---
title: "Superfast Likelihood Inference for Stationary Gaussian Time Series"
author: "Yun Ling, Martin Lysy"
date: "`r Sys.Date()`"
output:
  html_vignette:
    toc: true
bibliography: references.bib
csl: taylor-and-francis-harvard-x.csl
link-citations: true
vignette: >
  %\VignetteIndexEntry{Superfast Likelihood Inference for Stationary Gaussian Time Series}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

\newcommand{\bm}[1]{\boldsymbol{#1}}
\newcommand{\rv}[3][1]{#2_{#1},\ldots,#2_{#3}}
\newcommand{\X}{\bm{X}}
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\dt}{\Delta t}
\newcommand{\msd}{\mathrm{\scriptsize MSD}}
\newcommand{\acf}{\mathrm{\scriptsize ACF}}
\newcommand{\dX}{\Delta\X}
\newcommand{\VH}{\bm{V}_H}

This vignette illustrates the basic functionality of the **SuperGauss** package by simulating a few stochastic processes and estimating their parameters from regularly spaced data.

## 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 $\cov(X_t, X_s) = \tfrac 1 2 (|t|^{2H} + |s|^{2H} - |t-s|^{2H})$, for $0 < H < 1$.  fBM is not stationary but has stationary increments, such that $(X_{t+\dt} - X_t) \stackrel{D}{=} (X_{s+\dt} - X_s)$ for any $s,t$.  As such, its covariance function is completely determined its mean squared displacement (MSD)
$$
\msd_X(t) = E[(X_t - X_0)^2] = |t|^{2H}.
$$
When the Hurst parameter $H = \tfrac 1 2$, fBM reduces to ordinary Brownian motion.

```{r fbmsim, include = FALSE}
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)
})
# fast method (about 3x as slow)
system.time({
  rnormtz(n = npaths, acf = acf, fft = FALSE)
})
# unstructured variance method (much slower)
system.time({
  matrix(rnorm(N*npaths), npaths, N) %*% chol(toeplitz(acf))
})
```
The following **R** code generates `r npaths` independent fBM realizations of length $N = `r N`$ with $H = `r H`$.  The timing of the "superfast" method [@wood.chan94] provided in this package is compared to that of a "fast" method [e.g., @brockwell.davis91] and to the usual method (Cholesky decomposition of an unstructured variance matrix).
```{r fbmsim}
```
```{r, fig.width = 10, fig.height = 5, out.width = "90%"}
# 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 $\X = (\rv [0] X N)$ are equally spaced observations of an fBM process with $X_i = X(i \dt)$, and let $\dX = (\rv [0] {\Delta X} {N-1})$ denote the corresponding increments, $\Delta X_i = X_{i+1} - X_i$.  Then the loglikelihood function for $H$ is
$$
\ell(H \mid \dX) = -\tfrac 1 2 \big(\dX' \VH^{-1} \dX + \log |\VH|\big),
$$
where $V_H$ is a Toeplitz matrix,
$$
\VH = [\cov(\Delta X_i, \Delta X_j)]_{0 \le i,j < N} = \begin{bmatrix} \gamma_0 & \gamma_1 & \cdots & \gamma_{N-1} \\
                      \gamma_1 & \gamma_0 & \cdots & \gamma_{N-2} \\
					  \vdots & \vdots & \ddots & \vdots \\
					  \gamma_{N-1} & \gamma_{N-2} & \cdots & \gamma_0
      \end{bmatrix}.
$$
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.gragg88, which scales these computations as $\mathcal O(N \log^2 N)$.  With careful memory management and extensive use of the **FFTW** library [@frigo.johnson05], the **SuperGauss** implementation crosses over Durbin-Levinson at around $N = 300$.

### The `Toeplitz` Matrix Class

The bulk of the likelihood calculations in **SuperGauss** are handled by the `Toeplitz` matrix class.  A `Toeplitz` object is created as follows:
```{r}
# allocate and assign in one step
Tz <- Toeplitz$new(acf = acf)
Tz

# allocate memory only
Tz <- Toeplitz$new(N = N)
Tz
Tz$set_acf(acf = acf) # assign later
```
Its primary methods are illustrated below:
```{r}
all(acf == Tz$get_acf()) # extract acf

# 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)
range(x2-x3)

# 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)

# 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)
```

### Maximum Likelihood Calculation

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.
```{r}
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)
})
# fast method (about 10x slower)
system.time({
  DL_mle <- optimize(loglik_DL, interval = c(.01, .99), maximum = TRUE)
})
c(GS = GS_mle$max, DL = DL_mle$max)

# standard error calculation
require(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)
```

### Caution with `R6` Classes

In order to effectively manage memory in the underlying **C++** code, the `Toeplitz` class is implemented using [R6 classes](https://r6.r-lib.org/).  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:
```{r}
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
T2

fbm_logdet <- function(H) {
  T1$set_acf(acf = fbm_acf(H))
  T1$log_det()
}

# affects both objects
fbm_logdet(H = .3)
T1
T2
```
To avoid this behavior, it is necessary to make a deep copy of the object:
```{r}
T3 <- T1$clone(deep = TRUE)
T1
T3

# only affect T1
fbm_logdet(H = .7)
T1
T3
```

## 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
$$
\acf_X(t \mid \lambda, \sigma) = \sigma^2 \exp(- |t/\lambda|).
$$
The example uses `stats::nlm()` for optimization, which requires the derivatives to be passsed as attributes to the (negative) loglikelihood.

```{r}
# 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)
})

# display estimates with standard errors
rbind(true = c(lambda = lambda, sigma = sigma),
      est = mle_fit$estimate,
      se = sqrt(diag(solve(mle_fit$hessian))))

```

## References
