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Table A.1.
Description of the mathematical notations for the variables used in the Bayesian optimisation algorithm, the RSM detection map and Auto-RSM optimisation framework.
| Symbol | Dimension | Comments |
|---|---|---|
| Bayesian optimisation | ||
| 𝒪1 : t | t | Set of observations of the loss function |
| t | 1 | Number of tested sets of parameters |
| p1 : t | np × t | Tested sets of parameters |
| f(p) | 1 | Loss function evaluated with the set of parameters p |
| 𝒢𝒫(m(p1 : t),K) | 1 | Gaussian process returning the mean and variance of a Gaussian distribution |
| over the possible values of f at p | ||
| m(p) | 1 | Mean of the Gaussian distribution of the loss function at p |
| K | 1 | Covariance function of the tested set of parameters p1 : t |
| μ(pt + 1) | 1 | Mean of the Gaussian posterior distribution at pt + 1 |
| σ(pt + 1) | 1 | Variance of the Gaussian posterior distribution at pt + 1 |
 |
1 | Set of parameters providing the current known maximum value of the |
loss function  |
||
| RSM map | ||
| xia | θ × θ × TLa | Patch of residuals centred on pixel ia |
| Fia | TLa | Realisation of a two-state Markov chain representing the state of |
| the system for pixel ia | ||
| m | θ × θ | Cropped planetary signal (off-axis PSF) |
| εs, ia | 2θ × θ × TLa | Error terms associated with the two regimes |
| Sia | TLa | State of the system for every pixel ia |
| ξs, ia | 2 × TLa | Probability associated with state s for every pixel ia |
| ηs, ia | 2 × TLa | Likelihood of being in each state for every pixel ia |
| pq, s | 2 × 2 | Transition probabilities between the regimes |
| μ | 1 | Mean of the residuals contained in an annulus a, with width equal to θ |
| σ | 1 | Standard deviation of the residuals contained in an annulus a, with width |
| equal to θ | ||
| β | 1 | Intensity of the planetary signal in the cube of residuals |
| a | 1 | Annulus index |
| La | 1 | Number of pixels included in the annulus a |
| T | 1 | Number of frames in the cube of residuals |
| ia | 1 | Index associated with every pixel from every frame in the annulus a |
| (ranges from 1 to TLa) | ||
| θ | 1 | Angular size of the considered planetary signal ( set to 1λ/D) |
| δ | 1 | Multiplicative factor of the noise standard deviation providing the |
| intensity parameter β | ||
| Auto-RSM | ||
| i | Npix | Vectorised science image/annulus before speckle subtraction |
| m | Npix | Vectorised planetary signal (off-axis PSF) inside the selected annulus |
| amax | 1 | Largest angular separation considered for the detection map computation |
| Ca, m, c | La × Ntechnique × Nsequence | Average contrast obtained with the optimal set of parameters |
| Ma, m, c | La × Ntechnique × Nsequence | Position of the median contrast aperture within the selected annulus |
 |
La × Ntechnique × Nsequence | Optimal set of parameters for the reference PSF computation |
 |
La × Ntechnique × Nsequence | Optimal set of parameters for the reference PSF smoothed via moving average |
 |
La × Ntechnique × Nsequence | Optimal set of parameters for the computation of the RSM map |
 |
La × Ntechnique × Nsequence | Optimal set of parameters for the computation of the RSM map |
| interpolated via RBF | ||
| Ya | La | Set of likelihood time series available (one per couple of ADI sequence |
| and PSF-subtraction technique) for the computation of the RSM map | ||
 |
La × Ntechnique × Nsequence | Likelihood time series |
 |
La × T | Likelihood time series maximising the RSM performance index at a |
| given iteration | ||
| Za | La | Selected likelihood time series for the computation of the final detection map |
| Ta | La | Annulus-wise thresholds computed with flipped parallactic angles |
 |
La | Smoothed annulus-wise thresholds computed with flipped parallactic angles |
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