scholarly journals Augmented Gaussian random field: Theory and computation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sheng Zhang ◽  
Xiu Yang ◽  
Samy Tindel ◽  
Guang Lin
Keyword(s):  
Random Field ◽  
Gaussian Process ◽  
Gaussian Random Field ◽  
Computational Method ◽  
The Novel ◽  
Random Field Theory ◽  
Composite Function ◽  
Numerical Examples ◽  
Derivatives Of ◽  
Natural Way

<p style='text-indent:20px;'>We propose the novel augmented Gaussian random field (AGRF), which is a universal framework incorporating the data of observable and derivatives of any order. Rigorous theory is established. We prove that under certain conditions, the observable and its derivatives of any order are governed by a single Gaussian random field, which is the aforementioned AGRF. As a corollary, the statement "the derivative of a Gaussian process remains a Gaussian process" is validated, since the derivative is represented by a part of the AGRF. Moreover, a computational method corresponding to the universal AGRF framework is constructed. Both noiseless and noisy scenarios are considered. Formulas of the posterior distributions are deduced in a nice closed form. A significant advantage of our computational method is that the universal AGRF framework provides a natural way to incorporate arbitrary order derivatives and deal with missing data. We use four numerical examples to demonstrate the effectiveness of the computational method. The numerical examples are composite function, damped harmonic oscillator, Korteweg-De Vries equation, and Burgers' equation.</p>

scholarly journals Statistical multiscale mapping of IDH1, MGMT, and microvascular proliferation in human brain tumors from multiparametric MR and spatially-registered core biopsy

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Jason G. Parker ◽  
Emily E. Diller ◽  
Sha Cao ◽  
Jeremy T. Nelson ◽  
Kristen Yeom ◽  
...  
Keyword(s):  
Field Theory ◽  
Random Field ◽  
Classification Accuracy ◽  
Core Biopsy ◽  
Gaussian Random Field ◽  
Error Rates ◽  
Molecular Heterogeneity ◽  
K Nearest Neighbor ◽  
Random Field Theory ◽  
Microvascular Proliferation

AbstractWe propose a statistical multiscale mapping approach to identify microscopic and molecular heterogeneity across a tumor microenvironment using multiparametric MR (mp-MR). Twenty-nine patients underwent pre-surgical mp-MR followed by MR-guided stereotactic core biopsy. The locations of the biopsy cores were identified in the pre-surgical images using stereotactic bitmaps acquired during surgery. Feature matrices mapped the multiparametric voxel values in the vicinity of the biopsy cores to the pathologic outcome variables for each patient and logistic regression tested the individual and collective predictive power of the MR contrasts. A non-parametric weighted k-nearest neighbor classifier evaluated the feature matrices in a leave-one-out cross validation design across patients. Resulting class membership probabilities were converted to chi-square statistics to develop full-brain parametric maps, implementing Gaussian random field theory to estimate inter-voxel dependencies. Corrections for family-wise error rates were performed using Benjamini-Hochberg and random field theory, and the resulting accuracies were compared. The combination of all five image contrasts correlated with outcome (P < 10−4) for all four microscopic variables. The probabilistic mapping method using Benjamini-Hochberg generated statistically significant results (α ≤ 0.05) for three of the four dependent variables: (1) IDH1, (2) MGMT, and (3) microvascular proliferation, with an average classification accuracy of 0.984 ± 0.02 and an average classification sensitivity of 1.567% ± 0.967. The images corrected by random field theory demonstrated improved classification accuracy (0.989 ± 0.008) and classification sensitivity (5.967% ± 2.857) compared with Benjamini-Hochberg. Microscopic and molecular tumor properties can be assessed with statistical confidence across the brain from minimally-invasive, mp-MR.

Tumor Necrosis Factor-α Production-Enhancing Properties of Novel Adamantylalkylthio Derivatives of Some Heterocyclic Compounds

2005 ◽  
Vol 60 (4) ◽  
pp. 471-475 ◽  
Author(s):  
Barbara Orzeszko ◽  
Tomasz Świtaj ◽  
Anna B. Jakubowska-Mućka ◽  
Witold Lasek ◽  
Andrzej Orzeszko ◽  
...  
Keyword(s):  
Tumor Necrosis Factor ◽  
Heterocyclic Compounds ◽  
Tumor Necrosis ◽  
The Novel ◽  
Necrosis Factor Alpha ◽  
Tnf Α ◽  
Factor Alpha ◽  
Factor Α ◽  
Derivatives Of ◽  
Necrosis Factor

Certain adamantylated heterocycles were previously shown to enhance the secretion of tumor necrosis factor alpha (TNF-α) by murine melanoma cells that have been transduced with the gene for human TNF-α and constitutively expressed this cytokine. The stimulatory potency of those compounds depended, among other factors, on the structure of the linker between the adamantyl residue and the heterocyclic core. In the present study, a series of (1-adamantyl)alkylsulfanyl derivatives of heterocyclic compounds was prepared by alkylation of the corresponding thioheterocyles. Of the novel adamantylalkylthio compounds tested in the aforementioned cell line, 2-(2-adamantan-1-ylethylsulfanyl)- 4-methyl-pyrimidine was found to be the most active

scholarly journals ULTRAMETRIC RANDOM FIELD

2006 ◽  
Vol 09 (02) ◽  
pp. 199-213 ◽  
Author(s):  
A. YU. KHRENNIKOV ◽  
S. V. KOZYREV
Keyword(s):  
Wavelet Analysis ◽  
Random Field ◽  
Stochastic Equation ◽  
Gaussian Random Field ◽  
Ultrametric Space ◽  
Ultrametric Spaces

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

Thio derivatives of β-diketones and their metal chelates. XIX. Metal chelates of two new fluorinated Monothio-β-diketones and their mass spectra

1975 ◽  
Vol 28 (6) ◽  
pp. 1249 ◽  
Author(s):  
SE Livingstone ◽  
N Saha
Keyword(s):  
Mass Spectra ◽  
Metal Chelates ◽  
The Novel ◽  
Copper Chelates ◽  
Derivatives Of

The nickel(II), palladium(II), platinum(II), copper(II), zinc(II) and rhodium(III) chelates of the new fluorinated monothio-β-diketones RC(SH)=CHCOCF3 (R = Pri, Bui) and the iron(III), ruthenium(III) and cobalt(III) chelates of PriC(SH)=CHCOCF3 have been prepared. The mass spectra of all but the two copper chelates have been obtained. The novel features of the spectra are the occurrence of the ions M-(R-H), M-R, and M-LH and the loss of H2S from the ions M-LH and M-L.

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