Estimation Of Autocorrelation Function Research Articles

Estimation of 2D Stationary Random Medium Parameters from Post‐Stack Seismic Data

AbstractThe key to apply the random medium theory to seismic exploration is the estimation of random medium parameters (RMP) from seismic data. In this paper, we propose an approach to estimate parameters of 2D stationary random medium from post‐stack seismic data (PSD). Firstly, the relationship between PSD and the impedance model of random medium is established, and the principle and method of parameters estimation of autocorrelation function (ACF) are introduced. Then, by using power spectrum method, the algorithms and procedures for estimating RMP are discussed with an example; afterward, the feasibility and correctness of this approach is tested using several 2D theoretical models; finally, this approach is applied to the field seismic data, and the results indicate that the estimated parameters can provide an important reference for further dividing delta sedimentary facies, which reflects the estimation method of RMP is very promising. Compared with peer‐research work, the estimation method in this paper is a real 2D algorithm, and in particular it can estimate the autocorrelation angle, and this approach based on power spectrum method is obviously efficient and easily understandable. However, we could not ignore the relatively large errors in our approach, and it needs to be further improved and amended in the later researches.

Hypothesis Testing for Autocorrelated Short Climate Time Series

AbstractCommonly used statistical tests of hypothesis, also termed inferential tests, that are available to meteorologists and climatologists all require independent data in the time series to which they are applied. However, most of the time series that are usually handled are actually serially dependent. A common approach to handle such a serial dependence is to replace in those statistical tests the actual number of data by an estimated effective number of independent data that is computed from a classical and widely used formula that relies on the autocorrelation function. Despite being perfectly demonstrable under some hypotheses, this formula provides unreliable results on practical cases, for two different reasons. First, the formula has to be applied using the estimated autocorrelation function, which bears a large uncertainty because of the usual shortness of the available time series. After the impact of this uncertainty is illustrated, some recommendations of preliminary treatment of the time series prior to any application of this formula are made. Second, the derivation of this formula is done under the hypothesis of identically distributed data, which is often not valid in real climate or meteorological problems. It is shown how this issue is due to real physical processes that induce temporal coherence, and an illustration is given of how not respecting the hypotheses affects the results provided by the formula.

Identification of Time Series Models From Segments—Application to Scanning Transmission Electron Microscopy Images

Three important parametric models for describing the correlation functions and spectra of stationary stochastic processes are the autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models. Quite recently, the MATLAB toolbox “ARMASA” has been made publicly available. This toolbox provides state-of-the-art algorithms to perform automatic identification and selection between the models based on the estimated prediction error. ARMASA works on a single segment of data, whereas in some applications, the data are available as multiple segments. We could process each segment independently and average the estimated autocorrelation functions or spectra afterward. Better performance, however, can be expected when all segments are processed simultaneously, for two reasons. Initially, the bias in the estimated model parameters depends on the number of observations in a segment. Averaging models from segments does not reduce this bias, in contrast to identification from all the data at once. In either case, the variance of the estimated model parameters will decrease. By considering all the data at once this variance reduction can be considered in the order selection stage, decreasing the selection bias. In this paper, we extend the parameter estimation algorithms of the ARMASA toolbox to the case of multiple segments. AR parameter estimation from segments has been considered before, but not for the MA and ARMA models. In addition, in the AR case, model order selection can be done easily when the segments are independent. But, in the application we are primarily interested in, the segments are formed by lines of scanning transmission electron microscopy (STEM) images, which can be correlated with each other. Because the correlation among data segments influences the effective number of independent observations available it becomes necessary to estimate that number. This is not an easy problem in general. For observations from a stochastic process with an isotropic 2-D autocorrelation function we can, however, formulate an accurate estimator in terms of the line autocorrelation function. This situation arises, for example, in STEM images of amorphous specimens. From the AR model identification for segments, the extension to the MA and ARMA models is rather straightforward because MA and ARMA algorithms are based on Durbin's methods, whose first step is the estimation of a long AR model. We compare the segment-based algorithms to conventional ARMASA with simulations of stationary stochastic processes. An illustrative example involving simulated STEM images shows large potential benefits.

Evaluation of the type A uncertainty in measurements with autocorrelated observations

Described is the proposal of application of the GUM uncertainty type A evaluation to measurements with auto-correlated observations. The first step to it is the identification and cleaning of the raw sample data from the regularly variable components. Then formulas for standard deviation of the sample and of the mean value are expressed with the use correction coefficients or the so-called "effective number" of observations. These quantities depend on number of observations and on the sample autocorrelation function and allow to calculate the expanded uncertainty due to the GUM recommendations. The method of estimation of autocorrelation function for the sample data is also given. Considerations are illustrated by examples.

Computationally Efficient Single-Frequency Estimation Scheme in High-Rate WPAN Systems

Sub-optimal algorithms that avoid the complexity of the maximum likelihood scheme for estimating a frequency offset have been developed based on samples of the estimated auto-correlation function. However, their computation burden is still heavy for near-optimal performance. To overcome this problem, this paper proposes a reduced-complexity algorithm of single-frequency estimation for a high-rate wireless personal area network application. Accuracy and robustness of our frequency estimator are statistically assessed by Monte Carlo simulations. The results indicate that the proposed complexity effective algorithm closely conforms to the Cramer-Rao bound.

Variance Bound of ACF Estimation of One Block of fGn with LRD

This paper discusses the estimation of autocorrelation function (ACF) of fractional Gaussian noise (fGn) with long‐range dependence (LRD). A variance bound of ACF estimation of one block of fGn with LRD for a given value of the Hurst parameter (H) is given. The present bound provides a guideline to require the block size to guarantee that the variance of ACF estimation of one block of fGn with LRD for a given H value does not exceed the predetermined variance bound regardless of the start point of the block. In addition, the present result implies that the error of ACF estimation of a block of fGn with LRD depends only on the number of data points within the sample and not on the actual sample length in time. For a given block size, the error is found to be larger for fGn with stronger LRD than that with weaker LRD.

Finite-Sample Bias Propagation in Autoregressive Estimation With the Yule–Walker Method

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The Yule–Walker (YW) method for autoregressive (AR) estimation uses lagged-product (LP) autocorrelation estimates to compute an AR parametric spectral model. The LP estimates only have a small triangular bias in the estimated autocorrelation function and are asymptotically unbiased. However, using them in finite samples with the YW method for AR estimation can give a strong distortion in the weak parts of the power spectral density. The distortion is shown to be influential in an example without strong spectral peaks. The true biased AR model, which is computed by applying the triangular bias to the true autocorrelation function, has an infinite order. A new objective measure is introduced to determine the smallest sample size for which the unbiased asymptotic theory can be considered as a fair approximation. </para>

Preliminary Estimation in Gaussian Stationary Processes

Preliminary estimation of the kth Lag autocorrelation function in the Gaussian stationary processes is considered. An estimation procedure is derived from the ratio of the sum filter and the difference filter. The performance of this estimator is compared to the sample estimator through a Monte Carlo study.

Method of estimating pitch by using ratio of maximum peak to candidate for maximum of autocorrelation function and device using the method

A device and a method for estimating an open-loop pitch in a general speech CODEC are disclosed. The open-loop pitch estimation device includes an autocorrelation function calculation unit which calculates a normalized autocorrelation function from a perceptual weighing filtered speech signal, a maximum autocorrelation function and lag estimation unit which estimates a maximum autocorrelation function and candidates for the maximum autocorrelation function, a pitch candidate decision unit which decides candidates for a pitch by using the ratio of the estimated maximum autocorrelation function to the candidates for the estimated maximum autocorrelation function, and lags of which values are smaller than a predetermined threshold value, and a pitch estimation unit which estimates a pitch between the candidates for a pitch and the lags corresponding to the estimated maximum autocorrelation function by using a pitch of a previous frame of the speech signal.

A robust forward weighted Lagrange multiplier test for conditional heteroscedasticity

Statistical tests routinely adopted for detecting nonlinear components in time series rely on the auxiliary regression of ARMA lagged residuals, and the Lagrange multiplier test to detect ARCH components is an example. The size distortion of such test suggests adopting a weighted test, where the weights are computed through a forward search algorithm. Simulations show that the forward weighted robust test is preferable to the classical Lagrange test and to existing robust tests, which are based on backward weighted regression or on estimated autocorrelation function. The forward weighted robust test is applied to daily financial and quarterly macroeconomic time series, showing its usefulness in detecting ARCH effects, even when outliers are present.

New incoherent scatter diagnostic methods for the heated D-region ionosphere

Abstract. A new incoherent scatter (IS) diagnostic technique for the actively heated D-region ionosphere is presented. In this approach, an exponential autocorrelation function (ACF) was fitted to the data by using a Markov Chain Monte Carlo (MCMC) inversion and the Sodankylä Ion Chemistry model (SIC). The method was applied for a set of combined EISCAT heating and VHF radar experiments carried out in November 2006. A newly designed radar experiment, sippi, based on optimised phase codes and direct sampling of the transmitted and the received IS signal, was used to produce ACF estimates of the target plasma. Systematic features associated with the heating were found by comparing the mean ACFs corresponding to heated and unheated periods. However, the data analysis revealed reasonable electron temperature estimates at the altitudes of the expected maximum effect (70–75 km) only in three cases out of six, corresponding to the highest signal-to-noise ratio (SNR) conditions. In two of these cases, the electron temperature was increased by a factor of 5 to 7, which is in a good agreement with the theoretical heating modelling. In the case of the first presented data example, the model overestimates the effect. These are the first successful IS observations of the HF-induced maximum electron temperature enhancements in the D-region ionosphere.

Teaching Random Signals and Noise: An Experimental Approach

A practical approach for teaching random signals and noise is described, where theoretical aspects are complemented by several laboratory experiments enriching the student's understanding on basic topics, such as histograms and estimation of probability density function, autocorrelation function, and power spectral density. The equipment required is minimum and inexpensive. In fact, the existing equipment of laboratory benches employed for an electronic instrumentation course has been used. No investment from our institution has been necessary due to the full exploitation of the potentials of the existing instruments and their PC connectivity.

Systems and methods for improving power spectral estimation of speech signals

A system determines a power spectral density associated with an audio signal that includes a speech signal and/or a noise signal. The system updates an autocorrelation function of the audio signal from samples in the audio signal, estimates an autocorrelation function of the speech signal from the updated autocorrelation function of the audio signal, and calculates a power spectral density of the speech signal using the estimated autocorrelation function. The system then determines the power spectral density of the audio signal from the calculated power spectral density of the speech signal.

A novel auto-correlation function method for the determination of the decay ratio in BWR stability analysis

A novel auto-correlation function (ACF) method has been investigated for determining the oscillation frequency and the decay ratio in BWR stability analyses. The neutron signals are band-pass filtered to separate the oscillation peak in the power spectral density (PSD) from background. Two linear second-order oscillation models are considered. These models, corrected for signal filtering and including a background term under the peak in the PSD, are then least-squares fitted to the ACF of the previously filtered neutron signal, in order to determine the oscillation frequency and the decay ratio. Our method uses fast Fourier transform techniques with signal segmentation for filtering and ACF estimation. Gliding ‘short-term’ ACF estimates on a record allow the evaluation of uncertainties. Numerical results are given which have been obtained from neutron data of the recent Forsmark I and Forsmark II NEA benchmark project. Our results are compared with those obtained by other participants in the benchmark project.

Small-scale fluctuations of the soft X-ray background

We report the detection of the X-ray background fluctuations at small angular scales. ROSAT PSPC archive pointed observations are used to measure the autocorrelation function (ACF) of the background at scales of 0:03 0:3. The pointings have been selected from an area apparently free from galactic contamination and the ACF signal is generated mostly by the extragalactic component of the XRB. At separations below 0:1 known X-ray clusters of galaxies become a substantial source of the background fluctuations. Assuming a power law, the ACF for all the data has a slope of1, but is substantially flatter (with slope of0:6) when pointings containing bright clusters are removed. Flattening of the ACF at small separations could indicate existence of a hot, nonuniformly distributed gas accumulating in wells of gravitational potential. At separations 0:3 0: 4w here the ACF estimates based on the ROSAT pointings and All-Sky Survey are available, both data sets give consistent results.

On stochastic properties of daily river flow processes

A study of the structure of daily river flows is made by means of the correlation theory of stochastic processes, i.e. on the basis of time functions of the first two moments: mean, variance, and autocorrelation function. The experimental basis of the study are long (over 50 years) time series of daily flows, without any gaps from five stations, with an average length of 71.2 years and areas of drainage basins ranging from 25,083 to 131,959 km 2. As a result of the analysis, it can be stated that (i) days with high mean have also high variance; (ii) the coefficients of variations cannot be considered to be constant even approximately (iii) the lag 1 autocorrelation functions ρ i(1), i=1,2,…,365, estimated “over realizations” have at least half of their coefficients significantly different from other coefficients; (iv) autocorrelation functions estimated “over time” are greater in the absolute value for data which have not been standardized but were initially treated with the logarithmic transformation only; (v) autocorrelation functions estimated “over realizations” are preferred to autocorrelation functions estimated “over time” because the daily river process is not ergodic. It is suggested that the description of daily flows as ergodic stochastic processes should be abandoned and the autocorrelation functions should be estimated “over realizations”. The “over time” estimated autocorrelation functions may only be treated as a very rough approximation of the “over realizations” autocorrelation functions.

Vlbi rates with first order autoregressive disturbances

Repeated Very Long Baseline Interferometry, VLBI, measurements exhibitautocorrelation before the baseline rates are estimated and removed from the data.Autocorrelations reduce markedly once the baseline rates are estimated. This raises thepossibility that they may adversely influence the estimated baseline rates if the measurements arein fact autocorrelated but not modeled. We modeled and incorporated autoregressivedisturbances into the estimation of baseline rates using an iterative procedure to investigate theirpresence as well as to assess their impact on the estimated baseline rates and on thecorresponding statistics. Our findings indicate that, overall, the estimated baseline rates, theircorresponding standard deviations (at one sigma level, based on the a posteriori varianceof unit weight for each baseline model) are not markedly influenced by the introduction ofautocorrelated disturbances. On the other hand, the estimated autocorrelation functions of theresiduals show significantly higher order correlations that are due to stochastic and deterministiccyclic effects in the baseline measurements. Corellograms also have signatures of a slippageeffect on all baselines that involve HRAS station.

Autocorrelation function of the sum of normal noise and two harmonic oscillations for a limited process and after van vleck transformation

This paper presents a rigorous analysis of statistical characteristics for the sum of a Gaussian random process and two harmonic oscillations transmitted through an inertialess limiter, a typical unit in radio astronomical spectrometers. The analysis was done for the two most interesting cases, that in which the dominant intensity belongs to the continuous-spectrum component, and that in which, conversely, the dominant intensity belongs to the discrete-spectrum component. The resulting expressions for the autocorrelation functions make it possible to estimate the power spectrum of a completely limited process and the spectrum of its Van Vleck transform. In both cases, the transformation improved the estimates of the autocorrelation function and the spectrum and reduced the nonlinear products, including the components with combination frequencies.

Modeling of the Effect of Multiple Scattering in Photon Correlation Spectroscopy: Plane-Wave Approach

A technique for numerical modeling of the time autocorrelation function (ACF) of the electric field scattered from a concentrated dispersion, which is illuminated by a plane wave, has been developed as an approach to estimate the particle dynamics and multiple scattering effects in photon correlation spectroscopy measurements. Systematic error of the modeling of the particle dynamics and the error of the ACF estimation were investigated for Brownian free particles. It has been found that systematic error and the decreased dynamic delay-time range of the ACF exponential behavior for free particles are caused by low- and high-frequency oscillations of integrated functions. An optimization of the numerical model has been carried out in order to reach a required magnitude of the systematic error and a maximum of the dynamic delay-time range by a minimum of calculation expenditures. The dynamics of an ensemble of particles was generated by a stochastic technique. Practical application of the proposed technique was shown by modeling of the first-order ACF for the depolarized component of the backscattered electric field. Influence of multiple-scattering by interacting spheres on this ACF was investigated for different sphere diameters and dispersion concentrations by using a rigorous multiple-scattering technique. This model can be used to further develop photon correlation spectroscopy for the characterization of particles in concentrated and turbid media: in a following paper, the multiple-scattering suppression by one-beam cross correlation is simulated, taking into account geometrical parameters of a setup.

Randomization of alternating codes: Improving incoherent scatter measurements by reducing correlations of gated autocorrelation function estimates

Incoherent scatter radar autocorrelation function estimates produced by alternating codes are found to contain significant correlations between different lags and different heights. While no correlations are caused by the background thermal noise, the special structure of the alternating code sequences causes clutter or the self‐noise contribution of the codes themselves to introduce maximal positive correlations between the different lag estimates. Because of this, postintegration of adjacent heights does not improve the results as much as could be expected. For a high signal‐to‐noise ratio, alternating code results are worse than estimates produced by similar random codes. In this paper we show that we can correct the situation by randomizing the alternating codes. This is accomplished by multiplying each alternating code sequence in the alternating code cycle by a fixed random‐looking sequence. This multiplying sequence can be changed and randomly chosen at the end of each alternating code cycle. The multiplying sequence can also be kept constant with almost equally good results. If this is done, the special structure of the alternating code sequences actually causes the correlations to become less than the correlations produced by random codes. The improvement in accuracy is discussed in various situations. The reasons for the appearance of maximal positive correlations with alternating codes are explained in a mathematical appendix.