{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T08:44:05Z","timestamp":1763369045737,"version":"3.45.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T00:00:00Z","timestamp":1738627200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T00:00:00Z","timestamp":1738627200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100009799","name":"Stiftelsen L\u00e4nsf\u00f6rs\u00e4kringsbolagens Forskningsfond","doi-asserted-by":"publisher","award":["P9\/20"],"award-info":[{"award-number":["P9\/20"]}],"id":[{"id":"10.13039\/501100009799","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009244","name":"Stockholm University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009244","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The paper introduces a tree-based varying coefficient model (VCM) where the varying coefficients are modelled using the cyclic gradient boosting machine (CGBM) from Delong et al. (On cyclic gradient boosting machines, 2023). Modelling the coefficient functions using a CGBM allows for dimension-wise early stopping and feature importance scores. The dimension-wise early stopping not only reduces the risk of dimension-specific overfitting, but also reveals differences in model complexity across dimensions. The use of feature importance scores allows for simple feature selection and easy model interpretation. The model is evaluated on the same simulated and real data examples as those used in Richman and W\u00fcthrich (Scand Actuar J 2023:71\u201395, 2023), and the results show that it produces results in terms of out of sample loss that are comparable to those of their neural network-based VCM called LocalGLMnet.<\/jats:p>","DOI":"10.1007\/s00180-025-01603-8","type":"journal-article","created":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T14:17:23Z","timestamp":1738678643000},"page":"5105-5134","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A tree-based varying coefficient model"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-4248-6933","authenticated-orcid":false,"given":"Henning","family":"Zakrisson","sequence":"first","affiliation":[]},{"given":"Mathias","family":"Lindholm","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,4]]},"reference":[{"issue":"1","key":"1603_CR1","first-page":"7950","volume":"21","author":"M Al-Shedivat","year":"2020","unstructured":"Al-Shedivat M, Dubey A, Xing E (2020) Contextual explanation networks. J Mach Learn Res 21(1):7950\u20137993","journal-title":"J Mach Learn Res"},{"key":"1603_CR2","unstructured":"Alvarez\u00a0Melis D, Jaakkola T (2018) Towards robust interpretability with self-explaining neural networks. Adv Neural Inf Process Syst 31"},{"key":"1603_CR3","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/s11222-018-9804-8","volume":"29","author":"M Berger","year":"2019","unstructured":"Berger M, Tutz G, Schmid M (2019) Tree-structured modelling of varying coefficients. Stat Comput 29:217\u2013229","journal-title":"Stat Comput"},{"key":"1603_CR4","volume-title":"Classification and regression trees","author":"L Breiman","year":"1984","unstructured":"Breiman L, Friedman JH, Olshen RA, Stone CJ (1984) Classification and regression trees. Wadsworth Inc, Belmont, Calif"},{"key":"1603_CR5","unstructured":"B\u00fchlmann PL (2002) Consistency for L2boosting and matching pursuit with trees and tree-type basis functions. In Research report\/Seminar f\u00fcr Statistik, Eidgen\u00f6ssische Technische Hochschule (ETH), volume 109. Seminar f\u00fcr Statistik, Eidgen\u00f6ssische Technische Hochschule (ETH)"},{"key":"1603_CR6","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.csda.2015.01.003","volume":"86","author":"R B\u00fcrgin","year":"2015","unstructured":"B\u00fcrgin R, Ritschard G (2015) Tree-based varying coefficient regression for longitudinal ordinal responses. Comput Stat Data Anal 86:65\u201380","journal-title":"Comput Stat Data Anal"},{"key":"1603_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v080.i06","volume":"80","author":"R B\u00fcrgin","year":"2017","unstructured":"B\u00fcrgin R, Ritschard G (2017) Coefficient-wise tree-based varying coefficient regression with vcrpart. J Stat Softw 80:1\u201333","journal-title":"J Stat Softw"},{"issue":"4","key":"1603_CR8","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1080\/10618600.2020.1753530","volume":"29","author":"SB Chatla","year":"2020","unstructured":"Chatla SB, Shmueli G (2020) A tree-based semi-varying coefficient model for the com-Poisson distribution. J Comput Gr Stat 29(4):827\u2013846","journal-title":"J Comput Gr Stat"},{"key":"1603_CR9","unstructured":"Chen T, He T, Benesty M, Khotilovich V, Tang Y, Cho H, Chen K, Mitchell R, Cano I, Zhou T, et\u00a0al (2015) Xgboost: extreme gradient boosting. R package version 0.4-2 1(4):1\u20134"},{"key":"1603_CR10","unstructured":"Delong \u0141, Lindholm M, Zakrisson H (2023) On cyclic gradient boosting machines. Available at SSRN 4352505"},{"key":"1603_CR11","unstructured":"Duan T, Anand A, Ding DY, Thai KK, Basu S, Ng A, Schuler A (2020) Ngboost: natural gradient boosting for probabilistic prediction. In International conference on machine learning, 2690\u20132700. PMLR"},{"key":"1603_CR12","unstructured":"Dutang C, Charpentier A, Dutang MC (2020) CASdatasets: A Package of Insurance Datasets Provided by the Casualty Actuarial Society. Accessed: 2023-11-06"},{"key":"1603_CR13","doi-asserted-by":"crossref","unstructured":"Friedman JH (2001) Greedy function approximation: a gradient boosting machine. Ann Stat, 1189\u20131232","DOI":"10.1214\/aos\/1013203451"},{"issue":"4","key":"1603_CR14","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1111\/j.2517-6161.1993.tb01939.x","volume":"55","author":"T Hastie","year":"1993","unstructured":"Hastie T, Tibshirani R (1993) Varying-coefficient models. J R Stat Soc Ser B Stat Methodol 55(4):757\u2013779","journal-title":"J R Stat Soc Ser B Stat Methodol"},{"key":"1603_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-84858-7","volume-title":"The elements of statistical learning: data mining, inference, and prediction","author":"T Hastie","year":"2009","unstructured":"Hastie T, Tibshirani R, Friedman JH (2009) The elements of statistical learning: data mining, inference, and prediction. Springer, New York, NY, USA"},{"key":"1603_CR16","unstructured":"J\u00f8rgensen B (1997) The Theory of Dispersion Models. Chapman & Hall\/CRC Monographs on Statistics & Applied Probability. Taylor & Francis"},{"key":"1603_CR17","unstructured":"Ke G, Meng Q, Finley T, Wang T, Chen W, Ma W, Ye Q, Liu T-Y (2017) Lightgbm: a highly efficient gradient boosting decision tree. Adv Neural Inf Process Syst 30"},{"issue":"3","key":"1603_CR18","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1080\/10920277.2018.1431131","volume":"22","author":"SC Lee","year":"2018","unstructured":"Lee SC, Lin S (2018) Delta boosting machine with application to general insurance. North Am Actuar J 22(3):405\u2013425","journal-title":"North Am Actuar J"},{"key":"1603_CR19","doi-asserted-by":"crossref","unstructured":"Lindholm M, Lindskog F, Palmquist J (2023) Local bias adjustment, duration-weighted probabilities, and automatic construction of tariff cells. Scand Actuar J, 1\u201328","DOI":"10.2139\/ssrn.4256876"},{"key":"1603_CR20","doi-asserted-by":"crossref","unstructured":"Lindholm M, Nazar T (2023) On duration effects in non-life insurance pricing. Available at SSRN 4474908","DOI":"10.2139\/ssrn.4474908"},{"issue":"3","key":"1603_CR21","doi-asserted-by":"publisher","DOI":"10.1002\/widm.1493","volume":"13","author":"R Marcinkevi\u010ds","year":"2023","unstructured":"Marcinkevi\u010ds R, Vogt JE (2023) Interpretable and explainable machine learning: a methods-centric overview with concrete examples. WIREs Data Min Knowl Discov 13(3):e1493","journal-title":"WIREs Data Min Knowl Discov"},{"issue":"3","key":"1603_CR22","first-page":"370","volume":"135","author":"JA Nelder","year":"1972","unstructured":"Nelder JA, Wedderburn RW (1972) Generalized linear models. J R Stat Soc Ser A Stat Soc 135(3):370\u2013384","journal-title":"J R Stat Soc Ser A Stat Soc"},{"key":"1603_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-10791-7","volume-title":"Non-life insurance pricing with generalized linear models","author":"E Ohlsson","year":"2010","unstructured":"Ohlsson E, Johansson B (2010) Non-life insurance pricing with generalized linear models, vol 174. Springer"},{"key":"1603_CR24","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/s10985-019-09489-7","volume":"26","author":"M-T Puth","year":"2020","unstructured":"Puth M-T, Tutz G, Heim N, M\u00fcnster E, Schmid M, Berger M (2020) Tree-based modeling of time-varying coefficients in discrete time-to-event models. Lifetime Data Anal 26:545\u2013572","journal-title":"Lifetime Data Anal"},{"issue":"1","key":"1603_CR25","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1080\/03461238.2022.2081816","volume":"2023","author":"R Richman","year":"2023","unstructured":"Richman R, W\u00fcthrich MV (2023) LocalGLMnet: interpretable deep learning for tabular data. Scand Actuar J 2023(1):71\u201395","journal-title":"Scand Actuar J"},{"key":"1603_CR26","unstructured":"Sundberg R (2019) Statistical Modelling by Exponential Families, volume\u00a012 of Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, UK"},{"key":"1603_CR27","unstructured":"Thompson R, Dezfouli A, Kohn R (2023) The contextual lasso: sparse linear models via deep neural networks. arXiv preprint arXiv:2302.00878"},{"issue":"2","key":"1603_CR28","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1080\/10618600.2013.778777","volume":"23","author":"JC Wang","year":"2014","unstructured":"Wang JC, Hastie T (2014) Boosted varying-coefficient regression models for product demand prediction. J Comput Gr Stat 23(2):361\u2013382","journal-title":"J Comput Gr Stat"},{"key":"1603_CR29","doi-asserted-by":"crossref","unstructured":"W\u00fcthrich MV, Merz M (2023) Statistical foundations of actuarial learning and its applications. Springer Nature","DOI":"10.1007\/978-3-031-12409-9"},{"issue":"4","key":"1603_CR30","doi-asserted-by":"publisher","first-page":"1538","DOI":"10.1214\/009053605000000255","volume":"33","author":"T Zhang","year":"2005","unstructured":"Zhang T, Yu B (2005) Boosting with early stopping: convergence and consistency. Ann Stat 33(4):1538\u20131579","journal-title":"Ann Stat"},{"issue":"6","key":"1603_CR31","doi-asserted-by":"publisher","first-page":"2237","DOI":"10.1007\/s10618-022-00863-y","volume":"36","author":"Y Zhou","year":"2022","unstructured":"Zhou Y, Hooker G (2022) Decision tree boosted varying coefficient models. Data Min Knowl Discov 36(6):2237\u20132271","journal-title":"Data Min Knowl Discov"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01603-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-025-01603-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01603-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T08:40:40Z","timestamp":1763368840000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-025-01603-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,4]]},"references-count":31,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["1603"],"URL":"https:\/\/doi.org\/10.1007\/s00180-025-01603-8","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"type":"print","value":"0943-4062"},{"type":"electronic","value":"1613-9658"}],"subject":[],"published":{"date-parts":[[2025,2,4]]},"assertion":[{"value":"5 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}