2. Statistical foundations for CAE correlation

Occupant CAE correlation rarely deals with a single deterministic number. A dummy chest deflection from a sled test is one realization of a physical process that varies with build tolerances, material batch, fixture alignment, and measurement noise; the corresponding CAE value varies with mesh density, material card calibration, contact parameters, and hourglass control. Before any correlation metric (peak ratio, ISO/TS 18571 score, regression R²) can be trusted, the analyst must treat both sides as random variables and reason about their distributions, their expected values, their spread, and how they co-vary. This chapter establishes that statistical vocabulary in the specific context of occupant response signals.

Random variables in the occupant-response context

A random variable maps each outcome of an experiment to a real number. In occupant CAE correlation the "experiment" is a test campaign or a DOE run set, and the random variable is typically a scalar extracted from a time history: peak head acceleration, HIC₃₆, peak chest deflection, maximum femur force, or the time of peak pelvis acceleration. Formally, if a_head(t) is the acceleration signal, then X = max_t a_head(t) is a derived random variable. Distinguishing the signal (a function of time) from the extracted scalar (a random variable) is the first discipline: statistics apply to the scalar, while correlation metrics such as ISO/TS 18571 operate on the full curve.

Two categories matter. Discrete random variables arise from pass/fail or categorical outcomes — e.g., whether the HIC₃₆ exceeds 700, or which of three injury-risk bands a chest deflection falls into. Continuous random variables dominate: peak values, pulse durations, and integrated metrics are continuous. In practice, CAE outputs are continuous but reported at finite precision, so treat them as continuous and avoid binning unless the metric is genuinely categorical.

Distributions: what shape should we expect?

Occupant response metrics are rarely Gaussian in the tails. Peak acceleration and HIC are bounded below by zero and right-skewed; a lognormal or Weibull shape often fits better than a normal. Chest deflection under belt loading can be approximately symmetric, but femur force in a knee-bolster impact is frequently bimodal because the dummy can load the bolster or slip past it. The practical rule is: plot the histogram and the empirical CDF before assuming a family. With small test campaigns (n = 3 to 8 is common in sled testing), fitting a parametric distribution is fragile; the empirical distribution and robust summary statistics are safer.

For CAE, the distribution of interest is often the output distribution induced by input uncertainty. A 20-run Latin hypercube over friction coefficient, belt stiffness, and pad thickness produces 20 peak chest deflections whose histogram approximates the CAE response distribution. Comparing that histogram to the test histogram — not just comparing means — is the honest correlation question.

Expectation and variance: location and spread

The expectation E[X] is the probability-weighted average. For a test campaign with n runs, the sample mean x̄ estimates it. The variance Var(X) = E[(X − E[X])²] measures spread; its square root is the standard deviation σ, which carries the same units as the metric (g, mm, kN) and is therefore the number to quote in a correlation report. A common error is to compare a single CAE peak to a single test peak and declare "good correlation" when the test-to-test scatter alone is ±8%. The defensible statement is: the CAE value lies within one test standard deviation of the test mean, or the difference is small relative to σ_test.

Two variance facts are used constantly. First, Var(aX + b) = a²Var(X): scaling a signal (e.g., converting g to m/s²) scales the standard deviation by |a|, not by a². Second, for independent replicates, Var(x̄) = σ²/n, so the standard error of the mean shrinks as √n — which is why a 3-test campaign gives a mean with roughly 58% of the standard error of a single test, and why adding tests has diminishing returns for mean estimation but not for tail estimation.

Covariance and correlation: do test and CAE move together?

Covariance measures joint variation: Cov(X,Y) = E[(X − E[X])(Y − E[Y])]. Its sign tells direction — positive covariance means that when the test peak is high, the CAE peak also tends to be high. Its magnitude is unit-dependent, so we normalize to the Pearson correlation coefficient ρ = Cov(X,Y)/(σ_Xσ_Y), which lies in [−1, 1]. In occupant correlation, ρ is computed across a set of configurations (e.g., 6 impact conditions × 3 dummy sizes = 18 paired points), not across time samples of one signal. Computing ρ across time samples of a single crash pulse is a classic mistake: it measures waveform shape similarity, not correlation of responses across configurations, and it is dominated by the pulse's overall envelope.

Correlation is necessary but not sufficient. A CAE model can have ρ = 0.95 across configurations yet be biased high by 12% on every point; the covariance captures the trend, the mean difference captures the bias. Report both. Also note that ρ is sensitive to outliers and to range restriction: if all test configurations happen to produce similar peaks, σ_test is small and ρ becomes unstable.

Working through a paired comparison

Consider an illustrative campaign of 12 configurations (three impact severities × two dummy sizes × two restraint variants). For each configuration, extract the peak chest deflection from the filtered test signal and from the CAE history, using the same CFC 600 filter and the same time window. The paired values in the figure above are constructed for teaching; they are not measured data. The sample statistics are:

n        = 12
x̄_test   = 31.8 mm      σ_test = 6.4 mm
x̄_CAE    = 33.1 mm      σ_CAE  = 6.1 mm
Cov      = 37.2 mm²
ρ        = 37.2 / (6.4 × 6.1) ≈ 0.95
bias     = x̄_CAE − x̄_test = +1.3 mm  (+4.1 %)
slope    = Cov / σ_test² ≈ 0.91

Interpretation: the model tracks configuration-to-configuration variation well (ρ = 0.95), but it is biased high by about 1.3 mm, roughly 0.2 test standard deviations. A slope below 1 indicates the model compresses the range — it under-predicts the high-deflection cases and over-predicts the low ones. Neither the bias nor the slope is visible from ρ alone, which is why the report should carry mean difference, σ ratio, ρ, and slope together.

Practical cautions

• Small n. With n < 10, ρ has a wide confidence interval; a ρ of 0.8 from 6 points is not strong evidence. Report n alongside every statistic.
• Units and filtering. Covariance and σ carry units; state them. Apply identical CFC filtering and identical extraction windows to test and CAE before pairing.
• Outliers. A single mis-extracted peak (e.g., a contact spike at t = 0) can dominate Cov. Inspect the scatter plot, not just the number.
• Bias vs. scatter. A model with low bias but high σ is unreliable per configuration; a model with high bias but low σ is consistently wrong and easy to correct. Distinguish the two before tuning.

These four quantities — distribution shape, mean, standard deviation, and covariance/correlation — form the minimum statistical basis on which the regression and correlation metrics of later chapters are built. Establishing them on paired scalars first prevents the most common failure in occupant CAE correlation: reporting a single correlation score that hides bias, range compression, and test-to-test scatter.