1. Lag-1 autocorrelation coefficient (ρ1) and
2. Dynamic noise after orbital tuning (DYNOT)

This function conducts either single run or Monte Carlo simulations of lag-1 autocorrelation coefficient (ρ1) analysis using a sliding window approach.

Data requirement: either data in time domain or in depth domain works.
No interpolation is needed

The “Single run” requires the input of “window” and “interpolation sampling rate”.

5000 simulations could be sufficient for generating a publication-quality figure.

Default values have to modified!
If a (tuned) series in time domain is used, recommended sliding window sizes range from 300 kyr to 500 kyr; or fixed sliding window ranges from 400 kyr to 400 kyr.
If a series in depth domain is used, the sliding window size should be large enough to conatin sufficient number of datapoints to estimate the ρ1, for example, a sliding window size covers 30 or 50 data points is good.

Again, NO interpolation is needed
Either sr1 or sr2 should be no less than the mean sampling rate.
In our noise model paper (Li et al., 2018 nat. communn, https://doi.org/10.1038/s41467-018-03454-y), we said in page 10: "To avoid ultralow and ultrahigh, inappropriate sampling rates we set the 95th percentiles of sampling rates (sr1) as the lower limit of Monte Carlo-generated uniformly distributed sampling rates and 1.5–2.0 times sr2 as the upper limit."

Default value are usually good enough

See this paper for details about the parameters and significance of this method: (Li et al., 2018 nat. communn, https://doi.org/10.1038/s41467-018-03454-y)

Li et al. (2018a) developed a dynamic noise after orbital tuning, or DYNOT model for the sea-level changes based on the dynamic non-orbital signal in climate proxy records after subtracting orbital, i.e., astronomically forced climate signal.

The DYNOT model is supplemented by a second, independent lag-1 autocorrelation coefficient, or ρ1 model, which forms the basis of a statistical method for red noise estimation of time series.

DYNOT and ρ1 modeling of a GR series of ODP Site 1119 over the past 1.4 myr correlates with the classic low-passed δ18O sea-level curve, demonstrating the efficacy of the sedimentary noise model.

Data for the DYNOT model (support data in *.csv and *.txt format)

Length: m × 2 % must be a 2-column dataset

Column 1: time % unit must be in kyr

Column 2: value

Notes:
#1: Proxy data is assumed to be sensitive to water-depth related noise at your section/core.
#2: There is no requirement for interpolation, normalization, or removing long-term trend (i.e., pre-whitening) of the dataset.
#3: Extreme values should be removed.
#4: Both increasing-upward and decreasing-upward time series are valid.

1. Left-click to select a dataset file in Acycle main window.
2. Select “Timeseries” – “Sedimentary Noise Model” – “DYNOT”
3. The DYNOT sea-level model GUI (Fig. 2) will show below.

Fig. 1. MatLab workspace for the DYNOT model.

Fig. 2. The DYNOT model

4. Click Data ready button to load data or load data from *.txt or *.csv file

In the DYNOT menu:

Select “File” --> “Import Data (*.txt, *.csv)” --> Select data (chose “1119_gr_1400de_finetuned.txt” or “1119_gr_1400de_finetuned.csv”) --> Click “Open” button

Fig. 3. Load data to DYNOT model.

Yellow: load data and run the model.
Red: Key settings. Check before running the model.
Green: Optional settings. Default values are okay for most running.

• 5.3.0. Click on Data Ready (button) to load data into the DYNOT model.
  

• 5.3.1. Cut data (optional)
  These settings automatically show the beginning and the end of the time series, i.e., time span of dataset.
  Unit is ka.
  If you want to choose a different interval, just type two new ages and click the Cut button.

• 5.3.2. Sampling rates (optional)
  These show a range of sample rates covering 90% of sample rates
  (Green Box 20 in Fig. 4).
  Unit is ka.

  A Monte Carlo method of hypothesis testing and the multi-taper method (MTM) of power spectral analysis are to be undertaken, and so resampling must be applied. Sampling rates of proxy datasets in time are always greater than zero and so are non-normally distributed.
  

  Therefore, the Weibull distribution is used to represent sampling rate distributions for uncertainty analysis in the DYNOT model. To avoid an ultra-low or ultra-high, unrealistic sampling rate created by the Weibull distribution algorithm, we set the 5th and 95th percentiles of sampling rates of of the data as default, lower and upper limits of the generated, Weibull-distributed sampling rates.

• 5.3.3. Windows
  These values set sliding window range.
  Moving window length in units of time (<< total data length).
  Unit is ka.
  

  Different windows in the DYNOT model can affect results in two ways:

  • (1) The DYNOT model with a large window will shorten DYNOT results, and the model with a small window will generate longer DYNOT results, Nr = Ndata – window + 1, where Nr is total number of DYNOT values of each simulation, Ndata is total number of interpolated data points, and window is the running window employed.
  • (2) The DYNOT model with a small running window generates higher resolution results, however, the variance of low-frequency cycles and total variance diminish simultaneously, which leads to increased uncertainty in non-orbital signal ratio estimation.
    The DYNOT model with a small running window also increases the MTM power spectrum bandwidth (i.e., reduces frequency resolution). The expected sea-level variations of interest in the Early Triassic are 10^4 to 10^6 year-scale, i.e., the fifth to third-order sequences, therefore a comparable or shorter time window (e.g., 300-500 kyr, 400 kyr or shorter) should be adopted for DYNOT modeling.

• 5.3.4. Time-bandwidth product (optional)

  Time-bandwidth product of discrete prolate spheroidal sequences used for window.
  Typical choices are 2, 5/2, 3, 7/2, 4.

• 5.3.5. Zero-padding (optional)
  Zero-padding number, e.g., 1000.

• 5.3.6. Step (optional)
  Step of calculations; default is 5 ka.

• 5.3.7. Number of Monte Carlo Simulations
  Default is 1000.
  Maybe use 100 or 300 for a trial running.
  Recommended value for publication is >5000.

• 5.3.8. Age of the time series
  The age in Ma will be used to estimated target orbital cycles in 5.3.9.
  You can use either 5.3.8 or 5.3.9 section to tell the DYNOT model the target cycles.

• 5.3.9. Target orbital cycles (space delimited, in ka)
  Six orbital cycles of long-eccentricity (405), short-eccentricity (125 and 95), obliquity (40.9 or shorter), precession (23.6, 22.3, and 19.1 or shorter).
  
  This is age dependent (see 7.8).
  The 405, 125, and 95 kyr cycles are assumed to be invariant through time.
  While the
  obliquity = 41-0.0332*age;
  precession 1 = 23.75-0.0121*age;
  precession 2 = 22.43-0.0121*age;
  precession 3=19.18-0.0079*age.
  These calculations are from Yao et al. (2015), and are based on the La2004 astronomical model (Laskar et al., 2004).

Fig. 4. Settings of the DYNOT model.

Yellow: load data and run the model.
Red: Key settings. Check before running the model.
Green: Optional settings. Default values are okay for most running.

• 5.3.10. Frequency ranges (optional)
  For the definition of the non-orbital signal ratio by Li et al. (2018a), cutoff frequencies and their bandwidths are crucial for estimation of variances of eccentricity, obliquity and precession signals.
  
  We vary each cutoff frequency assuming a uniform distribution with cutoff frequency ranges at ± 90% to ± 120% bandwidth.
  
  Here the bandwidth (bw) equals nw/window, where nw is time-bandwidth product of discrete prolate spheroidal sequences, and window is the running window.

• 5.3.11. Cutoff frequencies (optional)
  lower cutoff frequency (> 0) for estimation of total variance and
  upper cutoff frequency (< Nyquist frequency) for estimation of total variance.

• 5.3.12. Confidence levels (optional) Default values show median and confidence levels (e.g., 50%, 68%, 80%, 90%, and 95%) of the DYNOT results.

• 5.3.13. Interpolation (optional)
  In 5.3.3 section, a smaller Nr compared to Ndata leads to a “no data” effect at the very beginning and/or very end of the DYNOT results.
  
  To avoid this problem and to provide a better constraint for noise estimation, technically, the DYNOT model is interpolated and randomly shifts and plots simulation results of a single iteration at the same time scale of the dataset, although the plots also generate relatively smoothed DYNOT spectra when a gap is shorter than 2 × window. Here 1000 is adequate for the DYNOT model.

• 5.3.14 Shift plot grids (optional)
  See 5.3.13 for interpretation.

  Default is 15. One can also use 15-30 for the better shape of the beginning and the end of the DYNOT spectra.

• 5.3.15. Number of physical cores (optional)
  This detects the physical cores of the CPU of the computer.

• 5.3.16. Number of itineraries to estimate the process time (optional)
  To estimate process time of the time-consuming DYNOT model, the model will run some itineraries. Default is 50.

• 5.3.17. Emergency note
  Press “Ctrl + C” to cease the DYNOT process before the parallel computing.
  Press “ Ctrl + X`” to cease the DYNOT process during the parallel computing.
  You may need to type the following script in the command window to quite parallel computing.

  delete(gcp(‘nocreate’))

• 5.3.18. Click the button to run the model.

• 5.3.19. A window shows the dataset.

• 5.3.20. A window shows sample rates of the dataset OR the DYNOT spectrum of the dataset.

Click the Let’s go button to run the DYNOT code.
In the command window, the estimated running time will appear:

16:21:20 Begin the process ...
16:22:54 First 50 iterations suggest: remain >= 0h:7m:27sec
% The model runs the first 50 iterations to estimate that the total running
% time will last ca. 7 minutes 27 seconds. The real run-time may be 10s seconds
% to several minutes longer than this estimate.
% Starting parallel pool (parpool) using the 'local' profile ... connected to 4 workers.
16:23:07 Current iteration takes 1.11 seconds
16:23:08 Current iteration takes 1.21 seconds
16:23:15 Current iteration takes 1.19 seconds
16:26:26 Current iteration takes 1.38 seconds
% Start parallel computing and show time of each iteration.
% Parallel pool using the 'local' profile is shutting down.
>> Done. % Stop parallel computing and display the DYNOT result (Fig. 5).

Fig. 5. DYNOT sea-level model of the gamma-ray series at ODP site 1119 from 0 to 1.4 Ma.

After running the DYNOT model, the median value of noise and percentiles of the outputs will be saved as text files.

The GUI menu (Fig. 6) can be used to:

• #1: save a MatLab-fig in the working directory entitled “plots_.fig”.
  
• #2: save a PDF file of the plots in the working directory entitled “plots_.pdf”
  
• #3: pop-up display the DYNOT spectrum in a new window.
  
• #4: save DYNOT output data in the working directory entitled “result_handles.mat”.
  

Caution: Change names of output files, or they will be overwritten by new files.

Fig. 6. Output files