Effect of nonlinear filters on detrended fluctuation analysis

We investigate how various linear and nonlinear transformations affect the scaling properties of a signal, using the detrended fluctuation analysis (DFA). Specifically, we study the effect of three types of transforms: linear, nonlinear polynomial and logarithmic filters. We compare the scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power of the polynomial filter and (c) the offset in the logarithmic filter. We further investigate the correlation properties of three analytic functions: exponential, logarithmic, and power-law. While these three functions have in general different correlation properties, we find that there is a broad range of variable values, common for all three functions, where they exhibit identical scaling behavior. We further note that the scaling behavior of a class of other functions can be reduced to these three typical cases. We systematically test the performance of the DFA method in accurately estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters, and the three analytic functions we consider.Comment: 12 pages, 7 figure

The Bartonian-Priabonian marine record of the eastern South Pyrenean foreland basin (NE Spain): a new calibration of the larger foraminifers and calcareous nannofossil biozonation

This study presents a combined biostratigraphic (calcareous nannofossils, larger foraminifers) and magnetostratigraphic study of the Middle and Late Eocene marine units of the Igualada area, eastern Ebro Basin. The studied sections of Santa Maria de Miralles and La Tossa encompass the complete marine succession of the Santa Maria Group, where assemblages rich in larger foraminifers have been studied since the early 1950’s. A total of 224 paleomagnetic sites and 62 biostratigraphic samples were collected along a 1350m-thick section that ranges from chron C20n to chron C16n (∼43Ma to ∼36Ma). The resulting magnetostratigraphy-based chronology challenges existing chronostratigraphic interpretations of these units and results in a new calibration of the biostratigraphic zonations. The base of calcareous nannofossil Zone NP19-20 is pinned down to an older age than its presently accepted attribution, whereas the time span assigned to Zone NP18 is significantly reduced. A revised calibration of larger foraminifers indicates that Zone SBZ18, formerly assigned exclusively to the late Bartonian, extends its range to the earlymost Priabonian, the Bartonian stage being almost entirely represented by Zone SBZ17. A division of Zone SBZ18 into two subzones is proposed

Bayesian Blocks, A New Method to Analyze Structure in Photon Counting Data

I describe a new time-domain algorithm for detecting localized structures (bursts), revealing pulse shapes, and generally characterizing intensity variations. The input is raw counting data, in any of three forms: time-tagged photon events (TTE), binned counts, or time-to-spill (TTS) data. The output is the most likely segmentation of the observation into time intervals during which the photon arrival rate is perceptibly constant -- i.e. has a fixed intensity without statistically significant variations. Since the analysis is based on Bayesian statistics, I call the resulting structures Bayesian Blocks. Unlike most, this method does not stipulate time bins -- instead the data themselves determine a piecewise constant representation. Therefore the analysis procedure itself does not impose a lower limit to the time scale on which variability can be detected. Locations, amplitudes, and rise and decay times of pulses within a time series can be estimated, independent of any pulse-shape model -- but only if they do not overlap too much, as deconvolution is not incorporated. The Bayesian Blocks method is demonstrated by analyzing pulse structure in BATSE $\gamma$-ray data. The MatLab scripts and sample data can be found on the WWW at: http://george.arc.nasa.gov/~scargle/papers.htmlComment: 42 pages, 2 figures; revision correcting mathematical errors; clarifications; removed Cyg X-1 sectio

New stopping criteria for segmenting DNA sequences

We propose a solution on the stopping criterion in segmenting inhomogeneous DNA sequences with complex statistical patterns. This new stopping criterion is based on Bayesian Information Criterion (BIC) in the model selection framework. When this stopping criterion is applied to a left telomere sequence of yeast Saccharomyces cerevisiae and the complete genome sequence of bacterium Escherichia coli, borders of biologically meaningful units were identified (e.g. subtelomeric units, replication origin, and replication terminus), and a more reasonable number of domains was obtained. We also introduce a measure called segmentation strength which can be used to control the delineation of large domains. The relationship between the average domain size and the threshold of segmentation strength is determined for several genome sequences.Comment: 4 pages, 4 figures, Physical Review Letters, to appea

FIELD TRIP TO THE YPRESIAN/LUTETIAN BOUNDARY AT THE GORRONDATXE BEACH SECTION (BASQUE COUNTRY, W PYRENEES)

One of the Paleogene Stage boundaries still needing official definition is the Ypresian/Lutetain (Early- Middle Eocene) boundary. With the aim of contributing to attain this definition, a high-resolution multidisciplinary study, including physical stratigraphy (lithostratigraphy, sequence stratigraphy and magnetostratigraphy) and biostratigraphy (calcareous nannofossil, planktic foraminifer and larger foraminifer), has been carried out over the 700 m thick uppermost Ypresian – lower Lutetian Gorrondatxe section. The results show that the different events traditionally used to place the Ypresian/Lutetian boundary, hitherto thought to be simultaneous (i.e., the planktic foraminifer P9 (=E7) / P10 (=E8) Zone boundary; the calcareous nannofossil CP12a / CP12b Subzone boundary; the larger foraminifer SBZ12 / SBZ13 Zone boundary; and the boundary between magnetic polarity chrons C22n and C21r), actually occur at very different levels. Therefore, before considering any section to place the Ypresian/Lutetian boundary stratotype, the criterion to precisely define this boundary should be selected. To this end, the succession of events pinpointed in the Ypresian/Lutetian boundary interval of the Gorrondatxe beach section might prove a useful database. The Gorrondatxe section fulfils most of the requirements demanded of a prospective stratotype section. In addition, the great sedimentary thickness, which implies a very high deep-marine sedimentation rate, provides the Gorrondatxe section an additional value, as it offers the opportunity to chronologically order successive biomagnetostratigraphic events more precisely than elsewhere. Therefore, we consider that, once the criterion to define the Ypresian/Lutetian boundary is selected, the Gorrondatxe beach section should be deemed a firm candidate to place the Global Stratotype Section and Point of the base of the Lutetian Stage

Phase Transition in a Random Fragmentation Problem with Applications to Computer Science

We study a fragmentation problem where an initial object of size x is broken into m random pieces provided x>x_0 where x_0 is an atomic cut-off. Subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x_0. The process stops when all the fragments have sizes smaller than x_0. We show that the fluctuation of the total number of splitting events, characterized by the variance, generically undergoes a nontrivial phase transition as one tunes the branching number m through a critical value m=m_c. For m<m_c, the fluctuations are Gaussian where as for m>m_c they are anomalously large and non-Gaussian. We apply this general result to analyze two different search algorithms in computer science.Comment: 5 pages RevTeX, 3 figures (.eps

Effect of extreme data loss on long-range correlated and anti-correlated signals quantified by detrended fluctuation analysis

We investigate how extreme loss of data affects the scaling behavior of long-range power-law correlated and anti-correlated signals applying the DFA method. We introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of correlations. These surrogate signals are characterized by: (i) the DFA scaling exponent $\alpha$ of the original correlated signal, (ii) the percentage $p$ of the data removed, (iii) the average length $\mu$ of the removed (or remaining) data segments, and (iv) the functional form of the distribution of the length of the removed (or remaining) data segments. We find that the {\it global} scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90%. In contrast, the global scaling of anti-correlated signals changes to uncorrelated behavior even when a very small fraction of the data is lost. These observations are confirmed on the examples of human gait and commodity price fluctuations. We systematically study the {\it local} scaling behavior of signals with missing data to reveal deviations across scales. We find that for anti-correlated signals even 10% of data loss leads to deviations in the local scaling at large scales from the original anti-correlated towards uncorrelated behavior. In contrast, positively correlated signals show no observable changes in the local scaling for up to 65% of data loss, while for larger percentage, the local scaling shows overestimated regions (with higher local exponent) at small scales, followed by underestimated regions (with lower local exponent) at large scales. Finally, we investigate how the scaling is affected by the statistics of the remaining data segments in comparison to the removed segments

The early/middle Eocene transition at the Ésera valley (South-Central Pyrenees): Implications in Shallow Benthic Zones (SBZ)

An integrated study including magnetostratigraphy, larger benthic foraminifera and calcareous nannofossil biostratigraphy is presented herein. This work was performed in shallow marine siliciclastics rich in larger foraminifera, around the Ypresian/Lutetian boundary in the Ésera valley (South-Central Pyrenees). Although the calcareous nannofossil content in the studied interval is low, not allowing a precise Y/L boundary to be recognised, the taxa found are enough to support the chronostratigraphic attribution. Data obtained in the Ésera valley section has improved the knowledge of larger benthic foraminifera (Nummulites and Assilina) distribution through chron C21. SBZ 11 to SBZ 12 transition took place at the lowermost C21r, as shown in previous works. SBZ 12 assemblages extend into C21n, where the SBZ 12 to SBZ 13 boundary occurs. These data, obtained in shallow marine siliciclastic facies, with in situ fauna, results in a shift of the SBZ 12/SBZ 13 boundary to the Lower Lutetian, younger than previously believed. Accordingly, the Ypresian/Lutetian boundary occurs in SBZ 12. © A. Rodríguez-Pintó, J. Serra-Kiel, G. Bernaola, A. Barnolas, E.L. Pueyo, J. Tosquella, P. Arbués, R. Scholger, 2022 CC BY-SA

Cyclostratigraphy of the Early/Middle Eocene transition: a Pyrenean perspective

An integrated bio-, magneto- and cyclostratigraphic study of the Ypresian/Lutetian (Early/Middle Eocene) transition along the Pyrenean Otsakar section (Payros et al., 2011) resulted in the identification of the C22n/C21r chron boundary and of the calcareous nannofossil CP12a/b zonal boundary; the latter is the main correlation criterion of the Lutetian Global Stratotype Section and Point (GSSP) recently defined at Gorrondatxe (Basque Country). By counting precession-related mudstone-marl couplets of 21 ka, the time lapse between both events was calculated to be of 819 ka. This suggests that the age of the CP12a/b boundary, and hence that of the Early/Middle Eocene boundary, is 47.76 Ma, 250 ka younger than previously thought. This age agrees with, and is supported by, estimates from Gorrondatxe based on the time lapse between the Lutetian GSSP and the C21r/C21n boundary. The duration of Chron C21r is estimated at 1.326 Ma. Given that the base of the Eocene is dated at 55.8 Ma, the duration of the Early Eocene is of 8 Ma, 0.8 Ma longer than in current time scales. The Otsakar results further show that the bases of planktic foraminiferal zones E8 and P10 are younger than the CP12a/b boundary. The first occurrence of Turborotalia frontosa, being approximately 550 ka older that the CP12a/b boundary, is the planktic foraminiferal event that lies closest to the Early/Middle Eocene boundary. The larger foraminiferal SBZ12/13 boundary is located close to the CP12a/b boundary and correlates with Chron C21r, not with the C22n/C21r boundary