get_MSD {AIUQ} | R Documentation |
Construct estimated mean squared displacement (MSD) for a given stochastic process.
get_MSD(theta, d_input, model_name, msd_fn = NA)
theta |
parameters in MSD function |
d_input |
sequence of lag times |
model_name |
model name for the process, options from ('BM','OU','FBM', 'OU+FBM','user_defined') |
msd_fn |
user defined mean squared displacement structure (MSD), a
function of |
For Brownian Motion, the MSD follows
MSD_{BM}(\Delta t) = \theta_1\Delta t= 4D\Delta t
where D
is the diffusion coefficient.
For Ornstein–Uhlenbeck process, the MSD follows
MSD_{OU}(\Delta t) = \theta_2(1-\theta_1^{\Delta t})
where \theta_1=\rho
is the correlation with previous steps.
For fractional Brownian Motion, the MSD follows
MSD_{FBM}(\Delta t) =\theta_1\Delta t^{\theta_2}
where \theta_2=2H
with H
is the the Hurst parameter.
For 'OU+FBM', the MSD follows
MSD_{OU+FBM}(\Delta t) = \theta_2(1-\theta_1^{\Delta t})+\theta_3\Delta t^{\theta_4}
A vector of MSD values for a given sequence of lag times.
Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]
Gu, M., He, Y., Liu, X., & Luo, Y. (2023). Ab initio uncertainty quantification in scattering analysis of microscopy. arXiv preprint arXiv:2309.02468.
Gu, M., Luo, Y., He, Y., Helgeson, M. E., & Valentine, M. T. (2021). Uncertainty quantification and estimation in differential dynamic microscopy. Physical Review E, 104(3), 034610.
library(AIUQ)
# Construct MSD for BM
get_MSD(theta=0.2,d_input=0:100,model_name='BM')