# Estimation of Interaction Locations in Super Cryogenic Dark Matter Search Detectors Using Genetic Programming-Symbolic Regression Method

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## Abstract

**:**

## 1. Introduction

#### 1.1. Application of ML Algorithms in CDSM/SuperCDSM Experiments

#### 1.2. Definition of Novelty and the Research Hypotheses

- Can an SE be obtained using GPSR that can accurately reconstruct the locations of the interactions in SuperCDMS detectors?
- Is it possible to obtain a set of robust SEs using GPSR with randomly tested hyperparameters and validated through k-fold cross-validation that can reconstruct the locations of the interactions in the SuperCDMS with high accuracy?
- Is it possible to achieve even higher estimation accuracy in the reconstruction of the locations’ interactions by combining multiple SEs that were obtained from different GPSR executions?
- Are all input variables required as model inputs to accurately reconstruct the interaction locations?

## 2. Materials and Methods

#### 2.1. Dataset Description

- B, C, D, and F start—the time at which the signal pulse rises to 20% of the signal peak to Channel A. The variables in the original dataset are labeled as PBstart, PCstart, PDstart, and PDstart, respectively.
- A, B, C, D, and F rise—the time required for the signal to rise from 20 % to 80% of its peak. These variables in the original dataset are labeled as PArise, PBrise, PCrise, PDrise, and PFrise,
- A, B, C, D, and F width—the width of the pulse at 80% of the pulse height. The height was measured in seconds. These variables in the original dataset are labeled as PAwidth, PBwidth, PCwidth, PDwidth, and PFwidth.
- A, B, C, D, and F fall—the time required for a pulse to fall from 40% to 20 % of its peak. These variables in the original dataset are labeled as PAfall, PBfall, PCfall, PDfall, and PFfall.

#### 2.2. Research Methodology

- Dataset preparation—Checking the dataset for null values and deleting the first column (“Row”), which is not relevant for the analysis. Separating the dataset into input and output variables and applying of StandardScaler method on the input dataset variables. Dividing the dataset into training and testing datasets in a 70:30 ratio.
- GPSR algorithm—The GPSR algorithm is combined with a RHS to find the hyperparameters that yield the best-performing models. Perform training of GPSR on the training dataset using cross-validation, for the evaluation of the testing dataset.
- Result comparison—Perform a comparison of the best sets of SEs in terms of the estimation accuracy.
- Customizing solution—Combining the five best SEs to obtain a robust estimator for the reconstruction of the interaction locations.
- Final evaluation—Perform the final evaluation of the customized solution on the entire dataset.

#### 2.3. Genetic Programming-Symbolic Regression

#### 2.4. Random Hyperparameter Search

#### 2.5. Training and Evaluation Process with the GPSR Algorithm

#### 2.6. Performance Metrics and Methodology

- During the 5-CV, calculate the performance metric values on the training and validation sets.
- After the 5-CV process is complete, calculate the performance metrics (mean values obtained on the training dataset).
- Perform the last evaluation of the trained model (five SEs) on the testing dataset, and calculate the performance (evaluation) metric values (values achieved on the testing dataset),
- Calculate the mean and $\sigma $ values of the performance metrics from the obtained values on the training and testing datasets.

#### 2.7. Computational Resources

## 3. Results

#### 3.1. Results Acquired Using GPSR with RHS and 5-CV

#### 3.2. Combination of the Best SE

## 4. Discussion

## 5. Conclusions

- Using the GPSR algorithm, it was possible to obtain SEs (mathematical equations) that can estimate the interaction locations in Super Cryogenic Matter Search detectors with high accuracy.
- Using a 5-CV process, the robust system of the five SEs has a more accurate estimation when compared to the estimation of only one SE. The RHS method proved to be very useful in finding the combination of the quality hyperparameters where the highest accuracy was achieved. From all the results, the best three cases of the SEs were selected and evaluated on the test set. The highest values of $\overline{{R}^{2}}\pm \sigma \left({R}^{2}\right)$, $\overline{MAE}\pm \sigma \left(MAE\right)$, $\overline{RMSE}\pm \sigma \left(RMSE\right)$ were achieved in Case 1 and are equal to $0.9876\pm 0.0048$, $1.0096\pm 0.1215$, $1.5506\pm 0.3050$.
- From the obtained results, three cases were selected based on the final mean ${R}^{2}$ score. From these cases, the SEs that achieved the highest estimation performance were selected as the main elements of the custom set of SEs. The results of the customized solution obtained on the entire dataset were equal to $0.991087\pm 0.001291$, $0.878362\pm 0.063662$, and $1.326449\pm 0.100901$, respectively. The final evaluation of these equations on the entire dataset showed that this system had slightly better performance when compared to Cases 1, 2, and 3.
- Unfortunately, the custom set of SEs required all 19 input variables to compute the output. However, if only Equation (7) was used, the highest estimation accuracy could be achieved, and to compute the output, only seven input variables were required.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Description of Mathematical Functions in SEs

- Square root:$${y}_{sqrt}\left(x\right)=\sqrt{\left|x\right|},$$
- Natural logarithm:$${y}_{log}\left(x\right)=\left(\right)open="\{"\; close>\begin{array}{cc}log\left(\right|x\left|\right)\hfill & \mathrm{if}\left|x\right|0.001\hfill \\ 0\hfill & \mathrm{if}\left|x\right|0.001\hfill \end{array}$$
- Division:$${y}_{div}({x}_{1},{x}_{2})=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{{x}_{1}}{{x}_{2}}\hfill & \mathrm{if}|{x}_{2}|0.001\hfill \\ 1\hfill & \mathrm{if}|{x}_{2}|0.001\hfill \end{array}$$

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**Figure 2.**Top view showing detector regions (A, B, C, D, E, and F) and interaction locations indicated with the “x” symbol in orange color.

**Figure 6.**The mean values of the obtained performance metric during GPSR 5-CV. The $\sigma $ values are presented as error bars.

**Figure 7.**The mean values of the performance metric obtained on the testing dataset. The $\sigma $ values are presented as error bars.

**Figure 8.**The graphical representation of the performance metric values obtained with the best combination of SEs on the entire dataset. $\sigma $ is shown in the case of the total mean values in the form of an error bar.

**Table 1.**The measured coordinates of the interaction locations shown in Figure 2.

Location | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Coordinate x | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Coordinate y | 0 | −3.96 | −9.98 | −12.502 | −17.992 | −19.7 | −21.034 | −24.077 | −29.5 | −36.116 | −39.4 | −41.01 | −41.9 |

**Table 2.**The results of the statistical investigation of dataset variables including the GSPR variable representation.

Dataset Variable | Data Points | Mean Value | $\mathit{\sigma}$ | Minimum Value | Maximum Value | GPSR Variable Symbol |
---|---|---|---|---|---|---|

PBstart | 7151 | $-2.484\times {10}^{-6}$ | $5.346\times {10}^{-6}$ | $-1.800\times {10}^{-5}$ | $1.613\times {10}^{-5}$ | ${X}_{0}$ |

PCstart | $2.491\times {10}^{-5}$ | $2.596\times {10}^{-5}$ | $-2.160\times {10}^{-5}$ | $6.359\times {10}^{-5}$ | ${X}_{1}$ | |

PDstart | $-1.316\times {10}^{-5}$ | $1.014\times {10}^{-5}$ | $-2.990\times {10}^{-5}$ | $5.149\times {10}^{-5}$ | ${X}_{2}$ | |

PFstart | $-5.240\times {10}^{-6}$ | $2.580\times {10}^{-5}$ | $-3.520\times {10}^{-5}$ | $4.211\times {10}^{-5}$ | ${X}_{3}$ | |

PArise | $1.710\times {10}^{-5}$ | $4.501\times {10}^{-6}$ | $8.464\times {10}^{-6}$ | $2.682\times {10}^{-5}$ | ${X}_{4}$ | |

PBrise | $1.973\times {10}^{-5}$ | $3.843\times {10}^{-6}$ | $9.338\times {10}^{-6}$ | $3.221\times {10}^{-5}$ | ${X}_{5}$ | |

PCrise | $3.012\times {10}^{-5}$ | $7.718\times {10}^{-6}$ | $1.157\times {10}^{-5}$ | $5.354\times {10}^{-5}$ | ${X}_{6}$ | |

PDrise | $1.298\times {10}^{-5}$ | $2.078\times {10}^{-6}$ | $8.705\times {10}^{-6}$ | $3.444\times {10}^{-5}$ | ${X}_{7}$ | |

PFrise | $1.675\times {10}^{-5}$ | $7.026\times {10}^{-6}$ | $8.060\times {10}^{-6}$ | $3.158\times {10}^{-5}$ | ${X}_{8}$ | |

PAfall | $2.148\times {10}^{-5}$ | $1.826\times {10}^{-5}$ | $1.521\times {10}^{-5}$ | $6.814\times {10}^{-5}$ | ${X}_{9}$ | |

PBfall | $2.516\times {10}^{-5}$ | $3.781\times {10}^{-5}$ | $1.596\times {10}^{-5}$ | $9.416\times {10}^{-5}$ | ${X}_{10}$ | |

PCfall | $2.712\times {10}^{-5}$ | $4.878\times {10}^{-5}$ | $1.388\times {10}^{-5}$ | $9.884\times {10}^{-5}$ | ${X}_{11}$ | |

PDfall | $2.588\times {10}^{-5}$ | $4.621\times {10}^{-5}$ | $1.415\times {10}^{-5}$ | $9.806\times {10}^{-5}$ | ${X}_{12}$ | |

PFfall | $2.472\times {10}^{-5}$ | $5.361\times {10}^{-5}$ | $1.163\times {10}^{-5}$ | $9.355\times {10}^{-5}$ | ${X}_{13}$ | |

PAwidth | $1.696\times {10}^{-5}$ | $3.659\times {10}^{-5}$ | $7.105\times {10}^{-5}$ | $2.271\times {10}^{-5}$ | ${X}_{14}$ | |

PBwidth | $2.071\times {10}^{-5}$ | $3.343\times {10}^{-5}$ | $7.133\times {10}^{-5}$ | $2.983\times {10}^{-5}$ | ${X}_{15}$ | |

PCwidth | $2.467\times {10}^{-5}$ | $2.453\times {10}^{-5}$ | $1.648\times {10}^{-5}$ | $3.443\times {10}^{-5}$ | ${X}_{16}$ | |

PDwidth | $1.494\times {10}^{-5}$ | $5.623\times {10}^{-5}$ | $3.603\times {10}^{-5}$ | $3.033\times {10}^{-5}$ | ${X}_{17}$ | |

PFwidth | $1.765\times {10}^{-5}$ | $7.636\times {10}^{-5}$ | $3.321\times {10}^{-5}$ | $2.665\times {10}^{-5}$ | ${X}_{18}$ | |

y | −21.758 | 14.091 | −41.9 | 0 | y |

Hyperparameter Name | Range |
---|---|

population_size | 1000–2000 |

number of generations | 100–200 |

tournament_selection | 10–500 |

init_depth | 3–15 |

crossover | 0.001–1.0 |

subtree_mutation | 0.001–1.0 |

hoist_mutation | 0.001–1.0 |

point_mutation | 0.001–1.0 |

stopping_criteria | 0–$1\times {10}^{-8}$ |

maximum_samples | 0.99–1 |

constant_range | −10,000–10,000 |

parsimony_coefficient | 0–$1\times {10}^{-4}$ |

**Table 4.**The combination of the GPSR hyperparameter values for which the highest estimation accuracy was achieved.

Case No. | GPSR Hyperparameters (Population Size, Number of Generations, Tournament Selection, Init_Depth, Crossover, Subtree_Mutation, Hoist_Mutation, Point_Mutation, Stopping_Criteria, Max_Samples, Constant_Range, Parsimony_Coefficient) |
---|---|

1 | 1978, 233, 254, (6, 15), 0.043, 0.22, 0.22, 0.43, $5.49\times {10}^{-7}$, 0.99, (−9793.08, 1402.72), $9.15\times {10}^{-6}$ |

2 | 1075, 196, 461, (3, 15), 0.41, 0.4, 0.05, 0.097, $9.95\times {10}^{-7}$, 0.99, (−8821.29, 3713.89), $9.2\times {10}^{-4}$ |

3 | 1500, 171, 108,(7, 11), 0.45, 0.469, 0.043, 0.0071, $8.17\times {10}^{-7}$, 0.99, (−4076.61, 4272.87), $7.16\times {10}^{-4}$ |

**Table 5.**The mean and $\sigma $ values of ${R}^{2}$, $MAE$, and $RMSE$ were achieved in a 5-fold cross-validation process.

Performance metric | Split 1 | Split 2 | Split 3 | Split 4 | Split 5 | Total | |

${\overline{R}}^{2}$ | 0.9864 | 0.9903 | 0.9821 | 0.9935 | 0.9909 | 0.9886 | |

$\sigma \left({R}^{2}\right)$ | 0.0009 | 0.0003 | 0.0013 | 0.0002 | 0.0001 | 0.0006 | |

Case 1 | $\overline{MAE}$ | 1.0599 | 0.9837 | 1.1479 | 0.8113 | 0.9342 | 0.9874 |

$\sigma \left(MAE\right)$ | 0.0057 | 0.0291 | 0.0354 | 0.0035 | 0.0003 | 0.0148 | |

$\overline{RMSE}$ | 1.6494 | 1.3905 | 1.8818 | 1.1445 | 1.3456 | 1.4824 | |

$\sigma \left(RMSE\right)$ | 0.0501 | 0.0222 | 0.0542 | 0.0230 | 0.0057 | 0.0311 | |

Performance metric | Split 1 | Split 2 | Split 3 | Split 4 | Split 5 | Total | |

${\overline{R}}^{2}$ | 0.9889 | 0.9905 | 0.9896 | 0.9835 | 0.9860 | 0.9877 | |

$\sigma \left({R}^{2}\right)$ | 0.0001 | 0.0006 | 0.0001 | 0.0001 | 0.0009 | 0.0004 | |

Case 2 | $\overline{MAE}$ | 1.0304 | 0.8973 | 0.9593 | 1.1713 | 1.0530 | 1.0223 |

$\sigma \left(MAE\right)$ | 0.0233 | 0.0115 | 0.0108 | 0.0006 | 0.0238 | 0.0140 | |

$\overline{RMSE}$ | 1.5057 | 1.3883 | 1.4369 | 1.7984 | 1.6564 | 1.5571 | |

$\sigma \left(RMSE\right)$ | 0.0331 | 0.0562 | 0.0062 | 0.0218 | 0.0772 | 0.0389 | |

Performance metric | Split 1 | Split 2 | Split 3 | Split 4 | Split 5 | Total | |

${\overline{R}}^{2}$ | 0.9899 | 0.9856 | 0.9896 | 0.9846 | 0.9868 | 0.9873 | |

$\sigma \left({R}^{2}\right)$ | 0.0004 | 0.0019 | 0.0001 | 0.0001 | 0.0013 | 0.0008 | |

Case 3 | $\overline{MAE}$ | 0.8020 | 1.0330 | 1.0266 | 1.2512 | 1.0365 | 1.0299 |

$\sigma \left(MAE\right)$ | 0.0041 | 0.0302 | 0.0170 | 0.0128 | 0.0328 | 0.0194 | |

$\overline{RMSE}$ | 1.4336 | 1.7058 | 1.4429 | 1.7375 | 1.6111 | 1.5862 | |

$\sigma \left(RMSE\right)$ | 0.0489 | 0.1311 | 0.0051 | 0.0170 | 0.1016 | 0.0608 |

Case No. | Symbolic Expression No. | Length | Depth | Average Length | Average Depth |
---|---|---|---|---|---|

1 | 220 | 33 | |||

2 | 257 | 41 | |||

1 | 3 | 106 | 13 | 206.6 | 28.4 |

4 | 351 | 36 | |||

5 | 99 | 19 | |||

1 | 75 | 17 | |||

2 | 87 | 16 | |||

2 | 3 | 109 | 18 | 81.4 | 16.2 |

4 | 58 | 15 | |||

5 | 78 | 15 | |||

1 | 145 | 31 | |||

2 | 156 | 19 | |||

3 | 3 | 127 | 27 | 161.6 | 24.8 |

4 | 230 | 17 | |||

5 | 150 | 30 |

Case Number | Evaluation Metric | Symbolic Expressions | Mean | $\mathit{\sigma}$ | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||

${R}^{2}$ | 0.9827 | 0.9892 | 0.9811 | 0.9934 | 0.9905 | 0.9876 | 0.0048 | |

1 | $MAE$ | 1.1180 | 0.9919 | 1.1645 | 0.8225 | 0.9619 | 1.0096 | 0.1215 |

$RMSE$ | 1.8649 | 1.4646 | 1.9488 | 1.1537 | 1.3810 | 1.5506 | 0.3050 | |

${R}^{2}$ | 0.9878 | 0.9896 | 0.9887 | 0.9836 | 0.9837 | 0.9867 | 0.0025 | |

2 | $MAE$ | 1.0615 | 0.9393 | 0.9894 | 1.1815 | 1.1450 | 1.0634 | 0.0911 |

$RMSE$ | 1.5664 | 1.4490 | 1.5057 | 1.8173 | 1.8116 | 1.6300 | 0.1551 | |

${R}^{2}$ | 0.9909 | 0.9865 | 0.9889 | 0.9828 | 0.9873 | 0.9873 | 0.0027 | |

3 | $MAE$ | 0.8032 | 1.0518 | 1.0513 | 1.3162 | 1.0666 | 1.0578 | 0.1623 |

$RMSE$ | 1.3529 | 1.6464 | 1.4958 | 1.8588 | 1.5978 | 1.5903 | 0.1677 |

**Table 8.**The performance metric values achieved with the best combination of SEs on the entire dataset.

Evaluation Metric | ${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | ${\mathit{y}}_{4}$ | ${\mathit{y}}_{5}$ | Mean | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|

${R}^{2}$ | 0.99022 | 0.990905 | 0.989798 | 0.993511 | 0.991003 | 0.991087 | 0.001291 |

$MAE$ | 0.795499 | 0.889602 | 0.954781 | 0.815055 | 0.936876 | 0.878362 | 0.063662 |

$RMSE$ | 1.393535 | 1.343819 | 1.423247 | 1.135059 | 1.336586 | 1.326449 | 0.100901 |

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**MDPI and ACS Style**

Anđelić, N.; Baressi Šegota, S.; Glučina, M.; Car, Z.
Estimation of Interaction Locations in Super Cryogenic Dark Matter Search Detectors Using Genetic Programming-Symbolic Regression Method. *Appl. Sci.* **2023**, *13*, 2059.
https://doi.org/10.3390/app13042059

**AMA Style**

Anđelić N, Baressi Šegota S, Glučina M, Car Z.
Estimation of Interaction Locations in Super Cryogenic Dark Matter Search Detectors Using Genetic Programming-Symbolic Regression Method. *Applied Sciences*. 2023; 13(4):2059.
https://doi.org/10.3390/app13042059

**Chicago/Turabian Style**

Anđelić, Nikola, Sandi Baressi Šegota, Matko Glučina, and Zlatan Car.
2023. "Estimation of Interaction Locations in Super Cryogenic Dark Matter Search Detectors Using Genetic Programming-Symbolic Regression Method" *Applied Sciences* 13, no. 4: 2059.
https://doi.org/10.3390/app13042059