Using symbolic expressions to get the Taylor expansion of the Lagrangian auto-covariance function

2021 ◽  
Author(s):  
Eduardo S. Schneider ◽  
Craig L. Zirbel
Keyword(s):  
Taylor Expansion ◽  
Covariance Function ◽  
Auto Covariance

scholarly journals Analysis of Air Pollution Parameters Using Covariance Function Theory

2020 ◽  
Vol 27 (4) ◽  
pp. 555-565
Author(s):  
Ignas Daugela ◽  
Jurate Suziedelyte Visockiene ◽  
Jonas Skeivalas
Keyword(s):  
Environmental Pollution ◽  
Time Scale ◽  
Covariance Function ◽  
Function Theory ◽  
Trace Gas ◽  
Covariance Functions ◽  
Intensity Changes ◽  
Oxygen Gas ◽  
Auto Covariance ◽  
Sentinel 2

Abstract The paper analyses the intensity changes of three pollution parameter vectors in space and time. The RGB raster pollution data of the Lithuanian territory used for the research were prepared according to the digital images of the Sentinel-2 Earth satellites. The numerical vectors of environmental pollution parameters CH4 (methane), NO2 (nitrogen dioxide) and for direct comparison O2 (oxygen gas) were used for the calculations. The covariance function theory was used to perform the analysis of intensity changes in digital vectors. Estimates of the covariance functions of the numerical vectors of pollution parameters and O2 or the auto-covariance functions of single vectors are calculated from random functions consisting of arrays of measurement parameters of all parameters vectors. Correlation between parameters vectors depends on the density of parameters and their structure. Estimates of covariance functions were calculated by changing the quantization interval on a time scale and using a compiled computer program using the Matlab procedure package. The probability dependence between the environmental pollution parameter vectors and trace gas of the territory in Lithuania and their change in time scale was determined.

Spontaneous activity in developing turtle retinal ganglion cells: Statistical analysis

2000 ◽  
Vol 17 (2) ◽  
pp. 229-241 ◽  
Author(s):  
NORBERTO M. GRZYWACZ ◽  
EVELYNE SERNAGOR
Keyword(s):  
Covariance Function ◽  
Ganglion Cells ◽  
Slow Component ◽  
Distributed Delay ◽  
Positive Phase ◽  
Retinal Ganglion ◽  
Temporal Properties ◽  
Cross Covariance ◽  
Two Factors ◽  
Auto Covariance

We report on the temporal properties of the spontaneous bursts of activity in the developing turtle retina. Quantitative statistical criteria were used to detect, cluster, and analyze the temporal properties of the bursts. The interburst interval, duration, firing rate, and number of spikes per burst varied widely among cells and from burst to burst in a single cell. Part of this variability was due to the positive correlation between a burst's duration and the interburst interval preceding that burst. This correlation indicated the influence of a refractory period on the bursts' properties. Further evidence of such a refractoriness came from the bursts' auto-covariance function, which gives the tendency of a spike to occur a given amount of time after another spike. This function showed a positive phase (between ≈10 ms and 10 s) followed by a negative one (between 10 s and more than 100 s), suggestive of burst refractoriness. The bursts seemed to be propagating from cell to cell, because there was a long (symmetrically distributed) delay between the activation of two neighbor cells (median absolute delay = 2.3 s). However, the activity often failed to propagate from one cell to the other (median safety factor = 0.59). The number of spikes per burst in neighbor cells was statistically positively correlated, indicating that the activity in the two cells was driven by the same excitatory process. At least two factors contribute to the excitability during bursts, because the positive phase of the cross-covariance function (similar to auto-covariance but for two cells) had a temporally asymmetric fast component (1–3 ms) followed by a temporally symmetric slow component (1 ms to 10 s).

A Stochastic Model for the Inheritance of the Cancer Proneness Phenotype

1987 ◽  
Vol 26 (03) ◽  
pp. 117-123
Author(s):  
P. Tautu ◽  
G. Wagner
Keyword(s):  
Stochastic Model ◽  
Gaussian Process ◽  
Covariance Function ◽  
Neoplastic Disease ◽  
Disease Phenotype ◽  
Probabilistic Representation ◽  
Temporal Behaviour ◽  
Continuous Trait ◽  
Continuous Parameter ◽  
Level Zero

SummaryA continuous parameter, stationary Gaussian process is introduced as a first approach to the probabilistic representation of the phenotype inheritance process. With some specific assumptions about the components of the covariance function, it may describe the temporal behaviour of the “cancer-proneness phenotype” (CPF) as a quantitative continuous trait. Upcrossing a fixed level (“threshold”) u and reaching level zero are the extremes of the Gaussian process considered; it is assumed that they might be interpreted as the transformation of CPF into a “neoplastic disease phenotype” or as the non-proneness to cancer, respectively.

scholarly journals Discussion on Competition for Spatial Statistics for Large Datasets

2021 ◽  
Author(s):  
Roman Flury ◽  
Reinhard Furrer
Keyword(s):  
Spatial Statistics ◽  
Covariance Function ◽  
Likelihood Function ◽  
Large Datasets ◽  
Missing Observations ◽  
Covariance Model ◽  
Compactly Supported ◽  
Covariance Tapering ◽  
Simple Kriging ◽  
Estimated Parameters

AbstractWe discuss the experiences and results of the AppStatUZH team’s participation in the comprehensive and unbiased comparison of different spatial approximations conducted in the Competition for Spatial Statistics for Large Datasets. In each of the different sub-competitions, we estimated parameters of the covariance model based on a likelihood function and predicted missing observations with simple kriging. We approximated the covariance model either with covariance tapering or a compactly supported Wendland covariance function.

scholarly journals Basis Expansions for Functional Snippets

Biometrika ◽  
2020 ◽  
Author(s):  
Zhenhua Lin ◽  
Jane-Ling Wang ◽  
Qixian Zhong
Keyword(s):  
Data Analysis ◽  
Convergence Rate ◽  
Functional Data Analysis ◽  
Functional Data ◽  
Covariance Function ◽  
Covariance Estimation ◽  
Simulation Studies ◽  
Finite Sample ◽  
Covariance Functions ◽  
The Mean

Summary Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. In this paper, we investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly (and often much) shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed estimator enjoys a convergence rate that is adaptive to the smoothness of the underlying covariance function, and has superior finite-sample performance in simulation studies.

scholarly journals Performance Degradation in Cross-Eye Jamming Due to Amplitude/Phase Instability between Jammer Antennas

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5027
Author(s):  
Je-An Kim ◽  
Joon-Ho Lee
Keyword(s):  
Performance Analysis ◽  
Phase Difference ◽  
Taylor Expansion ◽  
Amplitude Ratio ◽  
Random Variables ◽  
Performance Degradation ◽  
Analytical Performance ◽  
Phase Instability ◽  
Gaussian Random Variables ◽  
First Order

Cross-eye gain in cross-eye jamming systems is highly dependent on amplitude ratio and the phase difference between jammer antennas. It is well known that cross-eye jamming is most effective for the amplitude ratio of unity and phase difference of 180 degrees. It is assumed that the instabilities in the amplitude ratio and phase difference can be modeled as zero-mean Gaussian random variables. In this paper, we not only quantitatively analyze the effect of amplitude ratio instability and phase difference instability on performance degradation in terms of reduction in cross-eye gain but also proceed with analytical performance analysis based on the first order and second-order Taylor expansion.

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