bayesian modelling and computation for raman spectroscopy

Raman Spectroscopy Functional Model Bayesian Computation Experimental Results Conclusion

Bayesian modelling and computationfor Raman spectroscopy

Matt Moores

Department of StatisticsUniversity of Warwick

Oxford computational statistics & machine learning reading groupMarch 11, 2016


Acknowledgements

University of WarwickMark GirolamiJake CarsonKarla Monterrubio Gmez

University of StrathclydeKirsten GracieKaren FauldsDuncan Graham

Funded by the EPSRC grant In Situ Nanoparticle Assemblies forHealthcare Diagnostics and Therapy (ref: EP/L014165/1) and anAward for Postdoctoral Collaboration from the EPSRC Network onComputational Statistics & Machine Learning (ref: EP/K009788/2)


Outline

1 Raman Spectroscopy

2 Functional Model

3 Bayesian Computation

4 Experimental ResultsModel ChoiceMultivariate Calibration


Raman spectroscopy

Illustration courtesy Jake Carson (U. Warwick)


Statistical properties

Each Raman-active dye has a unique spectral signature:Peaks correspond to vibrational modes of the moleculeShift in wavenumber proportional to change in energy stateSmoothly-varying baseline (background fluorescence)

200 300 400 500 600 700 800 900 1100 1300 1500 1700

10000

20000

30000

40000

50000

~ cm

1

photo

n c

ounts

Replicate

A

B

C

D

E


Surface-enhanced Raman scattering (SERS)

Raman signal enhanced by proximity to nanoparticlesFunctionalisation using antibodies

Illustration courtesy Kirsten Gracie (U. Strathclyde)


Fluorescent background

200 400 600 800 1000 1200 1400 1600 1800 2000

02

00

40

06

00

80

0

(cm1)

ph

oto

n c

ou

nts

(a

.u.)

(a) surface-enhanced fluorescence(SEF)

200 400 600 800 1000 1200 1400 1600 1800 2000

02

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40

06

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0

(cm1)

ph

oto

n c

ou

nts

(a

.u.)

Well E1

Well E2

Well E3

background

(b) surface-enhanced Raman spec-troscopy (SERS)


Baseline correction

Existing methods estimate the baseline independently:Asymmetric least squaresBoelens, Eilers & Hankemeier (Anal. Chem., 2005)

Modified polynomial fitLieber & Mahadevan-Jansen (Appl. Spectrosc., 2003)

Robust baseline estimationRuckstuhl et al. (JQSRT, 2001)

WaveletsCai, Zhang & Ben-Amotz (Appl. Spectrosc., 2001)


Additive functional model of a SERS spectrum

Separate the hyperspectral signal into 3 components:

yi() = i() + s() + (1)

where:yi() is a an observed SERS spectrum, discretised at

multiple wavenumbers j Vi() is a smooth baseline functions() is the spectral signature of the dye molecule is additive, zero mean white noise with variance 2


Baseline

Penalised spline:

i() =

Mm=1

Bm()i,m (2)

(i,) NM (0,) (3)where Bm() are Demmler-Reinsch or B-spline basis functions

550 600 650 700 750 800

1

.5

0.5

0.5

1.5

~ (cm1

)

Bm(~

)


Spectral signature

An additive mixture of radial basis functions:

s() =P

p=1

f ( | `p,Ap, p) (4)

where:`p is the location of peak pAp is the amplitudep is the scale (broadening)


Squared exponential

Kernel function is the Gaussian density:

f (j | `p,Ap, p) = Ap exp

{(j `p)2

22p

}(5)

FWHM = 2

2 ln 2p (6)

1300 1400 1500 1600 1700 1800

05

00

01

00

00

15

00

0

~ (cm

1)

Inte

nsity (

a.u

.)


Lorentzian

Long-range dependence between peaks can be modelledusing the Cauchy density:

f (j | `p,Ap, p) = Ap2p

(j `p)2 + 2p(7)

FWHM = 2p (8)

1300 1400 1500 1600 1700 1800

05

00

01

00

00

15

00

0

~ (cm

1)

Inte

nsity (

a.u

.)


Informative priors

Obtained from manual peak fitting of independent data:

Density

1 2 5 10 20 50

0.0

00.0

50.1

00.1

5

kernel density

lognormal

(a) Scale parameters, (cm1)

amplitudes

Density

0 5000 10000 15000 20000 250000.0

0000

0.0

0010

0.0

0020

kernel density

truncated normal

gamma

(b) Amplitudes, A (arbitrary units)


Multivariate calibration

Signal intensity depends linearly on dye concentration, from thelimit of detection (LOD) up to monolayer coverage:

Ap = pci, cLOD ci < cMLC (9)

where:ci is the nanomolar (nM) concentration of the dye in

observation ip is a linear regression coefficient

cLOD is based on the signal-to-noise ratiocMLC is proportional to the surface area of the

nanoparticles

Jones, et al. (1999) Anal. Chem. 71(3): 596601


Markov chain Monte Carlo

MCMC targeting the joint posterior (,,, | yi())

Algorithm 1 Marginal Metropolis-Hastings1: Draw random walk proposals for the peaks: ,

2: Propose baseline i, q(i, | yi(),,,

)3: Propose q

( | yi(),,, i,

)4: Compute the marginal acceptance ratio:

=p(yi() | ,,, )

p(yi() | ,,, )q( | )q( | )(

)()()()

q( | )q( | )()()()( )

=p(yi() | ,)()()

p(yi() | ,)()()

5: Accept ,, i,, jointly with probability min(1, )

(assuming peak locations `p and number of peaks P are fixed)


Sequential Monte Carlo

Particle-based method targeting a sequence of partialposteriors i (,, 1:i,, | Y1:i,)

Algorithm 2 SMC

1: Initialise (q), (q), (q), (q) q {1, . . . ,Q}2: Initialise importance weights, w(q)0 =

1Q

3: for all observations i = 1, . . . , n do4: Update importance weights:

w(q)i w(q)i1 p

(yi() | (q),(q), (q)i, , (q)

)(10)

5: Resample particles if ESSi is below threshold6: for all particles q {1, . . . ,Q} do7: Update (q), (q), (q), (q) using Algorithm 18: end for9: end for

Chopin (2002) Biometrika 89(3): 539551


Model evidence

Algorithm 2 provides a consistent, unbiased estimate of themarginal likelihood:

ZiZi1

=

Qq=1

w(q)i1 p(

yi() | (q)k)

(11)

p(Y | Mk) Zn =n

i=1

ZiZi1

(12)

where:

(q)k =

{

(q)k ,

(q)k ,

(q)k ,

(q)k

}are the parameters of

model Mk for particle qZn is the normalising constantZ0 = 1

Del Moral, Doucet & Jasra (2006) JRSS B 68(3): 411436Pitt, Silva, Giordani & Kohn (2012) J. Econom. 171(2): 134151


Thermodynamic integration

An alternative approach is to use the path sampling identity:

log{ZnZ0

}=

10

E[

d log q()d

]d (13)

where = in identifies the sequence of partial posteriordistributions p =

qZi = i (,, 1:i,, | Y1:i,)

This equation cannot be solved exactly, so it must beapproximated using a numerical integration method.

The expectation E can be estimated using a weighted sumover the SMC particles.

Zhou, Johansen & Aston (2015) arXiv:1303.3123 [stat.ME]Gelman & Meng (1998) Statist. Sci. 13(2): 95208


R package serrsBayesl i b r a r y ( serrsBayes )l i b r a r y ( hyperSpec )

ramanSpectra read . spc ("mfutils.spc" )wavenumbers wl ( ramanSpectra )peakLoc wl2 i ( ramanSpectra , c (964 , 1138 , 1218 , . . . ) )

# i n f o r m a t i v e p r i o r s f o r the peaks and base l inel P r i o r s l i s t ( scale .mu=log (25 .27 ) ( 0 . 4 ^2 ) / 2 ,

scale . sd=0.4 , b l . smooth=1.25e3, b l . knots =200 ,amp.mu=3449 , amp. sd=5672 , noise . sd=3 ,noise . nu= length ( wavenumbers ) nrow ( ramanSpectra ) ,beta . sd=1000)

r e s u l t f i tPeaksWithBasel ineSMC ( wavenumbers ,ramanSpectra [ [ ] ] , peakLoc , l P r i o r s )


Synthetic data: Lorentzian peaks + Cauchy kernel10

000

4000

070

000

Inte

nsity

040

0080

0012

000

Inte

nsity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Lorentzian peaks + squared exponential kernel10

000

4000

070

000

Inte

nsity

040

0080

00In

tens

ity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Gaussian peaks + sq. exp. kernel

1000

040

000

7000

0In

tens

ity

040

0080

0012

000

Inte

nsity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Gaussian peaks + Cauchy kernel10

000

4000

070

000

Inte

nsity

050

0010

000

1500

0In

tens

ity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Real data: TAMRA + Cauchy kernel10

000

3000

050

000

7000

0In

tens

ity

050

0010

000

1500

0In

tens

ity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Real data: TAMRA + sq. exp. kernel

1000

040

000

7000

0In

tens

ity

040

0080

00In

tens

ity

1000

030

000

5000

0In

tens

ity

250 500 750 1000 1250 1500 1750Raman Shift


Real data: parameter estimates for

0 2000 4000 6000 8000 10000

200

210

220

230

(a) Cauchy

0 2000 4000 6000 8000 1000021

022

023

024

025

0

(b) Squared Exponential


Model Choice

The Bayes Factor (log10 BF) correctly identifies the generativemodel for simulated spectra with Gaussian (194) andLorentzian (-160) peaks:

Table: Simulation study

Data Mk log10 p(Y | Mk)Gaussian sq. exp. -2011Gaussian Cauchy -2205Lorentzian sq. exp. -2163Lorentzian Cauchy -2004


Model Choice for TAMRA

The Bayes Factor favours Lorentzian peaks (log10 BF = -32) fortetramethylrhodamine (TAMRA):

Table: Observed spectra (TAMRA)

Data Mk log10 p(Y | Mk)TAMRA sq. exp. -2257TAMRA Cauchy -2225

This indicates very strong evidence of long-range dependencebetween peaks in SERS spectra.

Kass & Raftery (1995) JASA 90(430): 773795


Dilution study for TAMRA

315 spectra at 21 different concentrations, from 0.13 to 24.7 nM

500 1000 1500 2000

05

00

01

00

00

15

00

02

00

00

25

00

0

~ (cm

1)

Inte

nsity (

a.u

.)24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM


Baseline correction: TAMRA

500 1000 1500 2000

05

00

01

00

00

15

00

02

00

00

25

00

0

~ (cm

1)

Inte

nsity (

a.u

.)

24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM


Spectral signature: TAMRA

500 1000 1500 2000

05

00

01

00

00

15

00

0

~ (cm

1)

Inte

nsity (

a.u

.)

24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM


95% HPD intervals: TAMRA`p (cm1) p (nM1) FWHM (cm1) LOD (nM)

460 [6.73; 17.23] [0.00; 14.43] [0.211; 0.784]505 [167.95; 177.83] [14.36; 15.52] [0.019; 0.047]632 [253.65; 263.16] [11.54; 12.13] [0.012; 0.032]725 [19.63; 29.52] [9.97; 16.43] [0.123; 0.316]752 [44.74; 54.81] [19.49; 23.45] [0.059; 0.156]843 [23.48; 33.73] [15.97; 22.26] [0.106; 0.297]965 [17.78; 28.09] [12.07; 20.32] [0.135; 0.355]

1140 [25.49; 36.11] [20.41; 27.92] [0.089; 0.253]1190 [18.67; 28.99] [11.27; 19.36] [0.110; 0.352]1220 [147.84; 158.30] [17.68; 19.20] [0.020; 0.051]1290 [31.19; 42.49] [15.72; 25.80] [0.080; 0.213]1358 [210.46; 221.96] [17.63; 18.97] [0.015; 0.035]1422 [53.13; 63.99] [15.34; 18.16] [0.059; 0.135]1455 [39.05; 53.87] [20.61; 41.11] [0.069; 0.175]1512 [146.07; 157.28] [20.81; 22.38] [0.019; 0.049]1536 [209.18; 221.03] [14.43; 15.80] [0.014; 0.036]1570 [38.54; 53.67] [23.87; 51.02] [0.073; 0.173]1655 [467.58; 477.91] [17.40; 17.92] [0.006; 0.016]


TAMRA at 0.13 nM

500 1000 1500 2000

020

40

60

~ (cm

1)

Inte

nsity (

a.u

.)

baselinecorrected spectra

posterior mean

3


TAMRA at 0.65 nM

500 1000 1500 2000

0100

200

300

400

~ (cm

1)

Inte

nsity (

a.u

.)

baselinecorrected spectra

posterior mean

3


Summary

Semi-parametric model of hyperspectral data:Joint estimation of baseline & peaksContinuous representation of discretised spectraData tempering using SMCBayesian model choice for long-range dependence

Ongoing work:Peak detection (estimation of `p & P)Informed source separation for multiplex spectraSpatio-temporal correlation between spectra

Appendix

For Further Reading I

Moores, Gracie, Carson, Faulds, Graham & GirolamiBayesian modelling and quantification of Raman spectroscopy.in prep.

Gracie, Moores, Smith, Harding, Girolami, Graham, & FauldsPreferential attachment of specific fluorescent dyes and dyelabelled DNA sequences in a SERS multiplex.Anal. Chem., 88(2): 11471153, 2016.

Zhong, Girolami, Faulds & GrahamBayesian methods to detect dye-labelled DNA oligonucleotides inmultiplexed Raman spectra.J. R. Stat. Soc. Ser. C, 60(2): 187206, 2011.

Zhou, Johansen & AstonTowards Automatic Model Comparison: An Adaptive SequentialMonte Carlo Approach.arXiv:1303.3123 [stat.ME], 2015.

Appendix

For Further Reading II

ChopinA Sequential Particle Filter Method for Static Models.Biometrika, 89(3): 539551, 2002.

Pitt, Silva, Giordani & KohnOn some properties of Markov chain Monte Carlo simulationmethods based on the particle filter.J. Econometrics 171(2): 134151, 2012.

Jones, McLaughlin, Littlejohn, Sadler, Graham & SmithQuantitative Assessment of Surface-Enhanced ResonanceRaman Scattering for the Analysis of Dyes on Colloidal Silver.Anal. Chem., 71(3): 596601, 1999.

Ramsay & SilvermanFunctional Data Analysis, 2nd ed.Springer, 2005.

Appendix

Raman scattering


Appendix

SERS

Surface-enhanced: proximity to a nanoplasmonic substrate(silver/gold colloid)


Appendix

SERRS

Surface-enhanced: proximity to a nanoplasmonic substrate(silver/gold colloid)

Resonance: tune excitation wavelength to an electronictransition of the molecule


Appendix

Dilution study for FAM

315 spectra at 21 different concentrations, from 0.13 to 24.7 nM

500 1000 1500 2000

02

00

04

00

06

00

08

00

01

00

00

~ (cm

1)

Co

un

ts

24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM

Appendix

Results: Baseline correction

500 1000 1500 2000

02

00

04

00

06

00

08

00

01

00

00

~ (cm

1)

Inte

nsity (

a.u

.)

24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM

(a) Posterior means of the baselines

500 1000 1500 2000

02

00

04

00

06

00

08

00

0

~ (cm1

)

Inte

nsity (

a.u

.)

24.7 nM

23.4 nM

22.1 nM

20.8 nM

19.5 nM

18.2 nM

16.9 nM

15.6 nM

14.3 nM

13 nM

11.7 nM

10.4 nM

9.1 nM

7.8 nM

6.5 nM

5.2 nM

3.9 nM

2.6 nM

1.3 nM

0.65 nM

0.13 nM

(b) Baseline-corrected spectra

Appendix

Results: Quantification

SERS peak intensities at 650cm1: 95% CI [257.7; 262.5] ci

0 5 10 15 20 25

02

00

04

00

06

00

08

00

0

Final target concentration (nM)

Inte

nsity (

a.u

.)

Expectation

95% HPD ivl+/ 2

(a) Linear regression for p

0 5 10 15 20 25

2

02

4

Final target concentration (nM)

Expectation

+/ 1.96

+/ 2.58

(b) Standardised residuals

Raman SpectroscopyFunctional ModelBayesian ComputationExperimental ResultsModel ChoiceMultivariate Calibration

bayesian modelling and computation for raman spectroscopy

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