Exponential Autocorrelation Research Articles

Statistical Analysis of Non‐Replicated Experiments in Farmers’ Fields. A Case of Balanced Fertilization Trials for Bean in Burundi

Core Ideas Often agronomic experiments in farmers’ fields do not allow replications. Between‐fields spatial variability can be used to model the error variance. The modeled error variance in a GLMM is used to test treatment‐related hypotheses. Using fields as replications can end in biased results due to fields’ autocorrelation. ABSTRACTVery often agricultural experiments in smallholder farms are conducted without actual replications due to land restrictions. Using fields as replications and performing the ANOVA with a randomized complete block design (RCBD) model can result in misleading conclusions about treatment’s performance due to a biased error variance matrix estimated under the wrong assumption of independence between fields. A better alternative is using the spatial variability modeling features of the GLMM to generate an error term to perform unbiased hypothesis tests. Non‐replicated trials to estimate bean response to three fertilization treatments were conducted across 175 farmers’ fields in Burundi. Yields were used to compare the performance of an ANOVA using a GLMM where the exponential autocorrelation pattern of the residuals was used to model the error variance‐covariance matrix against an ANOVA following a RCBD model where the fields are handled as blocks. The fields were grouped in three clusters prior to the ANOVA. AIC and the Generalized Chi‐Sq./DF ratio values for the model involving spatial modeling were 165.3 and 1, respectively. The same fit statistic values from the RCBD model were 330.1 and 0.1, respectively. The near‐half magnitude of AIC in the spatial model relative to the RCBD model indicates higher model goodness of fit for the spatial model. The 0.1 value for the Generalized Chi‐Sq./DF ratio in the RCBD model suggests underdispersion and violation of the independence‐between‐fields assumption. Detection of a significant CLUSTER*TREATMENT by the spatial model corroborates the superiority of the ANOVA model involving spatial variability.

Random field characterization of uniaxial compressive strength and elastic modulus for intact rocks

Rock properties exhibit spatial variabilities due to complex geological processes such as sedimentation, metamorphism, weathering, and tectogenesis. Although recognized as an important factor controlling the safety of geotechnical structures in rock engineering, the spatial variability of rock properties is rarely quantified. Hence, this study characterizes the autocorrelation structures and scales of fluctuation of two important parameters of intact rocks, i.e. uniaxial compressive strength (UCS) and elastic modulus (EM). UCS and EM data for sedimentary and igneous rocks are collected. The autocorrelation structures are selected using a Bayesian model class selection approach and the scales of fluctuation for these two parameters are estimated using a Bayesian updating method. The results show that the autocorrelation structures for UCS and EM could be best described by a single exponential autocorrelation function. The scales of fluctuation for UCS and EM respectively range from 0.3 m to 8.0 m and from 0.3 m to 8.4 m. These results serve as guidelines for selecting proper autocorrelation functions and autocorrelation distances for rock properties in reliability analyses and could also be used as prior information for quantifying the spatial variability of rock properties in a Bayesian framework.

On the numerical generation of positive-axis-defined distributions with an exponential autocorrelation function

Stochastic modeling commonly requires random process generation with an exponential autocorrelation function (ACF). These random processes may be represented as a solution of a stochastic differential equation (SDE) of the first order and usually have one-sided (positive-axis-defined) distributions. However, adoption of the SDE-based method faces serious limitations due to difficulties with the numerical solution. To overcome this issue we propose a tractable general numerical solution of the above-mentioned SDE that preserves solution positivity and accuracy, and validate it with numerical simulations.

Estimation of Conditional Expected Value for Exponentially Autocorrelated Data

AbstractAutocorrelation of signals and measurement data makes it difficult to estimate their statistical characteristics. However, the scope of usefulness of autocorrelation functions for statistical description of signal relation is narrowed down to linear processing models. The use of the conditional expected value opens new possibilities in the description of interdependence of stochastic signals for linear and non-linear models. It is described with relatively simple mathematical models with corresponding simple algorithms of their practical implementation.The paper presents a practical model of exponential autocorrelation of measurement data and a theoretical analysis of its impact on the process of conditional averaging of data. Optimization conditions of the process were determined to decrease the variance of a characteristic of the conditional expected value. The obtained theoretical relations were compared with some examples of the experimental results.

Interpretation of Brown Planthoper (Nilaparvata lugens Stal.) Attacks Using Exponential Smoothing and Spatial Autocorelation

Different ways of controlling and preventing for brown planthopper attacks (Nilaparvata lugen Stal.) called BPH in Boyolali regency has been done, but until this time the loss due to this pest occur periodically. This study aims to explore the interpretation of the results for prediction BPH attacks in Boyolali district, which is done using a combination of Exponential Smoothing methods and Spatial Autocorrelation. Training data for predicting the BPH attacks, has been taken from the events in 2001 until 2007, to predict the BPH attacks in 2008. Prediction for the spatial distribution of BPH attack using Local Moran’s and Local Geary, whereas for visualization is using choropleth Map and Local Moran’s Map. Rainfall data from 2001 until 2010 is used to determine the periods of wet and dry seasons in Boyolali. To visualizing the results of the analysis, was used a predictive maps for attacks of monthly periods and with the sub districts area as the smallest unit area. The results showed that the combination of methods can be used to predict the BPH attacks and can provide predictive information expansive dynamics and time attacks, include the deployment of attack pattern based on the sub district area, the initial attacks and the time peak of attacks. This prediction scan be used as a consideration to formulating early warning information and the priority how to overcome them by region.

Stochastic Stability Criteria for Second-Order Oscillator Parametrically Excited by Colored Noise

A second-order oscillator is considered having a random perturbation in its stiffness. This is given by a colored Gaussian or non-Gaussian process. In this way, the oscillator may be stochastically stable or unstable according to the intensity of the excitation. The almost sure (sample) stochastic stability and the stability in the first three response statistical moments are compared for different excitation processes: process with exponential autocorrelation, second-order Gaussian process, bounded noise process. Notable differences in the stability boundaries are found either according to the stability criteria or to the type of excitation. These comparisons are lacking in literature.

Modeling of rough surfaces with given roughness parameters

Modeling of rough surfaces with given roughness parameters is studied, where surfaces with Gaussian height distribution and exponential auto-correlation function (ACF) are concerned. A large number of micro topography samples are randomly generated first using the rough surface simulation method with FFT. Then roughness parameters of the simulated roughness profiles are calculated according to parameter definition, and the relationship between roughness parameters and statistical distribution parameters is investigated. The effects of high-pass filtering with different cut-off lengths on the relationship are analyzed. Subsequently, computing formulae of roughness parameters based on standard deviation and correlation length are constructed with mathematical regression method. The constructed formulae are tested with measured data of actual topographies, and the influences of auto-correlation variations at different lag lengths on the change of roughness parameter are discussed. The constructed computing formulae provide an approach to active modeling of rough surfaces with given roughness parameters.

A semiempirical error estimation technique for PWV derived from atmospheric radiosonde data

Abstract. A semiempirical method for estimating the error and optimum number of sampled levels in precipitable water vapour (PWV) determinations from atmospheric radiosoundings is proposed. Two terms have been considered: the uncertainties in the measurements and the sampling error. Also, the uncertainty has been separated in the variance and covariance components. The sampling and covariance components have been modelled from an empirical dataset of 205 high-vertical-resolution radiosounding profiles, equipped with Vaisala RS80 and RS92 sondes at four different locations: Güímar (GUI) in Tenerife, at sea level, and the astronomical observatory at Roque de los Muchachos (ORM, 2300 m a.s.l.) on La Palma (both on the Canary Islands, Spain), Lindenberg (LIN) in continental Germany, and Ny-Ålesund (NYA) in the Svalbard Islands, within the Arctic Circle. The balloons at the ORM were launched during intensive and unique site-testing runs carried out in 1990 and 1995, while the data for the other sites were obtained from radiosounding stations operating for a period of 1 year (2013–2014). The PWV values ranged between ∼ 0.9 and ∼ 41 mm. The method sub-samples the profile for error minimization. The result is the minimum error and the optimum number of levels. The results obtained in the four sites studied showed that the ORM is the driest of the four locations and the one with the fastest vertical decay of PWV. The exponential autocorrelation pressure lags ranged from 175 hPa (ORM) to 500 hPa (LIN). The results show a coherent behaviour with no biases as a function of the profile. The final error is roughly proportional to PWV whereas the optimum number of levels (N0) is the reverse. The value of N0 is less than 400 for 77 % of the profiles and the absolute errors are always < 0.6 mm. The median relative error is 2.0 ± 0.7 % and the 90th percentile P90 = 4.6 %. Therefore, whereas a radiosounding samples at least N0 uniform vertical levels, depending on the water vapour content and distribution of the atmosphere, the error in the PWV estimate is likely to stay below ≈ 3 %, even for dry conditions.

Comments on “Simple Generation of Gamma, Gamma–Gamma and K Distributions With Exponential Autocorrelation Function”

This paper is a comment on Simple generation of gamma, gamma-gamma and K distributions with exponential autocorrelation function”. The original article states that the simulation of a free-space optical (FSO) communication channel in the presence of strong turbulence typically requires the generation of channel states with a K or gamma-gamma distribution and a predefined autocorrelation function. In the comment it is concluded that the resulting autocorrelation function for gamma-gamma and K processes may be approximated as a simple exponent only under certain constraints on distribution parameters. Whenever these constraints are not met, the resulting autocorrelation function is still analytically tractable, but it is no longer a simple exponent.

Estimation of Bounded and Unbounded Trajectories in Diffusion MRI

Disentangling the tissue microstructural information from the diffusion magnetic resonance imaging (dMRI) measurements is quite important for extracting brain tissue specific measures. The autocorrelation function of diffusing spins is key for understanding the relation between dMRI signals and the acquisition gradient sequences. In this paper, we demonstrate that the autocorrelation of diffusion in restricted or bounded spaces can be well approximated by exponential functions. To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories. We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective. For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.

Construction of SDE-based wind speed models with exponentially decaying autocorrelation

This paper provides a systematic method to build wind speed models based on stochastic differential equations (SDEs). The resulting models produce stochastic processes with a given probability distribution and exponentially decaying autocorrelation function. The only information needed to build the models is the probability density function of the wind speed and its autocorrelation coefficient. Unlike other methods previously proposed in the literature, the proposed method leads to models able to reproduce an exact exponential autocorrelation even if the probability distribution is not Gaussian. A sufficient condition for the property above is provided. The paper includes the explicit formulation of SDE-based wind speed models obtained from several probability distributions used in the literature to describe different wind speed behaviors. All models are validated through numerical simulations. Finally, the proposed procedure is applied to model the wind speed observed at a meteorological station in New Zealand. A comparison of the statistical properties of the wind speed measurements and of the stochastic process generated by the SDE model is also provided.

Simple Generation of Gamma, Gamma–Gamma, and K Distributions With Exponential Autocorrelation Function

The simulation of a free-space optical (FSO) communication channel in the presence of strong turbulence typically requires the generation of channel states with a K or gamma–gamma distribution and a predefined autocorrelation function. In this paper, we propose a simple and effective simulator of the strong-turbulence FSO channel that addresses the influence of the temporal covariance effect. Specifically, the proposed simulator provides K and gamma–gamma distributed values with the exponential autocorrelation function and a prescribed correlation time. This simulator is based on the numerical solution of the first-order stochastic differential equation. The simulated channel states are generated by a simple discrete-time differential equation and the simulator performance is analyzed in the paper.

Recursive impedance inversion of ground-penetrating radar data in stochastic media

The travel time and amplitude of ground-penetrating radar (GPR) waves are closely related to medium parameters such as water content, porosity, and dielectric permittivity. However, conventional estimation methods, which are mostly based on wave velocity, are not suitable for real complex media because of limited resolution. Impedance inversion uses the reflection coefficient of radar waves to directly calculate GPR impedance and other parameters of subsurface media. We construct a 3D multiscale stochastic medium model and use the mixed Gaussian and exponential autocorrelation function to describe the distribution of parameters in real subsurface media. We introduce an elliptical Gaussian function to describe local random anomalies. The tapering function is also introduced to reduce calculation errors caused by the numerical simulation of discrete grids. We derive the impedance inversion workflow and test the calculation precision in complex media. Finally, we use impedance inversion to process GPR field data in a polluted site in Mongolia. The inversion results were constrained using borehole data and validated by resistivity data.

Numerical Investigation of Microbending Loss in Optical Fibres

Microbending plays a key role in the bend loss of optical fibres. To numerically investigate a microbending induced loss, an analytical model for microbending in optical fibres with arbitrary refractive index profiles is presented. In this model, random perturbations of the fibre core along the fibre axis are described by an analytical function whose power spectral density is derived from an exponential autocorrelation function. Using the model together with the beam propagation method, microbending loss is investigated for several different types of optical fibre, which include the traditional single-mode/multimode fibres (SMF/MMF) as used in existing optical networks, and typical few-mode/ring-core fibres (FMF/RCF) with the potential for future ultra high-speed optical networks. The validity of the proposed model is demonstrated by comparing the microbending loss of a SMF computed using coupled mode theory and the results. Simulation results show that, under the condition of launching only the fundamental mode into the optical fibres, the SMF and RCF supporting only one radial mode have nearly equal microbending loss, whilst the FMF and MMF have relatively low microbending loss. In addition, the microbending loss of the RCF is shown to be dependent on the ring core thickness.

Semi-Empirical Calibration of the Integral Equation Model for Co-Polarized L-Band Backscattering

The objective of this paper is to extend the semi-empirical calibration of the backscattering Integral Equation Model (IEM) initially proposed for Synthetic Aperture Radar (SAR) data at C- and X-bands to SAR data at L-band. A large dataset of radar signal and in situ measurements (soil moisture and surface roughness) over bare soil surfaces were used. This dataset was collected over numerous agricultural study sites in France, Luxembourg, Belgium, Germany and Italy using various SAR sensors (AIRSAR, SIR-C, JERS-1, PALSAR-1, ESAR). Results showed slightly better simulations with exponential autocorrelation function than with Gaussian function and with HH than with VV. Using the exponential autocorrelation function, the mean difference between experimental data and Integral Equation Model (IEM) simulations is +0.4 dB in HH and −1.2 dB in VV with a Root Mean Square Error (RMSE) about 3.5 dB. In order to improve the modeling results of the IEM for a better use in the inversion of SAR data, a semi-empirical calibration of the IEM was performed at L-band in replacing the correlation length derived from field experiments by a fitting parameter. Better agreement was observed between the backscattering coefficient provided by the SAR and that simulated by the calibrated version of the IEM (RMSE about 2.2 dB).

Random field-based modeling of size effect on the longitudinal tensile strength of clear timber

In the current study, a random field-based size effect model has been proposed for the longitudinal tensile strength of clear timber. Since the failure mode is brittle, the problem is basically an extreme value problem of finding the distribution of minima of strength fields for specimens of different volumes. The stochastic response has been evaluated based on the Monte Carlo method along with the weakest link theory. Within the framework of the spectral representation method, the Weibull distribution has been considered as the marginal distribution to generate realizations of the strength as a 3D random field. The squared exponential autocorrelation function has been used for the description of spatial variability. The error resulting from this model, as compared to existing experimental data in the literature, is much lower than that of the classical Weibull law. The results show that when one of the specimen’s dimensions decreases to less than 10 times the correlation length of the strength field, the size effect starts to deviate from the classical size effect law. Moreover, a simple analytical approximation, which includes the correlation length as length scale, has been presented that facilitates the application of the proposed model.

Golden Ratio Autocorrelation Function and the Exponential Decay

An autocorrelation function is obtained on the base of the recurrence relation formalism, whose continued fraction form corresponds to that of the golden ratio. It turns out that this Golden Ratio (GR) autocorrelation is known in science and obeys all necessary conditions, in contrast to the exponential autocorrelation function. Using the Kubo approach it is shown how exponential correlations appear in the linear response theory as a result of non-Hermitian relaxation of the system.

ESSA: Exponential Smoothing and Spatial Autocorrelation, Methods for Prediction of Outbreaks Pest in Indonesia

Geographically, Indonesia is one of the countries in Asia are at risk of disease pests of rice brown planthoppers or BPH (Nilaparvata lugen Stal.) through a cycle of long-distance migration along the years following the tropical Monsoon flow. BPH attacked this country since 1930 until now. It damaged ten thousand hectares of rice and made crop failure. As anticipated in the future surveillance system is needed that is able to predict the dynamics of migration of BPH early so that the concentration of BPH endemicity. The focus of the research is to develop a procedure Exponential Smoothing and Spatial Autocorrelation (ESSA), which includes a mechanism prediction using Exponential Smoothing method and mechanism analysis of BPH spatial patterns attack using Spatial Autocorrelation method. The research was conducted through four stages which include: (1) the identification and determination of areas experiencing high attack BPH in the study area, (2) grouping of BPH data attacks that followed local cropping patterns, (3) the prediction and analysis of spatial connectivity using the Local Indicator Spatial Association (LISA), and (4) visualization and interpretation of analytical results. Prototype was built using the programming language R (http://r-cran.project). The result of the research could be developed as geographical information system (GIS) tool to predict the migration pattern of BPH which has been done by observation center for Pest Plant Diseases Laboratory Observations Region V Surakarta, Central Java, Indonesia, Ministry of Agriculture Republic of Indonesia for many times.

The influence of crustal scattering on translational and rotational motions in regional and teleseismic coda waves

Monte Carlo solutions to the radiative transfer equations are used to model translational and rotational motion seismogram envelopes in random elastic media with deterministic background structure assuming multiple anisotropic scattering. Observation and modelling of the three additional components of rotational motions can provide independent information about wave propagation in the Earth's structure. Rotational motions around the vertical axis observed in the P-wave coda are of particular interest as they can only be excited by horizontally polarized shear waves and therefore indicate the conversion from P to SH energy by multiple scattering at 3-D heterogeneities. To investigate crustal scattering and attenuation parameters in south-east Germany beneath the Gräfenberg array multicomponent seismogram envelopes of rotational and translational motions are synthesized and compared to seismic data from regional swarm-earthquakes and of deep teleseismic events. In the regional case a nonlinear genetic inversion is used to estimate scattering and attenuation parameters at high frequencies (4–8 Hz). Our preferred model of crustal heterogeneity consists of a medium with random velocity and density fluctuations described by an exponential autocorrelation function with a correlation length of a few hundred metres and fluctuations in the range of 3 per cent. The quality factor for elastic S-waves attenuation |$Q_i^S$| is around 700. In a second, step simulations of teleseismic P-wave arrivals using this estimated set of scattering and attenuation parameters are compared to observed seismogram envelopes from deep events. Simulations of teleseismic events with the parameters found from the regional inversion show good agreement with the measured seismogram envelopes. This includes ringlaser observations of vertical rotations in the teleseismic P-wave coda that naturally result from the proposed model of wave scattering. The model also predicts, that the elastic energy recorded in the teleseismic P coda is not equipartitioned, unlike the coda of regional events, but contains an excess of shear energy. The results confirm that scattering generating the teleseismic P-wave coda mainly occurs in the crustal part of the lithosphere beneath the receiver. Our observations do not require scattering of high frequency waves in the mantle, but weak scattering in the lithospheric mantle cannot be ruled out.

The Influence of Characterization of Metallic Gasket Rough Surface on Gas Sealing Performance in Seal Structure

To investigate the influence of the roughness characteristics of the metallic gasket on the gas leakage rate, a mathematical model of a rough surface based on the Gauss distribution function and exponential auto-correlation function was built. According to this model, a 3D geometrical model of a sealing clearance with two rough surfaces was configured. Effect of the roughness parameters on the gas sealing performance was numerical analyzed, and through the comparison with the classical smooth parallel plate leakage model, the relationship between the leakage rate and the roughness parameters of the sealing surfaces was built. Furthermore, the relationship between contact pressure and the leakage rate for the different roughness surfaces was studied based on the current model. The results show the methodology proposed in current work is effective in predicting the actual sealing performance of static seal structure.