AIUQ (version 0.5.3)

get_MSD_with_grad: Construct MSD and MSD gradient with transformed parameters

Description

Construct mean squared displacement (MSD) and its gradient for a given stochastic process or a user defined MSD and gradient structure.

Usage

get_MSD_with_grad(theta, d_input, model_name, msd_fn = NA, msd_grad_fn = NA)

Value

A list of two variables, MSD and MSD gradient.

Arguments

theta

transformed parameters in MSD function for MLE estimation

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 MSD structure, a function of theta and d_input

msd_grad_fn

user defined MSD gradient structure, a function of theta and d_input

Author

tools:::Rd_package_author("AIUQ")

Details

Note for non user_defined model, msd_fn and msd_grad_fn are not needed. 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 $$MSD_{OU}(\Delta t) = \theta_2(1-\frac{\theta_1}{1+\theta_1}^{\Delta t})$$ where \(\frac{\theta_1}{1+\theta_1}=\rho\) is the correlation with previous steps.

For fractional Brownian Motion, the MSD follows $$MSD_{FBM}(\Delta t) =\theta_1\Delta t^{\frac{2\theta_2}{1+\theta_2}}$$ where \(\frac{2\theta_2}{1+\theta_2}=2H\) with H is the the Hurst parameter.

For 'OU+FBM', the MSD follows $$MSD_{OU+FBM}(\Delta t) = \theta_2(1-\frac{\theta_1}{1+\theta_1}^{\Delta t})+\theta_3\Delta t^{\frac{2\theta_4}{1+\theta_4}}$$

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)
msd_fn <- function(param, d_input){
  beta = 2*param[1]^2
  MSD = beta*d_input
}
msd_grad_fn <- function(param, d_input){
  MSD_grad = 4*param[1]*d_input
}

theta = 2
d_input = 0:10
model_name = "user_defined"

MSD_list = get_MSD_with_grad(theta=theta,d_input=d_input,
                             model_name=model_name,msd_fn=msd_fn,
                             msd_grad_fn=msd_grad_fn)
MSD_list$msd
MSD_list$msd_grad