Uncertainty Analysis

Uncertainty Analysis – Process of systematically quantifying error estimates

Measurement Errors

• Bias Error.
• Precision (or random) Error
• Blunders of Experimenter

Remember

Uncertainty analysis is a method to quantify u_x

Assumptions:

• Test objectives are known
• Measurement is clearly defined process, i.e., calibrations and bias error have been accounted for.
• Data are obtained under fixed conditions
• Knowledge of system components is available

Figure from Text: 5.1 pg.172: Distribution of errors upon repeated measurements

Bias Errors

A bias error remains constant during a given series of measurements

Bias errors are estimated by comparison. Various ways of quantifying are:

• Calibration
• Concomitant methodology – using different methods of estimating the same thing and then comparing the results.
• Inter-laboratory comparisons
• Experience

Table from Text: 5.1 pg.174: Calibration Error Source group

5.2 pg. 174: Data Acquisition Error source group

5.3 pg. 175: Data Reduction Error Source group

Precision Errors

Manifest themselves as scatter of the measured data.

Affected by:

• Measurement system - Repeatability and resolution
• Measurand - Temporal and spatial variations e.g. turbulence, random vibrations
• Process - Variations in operating and environmental conditions
• Measurement Procedure and technique - Repeatability

NOTE: Treat an error as a precision error if it can be statistically estimated; or otherwise treat is as a bias error.

Error Propagation

We will study uncertainty analysis for the following measurement situations:

• Design stage – initial analysis performed prior to the measurement
• Single measurement – used when statistical data is unavailable.
• Multiple measurement – used when a statistical database has been obtained.

Basic Idea:

• Each element of error present within a measurement will combine with all others errors to increase the uncertainty of the measurement. We are interested in measuring the variable . This measurement is subject to k sources of error, e_j, j = 1, 2, ...k. 
• Use Root-Sum-Squares method (RSS) to obtain the uncertainty in the measurement U_x

Or:

^*

VERY CONSERVATIVE ESTIMATE – Assumes Gaussian behavior and that errors will occur in worst possible way

Example from Text: 5.2 pg. 184

Propagation of Uncertainty to a result

• Many times functional relationships are used in conjunction with measured variables to determine a variable of interest.

• The True value of y depends on the sensitivity in the measurement of x

x

• Perform Taylor series expansion:

• Assume small changes so a linear approximation is valid:

• We can argue by inspection that 
• In general, uncertainty in x, u_x is manifested as an uncertainty in y, u_y as:

• What about multivariable problems? Let R be the result determined from several (L) independent variables x_i. i.e. 

• R’ is the true mean stated as 

and 

NOTE: Assign the uncertainties at the same probability level.

For the multivariable problem then:

Figure from Text: 5.2 pg. 178: Relationship between a measured variable and a resultant calculated using the value of that variable.

Design Stage Uncertainty Analysis

Used in initial stages for designing an experiment or experiments

• What do we know about the instruments? Perhaps we must select them!
• We can use uncertainty analysis to guide us in the selection of equipment & procedures.

Zero-order uncertainty of an instrument

Basic idea

• The value of the variable to be measured must be affected by the ability to resolve the information provided by the instruments, even when other sources of error are zero
• A general rule of thumb is to assign a numerical value to u₀, the zero order uncertainty, of ½ the instrument resolution at a probability of 95%.

i.e. 20 to 1 odds that interval u₀. That is only 1 value in 20 will exceed u₀.

• We can use a manufacturer statement concerning error.

Table from Text: 1.1 pg. 19: Manufacturer’s Specifications: Typical Pressure Transducer

• Can say that this is uncertainty due to the instrument u_c

NOTE: Use RSS method if several instruments are used.

Figure from Text: 5.3 pg. 184: Design-stage uncertainty procedure in combining uncertainties.

Example from Text: 5.3 pg. 185

Example from Text: 5.5 pg. 190

Multiple – Measurements Uncertainty analysis

Propagation of Elemental Errors.

Procedures for multiple measurement analysis are

• Identify the elemental errors in the following three source groups (Calibration, data acquisition, data reduction)
• Estimate the magnitude of bias and precision error in each of the elemental errors
• Estimate any propagation of uncertainty through to the result

Figure from Text: 5.6 pg. 195: Multiple-measurement uncertainty method separates elemental errors into precision and bias errors.

Figure from Text: 5.7 pg. 196: Multiple-measurement uncertainty procedure for combining uncertainties.

Note: For t_n,95 to compute

How do we find n, the degrees of freedom? Remember P^* is based on different elements which usually have different degrees of freedom. Use Welch-Satterthwaite formula

- Three source group errors

- Each elemental/error within each group

Note: The book mixes notation here; early in the chapter P stood for probability which we have picked at 95%, in this equation it denotes precision index

Example from Text: 5.12 pg. 200

Propagation of uncertainty to a result

What happens if we have functional relationships in this case? Remember, consider the result R

(95%)

Now however u_r is given as:

the propagation of precision through the variables yields:

The bias will propagate as

We combine B_R & P_R to yield an estimate of the uncertainty in the result u_r

Degrees of freedom n in each x_i is generally not the same. How do we find t_n,95?

NOTE: See text example 5.9 pg. 198, Look at it carefully!!

Example from Text: 5.# pg. ###

Single-Measurement Uncertainty Analysis

Used:

• In the advanced design stage of a test to estimate the expected uncertainty, beyond the initial design stage estimate.
• To report the results of a test program that involves measurements over a range of 1 or more parameters but with no or few replications at each test condition:

Zero-Order Uncertainty

• All variables and parameters that affect the outcome of the measurement, including time, are assumed fixed except for the physical act of the observation. i.e. Data scatter due to u₀, instrumentation resolution, alone.

Higher Order Uncertainty

• The controllability of the test operating conditions are considered. e.g. first-order level – time is included.

, Take N readings in time

If , time is not a factor

N_th-order Uncertainty

• Include calibration uncertainty u_c