4. Multi-output regression fundamentals

Occupant injury prediction is rarely a single-number problem. A crash pulse applied to a restrained occupant produces a vector of responses: head acceleration, chest deflection, pelvis acceleration, neck injury criterion, and so on. A model that predicts only one of these is of limited use in restraint design, because design decisions trade one response against another. Multi-output regression treats the whole response vector as the learning target, and this section establishes the formulation, the two standard loss functions, and the metrics that tell you whether the model is actually useful.

Problem formulation

Let each sample be a crash configuration described by an input vector x ∈ R^d — for example, impact velocity, pulse duration, seat belt load limiter force, pretensioner firing time, and occupant anthropometry. The target is a vector y ∈ R^k, where k is the number of injury responses. A multi-output regressor is a parameterized function f_θ: R^d → R^k trained on a dataset {(x_i, y_i)}_{i=1}^{N}.

The critical structural decision is whether the outputs share a representation. A shared-trunk architecture passes x through common hidden layers and then splits into k output heads. This is the natural choice when responses are physically coupled — chest deflection and chest acceleration both depend on belt force and loading rate — because the trunk can learn a shared latent description of the loading state. An independent architecture trains k separate models, which is simpler to debug but discards cross-output correlation and multiplies training cost. A middle path uses a shared trunk with per-output scaling and, optionally, a small output-specific head.

Because the outputs live on wildly different scales — head acceleration in hundreds of g, chest deflection in millimeters, HIC as a dimensionless number in the hundreds — the target vector must be standardized before training. Compute the mean μ_j and standard deviation σ_j for each output j on the training split only, then train on ỹ_j = (y_j − μ_j)/σ_j. At inference, invert the transform. Skipping this step lets the largest-magnitude output dominate the gradient and effectively turns a multi-output problem into a single-output one.

Loss functions: MSE and MAE

For a batch of N samples and k outputs, the two standard losses are:

MSE = (1 / (N·k)) · Σ_i Σ_j (ŷ_ij − y_ij)²
MAE = (1 / (N·k)) · Σ_i Σ_j |ŷ_ij − y_ij|

MSE penalizes errors quadratically, so a single large miss — a model that predicts 45 mm chest deflection when the reference is 70 mm — contributes disproportionately. Its gradient is proportional to the residual, which means large errors produce large updates and convergence is fast in the early epochs. The cost is sensitivity to outliers: a handful of severe, high-energy cases can pull the fit away from the bulk of the data. MAE penalizes linearly and its gradient has constant magnitude, making it robust to outliers but slower to converge near the optimum and non-differentiable at zero (in practice, frameworks use a smooth approximation or a subgradient).

A practical compromise is the Huber loss, quadratic for small residuals and linear beyond a threshold δ:

L_δ(r) = 0.5·r²            if |r| ≤ δ
       = δ·(|r| − 0.5·δ)   otherwise

With standardized targets, δ = 1.0 is a reasonable starting point, meaning residuals beyond one standard deviation are treated linearly. In injury modeling, where a small number of extreme crash cases are physically real and must not be ignored, Huber often outperforms both pure MSE and pure MAE.

Evaluation metrics

Training loss is not an evaluation metric. Report metrics on a held-out test split, per output, and always alongside the trivial baseline that predicts the training mean.

Per-output RMSE. For output j, RMSE_j = sqrt( (1/N) Σ_i (ŷ_ij − y_ij)² ). Report it in the original physical units so engineers can judge it against tolerance. A chest-deflection RMSE of 3.2 mm is interpretable; a standardized RMSE of 0.4 is not. Also report R²_j = 1 − SSE_j/SST_j; a negative R² means the model is worse than predicting the mean and signals a broken pipeline, not a tuning problem.

Correlation. Pearson correlation between predictions and references, computed per output, measures whether the model ranks cases correctly even if it is biased. This matters in design screening, where the goal is to identify which configurations are worse than others. Report Pearson r per output and, if the response distribution is skewed, Spearman rank correlation as well. High r with high RMSE indicates a systematic offset — often a missed inverse transform or a calibration issue — rather than a modeling failure.

Cross-output diagnostics. Compare the predicted correlation matrix between outputs against the reference correlation matrix. If chest deflection and pelvis acceleration are strongly correlated in the data but the model's predictions are nearly independent, the shared trunk is not capturing the coupling, and an independent-output model would perform just as well with less complexity.

Worked example: reading the metrics

Suppose a shared-trunk model is trained on 1,200 simulated sled tests with four outputs, using Huber loss on standardized targets. On a 240-case test split, the per-output results are summarized below. These numbers are illustrative, not measured.

The pattern is diagnostic. HIC has the lowest R² but a respectable correlation, which suggests the model ranks cases reasonably but compresses the extremes — HIC is a threshold-like quantity, and a smooth regressor tends to regress toward the mean near the threshold. The remedy is not more epochs but a different target transform (for example, predicting log(HIC)) or a loss that up-weights the high-HIC region. Pelvis acceleration's moderate R² with high r points to a scale bias; check whether the inverse standardization was applied with the correct per-output σ.

The formulation, loss, and metrics form one coherent chain: standardize the targets so no output dominates, choose MSE, MAE, or Huber according to how much you trust the extreme cases, and evaluate per output in physical units with both error magnitude and correlation. A model is only as good as its weakest output, so report all of them.