# Mann and Lees (1996) robust red noise MTM analysis

### Description

Mann and Lees (1996) robust red noise MTM analysis. This function implements several improvements to the algorithm used in SSA-MTM toolkit, including faster AR1 model optimization, and more appropriate 'edge-effect' treatment.

### Usage

1 2 3 |

### Arguments

`dat` |
Stratigraphic series for MTM spectral analysis. First column should be location (e.g., depth), second column should be data value. |

`tbw` |
MTM time-bandwidth product. |

`ntap` |
Number of DPSS tapers to use. By default, this is set to (2*tbw)-1. |

`padfac` |
Pad with zeros to (padfac*npts) points, where npts is the original number of data points. |

`demean` |
Remove mean from data series? (T or F) |

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

`medsmooth` |
ML96 median smoothing parameter (1 = use 100% of spectrum; 0.20 = use 20%) |

`opt` |
Optimization method for robust AR1 model estimation (1=Brent's method:fast, 2=Gauss-Newton:fast, 3=grid search:slow) |

`linLog` |
Optimize AR1 model fit using (1) linear power or (2) log(power)? |

`siglevel` |
Significance level for peak identification. (0-1) |

`output` |
What should be returned as a data frame? (0=nothing; 1= power spectrum + harmonic CL + AR1 CL + AR1 fit + 90%-99% AR1 power levels + median smoothed spectrum; 2=significant peak frequencies; 3=significant peak frequencies + harmonic CL) |

`CLpwr` |
Plot ML96 AR(1) noise confidence levels on power spectrum? (T or F) |

`xmin` |
Smallest frequency for plotting. |

`xmax` |
Largest frequency for plotting. |

`sigID` |
Identify significant frequencies on power and probabilty plots? (T or F) |

`pl` |
Plot logarithm of spectral power (1) or linear spectral power (2)? |

`genplot` |
Generate summary plots? (T or F) |

`verbose` |
Verbose output? (T or F) |

### Details

This function conducts the Mann and Lees (1996; ML96) "robust red noise" analysis, with an improved median smoothing approach. The original Mann and Lees (1996) approach applies a truncation of the median smoothing window to include fewer frequencies near the edges of the spectrum; while truncation is required, its implementation in the original method often results in an "edge effect" that can produce excess false positive rates at low frequencies, commonly within the eccentricity-band (Meyers, 2012).

To help address this issue, an alternative median smoothing approach is applied that implements Tukey's robust end-point rule and symmetrical medians (see the function runmed for details). Numerical experiments indicate that this approach produces an approximately uniform false positive rate across the spectrum. It should be noted that the false positive rates are still inflated with this method, but they are substantially reduced compared to the original ML96 approach. For example, simulations using rho=0.9 (using identical parameters to those in Meyers, 2012) yield median false positive rates of 1.7%, 7.3% and 13.4%, for the 99%, 95% and 90% confidence levels (respectively). This compares with 4.7%, 11.4% and 17.8% using the original approach (see Table 2 of Meyers, 2012).

Candidiate astronomical cycles are identified via isolation of those frequencies that achieve the required (e.g., 90 percent) "robust red noise" confidence level and MTM harmonic F-test confidence level. Allowance is made for the smoothing inherent in the MTM power spectral estimate as compared to the MTM harmonic spectrum. That is, an F-test peak is reported if it achieves the required MTM harmonic confidence level, while also achieving the required robust red noise confidence level within +/- half the power spectrum bandwidth resolution. One additional criterion is included to further reduce the false positive rate, a requirement that significant F-tests must occur on a local power spectrum high, which is parameterized as occurring above the local robust red noise background estimate. See Meyers (2012) for futher information.

NOTE: If the (fast) Brent or Gauss-Newton methods fail, use the (slow) grid search approach.

This version of the ML96 algorithm was first implemented in Patterson et al. (2014).

### References

Mann, M.E., and Lees, J.M., 1996, *Robust estimation of background noise and signal
detection in climatic time series*, Clim. Change, 33, 409-445.

Meyers, S.R., 2012, *Seeing red in cyclic stratigraphy: Spectral noise estimation for
astrochronology*, Paleoceanography, 27, PA3228.

Patterson, M.O., McKay, R., Naish, T., Escutia, C., Jimenez-Espejo, F.J.,
Raymo, M.E., Meyers, S.R., Tauxe, L., Brinkhuis, H., and IODP Expedition 318 Scientists,
2014, *Response of the East Antarctic Ice Sheet to orbital forcing during the
Pliocene and Early Pleistocene*, Nature Geoscience, v. 7, p. 841-847.

Thomson, D. J., 1982, *Spectrum estimation and harmonic analysis*, Proc. IEEE,
70, 1055-1096, doi:10.1109/PROC.1982.12433.

http://www.meteo.psu.edu/holocene/public_html/Mann/tools/MTM-RED/

Tukey, J.W., 1977, *Exploratory Data Analysis*, Addison.

### See Also

`eha`

, `lowspec`

, `mtm`

, `mtmAR`

, `periodogram`

, `runmed`

, and `spec.mtm`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
# generate example series with periods of 400 ka, 100 ka, 40 ka and 20 ka
ex = cycles(freqs=c(1/400,1/100,1/40,1/20),start=1,end=1000,dt=5)
# add AR1 noise
noise = ar1(npts=200,dt=5,sd=0.5)
ex[2] = ex[2] + noise[2]
# run ML96 analysis
pl(1, title="mtmML96")
mtmML96(ex)
# compare to analysis with conventional AR1 noise test
pl(1,title="mtm")
mtm(ex)
# compare to analysis with LOWSPEC
pl(1, title="lowspec")
lowspec(ex)
# compare to amplitudes from eha
pl(1,title="eha")
eha(ex,tbw=3,win=1000,pad=1000)
``` |