{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T04:29:52Z","timestamp":1778646592197,"version":"3.51.4"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T00:00:00Z","timestamp":1626048000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T00:00:00Z","timestamp":1626048000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001832","name":"Radboud Universiteit","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001832","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"crossref","award":["2020M671899"],"award-info":[{"award-number":["2020M671899"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Long-term forecasting involves predicting a horizon that is far ahead of the last observation. It is a problem of high practical relevance, for instance for companies in order to decide upon expensive long-term investments. Despite the recent progress and success of Gaussian processes (GPs) based on spectral mixture kernels, long-term forecasting remains a challenging problem for these kernels because they decay exponentially at large horizons. This is mainly due to their use of a mixture of Gaussians to model spectral densities. Characteristics of the signal important for long-term forecasting can be unravelled by investigating the distribution of the Fourier coefficients of (the training part of) the signal, which is non-smooth, heavy-tailed, sparse, and skewed. The heavy tail and skewness characteristics of such distributions in the spectral domain allow to capture long-range covariance of the signal in the time domain. Motivated by these observations, we propose to model spectral densities using a skewed Laplace spectral mixture (SLSM) due to the skewness of its peaks, sparsity, non-smoothness, and heavy tail characteristics. By applying the inverse Fourier Transform to this spectral density we obtain a new GP kernel for long-term forecasting. In addition, we adapt the lottery ticket method, originally developed to prune weights of a neural network, to GPs in order to automatically select the number of kernel components. Results of extensive experiments, including a multivariate time series, show the beneficial effect of the proposed SLSM kernel for long-term extrapolation and robustness to the choice of the number of mixture components.<\/jats:p>","DOI":"10.1007\/s10994-021-06031-5","type":"journal-article","created":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T19:14:13Z","timestamp":1626117253000},"page":"2213-2238","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Gaussian processes with skewed Laplace spectral mixture kernels for long-term forecasting"],"prefix":"10.1007","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4081-0687","authenticated-orcid":false,"given":"Kai","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Twan","family":"van Laarhoven","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elena","family":"Marchiori","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,7,12]]},"reference":[{"issue":"4","key":"6031_CR1","doi-asserted-by":"publisher","first-page":"630","DOI":"10.1093\/biostatistics\/kxj032","volume":"7","author":"D Bhowmick","year":"2006","unstructured":"Bhowmick, D., Davison, A., Goldstein, D. R., & Ruffieux, Y. (2006). A Laplace mixture model for identification of differential expression in microarray experiments. Biostatistics, 7(4), 630\u2013641.","journal-title":"Biostatistics"},{"key":"6031_CR2","volume-title":"Lectures on Fourier Integrals. (AM-42)","author":"S Bochner","year":"2016","unstructured":"Bochner, S. (2016). Lectures on Fourier Integrals. (AM-42) (Vol. 42). Princeton University Press."},{"issue":"Feb","key":"6031_CR3","first-page":"333","volume":"14","author":"K Chalupka","year":"2013","unstructured":"Chalupka, K., Williams, C. .K., & Murray, I. (2013). A framework for evaluating approximation methods for Gaussian process regression. Journal of Machine Learning Research, 14(Feb), 333\u2013350.","journal-title":"Journal of Machine Learning Research"},{"key":"6031_CR4","doi-asserted-by":"crossref","unstructured":"Chen, K., van Laarhoven, T., Chen, J., & Marchiori, E. (2019). Incorporating dependencies in spectral kernels for Gaussian processes. In Joint European conference on machine learning and knowledge discovery in databases (pp. 565\u2013581). Springer.","DOI":"10.1007\/978-3-030-46147-8_34"},{"key":"6031_CR5","unstructured":"Deisenroth, M., & Ng, J. W. (2015). Distributed Gaussian processes. In Proceedings of The 32nd international conference on machine learning (pp. 1481\u20131490)."},{"key":"6031_CR6","unstructured":"Duvenaud, D., Lloyd, J., Grosse, R., Tenenbaum, J., & Zoubin, G. (2013). Structure discovery in nonparametric regression through compositional kernel search, pp. 1166\u20131174. PMLR, Atlanta, Georgia, USA. http:\/\/proceedings.mlr.press\/v28\/duvenaud13.html."},{"key":"6031_CR7","unstructured":"Duvenaud, D., Lloyd, J.R., Grosse, R., Tenenbaum, J.B., & Ghahramani, Z. (2013). Structure discovery in nonparametric regression through compositional kernel search. arXiv preprint arXiv:1302.4922."},{"issue":"5","key":"6031_CR8","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1109\/LSP.2006.870353","volume":"13","author":"T Eltoft","year":"2006","unstructured":"Eltoft, T., Kim, T., & Lee, T. (2006). On the multivariate Laplace distribution. IEEE Signal Processing Letters, 13(5), 300\u2013303. https:\/\/doi.org\/10.1109\/LSP.2006.870353.","journal-title":"IEEE Signal Processing Letters"},{"key":"6031_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-01505-7","volume-title":"Long-range dependence and sea level forecasting","author":"A Ercan","year":"2013","unstructured":"Ercan, A., Kavvas, M. L., & Abbasov, R. K. (2013). Long-range dependence and sea level forecasting. Springer."},{"issue":"5\u20136","key":"6031_CR10","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1016\/j.mcm.2010.10.014","volume":"53","author":"K Fragiadakis","year":"2011","unstructured":"Fragiadakis, K., & Meintanis, S. G. (2011). Goodness-of-fit tests for multivariate Laplace distributions. Mathematical and Computer Modelling, 53(5\u20136), 769\u2013779.","journal-title":"Mathematical and Computer Modelling"},{"key":"6031_CR11","unstructured":"Frankle, J., & Carbin, M. (2019). The lottery ticket hypothesis: Finding sparse, trainable neural networks. In 7th international conference on learning representations, ICLR 2019, New Orleans, LA, USA, May 6\u20139, 2019. OpenReview.net https:\/\/openreview.net\/forum?id=rJl-b3RcF7."},{"issue":"7","key":"6031_CR12","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1109\/LSP.2003.813679","volume":"10","author":"S Gazor","year":"2003","unstructured":"Gazor, S., & Zhang, W. (2003). Speech probability distribution. Signal Processing Letters IEEE, 10(7), 204\u2013207. https:\/\/doi.org\/10.1109\/LSP.2003.813679.","journal-title":"Signal Processing Letters IEEE"},{"issue":"2","key":"6031_CR13","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1109\/TPAMI.2013.192","volume":"37","author":"E Gilboa","year":"2015","unstructured":"Gilboa, E., Saatci, Y., & Cunningham, J. P. (2015). Scaling multidimensional inference for structured Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(2), 424\u2013436. https:\/\/doi.org\/10.1109\/TPAMI.2013.192.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6031_CR14","unstructured":"Herlands, W., Wilson, A., Nickisch, H., Flaxman, S., Neill, D., Van\u00a0Panhuis, W., & Xing, E. (2016). Scalable Gaussian processes for characterizing multidimensional change surfaces. In Artificial intelligence and statistics (pp. 1013\u20131021)."},{"key":"6031_CR15","unstructured":"Hoffman, F. (1997). An introduction to Fourier theory. Extra\u00eddo el, 2."},{"key":"6031_CR16","unstructured":"Hyun Jin\u00a0Park, T. W. L. (2004). Modeling nonlinear dependencies in natural images using mixture of Laplacian distribution."},{"key":"6031_CR17","unstructured":"Jang, P. A., Loeb, A., Davidow, M., & Wilson, A. G. (2017). Scalable L\u00e9vy process priors for spectral kernel learning. In Advances in neural information processing systems (pp. 3943\u20133952)."},{"key":"6031_CR18","unstructured":"Kostantinos, N. (2000). Gaussian mixtures and their applications to signal processing. In Advanced signal processing handbook: theory and implementation for radar, sonar, and medical imaging real time systems (pp. 3-1)."},{"key":"6031_CR19","doi-asserted-by":"crossref","unstructured":"Kotz, S., Kozubowski, T.J., & Podg\u00f3rski, K. (2001). Asymmetric multivariate Laplace distribution. In The Laplace distribution and generalizations (pp. 239\u2013272). Springer.","DOI":"10.1007\/978-1-4612-0173-1_7"},{"issue":"4","key":"6031_CR20","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1007\/PL00022717","volume":"15","author":"TJ Kozubowski","year":"2000","unstructured":"Kozubowski, T. J., & Podg\u00f3rski, K. (2000). A multivariate and asymmetric generalization of Laplace distribution. Computational Statistics, 15(4), 531\u2013540. https:\/\/doi.org\/10.1007\/PL00022717.","journal-title":"Computational Statistics"},{"issue":"4","key":"6031_CR21","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1016\/S0169-2070(00)00057-1","volume":"16","author":"S Makridakis","year":"2000","unstructured":"Makridakis, S., & Hibon, M. (2000). The m3-competition: Results, conclusions and implications. International Journal of Forecasting, 16(4), 451\u2013476.","journal-title":"International Journal of Forecasting"},{"issue":"2","key":"6031_CR22","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1117\/1.1344592","volume":"10","author":"J Minguill\u00f3n","year":"2001","unstructured":"Minguill\u00f3n, J., & Pujol, J. (2001). JPEG standard uniform quantization error modeling with applications to sequential and progressive operation modes. Journal of Electronic Imaging, 10(2), 475\u2013485. https:\/\/doi.org\/10.1117\/1.1344592.","journal-title":"Journal of Electronic Imaging"},{"issue":"6","key":"6031_CR23","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1109\/79.543975","volume":"13","author":"TK Moon","year":"1997","unstructured":"Moon, T. K. (1997). The expectation-maximization algorithm. IEEE Signal Processing Magazine, 13(6), 47\u201360.","journal-title":"IEEE Signal Processing Magazine"},{"key":"6031_CR24","unstructured":"Nguyen, T., & Bonilla, E. (2014). Fast allocation of gaussian process experts. In Proceedings of The 31st international conference on machine learning (pp. 145\u2013153)."},{"key":"6031_CR25","unstructured":"Oliva, J. B., Dubey, A., Wilson, A. G., P\u00f3czos, B., Schneider, J., & Xing, E. P. (2016). Bayesian nonparametric kernel-learning. In Artificial intelligence and statistics (pp. 1078\u20131086)."},{"key":"6031_CR26","unstructured":"Pearce, T., Tsuchida, R., Zaki, M., Brintrup, A., & Neely, A. (2020). Expressive priors in Bayesian neural networks: Kernel combinations and periodic functions. In Uncertainty in artificial intelligence (pp. 134\u2013144). PMLR."},{"issue":"10","key":"6031_CR27","doi-asserted-by":"publisher","first-page":"e0221238","DOI":"10.1371\/journal.pone.0221238","volume":"14","author":"DJ Pedregal","year":"2019","unstructured":"Pedregal, D. J. (2019). Time series analysis and forecasting with ECOTOOL. PLoS One, 14(10), e0221238.","journal-title":"PLoS One"},{"key":"6031_CR28","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1016\/j.procs.2015.07.032","volume":"55","author":"JFM Pessanha","year":"2015","unstructured":"Pessanha, J. F. M., & Leon, N. (2015). Forecasting long-term electricity demand in the residential sector. Procedia Computer Science, 55, 529\u2013538.","journal-title":"Procedia Computer Science"},{"issue":"Dec","key":"6031_CR29","first-page":"1939","volume":"6","author":"J Qui\u00f1onero-Candela","year":"2005","unstructured":"Qui\u00f1onero-Candela, J., & Rasmussen, C. .E. (2005). A unifying view of sparse approximate Gaussian process regression. Journal of Machine Learning Research, 6(Dec), 1939\u20131959.","journal-title":"Journal of Machine Learning Research"},{"issue":"Nov","key":"6031_CR30","first-page":"3011","volume":"11","author":"C.E Rasmussen","year":"2010","unstructured":"Rasmussen, C. .E., & Nickisch, H. (2010). Gaussian processes for machine learning (GPML) toolbox. Journal of Machine Learning Research, 11(Nov), 3011\u20133015.","journal-title":"Journal of Machine Learning Research"},{"key":"6031_CR31","volume-title":"Gaussian process for machine learning","author":"C.E Rasmussen","year":"2006","unstructured":"Rasmussen, C. .E., & Williams, C. .K. (2006). Gaussian process for machine learning. MIT press."},{"key":"6031_CR32","unstructured":"Remes, S., Heinonen, M., & Kaski, S. (2017). Non-stationary spectral kernels. In Advances in neural information processing systems (pp. 4645\u20134654)."},{"key":"6031_CR33","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.jpowsour.2017.05.004","volume":"357","author":"RR Richardson","year":"2017","unstructured":"Richardson, R. R., Osborne, M. A., & Howey, D. A. (2017). Gaussian process regression for forecasting battery state of health. Journal of Power Sources, 357, 209\u2013219.","journal-title":"Journal of Power Sources"},{"key":"6031_CR34","unstructured":"Samo, Y. L. K., & Roberts, S. (2015). Generalized spectral kernels. arXiv preprint arXiv:1506.02236."},{"key":"6031_CR35","unstructured":"Shen, Z., Heinonen, M., & Kaski, S. (2019). Harmonizable mixture kernels with variational Fourier features. In The 22nd international conference on artificial intelligence and statistics (pp. 3273\u20133282). PMLR."},{"key":"6031_CR36","doi-asserted-by":"crossref","unstructured":"Stein, M. (1999). Interpolation of spatial data: Some theory for kriging.","DOI":"10.1007\/978-1-4612-1494-6"},{"key":"6031_CR37","unstructured":"Sun, S., Zhang, G., Wang, C., Zeng, W., Li, J., & Grosse, R. (2018). Differentiable compositional kernel learning for Gaussian processes. arXiv preprint arXiv:1806.04326."},{"issue":"4","key":"6031_CR38","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1080\/03610920802233945","volume":"38","author":"H Visk","year":"2009","unstructured":"Visk, H. (2009). On the parameter estimation of the asymmetric multivariate Laplace distribution. Communications in Statistics-Theory and Methods, 38(4), 461\u2013470.","journal-title":"Communications in Statistics-Theory and Methods"},{"key":"6031_CR39","unstructured":"Wilson, A., & Adams, R. (2013). Gaussian process kernels for pattern discovery and extrapolation. In Proceedings of the 30th international conference on machine learning (ICML-13) (pp. 1067\u20131075)."},{"key":"6031_CR40","volume-title":"Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes","author":"A.G Wilson","year":"2014","unstructured":"Wilson, A. .G. (2014). Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes. University of Cambridge."},{"key":"6031_CR41","unstructured":"Wilson, A. G., Gilboa, E., Nehorai, A., & Cunningham, J. P. (2014). Fast kernel learning for multidimensional pattern extrapolation. In Advances in neural information processing systems (pp. 3626\u20133634)."},{"key":"6031_CR42","unstructured":"Wilson, A. G., & Nickisch, H. (2015). Kernel interpolation for scalable structured Gaussian processes (KISS-GP). In: F. R. Bach & D. M. Blei (Eds.), Proceedings of the 32nd international conference on machine learning, ICML 2015, Lille, France, 6\u201311 July 2015, JMLR Workshop and Conference Proceedings, vol.\u00a037, pp. 1775\u20131784. JMLR.org . http:\/\/jmlr.org\/proceedings\/papers\/v37\/wilson15.html."},{"key":"6031_CR43","doi-asserted-by":"publisher","unstructured":"Yan, W., Qiu, H., & Xue, Y. (2009). Gaussian process for long-term time-series forecasting. In 2009 international joint conference on neural networks (pp. 3420\u20133427) . https:\/\/doi.org\/10.1109\/IJCNN.2009.5178729.","DOI":"10.1109\/IJCNN.2009.5178729"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-021-06031-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-021-06031-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-021-06031-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,17]],"date-time":"2021-08-17T16:23:18Z","timestamp":1629217398000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-021-06031-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,12]]},"references-count":43,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2021,8]]}},"alternative-id":["6031"],"URL":"https:\/\/doi.org\/10.1007\/s10994-021-06031-5","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,12]]},"assertion":[{"value":"22 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}