Stochastic differential equations mixed-effects models

This is a collection of resources pertaining so called stochastic differential equations mixed-effects models (SDEMEMs). SDEMEMs are powerful, dynamical hierarchical models with time-dependency driven by stochastic differential equations. SDEMEMs are useful for population estimation, where random variation between several experiments or between several subjects is explictly taken into account, together with subject-specific intrinsic random dynamics. SDEMEMs might result in state-space models when measurement error is considered.
The reason for compiling this list of resources is that publications in this area are sparse. Therefore having a list of references can be handy for the uninitiated reader. Software and other resources are also provided. Please contact me to help enlarge the list!
  1. M. Lavielle, “Mixed Effects Models for the Population Approach. Models, Tasks, Methods & Tools. ”, Chapman & Hall/CRC Biostatistics Series, 2014.
    - See section 6.3 for SDE models and mixed effects.

  2. Reviews
  3. Donnet, S., & Samson, A. (2013). A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models. Advanced Drug Delivery Reviews, 65(7), 929-939.

  4. Papers (most recent first)
  5. Picchini, U., & Forman, J. L. (2019). Bayesian inference for stochastic differential equation mixed effects models of a tumor xenography study. Journal of the Royal Statistical Society: Series C, doi:10.1111/rssc.12347.
  6. El Omari, M., El Maroufy, H., Fuchs C. (2019). Nonparametric estimation for fractional diffusion processes with random effects. arXiv:1901.05547. [Notice, unlike other entries in this list, here the stochastic noise is not Brownian motion, but instead fractional Brownian motion.]
  7. García, O. (2018). Estimating reducible stochastic differential equations by conversion to a least-squares problem. Computational Statistics, doi:10.1007/s00180-018-0837-4.
  8. Olafsdottir, H. K., Leander, J., Almquist, J., & Jirstrand, M. (2018). Exact Gradients Improve Parameter Estimation in Nonlinear Mixed Effects Models with Stochastic Dynamics. The AAPS Journal, 20(5), 88.
  9. Delattre, M., Genon-Catalot, V. and Larédo, C. (2018) Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion. Metrika, DOI 10.1007/s00184-018-0666-z
  10. Delattre, M., Genon-Catalot, V. and Larédo, C. (2017) Parametric inference for discrete observations of diffusion processes with mixed effects. Stochastic Processes and their Applications 128(6) p. 1929-1957
  11. Ruse, M. G., Samson, A., & Ditlevsen, S. (2017). Multivariate inhomogeneous diffusion models with covariates and mixed effects. arXiv preprint arXiv:1701.08284.
  12. Whitaker, G. A., Golightly, A., Boys, R. J., & Sherlock, C. (2017). Bayesian inference for diffusion-driven mixed-effects models. Bayesian Analysis, 12(2), 435-463.
  13. Dion, C. (2016). Nonparametric estimation in a mixed-effect Ornstein–Uhlenbeck model. Metrika, 79(8), 919-951.
  14. Dion, C., Hermann, S., & Samson, A. (2016). Mixedsde: a R package to fit mixed stochastic differential equations. 〈hal-01305574〉 [paper] [R code]
  15. Dion, C., & Genon-Catalot, V. (2016). Bidimensional random effect estimation in mixed stochastic differential model. Statistical Inference for Stochastic Processes, 19(2), 131-158.
  16. Delattre, M., Genon-Catalot, V., & Samson, A. (2016). Mixtures of stochastic differential equations with random effects: application to data clustering. Journal of Statistical Planning and Inference, 173, 109-124.
  17. Leander, J., Almquist, J., Ahlström, C., Gabrielsson, J., & Jirstrand, M. (2015). Mixed effects modeling using stochastic differential equations: illustrated by pharmacokinetic data of nicotinic acid in obese Zucker rats. The AAPS journal, 17(3), 586-596
  18. Rupšys, P. (2015). Generalized fixed-effects and mixed-effects parameters height–diameter models with diffusion processes. International Journal of Biomathematics, 8(05), 1550060.
  19. Delattre, M., Genon-Catalot, V. and Samson, A. (2015) Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient. ESAIM: Probability and Statistics 19 p. 671-688
  20. Donnet, S., & Samson, A. (2014). Using PMCMC in EM algorithm for stochastic mixed models: theoretical and practical issues. Journal de la Société Française de Statistique, Journal de la Société Française de Statistique, 155 (1), pp.49-72.
  21. Delattre, M., Genon-Catalot, V. and Samson, A. (2013) Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects. Scandinavian Journal of Statistics 40(2) p. 322-343
  22. Delattre, M., & Lavielle, M. (2013). Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models. Statistics and its interface, 6(4), 519-532.
  23. Donnet, S., & Samson, A. (2013). A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models. Advanced Drug Delivery Reviews, 65(7), 929-939.
  24. Comte, F., Genon-Catalot, V., & Samson, A. (2013). Nonparametric estimation for stochastic differential equations with random effects. Stochastic Processes and their Applications, 123(7), 2522-2551.
  25. Berglund, M., Sunnåker, M., Adiels, M., Jirstrand, M., & Wennberg, B. (2011). Investigations of a compartmental model for leucine kinetics using non-linear mixed effects models with ordinary and stochastic differential equations. Mathematical medicine and biology: a journal of the IMA, 29(4), 361-384.
  26. Picchini, U., & Ditlevsen, S. (2011). Practical estimation of high dimensional stochastic differential mixed-effects models. Computational Statistics & Data Analysis, 55(3), 1426-1444
  27. Picchini, U., De Gaetano, A., & Ditlevsen, S. (2010). Stochastic differential mixed‐effects models. Scandinavian Journal of statistics, 37(1), 67-90.
  28. Donnet, S., Foulley, J. L., & Samson, A. (2010). Bayesian analysis of growth curves using mixed models defined by stochastic differential equations. Biometrics, 66(3), 733-741.
  29. Klim, S., Mortensen, S. B., Kristensen, N. R., Overgaard, R. V., & Madsen, H. (2009). Population stochastic modelling (PSM)—an R package for mixed-effects models based on stochastic differential equations. Computer methods and programs in biomedicine, 94(3), 279-289.
  30. Picchini, U., Ditlevsen, S., De Gaetano, A., & Lansky, P. (2008). Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal. Neural Computation, 20(11), 2696-2714.
  31. Donnet, S., & Samson, A. (2008). Parametric inference for mixed models defined by stochastic differential equations. ESAIM: Probability and Statistics, 12, 196-218.
  32. Mortensen, S. B., Klim, S., Dammann, B., Kristensen, N. R., Madsen, H., & Overgaard, R. V. (2007). A Matlab framework for estimation of NLME models using stochastic differential equations. Journal of Pharmacokinetics and Pharmacodynamics, 34(5), 623-642.
  33. Ditlevsen, S., & De Gaetano, A. (2005). Mixed effects in stochastic differential equation models. REVSTAT-Statistical Journal, 3(2), 137-153.
  34. Tornøe, C. W., Overgaard, R. V., Agersø, H., Nielsen, H. A., Madsen, H., & Jonsson, E. N. (2005). Stochastic differential equations in NONMEM®: implementation, application, and comparison with ordinary differential equations. Pharmaceutical Research, 22(8), 1247-1258.
  35. Overgaard, R. V., Jonsson, N., Tornøe, C. W., & Madsen, H. (2005). Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm. Journal of pharmacokinetics and pharmacodynamics, 32(1), 85-107.
  36. - The last two references above are the oldest SDEMEM papers I am aware of. Also, notice that in an older paper the same authors write "A mixed-effect PK/PD modelling framework as implemented in NONMEM is not considered since the theory for hierarchical PK/PD modelling using stochastic differential equations has not been developed."

    Book chapters
  37. Strathe, A. B., Danfœr, A., Nielsen, B., Klim, S., & Sørensen, H. (2011). Population based growth curve analysis: a comparison between models based on ordinary or stochastic differential equations implemented in a nonlinear mixed effect framework. Book chapter in Modelling nutrient digestion and utilisation in farm animals (pp. 22-30). Wageningen Academic Publishers, Wageningen

  38. Software
  39. PSM: Non-Linear Mixed-Effects Modelling using Stochastic Differential Equations, [R package] [project page]
  40. mixedsde: Estimation Methods for Stochastic Differential Mixed Effects Models, [R package]
  41. MsdeParEst: Parametric Estimation in Mixed-Effects Stochastic Differential Equations, [R package]
  42. CTSM-R - Continuous Time Stochastic Modelling for R [R package]

  43. PhD theses
  44. Ruse, M.G. (2017). Inference from stochastic processes with application to birdsongs and biomedicine. University of Copenhagen, Denmark [PDF]
  45. Whitaker, G A. (2016). Bayesian inference for stochastic differential mixed-effects models. Newcastle University, UK [url]
  46. Dion, C. (2016). Estimation non-paramétrique de la densité de variables aléatoires cachées. Université Grenoble Alpes, France〈NNT : 2016GREAM031.〈tel-01685528〉[url]
  47. Delattre, M. (2012) Inférence statistique dans les modèles mixtes à dynamique Markovienne. Université Paris Sud - Paris XI, France〈tel-00765708〉[url]
  48. Picchini, U. (2007). Stochastic Differential Models with Applications to Physiology. Department of Statistics, Probability and Applied Statistics, University of Rome "La Sapienza", Italy.
  49. Tornøe, CW, Madsen, H & Nielsen, HA (2005), Population pharmacokinetic/pharmacodynamic modelling of the hypothalamic-pituitary-gonadal axis. DTU, Technical University of Denmark [url]