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coupled_osc.R
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coupled_osc.R
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# Coupled Oscillator Model parameter optimization with UKF on fMRI data
### Libraries
#- KernSmooth: Kernel smoothing on voxel data
#- deSolve: Solve ODEs
#- ggplot2: Plotting
# UKF library functions are imported locally for modifications. Original functions are designed to work on optimization of 2 variables, we modified the function for the new model. Plotting functions are implemented with ggplot2.
library(KernSmooth)
library(deSolve)
library(ggplot2)
#library(devtools)
#install_github("insilico/UKF")
# library(UKF)
#install.packages(c('KernSmooth'))
# hide warnings
options(warn=-1)
## Unscented Kalman Filter - UKF
# This code is taken from https://github.com/insilico/UKF/blob/main/R/UKF.R
# We modifed the UKF_blend and iterative_param_optim functions to work with new model.
UKF_dT <- function(t_dummy,ode_model,xhat,Pxx,y,N_p,N_y,R,dt,dT){
N_x <- N_p + N_y # number of augmented states
N_sigma <- 2*N_x # number sigma points
xsigma <- t(chol(N_x*Pxx, pivot=T))
Xa <- xhat %*% matrix(rep(1,length=N_sigma),nrow=1,ncol=N_sigma) + cbind(xsigma,-xsigma)
X <- propagate_model(t_dummy,ode_model,dt,dT,N_p,Xa)
xtilde <- t(t(rowSums(X))/N_sigma)
Pxx <- matrix(rep(0,length=N_x*N_x),nrow=N_x,ncol=N_x)
for(i in 1:N_sigma){
Pxx <- Pxx + ((X[,i] - xtilde) %*% t(X[,i] - xtilde))/N_sigma
}
Y <- X[(N_p+1):(N_p+N_y),]
ytilde <- t(t(rowSums(Y))/N_sigma)
Pyy <- R
for(i in 1:N_sigma){
Pyy <- Pyy + ((Y[,i] - ytilde) %*% t(Y[,i] - ytilde))/N_sigma
}
Pxy <- matrix(rep(0,length = N_x*N_y),nrow=N_x,ncol=N_y)
for(i in 1:N_sigma){
Pxy <- Pxy + ((X[,i] - xtilde) %*% t(Y[,i] - ytilde))/N_sigma
}
K <- Pxy %*% solve(Pyy) # Kalman Gain Matrix
xhat <- xtilde + K %*% (y - ytilde)
Pxx <- Pxx - K %*% t(Pxy)
return(list(xhat = xhat, Pxx = Pxx, K = K))
} # ENDFN UKF
propagate_model <- function(t,ode_model,dt,dT,N_p,x){
nn <- round(dT/dt)
p <- x[1:N_p,] # grab parameters
y <- x[(N_p+1):length(x[,1]),] # grab ind vars
# Runge-Kutta
for(n in 1:nn){
k1 <- dt*ode_model(t,y,p)
k2 <- dt*ode_model(t,y + k1/2,p)
k3 <- dt*ode_model(t,y + k2/2,p)
k4 <- dt*ode_model(t,y + k3,p)
y <- y + k1/6 + k2/ + k3/3 + k4/6
}
r <- rbind(x[(1:N_p),],y) # returns new augmented
return(r)
} # ENDFN propagate_model
UKF_blend <- function(t_dummy,ts_data,ode_model,N_p,N_y,param_guess,dt,dT,R_scale=0.3,Q_scale=0.015){
time_points <- ts_data[,1]
num_time <- length(time_points)
N_x <- N_p + N_y
# Modified the length of xhat to be N_x*num_time instead of 4*num_time
xhat <- matrix(rep(0,length=N_x*num_time),nrow=N_x,ncol=num_time)
Pxx <- vector(mode = "list", length = num_time)
for(i in 1:num_time){
Pxx[[i]] <- matrix(rep(0,length=N_x*N_x),nrow=N_x,ncol=N_x)
}
# Modified the length of xhat to be N_x*num_time instead of 4*num_time
errors <- matrix(rep(0,length=N_x*num_time), nrow=N_x,ncol=num_time)
Ks <- vector(mode = "list", length = num_time)
for(i in 1:num_time){
Ks[[i]] <- matrix(rep(0,length=N_x*N_y),nrow=N_x,ncol=N_y)
}
z <- t(t(param_guess)) %*% matrix(rep(1,length=num_time),nrow=1,ncol=num_time)
y0 <- t(ts_data[,-1])
x <- rbind(z,y0) # stack
xhat[,1] <- x[,1] # right now xhat is all data
R <- (R_scale)^2*cov(t(x[(N_p+1):(N_y+N_p),])) # only y
Pxx[[1]] <- pracma::blkdiag(Q_scale*diag(N_p),R)
set.seed(1)
y <- x[(N_p+1):(N_y+N_p),] + (pracma::sqrtm(R)$B) %*% # adding noise
matrix(rnorm(2*num_time),nrow=2,ncol=num_time)
UKF_kstep <- list(xhat = xhat[,1], Pxx = Pxx[[1]], K = Ks[[1]])
for(k in 2:num_time){
UKF_kstep <- tryCatch(
{
UKF_dT(t_dummy,ode_model,xhat[,k-1],Pxx[[k-1]],y[,k], N_p,N_y,R,dt,dT)
},
error=function(cond) {
message("By chance, matrix caused Cholesky to fail.")
message("Here's the original error message:")
message(cond)
message("Skipping this iteration.")
return(UKF_kstep)
}
) # end tryCatch
xhat[,k] <- UKF_kstep$xhat
Pxx[[k]] <- UKF_kstep$Pxx
Ks[[k]] <- UKF_kstep$K
errors[,k] <- sqrt(diag(Pxx[[k]]))
} # end application of UKF blend to all time points
param_estimated <- t(t(xhat[(1:N_p),num_time]))
chisq <- 0
for(i in 1:N_y){
chisq <- chisq + (x[(N_p+i),] - xhat[(N_p+i),])^2
}
chisq <- mean(chisq)
error <- t(errors[(1:N_p),num_time])
return(list(param_est=param_estimated, xhat=xhat,error=error,chisq=chisq))
} # END FN UKF_blend
optim_params <- function(param_guess,method="L-BFGS-B",lower_lim,upper_lim,maxit,temp=20,
t_dummy,ts_data,ode_model,N_p,N_y,dt,dT){
ukf_obj <- NULL
chisq_objective <- function(par_vec){
ukf_obj <<- UKF_blend(t_dummy,ts_data, ode_model, N_p,N_y,par_vec,dt,dT)
return(ukf_obj$chisq)
}
if (method=="SANN"){
# simulated annealing
opt <- optim(param_guess,chisq_objective,method="SANN",
control = list(maxit = maxit, temp = temp))
} else{
# Broyden-Fletcher-Goldfarb-Shannon
opt <- optim(param_guess,chisq_objective,method="L-BFGS-B",
lower=lower_lim,upper=upper_lim)
}
return(list(par=opt$par, value=opt$value, param_est=opt$par, xhat=ukf_obj$xhat))
}
iterative_param_optim <- function(param_guess, t_dummy,ts_data,ode_model, N_p,N_y,dt,dT, param_tol=.01,MAXSTEPS=30){
done <- F
steps <- 0
while (!done){
ukf_run <- UKF_blend(t_dummy,ts_data, ode_model, N_p,N_y,param_guess,dt,dT)
param_new <- ukf_run$param_est
steps <- steps + 1
param_norm <- abs(sum(param_new-param_guess))
converged <- param_norm < param_tol
done <- converged | steps >= MAXSTEPS
}
# Modified output varaiable par to param_est to match the UKF_blend output
return(list(param_est=param_new,value=ukf_run$chisq, xhat=ukf_run$xhat, param_norm=param_norm,steps=steps))
}
## Kernel Smoothing and Plot
# These functions are the modified versions of plot_voxels_and_smooth and plot_ukf_and_smoothed functions provided in https://github.com/insilico/UKF/blob/main/R/plot_and_smooth.R
# We created the ggplot2 versions of these functions and modified the plot_voxels_and_smooth_ggplot to utilize plot_ukf_and_smoothed_ggplot function.
plot_ukf_and_smoothed_ggplot <- function(ukf_out, smoothed_data, top_title="UKF estimate and kernel smooth"){
# Modification for the number of parameters, previously it was used as ukf_out$xhat[3,]
n_p <- length(ukf_out$param_est) + 1
# Create dataframes
df1 <- data.frame(Time = 1:length(ukf_out$xhat[n_p,]),
Value = c(ukf_out$xhat[n_p,], smoothed_data[,2]),
Type = rep(c("UKF", "Kernel"), each=length(ukf_out$xhat[n_p+1,])))
df2 <- data.frame(Time = 1:length(ukf_out$xhat[n_p+1,]),
Value = c(ukf_out$xhat[n_p+1,], smoothed_data[,3]),
Type = rep(c("UKF", "Kernel"), each=length(ukf_out$xhat[n_p+1,])))
# Plot for Voxel A
p1 <- ggplot(df1, aes(x = Time, y = Value, color = Type)) +
geom_point(data = subset(df1, Type == "UKF")) +
geom_line(data = subset(df1, Type == "Kernel")) +
labs(title = top_title, y = "Voxel A") +
theme_minimal() +
scale_color_manual(values = c("red", "darkgrey"))
param_title <- paste0("p", seq_along(ukf_out$param_est), " = ", round(ukf_out$param_est, digits=2), collapse=", ")
# Plot for Voxel B
p2 <- ggplot(df2, aes(x = Time, y = Value, color = Type)) +
geom_point(data = subset(df2, Type == "UKF")) +
geom_line(data = subset(df2, Type == "Kernel")) +
labs(title = param_title, y = "Voxel B") +
theme_minimal() +
scale_color_manual(values = c("red", "darkgrey"))
print(p1)
print(p2)
}
plot_voxels_and_smooth_ggplot <- function(voxel_input){
voxel_Time <- voxel_input[,1]
voxel_data <- cbind(voxel_Time,voxel_input[,2], voxel_input[,3])
voxel.df <- data.frame(times=voxel_data[,1], VoxA=voxel_data[,2], VoxB=voxel_data[,3])
y1_obs <- voxel.df[,2]
y2_obs <- voxel.df[,3]
n <- length(t(voxel_Time))
m_points <- seq(1,n,by=1)
h <- tryCatch( # kernel bandwidth using KernSmooth
{dpill(m_points, y1_obs)},
error=function(cond) {
message("Install/load KernSmooth for better smoothing.")
message(cond)
message("Using default kernel bandwidth h=.5.")
return(0.5)
}
) # end tryCatch
fit1 <- ksmooth(m_points, y1_obs, kernel="normal",bandwidth = h, n.points=176)
h <- dpill(m_points, y2_obs)
fit2 <- ksmooth(m_points, y2_obs, kernel="normal", bandwidth = h, n.points=176)
y1_obs.processed <- fit1$y
y2_obs.processed <- fit2$y
ukf_out <- list(xhat = rbind(y1_obs.processed, y2_obs.processed, y1_obs, y2_obs), param_est = c(0, 0)) # revised ukf_out
smoothed_data <- cbind(t(t(m_points)), t(t(y1_obs.processed)), t(t(y2_obs.processed)))
plot_ukf_and_smoothed_ggplot(ukf_out, smoothed_data, top_title="Real data and kernel smooth")
return(smoothed_data)
}
## New Coupled Oscillator Model - Two Coupled Pendulums
#The equation of motion of the combined system
#is then given by:
#$$L \ddot{\theta}_{1} =-g \sin \theta_{1}-k L\left(\sin \theta_{1}-\sin \theta_{2}\right)$$
#$$L \ddot{\theta}_{2} =-g \sin \theta_{2}+k L\left(\sin \theta_{1}-\sin \theta_{2}\right)$$
#There are 3 parameters in the equation:
#- g, gravity
#- L, length of pendulums
#- k, spring constant
#without the assumption of pendulums having same length, $L_1$ and $L_2$ would replace the L in corresponding equation:
#$$L_1 \ddot{\theta}_{1} =-g \sin \theta_{1}-k L_1\left(\sin \theta_{1}-\sin \theta_{2}\right)$$
#$$L_2 \ddot{\theta}_{2} =-g \sin \theta_{2}+k L_2\left(\sin \theta_{1}-\sin \theta_{2}\right)$$
new_coupled_osc_model <- function(t, x, p){
# Parameters
g <- p[1,]
L1 <- p[2,]
L2 <- p[3,]
k <- p[4,]
# State variables
theta1 <- x[1,]
theta2 <- x[2,]
# Equations of motion
theta1_dot_dot <- - g / L1 * sin(theta1) - k * (sin(theta1) - sin(theta2))
theta2_dot_dot <- - g / L2 * sin(theta2) + k * (sin(theta1) - sin(theta2))
return(rbind(theta1_dot_dot, theta2_dot_dot))
}
# Let’s put in some specific initial conditions: we leave pendulum number 2 at rest in its equilibrium position $(\theta_2(0)=\dot{\theta_2}(0)=0)$ and give pendulum number 1 a finite amplitude but also release it at rest $(\theta_1(0)=\theta_0,\dot{\theta_1}(0)=0)$.
times <- seq(0, 50, by = .1)
init <- c(theta1 = 0.1, theta1_dot = 0, theta2 = 0, theta2_dot = 0)
parameters <- c(g = 9.81, L1 = 1, L2 = 1, k = 1)
new_coupled_osc_model_ode <- function(t, state, parameters){
with(as.list(c(state, parameters)), {
theta1_dot <- theta1_dot # velocity of pendulum 1
theta2_dot <- theta2_dot # velocity of pendulum 2
theta1_dot_dot <- -g/L1*sin(theta1) - k*(sin(theta1) - sin(theta2)) # acceleration of pendulum 1
theta2_dot_dot <- -g/L2*sin(theta2) + k*(sin(theta1) - sin(theta2)) # acceleration of pendulum 2
return(list(c(theta1_dot, theta1_dot_dot, theta2_dot, theta2_dot_dot)))
})
}
out <- ode(y = init, times = times, func = new_coupled_osc_model_ode, parms = parameters)
out <- as.data.frame(out)
plot(out$time, out$theta1, type = "l", ylab = "Theta", xlab = "Time", col = "blue", main = "Theta1 and Theta2 over time")
lines(out$time, out$theta2, col = "red")
legend("topright", legend = c("Theta1", "Theta2"), fill = c("blue", "red"))
## fMRI Data
### Voxel A & B data
vox.A.B.data <- read.delim("data/voxel_A&B_data.txt",sep="",header=F)
# make column-wise
vox.A.B.data <- t(vox.A.B.data)
# dataset detailed information
head(vox.A.B.data)
tail(vox.A.B.data)
summary(vox.A.B.data)
str(vox.A.B.data)
dim(vox.A.B.data)
any(is.na(vox.A.B.data))
smoothed_data <- plot_voxels_and_smooth_ggplot(vox.A.B.data)
### Harvard-Oxford atlas dataset
# Read the ASD Subject 01 dataset and extract the voxel data for 18 (Left Amygdala) and 26th (Left Accumbens) voxels
subj1 <- read.csv("data/ASD Subject 01.csv")
vox.7.subj1 <- subj1[,7] # 18
vox.8.subj1 <- subj1[,8] # 26
vox.7.subj1 <- t(t(vox.7.subj1))
vox.8.subj1 <- t(t(vox.8.subj1))
# Read the Normal Subject 01 dataset and extract the voxel data for 18 (Left Amygdala) and 26th (Left Accumbens) voxels
subj2 <- read.csv("data/Normal Subject 01.csv")
vox.7.subj2 <- subj2[,7] # 18
vox.8.subj2 <- subj2[,8] # 26
vox.7.subj2 <- t(t(vox.7.subj2))
vox.8.subj2 <- t(t(vox.8.subj2))
# Combine columns and time vector
vox.A.B.subj1 <- cbind(vox.A.B.data[,1], vox.7.subj1, vox.8.subj1)
vox.A.B.subj2 <- cbind(vox.A.B.data[,1], vox.7.subj2, vox.8.subj2)
head(vox.A.B.subj1)
tail(vox.A.B.subj1)
summary(vox.A.B.subj1)
str(vox.A.B.subj1)
dim(vox.A.B.subj1)
any(is.na(vox.A.B.subj1))
smoothed_subj1 <- plot_voxels_and_smooth_ggplot(vox.A.B.subj1)
smoothed_subj2 <- plot_voxels_and_smooth_ggplot(vox.A.B.subj2)
# Inital Parameters
param_guess <- c(1,1,1,4)
t_vec <- vox.A.B.data[,1] # time is first column
dT <- t_vec[2]-t_vec[1] # assume uniform time steps dT=1
# smaller steps size for propagating model between dT steps
dt <- 0.1*dT
# num observed ind vars, first col is time, so -1
N_y <- ncol(vox.A.B.data)-1 # N_y=2
# number of unknown model parameters to be estimated
N_p <- 4
# size of augmented state vector
N_x <- N_p + N_y
# Run UKF with new coupled oscillator model and smoothed data as input one pass through the time series
ukf_out <- UKF_blend(t_dummy,vox.A.B.data,new_coupled_osc_model, N_p,N_y,param_guess,dt,dT)
ukf_out$param_est
ukf_out$chisq
plot_ukf_and_smoothed_ggplot(ukf_out, smoothed_data, top_title='One UKF Step, Raw Data')
## Parameter Optimization
### Iterative Approach
# Voxel A & B data
iter_opt <- iterative_param_optim(param_guess, t_dummy, smoothed_data, new_coupled_osc_model, N_p,N_y,dt,dT,
param_tol=.01,MAXSTEPS=100)
iter_opt$param_est # params
iter_opt$value # chi-square
iter_opt$steps
iter_opt$param_norm
plot_ukf_and_smoothed_ggplot(iter_opt, smoothed_data, top_title = 'Iterative Optimization')
# ASD Subject 01
iter_opt_subj1 <- iterative_param_optim(param_guess, t_dummy, smoothed_subj1, new_coupled_osc_model, N_p,N_y,dt,dT,
param_tol=.01,MAXSTEPS=100)
iter_opt_subj1$param_est # params
iter_opt_subj1$value # chi-square
iter_opt_subj1$steps
iter_opt_subj1$param_norm
plot_ukf_and_smoothed_ggplot(iter_opt_subj1, smoothed_subj1, top_title = 'Iterative Optimization')
# Normal Subject 01
iter_opt_subj2 <- iterative_param_optim(param_guess, t_dummy, smoothed_subj2, new_coupled_osc_model, N_p,N_y,dt,dT,
param_tol=.01,MAXSTEPS=100)
iter_opt_subj2$param_est # params
iter_opt_subj2$value # chi-square
iter_opt_subj2$steps
iter_opt_subj2$param_norm
plot_ukf_and_smoothed_ggplot(iter_opt_subj2, smoothed_subj2, top_title = 'Iterative Optimization')
### Simulated Annealing
# Simulated Annealing with smoothed data as input
# to optimize model parameters, chisquare goodness of fit
# Simulated Annealing, SANN ignores lower/upper limits
# Voxel A & B data
opt_sann <- optim_params(param_guess,method="SANN",lower_lim=-20,upper_lim=20,maxit=100,temp=20,
t_dummy,smoothed_data,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_sann$param_est # params
opt_sann$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_sann, smoothed_data, top_title = 'Simulated Annealing')
# ASD Subject 01
opt_sann_s1 <- optim_params(param_guess,method="SANN",lower_lim=-20,upper_lim=20,maxit=100,temp=20,
t_dummy,smoothed_subj1,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_sann_s1$param_est # params
opt_sann_s1$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_sann_s1, smoothed_subj1, top_title = 'Simulated Annealing')
# Normal Subject 01
opt_sann_s2 <- optim_params(param_guess,method="SANN",lower_lim=-20,upper_lim=20,maxit=100,temp=20,
t_dummy,smoothed_subj2,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_sann_s2$param_est # params
opt_sann_s2$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_sann_s2, smoothed_subj2, top_title = 'Simulated Annealing')
### Nelder-Mead
#########################################
# Voxel A & B data
opt_nm <- optim_params(param_guess,method="Nelder-Mead",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_data,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_nm$param_est # params
opt_nm$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_nm, smoothed_data, top_title = 'Nelder-Mead')
# ASD Subject 01
opt_nm_s1 <- optim_params(param_guess,method="Nelder-Mead",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_subj1,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_nm_s1$param_est # params
opt_nm_s1$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_nm_s1, smoothed_subj1, top_title = 'Nelder-Mead')
# Normal Subject 01
opt_nm_s2 <- optim_params(param_guess,method="Nelder-Mead",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_subj2,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_nm_s2$param_est # params
opt_nm_s2$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_nm_s2, smoothed_subj2, top_title = 'Nelder-Mead')
### L-BFGS-B
# Voxel A & B data
opt_lbfgsb <- optim_params(param_guess,method="L-BFGS-B",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_data,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_lbfgsb$param_est # params
opt_lbfgsb$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_lbfgsb, smoothed_data, top_title = 'L-BFGS-B')
# ASD Subject 01
opt_lbfgsb_s1 <- optim_params(param_guess,method="L-BFGS-B",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_subj1,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_lbfgsb_s1$param_est # params
opt_lbfgsb_s1$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_lbfgsb_s1, smoothed_subj1, top_title = 'L-BFGS-B')
# Normal Subject 01
opt_lbfgsb_s2 <- optim_params(param_guess,method="L-BFGS-B",lower_lim=-20,upper_lim=20,maxit=30,temp=20,
t_dummy,smoothed_subj2,new_coupled_osc_model,N_p,N_y,dt,dT)
opt_lbfgsb_s2$param_est # params
opt_lbfgsb_s2$value # objective function value, chi-square
plot_ukf_and_smoothed_ggplot(opt_lbfgsb_s2, smoothed_subj2, top_title = 'L-BFGS-B')