{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T17:56:32Z","timestamp":1754157392491,"version":"3.41.2"},"reference-count":19,"publisher":"Emerald","issue":"3","license":[{"start":{"date-parts":[[2011,10,20]],"date-time":"2011-10-20T00:00:00Z","timestamp":1319068800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2011,10,20]]},"abstract":"<jats:sec><jats:title content-type=\"abstract-heading\">Purpose<\/jats:title><jats:p>The purpose of this paper is to propose an uncertain regression model with an intrinsic error structure facilitated by an uncertain canonical process.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Design\/methodology\/approach<\/jats:title><jats:p>This model is suitable for dealing with expert's knowledge ranging from small to medium size data of impreciseness. In order to have a rigorous mathematical treatment on the new regression model, this paper establishes a series of new uncertainty concepts sequentially, such as uncertainty joint multivariate distribution, the uncertainty distribution of uncertainty product variables and uncertain covariance and correlation based on the axiomatic uncertainty theoretical foundation. Two examples are given for illustrating a small data regression analysis.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Findings<\/jats:title><jats:p>The uncertain regression model is formulated and the estimation of the model coefficients is developed.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Practical implications<\/jats:title><jats:p>The paper is devoted to a regression model to handle a small amount of data with mathematical rigor.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Originality\/value<\/jats:title><jats:p>The theory and the methodology of the uncertain canonical process regression is proposed for the first time. It addresses the practical challenges of small data size modelling.<\/jats:p><\/jats:sec>","DOI":"10.1108\/20439371111181215","type":"journal-article","created":{"date-parts":[[2011,11,12]],"date-time":"2011-11-12T07:02:39Z","timestamp":1321081359000},"page":"202-215","source":"Crossref","is-referenced-by-count":1,"title":["An uncertain regression model"],"prefix":"10.1108","volume":"1","author":[{"given":"Renkuan","family":"Guo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Danni","family":"Guo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YanHong","family":"Cui","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"140","reference":[{"key":"key2022031120321931500_b1","doi-asserted-by":"crossref","unstructured":"Billard, L. and Diday, E. (2000), \u201cRegression analysis for interval\u2010valued data\u201d, Data Analysis: Classification and Related Methods, paper presented at the 7th Conference of the International Federation of Classification Societies, Springer, Berlin, pp. 369\u201074.","DOI":"10.1007\/978-3-642-59789-3_58"},{"key":"key2022031120321931500_b2","unstructured":"Deng, J.L. (1984), \u201cGrey dynamic modeling and its application in long\u2010term prediction of food productions\u201d, Exploration of Nature, Vol. 3 No. 3, pp. 7\u201043."},{"key":"key2022031120321931500_b3","unstructured":"Draper, N. and Smith, H. (1966), Applied Regression Analysis, Wiley, New York, NY."},{"key":"key2022031120321931500_b4","doi-asserted-by":"crossref","unstructured":"D'Urso, P. and Gastaldi, T. (2000), \u201cA least\u2010squares approach to fuzzy linear regression analysis\u201d, Computational Statistics and Data Analysis, Vol. 34, pp. 427\u201040.","DOI":"10.1016\/S0167-9473(99)00109-7"},{"key":"key2022031120321931500_b5","unstructured":"Environmental Protection Agency (2010), \u201cParticulate matter\u201d, available at: www.epa.gov\/ttn\/naaqs\/pm\/pm10_index.html."},{"key":"key2022031120321931500_b7","doi-asserted-by":"crossref","unstructured":"Guo, R. and Guo, D. (2009), \u201cRandom fuzzy variable foundation for grey differential equation modeling\u201d, Soft Computing, Vol. 13 No. 2, pp. 185\u2010201.","DOI":"10.1007\/s00500-008-0301-4"},{"key":"key2022031120321931500_b9","unstructured":"Guo, R., Cui, Y.H. and Guo, D. (2011), \u201cUncertainty linear regression models\u201d, Journal of Uncertainty Systems, Vol. 4 No. 4."},{"key":"key2022031120321931500_b8","unstructured":"Guo, R., Guo, D., Dunne, T. and Thiart, C. (2009), \u201cDEAR model \u2013 the theoretical foundation\u201d, Journal of Uncertain Systems, Vol. 3 No. 1, pp. 36\u201051."},{"key":"key2022031120321931500_b10","doi-asserted-by":"crossref","unstructured":"Hojati, M., Bector, C.R. and Smimou, K. (2005), \u201cA simple method for computation of fuzzy linear regression\u201d, European Journal of Operation Research, No. 166, pp. 172\u201084.","DOI":"10.1016\/j.ejor.2004.01.039"},{"key":"key2022031120321931500_b11","unstructured":"Liu, B.D. (2007), Uncertainty Theory: An Introduction to Its Axiomatic Foundations, 2nd ed., Springer, Berlin."},{"key":"key2022031120321931500_b12","unstructured":"Liu, B.D. (2010), Uncertainty Theory: A Branch of Mathematics of Modelling Human Uncertainty, Springer, Berlin."},{"key":"key2022031120321931500_b13","unstructured":"Liu, S.F. and Lin, Y. (2006), Grey Information, Springer, London."},{"key":"key2022031120321931500_b14","unstructured":"Matheron, G. (1975), Random Sets and Integral Geometry, Wiley, New York, NY."},{"key":"key2022031120321931500_b15","unstructured":"Myers, R.H. (2000), Classical and Modern Regression with Applications, 2nd ed., Duxbury Press, Pacific Grove, CA."},{"key":"key2022031120321931500_b16","doi-asserted-by":"crossref","unstructured":"Rao, C.R. (1973), Linear Statistical Inference and Its Applications, Wiley, New York, NY.","DOI":"10.1002\/9780470316436"},{"key":"key2022031120321931500_b17","unstructured":"Searle, S.R. (1982), Matrix Algebra Useful for Statistics, Wiley, New York, NY."},{"key":"key2022031120321931500_b18","doi-asserted-by":"crossref","unstructured":"Tanaka, H. and Watada, J. (1988), \u201cPossibilistic linear systems and their application to the linear regression model\u201d, Fuzzy Sets and Systems, Vol. 27, pp. 275\u201089.","DOI":"10.1016\/0165-0114(88)90054-1"},{"key":"key2022031120321931500_b19","doi-asserted-by":"crossref","unstructured":"Zadeh, L.A. (1978), \u201cFuzzy sets as a basis for a theory of possibility\u201d, Fuzzy Sets and Systems, Vol. 1, pp. 3\u201028.","DOI":"10.1016\/0165-0114(78)90029-5"},{"key":"key2022031120321931500_frd1","doi-asserted-by":"crossref","unstructured":"Gil, M., L\u00f3pez\u2010Garc\u00eda, M., Lubiano, M. and Montenegro, M. (2001), \u201cRegression and correlation analysis of a linear relation between random intervals\u201d, Test, Vol. 10 No. 1, pp. 183\u2010201.","DOI":"10.1007\/BF02595831"}],"container-title":["Grey Systems: Theory and Application"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.emeraldinsight.com\/doi\/full-xml\/10.1108\/20439371111181215","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/20439371111181215\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/20439371111181215\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T23:52:34Z","timestamp":1753401154000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/gs\/article\/1\/3\/202-215\/91812"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,20]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,10,20]]}},"alternative-id":["10.1108\/20439371111181215"],"URL":"https:\/\/doi.org\/10.1108\/20439371111181215","relation":{},"ISSN":["2043-9377"],"issn-type":[{"type":"print","value":"2043-9377"}],"subject":[],"published":{"date-parts":[[2011,10,20]]}}}