ItoGaussPlot

An example of the calculation and displaying of Ito-Gauss plots. It is a measure of the variability of linear trends $B$ in a time-series observation, as a function of windowing length, $w$. Imagine splitting a time-series observation into chunks of length $w$, and fitting a line to each chunk

$A+Bt$

What is the variance of the collection of all the linear components, $B$? This statistic has applications in stochastic modelling, see:

• Variability of Millennial‐Scale Trends in the Geomagnetic Axial Dipole, Buffett et al. (2019)
• Decadal global temperature variability increases strongly with climate sensitivity, Nijsse et al. (2019)

Import libraries for calculations and getting data.

library(ItoGaussPlot)
library(rmatio)

For this example, we will consider the Ornstein-Ulhenbeck process,

$dx_t=-\gamma x_t dt + \sqrt{D}dW_t$

where $x_t$ is the amplitude as a function of time, $\gamma$ is the slope of the drift term, $\sqrt{D}$ is the amplitude of the noise, and $W_t$ is a Wiener process. A previously calculated realization of this process (calculated using the Euler-Maruyama method) is loaded.

data <- read.mat('tX_g1D1e0.dat')
time <- data$tX[1,]
amplitude <- data$tX[2,]
dataVars <- list(gamma=1.0, D=1.0, dt=0.001)
dataVars$Nt <- ceiling(tail(time, n=1)/dataVars$dt)

plot(time, amplitude, type = 'l',
     main='Ornstein-Uhlenbeck integration',
     xlab='Time', ylab='Amplitude')

Calculating the variability of trends as a function of windowing length is done with the stdWindowSlopeCast() function. The time vector, amplitude vector, and window length(s) are arguments.

windows <- 10^seq(-1,2,len=20)
slopeStds <- stdWindowSlopeCast(time, amplitude, windows)

For the Ornstein-Ulhenbeck process, a closed form expression of slope variability can be derived.

$\sigma_B(w) = \sqrt{24\gamma D f(\gamma w)}$

where $f(x)$ is the function

$f(x) = x^{-3}-3x^{-4} + 12x^{-6}-e^{-x}\Big(3x^{-4} + 12x^{-5} + 12x^{-6}\Big).$

windowsTheory <- 10^seq(-1,2,len=50)
slopeStdsTheory <- OUanalytic(windowsTheory, dataVars$gamma, dataVars$D)
slopeStdsTheoryLimit <- OUanalytic(windowsTheory, dataVars$gamma, dataVars$D, "smallW")

Below is a plot comparing the numerical calculations and the theoretical values.

plot(windowsTheory, slopeStdsTheory, 
     log='xy', type = 'l',
     main='Standard deviation of slope by windowing',
     xlab='Window length, w', ylab=expression('Std. of slope, σ'[b]))
lines(windowsTheory, slopeStdsTheoryLimit, lty=2)
points(windows, slopeStds, col="red", pch=16)
legend("topright", legend=c("Theory", "Small w limit", "Calculation"),
       col=c("black", "black", "red"), lty=c(1,3,0), pch=c(0,0,16), 
       pt.cex=c(0,0,1), lwd=c(1.5,1.5,1), cex=0.8)

Uneven time-steps

The package uses a different algorthm for unevenly spaces time-series.

data <- read.mat('tX_g1D1e0NU.dat')
timeNU <- data$tX[1,]
amplitudeNU <- data$tX[2,]
dataVarsNU <- list(gamma=1.0, D=1.0, dt=0.001)
dataVarsNU$Nt <- ceiling(tail(time, n=1)/dataVars$dt)

par(mfrow=c(1,2))    # set the plotting area into a 1*2 array
plot(timeNU, amplitudeNU, type = 'l',
     main='Ornstein-Uhlenbeck integration',
     xlab='Time', ylab='Amplitude')
hist(diff(timeNU),
     main='Histogram of time-steps',
     xlab='Time-step', ylab='Counts')

The stdWindowSlopeCast() function is used again, but theuneven timesteps in the time vector cause a different method to be called.

slopeStdsNU <- stdWindowSlopeCast(timeNU, amplitudeNU, windows)

Below is a plot comparing the numerical calculations and the theoretical values.

plot(windowsTheory, slopeStdsTheory, 
     log='xy', type = 'l',
     main='Standard deviation of slope by windowing',
     xlab='Window length, w', ylab=expression('Std. of slope, σ'[b]))
lines(windowsTheory, slopeStdsTheoryLimit, lty=2)
points(windows, slopeStdsNU, col="red", pch=16)
legend("topright", legend=c("Theory", "Small w limit", "Calculation"),
       col=c("black", "black", "red"), lty=c(1,3,0), pch=c(0,0,16), 
       pt.cex=c(0,0,1), lwd=c(1.5,1.5,1), cex=0.8)