White Noise Error Research Articles

Modelling rainfall trends in England and Wales

AbstractThe monthly England and Wales precipitation (EWP) series (once power transformed to induce symmetry and to stabilise variance) may be characterised as having linear seasonal trends with a white noise error process superimposed. However, these trends are not stable, for they are interrupted by four regime shifts occurring in 1828, 1871, 1917 and 1976. If these shifts are ignored then the series is consistent with a trend pattern in which winters are becoming increasingly wet and summers drier. If only the last regime from 1976 is considered, then summers are still becoming drier but winters have no trend, with spring becoming wetter. The unusually wet winter of 2014 is seen to have been a consequence of very high January and February rainfall relative to that predicted, the conjunction of which is unprecedented during the two and a half centuries over which the EWP series has been available, during which time such pairs of values have been essentially uncorrelated.

Random weighting method for estimation of error characteristics in SINS/GPS/SAR integrated navigation system

This paper presents a new random weighting method to estimate error characteristics in SINS/GPS/SAR (Strapdown Inertial Navigation System/Global Positioning System/Synthetic Aperture Radar) integrated navigation system. An error model is developed to represent and analyze white noise errors in SINS/GPS/SAR integrated navigation system. Based on the error model, random weighting theories and algorithms are established to estimate the statistical characteristics of the white noise errors. Experiments and comparison analysis have been conducted, demonstrating that the proposed method can effectively estimate error characteristics in SINS/GPS/SAR integrated navigation system.

Super-Resolution Reconstruction of Radar Tomographic Image Based on Image Decomposition

In this letter, the relationship between target scattering function and point spread function of radar tomographic imaging is discussed, and an inherent super-resolution reconstruction algorithm based on image decomposition (ID) is proposed. By removing the cross disturbance due to the resolution limitation, the exact radar images can be obtained. Furthermore, the total least-square regulation is applied to the ID-based algorithm in cases of additive white Gaussian noise and small parameter estimation error. Finally, the effectiveness of the proposed algorithm is demonstrated via numerical simulations.

Testing linear regression model with AR(1) errors against a first-order dynamic linear regression model with white noise errors: A point optimal testing approach

We know very little about the performance of point optimal (PO) and approximate point optimal (APO) tests in the presence of unavoidable nuisance parameters. Because marginal likelihood based tests are said to perform well in the presence of unavoidable nuisance parameters, this paper compares the performance of marginal likelihood based APO tests and classical tests using a testing problem which has been largely overlooked by econometric practitioners, namely testing for a static linear regression model with AR(1) errors against a dynamic linear regression model with white noise errors. It is well known that the classical tests are specifically designed for nested testing, they are applied to test for the significance of the dynamic coefficient of a dynamic linear regression model with AR(1) errors.A testing procedure is proposed, where the size and power comparisons used are based on near-exact non-similar critical values of tests obtained using the simulated annealing (SA) algorithm, as the near-exact non-similar critical values control the sizes of the tests well overall.Among marginal likelihood based classical tests, the likelihood ratio (LR) test and Lagrange multiplier (LM) test seem to perform well under the alternative hypothesis, particularly when the dynamic parameter is large and the sample size is reasonably big. The Wald (W) test is the worst performer overall. This concurs with previous observations that the W test performs poorly in small samples. Compared to the classical approach, APO tests appear to have good power properties, particularly in the neighborhood of the chosen parameter point under the alternative hypothesis. This finding may advance the use of PO and APO tests.

Evolving Perceptions of Central Bank Credibility: The European Central Bank Experience

What are the preferences of a central bank over inflation and output-gap stabilization objectives, and what is its preferred long-run inflation rate? While statements of priorities and goals are important, the credibility of such statements and the market perception of the policy reaction function of a central bank play a key role in determining economic outcomes. This point, early on described as the “credibility” of central bank policies, is a standard theoretical result with recent interpretation in the New Keynesian paradigms. It is also received wisdom among practitioners. The importance of the market perception of the central bank’s acceptable trade-offs between inflation and output goals as well its specific targets naturally leads to the question of how the market acquires this perception and whether and how it evolves over time. One view is that establishing an appropriate institutional structure is the key element in insulating the monetary authority from political pressure and thereby convincing markets that a central bank has a strong and unvarying aversion to inflation. A second, more dynamic, view focuses on the role that actual policy conduct plays in building the reputation of a central bank. These two different views have distinct implications for the relative importance of the institutional structure of a central bank, as compared to its conduct, for attaining and maintaining its credibility. A survey of the heads of central banks and prominent monetary economists reflects a belief that the credibility of a central bank is based more on its past actions than on institutional structures that afford it independence by insulating it from political concerns, although there is also a consensus that structurematters (Blinder 2000). Empirical research has found that institutional features related to central bank independence are associated with economic performance in cross sections of

A multiscale analysis of the temporal characteristics of resting-state fMRI data

In this paper, we conduct an investigation of the null hypothesis distribution for functional magnetic resonance imaging (fMRI) time series using multiscale analysis tools, SiZer (significance of zero crossings of the derivative) and wavelets. Most current approaches to the analysis of fMRI data assume simple models for temporal (short term or long term) dependence structure. Such simplifications are to some extent necessary due to the complex, high-dimensional nature of the data, but to date there have been few systematic studies of the dependence structures under a range of possible null hypotheses, using data sets gathered specifically for that purpose. We aim to address some of these issues by analyzing the detrended data with a long enough time horizon to study possible long-range temporal dependence. Our multiscale approach shows that even for resting-state data, data, i.e. “null” or ambient thought, some voxel time series cannot be modeled by white noise and need long-range dependent type error structure. This finding suggests the use of different time series models in different parts of the brain in fMRI studies.

TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTIC CHECKING FOR TIME SERIES MODELS

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

Global and local spectral-based tests for periodicities

We investigate tests for periodicity based on a spectral analysis of a time series, differentiating between global and local spectral-based tests. Global tests use information across the entire frequency band,whereas local tests are based on a window around the test frequency.We show that many spectral-based tests can be expressed in terms of a regression-based F test, which allows for approximate size and power calculations. Since global tests are usually derived assuming white noise errors, we extend to the correlated noise case. We demonstrate via a Monte Carlo study that although the global test may have better size and power, local tests are easier to use, and are comparable or better in terms of the power to detect periodicities, especially for spectra with a large dynamic range. We apply this methodology to a nonbehavioural test of hearing.

Hidden Markov models for alcoholism treatment trial data

In a clinical trial of a treatment for alcoholism, a common response variable of interest is the number of alcoholic drinks consumed by each subject each day, or an ordinal version of this response, with levels corresponding to abstinence, light drinking and heavy drinking. In these trials, within-subject drinking patterns are often characterized by alternating periods of heavy drinking and abstinence. For this reason, many statistical models for time series that assume steady behavior over time and white noise errors do not fit alcohol data well. In this paper we propose to describe subjects' drinking behavior using Markov models and hidden Markov models (HMMs), which are better suited to describe processes that make sudden, rather than gradual, changes over time. We incorporate random effects into these models using a hierarchical Bayes structure to account for correlated responses within subjects over time, and we estimate the effects of covariates, including a randomized treatment, on the outcome in a novel way. We illustrate the models by fitting them to a large data set from a clinical trial of the drug Naltrexone. The HMM, in particular, fits this data well and also contains unique features that allow for useful clinical interpretations of alcohol consumption behavior.

Nonparametric transformation to white noise

We consider a semiparametric distributed lag model in which the “news impact curve” m is nonparametric but the response is dynamic through some linear filters. A special case of this is a nonparametric regression with serially correlated errors. We propose an estimator of the news impact curve based on a dynamic transformation that produces white noise errors. This yields an estimating equation for m that is a type two linear integral equation. We investigate both the stationary case and the case where the error has a unit root. In the stationary case we establish the pointwise asymptotic normality. In the special case of a nonparametric regression subject to time series errors our estimator achieves efficiency improvements over the usual estimators, see Xiao, Linton, Carroll, and Mammen (2003). In the unit root case our procedure is consistent and asymptotically normal unlike the standard regression smoother. We also present the distribution theory for the parameter estimates, which is non-standard in the unit root case. We also investigate its finite sample performance through simulation experiments.

Stochastic response surface optimization via an enhanced Nelder–Mead simplex search procedure

There is abundant literature on response surface methodology about how the Nelder–Mead simplex search (NMSS) procedure can be applied to the determination of optimum operating conditions for deterministic response surface functions. However, searching for the optima of stochastic functions seems a more realistic task for many practical situations. This is particularly true when the response surface functions have not been fitted or the physical models may not even exist, so gradient information is not available. In such cases, it might be interesting to employ simplex-search-type methods ‘on-line’ to sequentially optimize the actual response of interest. Toward that end, an enhanced NMSS is proposed in this article to explore the terrains of empirical (or experimental) optimization adaptively where the known response surface function incorporates additional white-noise errors. Internal modifications to basic operations in NMSS are made primarily according to some statistical process control statistics in estimating response variation and confidence bands for mean responses. A series of graphical illustrations are presented to give an insight into the way the new simplex-search-type approach accurately anchors the true optimum point in noisy environments. As evidenced by a wide variety of simulation studies on the published response functions, the new method proves to perform much better than two recent modifications of NMSS in solution quality achieved while applied to the stochastic response surface optimization problems.

Spread Overreaction in International Bond Markets

This paper applies the Campbell-Shiller (1987) methodology to a study of the joint behaviour of a three-month and a five-year government yield in the United States, Canada, the United Kingdom, Germany and Japan. The period studied is for most countries the mid-1970s to the third quarter of 1997. The empirical results allow the rejection of the expectations theory of the term structure at high levels of statistical significance in every country except Japan. Furthermore, in every country where the expectations theory fails, the failure of the theory is consistent with the spread overreaction hypothesis of Froot (1989) and Campbell and Shiller (1991). This implies that the departures of long rates from levels predicted by the expectations theory in many major markets cannot be attributed to white noise error terms.

A general model and SINR analysis of low duty-cycle UWB access through multipath with narrowband interference and rake reception

Interference from coexisting narrowband services is a critical factor affecting performance of ultra-wideband (UWB) radio communications. There is clearly a need to quantify interference and compare UWB systems on that basis. In this paper, we develop a general Rake reception model along with a unifying transmission framework for low duty-cycle UWB multiple access encompassing existing and novel spreading codes, including direct sequence, digital singleand multi-carrier, time-hopping, and combinations of them. Our unifying framework relies on a digital model, which leads to closed-form performance analysis expressions. Different from existing alternatives that require oversampling, our general model is developed directly from the samples of the Rake receiver output and allows for various Rake finger delay selections. Signal-to-interference-plus-noise ratio analysis and simulations are carried out to assess the relative merits of several UWB systems in the presence of narrowband interference, multipath and additive white Gaussian noise, for both matched-filter and minimum mean square error Rake receivers.

Testing for Latent Factors in Models with Autocorrelation and Heteroskedasticity of Unknows Form

1. IntroductionEmpirical research in financial economics frequently suggests the existence of few latent factors driving the systematic component of asset returns. Existence of such latent factors makes it easier to understand the effect on asset returns of the many variables that comprise the systematic component. Results depend on the number and type of assets used and the number and types of instruments, which themselves serve as proxies for the latent factors (for examples, see Campbell 1987; Zhou 1995; and Costa, Gardini, and Paruolo 1997). In econometric terms, the existence of latent factors translates into a reduced rank restriction on the (array of) coefficients in an asset return regression system.The present work considers the problem of testing for latent factors in a broad class of (multivariate linear stationary) time-series models, wherein model errors have autocorrelation and heteroskedasticity of unknown form. The generality of error dynamics is well suited to financial models of bond and stock returns, as in the macro model of Chen, Roll, and Ross (1986). To deal with such generality, we consider heteroskedasticity and autocorrelation consistent (HAC) methods, a HAC version of Hansen's (1982) Generalized Method of Moments (GMM) test and a lesser-known but more user-friendly minimum distance or ratio of asymptotic densities (RAD) test.The primary benefit of using a HAC-type test of economic hypotheses, in time-series models, is a certain kind of increased robustness relative to tests that rely on parametric assumptions about error dynamics. This robustness implies that stated significance levels of HAC tests are frequently closer to their true values, at least in sufficiently large samples. So, for the financial economist who wants to know "Are there really multiple factors driving the link between the macroeconomy and the returns on bonds and stocks?" HAC tests (and the underlying mental exercise regarding error dynamics) give added insight. HAC tests may or may not agree with less robust tests. In our application, the HAC test results agree with the results of simpler (and more popular) implementation of Hansen's (1982) test for reduced rank, but the crucial point is that stated significance levels in the popular version of Hansen's test are not correct, in statistical terms, when the data have dynamics that cause residual serial correlation. Hence, to say that the HAC tests agree with the popular version of Hansen's test is really too liberal an interpretation; more accurate is to say that the nominal (but likely invalid) conclusions from the popular test coincide with that of the autocorrelation-robust tests.The HAC Hansen test and the RAD test each require some special calculation, and for this we do some programming. The computational complexity is due partly to the presence of corrections for residual autocorrelation and heteroskedasticity, and if instead we assumed that model errors were independent and identically distributed, then we could test for reduced rank via Anderson's (1951) convenient likelihood ratio (LR) test (see Reinsel and Velu [1998] for discussion and Zhou [1994] for a related test couched in GMM terms). It is, of course, possible to extend Anderson's LR test to (parametric) probability models with autocorrelated errors (see Reinsel and Velu 1998), but this approach relies on a correctly specified error dynamic. We instead take the nonparametric HAC approach, allowing a wider variety of error dynamics.Is special calculation really necessary for our testing purposes? In application to asset pricing models, Zhou (1994) gives an interesting modification of Hansen's (1982) testing approach, with an analytical (hence computationally convenient) test for latent factors in asset returns. However, to implement his analytical test, Zhou relies on parametric assumptions about the model's error dynamics. Specifically, he considers the case of white noise errors and also the case of errors that are uncorrelated but have a known (parametric) form of conditional heteroskedasticity. …

Estimation of a panel data model with parametric temporal variation in individual effects

This paper is an extension of Ahn et al. (J. Econom. 101 (2001) 219) to allow a parametric function for time-varying coefficients of the individual effects. It provides a fixed-effect treatment of models like those proposed by Kumbhakar (J. Econom. 46 (1990) 201) and Battese and Coelli (J. Prod. Anal. 3 (1992) 153). We present a number of GMM estimators based on different sets of assumptions. Least squares has unusual properties: its consistency requires white noise errors, and given white noise errors it is less efficient than a GMM estimator. We apply this model to the measurement of the cost efficiency of Spanish savings banks.

A Multivariate Model for Spatio‐temporal Health Outcomes with an Application to Suicide Mortality

This article considers models for multivariate mortality outcomes (e.g., bivariate, trivariate, or higher dimensional) observed over a set of areas and through time. The model outlined here allows for spatially structured and white noise errors and for their intercorrelation. It also includes possible temporal continuity in such types of error via structured temporal effects. An extension to spatially varying regression effects is considered, as well as the option of nonparametric specification of priors for spatial residuals and regression effects. Allowing for spatially correlated intercepts or regression effects may alter inferences regarding the changing impact on mortality of socioeconomic or environmental predictors. The modeling framework is illustrated by an application to male and female suicide mortality in London, focusing on the impact on suicide of deprivation and social fragmentation (“anomie”) in the 33 London boroughs during three periods: 1979–83, 1984–88 and 1989–93. Suicide trends by age group are also considered and show considerable differences in the trends in impacts of deprivation and social fragmentation.

A Multivariate Model for Spatio-temporal Health Outcomes with an Application to Suicide Mortality

This article considers models for multivariate mortality outcomes (e.g., bivariate, trivariate, or higher dimensional) observed over a set of areas and through time. The model outlined here allows for spatially structured and white noise errors and for their intercorrelation. It also includes possible temporal continuity in such types of error via structured temporal effects. An extension to spatially varying regression effects is considered, as well as the option of nonparametric specification of priors for spatial residuals and regression effects. Allowing for spatially correlated intercepts or regression effects may alter inferences regarding the changing impact on mortality of socioeconomic or environmental predictors. The modeling framework is illustrated by an application to male and female suicide mortality in London, focusing on the impact on suicide of deprivation and social fragmentation (“anomie”) in the 33 London boroughs during three periods: 1979–83, 1984–88 and 1989–93. Suicide trends by age group are also considered and show considerable differences in the trends in impacts of deprivation and social fragmentation.

Frequency and noise dependence of the image reconstruction of ground surfaces using the conjugate gradient based algorithm

The ground surface of the Earth is usually mapped using airborne microwave radar. However, it has been shown recently by the author that the Earth's ground surface can also be imaged using radio ground waves with antennas positioned on the ground surface. The image of the surface is obtained by reconstructing the distribution of the normalised surface impedance of the surface from the measurements of the scattered fields at a number of positions around the surface being imaged with electromagnetic radiation from different azimuthal directions. It has also been shown that the inverse problem can be solved iteratively using a conjugate gradient method, and the images of ground surfaces can be produced using the noiseless data of the measured scattered fields at an operating frequency at which the size of the surface is approximately one wavelength. In practice, the image reconstruction depends on the operating frequency, and the measured scattered fields are subject to white noise and measurement errors. The dependence of the image reconstruction using the conjugate gradient method on the operating frequency and noise in the measured data is thus studied and the results of reconstruction errors at different frequencies and signal-to-noise ratios and the reconstructed images are presented. It is shown that the image reconstruction using the conjugate gradient based algorithm has a strong dependence on the operating frequency and noise. However, the algorithm is able to reconstruct the images of ground surfaces in a small number of iterations when the signal-to-noise ratio is greater than 12 dB.

Theory & Methods: Residual Diagnostic Plots for Checking for Model Mis‐specification in Time Series Regression

This paper considers residuals for time series regression. Despite much literature on visual diagnostics for uncorrelated data, there is little on the autocorrelated case. To examine various aspects of the fitted time series regression model, three residuals are considered. The fitted regression model can be checked using orthogonal residuals; the time series error model can be analysed using marginal residuals; and the white noise error component can be tested using conditional residuals. When used together, these residuals allow identification of outliers, model mis‐specification and mean shifts. Due to the sensitivity of conditional residuals to model mis‐specification, it is suggested that the orthogonal and marginal residuals be examined first.

A Revised Simplex Search Procedure for Stochastic Simulation Response Surface Optimization

We develop a variant of the Nelder-Mead (NM) simplex search procedure for stochastic simulation optimization that is designed to avoid many of the weaknesses encumbering similar direct-search methods—in particular, excessive sensitivity to starting values, premature termination at a local optimum, lack of robustness against noisy responses, and computational inefficiency. The Revised Simplex Search (RSS) procedure consists of a three-phase application of the NM method in which: (a) the ending values for one phase become the starting values for the next phase; (b) the step size for the initial simplex (respectively, the shrink coefficient) decreases geometrically (respectively, increases linearly) over successive phases; and (c) the final estimated optimum is the best of the ending values for the three phases. To compare RSS versus NM and procedure RS+S9 due to Barton and Ivey, we summarize a simulation study based on four selected performance measures computed for six test problems that include additive white-noise error, with three levels of problem dimensionality and noise variability used in each problem. In the selected test problems, RSS yielded significantly more accurate estimates of the optimum than NM or RS+S9, and both RSS and RS+S9 required roughly four times as many function evaluations as NM.