Superfast Likelihood Inference for Stationary Gaussian Time Series

Simulation of Fractional Brownian Motion

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

# fast method (about 3x as slow)
system.time({
  rnormtz(n = npaths, acf = acf, fft = FALSE)
})

##    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)
}

Inference for the Hurst Parameter

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 \(N = 300\).

# allocate and assign in one step
Tz <- Toeplitz$new(acf = acf)
Tz

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

# allocate memory only
Tz <- Toeplitz$new(N = N)
Tz

## Toeplitz matrix of size 5000 
##  acf:  NULL

Tz$set_acf(acf = acf) # assign later

all(acf == Tz$get_acf()) # extract acf

## [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

range(x2-x3)

## [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

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

c(GS = GS_mle$max, DL = DL_mle$max)

##        GS        DL 
## 0.2950746 0.2950746

# standard error calculation
require(numDeriv)

## 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

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 ...

T2

## 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

T1

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

T2

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

T3 <- T1$clone(deep = TRUE)
T1

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

T3

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

# only affect T1
fbm_logdet(H = .7)

## [1] -29326.33

T1

## Toeplitz matrix of size 5000 
##  acf:  0.00324 0.00104 0.000612 0.000474 0.000397 0.000347 ...

T3

## Toeplitz matrix of size 5000 
##  acf:  0.0857 -0.0208 -0.00421 -0.00228 -0.0015 -0.00109 ...

Superfast Newton-Raphson

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