Minimum Mean Squared Error Estimation of the Noise in Unobserved Component Models

In model-based estimation of unobserved components, the minimum mean squared error estimator of the noise component is different from white noise. In this article, some of the differences are analyzed. It is seen how the variance of the component is always underestimated, and the smaller the noise variance, the larger the underestimation. Estimators of small-variance noise components will also have large autocorrelations. Finally, in the context of an application, the sample autocorrelation function of the estimated noise is seen to perform well as a diagnostic tool, even when the variance is small and the series is of relatively short length.     

Skew Gaussian Process for Nonlinear Regression

In this article, we extend the Gaussian process for regression model by assuming a skew Gaussian process prior on the input function and a skew Gaussian white noise on the error term. Under these assumptions, the predictive density of the output function at a new fixed input is obtained in a closed form. Also, we study the Gaussian process predictor when the errors depart from the Gaussianity to the skew Gaussian white noise. The bias is derived in a closed form and is studied for some special cases. We conduct a simulation study to compare the empirical distribution function of the Gaussian process predictor under Gaussian white noise and skew Gaussian white noise.     

Image processing through multiscale analysis and measurement noise modeling

We describe a range of powerful multiscale analysis methods. We also focus on the pivotal issue of measurement noise in the physical sciences. From multiscale analysis and noise modeling, we develop a comprehensive methodology for data analysis of 2D images, 1D signals (or spectra), and point pattern data. Noise modeling is based on the following: (i) multiscale transforms, including wavelet transforms; (ii) a data structure termed the multiresolution support; and (iii) multiple scale significance testing. The latter two aspects serve to characterize signal with respect to noise. The data analysis objectives we deal with include noise filtering and scale decomposition for visualization or feature detection.     

Inference for ergodic diffusions plus noise

We research an adaptive maximum‐likelihood–type estimation for an ergodic diffusion process where the observation is contaminated by noise. This methodology leads to the asymptotic independence of the estimators for the variance of observation noise, the diffusion parameter, and the drift one of the latent diffusion process. Moreover, it can lessen the computational burden compared to simultaneous maximum likelihood–type estimation. In addition to adaptive estimation, we propose a test to see if noise exists or not and analyze real data as the example such that the data contain observation noise with statistical significance.     

Estimating unobservable signal by Markovian noise induction

A sub threshold signal is transmitted through a channel and may be detected when some noise - with known structure and proportional to some level - is added to the data. There is an optimal noise level, called of stochastic resonance, that corresponds to the minimum variance of the estimators in the problem of recovering unobservable signals. For several noise structures it has been shown the evidence of stochastic resonance effect. Here we study the case when the noise is a Markovian process. We propose consistent estimators of the sub threshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.     

ESTIMATION OF REALIZED SIGNAL TO NOISE RATIO FOR A PAIR OF MULTIVARIATE SIGNALS

Consider the case of classifying an incoming message as one of two known p-dimension signals or as a pure noise. Let the noise co-variance matrix (assumed to be same in all the three cases) be unknown. We consider the problem of estimation of “realized signal to noise ratio matrix”, which is an index of discriminatory power, under various loss functions. Optimum estimators are obtained under these loss functions. Finally, an attempt is made to provide a lower confidence bound for the realized signal to noise ratio matrix. In the process, the probability distribution of the smaller eigenvalue of a 2 × 2 confluent hypergeometric random matrix is obtained.     

Correlation coefficient of response variances in compound noise and product array experiments

In robust parameter design, a compound noise experiment has been frequently used for reducing the number of experimental runs compared to a product array experiment. However, the results obtained by the compound noise experiment and the product array experiment are often much different. This paper derives an expression of the correlation coefficient of response variances in the compound noise and the product array experiments, which gives an explanation of that difference.     

ANASE: measuring aircraft noise annoyance very unreliably

Does anyone who lives under a flight-path like aircraft noise? It is a political hot potato as well as a peace-destroyer. Tens of thousands of people will hear the noise from any third runway at Heathrow. So, when a study commissioned by the government claimed that people are becoming less tolerant of aircraft noise, it made highly unpleasant reading for supporters of a third runway. But the Department for Transport rejected the report as unreliable. Peter Brooker senses the vibrations.     

Simultaneous bootstrap for all three parameters in random coefficient autoregressive models

In this paper we consider autoregressive processes with random coefficients and develop bootstrap approaches that asymptotically work for the distribution of estimated autoregressive parameter as well as for the distribution of estimated variances of the innovation noise and the disturbance noise. We discuss how to obtain approximative residuals of the process and how to separate between the innovation and the disturbance noise in order to be able to extend the classical residual bootstrap for autoregressive processes to the situation considered in this paper. Thereafter, we propose a wild bootstrap procedure as a variation of the residual bootstrap that uses estimated densities of the innovation and the disturbance noise to generate bootstrap replicates of the data generating process. The consistency of the bootstrap approaches is established and their performance is illustrated by a simulation study.     

Logspline Deconvolution in Besov Space

ABSTRACT. In this paper we consider logspline density estimation for random variables which are contaminated with random noise. In the logspline density estimation for data without noise, the logarithm of an unknown density function is estimated by a polynomial spline, the unknown parameters of which are given by maximum likelihood. When noise is present, B-splines and the Fourier inversion formula are used to construct the logspline density estimator of the unknown density function. Rates of convergence are established when the log-density function is assumed to be in a Besov space. It is shown that convergence rates depend on the smoothness of the density function and the decay rate of the characteristic function of the noise. Simulated data are used to show the finite-sample performance of inference based on the logspline density estimation.     

Random spectral measure for non Gaussian moving averages

We study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.     

Robust image segmentation via Bayesian type criterion

Image segmentation plays an important role in image processing before image recognition or compression. Many segmentation solutions follow the information theoretic criteria and often have excellent results; however, they are not robust to reduce the noise effect in contaminated image data. To guarantee the optimal segmentation with possible noise, a robust Bayesian information criterion is proposed to segment a grayscale image and it is less sensitive to noise. The asymptotic properties are also studied. Monte Carlo numerical experiments along with a brain magnetic resonance image are conducted to evaluate the performance of the new method.     

Effects of drift and noise on the optimal sliding window size for data stream regression models

The analysis of non stationary data streams requires a continuous adaption of the model to the relevant most recent data. This requires that changes in the data stream must be distinguished from noise. Many approaches are based on heuristic adaptation schemes. We analyze simple regression models to understand the joint effects of noise and concept drift and derive the optimal sliding window size for the regression models. Our theoretical analysis and simulations show that a near optimal window size can be crucial. Our models can be used as benchmarks for other models to see how they cope with noise and drift.     

New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms

The main objective of this paper is to establish a close relation between the fixed points of Poisson shot noise transforms and perpetuities of a special type. With this relation it is shown that the gamma distributions, the generalized positive Linnik distributions, and the S2 distributions are fixed points of Poisson shot noise transforms. The paper also proves that log‐convexity of the response functions is not needed for non‐negative Poisson shot noise distributions to be self‐decomposable. Finally, the problems of existence and uniqueness of the above mentioned perpetuities are investigated.     

Weighted Searching Probability for Classes of Equivalent Search Designs Comparison

Search design is searching and estimating for a few non zero effects in a large set of effects along with estimation of elements in a set of unknown parameters. In presence of noise, the probability of discrimination between the true non zero effect from an alternative one depends on the design and an unknown parameter, say ρ. We develop a new criterion for design comparison which is independent of ρ and for a family density weight function show that it discriminates and ranks the designs precisely. This criterion is invariance to the variable noise which may be present between designs due to noise factors. This allows us to extend the design comparison to classes of equivalent designs.     

An Exponential-Gamma Convolution Model for Background Correction of Illumina BeadArray Data

Illumina BeadArrays are becoming an increasingly popular Microarray platform due to its high data quality and relatively low cost. One distinct feature of Illumina BeadArrays is that each array has thousands of negative control bead types containing oligonucleotide sequences that are not specific to any target genes in the genome. This design provides a way of directly estimating the distribution of the background noise. In the literature of background correction for BeadArray data, the information from negative control beads is either ignored, used in a naive way that can lead to a loss in efficiency, or the noise is assumed to be normally distributed. However, we show with real data that the noise can be skewed. In this study, we propose an exponential-gamma convolution model for background correction of Illumina BeadArray data. Using both simulated and real data examples, we show that the proposed method can improve the signal estimation and detection of differentially expressed genes when the signal to noise ratio is large and the noise has a skewed distribution.     

Deconvolution Estimation of Onset of Pregnancy with Replicate Observations

In general, the precise date of onset of pregnancy is unknown and may only be estimated from ultrasound biometric measurements of the embryo. We want to estimate the density of the random variables corresponding to the interval between last menstrual period and true onset of pregnancy. The observations correspond to the variables of interest up to an additive noise. We suggest an estimation procedure based on deconvolution. It requires the knowledge of the density of the noise which is not available. But we have at our disposal another specific sample with replicate observations for twin pregnancies. This allows both to estimate the noise density and to improve the deconvolution step. Convergence rates of the final estimator are studied and compared with other settings. Our estimator involves a cut‐off parameter for which we propose a cross‐validation type procedure. Lastly, we estimate the target density in spontaneous pregnancies with an estimation of the noise obtained from replicate observations in twin pregnancies.     

Amplitude modulated model for analyzing non-stationary speech signals

Recently amplitude modulated (AM) model in presence of additive white noise was used to analyze certain non-stationary speech data. It is observed that the assumption of white noise may not be proper in many cases. In this article, we consider the AM signal model in presence of stationary noise. We consider the least squares estimators and the estimators obtained by maximizing the Periodogram function. The two estimators are asymptotically equivalent. We study the theoretical properties of both estimators and observe their performances through numerical simulations. One speech data is analyzed and it is observed that the performance of the proposed estimators is quite satisfactory.     

Monitoring for Increases in Process Variance

Process variation is made up of a wandering mean (signal) and the variation of the process around this wandering mean (noise). The paper advocates a new process monitoring method that uses various forms of moving variances to signal changes in process noise or variance. The monitoring plans are compared to the current approaches in terms of their average run length properties.