Perform Fourier Analysis to a Time Series in R

Question

I would like to perform fourier transform to a time series using R. I would like to:

1. Get the sum of the 5th to 18th harmonics
2. plot each wave
3. and output as a csv file.

Here's the link to the data: Link to Data

Here's my initial code.

dat   <- read.csv("Baguio.csv",header=FALSE)
y     <- dat$V1
ssp   <-spectrum(y)
t     <- 1:73
per   <- 1/ssp$freq[ssp$spec==max(ssp$spec)]
reslm <- lm(y ~ sin(2*pi/per*t)+cos(2*pi/per*t))
rg    <- diff(range(y))

#blue dashed line
plot(y~t,ylim=c(min(y)-0.1*rg,max(y)+0.1*rg))
lines(fitted(reslm)~t,col=4,lty=2)

#green line 2nd harmonics
reslm2 <- lm(y ~ sin(2*pi/per*t)+cos(2*pi/per*t)+sin(4*pi/per*t)+cos(4*pi/per*t))
lines(fitted(reslm2)~t,col=3)

Is there a way to simplify this code? If I have to get to the 18th harmonics the equation becomes very very long. Also, I still do not know how to add the harmonics here.

Many thanks in advance,

Answer 1

A much easier solution is to use the Fast Fourier Transform (fft)

dat   <- read.csv("Baguio.csv", header=FALSE)
y     <- dat$V1
t     <- 1:73
rg    <- diff(range(y))

nff = function(x = NULL, n = NULL, up = 10L, plot = TRUE, add = FALSE, main = NULL, ...){
  #The direct transformation
  #The first frequency is DC, the rest are duplicated
  dff = fft(x)
  #The time
  t = seq(from = 1, to = length(x))
  #Upsampled time
  nt = seq(from = 1, to = length(x)+1-1/up, by = 1/up)
  #New spectrum
  ndff = array(data = 0, dim = c(length(nt), 1L))
  ndff[1] = dff[1] #Always, it's the DC component
  if(n != 0){
    ndff[2:(n+1)] = dff[2:(n+1)] #The positive frequencies always come first
    #The negative ones are trickier
    ndff[length(ndff):(length(ndff) - n + 1)] = dff[length(x):(length(x) - n + 1)]
  }
  #The inverses
  indff = fft(ndff/73, inverse = TRUE)
  idff = fft(dff/73, inverse = TRUE)
  if(plot){
    if(!add){
      plot(x = t, y = x, pch = 16L, xlab = "Time", ylab = "Measurement",
        main = ifelse(is.null(main), paste(n, "harmonics"), main))
      lines(y = Mod(idff), x = t, col = adjustcolor(1L, alpha = 0.5))
    }
    lines(y = Mod(indff), x = nt, ...)
  }
  ret = data.frame(time = nt, y = Mod(indff))
  return(ret)
}

Then we need to call res, passing the timeseries as x, the number of harmonics as n and the upsampling (so we plot points in time beside the original ones) as up.

png("res_18.png")
res = nff(x = y, n = 18L, up = 100L, col = 2L)
dev.off()

---

To get the sum of the 5th to the 18th harmonics it's simply a difference between series

sum5to18 = nff(x = y, n = 18L, up = 10L, plot = FALSE)
sum5to18$y = sum5to18$y - nff(x = y, n = 4L, up = 10L, plot = FALSE)$y
png("sum5to18.png")
plot(sum5to18, pch = 16L, xlab = "Time", ylab = "Measurement", main = "5th to 18th harmonics sum", type = "l", col = 2)
dev.off()

---

Adding the arguments add and col allow us to plot multiple waves as well, with specific colors

colors = rainbow(36L, alpha = 0.3)
nff(x = y, n = 36L, up = 100L, col = colors[1])
png("all_waves.png")
for(i in 1:18){
  ad = ifelse(i == 1, FALSE, TRUE)
  nff(x = y, n = i, up = 100L, col = colors[i], add = ad, main = "All waves up to 18th harmonic")
}
dev.off()

---

Is there a way so extract the data of each series then save as a csv file. So in this example, I should have 18 csv files for the 18 waves.

I edited the code to allow a 0 harmonic (basically a mean), so now you extract the separate waves as:

sep = array(data = NA_real_, dim = c(7300L, 2 + 18), dimnames = list(NULL, c("t", paste0("H", 0:18))))
sep[,1:2] = as.matrix(nff(x = y, n = 0, up = 100L, plot = FALSE))

for(i in 1:18L){
  sep[,i+2] = nff(x = y, n = i, up = 100L, plot = FALSE)$y - nff(x = y, n = i-1, up = 100L, plot = FALSE)$y
} 

Then you can use write.table to write a csv file.