AIUQ: Ab Initio Uncertainty Quantification

Description

Uncertainty quantification and inverse estimation by probabilistic generative models from the beginning of the data analysis. An example is a Fourier basis method for inverse estimation in scattering analysis of microscopy videos. It does not require specifying a certain range of Fourier bases and it substantially reduces computational cost via the generalized Schur algorithm. See the reference: Mengyang Gu, Yue He, Xubo Liu and Yimin Luo (2023), doi:10.48550/arXiv.2309.02468.

Author(s)

Maintainer: Mengyang Gu mengyang@pstat.ucsb.edu

Authors:

• Yue He

• Xubo Liu

2D Fourier transformation and calculate wave number

Description

Perform 2D fast Fourier transformation on SS_T matrix, record frame size for each frame, total number of frames, and sequence of lag times. Calculate and record circular wave number for complete q ring.

Usage

FFT2D(intensity_list, pxsz, mindt)

Arguments

Value

A list object containing transformed intensity profile in reciprocal space and corresponding parameters.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Scattering analysis of microscopy

Description

Fast parameter estimation in scattering analysis of microscopy, using either AIUQ or DDM method.

Usage

SAM(
  intensity = NA,
  intensity_str = "T_SS_mat",
  pxsz = 1,
  sz = c(NA, NA),
  mindt = 1,
  AIUQ_thr = c(1, 1),
  model_name = "BM",
  sigma_0_2_ini = NaN,
  param_initial = NA,
  num_optim = 1,
  msd_fn = NA,
  msd_grad_fn = NA,
  num_param = NA,
  uncertainty = FALSE,
  M = 50,
  sim_object = NA,
  msd_truth = NA,
  method = "AIUQ",
  index_q_AIUQ = NA,
  index_q_DDM = NA,
  message_out = TRUE,
  A_neg = "abs",
  square = FALSE,
  output_dqt = FALSE,
  output_isf = FALSE,
  output_modeled_isf = FALSE,
  output_modeled_dqt = FALSE
)

Arguments

Details

For simulated data using simulation in AIUQ package, intensity will be automatically extracted from simulation class.

By default intensity_str is set to 'T_SS_mat', a time by space\timesspace matrix, which is the structure of intensity profile obtained from simulation class. For intensity_str='SST_array' , input intensity profile should be a space by space by time array, which is the structure from loading a tif file. For intensity_str='S_ST_mat', input intensity profile should be a space by space\timestime matrix.

By default AIUQ_thr is set to c(1,1), uses information from all complete q rings. The first element affects maximum wave number selected, and second element controls minimum proportion of wave number selected. By setting 1 for the second element, if maximum wave number selected is less than the wave number length, then maximum wave number selected is coerced to use all wave number unless user defined another index range through index_q_AIUQ.

If model_name equals 'user_defined', or NA (will coerced to 'user_defined'), then msd_fn and num_param need to be provided for parameter estimation.

Number of particles M is set to 50 or automatically extracted from simulation class for simulated data using simulation in AIUQ package.

By default, using all wave vectors from complete q ring, unless user defined index range through index_q_AIUQ or index_q_DDM.

Value

Returns an S4 object of class SAM.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
# Example 1: Estimation for simulated data
sim_bm = simulation(len_t=100,sz=100,sigma_bm=0.5)
show(sim_bm)
sam = SAM(sim_object = sim_bm)
show(sam)

SAM class

Description

S4 class for fast parameter estimation in scattering analysis of microscopy, using either AIUQ or DDM method.

Slots

pxsz

numeric. Size of one pixel in unit of micron with default value 1.

mindt

numeric. Minimum lag time with default value 1.

sz

vector. Frame size of the intensity profile in x and y directions, number of pixels contained in each frame equals sz_x by sz_y.

len_t

integer. Number of time steps.

len_q

integer. Number of wave vector.

q

vector. Wave vector in unit of um^-1.

d_input

vector. Sequence of lag times.

B_est_ini

numeric. Estimation of B. This parameter is determined by the noise in the system. See 'References'.

A_est_ini

vector. Estimation of A(q). Note this parameter is determined by the properties of the imaged material and imaging optics. See 'References'.

I_o_q_2_ori

vector. Absolute square of Fourier transformed intensity profile, ensemble over time.

q_ori_ring_loc_unique_index

list. List of location index of non-duplicate values for each q ring.

model_name

character. Fitted model, options from ('BM','OU','FBM','OU+FBM', 'user_defined').

param_est

vector. Estimated parameters contained in MSD.

sigma_2_0_est

numeric. Estimated variance of background noise.

msd_est

vector. Estimated MSD.

uncertainty

logical. A logical evaluating to TRUE or FALSE indicating whether parameter uncertainty should be computed.

msd_lower

vector. Lower bound of 95% confidence interval of MSD.

msd_upper

vector. Upper bound of 95% confidence interval of MSD.

msd_truth

vector. True MSD or reference MSD value.

sigma_2_0_truth

vector. True variance of background noise, non NA for simulated data using simulation.

param_truth

vector. True parameters used to construct MSD, non NA for simulated data using simulation.

index_q

vector. Selected index of wave vector.

Dqt

matrix. Dynamic image structure function D(q,delta t).

ISF

matrix. Empirical intermediate scattering function f(q,delta t).

I_q

matrix. Fourier transformed intensity profile with structure 'SS_T_mat'.

AIC

numeric. Akaike information criterion score.

mle

numeric. Maximum log likelihood value.

param_uq_range

matrix. 95% confidence interval for estimated parameters.

modeled_Dqt

matrix. Modeled dynamic image structure function D(q,delta t).

modeled_ISF

matrix. Modeled intermediate scattering function f(q,delta t).

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Compute dynamic image structure function

Description

Compute dynamic image structure function(Dqt) using Fourier transformed intensity profile and a selection of wave number(q) range.

Usage

SAM_Dqt(len_q, index_q, len_t, I_q_matrix, q_ori_ring_loc_unique_index, sz)

Arguments

Details

Dynamic image structure function(Dqt) can be obtained from ensemble average of absolute values squared of Four transformed intensity difference:

D(q,\Delta t) = \langle |\Delta \hat{I}(q,t,\Delta t)|^2\rangle

See 'References'.

Value

Matrix of dynamic image structure with dimension len_q by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Scattering analysis of microscopy for anisotropic processes

Description

Fast parameter estimation in scattering analysis of microscopy for anisotropic processes, using AIUQ method.

Usage

aniso_SAM(
  intensity = NA,
  intensity_str = "T_SS_mat",
  pxsz = 1,
  sz = c(NA, NA),
  mindt = 1,
  AIUQ_thr = c(1, 1),
  model_name = "BM",
  sigma_0_2_ini = NaN,
  param_initial = NA,
  num_optim = 1,
  msd_fn = NA,
  msd_grad_fn = NA,
  num_param = NA,
  uncertainty = FALSE,
  M = 50,
  sim_object = NA,
  msd_truth = NA,
  method = "AIUQ",
  index_q_AIUQ = NA,
  message_out = TRUE,
  square = FALSE
)

Arguments

Details

For simulated data using aniso_simulation in AIUQ package, intensity will be automatically extracted from aniso_simulation class.

By default intensity_str is set to 'T_SS_mat', a time by space\timesspace matrix, which is the structure of intensity profile obtained from aniso_simulation class. For intensity_str='SST_array' , input intensity profile should be a space by space by time array, which is the structure from loading a tif file. For intensity_str='S_ST_mat', input intensity profile should be a space by space\timestime matrix.

By default AIUQ_thr is set to c(1,1), uses information from all complete q rings. The first element affects maximum wave number selected, and second element controls minimum proportion of wave number selected. By setting 1 for the second element, if maximum wave number selected is less than the wave number length, then maximum wave number selected is coerced to use all wave number unless user defined another index range through index_q_AIUQ.

If model_name equals 'user_defined', or NA (will coerced to 'user_defined'), then msd_fn and num_param need to be provided for parameter estimation.

Number of particles M is set to 50 or automatically extracted from simulation class for simulated data using simulation in AIUQ package.

By default, using all wave vectors from complete q ring for both AIUQ, unless user defined index range through index_q_AIUQ.

Value

Returns an S4 object of class aniso_SAM.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
# Example 1: Estimation for simulated data
set.seed(1)
aniso_sim = aniso_simulation(sz=100,len_t=100, model_name="BM",M=100,sigma_bm=c(0.5,0.3))
show(aniso_sim)
plot_traj(object=aniso_sim)
aniso_sam = aniso_SAM(sim_object=aniso_sim, model_name="BM",AIUQ_thr = c(0.999,0))
show(aniso_sam)
plot_MSD(aniso_sam,msd_truth = aniso_sam@msd_truth)

Anisotropic SAM class

Description

S4 class for fast parameter estimation in scattering analysis of microscopy for anisotropic processes, using either AIUQ or DDM method.

Slots

pxsz

numeric. Size of one pixel in unit of micron with default value 1.

mindt

numeric. Minimum lag time with default value 1.

sz

vector. Frame size of the intensity profile in x and y directions, number of pixels contained in each frame equals sz_x by sz_y.

len_t

integer. Number of time steps.

len_q

integer. Number of wave vector.

q

vector. Wave vector in unit of um^-1.

d_input

vector. Sequence of lag times.

B_est_ini

numeric. Estimation of B. This parameter is determined by the noise in the system. See 'References'.

A_est_ini

vector. Estimation of A(q). Note this parameter is determined by the properties of the imaged material and imaging optics. See 'References'.

I_o_q_2_ori

vector. Absolute square of Fourier transformed intensity profile, ensemble over time.

q_ori_ring_loc_unique_index

list. List of location index of non-duplicate values for each q ring.

model_name

character. Fitted model, options from ('BM','OU','FBM','OU+FBM', 'user_defined').

param_est

matrix. Estimated parameters contained in MSD.

sigma_2_0_est

vector. Estimated variance of background noise.

msd_est

matrix. Estimated MSD.

uncertainty

logical. A logical evaluating to TRUE or FALSE indicating whether parameter uncertainty should be computed.

msd_truth

matrix. True MSD or reference MSD value.

sigma_2_0_truth

vector. True variance of background noise, non NA for simulated data using simulation.

param_truth

matrix. True parameters used to construct MSD, non NA for simulated data using aniso_simulation.

index_q

vector. Selected index of wave vector.

I_q

matrix. Fourier transformed intensity profile with structure 'SS_T_mat'.

AIC

numeric. Akaike information criterion score.

mle

numeric. Maximum log likelihood value.

msd_x_lower

vector. Lower bound of 95% confidence interval of MSD in x directions.

msd_x_upper

vector. Upper bound of 95% confidence interval of MSD in x directions.

msd_y_lower

vector. Lower bound of 95% confidence interval of MSD in y directions.

msd_y_upper

vector. Upper bound of 95% confidence interval of MSD in y directions.

param_uq_range

matrix. 95% confidence interval for estimated parameters.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Simulate anisotropic 2D particle movement

Description

Simulate anisotropic 2D particle movement from a user selected stochastic process, and output intensity profiles.

Usage

aniso_simulation(
  sz = c(200, 200),
  len_t = 200,
  M = 50,
  model_name = "BM",
  noise = "gaussian",
  I0 = 20,
  Imax = 255,
  pos0 = matrix(NaN, nrow = M, ncol = 2),
  rho = c(0.95, 0.9),
  H = c(0.4, 0.3),
  sigma_p = 2,
  sigma_bm = c(1, 0.5),
  sigma_ou = c(2, 1.5),
  sigma_fbm = c(2, 1.5)
)

Arguments

Value

Returns an S4 object of class anisotropic_simulation.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
# -------------------------------------------------
# Example 1: Simple diffusion for 200 images with
#            200 by 200 pixels and 50 particles
# -------------------------------------------------
aniso_sim_bm = aniso_simulation()
show(aniso_sim_bm)

# -------------------------------------------------
# Example 2: Simple diffusion for 100 images with
#            100 by 100 pixels and slower speed
# -------------------------------------------------
aniso_sim_bm = aniso_simulation(sz=100,len_t=100,sigma_bm=c(0.5,0.1))
show(aniso_sim_bm)

# -------------------------------------------------
# Example 3: Ornstein-Uhlenbeck process
# -------------------------------------------------
aniso_sim_ou = aniso_simulation(model_name="OU")
show(aniso_sim_ou)

Anisotropic simulation class

Description

S4 class for anisotropic 2D particle movement simulation.

Details

intensity should has structure 'T_SS_mat', matrix with dimension len_t by sz\timessz.

pos should be the position matrix with dimension M\timeslen_t. See bm_particle_intensity, ou_particle_intensity, fbm_particle_intensity, fbm_ou_particle_intensity.

Slots

sz

vector. Frame size of the intensity profile, number of pixels contained in each frame equals sz[1] by sz[2].

len_t

integer. Number of time steps.

noise

character. Background noise, options from ('uniform','gaussian').

model_name

character. Simulated stochastic process, options from ('BM','OU','FBM','OU+FBM').

M

integer. Number of particles.

pxsz

numeric. Size of one pixel in unit of micron, 1 for simulated data.

mindt

numeric. Minimum lag time, 1 for simulated data.

pos

matrix. Position matrix for particle trajectory, see 'Details'.

intensity

matrix. Filled intensity profile, see 'Details'.

num_msd

matrix. Numerical mean squared displacement (MSD).

param

matrix. Parameters used to construct MSD.

theor_msd

matrix. Theoretical MSD.

sigma_2_0

vector. Variance of background noise.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Simulate 2D particle trajectory follows anisotropic Brownian Motion

Description

Simulate 2D particle trajectory follows anisotropic Brownian Motion (BM) for M particles, with different step sizes in x, y-directions.

Usage

anisotropic_bm_particle_intensity(pos0, M, len_t, sigma)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = c(0.5,0.1)
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = anisotropic_bm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma)

Simulate 2D particle trajectory follows anisotropic fBM plus OU

Description

Simulate 2D particle trajectory follows anisotropic fraction Brownian Motion(fBM) plus a Ornstein–Uhlenbeck(OU) process for M particles, with different step sizes in x, y-directions.

Usage

anisotropic_fbm_ou_particle_intensity(
  pos0,
  M,
  len_t,
  sigma_fbm,
  sigma_ou,
  H,
  rho
)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma_fbm = c(2,1)
H = c(0.3,0.4)
sigma_ou = c(2,2.5)
rho = c(0.95,0.9)
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)

pos = anisotropic_fbm_ou_particle_intensity(pos0=pos0, M=M, len_t=len_t,
  sigma_fbm=sigma_fbm, sigma_ou=sigma_ou, H=H, rho=rho)

Simulate 2D particle trajectory follows anisotropic fBM

Description

Simulate 2D particle trajectory follows anisotropic fraction Brownian Motion(fBM) for M particles, with different step sizes in x, y-directions.

Usage

anisotropic_fbm_particle_intensity(pos0, M, len_t, sigma, H)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = c(2,1)
H = c(0.3,0.4)
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = anisotropic_fbm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma,H=H)

Log likelihood for anisotropic processes

Description

This function computes the natural logarithm of the likelihood of the latent factor model for selected range of wave vectors of anisotropic processes. See 'References'.

Usage

anisotropic_log_lik(
  param,
  I_q_cur,
  B_cur,
  index_q,
  I_o_q_2_ori,
  d_input,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  q1,
  q2,
  q1_unique_index,
  q2_unique_index,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

The numerical value of natural logarithm of the likelihood.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Gradient of log likelihood for anisotropic processes

Description

This function computes the gradient for natural logarithm of the likelihood for selected range of wave vectors of anisotropic processes. See 'References'.

Usage

anisotropic_log_lik_grad(
  param,
  I_q_cur,
  B_cur,
  index_q,
  I_o_q_2_ori,
  d_input,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  model_name,
  q1,
  q2,
  q1_unique_index,
  q2_unique_index,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

The numerical value of gradient for natural logarithm of the likelihood.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Compute anisotropic numerical MSD

Description

Compute numerical mean squared displacement(MSD) based on particle trajectory for anisotropic processes in x,y-directions separately.

Usage

anisotropic_numerical_msd(pos, M, len_t)

Arguments

Details

Input pos should be the position matrix with dimension M\timeslen_t. See bm_particle_intensity, ou_particle_intensity, fbm_particle_intensity, fbm_ou_particle_intensity.

Value

A matrix of numerical MSD for given lag times in x,y-directions, dimension 2 by len_t.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
# Simulate particle trajectory for BM
M = 10
len_t = 50
sigma = c(0.5,0.1)
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = anisotropic_bm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma)

# Compute numerical MSD
(num_msd = anisotropic_numerical_msd(pos=pos, M=M, len_t=len_t))

Simulate 2D particle trajectory follows anisotropic OU process

Description

Simulate 2D particle trajectory follows anisotropic Ornstein–Uhlenbeck process(OU) for M particles, with different step sizes in x, y-directions.

Usage

anisotropic_ou_particle_intensity(pos0, M, len_t, sigma, rho)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = c(2,2.5)
rho = c(0.95,0.9)
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = anisotropic_ou_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma, rho=rho)

Simulate 2D particle trajectory follows Brownian Motion

Description

Simulate 2D particle trajectory follows Brownian Motion (BM) for M particles.

Usage

bm_particle_intensity(pos0, M, len_t, sigma)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = 0.5
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = bm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma)

Transform Cartesian coordinates to polar coordinates

Description

Transform ordered pair (x,y), where x and y denotes the directed distance between the point and each of two perpendicular lines, the x-axis and the y-axis, to polar coordinate. Input x and y must have the same length.

Usage

cart2polar(x, y)

Arguments

Value

A data frame with 2 variables, where r is the directed distance from a point designed as the pole, and theta represents the angle, in radians, between the pole and the point.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)

# Input in Cartesian coordinates
(x <- rep(1:3,each = 3))
(y <- rep(1:3,3))

# Data frame with polar coordinates
(polar <- cart2polar(x, y))

Construct correlation matrix for FBM

Description

Construct correlation matrix for fractional Brownian motion.

Usage

corr_fBM(len_t, H)

Arguments

Value

Correlation matrix with dimension len_t-1 by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
len_t = 50
H = 0.3
m = corr_fBM(len_t=len_t,H=H)

Simulate 2D particle trajectory follows fBM plus OU

Description

Simulate 2D particle trajectory follows fraction Brownian Motion(fBM) plus a Ornstein–Uhlenbeck(OU) process for M particles.

Usage

fbm_ou_particle_intensity(pos0, M, len_t, sigma_fbm, sigma_ou, H, rho)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma_fbm = 2
H = 0.3
sigma_ou = 2
rho = 0.95
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)

pos = fbm_ou_particle_intensity(pos0=pos0, M=M, len_t=len_t,
  sigma_fbm=sigma_fbm, sigma_ou=sigma_ou, H=H, rho=rho)

Simulate 2D particle trajectory follows fBM

Description

Simulate 2D particle trajectory follows fraction Brownian Motion(fBM) for M particles.

Usage

fbm_particle_intensity(pos0, M, len_t, sigma, H)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = 2
H = 0.3
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = fbm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma,H=H)

fftshift

Description

Rearranges a 2D Fourier transform x by shifting the zero-frequency component to the center of the matrix.

Usage

fftshift(x, dim = -1)

Arguments

Details

By default, dim=-1, swaps the first quadrant of x with the third, and the second quadrant with the fourth. If dim=1, swaps rows 1 to middle with rows (middle+1) to end. If dim=2, swaps columns 1 to middle with columns (middle+1) to end. If dim=3, reverse fftshift.

Value

Shifted matrix.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)

(m <- matrix(0:8,3,3))
fftshift(m)

Construct intensity profile for a given particle trajectory

Description

Construct intensity profile with structure 'T_SS_mat' for a given particle trajectory, background intensity profile, and user defined radius of particle.

Usage

fill_intensity(len_t, M, I, pos, Ic, sz, sigma_p)

Arguments

Details

Input I should has structure 'T_SS_mat', matrix with dimension len_t by sz\timessz.

Input pos should be the position matrix with dimension M\timeslen_t. See bm_particle_intensity, ou_particle_intensity, fbm_particle_intensity, fbm_ou_particle_intensity.

Value

Intensity profile matrix with structure 'T_SS_mat' (matrix with dimension len_t by sz\timessz).

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

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)

Arguments

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}

Value

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

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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')

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)

Arguments

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}}

Value

A list of two variables, MSD and MSD gradient.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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

Compute observed dynamic image structure function

Description

Compute observed dynamic image structure function (Dqt) using object of SAM class.

Usage

get_dqt(object, index_q = NA)

Arguments

Value

A matrix of observed dynamic image structure function with dimension len_q by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

## Not run: 
library(AIUQ)
sim_bm = simulation(len_t=100,sz=100,sigma_bm=0.5)
show(sim_bm)
sam = SAM(sim_object = sim_bm)
show(sam)
Dqt = get_dqt(object=sam)

## End(Not run)

Transform parameters estimated in SAM class to parameters structure in MSD function

Description

Transform parameters estimated using Maximum Likelihood Estimation (MLE) in SAM class, to parameters contained in MSD with structure theta in get_MSD.

Usage

get_est_param(theta, model_name)

Arguments

Value

A vector of estimated parameters after transformation with structure theta in get_MSD.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Construct 95% confidence interval

Description

This function construct the lower and upper bound for 95% confidence interval of estimated parameters and mean squared displacement(MSD) for a given model. See 'References'.

Usage

get_est_parameters_MSD_SAM_interval(
  param_uq_range,
  model_name,
  d_input,
  msd_fn = NA
)

Arguments

Value

A list of lower and upper bound for 95% confidence interval of estimated parameters and MSD for a given model.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Construct 95% confidence interval for anisotropic processes

Description

This function construct the lower and upper bound for 95% confidence interval of estimated parameters and mean squared displacement(MSD) for a given anisotropic model. See 'References'.

Usage

get_est_parameters_MSD_SAM_interval_anisotropic(
  param_uq_range,
  model_name,
  d_input,
  msd_fn = NA
)

Arguments

Value

A list of lower and upper bound for 95% confidence interval of estimated parameters and MSD for a given model.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Construct parameter transformation for optimization using exact gradient

Description

Construct parameter transformation for parameters to be optimized over in AIUQ method of SAM class. See 'References'.

Usage

get_grad_trans(theta, d_input, model_name)

Arguments

Value

A vector of transformed parameters to be optimized over in AIUQ method of SAM class.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Construct initial values for the parameters to be optimized over

Description

Construct initial values for the parameters to be optimized over in AIUQ method of SAM class.

Usage

get_initial_param(model_name, sigma_0_2_ini = NA, num_param = NA)

Arguments

Details

If model_name equals 'user_defined', then num_param need to be provided to determine the length of the initial values vector.

Value

A matrix with one row of initial values for the parameters to be optimized over in AIUQ method of SAM class.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)
get_initial_param(model_name = "BM")

Compute empirical intermediate scattering function

Description

Compute empirical intermediate scattering function (ISF) using object of SAM class.

Usage

get_isf(object, index_q = NA, msd_truth = NA)

Arguments

Value

A matrix of empirical intermediate scattering function with dimension len_q by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

## Not run: 
library(AIUQ)
sim_bm = simulation(len_t=100,sz=100,sigma_bm=0.5)
show(sim_bm)
sam = SAM(sim_object = sim_bm)
show(sam)
ISF = get_isf(object=sam)

## End(Not run)

Transform parameters in anisotropic simulation class to parameters structure in MSD function

Description

Transform parameters in aniso_simulation class to parameters contained in MSD function with structure theta in get_MSD. Prepare for truth MSD construction.

Usage

get_true_param_aniso_sim(param_truth, model_name)

Arguments

Value

A vector of parameters contained in MSD with structure theta in get_MSD.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
# Simulate simple diffusion for 100 images with 100 by 100 pixels and
# distance moved per time step is 0.5
aniso_sim_bm = aniso_simulation(sz=100,len_t=100,sigma_bm=c(0.5,0.1))
show(aniso_sim_bm)
get_true_param_aniso_sim(param_truth=aniso_sim_bm@param,model_name=aniso_sim_bm@model_name)

Transform parameters in simulation class to parameters structure in MSD function

Description

Transform parameters in simulation class to parameters contained in MSD function with structure theta in get_MSD. Prepare for truth MSD construction.

Usage

get_true_param_sim(param_truth, model_name)

Arguments

Value

A vector of parameters contained in MSD with structure theta in get_MSD.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
# Simulate simple diffusion for 100 images with 100 by 100 pixels and
# distance moved per time step is 0.5
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)
get_true_param_sim(param_truth=sim_bm@param,model_name=sim_bm@model_name)

Transform intensity profile into SS_T matrix

Description

Transform intensity profile with different formats, ('SST_array','T_SS_mat', 'SS_T_mat','S_ST_mat'), space by space by time array, time by (space by space) matrix, (space by space) by time matrix, or space by (space by time) matrix, into 'SS_T_mat'. In addition, crop each frame with odd frame size.

Usage

intensity_format_transform(intensity, intensity_str, square = FALSE, sz = NA)

Arguments

Value

A matrix of transformed intensity profile.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
# -------------------------------------------------
# Example 1: Transform T_SS_mat into SS_T_mat, each
#             frame contains number 1-9
# -------------------------------------------------
(m <- matrix(rep(1:9,4),4,9,byrow=TRUE))
intensity_format_transform(m,intensity_str="T_SS_mat",sz=c(4,9))

# -------------------------------------------------
# Example 2: Transform SST_array into SS_T_mat, each
#             frame contains number 1-9
# -------------------------------------------------
(m <- array(rep(1:9,4),dim=c(3,3,4)))
intensity_format_transform(m,intensity_str="SST_array")

Compute l2 loss for Dqt

Description

Compute l2 loss for dynamic image structure function(Dqt) using A(q) and B are both estimated within the model.

Usage

l2_estAB(
  param,
  Dqt_cur,
  q_cur,
  d_input,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Details

Dynamic image structure function(Dqt) can be obtained from ensemble average of absolute values squared of Four transformed intensity difference:

D(q,\Delta t) = \langle |\Delta \hat{I}(q,t,\Delta t)|^2\rangle

See 'References'.

Value

Squared differences between the true Dqt and the predicted Dqt.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Compute l2 loss for Dqt with fixed A(q) and B

Description

Compute l2 loss for dynamic image structure function(Dqt) using fixed A(q) and B parameters.

Usage

l2_fixedAB(
  param,
  Dqt_cur,
  q_cur,
  A_est_q_cur,
  B_est,
  d_input,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Details

Dynamic image structure function(Dqt) can be obtained from ensemble average of absolute values squared of Four transformed intensity difference:

D(q,\Delta t) = \langle |\Delta \hat{I}(q,t,\Delta t)|^2\rangle

See 'References'.

Value

Squared differences between the true Dqt and the predicted Dqt.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Log likelihood of the model

Description

This function computes the natural logarithm of the likelihood of the latent factor model for selected range of wave vectors. See 'References'.

Usage

log_lik(
  param,
  I_q_cur,
  B_cur,
  index_q,
  I_o_q_2_ori,
  d_input,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  q,
  model_name,
  A_neg,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

The numerical value of natural logarithm of the likelihood.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Gradient of log likelihood

Description

This function computes the gradient for natural logarithm of the likelihood for selected range of wave vectors. See 'References'.

Usage

log_lik_grad(
  param,
  I_q_cur,
  B_cur,
  A_neg,
  index_q,
  I_o_q_2_ori,
  d_input,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  q,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

The numerical value of gradient for natural logarithm of the likelihood.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Compute modeled dynamic image structure function

Description

Compute modeled dynamic image structure function (Dqt) using object of SAM class.

Usage

modeled_dqt(object, index_q = NA, uncertainty = FALSE)

Arguments

Value

A matrix of modeled dynamic image structure function with dimension len_q by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
sim_bm = simulation(len_t=100,sz=100,sigma_bm=0.5)
show(sim_bm)
sam = SAM(sim_object = sim_bm)
show(sam)
modeled_Dqt = modeled_dqt(object=sam)

Compute modeled intermediate scattering function

Description

Compute modeled intermediate scattering function (ISF) using object of SAM class.

Usage

modeled_isf(object, index_q = NA)

Arguments

Value

A matrix of modeled intermediate scattering function with dimension len_q by len_t-1.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
sim_bm = simulation(len_t=100,sz=100,sigma_bm=0.5)
show(sim_bm)
sam = SAM(sim_object = sim_bm)
show(sam)
modeled_ISF = modeled_isf(object=sam)

Compute numerical MSD

Description

Compute numerical mean squared displacement(MSD) based on particle trajectory.

Usage

numerical_msd(pos, M, len_t)

Arguments

Details

Input pos should be the position matrix with dimension M\timeslen_t. See bm_particle_intensity, ou_particle_intensity, fbm_particle_intensity, fbm_ou_particle_intensity.

Value

A vector of numerical MSD for given lag times.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
# Simulate particle trajectory for BM
M = 10
len_t = 50
sigma = 0.5
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = bm_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma)

# Compute numerical MSD
(num_msd = numerical_msd(pos=pos, M=M, len_t = len_t))

Simulate 2D particle trajectory follows OU process

Description

Simulate 2D particle trajectory follows Ornstein–Uhlenbeck process(OU) for M particles.

Usage

ou_particle_intensity(pos0, M, len_t, sigma, rho)

Arguments

Value

Position matrix with dimension M\timeslen_t by 2 for particle trajectory. The first M rows being the initial position pos0.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
M = 10
len_t = 50
sigma = 2
rho = 0.95
pos0 = matrix(100/8+0.75*100*runif(M*2),nrow=M,ncol=2)
pos = ou_particle_intensity(pos0=pos0,M=M,len_t=len_t,sigma=sigma, rho=rho)

Compute 95% confidence interval

Description

This function construct the lower and upper bound for 95% confidence interval of estimated parameters for the given model, including parameters contained in the intermediate scattering function and background noise. See 'References'.

Usage

param_uncertainty(
  param_est,
  I_q_cur,
  B_cur = NA,
  A_neg,
  index_q,
  I_o_q_2_ori,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  d_input,
  q,
  model_name,
  estimation_method = "asymptotic",
  M,
  num_iteration_max,
  lower_bound,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

A matrix of lower and upper bound for natural logarithm of parameters in the fitted model using AIUQ method in SAM class

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Compute 95% confidence interval for anisotropic processes

Description

This function construct the lower and upper bound for 95% confidence interval of estimated parameters for the given anisotropic model, including parameters contained in the intermediate scattering function and background noise. See 'References'.

Usage

param_uncertainty_anisotropic(
  param_est,
  I_q_cur,
  B_cur = NA,
  index_q,
  I_o_q_2_ori,
  q_ori_ring_loc_unique_index,
  sz,
  len_t,
  d_input,
  q1,
  q2,
  q1_unique_index,
  q2_unique_index,
  model_name,
  estimation_method = "asymptotic",
  M,
  num_iteration_max,
  lower_bound,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Value

A matrix of lower and upper bound for natural logarithm of parameters in the fitted model using AIUQ method in SAM class

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Plot estimated MSD with uncertainty from SAM class

Description

Function to plot estimated MSD with uncertainty from SAM class, versus true mean squared displacement(MSD) or given reference values.

Usage

plot_MSD(object, msd_truth = NA, title = NA, log10 = TRUE)

Arguments

Value

A plot of estimated MSD with uncertainty versus truth/reference values.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)

## Simulate BM and get estimated parameters with uncertainty using BM model
# Simulation
set.seed(1)
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)

# AIUQ method: fitting using BM model
sam = SAM(sim_object=sim_bm, uncertainty=TRUE,AIUQ_thr=c(0.999,0))
show(sam)

plot_MSD(object=sam, msd_truth=sam@msd_truth) #in log10 scale
plot_MSD(object=sam, msd_truth=sam@msd_truth,log10=FALSE) #in real scale

Plot 2D intensity

Description

Function to plot 2D intensity profile for a certain frame, default is to plot the first frame. Input can be a matrix (2D) or an array (3D).

Usage

plot_intensity(
  intensity,
  intensity_str = "T_SS_mat",
  frame = 1,
  sz = NA,
  title = NA,
  color = FALSE
)

Arguments

Details

By default intensity_str is set to 'T_SS_mat', a time by space\timesspace matrix, which is the structure of intensity profile obtained from simulation class. For intensity_str='SST_array' , input intensity profile should be a space by space by time array, which is the structure from loading a tif file. For intensity_str='S_ST_mat', input intensity profile should be a space by space\timestime matrix. For intensity_str='SS_T_mat', input intensity profile should be a space\timesspace by time matrix.

Value

2D plot in gray scale (or with color) of selected frame.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)
plot_intensity(sim_bm@intensity, sz=sim_bm@sz)

Plot 2D particle trajectory

Description

Function to plot the particle trajectory after the simulation class has been constructed.

Usage

plot_traj(object, title = NA)

Arguments

Value

2D plot of particle trajectory for a given simulation from simulation class.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

Examples

library(AIUQ)
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)
plot_traj(sim_bm)

Show scattering analysis of microscopy for anisotropic processes (aniso_SAM) object

Description

Function to print the aniso_SAM class object after the aniso_SAM model has been constructed.

Usage

show.aniso_sam(object)

Arguments

Value

Show a list of important parameters in class aniso_SAM.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)

## Simulate BM and get estimated parameters using BM model
# Simulation
aniso_sim_bm = aniso_simulation(sz=100,len_t=100,sigma_bm=c(0.5,0.3))
show(aniso_sim_bm)

# AIUQ method: fitting using BM model
aniso_sam = aniso_SAM(sim_object=aniso_sim_bm, AIUQ_thr=c(0.99,0))
show(aniso_sam)

Show anisotropic simulation object

Description

Function to print the aniso_simulation class object after the aniso_simulation model has been constructed.

Usage

show.aniso_simulation(object)

Arguments

Value

Show a list of important parameters in class aniso_simulation.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)

# Simulate simple diffusion for 100 images with 100 by 100 pixels
aniso_sim_bm = aniso_simulation(sz=100,len_t=100,sigma_bm=c(0.5,0.1))
show(aniso_sim_bm)

Show scattering analysis of microscopy (SAM) object

Description

Function to print the SAM class object after the SAM model has been constructed.

Usage

show.sam(object)

Arguments

Value

Show a list of important parameters in class SAM.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)

## Simulate BM and get estimated parameters using BM model
# Simulation
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)

# AIUQ method: fitting using BM model
sam = SAM(sim_object=sim_bm)
show(sam)

Show simulation object

Description

Function to print the simulation class object after the simulation model has been constructed.

Usage

show.simulation(object)

Arguments

Value

Show a list of important parameters in class simulation.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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)

# Simulate simple diffusion for 100 images with 100 by 100 pixels
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)

Simulate 2D particle movement

Description

Simulate 2D particle movement from a user selected stochastic process, and output intensity profiles.

Usage

simulation(
  sz = c(200, 200),
  len_t = 200,
  M = 50,
  model_name = "BM",
  noise = "gaussian",
  I0 = 20,
  Imax = 255,
  pos0 = matrix(NaN, nrow = M, ncol = 2),
  rho = 0.95,
  H = 0.3,
  sigma_p = 2,
  sigma_bm = 1,
  sigma_ou = 2,
  sigma_fbm = 2
)

Arguments

Value

Returns an S4 object of class simulation.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Examples

library(AIUQ)
# -------------------------------------------------
# Example 1: Simple diffusion for 200 images with
#            200 by 200 pixels and 50 particles
# -------------------------------------------------
sim_bm = simulation()
show(sim_bm)

# -------------------------------------------------
# Example 2: Simple diffusion for 100 images with
#            100 by 100 pixels and slower speed
# -------------------------------------------------
sim_bm = simulation(sz=100,len_t=100,sigma_bm=0.5)
show(sim_bm)

# -------------------------------------------------
# Example 3: Ornstein-Uhlenbeck process
# -------------------------------------------------
sim_ou = simulation(model_name="OU")
show(sim_ou)

Simulation class

Description

S4 class for 2D particle movement simulation.

Details

intensity should has structure 'T_SS_mat', matrix with dimension len_t by sz\timessz.

pos should be the position matrix with dimension M\timeslen_t. See bm_particle_intensity, ou_particle_intensity, fbm_particle_intensity, fbm_ou_particle_intensity.

Slots

sz

vector. Frame size of the intensity profile, number of pixels contained in each frame equals sz[1] by sz[2].

len_t

integer. Number of time steps.

noise

character. Background noise, options from ('uniform','gaussian').

model_name

character. Simulated stochastic process, options from ('BM','OU','FBM','OU+FBM').

M

integer. Number of particles.

pxsz

numeric. Size of one pixel in unit of micron, 1 for simulated data.

mindt

numeric. Minimum lag time, 1 for simulated data.

pos

matrix. Position matrix for particle trajectory, see 'Details'.

intensity

matrix. Filled intensity profile, see 'Details'.

num_msd

vector. Numerical mean squared displacement (MSD).

param

vector. Parameters for simulated stochastic process.

theor_msd

vector. Theoretical MSD.

sigma_2_0

vector. Variance of background noise.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Minimize l2 loss for Dqt

Description

Minimize l2 loss function for dynamic image structure function(Dqt), and return estimated parameters and mean squared displacement(MSD).

Usage

theta_est_l2_dqt_estAB(
  param,
  A_ini,
  q,
  index_q,
  Dqt,
  d_input,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Details

Dynamic image structure function(Dqt) can be obtained from ensemble average of absolute values squared of Four transformed intensity difference:

D(q,\Delta t) = \langle |\Delta \hat{I}(q,t,\Delta t)|^2\rangle

See 'References'.

Value

A list of estimated parameters and MSD from minimizing the l2 loss function.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.

Minimize l2 loss for Dqt with fixed A(q) and B

Description

Minimize l2 loss function for dynamic image structure function(Dqt) with fixed A(q) and B, and return estimated parameters and mean squared displacement(MSD).

Usage

theta_est_l2_dqt_fixedAB(
  param,
  q,
  index_q,
  Dqt,
  A_est_q,
  B_est,
  d_input,
  model_name,
  msd_fn = NA,
  msd_grad_fn = NA
)

Arguments

Details

Dynamic image structure function(Dqt) can be obtained from ensemble average of absolute values squared of Four transformed intensity difference:

D(q,\Delta t) = \langle |\Delta \hat{I}(q,t,\Delta t)|^2\rangle

See 'References'.

Value

A list of estimated parameters and MSD from minimizing the l2 loss function.

Author(s)

Yue He [aut], Xubo Liu [aut], Mengyang Gu [aut, cre]

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.

Cerbino, R., & Trappe, V. (2008). Differential dynamic microscopy: probing wave vector dependent dynamics with a microscope. Physical review letters, 100(18), 188102.