# Interpretable Machine Learning for Assessing the Cumulative Damage of a Reinforced Concrete Frame Induced by Seismic Sequences

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## Abstract

**:**

## 1. Introduction

## 2. Feature Selection and Dataset Configuration and Preprocessing

#### 2.1. Ground Motion IMs and Damage Index

#### 2.1.1. Ground Motion IMs

#### 2.1.2. Damage Index

#### 2.2. Description and Modelling of the Examined Building

#### 2.3. Dataset Creation

#### 2.4. Data Preprocessing (Feature Scaling)

## 3. Machine Learning Methods, Hyperparameter Tuning and Interpretation Techniques

#### 3.1. Linear Models

**ℓ**1 regularization term to the cost function, leading to sparse solutions and potential feature selection but can aggressively shrink coefficients. Ridge adds an

**ℓ**2 regularization, causing coefficients to approach zero, offering stability and overfitting prevention. Elastic Net combines Lasso and Ridge, introducing both

**ℓ**1 and

**ℓ**2 penalties to the cost function ($\mathcal{J}$) (Equation (7)), controlled by a mixing parameter and regularization parameter. Adjusting these parameters can optimize the model balance.

#### 3.2. Non-Parametric Algorithms

#### 3.3. Ensemble Trees

#### 3.4. Feedforward Neural Networks

**ℓ**) performs computations on activations (${\mathrm{a}}_{\mathrm{j}}^{\mathbf{\ell}-1}$) received from the previous layer ($\mathbf{\ell}-1$) and passes them to the subsequent layer. The value of each neuron is estimated applying a non-linear activation function on the linear combination of previous layer units activations. The coefficients of the linear combination are divided into two tensors: (${W}_{\mathrm{i}\mathrm{j}}^{\mathbf{\ell}}$) and biases (${\mathrm{b}}_{\mathrm{i}}^{\mathbf{\ell}}$). The components of these tensors are the trainable parameters of the MLP. Propagation of information (Equation (9)), through forward succession, maps input features to the output target. During training, the partial derivatives with respect to trainable parameters of the cost function ($\mathcal{J}$) are estimated according to Equation (10), known as the back-propagation process [128,129]. The trainable parameters are updated in each step according to the gradient descent optimization algorithm (Equation (11)) to minimize losses. MLP complexity is affected by the number of hidden layers, units and regularization parameter ($\lambda $) (Equation (12)).

**ℓ**, m is the number neurons of layer $\mathbf{\ell}-1$, and $\lambda $ is the regularization parameter.

#### 3.5. Hyperparameter Tuning and K-Fold Cross-Validation

#### 3.6. Interpretation Methods

#### 3.6.1. Global Interpretation Methods

#### 3.6.2. Local Interpretation Methods

## 4. Results and Discussion

#### 4.1. Hyperparameter Tuning and ML Models Comparison

#### 4.2. Interpretation of the Best ML Model

#### 4.2.1. Features Importances

#### 4.2.2. Local Explanation Methods (LIME, SHAP)

#### 4.2.3. Global Explanation Methods (PDP and ALE)

## 5. Conclusions

- The most efficient model for predicting final structural damage under seismic sequences was an instance of the LightGBM method with an ${\mathrm{R}}^{2}$ greater than 0.95, while the method with the poorest performance was KNN, with an ${\mathrm{R}}^{2}$ value of approximately 0.4.
- Among the examined boosted trees, LightGBM and XGBoost demonstrated the most optimized and robust performance even against small changes in their hyperparameters. Moreover, they present great resistance to overfitting as the number of trees increases.
- In the case of Multi-Layer Perceptrons (MLPs), the ReLU activation function appeared to yield better performance, followed by the Tanh activation function. In addition, the MLP model presents slightly better bias-variance balance than the other advanced ML models.
- All the interpretation methods identified the initial damage $\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A},1\mathrm{s}\mathrm{t}}$ as the most significant feature followed by the IMs of the subsequent seismic shock. However, the ranking of the IMs importance is varying between the adopted approaches. The majority of interpretation methods indicate the ${\mathrm{I}}_{\mathrm{F}\mathrm{V}\mathrm{F}}$ as the most important IM, except for the impurity-based explanation, which identified $\mathrm{S}{\mathrm{I}}_{\mathrm{H}}$. As the second most important IM, LIME and SHAP ranked $\mathrm{S}{\mathrm{I}}_{\mathrm{H}}$, although permutation ranked $\frac{\mathrm{P}\mathrm{G}\mathrm{A}}{\mathrm{P}\mathrm{G}\mathrm{V}}$, and impurity ranked ${\mathrm{I}}_{\mathrm{A}\mathrm{S}}$.
- In case of examining the effect of all the IMs in total, both LIME and SHAP local explanation methods show that the contribution of the subsequent ground motion is larger than that of initial damage $\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A},1\mathrm{s}\mathrm{t}}$. In general, the effect of the initial damage tends to increase as the final increases. However, they differ in their estimation of contributions for higher damage states.
- The analysis of PDPs and ALE reveals key insights into the effects of damage predictors on the final damage. The pre-existing damage demonstrates a positive influence across the entire range of cumulative damage. Additionally, ${\mathrm{I}}_{\mathrm{F}\mathrm{V}\mathrm{F}}$ and $\mathrm{S}{\mathrm{I}}_{\mathrm{H}}$ present a notable positive impact on moderate final damages. In contrast, $\frac{\mathrm{P}\mathrm{G}\mathrm{A}}{\mathrm{P}\mathrm{G}\mathrm{V}}$ values smaller than 15 ${\mathrm{s}}^{-1}$ seems to have a negative impact on moderate final damages, while $\mathrm{C}\mathrm{A}\mathrm{V}$ and ${\mathrm{I}}_{\mathrm{R}\mathrm{G}}$ demonstrate more complex effects in a narrower range of the final damage.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ML | Machine learning |

RC | Reinforced Concrete |

IM | Intensity Measure |

NN | Neural Network |

ANN | Artificial Neural Network |

MLP | Multi-Layer Perceptron |

LIME | Local Interpretable Model-agnostic Explanations |

SHAP | SHapley Additive exPlanations |

ALE | Accumulated Local Effects |

PDP | Partial Dependence Plot |

${\mathrm{a}}_{\mathrm{g}}\left(\mathrm{t}\right)$ | ground acceleration signal |

${\mathrm{v}}_{\mathrm{g}}\left(\mathrm{t}\right)$ | ground velocity acceleration signal |

${\mathrm{d}}_{\mathrm{g}}\left(\mathrm{t}\right)$ | ground displacement signal |

${\mathrm{H}}_{\mathrm{d}}$ | Husid Diagram |

$\mathrm{P}\mathrm{S}\mathrm{V}$ | Pseudo-velocity spectrum |

PGA | Peak Ground Acceleration |

PGV | Peak Ground Velocity |

PGD | Peak Ground Displacement |

${\mathrm{I}}_{\mathrm{A}}$ | Arias intensity |

CAV | Cumulative Absolute Velocity |

${\mathrm{I}}_{\mathrm{A}\mathrm{S}}$ | Seismic intensity after Araya and Saragoni |

$\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}$ | Strong motion duration after Trifunac and Brady |

$\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{R}\mathrm{O}\mathrm{G}}$ | Strong motion duration after Reinoso, Ordaz and Guerrero |

$\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{B}\mathrm{o}\mathrm{l}\mathrm{t}}$ | Strong motion duration after Bolt |

${\mathrm{a}}_{\mathrm{r}\mathrm{m}\mathrm{s}}$ | Root-mean-squared of ground acceleration signal |

${\mathrm{I}}_{\mathrm{c}}$ | Characteristic Intensity |

${\mathrm{I}}_{\mathrm{F}\mathrm{V}\mathrm{F}}$ | Potential damage measure after Fajfar, Vidic and Fischinger |

${\mathrm{I}}_{\mathrm{R}\mathrm{G}}$ | Intensity measure after Riddel and Garcia |

$\mathrm{S}{\mathrm{I}}_{\mathrm{H}}$ | Spectral intensity after Housner |

$\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A},1\mathrm{s}\mathrm{t}}$ | The overall Park and Ang damage index after the first seismic shock (input feature) |

$\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A}}$ | The overall Park and Ang damage index after the second seismic shock (target) |

CV | Cross-Validation |

AdaBoost | Adaptive Boosting |

DT | Decision tree |

ERT | Extremely Randomized Trees |

GBoost | Gradient boosting |

KNN | K nearest neighbors |

LightGBM | Light Gradient Boosting Machine |

LR | Linear Regression |

Lasso | Lasso Regression |

RR | Ridge Regression |

EN | Elastic Net |

RF | Random forest |

NGBoost | Natural Gradient Boosting |

XGBoost | eXtreme Gradient Boosting |

CatBoost | Categorical Boosting |

## Appendix A

**Figure A1.**Histograms of input features ($\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A},1\mathrm{s}\mathrm{t}}$ and IMs) and target ($\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A}}$).

**Table A1.**Seismic metadata for the real sequences [54].

Region | First Shock | Second Shock | Station Code/Name | Component | PGA_{1st}(g) | PGA_{2nd}(g) | ||
---|---|---|---|---|---|---|---|---|

Date | M | Date | M | |||||

Ancona | 14-06-1972 | 4.2 | 21-06-1972 | 4.0 | ANP | N-S | 0.220 | 0.410 |

Friuli | 11-09-1976 | 5.8 | 15-09-1976 | 6.1 | BUI | N-S | 0.233 | 0.110 |

E-W | 0.108 | 0.093 | ||||||

GMN | N-S | 0.328 | 0.324 | |||||

E-W | 0.299 | 0.644 | ||||||

Montenegro | 15-04-1979 | 6.9 | 15-04-1979 | 5.8 | PETO | E-W | 0.304 | 0.089 |

24-05-1979 | 6.2 | BAR | N-S | 0.371 | 0.201 | |||

E-W | 0.360 | 0.267 | ||||||

HRZ | N-S | 0.215 | 0.066 | |||||

E-W | 0.254 | 0.076 | ||||||

ULO | N-S | 0.282 | 0.033 | |||||

E-W | 0.236 | 0.030 | ||||||

Imperial Valley | 15-10-1979 | 6.5 | 15-10-1979 | 5.0 | Holtville Post Office | 315 | 0.221 | 0.254 |

Mammoth Lakes | 25-05-1980 | 6.1 | 25-05-1980 | 5.7 | Convict Creek | 90 | 0.419 | 0.371 |

Irpinia | 23-11-1980 | 6.9 | 24-11-1980 | 5.0 | BGI | N-S | 0.129 | 0.031 |

E-W | 0.189 | 0.033 | ||||||

STR | N-S | 0.224 | 0.018 | |||||

E-W | 0.320 | 0.032 | ||||||

Gulf of Corinth | 24-02-1981 | 6.6 | 25-02-1981 | 6.3 | KORA | Trans | 0.296 | 0.121 |

Logn | 0.240 | 0.121 | ||||||

Coalinga | 22-07-1983 | 5.8 | 25-07-1983 | 5.2 | Elm (Old CHP) | 90 | 0.519 | 0.677 |

0 | 0.341 | 0.481 | ||||||

Kalamata | 13-09-1986 | 5.9 | 15-09-1986 | 4.8 | KAL1 | Trans | 0.269 | 0.140 |

Logn | 0.232 | 0.237 | ||||||

KALA | Trans | 0.296 | 0.152 | |||||

Logn | 0.216 | 0.334 | ||||||

Spitak | 07-12-1988 | 6.7 | 07-12-1988 | 5.9 | GUK | N-S | 0.181 | 0.144 |

E-W | 0.182 | 0.099 | ||||||

08-01-1989 | 4.0 | 08-01-1989 | 4.1 | NAB | E-W | 0.206 | 0.217 | |

Georgia | 03-05-1991 | 5.6 | 03-05-1991 | 5.2 | SAMB | N-S | 0.354 | 0.208 |

E-W | 0.504 | 0.122 | ||||||

Erzican | 13-03-1992 | 6.6 | 15-03-1992 | 5.9 | AI 178 ERC MET | N-S | 0.411 | 0.032 |

E-W | 0.487 | 0.039 | ||||||

Ilia | 26-03-1993 | 4.7 | 26-03-1993 | 4.9 | PYR1 | Logn | 0.109 | 0.100 |

Northridge | 17-01-1994 | 6.7 | 17-01-1994 | 5.9 | Moorpark—Fire Station | 90 | 0.193 | 0.139 |

180 | 0.291 | 0.184 | ||||||

17-01-1994 | 5.2 | Pacoima Kagel Canyon | 360 | 0.432 | 0.053 | |||

20-03-1994 | 5.3 | Rinaldi Receiving Station | 228 | 0.874 | 0.529 | |||

Sepulveda Hospital | 270 | 0.752 | 0.102 | |||||

Sylmar—Olive Med | 90 | 0.605 | 0.181 | |||||

Umbria Marche | 26-09-1997 | 5.7 | 26-09-1997 | 6.0 | CLF | N-S | 0.276 | 0.197 |

E-W | 0.256 | 0.227 | ||||||

NCR | N-S | 0.395 | 0.502 | |||||

Kalamata | 13-10-1997 | 6.5 | 18-11-1997 | 6.4 | KRN1 | Trans | 0.119 | 0.071 |

Logn | 0.118 | 0.092 | ||||||

Bovec | 12-04-1998 | 5.7 | 31-08-1998 | 4.3 | FAGG | N-S | 0.024 | 0.023 |

E-W | 0.023 | 0.026 | ||||||

Azores Islands | 09-07-1998 | 6.2 | 11-07-1998 | 4.7 | HOR | N-S | 0.405 | 0.082 |

E-W | 0.369 | 0.092 | ||||||

Izmit | 17-08-1999 | 7.6 | 12-11-1999 | 7.3 | ARC | N-S | 0.210 | 0.007 |

E-W | 0.132 | 0.007 | ||||||

ATK | N-S | 0.102 | 0.016 | |||||

E-W | 0.167 | 0.016 | ||||||

DHM | N-S | 0.090 | 0.017 | |||||

E-W | 0.084 | 0.017 | ||||||

FAT | N-S | 0.181 | 0.034 | |||||

E-W | 0.161 | 0.024 | ||||||

KMP | N-S | 0.102 | 0.014 | |||||

E-W | 0.127 | 0.017 | ||||||

ZYT | N-S | 0.119 | 0.021 | |||||

E-W | 0.109 | 0.029 | ||||||

Athens | 07-09-1999 | 5.9 | 07-09-1999 | 4.3 | SPLB | Trans | 0.324 | 0.059 |

Logn | 0.341 | 0.071 | ||||||

Chi-Chi | 20-09-1999 | 7.6 | 20-09-1999 | 6.2 | TCU071 | N-S | 0.651 | 0.382 |

E-W | 0.528 | 0.193 | ||||||

TCU129 | N-S | 0.624 | 0.398 | |||||

E-W | 1.005 | 0.947 | ||||||

25-09-1999 | 6.3 | TCU078 | N-S | 0.307 | 0.387 | |||

E-W | 0.447 | 0.266 | ||||||

TCU079 | N-S | 0.424 | 0.626 | |||||

E-W | 0.592 | 0.776 | ||||||

Duzce | 12-11-1999 | 7.3 | 12-11-1999 | 4.7 | AI 010 BOL | E-W | 0.820 | 0.060 |

Bingöl | 01-05-2003 | 6.3 | 01-05-2003 | 3.5 | AI 049 BNG | N-S | 0.519 | 0.147 |

E-W | 0.291 | 0.068 | ||||||

L Aquila | 06-04-2009 | 6.1 | 07-04-2009 | 5.5 | AQK | N-S | 0.353 | 0.081 |

E-W | 0.330 | 0.090 | ||||||

AQV | N-S | 0.545 | 0.146 | |||||

E-W | 0.657 | 0.129 | ||||||

AVZ | N-S | 0.069 | 0.021 | |||||

09-04-2009 | 5.4 | AQA | N-S | 0.442 | 0.057 | |||

Darfield | 03-09-2010 | 7.0 | 21-02-2011 | 6.2 | Botanical Gardens | S01W | 0.190 | 0.452 |

N89W | 0.155 | 0.552 | ||||||

Cashmere High School | S80E | 0.251 | 0.349 | |||||

Cathedral College | N26W | 0.194 | 0.384 | |||||

N64E | 0.233 | 0.478 | ||||||

Christchurch Hospital | N01W | 0.209 | 0.346 | |||||

S89W | 0.152 | 0.363 | ||||||

Emilia | 20-05-2012 | 6.1 | 29-05-2012 | 6.0 | MRN | N-S | 0.263 | 0.294 |

E-W | 0.262 | 0.222 | ||||||

03-06-2012 | 5.1 | 12-06-2012 | 4.9 | T0827 | N-S | 0.490 | 0.585 | |

E-W | 0.263 | 0.234 | ||||||

Central Italy | 24-08-2016 | 6.0 | 24-08-2016 | 5.4 | AQK | E-W | 0.050 | 0.010 |

26-08-2016 | 4.8 | AMT | N-S | 0.375 | 0.336 | |||

E-W | 0.867 | 0.325 | ||||||

26-10-2016 | 5.4 | 26-10-2016 | 5.9 | CMI | N-S | 0.341 | 0.308 | |

E-W | 0.720 | 0.651 | ||||||

CNE | E-W | 0.556 | 0.537 | |||||

30-10-2016 | 6.5 | CIT | N-S | 0.052 | 0.213 | |||

E-W | 0.092 | 0.325 | ||||||

26-10-2016 | 5.9 | 30-10-2016 | 6.5 | CLO | N-S | 0.193 | 0.582 | |

E-W | 0.183 | 0.427 | ||||||

CNE | N-S | 0.380 | 0.294 | |||||

MMO | N-S | 0.168 | 0.188 | |||||

E-W | 0.170 | 0.189 | ||||||

NOR | E-W | 0.215 | 0.311 | |||||

30-10-2016 | 6.5 | 31-10-2016 | 4.2 | T1213 | N-S | 0.867 | 0.185 | |

E-W | 0.794 | 0.212 | ||||||

18-01-2017 | 5.5 | 18-01-2017 | 5.4 | PCB | N-S | 0.586 | 0.561 | |

E-W | 0.408 | 0.388 | ||||||

Dodecanese Islands | 08-08-2019 | 4.8 | 30-10-2020 | 7.0 | GMLD | N-S | 0.450 | 0.899 |

E-W | 0.673 | 0.763 |

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**Figure 1.**Indicative seismic signal comprising two successive ground motions and the accompanying evolution of seismic damage.

**Figure 2.**(

**a**) The examined RC frame [54], (

**b**) concrete and (

**c**) steel stress-strain rules, (

**d**) the implemented hysteretic rule.

**Figure 3.**(

**a**) The machine learning workflow followed in this study and (

**b**) the K-fold CV schematic representation.

**Figure 4.**A comprehensive visualization of the ML methods and the interpretation techniques used in this study.

**Figure 5.**Side-by-side bar plot comparing the performance of different ML methods on training (10-fold), CV (10-fold), and test sets.

**Figure 6.**The evolution of ${\mathrm{R}}^{2}$ versus the number of base learner trees for boosted trees.

**Figure 7.**The evolution of ${\mathrm{R}}^{2}$ versus the number of base learner trees for random forest algorithms.

**Figure 8.**The evolution of ${\mathrm{R}}^{2}$ versus the total number of neurons, for each activation function.

**Figure 9.**Feature importances for the ML methods with ${\mathrm{R}}^{2}>0.9$, according to the permutation method.

**Figure 12.**Stacked bar plot of grouped (

**a**) LIME attributions (

**b**) SHAP values for IMs and $\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A}}$, normalized to unity.

**Figure 13.**The expected value of $\mathrm{D}{\mathrm{I}}_{\mathrm{G},\mathrm{P}\mathrm{A}}$ with respect to each examined input feature, according to the PDP and ALE methods.

**Table 1.**Mathematical formulas of the examined IMs [54].

Num | Name | Expression | Ref. | Num | Name | Expression | Ref. |
---|---|---|---|---|---|---|---|

1 | $\mathrm{P}\mathrm{G}\mathrm{A}$ | $\mathrm{m}\mathrm{a}\mathrm{x}|{\mathrm{a}}_{\mathrm{g}}\left(\mathrm{t}\right)|$ | [78] | 9 | $\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{R}\mathrm{O}\mathrm{G}}$ | $\mathrm{t}({\mathrm{H}}_{\mathrm{d}}=97.5\%)-\mathrm{t}({\mathrm{H}}_{\mathrm{d}}=2.5\%)$ * | [82] |

2 | $\mathrm{P}\mathrm{G}\mathrm{V}$ | $\mathrm{m}\mathrm{a}\mathrm{x}|{\mathrm{v}}_{\mathrm{g}}\left(\mathrm{t}\right)|$ | [78] | 10 | $\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{B}\mathrm{o}\mathrm{l}\mathrm{t}}$ | ${\mathrm{t}}_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}^{{\mathrm{a}}_{\mathrm{g}}>0.05\mathrm{g}}-{\mathrm{t}}_{1\mathrm{s}\mathrm{t}}^{{\mathrm{a}}_{\mathrm{g}}>0.05\mathrm{g}}$ | [83] |

3 | $\mathrm{P}\mathrm{G}\mathrm{D}$ | $\mathrm{m}\mathrm{a}\mathrm{x}|{\mathrm{d}}_{\mathrm{g}}\left(\mathrm{t}\right)|$ | [78] | 11 | ${\mathrm{P}}_{90}$ | $\frac{{\mathrm{I}}_{\mathrm{A}}({\mathrm{H}}_{\mathrm{d}}=95\%)-{\mathrm{I}}_{\mathrm{A}}({\mathrm{H}}_{\mathrm{d}}=5\%)}{\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}}$ | [78] |

4 | ${\mathrm{I}}_{\mathrm{A}}$ | $\frac{\pi}{2\mathrm{g}}{\int}_{0}^{{\mathrm{t}}_{\mathrm{e}\mathrm{n}\mathrm{d}}}{\mathrm{a}}_{\mathrm{g}}^{2}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}$ | [79] | 12 | ${\mathrm{a}}_{\mathrm{r}\mathrm{m}\mathrm{s}}$ | $\sqrt{\frac{1}{\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}}{\int}_{{\mathrm{t}}_{5\%}}^{{\mathrm{t}}_{95\%}}{\mathrm{a}}_{\mathrm{g}}{\left(\mathrm{t}\right)}^{2}\mathrm{d}\mathrm{t}}$ | [78] |

5 | $\mathrm{C}\mathrm{A}\mathrm{V}$ | ${\int}_{\mathrm{o}}^{{\mathrm{t}}_{\mathrm{e}\mathrm{n}\mathrm{d}}}\left|{\mathrm{a}}_{\mathrm{g}}\left(\mathrm{t}\right)\right|\mathrm{d}\mathrm{t}$ | [78] | 13 | ${\mathrm{I}}_{\mathrm{c}}$ | ${\mathrm{a}}_{\mathrm{r}\mathrm{m}\mathrm{s}}^{1.5}\xb7\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}^{0.5}$ | [78] |

6 | $\mathrm{P}\mathrm{G}\mathrm{A}/\mathrm{P}\mathrm{G}\mathrm{V}$ | $\frac{\mathrm{P}\mathrm{G}\mathrm{A}}{\mathrm{P}\mathrm{G}\mathrm{V}}$ | [78] | 14 | ${\mathrm{I}}_{\mathrm{F}\mathrm{V}\mathrm{F}}$ | $\mathrm{P}\mathrm{G}\mathrm{V}\xb7\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}^{0.25}$ | [84] |

7 | ${\mathrm{I}}_{\mathrm{A}\mathrm{S}}$ | $\frac{{\mathrm{I}}_{\mathrm{A}}}{{\mathrm{u}}_{\mathrm{o}}^{2}}$ | [80] | 15 | ${\mathrm{I}}_{\mathrm{R}\mathrm{G}}$ | $\mathrm{P}\mathrm{G}\mathrm{D}\xb7\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}^{\frac{1}{3}}$ | [85] |

8 | $\mathrm{S}\mathrm{M}{\mathrm{D}}_{\mathrm{T}\mathrm{B}}$ | $\mathrm{t}({\mathrm{H}}_{\mathrm{d}}=95\%)-\mathrm{t}({\mathrm{H}}_{\mathrm{d}}=5\%)$ * | [81] | 16 | $\mathrm{S}{\mathrm{I}}_{\mathrm{H}}$ | ${\int}_{0.1}^{2.5}\mathrm{P}\mathrm{S}\mathrm{V}(\mathrm{T},\xi =0.05)\mathrm{d}\mathrm{T}$ | [86] |

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## Share and Cite

**MDPI and ACS Style**

Lazaridis, P.C.; Kavvadias, I.E.; Demertzis, K.; Iliadis, L.; Vasiliadis, L.K.
Interpretable Machine Learning for Assessing the Cumulative Damage of a Reinforced Concrete Frame Induced by Seismic Sequences. *Sustainability* **2023**, *15*, 12768.
https://doi.org/10.3390/su151712768

**AMA Style**

Lazaridis PC, Kavvadias IE, Demertzis K, Iliadis L, Vasiliadis LK.
Interpretable Machine Learning for Assessing the Cumulative Damage of a Reinforced Concrete Frame Induced by Seismic Sequences. *Sustainability*. 2023; 15(17):12768.
https://doi.org/10.3390/su151712768

**Chicago/Turabian Style**

Lazaridis, Petros C., Ioannis E. Kavvadias, Konstantinos Demertzis, Lazaros Iliadis, and Lazaros K. Vasiliadis.
2023. "Interpretable Machine Learning for Assessing the Cumulative Damage of a Reinforced Concrete Frame Induced by Seismic Sequences" *Sustainability* 15, no. 17: 12768.
https://doi.org/10.3390/su151712768