Uncertainty Analysis Procedure

	Clean Air Engineering
	Kelly Aita

Overview

	Estimate the current measurement uncertainties that exist in testing
		Proves to clients that we are the leader in the field
		Gives clients additional tool to show compliance
	Creates quantitative tool that can be used to reduce these uncertainties
	Analysis evaluates the uncertainty in testing, not the method
		Assuming the EPA method has no uncertainty in itself
		Essentially the uncertainty measurement involves no Bias created by a faulty method.

Background

	Uncertainty quantification is very judgmental and requires experience and understanding of the testing method
	Continuously changing method which should be constantly monitored and updated

Introduction

	The quantity being determined is defined by Y
		Parameters (inputs) used to describe Y are X1, X2, X3 É Xi.
		These quantities are estimated by the terms y, x1, x2, x3 Éxi.
	Uncertainties
		The standard uncertainty of an input is u(xi)
		The combined standard uncertainty is uc(y),
		The expanded uncertainty is the value which is reported.

Procedure

	Write down equations that are used in calculations
	Determine uncertainty sources
		Variables being measured
		Contributions due to the testing process (some research might be necessary)
	Evaluate sensitivity of each parameter
		Can be done numerically using software

Procedure (contÕd)

	Evaluate covariance between all parameters
		Needs to be completed only once for each process
	Calculate combined uncertainty u(y)
	Calculate expanded uncertainty K*u(y)

Systematic Uncertainties

	Provided by manufacturer (x±a)
			Interpreted by own judgment according to probability distribution
	Experimentally
		Taking n measurements and using standard deviation s
	Theoretical
		Using coefficient of thermal expansion to derive volumetric uncertainty
	EPA methods as listed minimum accuracy

Probability Distribution

	Triangular distribution
	Rectangular distribution

Probability Distribution

	Normal distribution
		Manufacturer can report in different forms
			Standard Deviation u(x) = s
			Relative Standard Deviation
			Coefficient of Variance
			Uncertainty Interval (±c)
				95%  u(x) =c/2
				99%  u(x) =c/3

Sensitivity Parameter,

	The sensitivity quantifies the effect the parameter has on the output
		Solve analytically
		Solve Experimentally
			alter xi while observing the effect to y
		Solve numerically
			Assume linear relation or u/x is very small

Covariance

	Correlation between two input variables effect the combined standard uncertainty (The effect of one parameter might negate the effect of the other)

Determining Degree of Covariance, rik

	u(xi,xk) = u(xi)u(xk)rik for -1<rik<1
	Previous tests
		With enough data can calculate rik for a particular test/situation
	Analytical
	Experience
		Can make an educated guess about value of r

Random Uncertainty

	Involves the dispersion of data due to random effects
	Evaluated using sample variance s
		Normal Distribution (for n > 30)
		t- distribution (n < 30)

Combined Uncertainty

	Combine everything together
	u(y) essentially represents a standard deviation

Expanded Uncertainty, K*u(y)

	The reported uncertainty must include a description of what it is showing
		Confidence interval
			Probability that any data taken is within uncertainty limits
			Changes with distribution and degrees of freedom
		Certain limit (i.e. 3s)

Determining K

	t or z value taken from table or software
		Based on degrees of freedom (n-1)
		Must specify Confidence Interval (95% or 99% typically)
	Spreadsheet developed for this purpose

Conclusion

	Uncertainty analysis is conducted to allow the client to verify we provide the highest quality service
	Analysis is dynamic process and ever changing
	A rough Spreadsheet template has been created to begin process
	Process creates valuable way to audit ourselves and increase quality