Autoregressive Noise Research Articles

Linear Smoothing of Noisy Spatial Temporal Series

The main objective of the study is the development of a linear filter to extract the signal from a spatio-temporal series affected by measureme nt error. We assume that the evolution of the unobservable signal can be modelled by a space time autoregressive process. In its vectorial form, the model admits a state space representation allowing the direct application of the Kalman filter machinery to predict the unobservable state vector on the basis of the sample information. Having introduced the model, referred to as a STARG+Noise model, the study discusses Maximum Likelihood (ML) parameter estimation assuming knowledge of the variance of the noise process. Consistent method of moments estimators of the autoregressive coefficients and noise variance are also derived, primarily to be used as inputs in the ML estimation procedure. Finally, we consider some simulation studies and an investigation involving sulphur diox ide level monitoring.

Constrained linear basis sets for HRF modelling using Variational Bayes

FMRI modelling requires flexible haemodynamic response function (HRF) modelling, with the HRF being allowed to vary spatially and between subjects. To achieve this flexibility, voxelwise parameterised HRFs have been proposed; however, inference on such models is very slow. An alternative approach is to use basis functions allowing inference to proceed in the more manageable General Linear Model (GLM) framework. However, a large amount of the subspace spanned by the basis functions produces nonsensical HRF shapes. In this work we propose a technique for choosing a basis set, and then the means to constrain the subspace spanned by the basis set to only include sensible HRF shapes. Penny et al. [NeuroImage (2003)] showed how Variational Bayes can be used to infer on the GLM for FMRI. Here we extend the work of Penny et al. to give inference on the GLM with constrained HRF basis functions and with spatial Markov Random Fields on the autoregressive noise parameters. Constraining the subspace spanned by the basis set allows for far superior separation of activating voxels from nonactivating voxels in FMRI data. We use spatial mixture modelling to produce final probabilities of activation and demonstrate increased sensitivity on an FMRI dataset.

Measuring Period of Human Biological Clock: Infill Asymptotic Analysis of Harmonic Regression Parameter Estimates

Publisher Summary This chapter examines the infill asymptotic behavior of the maximum likelihood estimates of a harmonic regression plus continuous first-order autoregressive noise model in order to determine the precision with which the intrinsic period of the human biological clock can be estimated. It is found that the intrinsic period of the human biological clock is closer to 24 h instead of 25 h and that it can be estimated with a standard error of 1.4 to 2.1 min. It is shown that for the forced de-synchrony (FD) protocol, the intrinsic period of the clock can be estimated in a computationally efficient manner with near-maximal precision from a sampled series containing less than 2% of the original temperature data. The relation of the results to the classic formulas for the asymptotic covariance matrix of the harmonic regression parameter estimates is elaborated. The implications of these results for determining the precision with which the period of the human biological clock may be estimated are studied in an analysis of core temperature data from five male subjects monitored under FD.

Optimizing the bit aspect ratio of a recording system using an information-theoretic criterion

The objective of this work is to find the optimal symbol separation T and track width TW so as to maximize the inlormation density D in bits/inch'. Here D is defined as the information density per unit area of the recording medium where Cid is the i.u.d. channel capacity, i.e., the information rate (in bitslchannel-use) between the channel input and the channel output, assuming that the channel input is a sequence of independent and uniformly distributed (i.u.d.) binary random variables, We fit the datadependent autoregressive (AR) noise model in [I] to the recording channel, and then estimate the i.u.d. channel capacity C,u by computing the i.u.d. capacity ofthe model. The method used to compute the i.u.d. capacity of the data-dependent AR noise model was proposed in.

Autoregressive Noise, Deserialization, and Trend Detection and Quantification in Annual River Discharge Time Series

The evaluation of long-term trends in yearly discharge records, such as annual peak daily flow or total annual runoff, is important to a variety of issues including water resource planning, flood hazard studies, and the assessment of historical data for evidence of anthropogenic climate change effects. Prewhitening or deserialization procedures have recently been developed and applied to adjust statistical tests of monotonic trend, and the nonparametric Mann-Kendall test in particular, for sensitivity to serial dependence. Deserialization attributes much or all of the observed serial correlation in a time series to an autoregressive process; however, deterministic processes can also lead to a large lag-1 serial correlation coefficient, and the physical basis for autoregressive noise may be weaker for annual rather than more finely-discretized (e.g., daily) streamflow records. In this paper, the potential consequences of using such procedures are investigated through a suite of Monte Carlo simulations. We find that prewhitening can substantially and inappropriately reduce the power of trend significance tests and increase slope estimate errors. The choice of whether deserialization is applied is to some degree left to the judgement and conservatism of the individual practitioner. We suggest that such procedures not be applied to a given annual hydrologic time series unless there is a strong site-specific physical basis for the assumption of AR(1) noise and that if deserialization is performed, very recently-developed multi-stage techniques appear preferable. We also present a number of useful ancillary results regarding trend identification in streamflow-derived data.

Seasonal Rhythms of Neuronal Activity in the Human Biological Clock: A Mathematical Model

The mammalian suprachiasmatic nucleus (SCN) is implicated in the temporal organization of circadian and seasonal processes. Photic information may have a synchronizing effect on the endogenous clock of the SCN by inducing periodic changes in the functional activity of specific groups of neurons. The present study was performed to analyze the annual profiles of the arginine vasopressin (AVP)- and vasoactive intestinal polypeptide (VIP)-producing neurons in the human SCN. The populations of AVP- and VIP-expressing neurons in the SCN showed marked annual rhythms with an asymmetrical, bimodal waveform. Time series analysis indicated that these annual cycles in peptidergic activity could be described by a statistical model consisting of multiple-harmonic regression and ARMA components. The annual AVP cycle was adequately described by a two-harmonic model and a third-order autoregressive noise component, whereas the properties of the VIP cycle were best characterized by a two-harmonic model and a first-order autoregressive noise component. The models of both annual cycles reached a maximum in September–October and a minimum in May–June, and their estimated amplitudes, relative to the annual mean, were similar in size. These findings indicate that the biosynthesis of vasopressin and VIP in the human SCN exhibits an annual rhythmicity, and that the temporal organization of these processes is mainly controlled by environmental lighting conditions.

Optimal Disclosure Limitation Strategy in Statistical Databases: Deterring Tracker Attacks through Additive Noise

Disclosure limitation methods transform statistical databases to protect confidentiality, a practical concern of statistical agencies. A statistical database responds to queries with aggregate statistics. The database administrator should maximize legitimate data access while keeping the risk of disclosure below an acceptable level. Legitimate users seek statistical information, generally in aggregate form; malicious users—the data snoopers—attempt to infer confidential information about an individual data subject. Tracker attacks are of special concern for databases accessed online. This article derives optimal disclosure limitation strategies under tracker attacks for the important case of data masking through additive noise. Operational measures of the utility of data access and of disclosure risk are developed. The utility of data access is expressed so that trade-offs can be made between the quantity and the quality of data to be released. Application is made to Ohio data from the 1990 census. The article derives conditions under which an attack by a data snooper is better thwarted by a combination of query restriction and data masking than by either disclosure limitation method separately. Data masking by independent noise addition and data perturbation are considered as extreme cases in the continuum of data masking using positively correlated additive noise. Optimal strategies are established for the data snooper. Circumstances are determined under which adding autocorrelated noise is preferable to using existing methods of either independent noise addition or data perturbation. Both moving average and autoregressive noise addition are considered.

Detecting a global warming signal in hemispheric temperature series : A structural time series analysis

Non-stationary time series such as global and hemispheric temperatures, greenhouse gas concentrations, solar irradiance, and anthropogenic sulfate aerosols, may contain stochastic trends (the simplest stochastic trend is a random walk) which, due to their unique patterns, can act as a signal of the influence of other variables on the series in question. Two or more series may share a common stochastic trend, which indicates that either one series causes the behavior of the other or that there is a common driving variable. Recent developments in econometrics allow analysts to detect and classify such trends and analyze relationships among series that contain stochastic trends. We apply some univariate autoregression based tests to evaluate the presence of stochastic trends in several time series for temperature and radiative forcing. The temperature and radiative forcing series are found to be of different orders of integration which would cast doubt on the anthropogenic global warming hypothesis. However, these tests can suffer from size distortions when applied to noisy series such as hemispheric temperatures. We, therefore, use multivariate structural time series techniques to decompose Northern and Southern Hemisphere temperatures into stochastic trends and autoregressive noise processes. These results show that there are two independent stochastic trends in the data. We investigate the possible origins of these trends using a regression method. Radiative forcing due to greenhouse gases and solar irradiance can largely explain the common trend. The second trend, which represents the non-scalar non-stationary differences between the hemispheres, reflects radiative forcing due to tropospheric sulfate aerosols. We find similar results when we use the same techniques to analyze temperature data generated by the Hadley Centre GCM SUL experiment.

Extinction Risk and the 1/f Family of Noise Models

In order to predict extinction risk in the presence of reddened, or correlated, environmental variability, fluctuating parameters may be represented by the family of 1/f noises, a series of stochastic models with different levels of variation acting on different timescales. We compare the process of parameter estimation for three 1/f models (white, pink and brown noise) with each other, and with autoregressive noise models (which are not 1/f noises), using data from a model time-series (length, T) of population. We then calculate the expected increase in variance and the expected extinction risk for each model, and we use these to explore the implication of assuming an incorrect noise model. When parameterising these models, it is necessary to do so in terms of the measured (“sample”) parameters rather than fundamental (“population”) parameters. This is because these models are non-stationary: their parameters need not stabilize on measurement over long periods of time and are uniquely defined only over a specified “window” of timescales defined by a measurement process. We find that extinction forecasts can differ greatly between models, depending on the length, T, and the coefficient of variability, CV, of the time series used to parameterise the models, and on the length of time into the future which is to be projected. For the simplest possible models, ones with population itself the 1/f noise process, it is possible to predict the extinction risk based on CV of the observed time series. Our predictions, based on explicit formulae and on simulations, indicate that (a) for very short projection times relative to T, brown and pink noise models are usually optimistic relative to equivalent white noise model; (b) for projection timescales equal to and substantially greater than T, an equivalent brown or pink noise model usually predicts a greater extinction risk, unless CV is very large; and (c) except for very small values of CV, for timescales very much greater than T, the brown and pink models present a more optimistic picture than the white noise model. In most cases, a pink noise is intermediate between white and brown models. Thus, while reddening of environmental noise may increase the long-term extinction probability for stationary processes, this is not generally true for non-stationary processes, such as pink or brown noises.

Viability in a pink environment: why “white noise” models can be dangerous

Analysis of long time series suggests that environmental fluctuations may be accurately represented by 1/f noise (pink noise), where temporal correlation is found at several scales, and the range of fluctuations increases over time. Previous studies on the effects of coloured noise on population dynamics used first or second order autoregressive noise. I examined the importance of coloured noise for extinction risk using true 1/f noise. I also considered the problem of estimating extinction risk with a limited sample of environmental variation. Pink noise environments increased extinction risk in random walk models where environmental variation affected the growth rate. However, pink noise environments decreased extinction risk in the Ricker model where environmental variation modified the carrying capacity. Underestimation of environmental variance almost always yielded underestimation of extinction risk. For either population viability analysis or management, we should carefully consider the long‐term behaviour of the environment as well as how we include environmental noise in population models.

Long‐term variability of a stochastic forced pelagic ecosystem model

The long‐term natural variability of a pelagic ecosystem model is investigated using a simple numerical model which consists of six state variables: two nutrients, two phytoplankton species, detritus and benthos. For the investigation of natural variability the model is forced with first and second order autoregressive processes and noise. The interannual variability of the system with respect to phytoplankton mortality is investigated in different runs which are integrated over 300 years.

A minimum misadjustment adaptive FIR filter

The performance of an adaptive filter is limited by the misadjustment resulting from the variance of adapting parameters. This paper develops a method to reduce the misadjustment when the additive noise in the desired signal is correlated. Given a static linear model, the linear estimator that can achieve the minimum parameter variance estimate is known as the best linear unbiased estimator (BLUE). Starting from classical estimation theory and a Gaussian autoregressive (AR) noise model, a maximum likelihood (ML) estimator that jointly estimates the filter parameters and the noise statistics is established. The estimator is shown to approach the best linear unbiased estimator asymptotically. The proposed adaptive filtering method follows by modifying the commonly used mean-square error (MSE) criterion in accordance with the ML cost function. The new configuration consists of two adaptive components: a modeling filter and a noise whitening filter. Convergence study reveals that there is only one minimum in the error surface, and global convergence is guaranteed. Analysis of the adaptive system when optimized by LMS or RLS is made, together with the tracking capability investigation. The proposed adaptive method performs significantly better than a usual adaptive filter with correlated additive noise and tracks a time-varying system more effectively.

Efficient mixed-spectrum estimation with applications to target feature extraction

We present a decoupled parameter estimation (DPE) algorithm for estimating sinusoidal parameters from both 1-D and 2-D data sequences corrupted by autoregressive (AR) noise. In the first step of the DPE algorithm, we use a relaxation (RELAX) algorithm that requires simple fast Fourier transforms (FFTs) to obtain the estimates of the sinusoidal parameters. We describe how the RELAX algorithm may be used to extract radar target features from both 1-D and 2-D data sequences. In the second step of the DPE algorithm, a linear least-squares approach is used to estimate the AR noise parameters. The DPE algorithm is both conceptually and computationally simple. The algorithm not only provides excellent estimation performance under the model assumptions, in which case the estimates obtained with the DPE algorithm are asymptotically statistically efficient, but is also robust to mismodeling errors.

Selecting a Model for Detecting the Presence of a Trend

Abstract The authors consider the problem of determining whether the upward trending behavior in the global temperature anomaly series should be forecast to continue. To address this question, the generic problem of determining whether an observed trend in a time series realization is a random (i.e., short-term) trend or a deterministic (i.e., permanent) trend is considered. The importance of making this determination is that forecasts based on these two scenarios are dramatically different. Forecasts based on a series with random trends will not predict the observed trend to continue, while forecasts based on a model with deterministic trend will forecast the trend to continue into the future. In this paper, the authors consider an autoregressive integrated moving average (ARIMA) model and a “deterministic forcing function + autoregressive (AR) noise” model as possible random trend and deterministic trend models, respectively, for realizations displaying trending behavior. A bootstrap-based classificatio...

Detection of broadband planewaves in the presence of Gaussian noise of unknown covariance: asymptotically optimum tests using the 2-D autoregressive noise model

This paper addresses the problem of detecting a broadband planewave in noise of unknown spatial and temporal covariance at a linear array of sensors. Results of asymptotic detection theory are applied to derive detectors that approach optimal performance for large data records. A parametric approach is used to model the statistics of the data. A 2-D autoregressive (2DAR) model is chosen to model the noise process. Two broadband planewave signal models are considered. Both models represent the signal as a sum of monochromatic planewaves. In the Gaussian model, the amplitudes are assumed to be Gaussian with a single variance parameter, whereas in the deterministic assumption, they are individual unknown parameters. Detectors based on asymptotic theory are derived for both models. As part of the development of the asymptotically (AS) optimum detector, the Fisher information matrix (FIM) is derived. A proof of the locally asymptotic normal (LAN) property is provided for the Gaussian model probability density function (PDF). Both detectors, however, are AS equivalent to the generalized likelihood ratio test (GLRT), are AS of constant false alarm rate (CFAR), and perform AS as well as the GLRT constructed with full knowledge of the noise statistics. The performance of both detectors are compared with each other and to a standard spatially normalized beamformer in a computer simulation. >

Maximum likelihood estimation of signals in autoregressive noise

Time series modeling as the sum of a deterministic signal and an autoregressive (AR) process is studied. Maximum likelihood estimation of the signal amplitudes and AR parameters is seen to result in a nonlinear estimation problem. However, it is shown that for a given class of signals, the use of a parameter transformation can reduce the problem to a linear least squares one. For unknown signal parameters, in addition to the signal amplitudes, the maximization can be reduced to one over the additional signal parameters. The general class of signals for which such parameter transformations are applicable, thereby reducing estimator complexity drastically, is derived. This class includes sinusoids as well as polynomials and polynomial-times-exponential signals. The ideas are based on the theory of invariant subspaces for linear operators. The results form a powerful modeling tool in signal plus noise problems and therefore find application in a large variety of statistical signal processing problems. The authors briefly discuss some applications such as spectral analysis, broadband/transient detection using line array data, and fundamental frequency estimation for periodic signals. >

The Effect of Peripheral Visual Cues on the Control of Self-Stabilization in Roll

This paper reports a programme of laboratory research to assess the potential of a new form of aircraft attitude indicator. The Ambient Attitude Indicator (AAI), designed to exploit the characteristics of the ambient visual system, provides a continuous source of world-stabilized orientation information to the pilot's ambient visual system. In laboratory experiments, the first of which is reported here, the effects of peripheral visual cues on subjects' control of continuous self-stabilization in roll are assessed. Subjects, seated on a roll turntable, were required to null a continuous quasi-random forcing function using a velocity-control joystick. Their objective was to maintain a stable upright orientation. Three visual conditions were studied: no visual cues, world-stabilized peripheral visual cues, and turntable-stabilized peripheral visual cues. Significant effects of visual condition were found for measures of joystick RMS displacement, turntable RMS error, and frequency of changes of direction of the turntable. World-stabilized conditions resulted in the highest joystick RMS displacement, the lowest turntable RMS error, and the highest frequency of changes of direction of the turntable. A linear transfer function, with autoregressive noise process to represent the remnant, was fitted to the data using maximum likelihood estimation. Details of subjects' frequency response are presented and the effects of practice on these measures are also considered.

Stratospheric ozone profile and total ozone trends derived from the SAGE I and SAGE II data

SAGE I/II ozone data from the period 1979–1991 have been used to derive global trends in both stratospheric column ozone and as a function of altitude. A statistical model containing quasibiennial, seasonal, and semiannual oscillations, a linear component, and a first‐order autoregressive noise process was fit to the time series of SAGE I/II monthly zonal mean data. The linear trend in column ozone above 17 km altitude, averaged between 65°S and 65°N, is −0.30 ± 0.19%/year or −3.6% over the time period February 1979 through April 1991. The data further show that the column trend above 17 km is nearly zero in the tropics and increases towards the high latitudes with values of −0.6%/year at 60°S and −0.35%/year at 60°N. Both these results are in agreement with the recent TOMS results. Furthermore, the profile trend analyses show the column ozone losses are occurring below 25 km, with most of the loss coming from the region between 17 and 20 km. Negative trend values on the order of −2%/year are found at 17 km in mid‐latitudes.

Total Ozone trends deduced from Nimbus 7 Toms data

The Total Ozone Mapping Spectrometer (TOMS) on the Nimbus 7 satellite has been measuring the total column amount of ozone over the globe for more than 11 years. Recent improvements in the data analysis have led to a technique for determining and removing drift in the calibration such that the data at the end of the record are precise to ±1.3% (2σ) relative to the data at the beginning of the record. A statistical model, including terms for seasonal variation, linear trend, quasi‐biennial oscillation, solar cycle and second‐order autoregressive noise has been fit to the TOMS time series of total ozone data. The linear trend obtained when this statistical model is fit to the TOMS data averaged between 65N and 65S latitudes is −0.26±0.14%/year or −3% over the 11.6 year time period from November, 1978 through May, 1990. The trend is near zero (0.0002±0.2 %/year) at the equator and increases towards both poles. At 50N the annually averaged trend is −0.5±.21%/year. The 50N trend over the 11.6 year time period shows a strong seasonal variation from more than −0.8%/year in winter and early spring (February and March) to about −0.2%/year in summer (July and August).

The structure and design of realizable decision feedback equalizers for IIR channels with colored noise

A simple algorithm for optimizing decision feedback equalizers (DFEs) by minimizing the mean-square error (MSE) is presented. A complex baseband channel and correct past decisions are assumed. The dispersive channel may have infinite impulse response, and the noise may be colored. Consideration is given to optimal realizable (stable and finite-lag smoothing) forward and feedback filters in discrete time. They are parameterized as recursive filters. In the special case of transmission channels with finite impulse response and autoregressive noise, the minimum MSE can be attained with transversal feedback and forward filters. In general, the forward part should include a noise-whitening filter (the inverse noise model). The finite realizations of the filters are calculated using a polynomial equation approach to the linear quadratic optimization problem. The equalizer is optimized essentially by solving a system of linear equations Ax=B, where A contains transfer function coefficients from the channel and noise model. No calculation of correlations is required with this method. A simple expression for the minimal MSE is presented. The DFE is compared to MSE-optimal linear recursive equalizers. Expressions for the equalizer in the limiting case of infinite smoothing lags are also discussed. >