Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework
Abstract
:1. Introduction
2. Separating Spaces in Bayesian Context
3. The Bayesian Recipe
The Choice of Prior Probability Distributions
4. Numerical Simulation
5. Results
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
Algorithm A1 DRAM 

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Data (xspace)  $\hspace{1em}\hspace{1em}\hspace{1em}\overrightarrow{\mathit{t}},\phantom{\rule{4.pt}{0ex}}\overrightarrow{\mathit{x}}$ 
Parameters (vspace)  ${\overrightarrow{f}}_{v}$, ${\overrightarrow{\lambda}}_{v}$, ${\overrightarrow{\xi}}_{v},\phantom{\rule{4.pt}{0ex}}{E}_{v}$ 
Parameter  Proposal Distribution  Prior Distribution 

${f}_{{v}_{i}}$ for $i=1,\dots ,{E}_{v}$  Gaussian (${\mu}_{{v}_{i}},\phantom{\rule{4.pt}{0ex}}{\sigma}_{{v}_{i}}^{2}$) $\in (L{B}_{{v}_{i}},\phantom{\rule{4.pt}{0ex}}U{B}_{{v}_{i}})$  ${\mu}_{{\theta}_{{v}_{i}}},\phantom{\rule{4.pt}{0ex}}{\sigma}_{{\theta}_{{v}_{i}}}$ 
${\lambda}_{{v}_{i}}$ for $i=1,\dots ,{E}_{v}1$  Gaussian (${\mu}_{{\lambda}_{i}},\phantom{\rule{4.pt}{0ex}}{\sigma}_{{\lambda}_{i}}^{2}$) $\in (L{B}_{{\lambda}_{i}},\phantom{\rule{4.pt}{0ex}}U{B}_{{\lambda}_{i}})$  ${\mu}_{{\theta}_{{\lambda}_{i}}},\phantom{\rule{4.pt}{0ex}}{\sigma}_{{\theta}_{{\lambda}_{i}}}$ 
${\left\right\mathit{e}\left\right}^{2}$  

Noise Level  vFit ($\mathit{v}$Space)  Derivative of xFit ($\mathit{x}$Space) 
$0.001$  $0.2077$  $14.4107$ 
$0.3$  $1.9679$  $15.0792$ 
$0.7$  $3.1793$  $20.9974$ 
1  $3.1569$  $30.1122$ 
$1.3$  $3.8159$  $31.2719$ 
SSE with Noisy Data  SSE with True Data  

${\mathbf{\sigma}}_{\mathbf{e}}$  xFit ($\mathbb{U}$)  xFit ($\mathbb{V}$)  xFit ($\mathbb{U}$)  xFit ($\mathbb{V}$) 
$0.001$  $1.1743$  $0.6082$  $1.1753$  $0.6089$ 
$0.3$  $12.4677$  $12.6049$  $2.1946$  $0.8541$ 
$0.7$  $62.4582$  $63.4059$  $4.1855$  $0.8176$ 
$1.0$  $121.0864$  $120.8161$  $13.3478$  $1.1253$ 
$1.3$  $163.1339$  $264.3176$  $18.9582$  $3.1651$ 
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De Silva, K.; Cafaro, C.; Giffin, A. Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework. Entropy 2021, 23, 674. https://doi.org/10.3390/e23060674
De Silva K, Cafaro C, Giffin A. Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework. Entropy. 2021; 23(6):674. https://doi.org/10.3390/e23060674
Chicago/Turabian StyleDe Silva, Kushani, Carlo Cafaro, and Adom Giffin. 2021. "Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework" Entropy 23, no. 6: 674. https://doi.org/10.3390/e23060674