AOE 3054. Statistical Analysis of Experimental Data

AOE 3054. Statistical Analysis of Experimental Data

• Relevant to all experiments, IDLab and uncertainty analysis

• See lab manual, “Basic Concepts in Experiments”

1. Mean, Variance, Histograms2. Probability density functions and the normal

distribution3. Correlation and Regression

Analysis?

Separation

Transition to turbulence

Slowing of flow around stagnation point

Stall zone

Wake closes

Example

USS Jefferson City (SSN 759) navigates past Orote Point as the submarine enters Apra Harbor, Guam.

Wind Tunnel Model

Typical Velocity Signal

Mean, Variance, HistogramsDefinitions

Consider a series of measurements x1, x2, x3 … xN,

Mean

Variance

Standard deviation

For a continuous function x(t), where t could be anything ...

Mean

Variance

Standard deviation

Mean, Variance, HistogramsDefinitions

Histogram: A chart showing the distribution of values in a series of samples

• Divide up range of the measurements into a series of equal intervals (called bins)• Add up the number of measurements falling in each bin• The mean usually falls near the center of the histogram• The standard deviation is typically 1/4 to 1/6th of the spread• Histograms may be used to estimate probability

05

101520253035404550

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Wave height (ft)N

umbe

r of s

ampl

es

Typical Velocity Signal

Mean Velocity on Center-plane

‘Hull’

‘Con

ning

Tow

er L

eadi

ng E

dge’

Velocity Variance

‘Hull’

‘Con

ning

Tow

er L

eadi

ng E

dge’

Velocity

Num

ber o

f sam

ples

Velocity0-.5U∞

Velocity Histograms From in the Vortex

0-.5U∞

USS Seawolf (SSN 21), completing its initial sea trial

Probability Density FunctionsDefinition

Definition: The area under p(x) between two values x0 and x1 is the probability P that a given sample of x will fall between these values.

Mathematically:

(Add labels)

Normal DistributionDefinition

The probability P that a given sample of x will fall between x0 and x1 is:

Which can be re-written as

Where

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

p(x)

x

πσ=

σ−

−

2)(

2

2

2)( xx

exp

x

σ

= Matlab Excel

Normal DistributionExample of Use

A sensor is used to detect the flow rate of fuel to a jet engine. The flow rate fluctuates and the following are 21 successive readings readings (in arbitrary units),

(a) Determine the mean and standard deviation.

(b) Estimate the probability that a flow rate will have a value between .5 and .6.

(c) What percentage of the time is the flow rate likely to lie above a value of .8?

(d) What percentage of a large number of readings is likely to lie further than two standard deviations from the mean?

Reading Flow rate1 0.5122 0.4773 0.7944 0.6725 0.7136 0.5887 0.6218 0.7349 0.77110 0.48611 0.55912 0.61413 0.68714 0.72215 0.62716 0.70117 0.57318 0.72119 0.80220 0.55321 0.605

(a) Determine the mean and standard deviation644.0...)794.477.512(.21

1 =+++=x

098.0...))644.477(.)644.512((. 22201 =+−+−=xσ


(b) Estimate the probability that the flow rate will have a value between .5 and .7.

x = 0.644σ = 0.098


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(c) What percentage of the time is the flow rate likely to lie above a value of .8?

x = 0.644σ = 0.098


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(d) What percentage of a large number of readings will lie further than two standard deviations from the mean?

x = 0.644σ = 0.098


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So, on average, there is close to a 95% chance that the value will lie within 2 standard deviations of the mean. This result is important when we come to estimating experimental error.

Correlation (co-variance)Definition

Consider two series of measurements of related quantities x1, x2, x3 … xN, y1, y2, y3 … yN,

Correlation

For continuous functions x(t) and y(t), where t could be anything ...

Correlation

Correlation coefficientVaries from -1 to 1

We also define the Correlation function

where τ is a shift in time, distance of whatever t represents, and the

Correlation (co-variance)Application Example

• GE 90 Aircraft Engine• Eddies from fan tip entering stator row impinge in stators generating noise

∆y

∆z

τ

Zero time-delay correlation function, USpace Time Correlations∆y

/ca

Fan blade wakeFan blade wake

Casing

Fan blade tip vortices

Correlation with this point

Correlation coefficient

-1

-0.5

0

0.5

1

-τUe

/ c a

-0.4-0.3

-0.2-0.1

00.1

0.20.3

0.4

∆z / ca

0

0.1

0.2

0.3

0.4

∆y/ c

a

LSE of coherent structure based onU fluctuation.

Vorticity magnitude isosurface, andstreamtraces.

-τUe/ca

∆x3/ca

∆x2/c

a

Streamtraces

Vorticity isosurfaces

Note dominant elongated vortical structure with axis about 20o from the

time axis (i.e. the flow direction)

Reconstruction of 3D field from correlations

Linear stochastic estimate of flowfield around location 2

Linear RegressionConsider a series of measured points (xi,yi) describing a relationship between two quantities x and y. We wish to find a straight line of the form y = A+Bxthat lies as closely as possible to the data. That is, we wish to choose A and Bto minimize the error

This is done by setting

Which gives us two simultaneous equations for A and B, which may be solved to yield

Linear Regression• Using Excel:

– Under Tools>Data Analysis… (may have to add this feature by going to Tools>Add-ins… and checking Analsys Toolpak

– Alternatively …

• Using Matlab (preferred)

Matlab Example

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Matlab Example – Generalized Regression

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AOE 3054. Statistical Analysis of Experimental Data