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Uncertainty quantification in measurement and models

Build uncertainty budgets, propagate correlated inputs, compare Monte Carlo and linear methods, and separate numerical error from model discrepancy.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Measurement equation and sensitivity

Definitions & inputs. y=f(x),ui standard uncertainties,C covariance matrix,J gradient at input estimates.

  1. Taylor-expand about nominal input values.

    δy≃Jδx,Ji=∂f/∂xi\delta y\simeq J\delta x,\quad J_i=\partial f/\partial x_i

  2. Propagate both variance and covariance.

    uy2=JCJT=∑iJi2ui2+2∑i<jJiJjCiju_y^2=JCJ^T=\sum_iJ_i^2u_i^2+2\sum_{i<j}J_iJ_jC_{ij}

  3. A shared calibration error can prevent averaging benefits.

    y=x1+x2⇒uy2=u12+u22+2ρu1u2y=x_1+x_2\Rightarrow u_y^2=u_1^2+u_2^2+2\rho u_1u_2

Interpretation. Standard uncertainty is not an error bound. Correct known bias where possible and retain uncertainty in that correction.

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2. Nonlinear propagation and Monte Carlo

Definitions & inputs. X input random vector,Y=f(X),Ns draws,smc sampling standard error.

  1. Simulate the measurement or physical model with input draws.

    X(k)∼pX,Y(k)=f(X(k))X^{(k)}\sim p_X,\quad Y^{(k)}=f(X^{(k)})

  2. Estimate output moments and their simulation error for independent draws.

    E^[Y]=Ns−1∑kY(k),SE≃sY/Ns\widehat E[Y]=N_s^{-1}\sum_kY^{(k)},\quad SE\simeq s_Y/\sqrt{N_s}

  3. Quantiles describe asymmetric coverage intervals when a symmetric linear approximation is poor.

    qp=FY−1(p)q_p=F_Y^{-1}(p)

Interpretation. Check convergence of tails separately; rare-event probabilities can need importance sampling. Correlated MCMC samples need effective sample-size analysis.

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3. Sensitivity and surrogate models

Definitions & inputs. Vi variance attributable to input i,V output variance,STi total-effect So bol index.

  1. Orthogonal functional ANOVA separates main effects and interactions.

    V=∑iVi+∑i<jVij+⋯V=\sum_iV_i+\sum_{i<j}V_{ij}+\cdots

  2. First-order and total-effect indices answer different sensitivity questions.

    Si=Vi/V,STi=1−Var⁡(E[Y∣X−i])/VS_i=V_i/V,\quad S_{Ti}=1-\operatorname{Var}(E[Y|X_{-i}])/V

  3. Polynomial chaos is one surrogate option; Gaussian processes and response surfaces have different assumptions.

    f^(x)=∑αcαΨα(x)\widehat f(x)=\sum_\alpha c_\alpha\Psi_\alpha(x)

Interpretation. Local derivatives, global variance shares and causal effects are not interchangeable. Validate surrogates where decisions depend on their accuracy.

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4. Calibration, discrepancy and verification

Definitions & inputs. d observations,f(θ) simulator,δ discrepancy,ε measurement noise,h mesh size,p observed convergence order.

  1. Separate parameter uncertainty, missing physics and measurement error.

    d=f(θ)+δ+ϵd=f(\theta)+\delta+\epsilon

  2. Bayesian calibration must include the assumptions defining the likelihood and discrepancy.

    p(θ∣d)∝p(d∣θ)p(θ)p(\theta|d)\propto p(d|\theta)p(\theta)

  3. Extrapolate a leading discretization term when the convergence model applies.

    f∗≃fh/2+(fh/2−fh)/(2p−1)f_*\simeq f_{h/2}+(f_{h/2}-f_h)/(2^p-1)

Interpretation. Calibration is not validation; numerical convergence is not physical accuracy. Report input distributions, correlations, solver error, discrepancy assumptions, coverage meaning and sensitivity results.

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Graphical worked example

Output standard deviation 2 in the output variable’s units. Simulation error shrinks as N⁻¹/²; physical spread does not. X axis: Independent Monte Carlo samples (count). Y axis: Standard error of estimated mean.
Output standard deviation 2 in the output variable’s units. Simulation error shrinks as N⁻¹/²; physical spread does not. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Independent sum uncertainty

Definitions & inputs. u1.3,u2.4.

  1. Choose the governing model and isolate the requested quantity.

    u=u12+u22u=\sqrt{u_1^2+u_2^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .09+.16\sqrt{.09+.16}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Same input units.

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Example 02. Correlated sum

Definitions & inputs. u1.3,u2.4,ρ.5.

  1. Choose the governing model and isolate the requested quantity.

    u=.09+.16+2ρ(.3)(.4)u=\sqrt{.09+.16+2\rho(.3)(.4)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .37\sqrt{.37}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6082763 \mathrm{Result}=0.6082763\ {}

Interpretation. Positive correlation increases uncertainty.

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Example 03. Difference with common bias

Definitions & inputs. Equalu1=u2=.2,ρ1.

  1. Choose the governing model and isolate the requested quantity.

    u2=u12+u22−2ρu1u2u^2=u_1^2+u_2^2-2\rho u_1u_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .04+.04−.08.04+.04-.08

  3. Evaluate the expression; the result uses the units shown.

    Result=0 \mathrm{Result}=0\ {}

Interpretation. Perfect shared additive error cancels in this idealized difference.

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Example 04. Scale sensitivity

Definitions & inputs. y3x.

  1. Choose the governing model and isolate the requested quantity.

    J=dy/dxJ=dy/dx

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    33

  3. Evaluate the expression; the result uses the units shown.

    Result=3 \mathrm{Result}=3\ {}

Interpretation. Coefficient units follow the model.

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Example 05. Scaled uncertainty

Definitions & inputs. y3x,ux.2.

  1. Choose the governing model and isolate the requested quantity.

    uy=∣J∣uxu_y=|J|u_x

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3(.2)3(.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6 \mathrm{Result}=0.6\ {}

Interpretation. Linear propagation exact.

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Example 06. Relative product uncertainty

Definitions & inputs. Independent relative uncertainties.02,.03.

  1. Choose the governing model and isolate the requested quantity.

    ur=.022+.032u_r=\sqrt{.02^2+.03^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .0013\sqrt{.0013}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.03605551 \mathrm{Result}=0.03605551\ {}

Interpretation. First-order product law.

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Example 07. Area sensitivity

Definitions & inputs. Circle r2m.

  1. Choose the governing model and isolate the requested quantity.

    dA/dr=2πrdA/dr=2\pi r

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4π4\pi

  3. Evaluate the expression; the result uses the units shown.

    Result=12.56637 m\mathrm{Result}=12.56637\ \mathrm m

Interpretation. Gradient ofπr².

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Example 08. Area uncertainty

Definitions & inputs. r2m,ur.01m.

  1. Choose the governing model and isolate the requested quantity.

    uA=2πruru_A=2\pi r u_r

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4π(.01)4\pi(.01)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1256637 m2\mathrm{Result}=0.1256637\ \mathrm{m^2}

Interpretation. Small radius uncertainty.

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Example 09. Uniform tolerance standard uncertainty

Definitions & inputs. Uniform error±.1mm.

  1. Choose the governing model and isolate the requested quantity.

    u=a/3u=a/\sqrt3

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .1/3.1/\sqrt3

  3. Evaluate the expression; the result uses the units shown.

    Result=0.05773503 mm\mathrm{Result}=0.05773503\ \mathrm{mm}

Interpretation. Uniform distribution assumption explicit.

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Example 10. Quantization uncertainty

Definitions & inputs. Step.01V,uniform rounding.

  1. Choose the governing model and isolate the requested quantity.

    u=Δ/12u=\Delta/\sqrt{12}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .01/12.01/\sqrt{12}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.002886751 V\mathrm{Result}=0.002886751\ \mathrm V

Interpretation. Not all ADC errors are quantization.

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Example 11. Expanded uncertainty

Definitions & inputs. u.5,k2.

  1. Choose the governing model and isolate the requested quantity.

    U=kuU=ku

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2(.5)2(.5)

  3. Evaluate the expression; the result uses the units shown.

    Result=1 \mathrm{Result}=1\ {}

Interpretation. Approximately95percent only under appropriate coverage assumptions.

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Example 12. Monte Carlo mean SE

Definitions & inputs. Output standard deviation2,N10000independent draws.

  1. Choose the governing model and isolate the requested quantity.

    SE=s/NSE=s/\sqrt N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2/1002/100

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02 \mathrm{Result}=0.02\ {}

Interpretation. Simulation error,not physical output spread.

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Example 13. Draw count for target SE

Definitions & inputs. s2,target.01.

  1. Choose the governing model and isolate the requested quantity.

    N=(s/SE)2N=(s/SE)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (2/.01)2(2/.01)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=40000 \mathrm{Result}=40000\ {}

Interpretation. Estimated finite variance assumption.

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Example 14. Bernoulli probability SE

Definitions & inputs. p.01,N10000.

  1. Choose the governing model and isolate the requested quantity.

    SE=p(1−p)/NSE=\sqrt{p(1-p)/N}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .01(.99)/10000\sqrt{.01(.99)/10000}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0009949874 \mathrm{Result}=0.0009949874\ {}

Interpretation. Tail estimation can be relatively noisy.

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Example 15. Zero-event upper bound

Definitions & inputs. Zero events,N1000,one-sided95percent.

  1. Choose the governing model and isolate the requested quantity.

    pU=1−.051/Np_U=1-.05^{1/N}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−.051/10001-.05^{1/1000}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.00299125 \mathrm{Result}=0.00299125\ {}

Interpretation. Exact binomial upper bound under independence.

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Example 16. Nonlinear mean correction

Definitions & inputs. Xmean2,variance.25,Y=X².

  1. Choose the governing model and isolate the requested quantity.

    E[Y]=E[X]2+V[X]E[Y]=E[X]^2+V[X]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4+.254+.25

  3. Evaluate the expression; the result uses the units shown.

    Result=4.25 \mathrm{Result}=4.25\ {}

Interpretation. Plugging the mean into f would miss.25.

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Example 17. First-order Sobol share

Definitions & inputs. Vi2,totalV5.

  1. Choose the governing model and isolate the requested quantity.

    Si=Vi/VS_i=V_i/V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2/52/5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4 \mathrm{Result}=0.4\ {}

Interpretation. Independent-input decomposition.

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Example 18. Total-effect index

Definitions & inputs. Var conditional mean3,total5.

  1. Choose the governing model and isolate the requested quantity.

    ST=1−3/5S_T=1-3/5

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .4.4

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4 \mathrm{Result}=0.4\ {}

Interpretation. Includes interactions with that input.

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Example 19. Richardson extrapolation

Definitions & inputs. fh1.04,fh2=1.01,p2.

  1. Choose the governing model and isolate the requested quantity.

    f∗=fh/2+(fh/2−fh)/(2p−1)f_*=f_{h/2}+(f_{h/2}-f_h)/(2^p-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.01+(1.01−1.04)/31.01+(1.01-1.04)/3

  3. Evaluate the expression; the result uses the units shown.

    Result=1 \mathrm{Result}=1\ {}

Interpretation. Asymptotic leading error assumed.

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Example 20. Independent discrepancy combination

Definitions & inputs. Measurement u.2,model discrepancy sd.5.

  1. Choose the governing model and isolate the requested quantity.

    u=.22+.52u=\sqrt{.2^2+.5^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .29\sqrt{.29}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5385165 \mathrm{Result}=0.5385165\ {}

Interpretation. Assumed zero mean independent discrepancy;not universal.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.