What do noisy datapoints tell us about the true signal? · 1/17/2012 · Overview... . . . Bayesian Analysis. . . . .. . . . . Gaussian Processes. .. . .. . . Scattering Curves

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Bayesian Analysis. . . . .. . . . .

Gaussian Processes. .. . .. . .

Scattering Curves. . .Conclusions

What do noisy datapoints tell us about thetrue signal?

C. R. Hogg

November 16, 2011

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Outline

Overview

Bayesian Analysis

Gaussian Processes

Scattering Curves

Conclusions

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Overview

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Ernest Rutherford (1871-1937)

“If your experiment needs statistics,you ought to have done a better experiment”

(Or: use better statistics!)

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Goals for the talk

Uncertainty in single quantity:

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. . . in continuous functions:

1. Explain Bayesian analysis atconceptual level

2. Discuss quantifying uncertaintyin continuous functions

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Goals for the talk


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Goals for the talk


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Goals for the talk


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Goals for the talk


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Goals for the talk


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same shape

stays inside



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Why Bayes at NIST?

NIST’s mission:To promote U.S. innovation and industrial

competitiveness by advancing measurementscience, standards, and technology in ways

that enhance economic security and improveour quality of life.

NIST’s vision:NIST will be the world’s leader in creating

critical measurement solutions and promotingequitable standards. Our efforts stimulate

innovation, foster industrial competitiveness,and improve the quality of life.

NIST’s core competencies:• Measurement science• Rigorous traceability• Development and use of standards

• Measurement isextremely important atNIST

• Must quantifyuncertainty:“A measurement result is

complete only when accompanied

by a quantitative statement of its

uncertainty.”a

• Which language todiscuss uncertainty?

• If “probabilities”:Bayesian analysis

aNIST TN 1297

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Why Bayes at NIST?












uncertainty.”a



aNIST TN 1297

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Why Bayes at NIST?












uncertainty.”a



aNIST TN 1297

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Why Bayes at NIST?












uncertainty.”a



aNIST TN 12975/31

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What is Bayesian Analysis?

(Rev. Thomas Bayes, c. 1701 – 1761)

Essence of Bayes:• 2 questions for every

guess (i.e. every θ)1. How likely does it make

the actual data?

“LIKELIHOOD”

2. How plausible is it?

“PRIOR”

• Combine them to answerthe main question:

1. What is your newprobability, now thatyou’ve seen the data?“POSTERIOR”

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the actual data?

“LIKELIHOOD”


“PRIOR”



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the actual data?

“LIKELIHOOD”


“PRIOR”



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the actual data?

“LIKELIHOOD”


“PRIOR”


1. What is your newprobability, now thatyou’ve seen the data?

“POSTERIOR”

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the actual data?“LIKELIHOOD”

2. How plausible is it?“PRIOR”



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the actual data?“LIKELIHOOD”

2. How plausible is it?“PRIOR”



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Likelihood of function p(y |f )

Scattering vector Q (1/A)

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p(y |f ) = f ye−f

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• Assume independentpixels

• Problem: not plausible• (What makes a function

“plausible”?)

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“Plausibility” of function p(f )


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• Assume smooth andcontinuous

• No functional formassumed

• Naturally: unrelated to data

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Posterior probability p(f |y)


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• To represent uncertainty:show many guesses

• (Or, summarize them. . . )

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Posterior probability p(f |y)Quantitative uncertainty visuals

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Posterior probability p(f |y)Quantitative uncertainty visuals

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Recap: Bayesian denoisingPlausible curves


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Recap: Bayesian denoisingCurves which fit the data


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Recap: Bayesian denoisingPlausible curves which fit the data


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Random variables; Random functions

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• Random variable F :an uncertain quantity

• calculate probabilitiesfor its values

• take “random draws”(roll the die,flip the coin. . . )

• Random function F (x)?

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How to think about “random functions”?

x

F(x

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• Function: a collection ofindividual values

• Every value is a randomvariable, with. . .

1. variance2. correlation

• correlation × variance:covariance

• Gaussian Process:• Every point is a

Random Variable• Any (finite) subset has

Gaussian jointdistribution

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How to read a Covariance Matrix

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• How to read the matrix?

1. By individual entries2. As a whole

(central stripe)• Intensity:

height of features• Width:

width of features

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• How to read the matrix?1. By individual entries

2. As a whole(central stripe)

• Intensity:height of features

• Width:width of features

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• How to read the matrix?1. By individual entries2. As a whole



width of features

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How to read a Covariance MatrixS

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width of features

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How to read a Covariance MatrixS

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width of features

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Example 1: Hydrocarbon combustion(Dave Sheen and Wing Tsang, NIST, Div. 632)

Hydrocarbon burningsimulations

• Need (many!) reaction rateconstants

• Measured individually• Predictions are precise,

quantitative

, wrong

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Example 1: Hydrocarbon combustion(Dave Sheen and Wing Tsang, NIST, Div. 632)

Hydrocarbon burningsimulations

• Need (many!) reaction rateconstants

• Measured individually• Predictions are precise,

quantitative, wrong

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Hydrocarbon combustion: flame speed experiments

Flame speed

Fuel-to-oxygen ratio

Speed (cm/s)

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• Datapoints (fromseveral experiments)

• Model: lengthscales` and σf

• ±1σ range• See also: individual curves• But where did this model

come from. . . ?

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Flame speed


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Flame speed


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• ±1σ range

• See also: individual curves• But where did this model

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Flame speed


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• ±1σ range• See also: individual curves

• But where did this modelcome from. . . ?

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Flame speed


Speed (cm/s)

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40

0.6 0.8 1.0 1.2 1.4 1.6




come from. . . ?

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. .Overview

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Occam’s razor, intro

MODEL 1:

flat

• William of Occamc. 1288 - c. 1348

• Gave us Occam’s Razor

• (slightly paraphrased inthe name of science)

• Claim: use probability,get this automatically

• (And, quantitative, too!)

• Example: 3 models. . .

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MODEL 1:

flat


• Gave us Occam’s Razor• (slightly paraphrased in

the name of science)




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MODEL 1:

flat







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MODEL 1:

flat







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MODEL 2:

flat + line







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MODEL 3:

flat + line + wiggle







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Occam’s razor in action

• Some models can explainmore datasets

• Each model is probabilitydistribution:

• Same total probabilityto distribute

• Which data actuallyobserved?

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Occam’s razor and Gaussian Processes

ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1

• 9 models, varyingcomplexity

• Few datapoints (2):simple models preferred

• New data, some modelscan’t explain

• Three tiers1. Fit too poor2. Fit too good3. Just right

• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

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ℓℓℓ

σσ

σ

Probability

of Model:

0 1/9 1





• Clear winner

18/31

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Example 2: Metal Strain

(Adam Creuziger and Mark Iadicola, NIST, Div. 655)

• Testing stress/strain ofsteels (auto parts, etc.)

• Clamp flat plate; pushupwards on middle

• Measure:1. Stress: X-ray diffraction2. Strain: Digital imaging

of spray-paint pattern

• Can’t painteverywhere!

(Figures courtesy of Mark Iadicola)

19/31

. .Overview

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Example 2: Metal Strain

(Adam Creuziger and Mark Iadicola, NIST, Div. 655)

• Testing stress/strain ofsteels (auto parts, etc.)

• Clamp flat plate; pushupwards on middle

• Measure:1. Stress: X-ray diffraction2. Strain: Digital imaging

of spray-paint pattern• Can’t paint

everywhere!

(Figures courtesy of Mark Iadicola)

19/31

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Metal Strain: Preliminary Results

• Spheres representdatapoints

• Continuous surface• Uncertainty bounds ±1σ

• See also animations

• Competing model:anisotropic

• Occam’s razor lets uschoose!

• ∆ log(ML) = +183.4

• Suggestions forexperimental design

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• Continuous surface

• Uncertainty bounds ±1σ• See also animations



• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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• ∆ log(ML) = +183.4


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Need to extend the modelS

trip

e I

nte

nsity

= F

ea

ture

He

igh

t

X-1

00

0100

0 1 2 3X

01

23

0 1 2 3

Cov

2500

0

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

Str

ipe

Wid

th =

Fe

atu

re W

idth

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

• Recall: how to readcovariance matrices“as a whole”



• Not flexible enough forreal data!

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Need to extend the modelS

trip

e I

nte

nsity

= F

ea

ture

He

igh

t

X-1

00

0100

0 1 2 3X

01

23

0 1 2 3

Cov

2500

0

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

Str

ipe

Wid

th =

Fe

atu

re W

idth

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

• Recall: how to readcovariance matrices“as a whole”



• Not flexible enough forreal data!

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Two extensions

Va

ryin

g W

idth

s

X

-100

010

0

0 1 2 3X

01

23

0 1 2 3

Cov

2500

0

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

Mu

ltip

le C

on

trib

utio

ns

X

-100

01

00

0 1 2 3X

01

23

0 1 2 3

• Two main extensions. . .

1. Varying Feature widths• ` → `(X )

2. Multiple contributions• Background everywhere• Localized “peak” regions

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Two extensionsV

ary

ing

Wid

ths

X-1

00

0100

0 1 2 3X

01

23

0 1 2 3

Cov

2500

0

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

Mu

ltip

le C

on

trib

utio

ns

X

-100

01

00

0 1 2 3X

01

23

0 1 2 3

• Two main extensions. . .1. Varying Feature widths

• ` → `(X )


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Two extensionsV

ary

ing

Wid

ths

X-1

00

0100

0 1 2 3X

01

23

0 1 2 3

Cov

2500

0

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

Mu

ltip

le C

on

trib

utio

ns

X

-100

0100

0 1 2 3X

01

23

0 1 2 3

• Two main extensions. . .1. Varying Feature widths

• ` → `(X )


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Simulated XRD from 2 nm Au nanoparticles

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

I(Q

)

True function

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)

True function

• Core/shell structure• Shell atoms vibrate more• Correlated thermal

motion(Signature: hi-Qoscillations)

• Problem: Poisson noiseswamps these oscillations!

• Changing feature widths:use `(Q)

X

01

23

0 1 2 3

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0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

I(Q

)

True function

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120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)

True function





X

01

23

0 1 2 3

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0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

I(Q

)

True function

110

120

130

140

150

160

170

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Q (1/A)

True function





X

01

23

0 1 2 3

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0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

I(Q

)

Noisy dataTrue function

110

120

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140

150

160

170

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Q (1/A)






X

01

23

0 1 2 3

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0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

I(Q

)


110

120

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140

150

160

170

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Q (1/A)






X

01

23

0 1 2 3

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Denoising results

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)


AWS (raw data)• AWS: jagged; loses signal

at Q = 26A−1

• Wavelets: smooth, but stilllose signal

• Bayes: also smooth, butkeeps signal

• Uncertainty boundscapture true function

24/31

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Denoising results

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)


Wavelets• AWS: jagged; loses signal

at Q = 26A−1




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Denoising results

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)


Bayesian• AWS: jagged; loses signal

at Q = 26A−1




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Denoising results

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)

Noisy dataBayesian

True functionAWS (raw data)

Wavelets

• AWS: jagged; loses signalat Q = 26A−1




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Residuals

Residuals vs. noisy data

MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesi

an m

ean

Wa

vele

ts

Bayesi

an d

raw

s

0.240 0.245 0.250

Residuals vs. true curve

MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesi

an m

ean

Wa

vele

ts

Bayesi

an d

raw

s

Tru

e c

urv

e

0.000 0.005 0.010

• Need: global fidelitymeasure

• Mean square residuals . . .1. vs. noisy data

• AWS looks best2. vs. true curve

• Bayes is best• AWS “good” score:

was overfitting noise!

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Residuals


MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesi

an m

ean

Wa

vele

ts

Bayesi

an d

raw

s0.240 0.245 0.250


MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesi

an m

ean

Wa

vele

ts

Bayesi

an d

raw

s

Tru

e c

urv

e

0.000 0.005 0.010



• AWS looks best

2. vs. true curve• Bayes is best• AWS “good” score:


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Residuals


MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesi

an m

ean

Wa

vele

ts

Bayesi

an d

raw

s0.240 0.245 0.250


MSR

Pro

ba

bili

ty D

en

sity

0

AW

S

AW

S (

raw

data

)

Bayesia

n m

ean

Wa

vele

ts

Bayesi

an d

raw

s

Tru

e c

urv

e

0.000 0.005 0.010



• AWS looks best2. vs. true curve

• Bayes is best• AWS “good” score:


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TiO2 nanoparticles

NIST SRM 1898:

(Ohno et al., J. Catalysis, 2011)

• X-ray powder diffractionfrom 20 nm TiO2nanoparticles

• Motivations:1. Real-world example

(Violates ourassumptions)

2. More difficult data(contains feature-freebackground regions)

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Comparing the fits

Q (1/A)

0

1000

2000

3000

1 2 3 4 5 6 7

Q (1/A)

200

400

600

800

1000

3.2 3.3 3.4

• All handle sharp peaks• Every technique misses a

few features: AWS,wavelets, even Bayes

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Comparing the fits

Q (1/A)

0

1000

2000

3000

1 2 3 4 5 6 7

Q (1/A)

80

100

120

5.8 5.9



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Comparing the fits

Q (1/A)

0

1000

2000

3000

1 2 3 4 5 6 7

Q (1/A)

100

200

7.0 7.2



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Comparing the fits

Q (1/A)

0

1000

2000

3000

1 2 3 4 5 6 7

Q (1/A)

100

120

6.2 6.4



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Comparing the fits

Q (1/A)

0

1000

2000

3000

1 2 3 4 5 6 7

Q (1/A)

80

100

120

5.2 5.3



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ResidualsResiduals vs. quasi-true curve

MSR

Pro

ba

bili

ty D

en

sity

AW

S

Bayesi

an M

ean

Wa

vele

ts

0.025 0.030 0.035

Training on

Even Points

Mean MSR

Ove

rfitt

ing

0.0

00

0.0

20

Baye

sA

WS

Wa

vele

ts

0.170 0.180

Training on

Odd Points

Mean MSR

Ove

rfitt

ing

0.0

00

0.0

20

Baye

sA

WS

Wa

vele

ts

0.170 0.180

• Bayes single-curvecomparable to benchmarks

• Cross-validation:(Checking for overfitting)Bayes is best. . .

1. In both categories2. For both training sets

28/31

. .Overview

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. . . .




ResidualsResiduals vs. quasi-true curve

MSR

Pro

ba

bili

ty D

en

sity

AW

S

Bayesi

an M

ean

Wa

vele

ts

0.025 0.030 0.035

Training on

Even Points

Mean MSR

Ove

rfittin

g0.0

00

0.0

20

Baye

sA

WS

Wa

vele

ts

0.170 0.180

Training on

Odd Points

Mean MSR

Ove

rfittin

g0.0

00

0.0

20

Bayes

AW

S

Wa

vele

ts

0.170 0.180

• Bayes single-curvecomparable to benchmarks

• Cross-validation:(Checking for overfitting)Bayes is best. . .

1. In both categories2. For both training sets

28/31

. .Overview

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Recap: Bayesian Concepts


Inte

nsity

(n

um

be

r o

f co

un

ts)

02

04

0

0.0 0.5 1.0

• Bayesian analysis:using probabilitiesto describe uncertainty

• choose answers withboth plausibilityand data fit

• a natural framework formodel selectionconcepts(Occam’s razor)

29/31

. .Overview

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Recap: Uncertainty in continuous functions

110

120

130

140

150

160

170

23 24 25 26 27 28 29 30

Q (1/A)

Noisy dataBayesian

True functionAWS (raw data)

Wavelets

• Gaussian Processes: canstipulate smoothness,without worrying aboutfunctional form

• Open-source softwarepackage

• Very flexible: can help avariety of projects

30/31

. .Overview

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. . . .




Acknowledgements

• Team members: Igor Levin, Kate Mullen• Collaborators:

• Flame speed : Dave Sheen, Wing Tsang• Metal Strain: Adam Creuziger, Mark Iadicola

• WERB readers: Victor Krayzmann, Adam Pintar• Statistical guidance: Antonio Possolo, Blaza Toman

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Documents

What do noisy datapoints tell us about the true signal? · 1/17/2012 · Overview... . . . Bayesian Analysis. . . . .. . . . . Gaussian Processes. .. . .. . . Scattering Curves