TimeOptMCMC: Evaluation of eccentricity-related amplitude modulation and bundling in paleoclimate data ("TimeOpt"; Meyers, 2015), with uncertainties on all fitting parameters via Markov-Chain Monte Carlo (MCMC). This function follows the approach of Meyers and Malinverno (2018). MCMC is implemented using the Metropolis-Hastings algorithm. Optimization is conducted upon the sedimentation rate (constant within the study interval), the fundamental frequencies g1-g5, the precession constant k, and four hyperparameters associated with the residuals from the spectral and envelope fit. The priors for the k and g's are Gaussian, while other parameters (sedrate, hyperparameters) are uniform (uninformative).

```
timeOptMCMC(dat,iopt=1,sedmin=0.5,sedmax=5,sedstart=NULL,gAve=NULL,
gSd=NULL,gstart=NULL,kAve=NULL,kSd=NULL,kstart=NULL,
rhomin=0,rhomax=0.9999,rhostart=NULL,sigmamin=NULL,sigmamax=NULL,sigmastart=NULL,
ran=F,fit=1,ftol=0.01,roll=10^3,nsamples=1000,epsilon=NULL,test=F,burnin=-1,
detrend=T,output=1,savefile=F,genplot=1,verbose=T)
```

dat

Stratigraphic series for astrochronologic assessment. First column should be depth or height (in meters), second column should be data value.

iopt

(1) fit power and envelope, (2) fit power only.

sedmin

Minimum sedimentation rate for investigation (cm/ka).

sedmax

Maximum sedimentation rate for investigation (cm/ka).

sedstart

Initial sedimentation rate for MCMC search (cm/ka). Default is 0.5*(sedmin+sedmax). Alternatively, if set to negative number, a random value is selected from the prior distribution.

gAve

Vector which contains the average values for the g1 through g5 fundamental frequencies (arcsec/year). Must be in the following order: g1,g2,g3,g4,g5.

gSd

Vector which contains the standard deviation for the g1 through g5 fundamental frequencies (arcsec/year). Must be in the following order: g1,g2,g3,g4,g5.

gstart

Vector which contains the initial values for the g1 through g5 fundamental frequencies (arcsec/year). Must be in the following order: g1,g2,g3,g4,g5. Default is 0.5*(gmin+gmax). Alternatively, if set to negative number, a random value is selected from the prior distribution.

kAve

Average value for the precession constant (arcsec/year).

kSd

Standard deviation for the precession constant (arcsec/year).

kstart

Initial value for the precession constant (arcsec/year). Default is 0.5*(kmin+kmax). Alternatively, if set to negative number, a random value is selected from the prior distribution.

rhomin

Minimum value for residual lag-1 autocorrelation (for both spectral and envelope fit). Default is 0.

rhomax

Maximum value for residual lag-1 autocorrelation (for both spectral and envelope fit). Default is 0.9999

rhostart

Initial value for residual lag-1 autocorrelation (for both spectral and envelope fit). Default 0.5. Alternatively, if set to negative number, a random value is selected from the prior distribution.

sigmamin

Minimum value for residual sigma (for both spectral and envelope fit).

sigmamax

Maximum value for residual sigma (for both spectral and envelope fit).

sigmastart

Initial value for residual sigma (for both spectral and envelope fit). Default 0.5*(data standard deviation). Alternatively, if set to negative number, a random value is selected from the prior distribution.

ran

Would you like to randomly select the parameter for updating (T), or simultaneously update all the parameters (F)?

fit

Test for (1) precession amplitude modulation or (2) short eccentricity amplitude modulation? Option 2 is not yet functional!

ftol

Tolerance in cycles/ka used to define the precession bandpass. It is added to the highest precession frequency, and subtracted from the lowest precession frequency, to define the half power points for the Taner bandpass filter.

roll

Taner filter roll-off rate, in dB/octave.

nsamples

Number of candidate MCMC simluations to perform.

epsilon

Vector of dimension 11, which controls how large the jump is between each candidate value, e.g. sedimentation rate. For example, a value of 0.2 will yield maximum jump +/- 10 percent of sedimentation rate range. The vector must be arranged in the the following order: sedrate,k,g1,g2,g3,g4,g5,spec_rho,spec_sigma,env_rho,env_sigma. If NULL, all epsilon values will be assigned 0.2

test

Activate epsilon testing mode? This option will assign all MCMC samples a log-likelihood of unity. This provides a diagnostic check to ensure that the applied epsilon values are sampling the entire range of parameter values. (T or F)

burnin

Threshold for detection of MCMC stability.

detrend

Remove linear trend from data series? (T or F)

output

Which results would you like to return to the console? (0) no output; (1) return all MCMC candidates

savefile

Save MCMC samples to file MCMCsamples.csv? (T or F). If true, results are output after every 1000 iterations (last iterations will not be reported if you do not end on an even thousand!)

genplot

Generate summary plots? (0= none; 1=display all summary plots; 2=also include progress plot during iterations; 3=be quiet, but save all plots as pngs to the working directory)

verbose

Verbose output? (T or F)

TimeOpt is an astronomical testing algorithm for untuned (spatial) stratigraphic data. The algorithm identifies the sedimentation rate(s) that simultaneously optimizes: (1) eccentricity amplitude modulations within the precession band, and (2) the concentration of spectral power at specified target astronomical periods.

This version of TimeOpt uses MCMC via Metropolis-Hastings to estimate the parameters and their uncertainties. The priors for the k and g's are Gaussian, while the other parameters (sedrate, hyperparameters) are uniform (uninformative).

When ran=T, the following approach is used to select the parameter to modify:

0.25 probability of changing sedimentation rate

0.25 probability of changing k

0.30 probability of changing g1,g2,g3,g4,g5 (simultaneously)

0.10 probability of changing sigma_spec,rho_spec (simultaneously)

0.10 probability of changing sigma_env,rho_env (simultaneously)

This is motivated by sensitivity tests, and the fact that we are most interested in g, k and s; moving each group of parameters (sedrate, k or g's) has specific consequences we can isolate.

Here are some additional notes on the application of timeOptMCMC:

(1) Before conducting a timeOptMCMC analysis, run timeOpt to get a sense of the optimal sedimentation rate region(s).

(2) Make epsilon as large as you reasonably can, to maximize the chance of jumping between modes. Think of epsilon as analogous to a diffusion coefficient. A good strategy is to run a coarse resolution analysis (large epsilon) to identify the optimum region, then use as small an epsilon as possible to explore that optimum region. Note that larger epsilon yields less correlation in candidates. If you want to determine the time constant (thus number of independent samples) associated with a given epsilon, calculate the autocovariance function for accepted candidates (post-burnin). Decimation is useful for generating independent samples if desired.

(3) For greatest efficiency, the percentage of accepted candidates is typically expected to be between 23-44 percent (see Gelman et al., 1996, "Efficient Metropolis jumping rules"). However, the multimodal nature of the parameter space may require smaller acceptance rates.

(4) To ensure that the MCMC algorithm is exploring the full parameter space, run an analysis with 'test=T'. This option will accept all MCMC candidates. The histogram for each parameter value should approximate the prior distribution. If this is not the case, epsilon should be increased.

(5) It is expected that the MAP should be close to the mode when you have enough samples, although this is not guaranteed.

(6) There are different strategies for implementing the algorithm. One can run one long chain, or run multiple short chains and combine.

(7) If you run a very long test chain, you can decimate to conduct a rarefaction analysis (of the parameters).

(8) For testing, it is recommended to run at least 3 very long chains. Ideally they should be long enough that you can't tell the difference. Plot likelihood versus each candidate, and also sigma vs each candidate, for each run. This will allow identification of simulations that have gone into local minima.

(9) The following are useful estimates to consider: mean of candidate values (post-burnin), MAP, mode of kernel density estimate (post-burnin), 95 percent Credible Interval from kernel density estimate (post-burnin).

(10) Keep in mind that a parabolic plot of log-likelihood vs. parameter value (quadratic in log-likelihood) indicates a Gaussian distribution.

For additional information see Meyers & Malinverno (2018), Meyers (2015), Tarantola (2005), and Malinverno & Briggs (2004).

A. Gelman et al., 1996,
*Efficient Metropolis jumping rules*, Bayesian Statistics 5, p. 599-607.

A. Malinverno and V.A. Briggs, 2004,
*Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical bayes*: Geophysics, 69, doi:10.1190/1.1778243.

S.R. Meyers, 2015,
*The evaluation of eccentricity-related amplitude modulation and bundling in paleoclimate data: An inverse approach for astrochronologic testing and time scale optimization*: Paleoceanography, 30, doi:10.1002/2015PA002850.

S.R. Meyers and A. Malinverno, 2018,
*Proterozoic Milankovitch cycles and the history of the solar system*: Proceedings of the National Academy of Sciences, www.pnas.org/cgi/doi/10.1073/pnas.1717689115.

A. Tarantola, 2005,
*Inverse Problem Theory and Methods for Model Parameter Estimation*, Society for Industrial and Applied Mathematics, 339 pages.