Multitaper estimation on arbitrary domains

Multitaper estimators have enjoyed significant success in estimating spectral densities from finite samples using as tapers Slepian functions defined on the acquisition domain. Unfortunately, the numerical calculation of these Slepian tapers is only tractable for certain symmetric domains, such as rectangles or disks. In addition, no performance bounds are currently available for the mean squared error of the spectral density estimate. This situation is inadequate for applications such as cryo-electron microscopy, where noise models must be estimated from irregular domains with small sample sizes. We show that the multitaper estimator only depends on the linear space spanned by the tapers. As a result, Slepian tapers may be replaced by proxy tapers spanning the same subspace (validating the common practice of using partially converged solutions to the Slepian eigenproblem as tapers). These proxies may consequently be calculated using standard numerical algorithms for block diagonalization. We also prove a set of performance bounds for multitaper estimators on arbitrary domains. The method is demonstrated on synthetic and experimental datasets from cryo-electron microscopy, where it reduces mean squared error by a factor of two or more compared to traditional methods.Comment: 28 pages, 11 figure

A multitaper spectral estimator for time-series with missing data

Author Posting. © The Authors, 2019. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 218(3), (2019): 2165-2178, doi: 10.1093/gji/ggz280.A multitaper estimator is proposed that accommodates time-series containing gaps without using any form of interpolation. In contrast with prior missing-data multitaper estimators that force standard Slepian sequences to be zero at gaps, the proposed missing-data Slepian sequences are defined only where data are present. The missing-data Slepian sequences are frequency independent, as are the eigenvalues that define the energy concentration within the resolution bandwidth, when the process bandwidth is [−1/2,1/2) for unit sampling and the sampling scheme comprises integer multiples of unity. As a consequence, one need only compute the ensuing missing-data Slepian sequences for a given sampling scheme once, and then the spectrum at an arbitrary set of frequencies can be computed using them. It is also shown that the resulting missing-data multitaper estimator can incorporate all of the optimality features (i.e. adaptive-weighting, F-test and reshaping) of the standard multitaper estimator, and can be applied to bivariate or multivariate situations in similar ways. Performance of the missing-data multitaper estimator is illustrated using length of day, seafloor pressure and Nile River low stand time-series.The length of day utilized in Section 3 are available from http://hpiers.obspm.fr. The pressure data used in Section 4 are available from https://doi.org/10.1029/2018JC014586. A Matlab function MDmwps.m to compute missing-data power spectra is available from the Mathworks file exchange website. The author thanks Jeff Park and editor F.J. Simons for thorough reviews. This work was supported by an Internal Research and Development award at WHOI, and by the Walter A. and Hope Noyes Smith Chair for Excellence in Oceanography

Improving Power Spectral Estimation using Multitapering: Precise asteroseismic modeling of stars, exoplanets, and beyond

Asteroseismic time-series data have imprints of stellar oscillation modes, whose detection and characterization through time-series analysis allows us to probe stellar interiors physics. Such analyses usually occur in the Fourier domain by computing the Lomb-Scargle (LS) periodogram, an estimator of the \textit{power spectrum} underlying unevenly-sampled time-series data. However, the LS periodogram suffers from the statistical problems of (1) inconsistency (or noise) and (2) bias due to high spectral leakage. In addition, it is designed to detect strictly periodic signals but is unsuitable for non-sinusoidal periodic or quasi-periodic signals. Here, we develop a multitaper spectral estimation method that tackles the inconsistency and bias problems of the LS periodogram. We combine this multitaper method with the Non-Uniform Fast Fourier Transform (\texttt{mtNUFFT}) to more precisely estimate the frequencies of asteroseismic signals that are non-sinusoidal periodic (e.g., exoplanet transits) or quasi-periodic (e.g., pressure modes). We illustrate this using a simulated and the Kepler-91 red giant light curve. Particularly, we detect the Kepler-91b exoplanet and precisely estimate its period, $6.246 \pm 0.002$ days, in the frequency domain using the multitaper F-test alone. We also integrate \texttt{mtNUFFT} into the \texttt{PBjam} package to obtain a Kepler-91 age estimate of $3.96 \pm 0.48$ Gyr. This $36$\% improvement in age precision relative to the $4.27 \pm 0.75$ Gyr APOKASC-2 (uncorrected) estimate illustrates that \texttt{mtNUFFT} has promising implications for Galactic archaeology, in addition to stellar interiors and exoplanet studies. Our frequency analysis method generally applies to time-domain astronomy and is implemented in the public Python package \texttt{tapify}, available at \url{https://github.com/aaryapatil/tapify}.Comment: 32 pages (3 pages in the Appendix), 14 figures, 2 tables, Submitted to A

Relative stereociliary motion in a hair bundle opposes amplification at distortion frequencies

Direct gating of mechanoelectrical-transduction channels by mechanical force is a basic feature of hair cells that assures fast transduction and underpins the mechanical amplification of acoustic inputs. But the associated nonlinearity - the gating compliance - inevitably distorts signals. Because reducing distortion would make the ear a better detector, we sought mechanisms with that effect. Mimicking in vivo stimulation, we used stiff probes to displace individual hair bundles at physiological amplitudes and measured the coherence and phase of the relative stereociliary motions with a dual-beam differential interferometer. Although stereocilia moved coherently and in phase at the stimulus frequencies, large phase lags at the frequencies of the internally generated distortion products indicated dissipative relative motions. Tip links engaged these relative modes and decreased the coherence in both stimulated and free hair bundles. These results show that a hair bundle breaks into a highly dissipative serial arrangement of stereocilia at distortion frequencies, precluding their amplification.Comment: 33 pages in total, including the main article with one table and three figures, as well as the supplemental information that itself contains two figure

Comparing estimation techniques for temporal scaling in palaeoclimate time series

Characterizing the variability across timescales is important for understanding the underlying dynamics of the Earth system. It remains challenging to do so from palaeoclimate archives since they are more often than not irregular, and traditional methods for producing timescale-dependent estimates of variability, such as the classical periodogram and the multitaper spectrum, generally require regular time sampling. We have compared those traditional methods using interpolation with interpolation-free methods, namely the Lomb–Scargle periodogram and the first-order Haar structure function. The ability of those methods to produce timescale-dependent estimates of variability when applied to irregular data was evaluated in a comparative framework, using surrogate palaeo-proxy data generated with realistic sampling. The metric we chose to compare them is the scaling exponent, i.e. the linear slope in log-transformed coordinates, since it summarizes the behaviour of the variability across timescales. We found that, for scaling estimates in irregular time series, the interpolation-free methods are to be preferred over the methods requiring interpolation as they allow for the utilization of the information from shorter timescales which are particularly affected by the irregularity. In addition, our results suggest that the Haar structure function is the safer choice of interpolation-free method since the Lomb–Scargle periodogram is unreliable when the underlying process generating the time series is not stationary. Given that we cannot know a priori what kind of scaling behaviour is contained in a palaeoclimate time series, and that it is also possible that this changes as a function of timescale, it is a desirable characteristic for the method to handle both stationary and non-stationary cases alike

Relative occurrence rates and connection of discrete frequency oscillations in the solar wind density and dayside magnetosphere

[1] We present an analysis of the occurrence distributions of statistically significant apparent frequencies of periodic solar wind number density structures and dayside magnetospheric oscillations in the f = 0.5–5.0 mHz range. Using 11 years (1995–2005) of solar wind data, we identified all spectral peaks that passed both an amplitude test and a harmonic F test at the 95% confidence level in 6-hour data segments. We find that certain discrete frequencies, specifically f = 0.7, 1.4, 2.0, and 4.8 mHz, occur more often than do other frequencies over those 11 years. We repeat the analysis on discrete oscillations observed in 10 years (1996–2005) of dayside magnetospheric data. We find that certain frequencies, specifically f = 1.0, 1.5, 1.9, 2.8, 3.3, and 4.4 mHz, occur more often than do other frequencies over those 10 years. Many of the enhancements found in the magnetospheric occurrence distributions are similar to those found in the solar wind. Lastly, we counted the number of times the same discrete frequencies were identified as statistically significant using our two spectral tests on corresponding solar wind and magnetospheric 6-hour time series. We find that in 54% of the solar wind data segments in which we identified a spectral peak, at least one of the same discrete frequencies was statistically significant in the corresponding magnetospheric data segment. Our results argue for the existence of inherent apparent frequencies in the solar wind number density that directly drive global magnetospheric oscillations at the same discrete frequencies, although the magnetosphere also oscillates through other physical mechanisms

Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale

The detection of sound begins when energy derived from acoustic stimuli deflects the hair bundles atop hair cells. As hair bundles move, the viscous friction between stereocilia and the surrounding liquid poses a fundamental challenge to the ear's high sensitivity and sharp frequency selectivity. Part of the solution to this problem lies in the active process that uses energy for frequency-selective sound amplification. Here we demonstrate that a complementary part involves the fluid-structure interaction between the liquid within the hair bundle and the stereocilia. Using force measurement on a dynamically scaled model, finite-element analysis, analytical estimation of hydrodynamic forces, stochastic simulation and high-resolution interferometric measurement of hair bundles, we characterize the origin and magnitude of the forces between individual stereocilia during small hair-bundle deflections. We find that the close apposition of stereocilia effectively immobilizes the liquid between them, which reduces the drag and suppresses the relative squeezing but not the sliding mode of stereociliary motion. The obliquely oriented tip links couple the mechanotransduction channels to this least dissipative coherent mode, whereas the elastic horizontal top connectors stabilize the structure, further reducing the drag. As measured from the distortion products associated with channel gating at physiological stimulation amplitudes of tens of nanometres, the balance of forces in a hair bundle permits a relative mode of motion between adjacent stereocilia that encompasses only a fraction of a nanometre. A combination of high-resolution experiments and detailed numerical modelling of fluid-structure interactions reveals the physical principles behind the basic structural features of hair bundles and shows quantitatively how these organelles are adapted to the needs of sensitive mechanotransduction.Comment: 21 pages, including 3 figures. For supplementary information, please see the online version of the article at http://www.nature.com/natur

Gap filling and noise reduction of unevenly sampled data by means of the Lomb-Scargle periodogram

International audienceThe Lomb-Scargle periodogram is widely used for the estimation of the power spectral density of unevenly sampled data. A small extension of the algorithm of the Lomb-Scargle periodogram permits the estimation of the phases of the spectral components. The amplitude and phase information is sufficient for the construction of a complex Fourier spectrum. The inverse Fourier transform can be applied to this Fourier spectrum and provides an evenly sampled series (Scargle, 1989). We are testing the proposed reconstruction method by means of artificial time series and real observations of mesospheric ozone, having data gaps and noise. For data gap filling and noise reduction, it is necessary to modify the Fourier spectrum before the inverse Fourier transform is done. The modification can be easily performed by selection of the relevant spectral components which are above a given confidence limit or within a certain frequency range. Examples with time series of lower mesospheric ozone show that the reconstruction method can reproduce steep ozone gradients around sunrise and sunset and superposed planetary wave-like oscillations observed by a ground-based microwave radiometer at Payerne

CHARACTERIZATION OF SURFACE ROUGHNESS ALONG THE ROCKY COASTLINE USING STEREO PHOTOGRAPHY TECHNIQUES

High-resolution images have been used to estimate and characterize the roughness of the rocky seafloor in terms of small-scale roughness and power spectral density. The application of this work is acoustical modeling of scattering from the sea floor. Two camera systems were designed and built to collect images of the ten different types of surfaces along the rocky shoreline on the Monterey Peninsula at low tide. Using commercial photogrammetry software, the images were processed to calculate height Digital Elevation Maps, which were then used to estimate 1-D and 2-D roughness power spectra. A power-law model was fit to the spectrum and had two parameters, the spectral strength and spectral slope. These roughness power spectra parameters were compared to previously collected parameter data on sandy seafloor, and the scattering strength values were compared to recently collected data along the same rocky coastline of the Monterey Bay. The lower-frequency rocky seafloor spectral strength and slope showed overlap with some of the sand surfaces at varying spatial scales. These parameters were used as inputs into a small-scale roughness perturbation theory model to predict scattering strength of the ten different surfaces for a frequency of 200 kHz and three different grazing angles. The predictions using this scattering strength method were within 10 dB of measurements collected within the same area in the Monterey Bay.Lieutenant Commander, United States NavyApproved for public release. distribution is unlimite