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local outlier factor wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 basic idea 2 formal definition 3 advantages 4 disadvantages and extensions 5 references toggle the table of contents local outlier factor 12 languages català deutsch español فارسی bahasa indonesia 日本語 한국어 русский srpskohrvatski српскохрватски српски srpski українська tiếng việt edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia algorithm for anomaly detection part of a series on machine learning and data mining paradigms supervised learning unsupervised learning semi supervised learning self supervised learning reinforcement learning transfer learning meta learning few shot learning zero shot learning online learning batch learning curriculum learning rule based learning neuro symbolic ai neuromorphic engineering quantum machine learning problems classification generative modeling regression clustering dimensionality reduction density estimation anomaly detection data cleaning automl association rules semantic analysis structured prediction feature engineering feature learning learning to rank grammar induction ontology learning multimodal learning supervised learning classification regression apprenticeship learning decision trees ensembles bagging boosting random forest k nn linear regression naive bayes artificial neural networks logistic regression perceptron relevance vector machine rvm support vector machine svm clustering birch cure hierarchical k means fuzzy expectation maximization em dbscan optics mean shift dimensionality reduction factor analysis exploratory cca ica lda nmf pca pgd t sne sdl structured prediction graphical models bayes net conditional random field hidden markov anomaly detection ransac k nn local outlier factor isolation forest neural networks autoencoder deep learning feedforward neural network recurrent neural network lstm gru esn reservoir computing boltzmann machine restricted gan diffusion model som convolutional neural network u net lenet alexnet deepdream neural field neural radiance field physics informed neural networks transformer vision mamba spiking neural network memtransistor electrochemical ram ecram reinforcement learning q learning policy gradient sarsa temporal difference td multi agent self play learning with humans active learning crowdsourcing human in the loop mechanistic interpretability rlhf model diagnostics coefficient of determination confusion matrix learning curve roc curve mathematical foundations kernel machines bias variance tradeoff computational learning theory empirical risk minimization occam learning pac learning statistical learning vc theory topological deep learning journals and conferences aaai cvpr eccv ecml pkdd emnlp iccv neurips icml iclr ijcai ml jmlr related articles glossary of artificial intelligence list of datasets for machine learning research list of datasets in computer vision and image processing outline of machine learning v t e in anomaly detection the local outlier factor lof is an algorithm proposed by markus m breunig hans peter kriegel raymond t ng and jörg sander in 2000 for finding anomalous data points by measuring the local deviation of a given data point with respect to its neighbours 1 lof shares some concepts with dbscan and optics such as the concepts of core distance and reachability distance which are used for local density estimation 2 basic idea edit basic idea of lof comparing the local density of a point with the densities of its neighbors a has a much lower density than its neighbors the local outlier factor is based on a concept of a local density where locality is given by k nearest neighbors whose distance is used to estimate the density by comparing the local density of an object to the local densities of its neighbors one can identify regions of similar density and points that have a substantially lower density than their neighbors these are considered to be outliers the local density is estimated by the typical distance at which a point can be reached from its neighbors the definition of reachability distance used in lof is an additional measure to produce more stable results within clusters the reachability distance used by lof has some subtle details that are often found incorrect in secondary sources e g in the textbook of ethem alpaydin 3 formal definition edit let k distance a displaystyle k text distance a be the distance of the object a to the k th nearest neighbor note that the set of the k nearest neighbors includes all objects at this distance which can in the case of a tie be more than k objects we denote the set of k nearest neighbors as n k a displaystyle n_ k a illustration of the reachability distance objects b and c have the same reachability distance k 3 while d is not a k nearest neighbor this distance is used to define what is called reachability distance reachability distance k a b max k distance b d a b displaystyle text reachability distance _ k a b max k text distance b d a b in words the reachability distance of an object a from b is the true distance between the two objects but at least the k distance displaystyle k text distance of b objects that belong to the k nearest neighbors of b the core of b see dbscan cluster analysis are considered to be equally distant the reason for this is to reduce the statistical fluctuations between all points a close to b where increasing the value for k increases the smoothing effect 1 note that this is not a distance in the mathematical definition since it is not symmetric while it is a common mistake 4 to always use the k distance a displaystyle k text distance a this yields a slightly different method referred to as simplified lof 4 the local reachability density of an object a is defined by lrd k a n k a b n k a reachability distance k a b displaystyle text lrd _ k a frac n_ k a sum _ b in n_ k a text reachability distance _ k a b which is the inverse of the average reachability distance of the object a from its neighbors note that it is not the average reachability of the neighbors from a which by definition would be the k distance a displaystyle k text distance a but the distance at which a can be reached from its neighbors with duplicate points this value can become infinite the local reachability densities are then compared with those of the neighbors using lof k a 1 n k a b n k a lrd k b lrd k a 1 n k a lrd k a b n k a lrd k b displaystyle text lof _ k a frac 1 n_ k a sum _ b in n_ k a frac text lrd _ k b text lrd _ k a frac 1 n_ k a cdot text lrd _ k a sum _ b in n_ k a text lrd _ k b which is the average local reachability density of the neighbors divided by the object s own local reachability density a value of approximately 1 indicates that the object is comparable to its neighbors and thus not an outlier a value below 1 indicates a denser region which would be an inlier while values significantly larger than 1 indicate outliers lof k a 1 displaystyle text lof _ k a sim 1 means similar density as neighbors lof k a 1 displaystyle text lof _ k a 1 means higher density than neighbors inlier lof k a 1 displaystyle text lof _ k a 1 means lower density than neighbors outlier advantages edit lof scores as visualized by elki while the upper right cluster has a comparable density to the outliers close to the bottom left cluster they are detected correctly due to the local approach lof is able to identify outliers in a data set that would not be outliers in another area of the data set for example a point at a small distance to a very dense cluster is an outlier while a point within a sparse cluster might exhibit similar distances to its neighbors while the geometric intuition of lof is only applicable to low dimensional vector spaces the algorithm can be applied in any context a dissimilarity function can be defined it has experimentally been shown to work very well in numerous setups often outperforming the competitors for example in network intrusion detection 5 and on processed classification benchmark data 6 the lof family of methods can be easily generalized and then applied to various other problems such as detecting outliers in geographic data video streams or authorship networks 4 disadvantages and extensions edit the resulting values are quotient values and hard to interpret a value of 1 or even less indicates a clear inlier but there is no clear rule for when a point is an outlier in one data set a value of 1 1 may already be an outlier in another dataset and parameterization with strong local fluctuations a value of 2 could still be an inlier these differences can also occur within a dataset due to the locality of the method there exist extensions of lof that try to improve over lof in these aspects feature bagging for outlier detection 7 runs lof on multiple projections and combines the results for improved detection qualities in high dimensions this is the first ensemble learning approach to outlier detection for other variants see ref 8 local outlier probability loop 9 is a method derived from lof but using inexpensive local statistics to become less sensitive to the choice of the parameter k in addition the resulting values are scaled to a value range of 0 1 interpreting and unifying outlier scores 10 proposes a normalization of the lof outlier scores to the interval 0 1 using statistical scaling to increase usability and can be seen an improved version of the loop ideas on evaluation of outlier rankings and outlier scores 11 proposes methods for measuring similarity and diversity of methods for building advanced outlier detection ensembles using lof variants and other algorithms and improving on the feature bagging approach discussed above local outlier detection reconsidered a generalized view on locality with applications to spatial video and network outlier detection 4 discusses the general pattern in various local outlier detection methods including e g lof a simplified version of lof and loop and abstracts from this into a general framework this framework is then applied e g to detecting outliers in geographic data video streams and authorship networks references edit 1 2 breunig m m kriegel h p ng r t sander j 2000 lof identifying density based local outliers pdf proceedings of the 2000 acm sigmod international conference on management of data sigmod pp 93 104 doi 10 1145 335191 335388 isbn 1 58113 217 4 breunig m m kriegel h p ng r t sander j r 1999 optics of identifying local outliers pdf principles of data mining and knowledge discovery lecture notes in computer science vol 1704 pp 262 270 doi 10 1007 978 3 540 48247 5_28 isbn 978 3 540 66490 1 alpaydin ethem 2020 introduction to machine learning fourth ed cambridge massachusetts isbn 978 0 262 04379 3 oclc 1108782604 cite book cs1 maint location missing publisher link 1 2 3 4 schubert e zimek a kriegel h p 2012 local outlier detection reconsidered a generalized view on locality with applications to spatial video and network outlier detection data mining and knowledge discovery 28 190 237 doi 10 1007 s10618 012 0300 z s2cid 19036098 lazarevic a ozgur a ertoz l srivastava j kumar v 2003 a comparative study of anomaly detection schemes in network intrusion detection pdf proceedings of the 2003 siam international conference on data mining pp 25 36 doi 10 1137 1 9781611972733 3 isbn 978 0 89871 545 3 archived from the original pdf on 2013 07 17 retrieved 2010 05 14 campos guilherme o zimek arthur sander jörg campello ricardo j g b micenková barbora schubert erich assent ira houle michael e 2016 on the evaluation of unsupervised outlier detection measures datasets and an empirical study data mining and knowledge discovery 30 4 891 927 doi 10 1007 s10618 015 0444 8 issn 1384 5810 s2cid 1952214 lazarevic a kumar v 2005 feature bagging for outlier detection proceedings of the eleventh acm sigkdd international conference on knowledge discovery in data mining pp 157 166 doi 10 1145 1081870 1081891 isbn 1 59593 135 x s2cid 2054204 zimek a campello r j g b sander j r 2014 ensembles for unsupervised outlier detection acm sigkdd explorations newsletter 15 11 22 doi 10 1145 2594473 2594476 s2cid 8065347 kriegel h p kröger p schubert e zimek a 2009 loop local outlier probabilities proceedings of the 18th acm conference on information and knowledge management pdf cikm 09 pp 1649 1652 doi 10 1145 1645953 1646195 isbn 978 1 60558 512 3 kriegel h p kröger p schubert e zimek a 2011 interpreting and unifying outlier scores proceedings of the 2011 siam international conference on data mining pp 13 24 doi 10 1137 1 9781611972818 2 isbn 978 0 89871 992 5 schubert e wojdanowski r zimek a kriegel h p 2012 on evaluation of outlier rankings and outlier scores proceedings of the 2012 siam international conference on data mining pp 1047 1058 doi 10 1137 1 9781611972825 90 isbn 978 1 61197 232 0 retrieved from https en wikipedia org w index php title local_outlier_factor oldid 1374372579 categories statistical outliers data mining machine learning algorithms hidden categories articles with short description short description is different from wikidata cs1 maint location missing publisher this page was last edited on 11 september 2026 at 16 57 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 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