# Performance Test of QU-Fitting

## Abstract

**:**

## 1. Introduction

## 2. Model and Calculation

#### 2.1. RM Synthesis

#### 2.2. QU-Fitting

#### 2.3. Method and Models

- Gap: ${\varphi}_{2}-{\varphi}_{1}=$ 0.5 (g1), 1.0 (g2), 2.0 (g3), 5.0 (g4), and 10.0 (g5) in units of FWHM of the RMSF (${\varphi}_{1}$ is fixed).
- Width: ${\sigma}_{1}$ = ${\sigma}_{2}$ = 0.25 (w1) and ${\sigma}_{1}$ = 0.5, and ${\sigma}_{2}$ = 0.25 (w2) in units of FWHM of the RMSF.

## 3. Results

- Criterion (i), convergence of MCMC: This criterion checks whether all of the model parameters with the G2 fitting model satisfy Geweke’s diagnostics. Note that in the failure case, the MCMC reaches the maximum regulation number (100,000).
- Criterion (ii), calculated chi-squared values ${\chi}^{2}$: We check whether ${\chi}^{2}$ for the G2 fitting model is smaller than 2389 (within 3$\sigma $ of the chi-squared distribution with $n=2200$ degrees of freedom); in other words, this criterion checks whether the fitting model matches the mock data spectrum.
- Criterion (iii), model selection: This criterion checks whether AIC/BIC can select the G2 fitting model (correct fitting model).
- Criterion (iv), parameter estimation: We check whether all eight true parameters for the G2 fitting model (${\varphi}_{i}$, ${f}_{i}$, ${\chi}_{0,i}$, ${\sigma}_{i}$, i = 1, 2) are within 3$\sigma $ of their calculated $L\left(\theta \right)$. We regard parameter estimation to be not successful if any one of the eight correct parameters are out of 3$\sigma $ of their $L\left(\theta \right)$.

- Class (A): All four criteria satisfy the success line.
- Class (B): Criteria (i), (ii), and (iii) are satisfied, but Criterion (iv) does not satisfy the success line.
- Class (C): Criteria (i) and (ii) are satisfied, but Criteria (iii) and (iv) do not satisfy the success line.
- Class (D): Criteria (i) is satisfied, but Criteria (ii), (iii), and (iv) do not satisfy the success line.
- Class (E): None of the four criteria satisfy the success line.

## 4. Discussion

#### 4.1. Success Cases within an FWHM Structure

#### 4.2. Failure Case of the Large Gap Model

#### 4.3. QU-Fitting Simulation Using the Delta Function Fitting Model

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RM | Rotation Measure |

FDF | Faraday Dispersion Function |

LOS | Line Of Sight |

MCMC | Markov Chain Monte Carlo |

AIC | Akaike Information Criterion |

BIC | Bayesian Information Criterion |

RMSF | Rotation Measure Spread Function |

FWHM | Full Width at Half Maximum |

POSSUM | Polarization Sky Survey of the Universe’s Magnetism |

ASKAP | Australian SKA Pathfinder |

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**Figure 1.**Categorization into Classes (A)~(E) of 100 simulation results for the Width 1 (w1) (left) and w2 (right) source models, respectively.

**Figure 2.**The reconstructed FDF by QU-fitting (blue line) of the w2g1 (g1, Gap 1) source model, cleaned FDF (red line) constructed by RM CLEAN using the same simulation data of QU-fitting, and the source model FDF (black line).

**Figure 3.**Correlation of the estimated best-fit Faraday depths of the w2g4 source model in 100 realizations. Color indicates the chi-square values calculated by Equation (11).

**Figure 4.**The reconstructed FDF by QU-fitting (blue line) of the w2g4 source model, cleaned FDF (red line) constructed by RM CLEAN using the same simulation data of QU-fitting, and the source model FDF (black line).

**Figure 5.**The reconstructed FDF by the Delta 1 (d1) (red) and d2 (blue) fitting models for the w1g1 (left) and w1g5 (right) source models, respectively. The black lines show the source model FDF.

**Figure 6.**Estimated Stokes Q (top) and U (bottom) spectra by the d1 (red) and d2 (blue) fitting models for the w1g1 (left) and w1g5 (right) source models, respectively. These black lines show the source model Stokes Q,U spectra.

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**MDPI and ACS Style**

Miyashita, Y. Performance Test of QU-Fitting. *Galaxies* **2019**, *7*, 69.
https://doi.org/10.3390/galaxies7030069

**AMA Style**

Miyashita Y. Performance Test of QU-Fitting. *Galaxies*. 2019; 7(3):69.
https://doi.org/10.3390/galaxies7030069

**Chicago/Turabian Style**

Miyashita, Yoshimitsu. 2019. "Performance Test of QU-Fitting" *Galaxies* 7, no. 3: 69.
https://doi.org/10.3390/galaxies7030069