{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T07:42:05Z","timestamp":1784619725906,"version":"3.55.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T00:00:00Z","timestamp":1632441600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T00:00:00Z","timestamp":1632441600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Behav Res"],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Computational modeling plays an important role in a gamut of research fields. In affect research, continuous-time stochastic models are becoming increasingly popular. Recently, a non-linear, continuous-time, stochastic model has been introduced for affect dynamics, called the Affective Ising Model (AIM). The drawback of non-linear models like the AIM is that they generally come with serious computational challenges for parameter estimation and related statistical analyses. The likelihood function of the AIM does not have a closed form expression. Consequently, simulation based or numerical methods have to be considered in order to evaluate the likelihood function. Additionally, the likelihood function can have multiple local minima. Consequently, a global optimization heuristic is required and such heuristics generally require a large number of likelihood function evaluations. In this paper, a Julia software package is introduced that is dedicated to fitting the AIM. The package includes an implementation of a numeric algorithm for fast computations of the likelihood function, which can be run both on graphics processing units (GPU) and central processing units (CPU). The numerical method introduced in this paper is compared to the more traditional Euler-Maruyama method for solving stochastic differential equations. Furthermore, the estimation software is tested by means of a recovery study and estimation times are reported for benchmarks that were run on several computing devices (two different GPUs and three different CPUs). According to these results, a single parameter estimation can be obtained in less than thirty seconds using a mainstream NVIDIA GPU.<\/jats:p>","DOI":"10.3758\/s13428-021-01674-7","type":"journal-article","created":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T21:01:17Z","timestamp":1632517277000},"page":"1428-1443","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Efficient estimation of bounded gradient-drift diffusion models for affect on CPU and GPU"],"prefix":"10.3758","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7587-1922","authenticated-orcid":false,"given":"Tim","family":"Loossens","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kristof","family":"Meers","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Niels","family":"Vanhasbroeck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nil","family":"Anarat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stijn","family":"Verdonck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francis","family":"Tuerlinckx","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,9,24]]},"reference":[{"issue":"1","key":"1674_CR1","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1146\/annurev.psych.54.101601.145030","volume":"54","author":"N Bolger","year":"2003","unstructured":"Bolger, N., Davis, A., & Rafaeli, E. (2003). Diary Methods: Capturing Life as it is Lived. Annual Review of Psychology, 54(1), 579\u2013616. https:\/\/doi.org\/10.1146\/annurev.psych.54.101601.145030","journal-title":"Annual Review of Psychology"},{"key":"1674_CR2","volume-title":"Intensive Longitudinal Methods: An Introduction to Diary and Experience Sampling Research","author":"N Bolger","year":"2013","unstructured":"Bolger, N., & Laurenceau, J.-P. (2013) Intensive Longitudinal Methods: An Introduction to Diary and Experience Sampling Research, (1st edn.) New York: The Guilford Press.","edition":"1st edn."},{"issue":"1","key":"1674_CR3","doi-asserted-by":"publisher","first-page":"bcr2012006300","DOI":"10.1136\/bcr-2012-006300","volume":"2012","author":"EH Bos","year":"2012","unstructured":"Bos, E. H., Hoenders, R., & de Jonge, P. (2012). Wind direction and mental health: a time-series analysis of weather influences in a patient with anxiety disorder. Case Reports, 2012(1), bcr2012006300\u2013bcr2012006300. https:\/\/doi.org\/10.1136\/bcr-2012-006300","journal-title":"Case Reports"},{"issue":"4","key":"1674_CR4","doi-asserted-by":"publisher","first-page":"e60188","DOI":"10.1371\/journal.pone.0060188","volume":"8","author":"LF Bringmann","year":"2013","unstructured":"Bringmann, L. F., Vissers, N., Wichers, M., Geschwind, N., Kuppens, P., Peeters, F., ..., Tuerlinckx, F. (2013). A Network Approach to Psychopathology: New Insights into Clinical Longitudinal Data. Plos One, 8(4), e60188. https:\/\/doi.org\/10.1371\/journal.pone.0060188","journal-title":"Plos One"},{"issue":"1","key":"1674_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0021-9991(70)90001-X","volume":"6","author":"JS Chang","year":"1970","unstructured":"Chang, J. S., & Cooper, G. (1970). A practical difference scheme for Fokker-Planck equations. Journal of Computational Physics, 6(1), 1\u201316. https:\/\/doi.org\/10.1016\/0021-9991(70)90001-X","journal-title":"Journal of Computational Physics"},{"issue":"3","key":"1674_CR6","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1348\/0144665031752934","volume":"43","author":"JR Crawford","year":"2004","unstructured":"Crawford, J. R., & Henry, J. D. (2004). The Positive and Negative Affect Schedule (PANAS): Construct validity, measurement properties and normative data in a large non-clinical sample. British Journal of Clinical Psychology, 43(3), 245\u2013265. https:\/\/doi.org\/10.1348\/0144665031752934","journal-title":"British Journal of Clinical Psychology"},{"issue":"9","key":"1674_CR7","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1097\/00005053-198709000-00004","volume":"175","author":"M Csikszentmihalyi","year":"1987","unstructured":"Csikszentmihalyi, M., & Larson, R. (1987). Validity and reliability of the experience-sampling method. Journal of Nervous and Mental Disease, 175(9), 526\u2013536. https:\/\/doi.org\/10.1097\/00005053-198709000-00004","journal-title":"Journal of Nervous and Mental Disease"},{"key":"1674_CR8","doi-asserted-by":"publisher","unstructured":"de Haan-Rietdijk, S., Voelkle, M. C., Keijsers, L., & Hamaker, E. L. (2017). Discrete- vs. Continuous-Time Modeling of Unequally Spaced Experience Sampling Method Data. Frontiers in Psychology, 8. https:\/\/doi.org\/10.3389\/fpsyg.2017.01849","DOI":"10.3389\/fpsyg.2017.01849"},{"issue":"5","key":"1674_CR9","doi-asserted-by":"publisher","first-page":"478","DOI":"10.1038\/s41562-019-0555-0","volume":"3","author":"E Dejonckheere","year":"2019","unstructured":"Dejonckheere, E., Mestdagh, M., Houben, M., Rutten, I., Sels, L., Kuppens, P., & Tuerlinckx, F. (2019). Complex affect dynamics add limited information to the prediction of psychological well-being. Nature Human Behaviour, 3(5), 478\u2013491. https:\/\/doi.org\/10.1038\/s41562-019-0555-0","journal-title":"Nature Human Behaviour"},{"key":"1674_CR10","doi-asserted-by":"publisher","unstructured":"Driver, C. C., Oud, J. H. L., & Voelkle, M. C. (2017). Continuous Time Structural Equation Modeling with R Package ctsem. Journal of Statistical Software, 77(5). https:\/\/doi.org\/10.18637\/jss.v077.i05","DOI":"10.18637\/jss.v077.i05"},{"key":"1674_CR11","volume-title":"Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences","author":"CW Gardiner","year":"1986","unstructured":"Gardiner, C. W. (1986) Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences, (Subsequent). Berlin: Springer.","edition":"Subsequent"},{"key":"1674_CR12","doi-asserted-by":"publisher","unstructured":"Hamaker, E., Ceulemans, E., Grasman, R., & Tuerlinckx, F. (2015). Modeling Affect Dynamics: State of the Art and Future Challenges. Emotion Review, 7. https:\/\/doi.org\/10.1177\/1754073915590619","DOI":"10.1177\/1754073915590619"},{"key":"1674_CR13","doi-asserted-by":"publisher","unstructured":"Heininga, V. E., Dejonckheere, E., Houben, M., Obbels, J., Sienaert, P., Leroy, B., ..., Kuppens, P. (2019). The dynamical signature of anhedonia in major depressive disorder: positive emotion dynamics, reactivity, and recovery. BMC Psychiatry, 19(1). https:\/\/doi.org\/10.1186\/s12888-018-1983-5","DOI":"10.1186\/s12888-018-1983-5"},{"key":"1674_CR14","volume-title":"Experience Sampling Method: Measuring the Quality of Everyday Life","author":"MJ Hektner","year":"2006","unstructured":"Hektner, M. J., Schmidt, A. J., & Csikszentmihalyi, M. (2006) Experience Sampling Method: Measuring the Quality of Everyday Life, (1s). Thousand Oaks: SAGE Publications, Inc.","edition":"1s"},{"issue":"1","key":"1674_CR15","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1037\/per0000126","volume":"7","author":"M Houben","year":"2016","unstructured":"Houben, M., Vansteelandt, K., Claes, L., Sienaert, P., Berens, A., Sleuwaegen, E., & Kuppens, P. (2016). Emotional switching in borderline personality disorder: A daily life study. Personality Disorders: Theory, Research, and Treatment, 7(1), 50\u201360. https:\/\/doi.org\/10.1037\/per0000126","journal-title":"Personality Disorders: Theory, Research, and Treatment"},{"issue":"3-4","key":"1674_CR16","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1016\/S0009-2614(91)85070-D","volume":"185","author":"K Kikuchi","year":"1991","unstructured":"Kikuchi, K., Yoshida, M., Maekawa, T., & Watanabe, H. (1991). Metropolis Monte Carlo method as a numerical technique to solve the Fokker-Planck equation. Chemical Physics Letters, 185(3-4), 335\u2013338. https:\/\/doi.org\/10.1016\/S0009-2614(91)85070-D","journal-title":"Chemical Physics Letters"},{"key":"1674_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12616-5","volume-title":"Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability","author":"PE Kloeden","year":"1992","unstructured":"Kloeden, P. E., & Platen, E. (1992) Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability. Berlin: Springer."},{"issue":"4","key":"1674_CR18","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1037\/a0030811","volume":"139","author":"P Kuppens","year":"2013","unstructured":"Kuppens, P., Tuerlinckx, F., Russell, J. A., & Barrett, L. F. (2013). The relation between valence and arousal in subjective experience. Psychological Bulletin, 139(4), 917\u2013940. https:\/\/doi.org\/10.1037\/a0030811","journal-title":"Psychological Bulletin"},{"issue":"2","key":"1674_CR19","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1037\/emo0000231","volume":"17","author":"JT Larsen","year":"2017","unstructured":"Larsen, J. T., Hershfield, H. E., Stastny, B. J., & Hester, N. (2017). On the relationship between positive and negative affect: Their correlation and their co-occurrence. Emotion, 17(2), 323\u2013336. https:\/\/doi.org\/10.1037\/emo0000231","journal-title":"Emotion"},{"issue":"1","key":"1674_CR20","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1073\/pnas.1312114110","volume":"111","author":"IAvd Leemput","year":"2014","unstructured":"Leemput, I.A.v.d., Wichers, M., Cramer, A. O. J., Borsboom, D., Tuerlinckx, F., Kuppens, P., ..., Scheffer, M. (2014). Critical slowing down as early warning for the onset and termination of depression. Proceedings of the National Academy of Sciences, 111(1), 87\u201392. https:\/\/doi.org\/10.1073\/pnas.1312114110","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"1","key":"1674_CR21","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.jmp.2010.08.004","volume":"55","author":"T Lodewyckx","year":"2011","unstructured":"Lodewyckx, T., Tuerlinckx, F., Kuppens, P., Allen, N., & Sheeber, L. (2011). A hierarchical state space approach to affective dynamics. Journal of Mathematical Psychology, 55(1), 68\u201383. https:\/\/doi.org\/10.1016\/j.jmp.2010.08.004","journal-title":"Journal of Mathematical Psychology"},{"issue":"5","key":"1674_CR22","doi-asserted-by":"publisher","first-page":"e1007860","DOI":"10.1371\/journal.pcbi.1007860","volume":"16","author":"T Loossens","year":"2020","unstructured":"Loossens, T., Mestdagh, M., Dejonckheere, E., Kuppens, P., Tuerlinckx, F., & Verdonck, S. (2020). The Affective Ising Model: A computational account of human affect dynamics. PLOS Computational Biology, 16(5), e1007860. https:\/\/doi.org\/10.1371\/journal.pcbi.1007860","journal-title":"PLOS Computational Biology"},{"issue":"3","key":"1674_CR23","doi-asserted-by":"publisher","first-page":"508","DOI":"10.1177\/1745691616685863","volume":"12","author":"AM Mattek","year":"2017","unstructured":"Mattek, A. M., Wolford, G. L., & Whalen, P. J. (2017). A Mathematical Model Captures the Structure of Subjective Affect. Perspectives on Psychological Science: A Journal of the Association for Psychological Science, 12(3), 508\u2013526. https:\/\/doi.org\/10.1177\/1745691616685863","journal-title":"Perspectives on Psychological Science: A Journal of the Association for Psychological Science"},{"issue":"3","key":"1674_CR24","doi-asserted-by":"publisher","first-page":"942","DOI":"10.1016\/j.jad.2013.08.011","volume":"151","author":"EL Merz","year":"2013","unstructured":"Merz, E. L., Malcarne, V. L., Roesch, S. C., Ko, C. M., Emerson, M., Roma, V. G., & Sadler, G. R. (2013). Psychometric properties of Positive and Negative Affect Schedule (PANAS) original and short forms in an African American community sample. Journal of Affective Disorders, 151(3), 942\u2013949. https:\/\/doi.org\/10.1016\/j.jad.2013.08.011","journal-title":"Journal of Affective Disorders"},{"key":"1674_CR25","first-page":"26","volume":"69","author":"JHL Oud","year":"2002","unstructured":"Oud, J. H. L. (2002). Continuous Time Modeling of the Cross-Lagged Panel Design. Kwantitatieve Methoden, 69, 26.","journal-title":"Kwantitatieve Methoden"},{"key":"1674_CR26","doi-asserted-by":"crossref","unstructured":"Oud, J. H. L., & Delsing, M. J. M. H. (2010). Continuous Time Modeling of Panel Data by means of SEM. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.) Longitudinal Research with Latent Variables (pp. 201\u2013244). Berlin: Springer.","DOI":"10.1007\/978-3-642-11760-2_7"},{"issue":"2","key":"1674_CR27","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1177\/2167702614540645","volume":"3","author":"ML Pe","year":"2015","unstructured":"Pe, M. L., Kircanski, K., Thompson, R. J., Bringmann, L. F., Tuerlinckx, F., Mestdagh, M., ..., Gotlib, I. H. (2015). Emotion-Network Density in Major Depressive Disorder. Clinical Psychological Science, 3(2), 292\u2013300. https:\/\/doi.org\/10.1177\/2167702614540645","journal-title":"Clinical Psychological Science"},{"issue":"5","key":"1674_CR28","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1007\/s12671-014-0354-7","volume":"6","author":"E Snippe","year":"2015","unstructured":"Snippe, E., Bos, E. H., van der Ploeg, K. M., Sanderman, R., Fleer, J., & Schroevers, M. J. (2015). Time-Series Analysis of Daily Changes in Mindfulness, Repetitive Thinking, and Depressive Symptoms During Mindfulness-Based Treatment. Mindfulness, 6(5), 1053\u20131062. https:\/\/doi.org\/10.1007\/s12671-014-0354-7","journal-title":"Mindfulness"},{"issue":"4","key":"1674_CR29","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1023\/A:1008202821328","volume":"11","author":"R Storn","year":"1997","unstructured":"Storn, R., & Price, K. (1997). Differential Evolution \u2013 A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. Journal of Global Optimization, 11(4), 341\u2013359. https:\/\/doi.org\/10.1023\/A:1008202821328","journal-title":"Journal of Global Optimization"},{"issue":"3","key":"1674_CR30","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1037\/0033-295X.99.3.395","volume":"99","author":"HL Van der Maas","year":"1992","unstructured":"Van der Maas, H. L., & Molenaar, P. C. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99(3), 395\u2013417. https:\/\/doi.org\/10.1037\/0033-295X.99.3.395","journal-title":"Psychological Review"},{"issue":"1","key":"1674_CR31","doi-asserted-by":"publisher","first-page":"13","DOI":"10.3758\/s13428-015-0569-0","volume":"48","author":"S Verdonck","year":"2016","unstructured":"Verdonck, S., Meers, K., & Tuerlinckx, F. (2016). Efficient simulation of diffusion-based choice RT models on CPU and GPU. Behavior Research Methods, 48(1), 13\u201327. https:\/\/doi.org\/10.3758\/s13428-015-0569-0","journal-title":"Behavior Research Methods"},{"issue":"1","key":"1674_CR32","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1111\/j.2044-8317.2012.02043.x","volume":"66","author":"MC Voelkle","year":"2013","unstructured":"Voelkle, M. C., & Oud, J. H. L. (2013). Continuous time modelling with individually varying time intervals for oscillating and non-oscillating processes: Continuous time modelling. British Journal of Mathematical and Statistical Psychology, 66(1), 103\u2013126. https:\/\/doi.org\/10.1111\/j.2044-8317.2012.02043.x","journal-title":"British Journal of Mathematical and Statistical Psychology"},{"issue":"7","key":"1674_CR33","doi-asserted-by":"publisher","first-page":"1349","DOI":"10.1017\/S0033291713001979","volume":"44","author":"M Wichers","year":"2014","unstructured":"Wichers, M. (2014). The dynamic nature of depression: a new micro-level perspective of mental disorder that meets current challenges. Psychological Medicine, 44(7), 1349\u20131360. https:\/\/doi.org\/10.1017\/S0033291713001979","journal-title":"Psychological Medicine"},{"issue":"3","key":"1674_CR34","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1037\/\/0022-3514.77.3.600","volume":"77","author":"M Yik","year":"1999","unstructured":"Yik, M., Russell, J. A., & Barrett, L. F. (1999). Structure of Self-Reported Current Affect: Integration and Beyond. Journal of Personality and Social Psychology, 77 (3), 600\u2013619. https:\/\/doi.org\/10.1037\/\/0022-3514.77.3.600","journal-title":"Journal of Personality and Social Psychology"},{"issue":"2","key":"1674_CR35","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1080\/00273171.2013.763012","volume":"48","author":"Y Zheng","year":"2013","unstructured":"Zheng, Y., Wiebe, R. P., Cleveland, H. H., Molenaar, P. C. M., & Harris, K. S. (2013). An Idiographic Examination of Day-to-Day Patterns of Substance Use Craving, Negative Affect, and Tobacco Use Among Young Adults in Recovery. Multivariate Behavioral Research, 48(2), 241\u2013266. https:\/\/doi.org\/10.1080\/00273171.2013.763012","journal-title":"Multivariate Behavioral Research"}],"container-title":["Behavior Research Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.3758\/s13428-021-01674-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.3758\/s13428-021-01674-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.3758\/s13428-021-01674-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,6]],"date-time":"2022-06-06T18:12:23Z","timestamp":1654539143000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.3758\/s13428-021-01674-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,24]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1674"],"URL":"https:\/\/doi.org\/10.3758\/s13428-021-01674-7","relation":{},"ISSN":["1554-3528"],"issn-type":[{"value":"1554-3528","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,24]]},"assertion":[{"value":"16 July 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}