# Hilbert-Huang Transform-Based Seismic Intensity Parameters for Performance-Based Design of RC-Framed Structures

^{1}

^{2}

^{*}

## Abstract

**:**

_{PA,global}). The potential contribution of nine training algorithms to developing the most effective MFP was also investigated. The results confirm that the evolved MFP networks, utilizing the employed parameters, provide an accurate estimation of the target output of DI

_{PA,global}. As a result, the developed MFPs can constitute a reliable computational intelligence approach for determining the seismic damage induced on structures and, thus, a powerful tool for the scientific community for the performance-based design of buildings.

## 1. Introduction

## 2. Methods

#### 2.1. Hilbert-Huang Transform (HHT) Analysis

_{n}which was either a monotonic function or a constant

_{j}(t) was applied to each of them, as described in the following equation

_{j}(t) and the Hilbert transform y

_{j}(t) form an analytical signal z

_{j}(t) as follows:

_{j}(t) and the phase function θ

_{j}(t) were defined.

#### 2.2. HHT-Based Seismic Parameters

_{1(HHT)}occupied by each spectrum, which represents the released energy during a seismic excitation and was calculated as

_{max}and t

_{max}are the maximum instantaneous frequency calculated by the analytical signal and the total duration of the signal, respectively.

_{1(HHT)}obtained from every Hilbert spectrum was the second seismic parameter and was described as

_{1(mean,HHT)}value, was set (Figure 2). For the bounded Hilbert spectrum, the new volume V

_{1(Pos,HHT)}, the volume over the parallel layer, and the new upper surface S

_{1(Pos,HHT)}of the spectrum were defined as two more parameters.

_{1(HHT)}and V

_{1(Pos,HHT)}were divided by the corresponding values of surfaces S

_{1(HHT)}and S

_{1(Pos,HHT)}, respectively, and so, the parameters A

_{1(HHT)}and A

_{1(Pos,HHT)}were calculated.

_{1(max,HHT)}, VA

_{1(mean,HHT)}, and VA

_{1(dif,HHT)}were set by the multiplication of the volume V

_{1(HHT)}with the maximum, minimum values of amplitude and their difference correspondingly.

_{0}≤ f ≤ 1.10 ⋅ f

_{0}

_{2(HHT)}, S

_{2(HHT)}, A

_{2(max,HHT)}, A

_{2(mean,HHT)}, A

_{2(dif,HHT)}, V

_{2(Pos,HHT)}and S

_{2(Pos,HHT)}, VA

_{2(max,HHT)}, VA

_{2(mean,HHT)}, VA

_{2(dif,HHT)}, A

_{2(HHT)}, and A

_{2(Pos,HHT)}.

_{0}) value of a structure is presented by the calculation of the area S

_{EF(HHT)}of the amplitude-time section that intersects the Hilbert spectrum frequency-axis at the frequency value (f

_{0}) (Figure 3) and defined by Equation (9).

_{3(max,HHT)}and A

_{3(mean,HHT)}parameters, respectively.

_{1(mean,HHT)}, A

_{2(mean,HHT)}, and A

_{3(mean,HHT)}to A

_{1(max,HHT)}, A

_{2(max,HHT)}, and A

_{3(max,HHT)}resulted in the A

_{1(Ratio,HHT)}, A

_{2(Ratio,HHT)}and A

_{3(Ratio,HHT)}HHT-based seismic intensity parameters respectively.

#### 2.3. Global Damage Index of Park and Ang

_{PA,global}) to date, mainly due to its general applicability and the precise definition of different damage states. Its most used modification is the one proposed by Kunnath et al. [29,30], and it is described by the equation

_{m}is the maximum rotation in loading history, θ

_{u}is the ultimate rotation capacity, M

_{y}is the yield moment, dE

_{h}is the incremental absorbed hysteretic energy, and β is a non-negative parameter representing the effect of cyclic loading on structural damage.

_{PA,global}over 0.80 signifies total damage or complete collapse of the structure, while a value equal to zero signifies that the structure is under elastic response. According to the values of DI

_{PA,global}, classification of the structural damage is presented in Table 1.

## 3. Application

_{PA,global}). The evaluated overall structural damage indices of Park and Ang for every seismic vibration cover a broad spectrum of damage (low, medium, large, and total) for statistical reasons, as presented in Figure 5.

## 4. Results

#### 4.1. Evaluation of the HHT-Based Seismic Intensity Parameters

#### 4.2. Problem Formulation and ANN Framework Selection

#### 4.3. Configuration of ANNs

_{PA,global}). The DI

_{PA,global}values were derived from nonlinear dynamic analyses of the structure after applying every employed seismic accelerogram. Thus, the output layer of the MFPs consisted of one neuron presenting the value of DI

_{PA,global}.

_{H}, was employed for the hidden layer, while the choice of linear activation function was made for the output layer.

#### 4.4. Calculation of ANNs

**8**i9-9900k threads.

_{PA,global}, and the corresponding ones evaluated by the constructed ANNs.

## 5. Discussion

_{PA,global}damage index with MSE less than 0. Similarly, depending on the number of neurons in the hidden layer, up to 73.85% for the group 1 parameters and up to 50.92% for the group 2 can predict the DI

_{PA,global}damage index with MSE less than 0.02. This means that at least 66.30% of the first group and 37.04% for the second group of parameters are able to develop ANNs with excellent predictive accuracy (with R > 0.90 και MSE < 0.02 simultaneously).

## 6. Conclusions

_{PA,global}, with high accuracy.

_{PA,global}(with R > 0.90) and a very low MSE (MSE < 0.02), simultaneously with a percentage up to 66.30% for the first group and up to 37.04% for the second group of parameters. According to the classification table of the DI

_{PA,global}, an MSE coefficient with values lower than 0.02 cannot essentially change the class of structural damage caused by a seismic excitation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Hilbert spectrum (HS) for a seismic excitation; (

**b**) bounded HS with the layer that crosses the amplitude-axis of HS at A

_{mean,HHT}.

**Figure 3.**(

**a**) Limitation of the HS on the band of frequencies encompassed in the zone between 0.90 and 1.10 of the fundamental frequency f

_{0}; (

**b**) enlargement of the characteristic zone of HS.

Structural | Structural Damage Degree | |||
---|---|---|---|---|

Damage Index | Low | Medium | Large | Total |

DI_{PA,global} | ≤0.3 | 0.3 < DI_{PA,global} ≤ 0.6 | 0.6 < DI_{PA,global} ≤ 0.8 | DI_{PA,global} > 0.80 |

Parameters | Statistics | |||
---|---|---|---|---|

Min Value | Max Value | Average | Standard Deviation | |

S_{1(HHT)} (-) | 153.5914 | 4946.3096 | 1350.4544 | 1077.2957 |

V_{1(HHT)} (m/s) | 0.2050 | 27.8891 | 5.1527 | 4.8203 |

V_{1(Pos,HHT)} (m/s) | 0.0597 | 7.4880 | 1.5228 | 1.5227 |

S_{1(Pos,HHT)} (-) | 5.8971 | 548.5377 | 86.5171 | 95.3434 |

A_{1(max,HHT)} (m/s) | 0.0114 | 0.8559 | 0.2363 | 0.1848 |

A_{1(mean,HHT)} (m/s) | 0.0005 | 0.1044 | 0.0212 | 0.0200 |

A_{1(dif,HHT)} (m/s) | 0.0104 | 0.7850 | 0.2151 | 0.1707 |

A_{1(Pos,HHT)} (m/s) | 0.0016 | 0.1115 | 0.0247 | 0.0196 |

VA_{1(mean)} (m^{2}/s^{2}) | 0.0002 | 1.1788 | 0.1243 | 0.1637 |

VA_{1(max)} (m^{2}/s^{2}) | 0.0054 | 7.5185 | 1.4392 | 1.6682 |

VA_{1(dif,HHT)} (m^{2}/s^{2}) | 0.0053 | 7.1969 | 1.3150 | 1.5428 |

V_{2(HHT)} (m/s) | 0.0000 | 2.1061 | 0.2515 | 0.3048 |

S_{2(HHT)} (-) | 0.0024 | 33.9103 | 12.8771 | 9.7173 |

V_{2(Pos,HHT)} (m/s) | 0.0000 | 0.5207 | 0.1024 | 0.1009 |

S_{2(Pos,HHT)} (-) | 0.0012 | 14.8652 | 4.0702 | 3.2751 |

A_{2(max,HHT)} (m/s) | 0.0074 | 0.7622 | 0.1567 | 0.1526 |

A_{2(mean,HHT)} (m/s) | 0.0006 | 0.2554 | 0.0287 | 0.0456 |

S_{EF(HHT)} (-) | 0.0237 | 10.0491 | 1.2100 | 1.4582 |

A_{3(max,HHT)} (m/s) | 0.0056 | 0.7422 | 0.1410 | 0.1380 |

A_{3(mean,HHT)} (m/s) | 0.0006 | 0.2559 | 0.0292 | 0.0460 |

A_{1(Ratio,HHT)} (-) | 0.0125 | 0.2241 | 0.0946 | 0.0483 |

A_{2(Ratio,HHT)} (-) | 0.0339 | 0.4424 | 0.1748 | 0.1040 |

A_{3(Ratio,HHT)} (-) | 0.0358 | 0.4957 | 0.1950 | 0.1119 |

A_{1(HHT)} (m/s) | 0.0001 | 0.0259 | 0.0055 | 0.0052 |

A_{2(HHT)} (m/s) | 0.0006 | 0.2157 | 0.0275 | 0.0407 |

A_{2(Pos,HHT)} (m/s) | 0.0009 | 0.1490 | 0.0295 | 0.0266 |

S_{EF}A_{1(mean)} (m/s) | 0.0000 | 0.3947 | 0.0343 | 0.0614 |

S_{EF}A_{2(mean)} (m/s) | 0.0000 | 2.5669 | 0.0862 | 0.3145 |

S_{EF}A_{3(mean)} (m/s) | 0.0000 | 2.5718 | 0.0871 | 0.3149 |

S_{EF}A_{1(max)} (m/s) | 0.0006 | 3.0521 | 0.3912 | 0.6208 |

S_{EF}A_{2(max)} (m/s) | 0.0002 | 7.6594 | 0.3452 | 0.8981 |

S_{EF}A_{3(max)} (m/s) | 0.0002 | 7.4580 | 0.3188 | 0.8620 |

S_{1}A_{3(max)} (m/s) | 3.2803 | 1193.1771 | 172.0672 | 212.8759 |

S_{1}A_{1(mean)} (m/s) | 0.5230 | 121.8580 | 21.4915 | 21.1580 |

S_{1}A_{3(mean)} (m/s) | 0.7957 | 350.8138 | 27.3436 | 43.0174 |

S_{2}A_{2(mean)} (m/s) | 0.0000 | 2.4942 | 0.2603 | 0.3402 |

A_{2(dif,HHT)} (m/s) | 0.0044 | 0.5721 | 0.1280 | 0.1208 |

VA_{2(dif,HHT)} (m^{2}/s^{2}) | 0.0000 | 1.0673 | 0.0538 | 0.1251 |

VA_{2(mean)} (m^{2}/s^{2}) | 0.0000 | 0.5380 | 0.0180 | 0.0660 |

VA_{2(max)} (m^{2}/s^{2}) | 0.0000 | 1.6053 | 0.0719 | 0.1880 |

Backpropagation (BP) Training Algorithms | |
---|---|

Levenberg–Marquardt (LM) | Powell–Beale conjugate gradient (CGB) |

BFGS quasi-Newton (BFG) | Fletcher–Powell conjugate gradient (CGF) |

Resilient backpropagation (RP) | Polak–Ribiere conjugate gradient (CGP) |

Scaled conjugate gradient (BP) | One step secant (OSS) |

Gradient descent with momentum and adaptive linear (GDX) |

Group 1—R Statistics | ||||||||
---|---|---|---|---|---|---|---|---|

Training Algorithm | 7-Neuron Hidden Layer | 8-Neuron Hidden Layer | ||||||

Min | Max | Mean | st.dev. | Min | Max | Mean | st.dev. | |

trainlm | −0.6317 | 0.9841 | 0.9042 | 0.0514 | −0.5746 | 0.9882 | 0.9028 | 0.0528 |

trainbfg | −0.7944 | 0.9642 | 0.8435 | 0.1083 | −0.7071 | 0.9616 | 0.8459 | 0.1013 |

trainrp | −0.7916 | 0.9601 | 0.8173 | 0.1193 | −0.7038 | 0.9610 | 0.8182 | 0.1168 |

trainscg | −0.8194 | 0.9618 | 0.8376 | 0.1149 | −0.5916 | 0.9610 | 0.8391 | 0.1075 |

traincgb | −0.6282 | 0.9700 | 0.8524 | 0.1051 | −0.6858 | 0.9633 | 0.8530 | 0.1004 |

traincgf | −0.7216 | 0.9626 | 0.8393 | 0.1123 | −0.5956 | 0.9616 | 0.8432 | 0.1048 |

traincgp | −0.6849 | 0.9681 | 0.8410 | 0.1111 | −0.6858 | 0.9618 | 0.8416 | 0.1060 |

trainoss | −0.6953 | 0.9551 | 0.8346 | 0.1127 | −0.6866 | 0.9587 | 0.8366 | 0.1052 |

traingdx | −0.8512 | 0.9529 | 0.5966 | 0.3801 | −0.8566 | 0.9490 | 0.5958 | 0.3822 |

9-Neuron Hidden Layer | 10-Neuron Hidden Layer | |||||||

min | max | mean | st.dev. | min | max | mean | st.dev. | |

trainlm | −0.6315 | 0.9861 | 0.9017 | 0.0537 | −0.5106 | 0.9838 | 0.9008 | 0.0548 |

trainbfg | −0.6329 | 0.9622 | 0.8479 | 0.0956 | −0.6748 | 0.9674 | 0.8495 | 0.0918 |

trainrp | −0.6880 | 0.9571 | 0.8191 | 0.1150 | −0.7202 | 0.9622 | 0.8191 | 0.1147 |

trainscg | −0.7171 | 0.9721 | 0.8398 | 0.1032 | −0.6387 | 0.9656 | 0.8403 | 0.1005 |

traincgb | −0.6102 | 0.9661 | 0.8536 | 0.0963 | −0.6650 | 0.9692 | 0.8538 | 0.0937 |

traincgf | −0.7120 | 0.9618 | 0.8455 | 0.1000 | −0.6384 | 0.9627 | 0.8470 | 0.0963 |

traincgp | −0.6248 | 0.9616 | 0.8420 | 0.1017 | −0.6650 | 0.9658 | 0.8423 | 0.0990 |

trainoss | −0.7067 | 0.9607 | 0.8379 | 0.0994 | −0.6939 | 0.9584 | 0.8386 | 0.0959 |

traingdx | −0.8554 | 0.9524 | 0.5910 | 0.3853 | −0.8609 | 0.9512 | 0.5829 | 0.3904 |

Group 1—MSE Statistics | ||||||||
---|---|---|---|---|---|---|---|---|

Training Algorithm | 7-Neuron Hidden Layer | 8-Neuron Hidden Layer | ||||||

Min | Max | Mean | st.dev. | Min | Max | Mean | st.dev. | |

trainlm | 0.0029 | 0.3214 | 0.0183 | 0.0103 | 0.0023 | 0.3745 | 0.0187 | 0.0107 |

trainbfg | 0.0066 | 0.4275 | 0.0266 | 0.0154 | 0.0070 | 0.4322 | 0.0264 | 0.0149 |

trainrp | 0.0072 | 0.5305 | 0.0309 | 0.0180 | 0.0071 | 0.5626 | 0.0309 | 0.0182 |

trainscg | 0.0069 | 0.5355 | 0.0272 | 0.0156 | 0.0070 | 0.4774 | 0.0271 | 0.0151 |

traincgb | 0.0054 | 0.3952 | 0.0249 | 0.0146 | 0.0067 | 0.4553 | 0.0249 | 0.0143 |

traincgf | 0.0067 | 0.5486 | 0.0270 | 0.0155 | 0.0070 | 0.5340 | 0.0265 | 0.0149 |

traincgp | 0.0058 | 0.4636 | 0.0267 | 0.0153 | 0.0070 | 0.4553 | 0.0267 | 0.0150 |

trainoss | 0.0081 | 0.4828 | 0.0280 | 0.0158 | 0.0074 | 0.4388 | 0.0279 | 0.0153 |

traingdx | 0.0084 | 0.7099 | 0.0595 | 0.0534 | 0.0091 | 1.0021 | 0.0618 | 0.0579 |

9-Neuron Hidden Layer | 10-Neuron Hidden Layer | |||||||

min | max | mean | st.dev. | min | max | mean | st.dev. | |

trainlm | 0.0026 | 0.4003 | 0.0190 | 0.0110 | 0.0030 | 0.4343 | 0.0192 | 0.0114 |

trainbfg | 0.0069 | 0.4309 | 0.0262 | 0.0145 | 0.0059 | 0.4576 | 0.0260 | 0.0141 |

trainrp | 0.0077 | 0.9272 | 0.0309 | 0.0183 | 0.0069 | 0.6747 | 0.0310 | 0.0187 |

trainscg | 0.0050 | 0.6328 | 0.0271 | 0.0148 | 0.0062 | 0.7438 | 0.0271 | 0.0148 |

traincgb | 0.0062 | 0.4318 | 0.0249 | 0.0141 | 0.0056 | 0.6200 | 0.0249 | 0.0140 |

traincgf | 0.0069 | 0.4737 | 0.0262 | 0.0146 | 0.0068 | 0.5245 | 0.0260 | 0.0144 |

traincgp | 0.0069 | 0.4597 | 0.0267 | 0.0147 | 0.0062 | 0.6200 | 0.0268 | 0.0147 |

trainoss | 0.0071 | 0.5018 | 0.0278 | 0.0149 | 0.0075 | 0.6408 | 0.0277 | 0.0147 |

traingdx | 0.0086 | 0.9054 | 0.0648 | 0.0626 | 0.0087 | 0.9933 | 0.0685 | 0.0677 |

Group 2—R Statistics | ||||||||
---|---|---|---|---|---|---|---|---|

Training Algorithm | 7-Neuron Hidden Layer | 8-Neuron Hidden Layer | ||||||

Min | Max | Mean | st.dev. | Min | Max | Mean | st.dev. | |

trainlm | −0.7834 | 0.9730 | 0.8824 | 0.0494 | −0.8028 | 0.9706 | 0.8818 | 0.0502 |

trainbfg | −0.7816 | 0.9357 | 0.8439 | 0.0594 | −0.7483 | 0.9396 | 0.8443 | 0.0590 |

trainrp | −0.7842 | 0.9248 | 0.8305 | 0.0692 | −0.7366 | 0.9312 | 0.8298 | 0.0709 |

trainscg | −0.7808 | 0.9317 | 0.8399 | 0.0655 | −0.7462 | 0.9397 | 0.8394 | 0.0657 |

traincgb | −0.7319 | 0.9450 | 0.8475 | 0.0604 | −0.8165 | 0.9441 | 0.8474 | 0.0607 |

traincgf | −0.7618 | 0.9350 | 0.8431 | 0.0640 | −0.7744 | 0.9364 | 0.8433 | 0.0640 |

traincgp | −0.7618 | 0.9350 | 0.8420 | 0.0629 | −0.7661 | 0.9327 | 0.8415 | 0.0636 |

trainoss | −0.7750 | 0.9289 | 0.8397 | 0.0607 | −0.8062 | 0.9312 | 0.8396 | 0.0598 |

traingdx | −0.8783 | 0.9161 | 0.6750 | 0.3145 | −0.8764 | 0.9184 | 0.6622 | 0.3236 |

9-Neuron Hidden Layer | 10-Neuron Hidden Layer | |||||||

min | max | mean | st.dev. | min | max | mean | st.dev. | |

trainlm | −0.7400 | 0.9698 | 0.8812 | 0.0513 | −0.7718 | 0.9668 | 0.8807 | 0.0519 |

trainbfg | −0.8010 | 0.9400 | 0.8444 | 0.0595 | −0.6828 | 0.9406 | 0.8445 | 0.0595 |

trainrp | −0.7421 | 0.9271 | 0.8286 | 0.0735 | −0.7388 | 0.9299 | 0.8275 | 0.0756 |

trainscg | −0.7310 | 0.9326 | 0.8387 | 0.0662 | −0.7900 | 0.9337 | 0.8377 | 0.0680 |

traincgb | −0.7702 | 0.9417 | 0.8469 | 0.0618 | −0.7683 | 0.9428 | 0.8463 | 0.0626 |

traincgf | −0.8038 | 0.9417 | 0.8429 | 0.0655 | −0.7986 | 0.9383 | 0.8426 | 0.0663 |

traincgp | −0.7792 | 0.9351 | 0.8409 | 0.0640 | −0.7785 | 0.9394 | 0.8403 | 0.0649 |

trainoss | −0.7717 | 0.9373 | 0.8391 | 0.0601 | −0.7550 | 0.9388 | 0.8385 | 0.0602 |

traingdx | −0.8769 | 0.9152 | 0.6483 | 0.3341 | −0.8766 | 0.9224 | 0.6326 | 0.3464 |

Group 2—MSE Statistics | ||||||||
---|---|---|---|---|---|---|---|---|

Training Algorithm | 7-Neuron Hidden Layer | 8-Neuron Hidden Layer | ||||||

Min | Max | Mean | st.dev. | Min | Max | Mean | st.dev. | |

trainlm | 0.0049 | 0.3890 | 0.0225 | 0.0113 | 0.0055 | 0.6813 | 0.0228 | 0.0119 |

trainbfg | 0.0115 | 0.6292 | 0.0272 | 0.0097 | 0.0108 | 0.6749 | 0.0273 | 0.0100 |

trainrp | 0.0134 | 0.4977 | 0.0298 | 0.0117 | 0.0122 | 0.4258 | 0.0300 | 0.0122 |

trainscg | 0.0122 | 0.3255 | 0.0276 | 0.0100 | 0.0109 | 0.6220 | 0.0278 | 0.0104 |

traincgb | 0.0099 | 0.3728 | 0.0263 | 0.0093 | 0.0100 | 0.4400 | 0.0264 | 0.0096 |

traincgf | 0.0116 | 0.3831 | 0.0270 | 0.0098 | 0.0113 | 0.4416 | 0.0270 | 0.0100 |

traincgp | 0.0119 | 0.3452 | 0.0272 | 0.0096 | 0.0119 | 0.4335 | 0.0274 | 0.0099 |

trainoss | 0.0126 | 0.5109 | 0.0280 | 0.0099 | 0.0122 | 0.7328 | 0.0281 | 0.0100 |

traingdx | 0.0149 | 1.0202 | 0.0500 | 0.0407 | 0.0144 | 1.3593 | 0.0529 | 0.0446 |

9-Neuron Hidden Layer | 10-Neuron Hidden Layer | |||||||

min | max | mean | st.dev. | min | max | mean | st.dev. | |

trainlm | 0.0055 | 0.4265 | 0.0230 | 0.0124 | 0.0060 | 0.5154 | 0.0233 | 0.0129 |

trainbfg | 0.0107 | 0.4202 | 0.0273 | 0.0102 | 0.0106 | 6.7141 | 0.0274 | 0.0124 |

trainrp | 0.0129 | 0.4869 | 0.0303 | 0.0129 | 0.0124 | 0.3849 | 0.0306 | 0.0135 |

trainscg | 0.0119 | 0.3476 | 0.0280 | 0.0107 | 0.0119 | 0.4225 | 0.0282 | 0.0113 |

traincgb | 0.0105 | 0.6390 | 0.0265 | 0.0100 | 0.0102 | 0.6842 | 0.0267 | 0.0104 |

traincgf | 0.0105 | 0.4828 | 0.0272 | 0.0105 | 0.0110 | 0.7971 | 0.0273 | 0.0109 |

traincgp | 0.0115 | 0.3930 | 0.0275 | 0.0103 | 0.0108 | 0.4685 | 0.0277 | 0.0107 |

trainoss | 0.0111 | 0.3888 | 0.0283 | 0.0103 | 0.0112 | 0.6655 | 0.0285 | 0.0107 |

traingdx | 0.0150 | 1.1226 | 0.0563 | 0.0494 | 0.0139 | 1.0971 | 0.0600 | 0.0541 |

Group 1_ Classification of R | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 Neurons in the Hidden Layer | Training Function of ANNs | |||||||||

Train-lm | Train-bfg | Train-rp | Train-scg | Train-cgb | Train-cgf | Train-cgp | Train-oss | Train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 3.824 | 0.025 | 0.003 | 0.012 | 0.065 | 0.027 | 0.026 | 0.003 | 0.000 |

0.92 ≤ R < 0.95 | (%) of ANNs | 39.795 | 5.825 | 1.796 | 4.789 | 8.921 | 6.053 | 5.833 | 2.933 | 2.433 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.681 | 18.798 | 9.619 | 17.356 | 22.611 | 18.198 | 18.511 | 14.854 | 11.814 |

Total (%) | 69.300 | 24.623 | 11.415 | 22.145 | 31.532 | 24.251 | 24.344 | 17.787 | 14.247 | |

8 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 3.567 | 0.026 | 0.004 | 0.013 | 0.054 | 0.025 | 0.021 | 0.003 | 0.000 |

0.92 ≤ R < 0.95 | (%) of ANNs | 38.888 | 6.099 | 2.105 | 4.828 | 8.836 | 6.400 | 5.744 | 3.062 | 2.542 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.782 | 18.693 | 9.964 | 16.728 | 21.995 | 18.488 | 17.843 | 14.663 | 11.897 |

Total (%) | 68.237 | 24.792 | 12.069 | 21.556 | 30.831 | 24.888 | 23.587 | 17.725 | 14.439 | |

9 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 3.429 | 0.028 | 0.007 | 0.014 | 0.051 | 0.028 | 0.025 | 0.003 | 0.000 |

0.92 ≤ R < 0.95 | (%) of ANNs | 38.294 | 6.350 | 2.445 | 4.895 | 8.871 | 6.681 | 5.622 | 3.176 | 2.639 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.803 | 18.640 | 10.333 | 16.402 | 21.433 | 18.632 | 17.407 | 14.492 | 11.871 |

Total (%) | 67.526 | 24.990 | 12.778 | 21.297 | 30.304 | 25.313 | 23.029 | 17.668 | 14.510 | |

10 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 3.400 | 0.029 | 0.009 | 0.015 | 0.053 | 0.030 | 0.023 | 0.004 | 0.000 |

0.92 ≤ R < 0.95 | (%) of ANNs | 37.896 | 6.681 | 2.720 | 4.964 | 8.863 | 6.982 | 5.654 | 3.301 | 2.672 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.542 | 18.710 | 10.626 | 16.184 | 21.061 | 18.586 | 16.933 | 14.224 | 11.744 |

Total (%) | 66.838 | 25.391 | 13.346 | 21.148 | 29.924 | 25.568 | 22.587 | 17.525 | 14.416 |

Group 1_ Classification of MSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 Neurons in the Hidden Layer | Training Function of ANNs | |||||||||

Train-lm | Train-bfg | Train-rp | Train-scg | Train-cgb | Train-cgf | Train-cgp | Train-oss | Train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 73.845 | 39.738 | 22.742 | 37.373 | 47.960 | 39.114 | 39.759 | 32.559 | 24.172 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 24.297 | 53.256 | 67.244 | 55.197 | 46.043 | 53.267 | 53.122 | 59.590 | 35.775 |

MSE > 0.05 | (%) of ANNs | 1.858 | 7.006 | 10.013 | 7.430 | 5.997 | 7.619 | 7.120 | 7.851 | 40.053 |

8 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 72.489 | 39.623 | 23.289 | 36.359 | 46.871 | 39.802 | 38.576 | 32.031 | 24.230 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 25.411 | 53.858 | 66.622 | 56.650 | 47.445 | 53.435 | 54.580 | 60.565 | 35.736 |

MSE > 0.05 | (%) of ANNs | 2.100 | 6.519 | 10.089 | 6.991 | 5.684 | 6.763 | 6.844 | 7.404 | 40.033 |

9 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 71.505 | 39.575 | 23.939 | 35.687 | 46.013 | 40.113 | 37.558 | 31.539 | 24.049 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 26.190 | 54.330 | 65.894 | 57.610 | 48.541 | 53.649 | 55.862 | 61.414 | 35.374 |

MSE > 0.05 | (%) of ANNs | 2.305 | 6.094 | 10.167 | 6.703 | 5.446 | 6.238 | 6.580 | 7.048 | 40.578 |

10 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 70.517 | 39.710 | 24.408 | 35.199 | 45.344 | 40.257 | 36.821 | 31.122 | 23.675 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 26.923 | 54.526 | 65.203 | 58.330 | 49.395 | 53.846 | 56.803 | 62.073 | 34.845 |

MSE > 0.05 | (%) of ANNs | 2.560 | 5.765 | 10.389 | 6.471 | 5.261 | 5.897 | 6.376 | 6.805 | 41.480 |

Group 2_ Classification of R | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 Neurons in the Hidden Layer | Training Function of ANNs | |||||||||

Train-lm | Train-bfg | Train-rp | Train-scg | Train-cgb | Train-cgf | Train-cgp | Train-oss | Train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 0.024 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.92 ≤ R < 0.95 | (%) of ANNs | 11.006 | 0.061 | 0.001 | 0.011 | 0.102 | 0.046 | 0.023 | 0.003 | 0.000 |

0.90 ≤ R < 0.92 | (%) of ANNs | 26.009 | 1.431 | 0.627 | 0.839 | 2.351 | 1.709 | 1.094 | 0.527 | 0.204 |

Total (%) | 37.039 | 1.492 | 0.628 | 0.850 | 2.453 | 1.755 | 1.117 | 0.530 | 0.204 | |

8 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 0.031 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.92 ≤ R< 0.95 | (%) of ANNs | 10.794 | 0.067 | 0.004 | 0.012 | 0.113 | 0.057 | 0.025 | 0.004 | 0.000 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.706 | 1.687 | 0.874 | 1.021 | 2.586 | 2.031 | 1.250 | 0.616 | 0.268 |

Total (%) | 36.531 | 1.754 | 0.878 | 1.033 | 2.699 | 2.088 | 1.275 | 0.620 | 0.268 | |

9 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 0.030 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.92 ≤ R< 0.95 | (%) of ANNs | 10.717 | 0.073 | 0.006 | 0.014 | 0.123 | 0.067 | 0.029 | 0.004 | 0.000 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.692 | 1.943 | 1.107 | 1.155 | 2.871 | 2.373 | 1.415 | 0.737 | 0.328 |

Total (%) | 36.439 | 2.016 | 1.113 | 1.169 | 2.994 | 2.440 | 1.444 | 0.741 | 0.328 | |

10 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

R ≥ 0.95 | (%) of ANNs | 0.032 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.92 ≤ R< 0.95 | (%) of ANNs | 10.723 | 0.086 | 0.007 | 0.017 | 0.141 | 0.080 | 0.031 | 0.006 | 0.000 |

0.90 ≤ R < 0.92 | (%) of ANNs | 25.727 | 2.238 | 1.367 | 1.325 | 3.160 | 2.643 | 1.577 | 0.849 | 0.377 |

Total (%) | 36.482 | 2.324 | 1.374 | 1.342 | 3.301 | 2.723 | 1.608 | 0.855 | 0.377 |

Group 2_ Classification of MSE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

7 Neurons in the Hidden Layer | Training Function of ANNs | |||||||||

Train-lm | Train-bfg | Train-rp | Train-scg | Train-cgb | Train-cgf | Train-cgp | Train-oss | Train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 50.924 | 9.995 | 6.220 | 8.388 | 13.713 | 11.564 | 9.326 | 6.209 | 3.938 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 46.544 | 86.798 | 88.399 | 88.132 | 83.578 | 85.282 | 87.567 | 90.324 | 63.407 |

MSE > 0.05 | (%) of ANNs | 2.533 | 3.207 | 5.381 | 3.480 | 2.709 | 3.153 | 3.107 | 3.467 | 32.655 |

8 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 50.028 | 10.902 | 7.247 | 8.975 | 14.576 | 12.550 | 9.914 | 6.722 | 4.214 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 47.193 | 85.896 | 86.977 | 87.449 | 82.668 | 84.300 | 86.904 | 89.805 | 60.522 |

MSE > 0.05 | (%) of ANNs | 2.778 | 3.202 | 5.776 | 3.575 | 2.757 | 3.150 | 3.182 | 3.473 | 35.264 |

9 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 49.432 | 11.812 | 8.166 | 9.455 | 15.237 | 13.451 | 10.431 | 7.158 | 4.412 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 47.556 | 84.887 | 85.491 | 86.791 | 81.856 | 83.177 | 86.251 | 89.213 | 57.669 |

MSE > 0.05 | (%) of ANNs | 3.012 | 3.301 | 6.344 | 3.754 | 2.907 | 3.372 | 3.318 | 3.628 | 37.919 |

10 Neurons in the Hidden Layer | Training function of ANNs | |||||||||

train-lm | train-bfg | train-rp | train-scg | train-cgb | train-cgf | train-cgp | train-oss | train-gdx | ||

MSE ≤ 0.02 | (%) of ANNs | 48.984 | 12.710 | 8.854 | 9.912 | 15.907 | 14.177 | 10.927 | 7.529 | 4.492 |

0.02 < MSE ≤ 0.05 | (%) of ANNs | 47.765 | 83.859 | 84.351 | 86.051 | 80.954 | 82.254 | 85.507 | 88.662 | 54.944 |

MSE > 0.05 | (%) of ANNs | 3.251 | 3.431 | 6.795 | 4.037 | 3.139 | 3.568 | 3.567 | 3.809 | 40.564 |

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## Share and Cite

**MDPI and ACS Style**

Tyrtaiou, M.; Elenas, A.; Andreadis, I.; Vasiliadis, L.
Hilbert-Huang Transform-Based Seismic Intensity Parameters for Performance-Based Design of RC-Framed Structures. *Buildings* **2022**, *12*, 1301.
https://doi.org/10.3390/buildings12091301

**AMA Style**

Tyrtaiou M, Elenas A, Andreadis I, Vasiliadis L.
Hilbert-Huang Transform-Based Seismic Intensity Parameters for Performance-Based Design of RC-Framed Structures. *Buildings*. 2022; 12(9):1301.
https://doi.org/10.3390/buildings12091301

**Chicago/Turabian Style**

Tyrtaiou, Magdalini, Anaxagoras Elenas, Ioannis Andreadis, and Lazaros Vasiliadis.
2022. "Hilbert-Huang Transform-Based Seismic Intensity Parameters for Performance-Based Design of RC-Framed Structures" *Buildings* 12, no. 9: 1301.
https://doi.org/10.3390/buildings12091301