Anomaly detection, finding patterns that substantially deviate from those seen previously, is one of the fundamental problems of artificial intelligence. Recently, classification-based methods were shown to achieve superior results on this task. In this work, we present a unifying view and propose an open-set method, GOAD, to relax current generalization assumptions. Furthermore, we extend the applicability of transformation-based methods to non-image data using random affine transformations. Our method is shown to obtain state-of-the-art accuracy and is applicable to broad data types. The strong performance of our method is extensively validated on multiple datasets from different domains.
University of Adelaide（阿德莱德大学）；University of Technology Sydney（悉尼科技大学）
Anomaly detection, a.k.a. outlier detection or novelty detection, has been a lasting yet active research area in various research communities for several decades. There are still some unique problem complexities and challenges that require advanced approaches. In recent years, deep learning enabled anomaly detection, i.e., deep anomaly detection, has emerged as a critical direction. This paper surveys the research of deep anomaly detection with a comprehensive taxonomy, covering advancements in three high-level categories and 11 fine-grained categories of the methods. We review their key intui…
NTT Communication Science Laboratories（NTT通信科学实验室）；NTT Software Innovation Center（NTT软件创新中心）
We propose a supervised anomaly detection method for data with inexact anomaly labels, where each label, which is assigned to a set of instances, indicates that at least one instance in the set is anomalous. Although many anomaly detection methods have been proposed, they cannot handle inexact anomaly labels. To measure the performance with inexact anomaly labels, we define the inexact AUC, which is our extension of the area under the ROC curve (AUC) for inexact labels. The proposed method trains an anomaly score function so that the smooth approximation of the inexact AUC increases while anom…
We develop a supervised machine learning model that detects anomalies in systems in real time. Our model processes unbounded streams of data into time series which then form the basis of a low-latency anomaly detection model. Moreover, we extend our preliminary goal of just anomaly detection to simultaneous anomaly prediction. We approach this very challenging problem by developing a Bayesian Network framework that captures the information about the parameters of the lagged regressors calibrated in the first part of our approach and use this structure to learn local conditional probability dis…
Anomaly detection seeks to identify unusual phenomena, a central task in science and industry. The task is inherently unsupervised as anomalies are unexpected and unknown during training. Recent advances in self-supervised representation learning have directly driven improvements in anomaly detection. In this position paper, we first explain how self-supervised representations can be easily used to achieve state-of-the-art performance in commonly reported anomaly detection benchmarks. We then argue that tackling the next generation of anomaly detection tasks requires new technical and conceptu…
Nearest neighbor (kNN) methods utilizing deep pre-trained features exhibit very strong anomaly detection performance when applied to entire images. A limitation of kNN methods is the lack of segmentation map describing where the anomaly lies inside the image. In this work we present a novel anomaly segmentation approach based on alignment between an anomalous image and a constant number of the similar normal images. Our method, Semantic Pyramid Anomaly Detection (SPADE) uses correspondences based on a multi-resolution feature pyramid. SPADE is shown to achieve state-of-the-art performance on u…
NTT Communication Science Laboratories（NTT通信科学实验室）；NTT Secure Platform Laboratories（NTT安全平台实验室）
We propose a supervised anomaly detection method based on neural density estimators, where the negative log likelihood is used for the anomaly score. Density estimators have been widely used for unsupervised anomaly detection. By the recent advance of deep learning, the density estimation performance has been greatly improved. However, the neural density estimators cannot exploit anomaly label information, which would be valuable for improving the anomaly detection performance. The proposed method effectively utilizes the anomaly label information by training the neural density estimator so th…
Log messages are now widely used in software systems. They are important for classification as millions of logs are generated each day. Most logs are unstructured which makes classification a challenge. In this paper, Deep Learning (DL) methods called Auto-LSTM, Auto-BLSTM and Auto-GRU are developed for anomaly detection and log classification. These models are used to convert unstructured log data to extracted features which is suitable for classification algorithms. They are evaluated using four data sets, namely BGL, Openstack, Thunderbird and IMDB. The first three are popular log data sets…
General Electric Global Research Center（通用电气全球研究中心）；General Electric Power & Water Engineering（通用电气电力与水务工程部门）
Monitoring gas turbine combustors health, in particular, early detecting abnormal behaviors and incipient faults, is critical in ensuring gas turbines operating efficiently and in preventing costly unplanned maintenance. One popular means of detecting combustor abnormalities is through continuously monitoring exhaust gas temperature profiles. Over the years many anomaly detection technologies have been explored for detecting combustor faults, however, the performance (detection rate) of anomaly detection solutions fielded is still inadequate. Advanced technologies that can improve detection pe…
Anomaly detection is a significant problem faced in several research areas. Detecting and correctly classifying something unseen as anomalous is a challenging problem that has been tackled in many different manners over the years. Generative Adversarial Networks (GANs) and the adversarial training process have been recently employed to face this task yielding remarkable results. In this paper we survey the principal GAN-based anomaly detection methods, highlighting their pros and cons. Our contributions are the empirical validation of the main GAN models for anomaly detection, the increase of …
Australian Institute for Machine Learning（澳大利亚机器学习研究所）；The University of Adelaide（阿德莱德大学）
Although deep learning has been applied to successfully address many data mining problems, relatively limited work has been done on deep learning for anomaly detection. Existing deep anomaly detection methods, which focus on learning new feature representations to enable downstream anomaly detection methods, perform indirect optimization of anomaly scores, leading to data-inefficient learning and suboptimal anomaly scoring. Also, they are typically designed as unsupervised learning due to the lack of large-scale labeled anomaly data. As a result, they are difficult to leverage prior knowledge …
Anomaly detection for time-series data has been an important research field for a long time. Seminal work on anomaly detection methods has been focussing on statistical approaches. In recent years an increasing number of machine learning algorithms have been developed to detect anomalies on time-series. Subsequently, researchers tried to improve these techniques using (deep) neural networks. In the light of the increasing number of anomaly detection methods, the body of research lacks a broad comparative evaluation of statistical, machine learning and deep learning methods. This paper studies …
This paper introduces anomalib, a novel library for unsupervised anomaly detection and localization. With reproducibility and modularity in mind, this open-source library provides algorithms from the literature and a set of tools to design custom anomaly detection algorithms via a plug-and-play approach. Anomalib comprises state-of-the-art anomaly detection algorithms that achieve top performance on the benchmarks and that can be used off-the-shelf. In addition, the library provides components to design custom algorithms that could be tailored towards specific needs. Additional tools, includin…
University at Buffalo（布法罗大学）；Tufts University（塔夫茨大学）
Most current anomaly detection methods suffer from the curse of dimensionality when dealing with high-dimensional data. We propose an anomaly detection algorithm that can scale to high-dimensional data using concepts from the theory of large deviations. The proposed Large Deviations Anomaly Detection (LAD) algorithm is shown to outperform state of art anomaly detection methods on a variety of large and high-dimensional benchmark data sets. Exploiting the ability of the algorithm to scale to high-dimensional data, we propose an online anomaly detection method to identify anomalies in a collecti…
University of Edinburgh（爱丁堡大学）；Technische Universität Berlin（柏林工业大学）
Deep anomaly detection is a difficult task since, in high dimensions, it is hard to completely characterize a notion of "differentness" when given only examples of normality. In this paper we propose a novel approach to deep anomaly detection based on augmenting large pretrained networks with residual corrections that adjusts them to the task of anomaly detection. Our method gives rise to a highly parameter-efficient learning mechanism, enhances disentanglement of representations in the pretrained model, and outperforms all existing anomaly detection methods including other baselines utilizing…
National Research University Higher School of Economics（国立研究型大学高等经济学院）；Laboratory of Methods for Big Data Analysis（大数据分析方法实验室）
Anomaly detection is not an easy problem since distribution of anomalous samples is unknown a priori. We explore a novel method that gives a trade-off possibility between one-class and two-class approaches, and leads to a better performance on anomaly detection problems with small or non-representative anomalous samples. The method is evaluated using several data sets and compared to a set of conventional one-class and two-class approaches.
Anomaly detection methods strive to discover patterns that differ from the norm in a semantic way. This goal is ambiguous as a data point differing from the norm by an attribute e.g., age, race or gender, may be considered anomalous by some operators while others may consider this attribute irrelevant. Breaking from previous research, we present a new anomaly detection method that allows operators to exclude an attribute from being considered as relevant for anomaly detection. Our approach then learns representations which do not contain information over the nuisance attributes. Anomaly scorin…
Anomaly detection in data analysis is an interesting but still challenging research topic in real world applications. As the complexity of data dimension increases, it requires to understand the semantic contexts in its description for effective anomaly characterization. However, existing anomaly detection methods show limited performances with high dimensional data such as ImageNet. Existing studies have evaluated their performance on low dimensional, clean and well separated data set such as MNIST and CIFAR-10. In this paper, we study anomaly detection with high dimensional and complex norma…
University of Oslo（奥斯陆大学）；SimulaMet；OsloMet（奥斯陆城市大学）；UiT（特罗姆瑟大学）
The interest for video anomaly detection systems has gained traction for the past few years. The current approaches use deep learning to perform anomaly detection in videos, but this approach has multiple problems. For starters, deep learning in general has issues with noise, concept drift, explainability, and training data volumes. Additionally, anomaly detection in itself is a complex task and faces challenges such as unknowness, heterogeneity, and class imbalance. Anomaly detection using deep learning is therefore mainly constrained to generative models such as generative adversarial networ…
CentraleSupélec（中央理工-高等电力学院）；École Polytechnique（巴黎综合理工学院）；Télécom ParisTech（巴黎电信学院）；Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）
Anomaly detection is a significant and hence well-studied problem. However, developing effective anomaly detection methods for complex and high-dimensional data remains a challenge. As Generative Adversarial Networks (GANs) are able to model the complex high-dimensional distributions of real-world data, they offer a promising approach to address this challenge. In this work, we propose an anomaly detection method, Adversarially Learned Anomaly Detection (ALAD) based on bi-directional GANs, that derives adversarially learned features for the anomaly detection task. ALAD then uses reconstruction…
Airbus AI Research（空客人工智能研究院）；ISAE Supaero（法国高等航空航天学院）
Generative adversarial networks are a class of generative algorithms that have been widely used to produce state-of-the-art samples. In this paper, we investigate GAN to perform anomaly detection on time series dataset. In order to achieve this goal, a bibliography is made focusing on theoretical properties of GAN and GAN used for anomaly detection. A Wasserstein GAN has been chosen to learn the representation of normal data distribution and a stacked encoder with the generator performs the anomaly detection. W-GAN with encoder seems to produce state of the art anomaly detection scores on MNIS…
X-ray baggage security screening is in widespread use and crucial to maintaining transport security for threat/anomaly detection tasks. The automatic detection of anomaly, which is concealed within cluttered and complex electronics/electrical items, using 2D X-ray imagery is of primary interest in recent years. We address this task by introducing joint object sub-component level segmentation and classification strategy using deep Convolution Neural Network architecture. The performance is evaluated over a dataset of cluttered X-ray baggage security imagery, consisting of consumer electrical an…
Anomaly detection aims to identify observations that deviate from the typical pattern of data. Anomalous observations may correspond to financial fraud, health risks, or incorrectly measured data in practice. We show detecting anomalies in high-dimensional mixed data is enhanced through first embedding the data then assessing an anomaly scoring scheme. We focus on unsupervised detection and the continuous and categorical (mixed) variable case. We propose a kurtosis-weighted Factor Analysis of Mixed Data for anomaly detection, FAMDAD, to obtain a continuous embedding for anomaly scoring. We ill…
Reliable detection of anomalies is crucial when deploying machine learning models in practice, but remains challenging due to the lack of labeled data. To tackle this challenge, contrastive learning approaches are becoming increasingly popular, given the impressive results they have achieved in self-supervised representation learning settings. However, while most existing contrastive anomaly detection and segmentation approaches have been applied to images, none of them can use the contrastive losses directly for both anomaly detection and segmentation. In this paper, we close this gap by maki…
We introduce a new unsupervised anomaly detection ensemble called SPI which can harness privileged information - data available only for training examples but not for (future) test examples. Our ideas build on the Learning Using Privileged Information (LUPI) paradigm pioneered by Vapnik et al. [19,17], which we extend to unsupervised learning and in particular to anomaly detection. SPI (for Spotting anomalies with Privileged Information) constructs a number of frames/fragments of knowledge (i.e., density estimates) in the privileged space and transfers them to the anomaly scoring space through…
System states that are anomalous from the perspective of a domain expert occur frequently in some anomaly detection problems. The performance of commonly used unsupervised anomaly detection methods may suffer in that setting, because they use frequency as a proxy for anomaly. We propose a novel concept for anomaly detection, called relative anomaly detection. It is tailored to be robust towards anomalies that occur frequently, by taking into account their location relative to the most typical observations. The approaches we develop are computationally feasible even for large data sets, and the…
Fine-grained anomaly detection has recently been dominated by segmentation based approaches. These approaches first classify each element of the sample (e.g., image patch) as normal or anomalous and then classify the entire sample as anomalous if it contains anomalous elements. However, such approaches do not extend to scenarios where the anomalies are expressed by an unusual combination of normal elements. In this paper, we overcome this limitation by proposing set features that model each sample by the distribution its elements. We compute the anomaly score of each sample using a simple dens…
The utilization of prior knowledge about anomalies is an essential issue for anomaly detections. Recently, the visual attention mechanism has become a promising way to improve the performance of CNNs for some computer vision tasks. In this paper, we propose a novel model called Layer-wise External Attention Network (LEA-Net) for efficient image anomaly detection. The core idea relies on the integration of unsupervised and supervised anomaly detectors via the visual attention mechanism. Our strategy is as follows: (i) Prior knowledge about anomalies is represented as the anomaly map generated b…
Shanghai AI Laboratory（上海人工智能实验室）；Huawei Cloud & AI（华为云与人工智能）
This paper explores semi-supervised anomaly detection, a more practical setting for anomaly detection where a small additional set of labeled samples are provided. We propose a new KL-divergence based objective function for semi-supervised anomaly detection, and show that two factors: the mutual information between the data and latent representations, and the entropy of latent representations, constitute an integral objective function for anomaly detection. To resolve the contradiction in simultaneously optimizing the two factors, we propose a novel encoder-decoder-encoder structure, with the …
MindLab（思维实验室）；Universidad Nacional de Colombia（哥伦比亚国立大学）
Streaming anomaly detection refers to the problem of detecting anomalous data samples in streams of data. This problem poses challenges that classical and deep anomaly detection methods are not designed to cope with, such as conceptual drift and continuous learning. State-of-the-art flow anomaly detection methods rely on fixed memory using hash functions or nearest neighbors that may not be able to constrain high-frequency values as in a moving average or remove seamless outliers and cannot be trained in an end-to-end deep learning architecture. We present a new incremental anomaly detection m…
Anomalies in images occur in various scales from a small hole on a carpet to a large stain. However, anomaly detection based on sparse coding, one of the widely used anomaly detection methods, has an issue in dealing with anomalies that are out of the patch size employed to sparsely represent images. A large anomaly can be considered normal if seen in a small scale, but it is not easy to determine a single scale (patch size) that works well for all images. Then, we propose to incorporate multi-scale features to sparse coding and improve the performance of anomaly detection. The proposed method…
Self-supervised learning allows for better utilization of unlabelled data. The feature representation obtained by self-supervision can be used in downstream tasks such as classification, object detection, segmentation, and anomaly detection. While classification, object detection, and segmentation have been investigated with self-supervised learning, anomaly detection needs more attention. We consider the problem of anomaly detection in images and videos, and present a new visual anomaly detection technique for videos. Numerous seminal and state-of-the-art self-supervised methods are evaluated…
CentraleSupélec（中央理工-高等电力学院）；Institute for Infocomm Research（资讯通信研究所）；Télécom ParisTech（巴黎高等电信学院）；Carnegie Mellon University（卡内基梅隆大学）；Nanyang Technological University（南洋理工大学）
Generative adversarial networks (GANs) are able to model the complex highdimensional distributions of real-world data, which suggests they could be effective for anomaly detection. However, few works have explored the use of GANs for the anomaly detection task. We leverage recently developed GAN models for anomaly detection, and achieve state-of-the-art performance on image and network intrusion datasets, while being several hundred-fold faster at test time than the only published GAN-based method.
Anomaly detection in time series has been widely researched and has important practical applications. In recent years, anomaly detection algorithms are mostly based on deep-learning generative models and use the reconstruction error to detect anomalies. They try to capture the distribution of normal data by reconstructing normal data in the training phase, then calculate the reconstruction error of test data to do anomaly detection. However, most of them only use the normal data in the training phase and can not ensure the reconstruction process of anomaly data. So, anomaly data can also be we…
Usage-based insurance is becoming the new standard in vehicle insurance; it is therefore relevant to find efficient ways of using insureds' driving data. Applying anomaly detection to vehicles' trip summaries, we develop a method allowing to derive a "routine" and a "peculiarity" anomaly profile for each vehicle. To this end, anomaly detection algorithms are used to compute a routine and a peculiarity anomaly score for each trip a vehicle makes. The former measures the anomaly degree of the trip compared to the other trips made by the concerned vehicle, while the latter measures its anomaly de…
Anomaly detection is essential for preventing hazardous outcomes for safety-critical applications like autonomous driving. Given their safety-criticality, these applications benefit from provable bounds on various errors in anomaly detection. To achieve this goal in the semi-supervised setting, we propose to provide Probably Approximately Correct (PAC) guarantees on the false negative and false positive detection rates for anomaly detection algorithms. Our method (PAC-Wrap) can wrap around virtually any existing semi-supervised and unsupervised anomaly detection method, endowing it with rigoro…
Constructing an ensemble from a heterogeneous set of unsupervised anomaly detection methods is challenging because the class labels or the ground truth is unknown. Thus, traditional ensemble techniques that use the response variable or the class labels cannot be used to construct an ensemble for unsupervised anomaly detection. We use Item Response Theory (IRT) -- a class of models used in educational psychometrics to assess student and test question characteristics -- to construct an unsupervised anomaly detection ensemble. IRT's latent trait computation lends itself to anomaly detection becau…
African Institute of Mathematical Sciences（非洲数学科学研究所）；South African Radio Astronomical Observatory（南非射电天文台）
Statistical uncertainties are rarely incorporated in machine learning algorithms, especially for anomaly detection. Here we present the Bayesian Anomaly Detection And Classification (BADAC) formalism, which provides a unified statistical approach to classification and anomaly detection within a hierarchical Bayesian framework. BADAC deals with uncertainties by marginalising over the unknown, true, value of the data. Using simulated data with Gaussian noise, BADAC is shown to be superior to standard algorithms in both classification and anomaly detection performance in the presence of uncertain…
Anomaly detection aims to find instances that are considered unusual and is a fundamental problem of data science. Recently, deep anomaly detection methods were shown to achieve superior results particularly in complex data such as images. Our work focuses on deep one-class classification for anomaly detection which learns a mapping only from the normal samples. However, the non-linear transformation performed by deep learning can potentially find patterns associated with social bias. The challenge with adding fairness to deep anomaly detection is to ensure both making fair and correct anomaly…
We present five methods to the problem of network anomaly detection. These methods cover most of the common techniques in the anomaly detection field, including Statistical Hypothesis Tests (SHT), Support Vector Machines (SVM) and clustering analysis. We evaluate all methods in a simulated network that consists of nominal data, three flow-level anomalies and one packet-level attack. Through analyzing the results, we point out the advantages and disadvantages of each method and conclude that combining the results of the individual methods can yield improved anomaly detection results.
This paper presents a novel framework for unsupervised anomaly detection on masked objects called ODDObjects, which stands for Out-of-Distribution Detection on Objects. ODDObjects is designed to detect anomalies of various categories using unsupervised autoencoders trained on COCO-style datasets. The method utilizes autoencoder-based image reconstruction, where high reconstruction error indicates the possibility of an anomaly. The framework extends previous work on anomaly detection with autoencoders, comparing state-of-the-art models trained on object recognition datasets. Various model archi…
In this paper, we address the problem of image anomaly detection and segmentation. Anomaly detection involves making a binary decision as to whether an input image contains an anomaly, and anomaly segmentation aims to locate the anomaly on the pixel level. Support vector data description (SVDD) is a long-standing algorithm used for an anomaly detection, and we extend its deep learning variant to the patch-based method using self-supervised learning. This extension enables anomaly segmentation and improves detection performance. As a result, anomaly detection and segmentation performances measu…
Anomalies represent deviations from the intended system operation and can lead to decreased efficiency as well as partial or complete system failure. As the causes of anomalies are often unknown due to complex system dynamics, efficient anomaly detection is necessary. Conventional detection approaches rely on statistical and time-invariant methods that fail to address the complex and dynamic nature of anomalies. With advances in artificial intelligence and increasing importance for anomaly detection and prevention in various domains, artificial neural network approaches enable the detection of…
Anomaly detection is the practice of identifying items or events that do not conform to an expected behavior or do not correlate with other items in a dataset. It has previously been applied to areas such as intrusion detection, system health monitoring, and fraud detection in credit card transactions. In this paper, we describe a new method for detecting anomalous behavior over network performance data, gathered by perfSONAR, using two machine learning algorithms: Boosted Decision Trees (BDT) and Simple Feedforward Neural Network. The effectiveness of each algorithm was evaluated and compared…
Utah State University（犹他州立大学）；University of Arkansas（阿肯色大学）；Clemson University（克莱姆森大学）
Ensuring fairness in anomaly detection models has received much attention recently as many anomaly detection applications involve human beings. However, existing fair anomaly detection approaches mainly focus on association-based fairness notions. In this work, we target counterfactual fairness, which is a prevalent causation-based fairness notion. The goal of counterfactually fair anomaly detection is to ensure that the detection outcome of an individual in the factual world is the same as that in the counterfactual world where the individual had belonged to a different group. To this end, we…
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Information Technology University（信息技术大学）
In modern intelligent video surveillance systems, automatic anomaly detection through computer vision analytics plays a pivotal role which not only significantly increases monitoring efficiency but also reduces the burden on live monitoring. Anomalies in videos are broadly defined as events or activities that are unusual and signify irregular behavior. The goal of anomaly detection is to temporally or spatially localize the anomaly events in video sequences. Temporal localization (i.e. indicating the start and end frames of the anomaly event in a video) is referred to as frame-level detection.…
Nanjing University of Science and Technology（南京理工大学）
Video anomaly detection (VAD) is currently a challenging task due to the complexity of anomaly as well as the lack of labor-intensive temporal annotations. In this paper, we propose an end-to-end Global Information Guided (GIG) anomaly detection framework for anomaly detection using the video-level annotations (i.e., weak labels). We propose to first mine the global pattern cues by leveraging the weak labels in a GIG module. Then we build a spatial reasoning module to measure the relevance between vectors in spatial domain with the global cue vectors, and select the most related feature vector…
DAS3D: Dual-modality Anomaly Synthesis for 3D Anomaly Detection
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Microsoft Research Asia（微软亚洲研究院）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Tsinghua Shenzhen International Graduate School（清华大学深圳国际研究生院）
Jiangxi University of Finance and Economics（江西财经大学）
Anomaly detection has attracted considerable search attention. However, existing anomaly detection databases encounter two major problems. Firstly, they are limited in scale. Secondly, training sets contain only video-level labels indicating the existence of an abnormal event during the full video while lacking annotations of precise time durations. To tackle these problems, we contribute a new Large-scale Anomaly Detection (LAD) database as the benchmark for anomaly detection in video sequences, which is featured in two aspects. 1) It contains 2000 video sequences including normal and abnorma…
Jiangxi University of Finance and Economics（江西财经大学）
Anomaly detection in surveillance videos is a challenging task due to the diversity of anomalous video content and duration. In this paper, we consider video anomaly detection as a regression problem with respect to anomaly scores of video clips under weak supervision. Hence, we propose an anomaly detection framework, called Anomaly Regression Net (AR-Net), which only requires video-level labels in training stage. Further, to learn discriminative features for anomaly detection, we design a dynamic multiple-instance learning loss and a center loss for the proposed AR-Net. The former is used to …
University of Ljubljana（卢布尔雅那大学）；Faculty of Computer Information Science（计算机信息科学学院）
Visual surface anomaly detection aims to detect local image regions that significantly deviate from normal appearance. Recent surface anomaly detection methods rely on generative models to accurately reconstruct the normal areas and to fail on anomalies. These methods are trained only on anomaly-free images, and often require hand-crafted post-processing steps to localize the anomalies, which prohibits optimizing the feature extraction for maximal detection capability. In addition to reconstructive approach, we cast surface anomaly detection primarily as a discriminative problem and propose a …
