Correction of systematic disturbances in latent-variable ... · STEP 2: DRIFT-CORRECTION Shrinking Orthogonal projection Subtraction GLSW CC, IIR, EPO, DOP, TOP, EROS DCPS Calibrate

Correction of systematic disturbances in latent-variable

calibration models

Paman Gujral, Michael Amrhein, Barry M. Wise, Enrique Guzman,

Davyd Chivala and Dominique Bonvin

Laboratoire d’AutomatiqueEcole Polytechnique Federale de Lausanne

ÉCOLE POLYTECHNIQUEFÉDÉRALE DE LAUSANNE

Gujral (LA-EPFL) SSC11 2009 08/06/2009 1 / 15

Agenda

Introduction

Calibration and predictionConstituents of prediction error

Unifying framework for different correction methodologies

Illustrative examples

Simulation exampleTwo real data examples


Introduction

Background

Calibration: {Xc , yc} → b Prediction: yp = Xpb


Introduction

Background


xT

c = ycsT + (spectra from other species) + noisec


Introduction

Background


xT

c = ycsT + (spectra from other species) + noisec

but

xT

p = ypsT + (spectra from other species) + dT + noisep


Introduction

Constituents of prediction error

yp = xT

p b

= (ypsT + spectra from other species)b + dTb + (noise) b

Prediction error (yp − yp) has three constituents:

1 due to noise (variance)

2 due to systematic disturbance (bias)

3 due to the PCR/PLSR modeling error (bias)


Unifying frameworkEXPLICIT CORRECTION METHODS USING ADDITIONAL MEASUREMENTS

CC: component correction, 2000

IIR: independent interference reduction, 2001

GLSW: generalized least squares weighting, 2003

EPO: external parameter orthogonalization, 2003

TOP: calibration transfer by orthogonal projection, 2004

DCPS: difference correction of prediction samples, 2005

DOP: dynamic orthogonal projection, 2006

EROS: error removal by orthogonal subtraction, 2008


Unifying frameworkSTEP 1: ESTIMATION OF DRIFT SPACE

nτ replicate measurements nτ reference measurements

Matched y-values Non-matched y-values (e.g. uncontrolledonline measurements)

D approximated as

D = Xτ,2|{z}

slave

− Xτ,1|{z}

master

D approximated as

D = Xτ|{z}

slave

−AXc| {z }

master

GLSW & TOP (calibration transfer),EPO, DCPS & EROS (temperaturechanges), IIR (unknown variation), CC(unknown drift)

DOP (unknown drift)


Unifying frameworkSTEP 2: DRIFT-CORRECTION

Shrinking Orthogonal projection Subtraction

GLSW CC, IIR, EPO, DOP, TOP,

EROS

DCPS

Calibrate with {XcW, yc},where

W =

DTD

nτ − 1+ α

2I

!−

12

Calibrate with {XcN, yc}

D = TPT + E

N = (I − PPT)

Calibrate with {Xc , yc}Assuming one drift factor, d,correct the prediction sample:

xp∗ = xp − β d

β optimized to minimize the2-norm ||xp∗||.

r





EROS

DCPS


W =

DTD

nτ − 1+ α

2I

!−

12


D = TPT + E

N = (I − PPT)


xp∗ = xp − β d


r

ANALYTICALRESULTS

{(i) Equivalent for

one drift component





EROS

DCPS


W =

DTD

nτ − 1+ α

2I

!−

12


D = TPT + E

N = (I − PPT)


xp∗ = xp − β d


r

ANALYTICALRESULTS

{(i) Equivalent for

one drift component

{(ii) Equivalent when

r = nτ and α → 0


Step 2: Drift correctionCHOICE OF META-PARAMETERS (α, r)

More complex than determining pseudo-rank of D

Wilks’ f test, Malinowski’s F-test, Faber-Kowalski F-testResults based on random matrix theory and perturbation theory (Nadler et al.)






yp = xT

p b

= (ypsT + spectra from other species)b + dTb + (noise)p b






yp = xT

p b


Prediction error constituents after drift correctionb ⊥ estimated drift-space bias due to drift |dTb| ↓ ✔






yp = xT

p b


Prediction error constituents after drift correctionb ⊥ estimated drift-space

RMSECV ↑

bias due to drift |dTb| ↓

bias in PCR/PLSR ↑

✔

✘






yp = xT

p b


Prediction error constituents after drift correctionb ⊥ estimated drift-space

RMSECV ↑

||b||2 ↑

bias due to drift |dTb| ↓

bias in PCR/PLSR ↑

variance due to noise ↑

✔

✘

✘


Step 2: Shrinkage vs orthogonal projectionEXAMPLE 1: SIMULATION

Data generation

Using Beer’s law and known pure component spectra of 4 speciesDrift in 7-dimensional loading space S(Pd), overlapping with signal loadingspace S(Ps)



Data generation


Estimated drift-space for one (out of 5) reference sample

dT = σdT

pT

d,1

pT

d,2

pT

d,3

...pT

d,7

+



Data generation



dT = σdT

pT

d,1

pT

d,2

pT

d,3

...pT

d,7

+ σsT

pT

s,1

...pT

s,4

+



Data generation



dT = σdT

pT

d,1

pT

d,2

pT

d,3

...pT

d,7

+ σsT

pT

s,1

...pT

s,4

+ σn randn(1, L)



Data generation



dT = σdT

pT

d,1

pT

d,2

pT

d,3

...pT

d,7

+ σsT

pT

s,1

...pT

s,4

+ σn randn(1, L)

σdT = [1 × randn(1, 2) 0.1 × randn(1, 5)]

σsT = [0.3 × randn(1, 4)]

σn = 0.1



−6 −4 −2 0

−6 −4 −2 0

0.04

0.06

0.08

0.1

log( α )

RR

MS

EP

Without correction

With shrinkage

With OP r = 1

r = 2,3,4,5



−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

−6 −4 −2 0

−6 −4 −2 0

log( α)

RR

MS

EP

High Signal (σ = 3)

Low Noise (σ = 0.1)

Medium Signal (σ = 2)


Low Signal (σ = 0.3)



High Noise (σ = 0.3)



s

n

s

s

s

s


High Noise (σ = 0.3)s

n

n

n

n

n



−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

−6 −4 −2 0

−6 −4 −2 0

log( α)

RR

MS

EP











s

n

s

s

s

s



n

n

n

n

n



−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

−6 −4 −2 0

−6 −4 −2 0

log( α)

RR

MS

EP











s

n

s

s

s

s



n

n

n

n

n



−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

0.04

0.06

0.08

0.1

−6 −4 −2 0

−6 −4 −2 0

−6 −4 −2 0

log( α)

RR

MS

EP











s

n

s

s

s

s



n

n

n

n

n


Step 2: Shrinkage vs orthogonal projectionEXAMPLE 2: REAL DATA WITH TEMPERATURE EFFECTS

NIR, 3 species

11 x 2 = 22@ 30, 40 C

11 x 3 = 33@ 50, 60, 70 C

Monte Carlo5 reference samples

Calibration

Predictionο

ο

−10 −8 −6 −4 −2 0 20.02

0.03

0.04

0.05

0.06

r = 1

r = 2 r = 3

r = 4 r = 5

log(α)

RR

MS

EP

Without correctionWith shrinkageWith OP


Step 2: Shrinkage vs orthogonal projectionEXAMPLE 3: REAL DATA FOR CALIBRATION TRANSFER

NIR, 4 measured properties

40on Instrument 1

40on Instrument 2

Monte Carlo5 reference samples

Calibration

Prediction

−8 −6 −4 −2 0 20.1

0.11

0.12

0.13

0.14

r = 1

log(α)

RR

MS

EP

r = 2

r = 4 r = 5

r = 3

Without correctionWith shrinkageWith OP


Summary

Unifying framework

Many drift-correction methods proceed in two steps: (i) drift estimation, (ii)drift correction


Summary

Unifying framework



Shrinkage performs better than orthogonal projection typically whensignificant signal present in estimated drift


Summary

Unifying framework




Open issues

Statistically sound approach to choose meta-parameters


Summary

Unifying framework




Open issues

Statistically sound approach to choose meta-parametersMulti shrinkage parameters {α1, α2, . . . }


Documents

Correction of systematic disturbances in latent-variable ... · STEP 2: DRIFT-CORRECTION Shrinking Orthogonal projection Subtraction GLSW CC, IIR, EPO, DOP, TOP, EROS DCPS Calibrate