed procedure consists of three key processes. First, we obtained smoothed nonlinear profiles using the spline smoothing method. Second, we proposed a method for estimating the proportion of outliers in the dataset. A distance-based decision function was developed to identify potential outliers and provide a rough estimate of the contamination rate. Finally, PCA was used as a dimensionality reduction method. An outlier-detection algorithm was then employed to identify outlying profiles based on the estimated contamination rate. The algorithms considered in this study included Local Outlier Factor (LOF), Elliptic Envelope (EE), and Isolation Forest (IF). The proposed procedure was evaluated using a nonlinear profile that has been studied by various researchers. We compared various competing methods based on commonly used metrics such as type I error, type II error, and F2 score. Based on the evaluation metrics, our experimental results indicate that the performance of the proposed method is better than other existing methods. When considering the smallest and hardest-to-detect variation, the LOF algorithm, with the contamination rate determined by the method proposed in this study, achieved type I errors, type II errors, and F2 scores of 0.049, 0.001, and 0.951, respectively, while the performance metrics of the current best method were 0.081, 0.015, and 0.899, respectively.