Please note this is a comparison between Version 1 by Szymon Łukaszyk and Version 2 by Jason Zhu.

The **Łukaszyk–Karmowski metric** (LK-metric) defines a distance between two random variables or vectors. LK-metric is not a metric as it does not satisfy the identity of indiscernibles axiom of the metric; for the same random variables, its value is greater than zero, providing they are not both degenerated.

- distance functions
- identity of indiscernibles
- ugly duckling theorem

LK-metric^{[1]} between two continuous random variables X and Y having a joint probability density function (PDF) F(x,y) is defined as

If X and Y are independent, then

where f(x) and g(y) are PDFs of X and Y, and subscripts denote their types. For example, if X and Y have normal PDFs having the same standard deviation σ but different means μ_{x}, μ_{y}, then

LK-metric between two random variables having normal PDFs and the same

standard deviations σ = {0, 0.2, 0.4, 0.6, 0.8, 1}.

where μ_{xy}=|μ_{x}-μ_{y}|. For discrete X and Y, LK-metric has a form

and for random vectors **X** and **Y**, LK-metric becomes

where d(**x**,**y**) is a metric function, such as the Euclidean metric. In case, **X** and **Y **are mutually and internally independent, a simplified form of LK-metric can also be defined as

If X and Y are degenerated, almost sure variables having the Dirac delta (or one-point, in the discrete case) PDFs, then LK-metric becomes the metric between their mean values.

and obviously

However, in any other case

LK-metric satisfies all the remaining axioms of the metric. It is symmetric by definition, and it satisfies the triangle inequality

Thus

since

LK-metric is not the only distance function that does not satisfy the identity of indiscernibles axiom^{[2]}. For example, the partial metric^{[3]} also allows each object not necessarily to have zero distance from itself. However, the partial metric satisfies two additional axioms of small self-distances and modified triangle inequality, which are not satisfied by LK-metric^{[4]}. Remarkably, the identity of indiscernibles ontological axiom, introduced to philosophy by Gottfried Wilhelm Leibniz around 1686, is also invalidated by the ugly duckling theorem^{[5]} stated in 1969 and asserting that every two objects one perceives are equally similar (or equally dissimilar). Consequently, the identity of indiscernibles is neither a logical nor empirical principle.

This characteristic non-zero distance effect built in LK-metric allows to avoid ill-conditioning problems in radial basis function interpolation^{[6][7]} and inverse distance weighting^{[8][9][10][11]}, where the interpolation accuracy can be improved by choosing the type of distance metric^{[12][11]} and leads to a smooth interpolation function^{[13]}. By preventing zero distances based on parameter uncertainty, LK-metric can, furthermore, be used in analysis of nondeterministic dynamical systems with competing attractors^{[14]}. Since LK-metric represents the mean of distances between all the outcomes of the two uncertain objects, it can also be used in uncertain nearest neighbor classification^{[15]}. The actual value of an uncertain object is modeled by a probability density function^{[16]}. LK-metric has been successfully applied in various fields of science and technology^{[17][18][19][20][21][22][23][24][13][25][26][27][28][29][30][7][31][32][33][34][35][10][36][37][38][39][40][41][42][43][44][45][11][46][47][48][49][14][50][51]}.

- S. Lukaszyk; A new concept of probability metric and its applications in approximation of scattered data sets.
*Comput. Mech.*.**2004**,*33*, 299-304. - Andrzej Tomski; Szymon Łukaszyk; Reply to "Various issues around the L1-norm distance".
*Ipi Lett.*.**2024**,*2*, 1-8. - S. G. Matthews; Partial Metric Topology.
*Ann. New York Acad. Sci.*.**1994**,*728*, 183-197. - Castro, Pablo Samuel and Kastner, Tyler and Panangaden, Prakash and Rowland, Mark. A Kernel Perspective on Behavioural Metrics for Markov Decision Processes.
*Transactions on Machine Learning Research*.**2022**,*https://openreview.net/forum?id=nHfPXl1ly7*, 2835-8856. - Satosi Watanabe; Epistemological Relativity.
*Ann. Jpn. Assoc. Philos. Sci.*.**1986**,*7*, 1-14. - You-Long Zou; Fa-Long Hu; Can-Can Zhou; Chao-Liu Li; Keh-Jim Dunn; Analysis of radial basis function interpolation approach.
*Appl. Geophys.*.**2013**,*10*, 397-410. - Abbas M. Abd; Suhad M. Abd; Modelling the strength of lightweight foamed concrete using support vector machine (SVM).
*Case Stud. Constr. Mater.*.**2017**,*6*, 8-15. - M. Gentile; F. Courbin; G. Meylan; Interpolating point spread function anisotropy.
*Astron. Astrophys.*.**2012**,*549*, A1. - İ. Bedii Özdemir; A modification to temperature-composition pdf method and its application to the simulation of a transitional bluff-body flame.
*Comput. Math. Appl.*.**2018**,*75*, 2574-2592. - Christos Anagnostopoulos; Edge-centric inferential modeling & analytics.
*J. Netw. Comput. Appl.*.**2020**,*164*, 102696. - Zhanglin Li; An enhanced dual IDW method for high-quality geospatial interpolation.
*Sci. Rep.*.**2021**,*11*, 1-17. - You, Hojun and Kim, Dongsu. Development of an anisotropic spatial interpolation method for velocity in meandering river channel.
*J. Korea Water Resour. Assoc.*.**2017**,*50*, 1023. - Petar Durdevic; Leif Hansen; Christian Mai; Simon Pedersen; Zhenyu Yang; Cost-Effective ERT Technique for Oil-in-Water Measurement for Offshore Hydrocyclone Installations.
*IFAC-PapersOnLine*.**2015**,*48*, 147-153. - Kaio C. B. Benedetti; Paulo B. B. Goncalves; Stefano Lenci; Giuseppe Rega; Global analysis of stochastic and parametric uncertainty in nonlinear dynamical systems: adaptative phase-space discretization strategy, with application to Helmholtz oscillator.
*Nonlinear Dyn.*.**2023**,*111*, 15675-15703. - Fabrizio Angiulli; Fabio Fassetti; Nearest Neighbor-Based Classification of Uncertain Data.
*ACM Trans. Knowl. Discov. Data*.**2013**,*7*, 1-35. - Angiulli, Fabrizio and Fassetti, Fabio. Indexing Uncertain Data in General Metric Spaces.
*IEEE Transactions on Knowledge and Data Engineering*.**2012**,*24*, 1640-1657. - James C. Davidson; Seth A. Hutchinson. A Sampling Hyperbelief Optimization Technique for Stochastic Systems; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2009; pp. 217-231.
- Gang Meng; Jane Law; Mary E Thompson; Small-scale health-related indicator acquisition using secondary data spatial interpolation.
*Int. J. Heal. Geogr.*.**2010**,*9*, 50-50. - Yingjie Xia; Xingmin Shi; Li Kuang; Junhua Xuan. Parallel geospatial analysis on windows HPC platform; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2010; pp. 210-213.
- V. Roshan Joseph; Lulu Kang; Regression-Based Inverse Distance Weighting With Applications to Computer Experiments.
*Technometrics*.**2011**,*53*, 254-265. - Fengxiang Xu; Guangyong Sun; Guangyao Li; Qing Li; Crashworthiness design of multi-component tailor-welded blank (TWB) structures.
*Struct. Multidiscip. Optim.*.**2013**,*48*, 653-667. - Chanyoung Jun; Subhrajit Bhattacharya; Robert Ghrist. Pursuit-evasion game for normal distributions; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, United States, 2014; pp. 83-88.
- Simon Pedersen; Christian Mai; Leif Hansen; Petar Durdevic; Zhenyu Yang; Online Slug Detection in Multi-phase Transportation Pipelines Using Electrical Tomography∗∗Supported by the Danish National Advanced Technology Foundation through PDPWAC Project (J.nr. 95-2012-3)..
*IFAC-PapersOnLine*.**2015**,*48*, 159-164. - Branislav Brutovsky; Denis Horvath; Towards inverse modeling of intratumor heterogeneity.
*Open Phys.*.**2015**,*13*, 103. - S. Mohammed Hosseini; Samira Smaeili; Numerical integration of multi-dimensional highly oscillatory integrals, based on eRPIM.
*Numer. Algorithms*.**2014**,*68*, 423-442. - Luciana Balsamo; Suparno Mukhopadhyay; Raimondo Betti; A statistical framework with stiffness proportional damage sensitive features for structural health monitoring.
*Smart Struct. Syst.*.**2015**,*15*, 699-715. - Bing Han; Xinbo Gao; Hui Liu; Ping Wang. Auroral Oval Boundary Modeling Based on Deep Learning Method; Springer Science and Business Media LLC: Dordrecht, GX, Netherlands, 2015; pp. 96-106.
- J. Park; V. Sreeja; M. Aquino; C. Cesaroni; L. Spogli; A. Dodson; G. De Franceschi; Performance of ionospheric maps in support of long baseline GNSS kinematic positioning at low latitudes.
*Radio Sci.*.**2016**,*51*, 429-442. - Grzegorz Lenda; Marcin Ligas; Paulina Lewińska; Anna Szafarczyk; The use of surface interpolation methods for landslides monitoring.
*KSCE J. Civ. Eng.*.**2015**,*20*, 188-196. - Jan Koloda; Jurgen Seiler; Andre Kaup; Frequency-Selective Mesh-to-Grid Resampling for Image Communication.
*IEEE Trans. Multimedia*.**2017**,*19*, 1689-1701. - Pijus Kasparaitis; Kipras Kančys; Phoneme vs. Diphone in Unit Selection TTS of Lithuanian.
*Balt. J. Mod. Comput.*.**2018**,*6*, 162-172. - Jorge Vicent; Jochem Verrelst; Juan Pablo Rivera-Caicedo; Neus Sabater; Jordi Munoz-Mari; Gustau Camps-Valls; Jose Moreno; Emulation as an Accurate Alternative to Interpolation in Sampling Radiative Transfer Codes.
*IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.*.**2018**,*11*, 4918-4931. - Matthew J. Lake; Marek Miller; Ray F. Ganardi; Zheng Liu; Shi-Dong Liang; Tomasz Paterek; Generalised uncertainty relations from superpositions of geometries.
*Class. Quantum Gravity*.**2019**,*36*, 155012. - Aimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; Investigation of thermal stratification in premixed homogeneous charge compression ignition engines: A Large Eddy Simulation study.
*Int. J. Engine Res.*.**2018**,*20*, 931-944. - D. J. Tan; D. Honnery; A. Kalyan; V. Gryazev; S. A. Karabasov; D. Edgington-Mitchell; Equivalent Shock-Associated Noise Source Reconstruction of Screeching Underexpanded Unheated Round Jets.
*AIAA J.*.**2019**,*57*, 1200-1214. - Xiaohui Chen; Qing Zhao; Fei Huang; Rongzu Qiu; Yuhong Lin; Lanyi Zhang; Xisheng Hu; Understanding spatial variation in the driving pattern of carbon dioxide emissions from taxi sector in great Eastern China: evidence from an analysis of geographically weighted regression.
*Clean Technol. Environ. Policy*.**2020**,*22*, 979-991. - Helin Gong; Yingrui Yu; Qing Li; Chaoyu Quan; An inverse-distance-based fitting term for 3D-Var data assimilation in nuclear core simulation.
*Ann. Nucl. Energy*.**2020**,*141*, 107346. - Liming Xu; Xianhua Zeng; He Zhang; Weisheng Li; Jianbo Lei; Zhiwei Huang; BPGAN: Bidirectional CT-to-MRI prediction using multi-generative multi-adversarial nets with spectral normalization and localization.
*Neural Networks*.**2020**,*128*, 82-96. - Zhanglin Li; Xialin Zhang; Rui Zhu; Zhiting Zhang; Zhengping Weng; Integrating data-to-data correlation into inverse distance weighting.
*Comput. Geosci.*.**2019**,*24*, 203-216. - Wei Wei; Zecheng Guo; Liang Zhou; Binbin Xie; Junju Zhou; Assessing environmental interference in northern China using a spatial distance model: From the perspective of geographic detection.
*Sci. Total. Environ.*.**2020**,*709*, 136170. - Yiyuan Han; Bing Han; Zejun Hu; Xinbo Gao; Lixia Zhang; Huigen Yang; Bin Li; Prediction and variation of the auroral oval boundary based on a deep learning model and space physical parameters.
*Nonlinear Process. Geophys.*.**2020**,*27*, 11-22. - Yuying Lin; Xisheng Hu; Mingshui Lin; Rongzu Qiu; Jinguo Lin; Baoyin Li; Spatial Paradigms in Road Networks and Their Delimitation of Urban Boundaries Based on KDE.
*ISPRS Int. J. Geo-Information*.**2020**,*9*, 204. - Aimilios Sofianopoulos; Mozhgan Rahimi Boldaji; Benjamin Lawler; Sotirios Mamalis; John E Dec; Effect of engine size, speed, and dilution method on thermal stratification of premixed homogeneous charge compression–ignition engines: A large eddy simulation study.
*Int. J. Engine Res.*.**2019**,*21*, 1612-1630. - Panagiotis Pergantas; Nikos E. Papanikolaou; Chrisovalantis Malesios; Andreas Tsatsaris; Marios Kondakis; Iokasti Perganta; Yiannis Tselentis; Nikos Demiris; Towards a Semi-Automatic Early Warning System for Vector-Borne Diseases.
*Int. J. Environ. Res. Public Heal.*.**2021**,*18*, 1823. - Mobeen Akhtar; Yuanyuan Zhao; Guanglei Gao; An analytical approach for assessment of geographical variation in ecosystem service intensity in Punjab, Pakistan.
*Environ. Sci. Pollut. Res.*.**2021**,*28*, 38145-38158. - Bogdan-Mihai Negrea; Valeriu Stoilov-Linu; Cristian-Emilian Pop; György Deák; Nicolae Crăciun; Marius Mirodon Făgăraș; Expansion of the Invasive Plant Species Reynoutria japonica Houtt in the Upper Bistrița Mountain River Basin with a Calculus on the Productive Potential of a Mountain Meadow.
*Sustain.*.**2022**,*14*, 5737. - Fabio Saggese; Vincenzo Lottici; Filippo Giannetti; Rainfall Map from Attenuation Data Fusion of Satellite Broadcast and Commercial Microwave Links.
*Sensors*.**2022**,*22*, 7019. - Zhiguang Han; Jianzhang Xiao; Yingqi Wei; Spatial Distribution Characteristics of Microbial Mineralization in Saturated Sand Centrifuge Shaking Table Test.
*Mater.*.**2022**,*15*, 6102. - Oludayo Ayodeji Akintunde; Chidi Vitalis Ozebo; Kayode Festus Oyedele; Groundwater Quality around upstream and downstream area of the Lagos lagoon using GIS and Multispectral Analysis.
*Sci. Afr.*.**2022**,*16*, e01126. - A. A. El-Atik; Y. Tashkandy; S. Jafari; A. A. Nasef; W. Emam; M. Badr; Mutation of DNA and RNA sequences through the application of topological spaces.
*AIMS Math.*.**2023**,*8*, 19275-19296. - David Hernández-López; Esteban Ruíz de Oña; Miguel A. Moreno; Diego González-Aguilera; SunMap: Towards Unattended Maintenance of Photovoltaic Plants Using Drone Photogrammetry.
*Drones*.**2023**,*7*, 129.

More