# Preamble

This file provides replication materials for examples and analysis conducted in the paper, DiffusionRgqd: An R Package for Performing Inference and Analysis on Time-Inhomogeneous Quadratic Diffusion Processes, as submitted to the Journal of Statistical Software (JSS). As such, this file may not be useful as a stand-alone document, and should be read in conjunction with the paper. Note further:

• Since this document exists as a reference document for the paper, we reffrain from producing output for some of the examples here. However, all examples should run as is, and correspond to the figures presented in the paper itself.

• Some of the examples provided here are given in greater detail within the package vignettes (start here, or skip to the further reading section).

# Example 7.1: Generate the transitional density of a time-inhomogeneous GQD.

# Load the package:
library("DiffusionRgqd")

# Remove any pre-existing models:
GQD.remove()

# Define the model wihin the GQD-framework:
G0 <- function(t){2 * (10 + sin(2 * pi * (t - 0.5)))}
G1 <- function(t){-2}
Q1 <- function(t){0.25 * (1 + 0.75 * (sin(4 * pi * t)))}

# Some peripheral parameters:
states    <- seq(5, 15, 1 / 10)
initial   <- 8
Tmax      <- 5
Tstart    <- 1
increment <- 1 / 100

# Approximate the transitional density:
M <- GQD.density(Xs = initial, Xt = states, s = Tstart, t = Tmax,
delt = increment)

# Plot the transitional density:
library("rgl")
open3d(windowRect = c(50, 50, 690, 690), zoom=0.95)
persp3d(x = M$Xt,y = M$time,z = M$density, col = 'white', box = F, xlab = 'State (X_t)', ylab = 'Time(t)',zlab = 'Density f(X_t|X_s)') # Example 7.2: Time-inhomogeneous Jacobi diffusion. # Remove pre-existing models and define the model GQD.remove() a <- 0.5; b <- 0.6; cc <- 1; X0 <- 0.5; # Define the model: G0 <- function(t){a * (b * (1 + 0.25 * sin(pi * t)))} G1 <- function(t){-a} Q1 <- function(t){cc} Q2 <- function(t){-cc} # Approximate the transitional density using various truncation orders: states <- seq(0.001, 0.999, 0.001) res1 <- GQD.density(X0, states, 0, 2, 0.01, Dtype='Beta', Trunc=c(4, 4)) res2 <- GQD.density(X0, states, 0, 2, 0.01, Dtype='Beta', Trunc=c(6, 6)) res3 <- GQD.density(X0, states, 0, 2, 0.01, Dtype='Beta', Trunc=c(8, 8)) # Set up a simulation algortihm for comparing the approximate trans. dens. delt <- 0.001 N <- 100000 d <- 0 X <- rep(X0,N) when <- c(0, 0.25, 0.5, 1, 1.75) for(i in 2:2000) { X <- pmax(pmin(X + (G0(d)+ G1(d) * X) * delt + sqrt(Q1(d) * X + Q2(d) * X^2) * rnorm(length(X), sd = sqrt(delt)), 1), 0) d <- d + delt if(any(when == round(d, 3))) { index <- which(res1$time == round(d, 3))
hist(X, col = 'grey75', freq = F, breaks = 30,
main = paste0('Transitional Density at t  =  ', round(d, 3)),
ylim = c(0, 3), border = 'white')
lines(res1$density[, index] ~ res1$Xt, col = 'black'  , lty = 'dotted',lwd=1)
lines(res2$density[, index] ~ res2$Xt, col = '#BBCCEE', lty = 'solid' ,lwd=2)
lines(res3$density[, index] ~ res3$Xt, col = '#222299', lty = 'dashed',lwd=2)
}
}

# Now plot the transitional density using a built-in function:
GQD.plot(res2)

# Example 7.3 Bivariate non-linear dynamics - The stochastic Lotka-Volterra equations.

# Remove pre-existing models:
GQD.remove()

# Define the model:
a10 <- function(t){1.5}
a11 <- function(t){-0.4}
c10 <- function(t){0.05}

b01 <- function(t){-1.5}
b11 <- function(t){0.4}
b02 <- function(t){-0.2}
f01 <- function(t){0.1}

# Approximate the transitional density:
res <- BiGQD.density(Xs = 5, Ys = 5, Xt = seq(3, 8, length = 50),
Yt = seq(2, 6, length = 50), s = 0, t = 10, delt = 0.01)

# Visualize the evolution of the transitional density:
data("SDEsim3")
attach(SDEsim3)
library("colorspace")
colpal=function(n){rev(sequential_hcl(n,power=1,l=c(40,100)))}
time.index <- c(10, 300, 750, 1000) +1
for(i in time.index)
{
filled.contour(res$Xt, res$Yt, res$density[, , i], main = paste0('Transition Density \n (t = ', res$time[i], ')'),
color.palette = colpal,
xlab = 'Prey', ylab = 'Predator', plot.axes =
{
lines(my ~ mx, col = 'black', lty = 'dashed', lwd = 2)
points(res$cumulants[5, i] ~ res$cumulants[1, i], bg = 'white',
pch = 21, cex=1.5)
axis(1); axis(2);
legend('topright', lty = c('dashed', NA), pch = c(NA, 21),
lwd = c(2, NA), legend = c('Simulated Expectation',
'Predicted Expectation'))
})

}

# Example 7.4 Maximum likelihood estimation - Stochastic volatility models.

# Source data using the Quandl package:
library("Quandl")
quandldata1 <- Quandl("YAHOO/INDEX_GSPC", collapse = "weekly",
start_date = "1990-01-01", end_date = "2015-01-01",  type = "raw")
St <- rev(quandldata1[, names(quandldata1) == 'Close'])
time1 <- rev(quandldata1[, names(quandldata1) == 'Date'])

quandldata2 <- Quandl("YAHOO/INDEX_VIX", collapse = "weekly",
start_date = "1990-01-01", end_date = "2015-01-01",  type = "raw")
Vt <- rev(quandldata2[, names(quandldata2) == 'Close'])

# Fit the Heston Model:
GQD.remove()
a00 <- function(t){theta[1]}
a01 <- function(t){-0.5 * theta[2] * theta[2]}
c01 <- function(t){theta[2] * theta[2]}
d01 <- function(t){theta[2] * theta[5] * theta[6]}
b00 <- function(t){theta[3]}
b01 <- function(t){-theta[4]}
e01 <- function(t){theta[2] * theta[5] * theta[6]}
f01 <- function(t){theta[5] * theta[5]}

#Give some starting parameters and calculate MLEs:
X    <- cbind(log(St), (Vt / 100)^2)
time <- cumsum(c(0, diff(as.Date(time1)) * (1 / 365)))
theta.start <- c(0, 1, 1, 0.5, 1, 0)
model_1     <- BiGQD.mle(X, time, mesh = 10, theta = theta.start)

# Print the parameter estimates:
GQD.estimates(model_1)

# Define a new model:
GQD.remove()
a00 <- function(t){theta[1]}
a02 <- function(t){-0.5 * theta[2] * theta[2]}
c02 <- function(t){theta[2] * theta[2]}
d02 <- function(t){theta[2] * theta[5] * theta[6]}
b00 <- function(t){theta[3]}
b01 <- function(t){-theta[4]}
e02 <- function(t){theta[2] * theta[5] * theta[6]}
f02 <- function(t){theta[5] * theta[5]}

# Fit the revised model:
theta.start <- c(0, 1, 1, 1, 1, 0)
model_2     <- BiGQD.mle(X, time, mesh = 10, theta = theta.start)

# Compare aic statistics for the two models:
GQD.aic(list(model_1, model_2))

# Fit another model:
GQD.remove()
a02 <- function(t){-0.5 * theta[1] * theta[1]}
c02 <- function(t){theta[1] * theta[1]}
d02 <- function(t){theta[1] * theta[4] * theta[5]}
b00 <- function(t){theta[2]}
b01 <- function(t){-theta[3]}
e02 <- function(t){theta[1] * theta[4] * theta[5]}
f02 <- function(t){theta[4] * theta[4]}

theta.start <- c(1, 1, 1, 1, 0)
model_3 <- BiGQD.mle(X, time, mesh = 10, theta = theta.start)

# Fit another model:
GQD.remove()
a00 <- function(t){theta[1]}
a10 <- function(t){theta[7]}
a02 <- function(t){-0.5 * theta[2] * theta[2]}
c02 <- function(t){theta[2] * theta[2]}
d02 <- function(t){theta[2] * theta[5] * theta[6]}
b00 <- function(t){theta[3]}
b01 <- function(t){-theta[4]}
e02 <- function(t){theta[2] * theta[5] * theta[6]}
f02 <- function(t){theta[5] * theta[5]}

theta.start <- c(0, 1, 1, 1, 1, 0, 0)
model_4 <- BiGQD.mle(X, time, mesh = 10, theta = theta.start)

# Fit another model:
GQD.remove()
a00 <- function(t){theta[1]}
a02 <- function(t){-0.5 * theta[2] * theta[2]}
c02 <- function(t){theta[2] * theta[2]}
d02 <- function(t){theta[2] * theta[5] * theta[6]}
b00 <- function(t){theta[3]}
b10 <- function(t){theta[7]}
b01 <- function(t){-theta[4]}
e02 <- function(t){theta[2] * theta[5] * theta[6]}
f02 <- function(t){theta[5] * theta[5]}
theta.start <- c(0, 1, 1, 1, 1, 0, 0)
model_5 <- BiGQD.mle(X, time, mesh = 10, theta = theta.start)

# Compare all of the models' AIC statistics:
GQD.aic(list(model_1, model_2, model_3, model_4, model_5))

# Have a look at model_4's parameter estimates:
GQD.estimates(model_4)

# Example 7.5 Model selection for 2D diffusions via DIC.

data("SDEsim4")
attach(SDEsim4)
plot(Xt~time, type = 'l', col = '#BBCCEE', ylim = c(0, 30),
main = 'Simulated Data', xlab = 'Time (t)', ylab = 'State')
lines(Yt~time, col = '#222299')

GQD.remove()
a00 <- function(t){theta[1] * theta[2]}
a10 <- function(t){-theta[1]}
c11 <- function(t){theta[3] * theta[3]}
b00 <- function(t){theta[4] * theta[5]+theta[7] * sin(0.25 * pi * t)}
b01 <- function(t){-theta[4]}
f01 <- function(t){theta[6] * theta[6]}

priors <- function(theta){dnorm(theta[1], 1, 5) * dnorm(theta[4], 1, 5)}

mesh    <- 10
burns   <- 50000
X <- cbind(Xt, Yt)
th <- c(5, 5, 5, 5, 5, 5, 5)
par.sds <- c(0.22, 0.30, 0.02, 0.11, 0.04, 0.01, 0.21)
m1 <- BiGQD.mcmc(X, time, mesh, th, par.sds, updates, burns)

GQD.estimates(m1, thin = 200)

M  <- 4
SaveOutput  <- list()
M.counter   <- 1
Tot.counter <- 1
kill.count  <- 1
while((M.counter <= M) & (kill.count < 20))
{
th <- runif(7, 1, 10)
m1 <- BiGQD.mcmc(X, time, mesh, th, par.sds, updates, burns,
Tag = paste0('Model_A_run_', M.counter))
if(!m1$failed.chain) { SaveOutput[[Tot.counter]] <- m1 Tot.counter <- Tot.counter+1 M.counter <- M.counter +1 kill.count <- 1 }else { kill.count <- kill.count+1 } } GQD.remove() a00 <- function(t){theta[1] * theta[2]} a10 <- function(t){-theta[1]} c00 <- function(t){theta[7] * (1+sin(0.25 * pi * t))} c10 <- function(t){theta[3] * theta[3]} b00 <- function(t){theta[4] * theta[5]+theta[8] * sin(0.25 * pi * t)} b01 <- function(t){-theta[4]} f01 <- function(t){theta[6] * theta[6]} priors <- function(theta){dnorm(theta[1], 1, 5) * dnorm(theta[4], 1, 5)} par.sds <- c(0.17, 0.29, 0.07, 0.10, 0.04, 0.01, 0.83, 0.19) M.counter <- 1 kill.count <- 1 while((M.counter <= M) & (kill.count < 20)) { th <- runif(8, 1, 10) m2 <- BiGQD.mcmc(X, time, mesh, th, par.sds, updates, burns, Tag = paste0('Model_B_run_', M.counter)) if(!m2$failed.chain)
{
SaveOutput[[Tot.counter]]  <- m2
Tot.counter <- Tot.counter+1
M.counter   <- M.counter +1
kill.count  <- 1
}else
{
kill.count <- kill.count+1
}
}

GQD.remove()
a00 <- function(t){theta[1] * theta[2]}
a10 <- function(t){-theta[1]}
a01 <- function(t){theta[7]}
c11 <- function(t){theta[3] * theta[3]}

b00 <- function(t){theta[4] * theta[5]+theta[8] * sin(0.25 * pi * t)}
b01 <- function(t){-theta[4]}
f01 <- function(t){theta[6] * theta[6]}

priors <- function(theta){dnorm(theta[1], 1, 5) * dnorm(theta[4], 1, 5)}
par.sds  <- c(0.18, 0.95, 0.02, 0.10, 0.03, 0.01, 0.23, 0.18)
M.counter  <- 1
kill.count <- 1
while((M.counter <= M)&(kill.count < 20))
{
th <- runif(8, 1, 5)
m3 <- BiGQD.mcmc(X, time, mesh, th, par.sds, updates, burns,
Tag = paste0('Model_C_run_', M.counter))
if(!m3$failed.chain) { SaveOutput[[Tot.counter]] <- m3 Tot.counter <- Tot.counter+1 M.counter <- M.counter +1 kill.count <- 1 }else { kill.count <- kill.count+1 } } GQD.dic(SaveOutput) GQD.estimates(SaveOutput[[12]], 200) # Example 7.6 Scalar first passage times # Compare our scheme to an existing package: # Use the fptApprox package to calculate a FPT density: library("fptdApprox") OU <- diffproc(c("alpha * x + beta", "sigma^2", "dnorm((x-(y * exp(alpha * (t-s)) - beta * (1 - exp(alpha * (t-s))) / alpha)) / (sigma * sqrt((exp(2 * alpha * (t-s)) - 1) / (2 * alpha))), 0, 1) / (sigma * sqrt((exp(2 * alpha * (t-s)) - 1) / (2 * alpha)))", "pnorm(x, y * exp(alpha * (t-s)) - beta * (1 - exp(alpha * (t-s))) / alpha, sigma * sqrt((exp(2 * alpha * (t-s)) - 1) / (2 * alpha)))")) # Calculate the density: res1 <- Approx.fpt.density(OU, 0, 10, 3, "5+0.25 * sin(2 * pi * t)", list(alpha = -0.5, beta = 0.5 * 5, sigma = 1)) # Now do the same using DiffusionRgqd: # Define the model (after accounting for the static-barrier transform): GQD.remove() G0 <- function(t) { 0.5 * 5 - 0.5 * pi * cos(2 * pi * t)-0.5 * 0.25 * sin(2 * pi * t) } G1 <- function(t){-0.5} Q0 <- function(t){1} # Calculate the FPT density: res2 <- GQD.TIpassage(Xs = 3, B = 5, s = 0, t = 10, delt = 1 / 200) # PLot the resulting densities: plot(res1$y ~ res1$x, type = 'l', col = '#BBCCEE', xlab = 'Time(t)', ylab = 'FPT Density', main = 'DiffusionRgqd vs. fptdApprox.', lwd = 2) lines(res2$density ~ res2$time, col = '#222299', lwd = 2, lty = 'dashed') legend('topright', lty = c('solid', 'dashed'), col = c('#BBCCEE', '#222299'), legend = c('fptdApprox', 'DiffusionRgqd'), lwd = c(2, 2), bty = 'n') # Note, although the paper uses a stored dataset for illustrating the simulated # FPT density, we show here how to simulate such a density: theta <- 0.5 mu <- function(X, t) { theta[1] * (10+0.2 * sin(2 * pi * t)+0.3 * sqrt(t) * (1+cos(3 * pi * t))) * X-theta[1] * X^2 } sigma <- function(X, t){sqrt(0.1) * X} simulate.fpt = function(N, S, B, delt) { X = rep(S, N) Ndim = N time.vector = rep(0, N) t = 1 k = 0 while(Ndim >= 1) { X = X+mu(X, t) * delt+sigma(X, t) * rnorm(Ndim, sd = sqrt(delt)) I0 = X<B X = X[I0] count = sum(!I0) Ndim = length(X) t = t+delt if(count>0) { time.vector[k+1:count] = t k = k+count } } return(time.vector) } # res.sim <- simulate.fpt(500000, 8, 12, 1 / 2000) # Calculate the FPT density using a parameterised model: GQD.remove() G1 <- function(t) { theta[1] * (10+0.2 * sin(2 * pi * t)+0.3 * prod(sqrt(t), 1+cos(3 * pi * t))) } G2 <- function(t){-theta[1]} Q2 <- function(t){0.1} res3 = GQD.TIpassage(8, 12, 1, 4, 1 / 100, theta = c(0.5)) # Vary the model parameter to see its effect on the FPT density: # Load simulated first passage times: data("SDEsim6") hist(fpt.sim.times, freq = F, col = 'gray85', border = 'white', main = 'First Passage Time Density', ylab = 'Density', xlab = 'Time', ylim = range(res3$density), xlim = range(res3$time), breaks = 100) library("colorspace") colpal = function(n){rev(sequential_hcl(n, power = 1, l = c(20, 60)))} th.seq = seq(0.1, 0.5, 0.05) for(i in 2:length(th.seq)) { res3 = GQD.TIpassage(8, 12, 1, 4, 0.01, theta = c(th.seq[i])) lines(res3$density ~ res3$time, type = 'l', col = colpal(10)[i], lty = 11 - i, lwd =1.5) } lines(res3$density ~ res3\$time, type = 'l', col = colpal(10)[i], lwd = 2)
legend('topright', legend = th.seq, col = colpal(10), lty = 9:1,
lwd = c(rep(1.5, 8), 2), title = expression(theta[1]), bty = 'n')

browseVignettes('DiffusionRgqd')