Measured vs Predicted Strain: Correlation Methods

The Purpose of Correlation Methods

Once the FE strain has been extracted, transformed, averaged and scaled to match the gauge measurement, the engineer has, for each gauge, a predicted value and a measured value. The correlation method is how these pairs are compared and presented. The goal is not to produce a single number that summarises the correlation — it is to reveal the pattern of agreement and disagreement across all gauges, in a form that supports physical interpretation. Different methods — absolute difference, ratio, slope, scatter plot, regression — reveal different aspects of the pattern. The engineer should use several together, because each method is blind to something the others see.

Why It Matters

The correlation method determines what the engineer sees. A table of absolute differences shows which gauges match and which do not, but it hides whether the mismatches are proportional or systematic. A scatter plot shows the pattern but hides the individual gauge identities. A ratio analysis shows proportional error but hides the absolute magnitude. No single method gives the complete picture. An engineer who uses only one method will miss aspects of the correlation that another method would reveal. The discipline is to use multiple complementary methods and to read them together, building a complete picture of the correlation quality.

No single correlation metric tells the whole story. Use absolute difference, ratio, slope and scatter plot together — each reveals a different aspect of the agreement pattern.

Absolute Difference

The absolute difference is the simplest comparison: the measured strain minus the predicted strain, in microstrain. It tells the engineer the magnitude of the discrepancy at each gauge. It is useful for identifying which gauges match well (small difference) and which do not (large difference). However, the absolute difference is not normalised — a 100 microstrain difference at a gauge reading 200 microstrain is a 50% error, while a 100 microstrain difference at a gauge reading 2000 microstrain is a 5% error. The absolute difference must be interpreted in the context of the strain magnitude. Despite this limitation, the absolute difference is valuable because it is in the same units as the measurement and can be compared directly to the measurement uncertainty.

Absolute difference:

Δε  =  ε_test  −  ε_FE

where:
Δε     =  absolute difference (με)
ε_test =  measured strain (με)
ε_FE   =  predicted strain (με)

Interpret in context of magnitude:
  |Δε| / |ε_test|  =  relative difference (see ratio method)

Ratio (Test/FE)

The ratio of measured to predicted strain normalises the difference by the predicted magnitude. A ratio of 1.0 means perfect agreement. A ratio of 1.1 means the test is 10% higher than the prediction. A ratio of 0.9 means the test is 10% lower. The ratio is useful because it reveals proportional error: if the ratio is consistently 1.1 across many gauges, there is a systematic 10% error — most likely in the load, the modulus or a global stiffness. If the ratio varies randomly around 1.0, the agreement is good with scatter. If the ratio varies systematically with location — high in one region, low in another — there is a regional modelling issue. The ratio strips out the magnitude and focuses on the proportional agreement.

Ratio (test/predicted):

R  =  ε_test  /  ε_FE

where:
R      =  test/FE ratio (dimensionless)
ε_test =  measured strain (με)
ε_FE   =  predicted strain (με)

R = 1.0  → perfect agreement
R > 1.0  → test higher than prediction (structure less stiff than model)
R < 1.0  → test lower than prediction (structure stiffer than model)
Consistent R across gauges → systematic error (load, modulus, stiffness)

The Measured-Versus-Predicted Scatter Plot

The scatter plot is the primary correlation tool. Each gauge is one point: predicted strain on the x-axis, measured strain on the y-axis. Perfect correlation puts all points on the 45° line (y = x). The pattern of deviations from this line is the diagnostic. A uniform offset (all points above or below the line) suggests a systematic error. A slope different from 1.0 (points fan away from the 45° line as strain increases) suggests a stiffness error that scales with magnitude. A single point far from the line suggests a local gauge or modelling issue. A cluster of points off the line suggests a regional problem. Random scatter about the line suggests measurement noise or local detail variation. The scatter plot shows all of this at a glance and should be the first and last thing the engineer looks at in a correlation analysis.

Measured vs predicted scatter plot — reading the pattern:

  Measured (test)  με
   ↑
   |        /  y = x  (perfect correlation)
   |      /
   |    /     ·
   |  /    ·       ·
   |/  ·        ·        ·
   |——————————————→ Predicted (FE)  με
   0

  Pattern interpretations:
  · All points on line          → excellent correlation
  · All points above line        → test consistently high (load/modulus/stiffness)
  · Points fan from line (slope) → stiffness error scaling with strain
  · Single point off line        → local gauge or model issue
  · Cluster of points off line   → regional model deficiency
  · Random scatter about line    → measurement noise / local detail

Regression: Slope and Intercept

A linear regression on the scatter plot quantifies the pattern. The regression line y = m·x + b has a slope m and an intercept b. Perfect correlation gives m = 1.0 and b = 0. A slope greater than 1.0 means the test strain increases faster than the prediction — the structure is less stiff than the model. A slope less than 1.0 means the test strain increases slower — the structure is stiffer. A non-zero intercept means a constant offset — a preload, zero or thermal offset. The regression also gives an R-squared value that quantifies how tightly the points follow the regression line — a low R-squared means the correlation is poor even if the slope and intercept are right, because the scatter is large. The regression is a useful summary, but it should not replace the scatter plot itself — the plot shows outliers and clusters that the regression summary hides.

Slope as a Stiffness Indicator

The regression slope is a powerful global stiffness indicator. If the slope is consistently different from 1.0 across the gauges, the overall stiffness of the model is wrong. A slope of 1.2 means the test strains are 20% higher than predicted across the board — the structure is 20% less stiff than the model. This points to a global cause: the material modulus is too high, a joint is modelled too stiff, a boundary is too rigid, or a load path is missing compliance. The slope, combined with the deflection correlation, narrows the cause. A slope of 1.0 with large scatter, by contrast, means the global stiffness is right but the local distribution is wrong — the load is being distributed differently in the test than in the model. These are very different problems, and the slope distinguishes them.

The regression slope is a global stiffness indicator. A slope consistently different from 1.0 across gauges points to a global stiffness error — modulus, joint, boundary. A slope of 1.0 with scatter points to a local distribution error.

Correlation Metrics and What They Miss

Every correlation metric has blind spots. The absolute difference misses proportional error. The ratio misses the absolute magnitude and can be unstable at low strains (a ratio at 50 microstrain is noisy). The scatter plot shows the pattern but does not quantify it. The regression gives numbers but hides outliers. The R-squared can be high with a wrong slope. The engineer must be aware of these blind spots and use the methods in combination. The recommended set is: the scatter plot (primary visual), the regression slope and intercept (global summary), the ratio per gauge (proportional error), and the absolute difference (comparison with uncertainty). Together, these give a complete picture; individually, each misses something important.

• Absolute difference — shows magnitude, misses proportional error, compare to uncertainty
• Ratio — shows proportional error, unstable at low strain, misses absolute magnitude
• Scatter plot — shows pattern, misses quantification, the primary visual tool
• Regression slope — shows global stiffness error, misses outliers and local patterns
• R² — shows fit quality, can be high with wrong slope, always report with slope

Presenting the Correlation

The correlation should be presented in a form that allows the reader to see the pattern and assess the quality. The recommended presentation includes: the scatter plot with the 45° reference line and the regression line; a table of all gauges with predicted, measured, difference, ratio and comment; the regression statistics (slope, intercept, R²); and the global checks (reaction, deflection). The presentation should not reduce the correlation to a single number — "the correlation is 85%" is meaningless because it hides the pattern. The engineer should present the evidence and state the interpretation, not a summary statistic. The reader should be able to see which gauges match, which do not, what the pattern is, and what the engineer concludes from it.

CONSIDERATION: Do not reduce a correlation to a single percentage or score. Present the scatter plot, the per-gauge table, the regression and the global checks. Let the reader see the pattern and follow the interpretation.

Key Takeaways

• The measured-versus-predicted scatter plot is the primary correlation tool — it shows the whole pattern at a glance
• Absolute difference, ratio and slope each reveal a different aspect — use them in combination, not alone
• Regression slope quantifies global stiffness error; intercept quantifies systematic offset; R² quantifies scatter
• A consistent ratio across gauges suggests a systematic error; random scatter suggests noise or local detail
• Present the full evidence — scatter plot, table, regression, global checks — not a single summary number

Key takeaways

• The measured-versus-predicted scatter plot is the primary tool — it shows the whole correlation pattern at a glance and reveals offsets, slopes, outliers and scatter.
• Absolute difference, ratio and slope each interrogate a different aspect of the correlation — use them together, not in isolation.
• Regression on the scatter plot gives a slope and an intercept that quantify systematic error — a slope of 1.0 and intercept of 0.0 is perfect correlation.
• A consistent ratio across gauges suggests a systematic error (load or modulus); random scatter suggests noise or local detail issues.
• The correlation quality is in the pattern across many gauges, not in any single comparison — present and read the pattern.