Wavelet Lecture

Wavelet lecture ‘Analysis of climate variability’

This lecture was part of the winter school ‘Analysis of climate variability’ at the Institute of Baltic Sea Research. The students were participating by using their on jupyter notebook similar to the one below.

If you are interested in wavelets a use R I strongly encourage you to read

WaveletComp 1.1: A guided tour through the R package Angi R̈osch and Harald Schmidbauer

for the programming part (many of the example come from that pdf)

and for the theory part:

A practical guide to wavelet analysis

Introduction to wavelet analysis

Why not use Fourier analysis?

Although Fourier analysis is well suited to analyze constant periodic components in time series, it cannot characterize signals whose frequency content changes with time. So, a Fourier decomposition may give all spectral components of a signal, but it does not provide any information when they are present.

Task 1: Compute the Power Spectrum of a time series

library("WaveletComp")
library("fields")

Loading required package: spam
Loading required package: dotCall64
Loading required package: grid

Attaching package: ‘grid’

[...]

x = periodic.series(start.period = 50, length = 1000)
x = x + 0.2*rnorm(1000)  # add some noise

plot(x, type="l")

Power spectrum

The power spectrum is used to examine the main characteristics of a time series. For example, it can be used to detect if seasonality is present in the data, if so, the spectrum will show peaks at the seasonal frequencies.

The power spectrum S_xx of a time series x(t) can be used to describe the distribution of power into frequency components composing that singal.

Using fourier analysis we can decompose any physical signal into a spectrum of frequencies.

So:

$ E = \int_{-\infty}^{\infty}

x (t)

^2 dt $

\[\int_{-\infty}^{\infty} | x (t) |^2 dt = \int_{-\infty}^{\infty} | \hat{x} (f) |^2 df\]

where \(\hat{x} (f) = \int_{-\infty}^{\infty} e^{-2\pi i f t} x(t) dt\)

\[S_{xx} (f) = | \hat{x} (f) |^2\]

Tip: Fourier transformations in R can be applied with:

fft()

#
# Task 1
#

power_spectrum <- function(x)
    {
    x.fft <- fft(x)# ADD HERE
    sym.x <- floor(length(x.fft)/2) # Symetric part of fft
    range <- seq(1, sym.x, 1)
    x.fft <- x.fft[range]
    x.fft.power <- abs(x.fft)**2# ADD HERE
    freq <- 0:(length(x.fft)-1) * 1 / length(x.fft) / 2 
    our.own.power <- list("freq" = freq, "power" = x.fft.power)
    return(our.own.power)
}

a <- power_spectrum(x)
plot(1/a$freq,a$power,xlim=c(0,100), type = "l")

Of course someone already implemented this in R.

x.spec <- spec.pgram(x,plot = FALSE)
plot(1/x.spec$freq, x.spec$spec, type = "l", xlim = c(0,100))

Why do we need time information about our time series

Task 2:

plot and compare these two time series:

x1 <- periodic.series(start.period = 90, length = 500)
x2 <- 1.2*periodic.series(start.period = 40, length = 500)
x <- c(x1, x2) + 0.3*rnorm(1000)

y1 <- periodic.series(start.period = 90, length = 1000)
y2 <- 1.2*periodic.series(start.period = 40, length = 1000)
y <- (y1 + y2)/2  + 0.3*rnorm(1000)

par(mfrow=c(2,1))    # set the plotting area into a 2*1 array

plot(x, type = "l")
plot(y, type = "l")

x.spec <- spec.pgram(x, plot = FALSE)
y.spec <- spec.pgram(y, plot = FALSE)

par(mfrow=c(1,2))    # set the plotting area into a 2*1 array

# xlim setzen
plot(1/y.spec$freq, y.spec$spec, type = "l", xlim=c(0, 200))
plot(1/x.spec$freq, x.spec$spec, type = "l", xlim=c(0, 200))

Finally, wavelets …

my.data <- data.frame(x = x)
my.w.x <- analyze.wavelet(my.data, "x",loess.span = 0,
                        dt = 1, dj = 1/250,lowerPeriod = 16,
                        upperPeriod = 128,make.pval = TRUE, n.sim = 10)


my.data <- data.frame(x = y)
my.w.y <- analyze.wavelet(my.data, "x",loess.span = 0,
                        dt = 1, dj = 1/250,lowerPeriod = 16,
                        upperPeriod = 128,make.pval = TRUE, n.sim = 10)

wt.image(my.w.x, n.levels = 250,legend.params = list(lab = "wavelet power levels"))

wt.image(my.w.y, n.levels = 250,legend.params = list(lab = "wavelet power levels"))

One more example

A series with linearly increasing trend

x = periodic.series(start.period = 1, end.period = 100, length = 1000)
x = x + 0.2*rnorm(1000)
plot(x, type="l")

my.data <- data.frame(x = x)
my.w <- analyze.wavelet(my.data, "x",loess.span = 0,dt = 1, dj = 1/250,make.pval = TRUE, n.sim = 10)
wt.image(my.w, n.levels = 250,legend.params = list(lab = "wavelet power levels"))

Starting wavelet transformation...
... and simulations... 
  |======================================================================| 100%
Class attributes are accessible through following names:
series loess.span dt dj Wave Phase Ampl Power Power.avg Power.pval Power.avg.pval Ridge Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tz 

Wavelets are a good choice, if you want to study periodic phenomena in a time series. Especially, when the frequency changes across time. Wavelets provide a reasonable compromise in the time and frequency resolution dilemma.

Remember your last physics class and think about Heisenbergs uncertainty principle

Wavelet analysis is being used since the 1980s a find a broad use image processing, medicine, geophysics or astronomy. But recently it is also applied to economics:

So in this course we will analyze the frequency structure on a time series using the Morlet wavelet. The continous, complex-valued wavelet leads to a continous, complex valued wavelet transform of the time series. We can seperate the wavelet transform into its real part and its imaginary part. Thus, providing information on both, local amplitude and instantaneous phase of any periodic process across time.

The “mother” Morlet wavelet we will use is defined as

\psi (t) = \pi^{-1/4} \exp(i\omega t) \exp(-t^2 /2)

with the “angular frequency” $\omega$ set to 6.

Task 1:

Define the wavelet function mentioned above and seperate its real and imaginary part using

Re() and Im().

Note: imaginary numbers are denoted as:

R-code snippet:

imaginary_number <- 1i

morlet_wavelet <- function(t){
return(pi**-0.25 * exp(1i*6*t) * exp(-0.5*t**2))
}

t <- seq(-6,6,0.01)

plot(Re(morlet_wavelet(t)), col="black",xlab="", ylab="", type="l")
par(new=TRUE)
lines(Im(morlet_wavelet(t)), col="green",xlab="", ylab="", lty="solid")
title("The Morlet mother wavelet")
legend("topright",legend=c("Real","Imag"), col = c("black", "green"),
       pch=rep(c(16,18),each=4),ncol=2,cex=0.7,pt.cex=0.7)

The morlet wavelet transform of a time series x_t is defined as the convolution of the series with a set of “wavelet daughters” generated by the mother wavelet, by shifting it in time by \tau and scaling it by s.

Wave(\tau , s) = \sum_t x_t \frac{1}{\sqrt(s)} \psi^{*} (\frac{t-\tau}{s})

where * denotes the complex conjugate.

The position of the particular daugher wavelet in the time domain is determined by the localizing time parameter $\tau$ being shift by a time increment of $dt$. The choice of the set of scales $s$ determines the wavelet coverage of the series in the frequncey domain.

morlet_wavelet <- function(t){
    return(pi**-0.25 * exp(1i*6*t) * exp(-0.5*t**2))
}

morlet_wavelet_fft <- function(f){
    wave <- (pi**-0.25) * exp(-0.5 * (f - 6)**2) 
    return(wave)
}

x = periodic.series(start.period = 1, end.period = 100, length = 1000)
x = x + 0.2*rnorm(1000)

f <- fft(x)# ADD HERE

dt = 1
dj = 1/12
s0 = 2

n <- NROW(x)

# FFT frequencies
freq <- 0:(length(f)-1) * 1 / length(f) * 2 * pi

# Scaling of the wavelet 
J1 <- round(log2(n * dt / s0) / dj)
scale <- s0 * 2 ^ ((0:J1) * dj)
wave <- matrix(0, nrow = J1 + 1, ncol = n)

for (a1 in seq_len(J1 + 1)) {
psi.star = Conj(morlet_wavelet_fft(scale[a1] * freq))
psi.ft.bar = ((scale[a1] * freq[2] * n)^0.5)* psi.star
wave[a1, ] <- fft(f * psi.ft.bar, inverse = TRUE)
}

power <- abs(wave)^2

new.data <- aperm(power, c(2,1))

range <- seq(1, length(f), 1)

image.plot(range, scale, new.data, ylim=c(16,128))

El Nino Southern Oscillation

sst <- read.table("http://paos.colorado.edu/research/wavelets/wave_idl/nino3sst.txt", header = F, skip = 19)

date <- seq(ISOdate(1871,1,1), ISOdate(1996,12,1), by = "quarter")

my.data <- data.frame(x = sst, date = date)
plot(date, my.data$V1, type = "l", ylab="[°C]", main = "NINO3 SST")

my.w <- analyze.wavelet(my.data, "V1",loess.span = 0,dt = 0.25, dj = 1/250,make.pval = TRUE, n.sim = 30)
wt.image(my.w,color.key="i", n.levels = 250,legend.params = list(lab = "wavelet power levels"))

Starting wavelet transformation...
... and simulations... 
  |======================================================================| 100%
Class attributes are accessible through following names:
series loess.span dt dj Wave Phase Ampl Power Power.avg Power.pval Power.avg.pval Ridge Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tz 

First, we can see that most of the power is concentrad withing the ENSO band of 2-8 yr band.

With wavelet analysis, one can see variations in the frequency of occurrence and amplitude of El Niño (warm) and La Niña (cold) events. During 1875–1920 and 1960–90 there were many warm and cold events of large amplitude, while during 1920–60 there were few events (Torrence and Webster 1997). From 1875–1910, there was a slightshift from a period near 4 yr to a period closer to 2 yr,while from 1960–90 the shift is from shorter to longer periods.

Weather and Radiation

data(weather.radiation.Mannheim)
head(weather.radiation.Mannheim)

date	temperature	humidity	radiation
2005-01-01	6,5	90	0,096
2005-01-02	6,6	71	0,095
2005-01-03	4,8	75	0,094
2005-01-04	5,0	80	0,094
2005-01-05	4,5	83	0,097
2005-01-06	7,2	81	0,096

par(mfrow=c(3,1))

date <- as.POSIXct(strptime(weather.radiation.Mannheim$date, format="%Y-%m-%d"))

plot(date, weather.radiation.Mannheim$temperature,
     ylab = "Temp in Degree C", type="l")
plot(date, weather.radiation.Mannheim$humidity,
     type="l", ylab = "relative humidity [%]")
plot(date, weather.radiation.Mannheim$radiation,
     type="l", ylab ="radiation ( µ SV / h)")

For a systematic analysis, we compute the wavelet transform and plot the wavelet power spectrum of each series.

do.wavelet <- function(x, col.name){
    my.w <- analyze.wavelet(x,
                            col.name,
                            loess.span = 0,
                            dt = 1,
                            dj = 1/50,
                            make.pval = TRUE,
                            n.sim = 30)
    
    max.power <- max(my.w$Power)
    wt.image(my.w,
             color.key ="i",
             maximum.level = sqrt(max.power) * 1.001,
             exponent = 0.5,
             n.levels = 250,
             legend.params = list(lab = "wavelet power levels"),
             show.date = TRUE, date.format= "%F", timelab = "")
}

# wavelet for temperature

do.wavelet(weather.radiation.Mannheim, "temperature")

Starting wavelet transformation...
... and simulations... 
  |======================================================================| 100%
Class attributes are accessible through following names:
series loess.span dt dj Wave Phase Ampl Power Power.avg Power.pval Power.avg.pval Ridge Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tz 

# wavelet for radiation

do.wavelet(weather.radiation.Mannheim, "radiation")

Starting wavelet transformation...
... and simulations... 
  |===================================================                   |  73%

The same settings for the power level range have been used for the three plots, this makes it easier to compare the features of the three series. Indeed, there is a strong 365-day period in the temperature series. To a lesser extent, this is also true for the other two series. Radiation is a more complex phenomenon: many other periods are significant too.

Cross-wavelet

There are three observations for each day, and we can use cross-wavelet analysis to investigate the periodic linkages between pairs of the three series. To that end, we need a scheme which helps interpret cross-wavelet power spectra.

[placeholder]

Each pair from the series temperature, humidity and radiation can be analyzed jointly with respect to its wavelet coherency; this will reveal which series is leading at given time and period (in case of joint significance):

my.wc <- analyze.coherency(weather.radiation.Mannheim,
my.pair = c("temperature", "humidity"),
loess.span = 0,
dt = 1, dj = 1/50,
lowerPeriod = 32, upperPeriod = 1024,
make.pval = TRUE, n.sim = 10)

wc.image(my.wc, n.levels = 250,
legend.params = list(lab = "cross-wavelet power levels"),
color.key = "interval",
# time axis:
label.time.axis = TRUE, show.date = TRUE,
spec.time.axis = list(at = paste(2005:2014, "-01-01", sep = ""),
labels = 2005:2014),
timetcl = -0.5, # outward ticks
# period axis:
periodlab = "period (days)",
spec.period.axis = list(at = c(32, 64, 128, 365, 1024)),
periodtck = 1, periodtcl = NULL)

Temperature and humidity are out of phase, with humidity leading by roughly 1 / 8 year.

In the atmosphere the relative humidity of the air is increased, and condensation results when air temperature is reduced to the dew point or when sufficient water vapor is added to saturate the air