# Adult Skeletal Age-at-Death Estimation through Deep Random Neural Networks: A New Method and Its Computational Analysis

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.1.1. Sampled Identified Skeletal Collections

#### 2.1.2. Data Management and Processing

#### 2.2. A Novel Technique for Macroscopic Age-At-Death Estimation

#### 2.2.1. Cranial and Palatine Suture Scoring

#### 2.2.2. Vertebrae Development and Degeneration Scoring

#### 2.2.3. Joint and Musculoskeletal Degeneration Scoring

#### 2.2.4. Clavicle Sternal and Acromial Ends Scoring

#### 2.2.5. First Rib Costal Face and Tubercle Scoring

#### 2.2.6. Pubic Symphysis Scoring

#### 2.2.7. Sacral and Iliac Auricular Surfaces (Sacroiliac Joint) Scoring

#### 2.2.8. Acetabulum Scoring

#### 2.2.9. Scoring Reliability: Intra-Observer Error

#### 2.3. Feature Analysis Via Sphering and Marginal Correlation Analysis

^{2}). In addition to these two coefficients, we also computed marginal correlations adjusted for inter-trait correlation following Zuber and Strimmer [97]. This technique aims to cope with the myopy of univariate feature selection methods by computing marginal correlations of decorrelated predictors with the target class. First, the data centered and scaled, and then transformed by applying a linear basis that enforces orthogonality among predictors while maintaining the maximum relationship with the original standardized predictors. After this transformation, also known as the Mahalanobis transform or sphering, the predictors covariance matrix is the identity matrix (no correlation). The authors called the adjusted marginal correlations CAR scores and proved that ranking based on these quantities provides a fast and optimal procedure for feature ranking and selection. We suggest [97,98] as primers on feature selection and data sphering based on this approach.

#### 2.4. Randomized Neural Networks: Theory and Implementation

#### 2.4.1. Efficient Training and Regularization in Randomized Neural Networks

_{ridge}, is obtained by substituting Equation (3) as follows:

_{ii}is the i-th diagonal element of the hat or projection matrix, which is the matrix that maps the hidden layer parameters to the predicted values of the network, in our case age-at-death. Shao and Er [112] have demonstrated that computing the projection matrix of the network and finding the optimal regularization parameter, C, under leave-one-out cross-validation (LOO-CV), can be achieved with computational efficiency by performing a singular value decomposition (SVD) of the hidden layer, which, given such an operation, is written as $H=U\mathsf{\Sigma}{V}^{T}$. Using SVD, the network estimate can be written as:

_{ii}, can be obtained by performing a column-wise sum of the elements of $\gamma $. The LOO predictions of the network can be obtained analytically as follows:

#### 2.4.2. From Shallow to Deep Randomized Neural Networks

^{(j−1)}is the previous layer. One can also allow connections from the input to all hidden layers and define the hidden layer as:

#### 2.4.3. Deep Random Neural Networks as Implicit Ensemble Models

#### 2.5. Regression Uncertainty Modeling and Prediction Intervals

#### 2.6. Computational Analysis: Design, Parameterization, Metrics, and Software

#### 2.6.1. Experimental Design

- (A)
- The first experiment we conducted was designed to provide a baseline of the accuracy obtained by fitting DRNN models to blocks of traits that have standard or traditional analytical framing. For instance, we fitted models to different anatomical complexes or sets of traits that mimic existing aging standards, i.e., a model for the sutures or the pubis symphysis.
- (B)
- Our second computational experiment consisted of simulated different proportions of available traits from 90% to 10%. The objective of this experiment was to assess model performance in a more realistic scenario where the forensic anthropologist has skeletal traits available on a case-by-case basis.

#### 2.6.2. Network Parameterization

#### 2.6.3. Performance Metrics

#### 2.6.4. Software

## 3. Results

#### 3.1. Intra-Observer Scoring Error

#### 3.2. Marginal Correlation Analysis

^{2}: 0.088–0.249), with palatine sutures explaining less than 10% of the variation in observed age-at-death. The axial traits—cervical and lumbar vertebrae—exhibited a moderate to strong monotonic relationship and explained variation with age-at-death (ρ: 0.794–0.845, η

^{2}: 0.639–0.725). A similar correlation and explained variation pattern were observed for the clavicle traits (ρ: 0.710–0.851, η

^{2}: 0.507–0.729), first rib traits (ρ: 0.763–0.776, η

^{2}: 0.590–0.607), iliac auricular surface traits (ρ: 0.731–0.789, η

^{2}: 0.539–0.631), and the acetabular traits (ρ: 0.782–0.818, η

^{2}: 0.625–0.674). A slightly lower marginal correlation was observed for the pubic symphysis traits (ρ: 0.711–0.731, η

^{2}: 0.523–0.549) and sacral auricular surface traits (ρ: 0.632–0.704, η

^{2}: 0.398–0.499). Traits from the upper and lower limbs presented a wider range of correlation (ρ: 0.380–0.789, η

^{2}: 0.145–0.628). When analyzed in the context of feature ranking based on marginal correlations adjusted for inter-trait correlation (CAR scores), the suture traits score was among the worst predictors and its decorrelated components showed no statistically significant relationship with age-at-death. The several appendicular degenerative traits—HM04, UL01, RD01, FM01, FM02, and TB01—also showed no statistically significant correlation when assessed on a Mahalanobis transformed space. Ranking based on CAR scores showed that the top-ranking traits came from all anatomical regions rather than a specific indicator.

#### 3.3. Computational Model Assessment

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Predictive efficiency of degenerative traits of the axial and appendicular skeleton, α = 0.1.

**Table 1.**Demographic characterization of reference data sampled from the CISC and XXI-ISC collections.

CISC | XXI-ISC | Pooled Collections | Pooled Sex | |||||
---|---|---|---|---|---|---|---|---|

Female | Male | Female | Male | Female | Male | |||

n | 168 | 166 | 82 | 84 | 250 | 250 | 500 | |

Age-at-Death | Mean | 48.482 | 45.331 | 81.841 | 74.881 | 59.424 | 55.260 | 57.34 |

(AGE) | Std. Dev. | 19.483 | 18.171 | 12.889 | 15.082 | 23.556 | 22.141 | 22.93 |

Min. | 19 | 19 | 38 | 25 | 19 | 19 | 19 | |

Max. | 95 | 96 | 101 | 96 | 101 | 96 | 101 | |

Year of Birth | Mean | 1877.286 | 1879.994 | 1923.866 | 1930.560 | 1892.564 | 1896.984 | 1894.774 |

(YOB) | Std. Dev. | 21.252 | 19.948 | 13.137 | 14.424 | 28.969 | 30.096 | 29.591 |

Min. | 1830 | 1836 | 1904 | 1908 | 1830 | 1836 | 1830 | |

Max. | 1911 | 1917 | 1970 | 1982 | 1970 | 1982 | 1982 | |

Year of Death | Mean | 1925.768 | 1925.325 | 2005.707 | 2005.440 | 1951.988 | 1952.244 | 1952.116 |

(YOD) | Std. Dev. | 6.597 | 7.343 | 3.707 | 3.919 | 38.051 | 38.452 | 38.214 |

Min. | 1910 | 1910 | 2000 | 1995 | 1910 | 1910 | 1910 | |

Max. | 1936 | 1936 | 2012 | 2011 | 2012 | 2011 | 2012 |

**Table 2.**Monte Carlo cross-validation metrics for DRNN models built on pre-specified skeletal traits sets.

Accuracy | Bias | Validity | Efficiency | ||||
---|---|---|---|---|---|---|---|

Traits | MAE | ${\widehat{\mathit{\beta}}}_{\mathit{e}}$ | $\mathit{P}\mathbf{\left(}\mathbf{\alpha}\mathbf{\right)}$ | PIW | PIW 95% CI | ||

Sutures | Median | 15.300 | 0.656 | 0.950 | 68.144 | 51.699 | 69.759 |

(m = 9) | 95% CI | 13.586 | 0.590 | 0.900 | 66.054 | 46.361 | 68.312 |

17.206 | 0.732 | 0.990 | 69.741 | 55.776 | 70.963 | ||

Axial | Median | 8.185 | 0.198 | 0.960 | 38.754 | 33.732 | 40.842 |

(m = 16) | 95% CI | 7.365 | 0.137 | 0.920 | 37.102 | 32.272 | 39.215 |

9.139 | 0.260 | 0.990 | 40.091 | 35.029 | 42.191 | ||

Appendicular | Median | 7.583 | 0.167 | 0.960 | 37.378 | 29.109 | 39.541 |

(m = 23) | 95% CI | 6.678 | 0.103 | 0.910 | 35.412 | 27.613 | 38.014 |

8.523 | 0.231 | 0.990 | 39.079 | 30.399 | 41.061 | ||

Clavicle | Median | 8.949 | 0.244 | 0.960 | 49.234 | 17.354 | 51.610 |

(m = 2) | 95% CI | 7.798 | 0.169 | 0.920 | 39.064 | 15.981 | 49.962 |

10.192 | 0.307 | 0.990 | 52.688 | 18.617 | 53.098 | ||

First Rib | Median | 9.500 | 0.277 | 0.950 | 48.936 | 24.334 | 49.637 |

(m = 2) | 95% CI | 8.138 | 0.204 | 0.900 | 46.879 | 22.499 | 47.687 |

10.831 | 0.351 | 0.990 | 50.903 | 26.078 | 51.533 | ||

Pubic symphysis | Median | 10.897 | 0.370 | 0.940 | 51.210 | 26.905 | 56.954 |

(m = 3) | 95% CI | 9.371 | 0.280 | 0.870 | 48.688 | 24.520 | 54.799 |

12.542 | 0.459 | 0.980 | 55.558 | 29.058 | 58.802 | ||

Sacroiliac complex | Median | 8.523 | 0.223 | 0.950 | 44.668 | 20.378 | 47.969 |

(m = 6) | 95% CI | 7.380 | 0.145 | 0.890 | 39.350 | 18.596 | 46.017 |

9.742 | 0.288 | 0.990 | 47.547 | 21.915 | 49.720 | ||

Acetabulum | Median | 8.886 | 0.229 | 0.970 | 42.978 | 31.727 | 45.742 |

(m = 3) | 95% CI | 7.758 | 0.162 | 0.920 | 41.201 | 29.897 | 43.891 |

10.006 | 0.287 | 1.000 | 44.509 | 33.240 | 47.304 | ||

Degenerative traits | Median | 6.962 | 0.147 | 0.970 | 33.732 | 28.882 | 35.122 |

(m = 39) | 95% CI | 6.084 | 0.085 | 0.920 | 32.460 | 27.570 | 33.488 |

7.814 | 0.200 | 1.000 | 34.935 | 30.019 | 36.656 | ||

Standard traits | Median | 6.609 | 0.147 | 0.950 | 34.245 | 12.927 | 41.087 |

(m = 16) | 95% CI | 5.561 | 0.087 | 0.890 | 29.701 | 11.833 | 39.097 |

7.598 | 0.202 | 0.990 | 37.857 | 14.169 | 42.833 | ||

All | Median | 5.925 | 0.117 | 0.950 | 30.010 | 15.631 | 36.081 |

(m = 64) | 95% CI | 5.101 | 0.060 | 0.900 | 26.817 | 14.464 | 34.612 |

6.728 | 0.170 | 0.990 | 33.191 | 16.811 | 37.515 |

**Table 3.**Leave-one-out cross-validation metrics for DRNN models built on pre-specified skeletal traits sets.

Accuracy | Bias | Validity | Efficiency | ||||
---|---|---|---|---|---|---|---|

Traits | MAE | ${\widehat{\mathit{\beta}}}_{\mathit{e}}$ | $\mathit{P}\mathbf{\left(}\mathbf{\alpha}\mathbf{\right)}$ | PIW | PIW 95% CI | ||

Sutures | Median | 15.245 | 0.655 | 0.953 | 68.120 | 51.782 | 69.796 |

(m = 9) | 95% CI | 14.683 | 0.616 | 0.940 | 66.377 | 46.429 | 68.371 |

15.751 | 0.692 | 0.963 | 69.708 | 55.878 | 70.996 | ||

Axial | Median | 8.156 | 0.200 | 0.960 | 38.825 | 33.594 | 40.881 |

(m = 16) | 95% CI | 7.896 | 0.184 | 0.953 | 37.468 | 32.131 | 39.279 |

8.394 | 0.213 | 0.968 | 39.872 | 34.902 | 42.234 | ||

Appendicular | Median | 7.557 | 0.169 | 0.960 | 37.534 | 29.035 | 39.599 |

(m = 23) | 95% CI | 7.278 | 0.155 | 0.948 | 35.996 | 27.542 | 38.082 |

7.823 | 0.184 | 0.970 | 38.920 | 30.319 | 41.109 | ||

Clavicle | Median | 8.943 | 0.245 | 0.963 | 49.216 | 17.336 | 51.768 |

(m = 2) | 95% CI | 8.606 | 0.228 | 0.953 | 47.184 | 15.969 | 50.112 |

9.248 | 0.263 | 0.970 | 51.238 | 18.597 | 53.252 | ||

First Rib | Median | 9.409 | 0.275 | 0.950 | 48.897 | 24.356 | 49.811 |

(m = 2) | 95% CI | 9.067 | 0.255 | 0.938 | 47.036 | 22.502 | 47.862 |

9.751 | 0.296 | 0.960 | 50.829 | 26.102 | 51.724 | ||

Pubic symphysis | Median | 10.898 | 0.370 | 0.932 | 51.113 | 27.029 | 57.040 |

(m = 3) | 95% CI | 10.436 | 0.343 | 0.922 | 48.668 | 24.616 | 54.949 |

11.315 | 0.398 | 0.945 | 53.003 | 29.217 | 58.909 | ||

Sacroiliac complex | Median | 8.438 | 0.220 | 0.950 | 44.765 | 20.350 | 48.037 |

(m = 6) | 95% CI | 8.075 | 0.200 | 0.940 | 42.461 | 18.607 | 46.091 |

8.741 | 0.239 | 0.960 | 46.755 | 21.893 | 49.800 | ||

Acetabulum | Median | 8.833 | 0.229 | 0.965 | 43.051 | 31.541 | 45.832 |

(m = 3) | 95% CI | 8.490 | 0.210 | 0.955 | 41.302 | 29.726 | 43.995 |

9.116 | 0.247 | 0.975 | 44.535 | 33.054 | 47.395 | ||

Degenerative traits | Median | 6.929 | 0.147 | 0.963 | 33.744 | 28.816 | 35.194 |

(m = 39) | 95% CI | 6.694 | 0.133 | 0.953 | 32.530 | 27.499 | 33.566 |

7.154 | 0.157 | 0.973 | 34.829 | 29.946 | 36.715 | ||

Standard traits | Median | 6.561 | 0.145 | 0.948 | 34.283 | 12.952 | 41.170 |

(m = 16) | 95% CI | 6.277 | 0.132 | 0.935 | 32.464 | 11.853 | 39.222 |

6.855 | 0.157 | 0.960 | 36.027 | 14.122 | 42.921 | ||

All | Median | 5.899 | 0.118 | 0.950 | 30.057 | 15.558 | 36.141 |

(m = 64) | 95% CI | 5.677 | 0.110 | 0.940 | 28.758 | 14.403 | 34.644 |

6.121 | 0.127 | 0.963 | 31.485 | 16.668 | 37.620 |

**Table 4.**Monte Carlo cross-validation metrics for DRNN models built on different fractions of available skeletal traits.

Accuracy | Bias | Validity | Efficiency | ||||
---|---|---|---|---|---|---|---|

Available Traits % | MAE | ${\widehat{\mathit{\beta}}}_{\mathit{e}}$ | $\mathit{P}\mathbf{\left(}\mathbf{\alpha}\mathbf{\right)}$ | PIW | PIW 95% CI | ||

90% | Median | 5.964 | 0.120 | 0.950 | 30.354 | 15.851 | 36.215 |

(m ≈ 57) | 95% CI | 5.136 | 0.062 | 0.900 | 27.067 | 14.466 | 34.554 |

6.773 | 0.169 | 0.990 | 33.422 | 18.081 | 37.705 | ||

80% | Median | 6.026 | 0.121 | 0.950 | 30.498 | 16.004 | 36.261 |

(m ≈ 51) | 95% CI | 5.211 | 0.061 | 0.900 | 27.183 | 14.213 | 34.498 |

6.851 | 0.172 | 0.990 | 33.584 | 18.492 | 37.902 | ||

70% | Median | 6.072 | 0.125 | 0.950 | 30.805 | 16.206 | 36.454 |

(m ≈ 44) | 95% CI | 5.152 | 0.062 | 0.900 | 27.528 | 14.001 | 34.600 |

6.924 | 0.180 | 0.990 | 34.004 | 19.666 | 38.405 | ||

60% | Median | 6.131 | 0.125 | 0.950 | 30.964 | 16.352 | 36.649 |

(m ≈ 38) | 95% CI | 5.316 | 0.065 | 0.900 | 27.513 | 13.893 | 34.672 |

7.049 | 0.179 | 0.990 | 34.320 | 20.532 | 38.692 | ||

50% | Median | 6.237 | 0.129 | 0.950 | 31.479 | 16.717 | 36.969 |

(m ≈ 32) | 95% CI | 5.293 | 0.064 | 0.900 | 27.820 | 13.757 | 34.930 |

7.180 | 0.179 | 0.990 | 34.854 | 22.119 | 39.250 | ||

40% | Median | 6.360 | 0.134 | 0.950 | 32.125 | 17.165 | 37.429 |

(m ≈ 25) | 95% CI | 5.441 | 0.074 | 0.900 | 28.500 | 13.910 | 35.075 |

7.380 | 0.193 | 0.990 | 35.636 | 23.292 | 40.166 | ||

30% | Median | 6.570 | 0.140 | 0.950 | 33.163 | 17.933 | 38.137 |

(m ≈ 19) | 95% CI | 5.565 | 0.075 | 0.900 | 29.036 | 13.905 | 35.393 |

7.651 | 0.201 | 0.990 | 36.916 | 25.407 | 40.861 | ||

20% | Median | 6.951 | 0.153 | 0.950 | 35.263 | 19.946 | 39.694 |

(m ≈ 12) | 95% CI | 5.857 | 0.086 | 0.900 | 31.082 | 14.074 | 36.427 |

8.139 | 0.218 | 0.990 | 39.625 | 28.892 | 43.619 | ||

10% | Median | 8.026 | 0.196 | 0.950 | 39.618 | 26.914 | 43.025 |

(m ≈ 6) | 95% CI | 6.592 | 0.119 | 0.900 | 34.681 | 15.495 | 38.368 |

9.683 | 0.276 | 0.990 | 46.043 | 34.276 | 49.479 |

**Table 5.**Leave-one-out cross-validation metrics for DRNN models built on different fractions of available skeletal traits.

Accuracy | Bias | Validity | Efficiency | ||||
---|---|---|---|---|---|---|---|

Available Traits % | MAE | ${\widehat{\mathit{\beta}}}_{\mathit{e}}$ | $\mathit{P}\mathbf{\left(}\mathbf{\alpha}\mathbf{\right)}$ | PIW | PIW 95% CI | ||

90% | Median | 5.942 | 0.121 | 0.953 | 30.276 | 15.745 | 36.278 |

(m ≈ 57) | 95% CI | 5.699 | 0.110 | 0.940 | 28.748 | 14.339 | 34.599 |

6.198 | 0.131 | 0.965 | 31.797 | 18.048 | 37.772 | ||

80% | Median | 5.970 | 0.122 | 0.953 | 30.476 | 15.941 | 36.332 |

(m ≈ 51) | 95% CI | 5.702 | 0.108 | 0.940 | 28.860 | 14.162 | 34.574 |

6.235 | 0.132 | 0.965 | 31.963 | 18.470 | 37.938 | ||

70% | Median | 6.028 | 0.124 | 0.953 | 30.711 | 16.182 | 36.518 |

(m ≈ 44) | 95% CI | 5.737 | 0.108 | 0.938 | 28.960 | 14.013 | 34.697 |

6.376 | 0.137 | 0.965 | 32.583 | 19.643 | 38.435 | ||

60% | Median | 6.078 | 0.125 | 0.953 | 30.975 | 16.342 | 36.716 |

(m ≈ 38) | 95% CI | 5.768 | 0.108 | 0.938 | 29.070 | 13.872 | 34.756 |

6.441 | 0.140 | 0.965 | 33.017 | 20.569 | 38.732 | ||

50% | Median | 6.173 | 0.128 | 0.953 | 31.502 | 16.684 | 37.040 |

(m ≈ 32) | 95% CI | 5.819 | 0.111 | 0.938 | 29.410 | 13.724 | 34.989 |

6.648 | 0.146 | 0.968 | 33.900 | 22.110 | 39.305 | ||

40% | Median | 6.305 | 0.132 | 0.953 | 32.146 | 17.153 | 37.511 |

(m ≈ 25) | 95% CI | 5.903 | 0.114 | 0.935 | 29.839 | 13.905 | 35.130 |

6.797 | 0.153 | 0.968 | 34.565 | 23.287 | 40.214 | ||

30% | Median | 6.501 | 0.138 | 0.953 | 33.097 | 17.923 | 38.203 |

(m ≈ 19) | 95% CI | 6.046 | 0.118 | 0.935 | 30.583 | 13.899 | 35.468 |

7.096 | 0.163 | 0.965 | 35.986 | 25.377 | 40.943 | ||

20% | Median | 6.957 | 0.154 | 0.953 | 35.321 | 19.986 | 39.742 |

(m ≈ 12) | 95% CI | 6.316 | 0.127 | 0.935 | 32.096 | 14.117 | 36.479 |

7.674 | 0.184 | 0.968 | 38.931 | 28.768 | 43.707 | ||

10% | Median | 7.952 | 0.192 | 0.955 | 39.733 | 26.846 | 43.076 |

(m ≈ 6) | 95% CI | 6.968 | 0.154 | 0.940 | 35.229 | 15.515 | 38.419 |

9.214 | 0.256 | 0.973 | 46.437 | 34.087 | 49.551 |

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**MDPI and ACS Style**

Navega, D.; Costa, E.; Cunha, E.
Adult Skeletal Age-at-Death Estimation through Deep Random Neural Networks: A New Method and Its Computational Analysis. *Biology* **2022**, *11*, 532.
https://doi.org/10.3390/biology11040532

**AMA Style**

Navega D, Costa E, Cunha E.
Adult Skeletal Age-at-Death Estimation through Deep Random Neural Networks: A New Method and Its Computational Analysis. *Biology*. 2022; 11(4):532.
https://doi.org/10.3390/biology11040532

**Chicago/Turabian Style**

Navega, David, Ernesto Costa, and Eugénia Cunha.
2022. "Adult Skeletal Age-at-Death Estimation through Deep Random Neural Networks: A New Method and Its Computational Analysis" *Biology* 11, no. 4: 532.
https://doi.org/10.3390/biology11040532