Non-zero Lags Research Articles

SAMPLE VARIANCE IN LARGE-SCALE COSMIC MICROWAVE BACKGROUND ANISOTROPY MEASUREMENTS

The effects of limited sky coverage in large-angle cosmic microwave background anisotropy experiments are investigated by computing the variance of the angular two-point correlation function with an incomplete sphere. We find that, assuming a power spectrum of density fluctuations with spectral index n = 1, the Galactic cut of half-width 20° (40°) about the Equator made by the COBE DMR experiment would induce a sample variance on the rms temperature fluctuation [(ΔT/T) rms ]2 (or equivalently, the correlation function at zero lag), which is 12% (38%) greater than the cosmic variance with a whole sky coverage. This result is about two times smaller than the naive expectation that the cosmic variance is enhanced by a factor of [Formula: see text], where A is the solid angle sampled by the experiment. We also find that the sample variance of the correlation function at nonzero lag can approach the cosmic variance limit. Our approach provides an analytic way of finding a theoretical error to the theoretical prediction for a particular experiment (either large- or small-scale), without having recourse to computationally intensive Monte Carlo or maximum likelihood methods.

Heavy tailed hidden semi-markov models

Hidden semi-Markov models have been proposed in Meier-Hellstern et al (1991) to model the times between transmission of packets at a source. In this paper, we specifically investigate such models where the distributions associated with one or more of the states are heavy-tailed. We study the asymptotic properties of the sample covariance and correlation functions and show that the correlation function at nonzero lags is asymptotically equal to zero. Behavior of the sample autocorrelation function (acf) for a simulated five state semi-Markov model is contrasted with the behavior of the sample acf for a real data set in order to infer that the real data cannot be modeled by a hidden semi-Markov scheme

Detailed Analysis of the Cross‐Correlation Function between the X‐Ray Background and Foreground Galaxies

Recent ROSAT surveys suggest that galaxies constitute the new class of X-ray sources required to explain the full phenomenology of the cosmic X-Ray Background (XRB). To test this hypothesis, we compute the two-point cross-correlation function Wxg(theta) between 62 Einstein-IPC fields (.81-3.5 keV) and the APM Northern galaxy catalog (13.5<E<19.0). At zero-lag (theta=0), we detect a 3.5sigma correlation signal with an amplitude of Wxg(0)=.045+/-.013. At non-zero lag (theta>0), the angular dependence of Wxg has two main features: the main signal for theta<4', and an almost flat plateau with an amplitude of Wxg(theta>4')=.015, likely to result from the diffuse X-ray emission by clusters of galaxies. Taking the point-spread function of the X-ray instrument into account, we find that the mean X-ray flux of galaxies with <E>=17.5+/-.3 is (8.1+/-4.7) 10^(-16) ergs/cm^2/s. This agrees with the X-ray flux expected from earlier direct studies of brighter galaxies, which were shown to result in a total galaxy contribution to the XRB of about 13%. We discuss how this powerful cross-correlation method can be used to measure the flux of X-ray sources well below the detection limit of X-ray instruments, and, perhaps, to probe otherwise undetectable faint diffuse X-ray emission.

Spectral Relative Motion of Two Structures due to Seismic Travel Waves

This paper proposes a spectral difference method to estimate the maximum relative displacement between adjacent elastic structures subjected to seismic traveling waves. The method, based on random vibration theory, seeks the cross-correlation between the responses of adjacent structures subjected to differential earthquake excitation with time lag. The results of computer simulation with 15 earthquake records show a good agreement between the proposed spectral analysis and time-history analysis. The relations between the maximum relative displacement of adjacent structures and their vibration periods, damping, as well as the time lag of earthquake excitations are clarified using the closed form solution obtained. Time lag of the excitations among adjacent structures alters the correlation and phase of vibration of the adjacent structures. The cross-correlation coefficient explained herein captures this aspect. Its magnitude under nonzero time lag can be very different from that under zero time lag. In some instances, these coefficients can have different signs. This leads to a significant change in the peak relative displacement of adjacent structures. These are especially true when the structures are subjected to surface waves; they rest on soft soils where low wave velocities develop, and their supports are remotely located.

The Effect of Convective Response Time on WISHE Modes

Abstract Recent research has shown that a variety of wavelike oscillations in the tropics may be explained by instabilities driven by wind-induced surface heat exchange (WISHE). All such studies to date have implicitly assumed that moist convection is in quasi equilibrium with the flow in question. Here that assumption is relaxed by accounting for a small but nonzero lag between the large-scale forcing of convection and its response. Reaction times as short as 30 minutes damp the higher-frequency Kelvin-like equatorial modes, favoring zonal wavenumbers 1–4, and strongly bias the higher-order modes to westward-propagating disturbances of synoptic scale. An analysis of off-equatorial disturbances reveals a preference for poleward- and westward-propagating modes with wavelengths of the order of 1000 km.

Use of cross correlations in the search for teleconnections

AbstractThe search for teleconnections usually involves the calculation of sample cross correlation functions for pairs of atmospheric/oceanic time series. Properties of the theoretical cross correlation function are derived by means of analytical techniques, and properties of the sample cross correlation function are obtained by means of simulations. In particular, the effect of autocorrelation of the individual time series is to ‘smear out’ any contemporaneous cross correlation that might be present, producing nonzero cross correlations at nonzero lags. To combat this problem, the advantages of ‘prewhitening’ before computing cross correlations are demonstrated. Moreover, the operation of smoothing individual time series has the effect of inflating the values of the cross correlation coefficients at nonzero lags. Finally, the behaviour of running cross correlation coefficients is somewhat counterintuitive, making this technique potentially very misleading for detecting changes in relationships over time. For illustrative purposes, these calculations are also performed for Darwin and Tahiti time series of monthly sea level pressure.

Interpretation of the Autocovariances and Cross-Covariance from a Polarization Diversity Radar

A theory is presented for interpretation of the autocovariances and cross-covariance derivable from coherent polarization diversity radars. At zero lag time these quantities yield the “main” received power, the depolarization ratio, the cross-correlation, and the cross-variance amplitude ratio, which have been used previously to deduce shape, orientation, and differential propagation effects. The significance of the covariances at nonzero lag times is developed here in terms of traditional models of the backscattering and propagation media. In particular, the autocovariance of the “depolarized” signal and the cross-covariance of the two received signals are shown to yield Doppler velocity means and variances which can be compared to the mean and variance derived from the “main” backscattered signal. If the velocity means and variances thus derived differ from one another, they can then be interpreted jointly to identify variations in the shapes and orientations of the scatterers across the Doppler velocity domain. The interpretation is illustrated by model calculations for rain and mixed rain and ice-phase scattering media and by a radar observation of mixed-phase precipitation.

Presence of cepstral peak in random reflected ultrasound signals.

A peak in the power cepstrum, or the squared magnitude of the Fourier transform of the data log-power spectrum, is commonly observed when processing reflections from plate-like structures, such as membranes. In this case, the cepstral peak at the smallest nonzero time lag, or quefrency, value can be used to determine the thickness of the plate. For reflections from a medium composed of randomly distributed scatterers, such as liver tissue, a cepstral peak is also commonly observed, but cannot be so intuitively explained as in the deterministic case above. In this paper, it is demonstrated that the presence of a cepstral peak depends on the form of the probability density function (pdf) of the separation between reflectors. In the case where the pdf is uniform from O to SM, the cepstral peak is found to occur at the quefrency corresponding to SM. For simple unimodal pdfs, a cepstral peak will occur at the location of the maximum probability. These observations are shown analytically and verified through simulations. The diagnostic value of these results lies in the interpretation of the relation of the cepstral peak location to the spacing of the scattering elements in the tissue.

Presence of cepstral peak in random reflected ultrasound signals

A peak in the power cepstrum, or the squared magnitude of the Fourier transform of the data log-power spectrum, is commonly observed when processing reflections from plate-like structures, such as membranes. In this case, the cepstral peak at the smallest nonzero time lag, or quefrency, value can be used to determine the thickness of the plate. For reflections from a medium composed of randomly distributed scatterers, such as liver tissue, a cepstral peak is also commonly observed, but cannot be so intuitively explained as in the deterministic case above. In this paper, it is demonstrated that the presence of a cepstral peak depends on the form of the probability density function (pdf) of the separation between reflectors. In the case where the pdf is uniform from 0 to S M, the cepstral peak is found to occur at the quefrency corresponding to S M. For simple unimodal pdfs, a cepstral peak will occur at the location of the maximum probability. These observations are shown analytically and verified through simulations. The diagnostic value of these results lies in the interpretation of the relation of the cepstral peak location to the spacing of the scattering elements in the tissue.

The theory of the electromagnetic self-force according to conformal symmetry II: Initial data and stability problems

The problem of well-posed initial data, those guaranteeing existence and uniqueness of solutions, is studied in this Part II. The motion equations, including the new self-force, are second-order differential-difference equations with a variable lag, due to the nonzero minimum time lag in causal signal propagation enforced ultimately by the conformal group symmetry. An initial-data theorem for a quite general class of solutions is conjectured on the basis of recent work in the literature. But these may well be too general, showing physical anomalies like «self-collision», infinite impulsive forces or others. We conjecture that the subclass of physical («doubly analytic») solutions is obtained precisely by the traditional initial data of space-time position and velocity specified at the initial instant. The idea of the proof is sketched, and simple solvable models are shown to support it. The stability problem is formulated and verified in the free (no external field) case.

The theory of the electromagnetic self-force according to conformal symmetry

Writing a conformal invariant electrodynamics over space-time sphere space, suggested originally by the conformal invariance of the Maxwell theory, leads unambiguously to a mathematically nonsingular theory of the classical self-force. The new degree of freedom λ, geometrically the sphere radius, has the physical meaning of a minimum nonzero time lag in causal signal propagation. The nonsingular self-force, the absence of Coulomb infinities, the finite e.m. mass renormalization are directly due to the presence of λ. Conformal invariance essentially alone fixes the model of a finite particle, which is both «point» and «extended» at the same time, however «extended in time» rather than spatially. Only retarded fields are used, and no approximations are made. In the lowest orders of a perturbation theory in powers of the classical radius, one recovers the Lorentz equation and (with an important difference) the Lorentz-Dirac equation.

Polyphase codes with good periodic correlation properties (Corresp.)

This correspondence describes the construction of complex codes of the form exp i \alpha_k whose discrete circular autocorrelations are zero for all nonzero lags. There is no restriction on code lengths.