# Markov Chain Monte Carlo Used in Parameter Inference of Magnetic Resonance Spectra

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Intoduction

**Figure 2.**(

**a**) The simulated signal for the optimum parameter set with the addition of noise (red) and the noiseless lineshape (blue). The red signal corresponds to a signal to noise ratio of 7. (

**b**) The simulated signal for the optimum parameter set with the addition of noise (red) and the noiseless lineshape (blue). The red signal corresponds to a signal to noise ratio of 70.

_{k}/n, where N

_{k}is the normal noise generated in MATLAB (version R2013a), which has a width = 1 which is then scaled by n to obtain the desired SNR), {X} represents a set of parameters (in this case α and β), our signal ${D}_{k}={F}_{k}\left(\left\{{X}_{o}\right\}\right)+\raisebox{1ex}{${N}_{k}$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$, and the k subscript indicates a specific observation frequency in the bandwidth of the spectrum. Here, {X

_{o}} is the optimum parameter set, {α

_{o}, β

_{o}}.

_{k}. We can normalize Equation (4) by constructing a probability mass function whose evidence (or partition function) may be defined by using Equations (4) and (6):

_{i}} is the i-th set of parameters being evaluated, {α

_{i}, β

_{i}}.

_{0}}.

**Figure 4.**The simulated signal for the optimum parameter set with SNR = 70 in red and the model lineshape evaluated at α = 5.2, β = 1.49 in blue.

## 2. Methods

#### 2.1. MCMC Simulation

#### 2.2. Computation of Parameter Uncertainties

^{m}, X

^{n}, where X

^{m}, X

^{n}are chosen from the set {α, β}. In this two parameter model, one obtains a 2 × 2 matrix. The (1,1) component of this matrix is the second partial derivative of the residual (9) with respect to parameter 1 (in this case derivatives with respect to α); off diagonal components are partial derivatives to α and β.

## 3. Results

#### MCMC Simulation Results

**Figure 7.**(

**a**) The partition function (7) at SNR = 7 for α values from 4.5 to 6.5 and β values from 0.5 to 1.5. (

**b**) The partition function (7) at SNR = 70 for α values from 4.5 to 6.5 and β from 0.5 to 1.5. For both plots, the red vertical line indicates the optimum parameter set.

**Figure 8.**(

**a**) Results of the algorithm described in this work for SNR = 7 yield α = 5.4067 ± 0.26G. (

**b**) Results of the algorithm described in this work for SNR = 7 yield β = 1.5099 ± 0.77G.

**Figure 9.**(

**a**) Results of the algorithm described in this work for SNR = 70 yield α = 5.3996 ± 2.09 × 10

^{−4}G. (

**b**) Results of the algorithm described in this work for SNR = 70 yield β = 1.5391 ± 1.4 × 10

^{−3}G.

**Figure 11.**(

**a**) A scan comparing the Nelder–Mead optimization algorithm (red) with the MCMC algorithm discussed in this paper (blue) for various starting parameters of alpha and beta and the final value for alpha. (

**b**) A scan comparing the Nelder-Mead optimization algorithm (red) with the MCMC algorithm discussed in this paper (blue) for various starting parameters of alpha and beta and the final value for beta.

**Figure 12.**(

**a**) Gaussian curvature K as a function of model parameters for SNR = 7. (

**b**) Gaussian curvature K as a function of model parameters for SNR = 70.

^{−3}for $\alpha $, and 7.8×10

^{−3}for $\beta $, while the curvature corrected variances are found to be 2.8×10

^{−3}for $\alpha $, and 7.9×10

^{−3}for $\beta $. Note that the curvature corrections are more significant in the low SNR case than the high SNR case.

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## References

- Bretthorst, G.L. Bayesian Analysis I. Parameter Estimation Using Quadrature NMR Models. J. Magn. Reson.
**1990**, 88, 533–551. [Google Scholar] [CrossRef] - Bretthorst, G.L. Bayesian Analysis II. Parameter Estimation Using Quadrature NMR Models. J. Magn. Reson.
**1990**, 88, 552–570. [Google Scholar] - Bretthorst, G.L. Bayesian Analysis III. Parameter Estimation Using Quadrature NMR Models. J. Magn. Reson.
**1990**, 88, 571–595. [Google Scholar] - Berliner, L.J.; Eaton, S.S.; Eaton, G.R. (Eds.) Distance Measurements in Biological Systems by EPR. In Biological Magnetic Resonance; Kluwer Academic/Plenum Publishers: New York, NY, USA, 2000.
- Pake, G.E. Nuclear Resonance Absorption in Hydrated Crystals: Fine Structure of the Proton Line. J. Chem. Phys.
**1948**, 16, 327–336. [Google Scholar] [CrossRef] - Abragam, A. The Principles of Nuclear Magnetism; Oxford University Press: London, UK, 1961. [Google Scholar]
- Earle, K.A.; Mainali, L.; Dev Sahu, I.; Schneider, D.J. Magnetic Resonance Spectra and Statistical Geometry. Appl. Magn. Reson.
**2010**, 37, 865–880. [Google Scholar] [CrossRef] [PubMed] - Sivia, D.S.; Skilling, J. Data Analysis: A Bayesian Tutorial; Oxford University Press: New York, NY, USA, 2010. [Google Scholar]
- Ben-Naim, A. A Farewell to Entropy: Statistical Thermodynamics Based on Information; World Scientific Publishing: Singapore, 2011. [Google Scholar]
- Hock, K.; Earle, K. Information Theory Applied to Parameter Inference of Pake Doublet Spectra. Appl. Magn. Reson.
**2014**, 45, 859–879. [Google Scholar] [CrossRef] - Rao, C.R. Information and the Accuracy Attainable in the Estimation of Statistical Parameters. In Breakthroughs in Statistics; Springer: New York, NY, USA, 1992. [Google Scholar]
- Lee, J.M. Riemannian Manifolds: An Introduction to Curvature; Springer: New York, NY, USA, 1997. [Google Scholar]
- Lagaris, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E. Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM J. Optim.
**1998**, 9, 112–147. [Google Scholar] [CrossRef] - Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity; Wiley: New York, NY, USA, 1972. [Google Scholar]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hock, K.; Earle, K.
Markov Chain Monte Carlo Used in Parameter Inference of Magnetic Resonance Spectra. *Entropy* **2016**, *18*, 57.
https://doi.org/10.3390/e18020057

**AMA Style**

Hock K, Earle K.
Markov Chain Monte Carlo Used in Parameter Inference of Magnetic Resonance Spectra. *Entropy*. 2016; 18(2):57.
https://doi.org/10.3390/e18020057

**Chicago/Turabian Style**

Hock, Kiel, and Keith Earle.
2016. "Markov Chain Monte Carlo Used in Parameter Inference of Magnetic Resonance Spectra" *Entropy* 18, no. 2: 57.
https://doi.org/10.3390/e18020057