Nonstationary Mean Research Articles

Kriging Nonstationary Data

Abstract Spatial data modeled to have come from a random function with a nonstationary mean are considered. The spatial prediction method known as kriging exploits second-order spatial correlation structure to obtain minimum variance predictions of certain average values of the random function. But to do so, it must be assumed that either the mean function (the drift) is known up to a constant or the second-order structure (the variogram) is known exactly. Knowledge of the drift allows the (stationary) variogram to be estimated and leads to ordinary kriging. Knowledge of the variogram allows the drift to be estimated and leads to universal kriging. More usually, neither is known. This article shows how median polish of gridded spatial data provides a resistant and relatively bias-free way of kriging in the presence of drift, yet yields results as good as the mathematically optimal (but operationally difficult) universal kriging. Comparisons are performed on two data sets.

Kriging Nonstationary Data

Abstract Spatial data modeled to have come from a random function with a nonstationary mean are considered. The spatial prediction method known as kriging exploits second-order spatial correlation structure to obtain minimum variance predictions of certain average values of the random function. But to do so, it must be assumed that either the mean function (the drift) is known up to a constant or the second-order structure (the variogram) is known exactly. Knowledge of the drift allows the (stationary) variogram to be estimated and leads to ordinary kriging. Knowledge of the variogram allows the drift to be estimated and leads to universal kriging. More usually, neither is known. This article shows how median polish of gridded spatial data provides a resistant and relatively bias-free way of kriging in the presence of drift, yet yields results as good as the mathematically optimal (but operationally difficult) universal kriging. Comparisons are performed on two data sets.

Noise updating repeated Wiener filter and other adaptive noise smoothing filters using local image statistics

We consider the restoration of images degraded by a class of signal-uncorrelated noise, which is possibly signal-dependent. Some adaptive noise smoothing filters, which assume a nonstationary mean, nonstationary variance image model implicitly or explicitly, are reviewed, and their performances are compared by the mean-squares errors (MSES) and by the human subjective judgment. We also present a new noise smoothing technique which is called the noise updating repeated Wiener (NURW) filter. Explicit noise variance updating formulas are derived for the NURW filter. The performance is improved both in the MSE sense and in the vicinity of edges by subjective observation.

Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise

In this paper, we consider the restoration of images with signal-dependent noise. The filter is noise smoothing and adapts to local changes in image statistics based on a nonstationary mean, nonstationary variance (NMNV) image model. For images degraded by a class of uncorrelated, signal-dependent noise without blur, the adaptive noise smoothing filter becomes a point processor and is similar to Lee's local statistics algorithm [16]. The filter is able to adapt itself to the nonstationary local image statistics in the presence of different types of signal-dependent noise. For multiplicative noise, the adaptive noise smoothing filter is a systematic derivation of Lee's algorithm with some extensions that allow different estimators for the local image variance. The advantage of the derivation is its easy extension to deal with various types of signal-dependent noise. Film-grain and Poisson signal-dependent restoration problems are also considered as examples. All the nonstationary image statistical parameters needed for the filter can be estimated from the noisy image and no a priori information about the original image is required.

Non-stationary response of non-linear liquid motion in A cylindrical tank subjected to random base excitation

The non-stationary response of non-linear liquid motion in a cylindrical tank subjected to lateral earthquake excitation is investigated by modeling the earthquake excitation as an amplitude modulated non-white random process having a dominat frequency. The non-stationary standard deviation and mean responses of the liquid surface displacement are calculated and it is shown that the linear liquid motion theory is not necessarly sufficient to evaluate the safety of tanks, since the non-linear analysis leads to a positive value of the mean response for the liquid surface displacement at the tank wall, while the linear theory predicts zero mean response.

Bayesian approach to limited-angle reconstruction in computed tomography

An arbitrary source function cannot be determined fully from projection data that are limited in number and range of viewing angle. There exists a null subspace in the Hilbert space of possible source functions about which the available projection measurements provide no information. The null-space components of deterministic solutions are usually zero, giving rise to unavoidable artifacts. It is demonstrated that these artifacts may be reduced by a Bayesian maximum a posteriori (MAP) reconstruction method that permits the use of significant a priori information. Since normal distributions are assumed for the a priori and measurement-error probability densities, the MAP reconstruction method presented here is equivalent to the minimum-variance linear estimator with nonstationary mean and covariance ensemble characterizations. A more comprehensive Bayesian approach is suggested in which the ensemble mean and covariance specifications are adjusted on the basis of the measurements.

Optical Characteristics at 3.4 μm Wavelength of Munition-Produced Phosphorus Smoke as a Function of Relative Humidity

Data selected from major U.S. Army field tests conducted during the last four years are analyzed from five different instruments to determine the 3.4–μm wavelength extinction cross-section and mass extinction coefficient of phosphorus smoke. The optical properties of the smokes are found to behave as stochastic spatial and temporal variables with nonstationary means. Data comparisons from different instruments demonstrate a reasonable degree of internal consistency and clearly show the increase of the mass extinction coefficient when the relative humidity is greater than 50%.

Random Hydrodynamic Force on Dams from Earthquakes

A new solution for the hydrodynamic force on dams due to random earthquake accelerations is obtained in terms of nonstationary power spectra and mean square functions. Numerical results and special analytic solutions are obtained and presented in nondimensional forms. The earthquake acceleration is modeled by a stationary white noise with a unit step modulation function. The response spectra are narrowly banded and sharply peaked at resonant frequencies.

Calculation of the non-stationary mean square response of a non-linear system subjected to non-white excitation

An approximate method is developed to calculate the non-stationary response of a non-linear system subjected to non-white excitation. The method consists in modification of the equivalent linearization technique and the use of the moment equations of the equivalent linear system to evaluate the mean square response. The results obtained by the present method are compared with the corresponding digital simulation results for a Duffing system or a dry friction system.

Real-time estimation of velocity and covariance structure of rainfall events using telemetered raingage data — A comparison of methods

Short-term rainfall prediction in time and space requires parameter estimation in real time. This paper discusses several methods of using telemetric raingage data to estimate the non-stationary mean and variance of a rainfall event. Similarly, real-time estimation of residual covariance structure and storm velocity is discussed. The performance of the suggested techniques are illustrated with examples. The parameter estimation corresponds to a forecasting model developed by Johnson and Bras. The goal of this article is to inform other researchers of the advantages and disadvantages of different methodologies for real-time estimation of rainfall statistical characteristics.

MITOTIC INDEX, INFLUX AND MEAN TRANSIT TIME IN THE HAMSTER CHEEK POUCH EPITHELIUM, A PARTIALLY SYNCHRONIZED CELL SYSTEM*†

Mitotic activity was followed in the epithelium of the hamster cheek pouch for about 12 hr in two experiments under different conditions of noise and light intensity. A mathematical model based on a non-stationary probability density function for the transit time through mitosis was developed, making an analysis of this partially synchronized cell system possible. The most important result of the two experiments is the indication of a non-stationary mean transit time for cells in mitosis. In the first experiment (70 dB, 103 lux), which produced high mitotic indices (maximum 2.0%), the influx changed from 0. to 0.9%/hr (mean of 7-hourly determinations: 0.4%/hr), and the mean transit time from 1.9 to 5.5 hr. In the second experiment (70 dB, 265 lux), which had a lower level of MI (maximum 1.3%), the influx changed from 0 to 1.8%/hr (mean of 13-hourly determinations: 0.4%/hr), and the mean transit time from 0.3 to 3.3 hr. It was difficult to say precisely whether the variation in influx or in mean transit time was the main factor in the increase and decrease of MI. The data could not be simulated with a stationary mean transit time. It is suggested that changes in MI due to different external circumstances are mainly the result of variation in the mean transit time.

Effects of Uncertain and Nonstationary Parameters Upon Capital Market Equilibrium Conditions

Formal models for portfolio analysis, such as Markowitz [13], are frequently based upon mean-variance analyses and involve the estimation of a mean vector and a variance-covariance matrix describing expected returns and variability of returns for all securities under consideration. These parameter estimates play a major role in the selection of a single, optimal portfolio. Kalymon [9] and Barry [1] have considered the effects of parameter uncertainty upon individual investors' inferences and decisions in the context of portfolio selection, and Barry and Winkler [2] have similarly considered the impact of nonstationary means upon portfolio selection decisions by individual investors.

Climate change: Detection and its impact on hydrologic design

Controversy regarding origins of the so‐called Hurst phenomenon has continued since the first appearance of Hurst's work. There appear to be at least two general mechanisms which might generate geophysical time series displaying the Hurst phenomenon. The first is nonstationarity of process mean level, perhaps owing to dynamic characteristics of the entire earth geophysical system. The second generating mechanism is a stationary model structure such as the Box‐Jenkins models with parameters such that substantial low‐frequency effects are present. A series of Monte Carlo tests have been performed which show that it will generally be very difficult to distinguish between the two generating mechanisms on the basis of geophysical records of lengths usually available. The Monte Carlo experiments are augmented by analysis of several series of tree ring growth indices ranging in length from 506 to 1164 years. Analysis of these sequences for nonstationarity in mean level showed that the sequences were entirely compatible with stationary Box‐Jenkins models earlier fit to the data by Hipel (1975). However, a similar analysis of the variance of the time series showed that there was evidence of nonstationarity in the variances of the records such that the time series were not compatible with the constant variance assumption of the stationary Box‐Jenkins models. Finally, several reservoir simulations were made by using the sequent peak algorithm for type A (nonstationary mean) and type B (constant mean) models which had been found to be statistically indistinguishable in the Monte Carlo experiments. The results showed that so long as nonstationarity in mean level was modest and demand levels were not too high the models give very similar results; however, at high demand levels or large nonstationarity in process mean, substantial differences in storage requirements may result.

Capacity expansion for water resource facilities: a control theoretic algorithm for nonstationary uncertain demands

Municipal water and waste water investments represent a large and important segment of the expenditures made by local governments. The response of designers to the problem of sizing increments to capacity has been to overbuild in order to assure safe and adequate supplies. A growing body of evidence suggests that the magnitude of overbuilding has been excessive. Earlier research has neglected to model demand growth as uncertain with the possibility of both nonstationary mean and variance. Also, failure to recognize the problem as one of closed loop control has resulted in erroneous modeling efforts. The basic approach in this research is to model using stochastic dynamic programming. The criterion was minimization of expected cost. Excess capacity, the state of the system, increases due to investment decisions. Negative excess capacity implies shortage. As time progresses known, but possibly uncertain demands impinge upon the system. Overall, several highly flexible capacity planning models for nonstationary demands can be formulated, are computationally feasible, and produce excellent approximations to known solutions. However, these models should only be used after sensitivity testing in conjunction with simpler approaches.

Nonstationary Response of Structural Systems

This paper presents a procedure for calculating the nonstationary mean square response of multidegree-of-freedom linearly elastic structural systems to multiple segmented nonstationary random forcing functions. Each forcing function is a stochastic process generated by adding a series of covariance stationary zero mean random forcing functions which are each shared by deterministic functions that do not overlap in the time domain. The procedure is used to compare the transient and stationary aspects of the response of structural systems to forcing functions that are time-variant in intensity and frequency content. The magnitudes of the modal displacement autocorrelation functions are studied in comparison with the cross-correlation functions between any two normal modes.

Theory and Application of an Estimation Model for Time Series with Nonstationary Means

Time series models of a complex nature, such as consumer brand switching analyses, have required assumptions of parameter stability because statistical models were not available to deal with parameter change. A model is developed here to estimate a stepwise change in the mean process of a Gaussian time series. Estimators which are small-sample efficient in a special sense are presented, along with examples and suggested applications of the method to brand switching problems.