Computer Model Calibration to EnableDisaggregation of Large Parameter Spaces, with

Application to Mars Rover Data

David C. Stenning

SAMSI Transition Workshop May 9 2017

Motivation: Mars! Robots! Lasers!

Image credit: NASA/JPL-Caltech

Laser Induced Breakdown Spectroscopy

I ChemCam performs laser

induced breakdown spectroscopy

(LIBS) by firing a laser intorocks and soils on Mars togenerate a plasma and atomizeand ionize the target.

I It collects photons emitted asthe plasma cools and expands,yielding chemical spectra thatprovide information about thetarget’s composition.

Laser Induced Breakdown Spectroscopy

I Spectra are difficult to interpretdue to matrix effects—interactions between compoundsin the target that amplify orsuppress spectral peaks.

I We study these using computermodels of LIBS plasmas.

I Given a plasma’s temperatureand density, the models build onatomic structure calculations tocompute the emission spectrum.

I Each run of the model takesminutes to hours on parallelcomputing platforms.

Scientific Goals

I Provide the first statistical characterization of matrix effects.I Develop the first capability in uncertainty quantification (UQ)

that addresses the unique challenges of chemical spectra.

Statistical Formulation: Single Compound MgCl2

I The data y is a noisy version of the model/simulator h (·) atthe “correct” input qs :

y = h (qs)+ e

e ⇠ N

⇣~0,⌃y

⌘

I y is a measured spectrum from a MgCl2 sample:

300 400 500 600 700 800 900

0e+00

2e+13

4e+13

wavelength

intensity

I qs = {temperature, density} of the sample, which is unknown

Bayesian Model Calibration

I Given a prior dist’n p (qs) for the true parameter vector qs , wecan write down the posterior distribution for qs :

p (qs | y) µ L(y | h (qs))p (qs) ,

where L(y | h (qs)) is the Gaussian sampling model:

L(y | h (qs)) = exp

⇢�1

2(y �h (qs))

0⌃�1y (y �h (qs))

�

I h (·) is often computationally intensive ! bad for MCMCI Cannot directly embed h (·) into likelihood

Reference: Higdon et al. (2012)

Bayesian Model Calibration

ISolution: treat h (·) as an unknown function for which wehave a fixed collection of simulator runs h (t1) , . . . ,h (tm)carried out at input settings t1, . . . , tm.

I Need a prior dist’n p (h (·)) on the unknown function h (·).

I We treat the simulation runs h? = (h (t1) , . . . ,h (tm))0 as data

during the analysis.I The resulting posterior dist’n is given by

p (qs ,h (·) | y ,h?) µ L(y | h (qs))L(h? | h (·))p (h (·))p (qs) .

IKey: Exchange direct evaluations of h (·) for a morecomplicated form that depends on p (h (·)).

Reference: Higdon et al. (2012)

Emulation (with Multivariate Output)

I We require an emulator�a probability model of the simulatoroutput at untried settings

I The emulator models the simulator output using aq-dimensional basis representation:

h (t) =q

Âi=1

fiwi (t)+ e, t 2 [0,1]p

I {f1, . . . ,fq} is a collection of orthogonal nh -dim basis vectorsI wi (t) are weights that depend on the inputI e is an nh -dimensional error term

IKey: instead of building an emulator that maps [0,1]p to Rnh ,we model each wi (t) with an indep. Gaussian process (GP)

Reference: Higdon et al. (2008, 2012)

Emulating the Simulator Output

I We have m = 25 runs of the simulator on a grid oftemperature and density

I Simulator output at each setting t is a spectrum atnh = 10,000 wavelengths

I Take log transform of the simulator runs and center/scale

I Apply the singular value decomposition (SVD) to simulationoutput matrix:

[h1; . . . ;hm] = UDV 0

I The basis matrix is the first q = 5 columns of [UD/pm]

SVD●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

principal component

varia

nce

expl

aine

d

●

●

● ● ● ● ● ● ● ● ●

2 4 6 8 10

0.99

00.

994

0.99

8

total # of components

tota

l var

ianc

e ex

plai

ned

500 600 700 800

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

photon energy (eV)

ampl

itude

simulatorPCA reconstruction

500 600 700 800

−0.0

15−0

.005

0.00

50.

015

photon energy (eV)

resi

dual

500 600 700 800

4648

5052

photon energy (eV)

ampl

itude

simulatorPCA reconstruction

500 600 700 800

−0.0

2−0

.01

0.00

0.01

0.02

photon energy (eV)

resi

dual

Statistical Model for Simulated Spectra

I Simulations h1, . . . ,h25 run over the training set t1, . . . , t25.

[h1; . . . ;h25] = UDV 0

�h = UD/p

25

h (t) =5

Âi=1

fiwi (t)+ e

500 600 700 800

4648

5052

simulations

photon energy (eV)

ampl

itude

500 600 700 800

4748

4950

5152

mean

photon energy (eV)

ampl

itude

500 600 700 800−1

.0−0

.50.

00.

51.

0

principal component bases

photon energy (eV)

ampl

itude

GP Model for Weights

I Each vector of basis weights wi (t) is modeled as a mean zeroGP

wi (t)⇠ N

�0,l�1

wi R (t;ri )�

I lwi is the marginal precision of the processI R (t;ri ) is a correlation matrix whose entries depend on the

inputs and the parameters of the correlation function

Corr

�wi (t) ,wi

�t 0��

=p

’k=1

r4(tk�t 0k)2

ik

Reference: Higdon et al. (2008, 2012)

GP Model for Weights

I The m ·q vector w has prior distribution0

B@w1...wq

1

CA⇠N

0

B@

0

B@0...0

1

CA ,

0

B@l�1w1R (t;r1) 0 0

0. . . 0

0 0 l�1wq R (t;rq)

1

CA

1

CA

which is determined by q precision parameters contained in lw

and q ·p spatial correlation parameters in r .I More succinctly,

w | lw ,r ⇠ N(0,Sw ) ,

where Sw is the block-diagonal covariance matrix given above.

Reference: Higdon et al. (2008, 2012)

Sampling Model for h

I Recall our model for the emulator:

h (t) =q

Âi=1

fiwi (t)+ e, t 2 [0,1]p

I Assuming e is independent Gaussian with common precisionlh , the sampling model for h can be written as

h | w ,lh ⇠ N

✓�w ,

1lh

Imq

◆,

where h = vec(h1; . . . ;hm), �= [Im ✏f1; . . . ; Im ✏fq], and fi ,i = 1, . . . ,q, are the q = 5 basis vectors computed via SVD.

Reference: Higdon et al. (2008, 2012)

Prior Distributions

lwi ⇠ Gamma(aw = 5,bw = 5)I Encourages lwi to be close to 1

lh ⇠ Gamma

⇣ah = 25,bh = 5

100,000

⌘

I We’re rather ignorant about this term...

rik ⇠ Beta

�ar = 1,br = 0.1

�

I Pr(rik < 0.98)⇡ 13 a priori; encourages effect sparsity because

input k is inactive for PC i if rik = 1

Emulator Model

Putting it all together...

h | w ,lh ⇠ N

⇣�w , 1

lhImq

⌘

w | lw ,r ⇠ N(0,Sw )

lwi ⇠ Gamma(aw = 5,bw = 5)

rik ⇠ Beta

�ar = 1,br = 0.1

�

lh ⇠ Gamma

⇣ah = 25,bh = 5

100,000

⌘

I Parameters estimated with one-at-a-time Metropolis MCMC

Fitted Emulator: Spatial Correlation Parameters

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ρ11 ρ12 ρ21 ρ22 ρ31 ρ32 ρ41 ρ42 ρ51 ρ52

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

IInterpretation: Input k is inactive for PC i if rik = 1

I Used gamma prior dist’n with shape = 1 and rate = 0.1 s.t.Pr(rik < 0.98)⇡ 1

3 a priori.

Recovering Simulated Spectrum h (t18)

−0.4

0.0

0.4

0.8

ampl

itude

500 600 700 800

−0.0

2−0

.01

0.00

0.01

photon energy (eV)

resi

dual

I Held out the spectrum corresponding to input settings t18I Predicted the held-out spectrum by drawing weights given the

posterior samples (and using conditional normal rules)

Bringing in the Data

I Recall: spectrum, y , is modeled as a noisy version of thesimulated spectrum h (qs) run at the true/unknown qs . Thus

y = h (q)+ e,

where e ⇠ N(0,⌃y ). Also, let ⌃�1y = lyWy .

I With the basis representation of the simulator

y = �yw (q)+ e,

and we have data sampling model

y | w (q) ,ly ⇠ N

⇣�yw (q) ,(lyWy )

�1⌘.

I We specify a Gamma prior on the precision parameter ly :

ly ⇠ Gamma(ay = 5,by = 5) .

Full Model

Putting it all together (again)...

y | w (q) ,ly ⇠ N

⇣�yw (q) ,(lyWy )

�1⌘

h | w ,lh ⇠ N

⇣�w , 1

lhImq

⌘

w | lw ,r ⇠ N(0,Sw )

ly ⇠ Gamma(ay = 5,by = 5)

lwi ⇠ Gamma(aw = 5,bw = 5)

rik ⇠ Beta

�ar = 1,br = 0.1

�

lh ⇠ Gamma

⇣ah = 25,bh = 5

100,000

⌘

Parameters estimated with one-at-a-time Metropolis MCMC

Fitting the full model

Computer Model Calibration for LIBS Redux

IIdea: Include all candidate compounds in q .

I Estimation of augmented q directly solves the scientificquestion of interest.

I Rely on a multi-stage emulation approach that combinesingle-compound emulators in a hierarchical model.

Bayesian Variable Selection

I Augmented q is sparseI We want to embed Bayesian variable selection in the

calibration procedureI Spike-and-slab?

I Reversible-jump MCMC?

ReferencesI Sacks et al. “Design and Analysis of Computer Experiments.”

Statistical Science. 1989.I Jones et al. “Efficient Global Optimization of Expensive Black-Box

Functions.” Journal of Global Optimization. 1998.I Kennedy and O’Hagan.“Predicting the output from a complex

computer code when fast approximations are available.” Biometrika.2000.

I Kennedy and O’Hagan. “Bayesian Calibration of Computer Models.”Journal of the Royal Statistical Society, Series B. 2001.

I Higdon et al. “Combining Field Data and Computer Simulations forCalibration and Prediction.” SIAM Journal of Scientific Computing.2004.

I Higdon et al. “Computer Model Calibration Using High-DimensionalOutput.” Journal of the American Statistical Association. 2008.

I Lawrence et al. “The Coyote Universe III: Simulation Suite andPrecision Emulator for the Nonlinear Matter Power Spectrum.” TheAstrophysical Journal. 2010.

I Higdon et al. “Simulation-Aided Inference in Cosmology.” StatisticalChallenges in Modern Astronomy V. 2012.