‘fourierin’ package

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‘fourierin’ package

Guillermo Basulto-Elias

2023-12-08

This is a package in R to numerically calculate Fourier-type integrals of multivariate functions with compact support evaluated at regular grids. Specifically, integrals of the type

\[ I \left[f(t), \boldsymbol{a}, \boldsymbol{b};r, s \right] = \left[ \frac{|s|}{(2\pi)^{1 - r}}\right]^{n/2} \int_{a_1}^{b_1}\int_{a_2}^{b_2}\cdots\int_{a_n}^{b_n} f(\boldsymbol{t})e^{\imath s \langle \boldsymbol{w}, \boldsymbol{t}\rangle} \text{d}\boldsymbol{t}, \]

\(\boldsymbol{a} = (a_1, \ldots, a_n)\), \(\boldsymbol{b} = (b_1, \ldots, b_n)\), \(\boldsymbol{t} = (t^{(1)}, \ldots, t^{(n)})\), \(\boldsymbol{w} = (w^{(1)}, \ldots, w^{(n)})\), \(a_l \leq t^{(l)} \leq b_l\), \(c_l \leq w^{(l)} \leq c_l\).

Common values for r are -1, 0 and -1, while common values for s are -2, -1, 1 and 2. For example, if f is a density function, s = 1 and r = 1 could be used to obtain the characteristic function of f. Conversely, if f is the characteristic function of a probability density function, then r = -1 and s = -1 could be used to recover the density.

The implementation of this algorithm is the one described in Inverarity (2002), Fast Computation of multidimensional Integrals (at ‘epubs.siam.org/doi/abs/10.1137/S106482750138647X’).

Example 0: Speed tests

Below it is an example of univariate continuous Fourier transform.

library(fourierin)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(purrr)
library(ggplot2)

## Test speed at several resolutions
resolution <- 2^(3:8)

## Function to be tested
myfnc <- function(t) exp(-t^2/2)

## Aux. function
compute_times <- function(resol){
  reps <- 100
  rbenchmark::benchmark(
    yes = fourierin_1d(f = myfnc, -5, 5, -3, 3, -1, -1, resol),
    no = fourierin_1d(f = myfnc, -5, 5, -3, 3, -1, -1, resol,
                      use_fft = FALSE),
    replications = reps) %>%
    as.data.frame() %>%
    select(FFT = test, time = elapsed) %>%
    mutate(time = time*1e3/reps, resolution = resol)
}

resolution %>%
    map_df(compute_times) %>%
    mutate(resolution = as.factor(resolution)) %>%
    ggplot(aes(resolution, time, color = FFT)) +
    geom_point(size = 2, aes(shape = FFT)) +
    geom_line(aes(linetype = FFT, group = FFT)) +
    ylab("time (in milliseconds)")

Now we present an example using a bivariate function.

## Load packages
library(fourierin)
library(dplyr)
library(purrr)
library(ggplot2)

## Test speed at several resolutions
resolution <- 2^(3:7)

## Bivariate function to be tested
myfnc <- function(x) dnorm(x[, 1])*dnorm(x[, 2])

## Aux. function
compute_times <- function(resol){
  reps <- 1
  resol <- rep(resol, 2)
  rbenchmark::benchmark(
    yes = fourierin(myfnc,
                      lower_int = c(-8, -6), upper_int = c(6, 8),
                      lower_eval = c(-4, -4), upper_eval = c(4, 4),
                      const_adj = 1, freq_adj = 1,
                      resolution = resol),
    no = fourierin(myfnc,
                      lower_int = c(-8, -6), upper_int = c(6, 8),
                      lower_eval = c(-4, -4), upper_eval = c(4, 4),
                      const_adj = 1, freq_adj = 1,
                      resolution = resol, use_fft = FALSE),
    replications = reps) %>%
    as.data.frame() %>%
    select(FFT = test, time = elapsed) %>%
    mutate(time = time/reps, resolution = resol)
}

## Values
comparison <- 
    resolution %>%
    map_df(compute_times) 

fctr_order <-
    unique(comparison$resolution) %>%
    paste(., ., sep = "x")

## Plot
comparison %>%
    mutate(resolution = paste(resolution, resolution, sep = "x"),
           resolution = ordered(resolution, levels = fctr_order)) %>%
    ggplot(aes(resolution, time, color = FFT)) +
    geom_point(size = 2, aes(shape = FFT)) +
    geom_line(aes(linetype = FFT, group = FFT)) +
    ylab("time (in seconds)")

Example 1: Recovering standard normal from its characteristic function

The density function of a random variable can be recovered using the Fourier inversion theorem. The characteristic function of a standard normal distribution is \[ \phi(t) = \exp(-t^2/2) \;\; t \in \mathbb{R} \]

The due to the smoothness and symmetry of both, the characteristic function and the standard normal density, it can be well approximated even with a low resolution.

library(fourierin)
library(dplyr)
library(ggplot2)

## Characteristic function of std. normal.
fnc <- function(t) exp(-t^2/2)

## Compute integral
out <- fourierin(f = fnc, lower_int = -5, upper_int = 5,
                 lower_eval = -5, upper_eval = 5,
                 const_adj = -1, freq_adj = -1, resolution = 64)

## Extract values and compute true values of the density
df1 <- out %>%
    as_tibble() %>%
    mutate(values = Re(values),
           density = "approximated")

df2 <- tibble(w = seq(min(df1$w), max(df1$w), len = 150),
                  values = dnorm(w),
                  density = "true")

bind_rows(df1, df2) %>%
    ggplot(aes(w, values, color = density)) +
    geom_line(aes(linetype = density)) +
    xlab("x") + ylab("f(x)")

We now present a second example: recovering the density of a \(\chi^2\) distribution using its characteristic function. The corresponding density is not symmetric and its support is not the entire real line, thus a higher resolution might be required to achieve a good approximation.

library(fourierin)
library(dplyr)
library(purrr)
library(ggplot2)

## Set functions
df <- 5                                        
cf <- function(t) (1 - 2i*t)^(-df/2)
dens <- function(x) dchisq(x, df)


## Set resolutions
resolutions <- 2^(6:8)

## Compute integral given the resoltion
recover_f <- function(resol){
    ## Get grid and density values
    out <- fourierin(f = cf, lower_int = -10, upper_int = 10,
                     lower_eval = -3, upper_eval = 20,
                     const_adj = -1, freq_adj = -1,
                     resolution = resol)
    ## Return in dataframe format
    out %>%
        as_data_frame() %>%
        transmute(
            x = w,
            values = Re(values),
            resolution = resol)
}

## Density approximations
vals <- map_df(resolutions, recover_f)

## Warning: `as_data_frame()` was deprecated in tibble 2.0.0.
## ℹ Please use `as_tibble()` (with slightly different semantics) to convert to a
##   tibble, or `as.data.frame()` to convert to a data frame.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

## True values
true <- data_frame(x = seq(min(vals$x), max(vals$x), length = 150),
                   values = dens(x))

## Warning: `data_frame()` was deprecated in tibble 1.1.0.
## ℹ Please use `tibble()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

vals %>%
    mutate(resolution = ordered(resolution,
                                labels = resolutions)) %>%
    ggplot(aes(x, values)) +
    geom_line(aes(color = resolution)) +
    geom_line(data = true, aes(color = "true values"))

Example 2: Computing characteristic function of a gamma density

library(fourierin)
library(tidyr)
library(dplyr)
library(purrr)
library(ggplot2)

## Compute integral
shape <- 5
rate <- 3
myfnc <- function(t) dgamma(t, shape, rate)

## Function to compute characteristic function for a given resolution.
approximate_CF <- function (resol) {
    out <- fourierin(f = myfnc, -0, 8, -5, 5, 1, 1, resol)
    
    out %>%
        as_data_frame() %>%
        transmute(t = w,
                  real = Re(values),
                  imaginary = Im(values),
                  resolution = as.character(resol)) %>%
        gather(CF, values, -t, -resolution)
}

## Evaluate approxs. to CF for different resulutions
resolution <- 2^c(6, 7)
CF <- map_df(resolution, approximate_CF)

## Evaluate true values of characteristic function
true_cf <- function(t, shape, rate) (1 - 1i*t/rate)^-shape
true <- data_frame(t = seq(min(CF$t), max(CF$t), length = 150),
                   values = true_cf(t, shape, rate),
                   real = Re(values),
                   imaginary = Im(values),
                   resolution = "true values") %>%
    select(-values) %>%
    gather(CF, values, -t, -resolution)

## Plot values
CF %>%
    ggplot(aes(t, values, color = resolution,
               group = resolution)) +
    geom_line(aes()) +
    geom_line(data = true) + 
    facet_grid(~CF) + 
    theme(legend.position = "bottom")

Example 3: Recovering std. normal from its characteristic function

Testing example.

## Load packages
library(fourierin)
library(tidyr)
library(dplyr)
library(purrr)
library(lattice)
library(ggplot2)

## Set functions to be tested with their corresponding parameters.
mu <- c(-1, 1)
sig <- matrix(c(3, -1, -1, 2), 2, 2)

## Multivariate normal density, x is n x d
f <- function(x) {
    ## Auxiliar values
    d <- ncol(x)
    z <- sweep(x, 2, mu, "-")
    ## Get numerator and denominator of normal density
    num <- exp(-0.5*rowSums(z * (z %*% solve(sig))))
    denom <- sqrt((2*pi)^d*det(sig))
    return(num/denom)
}

## Characteristic function, s is n x d
phi <- function (s) {
    complex(modulus = exp(-0.5*rowSums(s*(s %*% sig))),
            argument = s %*% mu)
}

## Evaluate characteristic function for a given resolution.
eval <- fourierin(f,
                  lower_int = c(-8, -6), upper_int = c(6, 8),
                  lower_eval = c(-4, -4), upper_eval = c(4, 4),
                  const_adj = 1, freq_adj = 1, resolution = 2*c(64, 64),
                  use_fft = T)

## --- Little test
dat <- eval %>%
    with(crossing(y = w2, x = w1) %>%
         mutate(approximated = c(values))) %>%
    mutate(true = phi(matrix(c(x, y), ncol = 2)),
           difference = approximated - true) %>%
    gather(value, z, -x, -y) %>%
    mutate(real = Re(z), imaginary = Im(z)) %>%
    select(-z) %>%
    gather(part, z, -x, -y, -value)


## Surface plot
wireframe(z ~ x*y | value*part, data = dat,          
          scales =
              list(arrows=FALSE, cex= 0.45,
                   col = "black", font = 3, tck = 1),
          screen = list(z = 90, x = -74),
          colorkey = FALSE,
          shade=TRUE,
          light.source= c(0,10,10),
          shade.colors = function(irr, ref,
                                  height, w = 0.4)
              grey(w*irr + (1 - w)*(1 - (1 - ref)^0.4)),
          aspect = c(1, 0.65))


## Contours of values
dat %>%
    filter(value != "difference") %>%
    ggplot(aes(x, y, z = z)) +
    geom_tile(aes(fill = z)) +
    facet_grid(part ~ value) +
    scale_fill_distiller(palette = "Reds")

## Contour of differences
dat %>%
    filter(value == "difference") %>%
    ggplot(aes(x, y, z = z)) +
    geom_tile(aes(fill = z)) +
    facet_grid(part ~ value) +
    scale_fill_distiller(palette = "Spectral")

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.