This vignette illustrates the basic functionality of the SuperGauss package by simulating a few stochastic processes and estimating their parameters from regularly spaced data.
A one-dimensional fractional Brownian motion (fBM) \(X_t = X(t)\) is a continuous Gaussian process with \(E[X_t] = 0\) and \(\mathrm{cov}(X_t, X_s) = \tfrac 1 2 (|t|^{2H} + |s|^{2H} - |t-s|^{2H})\), for \(0 < H < 1\). fBM is not stationary but has stationary increments, such that \((X_{t+\Delta t} - X_t) \stackrel{D}{=} (X_{s+\Delta t} - X_s)\) for any \(s,t\). As such, its covariance function is completely determined its mean squared displacement (MSD) \[ \mathrm{\scriptsize MSD}_X(t) = E[(X_t - X_0)^2] = |t|^{2H}. \] When the Hurst parameter \(H = \tfrac 1 2\), fBM reduces to ordinary Brownian motion.
The following R code generates 5 independent fBM realizations of length \(N = 5000\) with \(H = 0.3\). The timing of the “superfast” method (Wood and Chan 1994) provided in this package is compared to that of a “fast” method (e.g., Brockwell and Davis 1991) and to the usual method (Cholesky decomposition of an unstructured variance matrix).
require(SuperGauss)
N <- 5000 # number of observations
dT <- 1/60 # time between observations (seconds)
H <- .3 # Hurst parameter
tseq <- (0:N)*dT # times at which to sample fBM
npaths <- 5 # number of fBM paths to generate
# to generate fbm, generate its increments, which are stationary
msd <- fbm_msd(tseq = tseq[-1], H = H)
acf <- msd2acf(msd = msd) # convert msd to acf
# superfast method
system.time({
dX <- rnormtz(n = npaths, acf = acf, fft = TRUE)
})
## user system elapsed
## 0.011 0.002 0.014
## user system elapsed
## 0.057 0.000 0.057
# unstructured variance method (much slower)
system.time({
matrix(rnorm(N*npaths), npaths, N) %*% chol(toeplitz(acf))
})
## user system elapsed
## 17.708 0.276 18.047
# convert increments to position measurements
Xt <- apply(rbind(0, dX), 2, cumsum)
# plot
clrs <- c("black", "red", "blue", "orange", "green2")
par(mar = c(4.1,4.1,.5,.5))
plot(0, type = "n", xlim = range(tseq), ylim = range(Xt),
xlab = "Time (s)", ylab = "Position (m)")
for(ii in 1:npaths) {
lines(tseq, Xt[,ii], col = clrs[ii], lwd = 2)
}
Suppose that \(\boldsymbol{X}= (N_{X},\ldots,N_{)}\) are equally spaced observations of an fBM process with \(X_i = X(i \Delta t)\), and let \(\Delta\boldsymbol{X}= (N-1_{\Delta X},\ldots,N-1_{)}\) denote the corresponding increments, \(\Delta X_i = X_{i+1} - X_i\). Then the loglikelihood function for \(H\) is \[ \ell(H \mid \Delta\boldsymbol{X}) = -\tfrac 1 2 \big(\Delta\boldsymbol{X}' \boldsymbol{V}_H^{-1} \Delta\boldsymbol{X}+ \log |\boldsymbol{V}_H|\big), \] where \(V_H\) is a Toeplitz matrix, \[ \boldsymbol{V}_H= [\mathrm{cov}(\Delta X_i, \Delta X_j)]_{0 \le i,j < N} = \begin{bmatrix} \gamma_0 & \gamma_1 & \cdots & \gamma_{N-1} \\ \gamma_1 & \gamma_0 & \cdots & \gamma_{N-2} \\ \vdots & \vdots & \ddots & \vdots \\ \gamma_{N-1} & \gamma_{N-2} & \cdots & \gamma_0 \end{bmatrix}. \] Thus, each evaluation of the loglikelihood requires the inverse and log-determinant of a Toeplitz matrix, which scales as \(\mathcal O(N^2)\) with the Durbin-Levinson algorithm. The SuperGauss package implements an extended version of the Generalized Schur algorithm of Ammar and Gragg (1988), which scales these computations as \(\mathcal O(N \log^2 N)\). With careful memory management and extensive use of the FFTW library (Frigo and Johnson 2005), the SuperGauss implementation crosses over Durbin-Levinson at around \(N = 300\).
Toeplitz
Matrix ClassThe bulk of the likelihood calculations in SuperGauss are handled by the Toeplitz
matrix class. A Toeplitz
object is created as follows:
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
## Toeplitz matrix of size 5000
## acf: NULL
Its primary methods are illustrated below:
## [1] TRUE
# matrix multiplication
z <- rnorm(N)
x1 <- toeplitz(acf) %*% z # regular way
x2 <- Tz$prod(z) # with Toeplitz class
x3 <- Tz %*% z # with Toeplitz class overloading the `%*%` operator
range(x1-x2)
## [1] -1.526557e-15 1.665335e-15
## [1] 0 0
# system of equations
y1 <- solve(toeplitz(acf), z) # regular way
y2 <- Tz$solve(z) # with Toeplitz class
y2 <- solve(Tz, z) # same thing but overloading `solve()`
range(y1-y2)
## [1] -1.280398e-11 8.704149e-12
# log-determinant
ld1 <- determinant(toeplitz(acf))$mod
ld2 <- Tz$log_det() # with Toeplitz class
ld2 <- determinant(Tz) # same thing but overloading `determinant()`
# note: no $mod
c(ld1, ld2)
## [1] -12737.89 -12737.89
The following code shows how to obtain the maximum likelihood of \(H\) and its standard error for a given fBM path. The log-PDF of the Gaussian with Toeplitz variance matrix is obtained either with SuperGauss::dnormtz()
, or using the NormalToeplitz
class. The advantage of the latter is that it does not reallocate memory for the underlying Toeplitz
object at every likelihood evaulation.
For speed comparisons, the optimization underlying the MLE calculation is done both using the superfast Generalized Schur algorithm and the fast Durbin-Levinson algorithm.
dX <- diff(Xt[,1]) # obtain the increments of a given path
N <- length(dX)
# autocorrelation of fBM increments
fbm_acf <- function(H) {
msd <- fbm_msd(1:N*dT, H = H)
msd2acf(msd)
}
# loglikelihood using generalized Schur algorithm
NTz <- NormalToeplitz$new(N = N) # pre-allocate memory
loglik_GS <- function(H) {
NTz$logdens(z = dX, acf = fbm_acf(H))
}
# loglikelihood using Durbin-Levinson algorithm
loglik_DL <- function(H) {
dnormtz(X = dX, acf = fbm_acf(H), method = "ltz", log = TRUE)
}
# superfast method
system.time({
GS_mle <- optimize(loglik_GS, interval = c(.01, .99), maximum = TRUE)
})
## user system elapsed
## 0.067 0.003 0.070
# fast method (about 10x slower)
system.time({
DL_mle <- optimize(loglik_DL, interval = c(.01, .99), maximum = TRUE)
})
## user system elapsed
## 0.851 0.006 0.857
## GS DL
## 0.2950746 0.2950746
## Loading required package: numDeriv
Hmle <- GS_mle$max
Hse <- -hessian(func = loglik_GS, x = Hmle) # observed Fisher Information
Hse <- sqrt(1/Hse[1])
c(mle = Hmle, se = Hse)
## mle se
## 0.295074650 0.002584653
R6
ClassesIn order to effectively manage memory in the underlying C++ code, the Toeplitz
class is implemented using R6 classes. Among other things, this means that when a Toeplitz
object is passed to a function, the function does not make a copy of it: all modifications to the object inside the object are reflected on the object outside the function as well, as in the following example:
T1 <- Toeplitz$new(N = N)
T2 <- T1 # shallow copy: both of these point to the same memory location
# affects both objects
T1$set_acf(fbm_acf(.5))
T1
## Toeplitz matrix of size 5000
## acf: 0.0167 0 1.73e-18 -3.47e-18 0 6.94e-18 ...
## Toeplitz matrix of size 5000
## acf: 0.0167 0 1.73e-18 -3.47e-18 0 6.94e-18 ...
fbm_logdet <- function(H) {
T1$set_acf(acf = fbm_acf(H))
T1$log_det()
}
# affects both objects
fbm_logdet(H = .3)
## [1] -12737.89
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
To avoid this behavior, it is necessary to make a deep copy of the object:
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
## [1] -29326.33
## Toeplitz matrix of size 5000
## acf: 0.00324 0.00104 0.000612 0.000474 0.000397 0.000347 ...
## Toeplitz matrix of size 5000
## acf: 0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...
In addition to the superfast algorithm for Gaussian likelihood evaluations, SuperGauss provides such algorithms for the loglikelihood gradient and Hessian functions, leading to superfast versions of many inference algorithms such as Newton-Raphson and Hamiltonian Monte Carlo. These are provided by the NormalToeplitz$grad()
and NormalToeplitz$hess()
methods. Both of these methods optionally return the lower order derivatives as well, reusing common computations to improve performance. An example of Newton-Raphson is given below using the two-parameter exponential autocorrelation model \[
\mathrm{\scriptsize ACF}_X(t \mid \lambda, \sigma) = \sigma^2 \exp(- |t/\lambda|).
\] The example uses stats::nlm()
for optimization, which requires the derivatives to be passsed as attributes to the (negative) loglikelihood.
# autocorrelation function
exp_acf <- function(t, lambda, sigma) sigma^2 * exp(-abs(t/lambda))
# gradient, returned as a 2-column matrix
exp_acf_grad <- function(t, lambda, sigma) {
ea <- exp_acf(t, lambda, 1)
cbind(abs(t)*(sigma/lambda)^2 * ea, # d_acf/d_lambda
2*sigma * ea) # d_acf/d_sigma
}
# Hessian, returned as an array of size length(t) x 2 x 2
exp_acf_hess <- function(t, lambda, sigma) {
ea <- exp_acf(t, lambda, 1)
sl2 <- sigma/lambda^2
hess <- array(NA, dim = c(length(t), 2, 2))
hess[,1,1] <- sl2^2*(t^2 - 2*abs(t)*lambda) * ea # d2_acf/d_lambda^2
hess[,1,2] <- 2*sl2 * abs(t) * ea # d2_acf/(d_lambda d_sigma)
hess[,2,1] <- hess[,1,2] # d2_acf/(d_sigma d_lambda)
hess[,2,2] <- 2 * ea # d2_acf/d_sigma^2
hess
}
# simulate data
lambda <- runif(1, .5, 2)
sigma <- runif(1, .5, 2)
tseq <- (1:N-1)*dT
acf <- exp_acf(t = tseq, lambda = lambda, sigma = sigma)
Xt <- rnormtz(acf = acf)
NTz <- NormalToeplitz$new(N = N) # storage space
# negative loglikelihood function of theta = (lambda, sigma)
# include attributes for gradient and Hessian
exp_negloglik <- function(theta) {
lambda <- theta[1]
sigma <- theta[2]
# acf, its gradient, and Hessian
acf <- exp_acf(tseq, lambda, sigma)
dacf <- exp_acf_grad(tseq, lambda, sigma)
d2acf <- exp_acf_hess(tseq, lambda, sigma)
# derivatives of NormalToeplitz up to order 2
derivs <- NTz$hess(z = Xt,
dz = matrix(0, N, 2),
d2z = array(0, dim = c(N, 2, 2)),
acf = acf,
dacf = dacf,
d2acf = d2acf,
full_out = TRUE)
# negative loglikelihood with derivatives as attributes
nll <- -1 * derivs$ldens
attr(nll, "gradient") <- -1 * derivs$grad
attr(nll, "hessian") <- -1 * derivs$hess
nll
}
# optimization
system.time({
mle_fit <- nlm(f = exp_negloglik, p = c(1,1), hessian = TRUE)
})
## user system elapsed
## 0.431 0.009 0.439
# display estimates with standard errors
rbind(true = c(lambda = lambda, sigma = sigma),
est = mle_fit$estimate,
se = sqrt(diag(solve(mle_fit$hessian))))
## lambda sigma
## true 1.3914065 0.71412532
## est 1.7019320 0.79757410
## se 0.3478682 0.08112464
Ammar, G.S. and Gragg, W.B., 1988. Superfast solution of real positive definite Toeplitz systems. SIAM Journal on Matrix Analysis and Applications, 9 (1), 61–76.
Brockwell, P.J. and Davis, R.A., 1991. Time series: Theory and methods. Springer: New York.
Frigo, M. and Johnson, S.G., 2005. The design and implementation of FFTW3. Proceedings of the IEEE, 93 (2), 216–231.
Wood, A.T. and Chan, G., 1994. Simulation of stationary gaussian processes in \([0, 1]^d\). Journal of Computational and Graphical Statistics, 3 (4), 409–432.