🎓 Lesson 21
D5
Uncertainty Band Visualization Lab
An uncertainty band is a visual range around a predicted value that shows how much the actual result might vary due to unknown or variable factors.
🎯 Learning Objectives
- ✓ Calculate uncertainty bands for unit mining cost using Monte Carlo simulation outputs
- ✓ Design visualization plots that overlay uncertainty bands on time-series production cost forecasts
- ✓ Analyze how input parameter sensitivities (e.g., ore grade variance, diesel price volatility) drive band width
- ✓ Explain the difference between prediction intervals and tolerance intervals in cost modeling contexts
- ✓ Apply ISO 16269-6 and AS/NZS ISO 31010 standards to validate uncertainty band reporting
📖 Why This Matters
In open-pit mine planning, a $5/ton error in estimated operating cost can swing NPV by tens of millions over a 15-year life-of-mine. Traditional 'single-point' cost estimates hide risk — uncertainty bands make it visible, actionable, and communicable to executives and investors. This lab teaches you to translate statistical uncertainty into engineering decisions: e.g., whether to pre-strip earlier, hedge fuel contracts, or adjust ROM stockpile strategy.
📘 Core Principles
Uncertainty bands emerge from two primary sources: aleatory (inherent randomness, e.g., rock strength variability) and epistemic (knowledge limitations, e.g., sparse drill hole data). In cost modeling, bands are constructed by propagating distributions of input parameters (ore grade, equipment availability, energy prices) through a deterministic cost model using stochastic methods. Band width reflects both input uncertainty magnitude and model nonlinearity — narrow bands indicate robust predictions; widening bands signal high leverage points where additional data collection yields outsized ROI. Confidence-based bands (e.g., 90% prediction interval) differ fundamentally from tolerance intervals (which cover proportion of future observations) — selecting the correct type is critical for contractual and regulatory reporting.
📐 Prediction Interval for Unit Cost
For normally distributed residuals and linearized cost models, the 90% prediction interval for unit cost C at a given production scenario is computed using the standard error of prediction. This formula anchors visualization axes and informs contingency allowances.
90% Prediction Interval (PI)
PI_{0.90} = \hat{y}_0 \pm t_{\alpha/2, df} \cdot \sqrt{SE(\hat{y}_0)^2 + \hat{\sigma}^2}Computes the interval within which a new observation of unit cost is expected to fall with 90% confidence.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \hat{y}_0 | Predicted unit cost | $/ton | Mean output of cost model for scenario 0 |
| t_{\alpha/2, df} | Critical t-value | dimensionless | From Student's t-distribution for α = 0.10 and degrees of freedom from regression |
| SE(\hat{y}_0) | Standard error of prediction | $/ton | Uncertainty in the model's mean prediction due to parameter estimation |
| \hat{\sigma} | Residual standard deviation | $/ton | Model inadequacy — unexplained variation after calibration |
Typical Ranges:
Base-case LOM cost forecast: $1.10 – $3.40
Early-stage feasibility study: $4.20 – $9.80
💡 Worked Example
Problem: Given: mean unit cost estimate = $18.40/ton, standard error of prediction = $1.22/ton, t-value for 90% PI with df=24 = 1.711 (from t-distribution), and model residual standard deviation = $1.18/ton.
1.
Step 1: Compute margin of error = t × √[SE² + σ²_residual] = 1.711 × √[(1.22)² + (1.18)²]
2.
Step 2: Evaluate √[1.488 + 1.392] = √2.880 ≈ 1.697
3.
Step 3: Margin of error = 1.711 × 1.697 ≈ $2.90/ton
4.
Step 4: Lower bound = $18.40 − $2.90 = $15.50/ton; Upper bound = $18.40 + $2.90 = $21.30/ton
Answer:
The 90% prediction interval is [$15.50, $21.30]/ton, which exceeds typical industry contingency allowance of ±12% ($2.21), indicating need for either model refinement or explicit risk mitigation.
🏗️ Real-World Application
At Newmont’s Boddington Mine (Western Australia), uncertainty band visualization was deployed during the 2022 pit expansion review. Input distributions for haul truck payload (±8.3% CV), crushing circuit throughput (lognormal, μ=12.4 Mt/y, σ=1.7 Mt/y), and electricity cost ($0.11–$0.29/kWh, triangular) were propagated via 10,000 Monte Carlo iterations. The resulting 90% uncertainty band for G&A cost per ton widened from ±$0.85 (baseline) to ±$2.10 when including tailings storage facility escalation risk — triggering a $42M contingency reserve increase and revised bond structure with lenders.
✏️ Student Exercise
Using the provided Excel dataset (simulated cost model outputs: 500 iterations of unit mining cost in $/ton), students will: (1) Calculate the 80% and 95% prediction intervals using bootstrap resampling; (2) Plot the median forecast with shaded uncertainty bands overlaid on a 5-year production schedule; (3) Identify the year with maximum band width and explain, using partial correlation coefficients, which two inputs contribute >70% of its variance.