https://github.com/williamjsdavis/itogaussplot
A measure of the variability of linear trends in a time-series observation, as a function of windowing length
https://github.com/williamjsdavis/itogaussplot
r-language statistics stochastic-processes time-series
Last synced: 22 days ago
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A measure of the variability of linear trends in a time-series observation, as a function of windowing length
- Host: GitHub
- URL: https://github.com/williamjsdavis/itogaussplot
- Owner: williamjsdavis
- License: mit
- Created: 2020-08-04T03:56:00.000Z (almost 6 years ago)
- Default Branch: master
- Last Pushed: 2023-10-17T18:08:51.000Z (over 2 years ago)
- Last Synced: 2023-10-17T19:25:50.152Z (over 2 years ago)
- Topics: r-language, statistics, stochastic-processes, time-series
- Language: R
- Homepage:
- Size: 120 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# 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)](https://doi.org/10.1029/2019GL085909)
- [Decadal global temperature variability increases strongly with climate sensitivity, Nijsse et al. (2019)](https://doi.org/10.1038/s41558-019-0527-4)
Import libraries for calculations and getting data.
```R
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](https://en.wikipedia.org/wiki/Wiener_process). A previously calculated realization of this process (calculated using the [Euler-Maruyama method](https://en.wikipedia.org/wiki/Euler–Maruyama_method)) is loaded.
```R
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.
```R
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
$f(x) = x^{-3}-3x^{-4} + 12x^{-6}-e^{-x}\Big(3x^{-4} + 12x^{-5} + 12x^{-6}\Big).$
```R
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.
```R
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.
```R
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.
```R
slopeStdsNU <- stdWindowSlopeCast(timeNU, amplitudeNU, windows)
```
Below is a plot comparing the numerical calculations and the theoretical values.
```R
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)
```
