AIUQ (version 0.5.3)

get_MSD: Construct MSD

Description

Construct estimated mean squared displacement (MSD) for a given stochastic process.

Usage

get_MSD(theta, d_input, model_name, msd_fn = NA)

Value

A vector of MSD values for a given sequence of lag times.

Arguments

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 param parameters and d_input lag times

Author

tools:::Rd_package_author("AIUQ")

Details

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 $$MSD_{OU+FBM}(\Delta t) = \theta_2(1-\theta_1^{\Delta t})+\theta_3\Delta t^{\theta_4}$$

References

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.

Examples

library(AIUQ)
# Construct MSD for BM
get_MSD(theta=0.2,d_input=0:100,model_name='BM')