BIOSYST-MeBioS www.biw.kuleuven.be

The potential of Functional Data

Analysis for Chemometrics

Dirk De Becker, Wouter Saeys,

Bart De Ketelaere and Paul Darius

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The Potential of FDA for Chemometrics

Introduction to FDA

Introduction to Chemometrics

Using FDA in chemometrics

For prediction

For Analysis Of Variance

Conclusions

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What is Functional Data Analysis?

Developed by Ramsay & Silverman (1997)

Analyse Data

By approximating it

Using some kind of functional basis

Mainly for longitudinal data

High correlation between neighbouring datapoints

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Why use FDA?

Data as single entity <-> individual observations

Make a function of your data

Derivatives

Reduce the amount of data

Noise -> smoothing

Impose some known properties on the data

Monotonicity, non-negativeness, smoothness, ...

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Basis Functions?

Polynomials: 1, t, t², t³, ...

Fourier: 1, sin(ωt), cos(ωt), sin(2ωt),

cos(2ωt)

Splines

Wavelets

Depends on your data

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Chemometrics

Measure optical properties of material

Transmission or reflection of light

At a large number of wavelengths

Use these properties to predict something else

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Why Chemometrics?

Fast

Cheap

Non-destructive

Environment-friendly

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Classical methods

Ignore correlation between neighbouring

wavelengths:

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FDA in chemometrics

NIR spectra

Absorption peaks

Width and height

Basis: B-splines

~ shape of absorption peaks

Preserve the vicinity constraint

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Spline Functions

Piecewise joining polynomials of order m

Fast evaluation

Continuity of derivatives

Up to order m-2

In L interior knots

Degrees of freedom: L + m

Flexible

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Constructing a spline basis

Order

What to use the model for

Mostly cubic splines (order 4)

Number and position of knots

Use enough

Look at the data

!Overfitting

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Position of knots

More variation -> more knots

0 500 1000 1500 2000

12

34

5

valu

es

54 knots, equally spaced

0 500 1000 1500 2000

12

34

5

valu

es

54 knots, tuned

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B-spline approximation

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FDA for prediction

Functional regression models

P-Spline Regression (Marx and Eilers)

Non-Parametric Functional Data Analysis

(Ferraty and Vieu)

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Functional Regression Models

Project spectra to spline basis

Apply Multivariate Linear Regression to the spline

coefficients

Great reduction in system complexity

Natural shape of absorption peaks is used

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Functional Regression Models: case study

420 samples of hog manure

Reflectance spectra

Total nitrogen (TN) and dry matter (DM) content

PLS and Functional Regression applied

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Functional Regression: case study (ct'd)

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Functional Regression: case study: results

FDA PLS # B-splines # lat varDataset 1 10,4069 10,3282 22 6Dataset 2 9,9084 10,565 20 6Dataset 3 10,4921 10,4857 22 6Dataset 4 10,4533 10,3236 22 6Dataset 5 9,1203 10,6019 23 6

Dry matter content

FDA PLS # B-splines # lat varDataset 1 1,1922 1,2603 25 6Dataset 2 1,1582 1,1826 25 6Dataset 3 1,1806 1,2325 25 6Dataset 4 1,253 1,2852 25 6Dataset 5 1,1562 1,2664 25 6

Total nitrogen content

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P-Spline Regression (PSR)

By Marx and Eilers

Construct with B-splines:

Use roughness parameter on

Minimize

Full spectra are used for regression

BD

22 DXByS

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P-Spline Regression: case study

121 samples of seed pills

y is % humidity

PLS: RMSEP = 1,19

PSR: RMSEP = 1,115

# B-spline coefficients = 7

λ = 0.001

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Non-Parametric Functional Data Analysis

By F. Ferraty and P. Vieu

No regression model is involved

Prediction by applying local kernel functions in

function space

So far, no good results yet

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FDA in Anova setting: FANOVA

ANOVA:

“Study the relation between a response variable and

one or more explanatory variables”

is overall mean

are the effects of belonging to a group g

are residuals

)()()()( iggigx

)()( g

)( ig

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FANOVA: theory

Constraint:

Introduce so that

Introduce functional aspect:

Constraint: introduce

],[,0)( 1 mb

Z

Tg ],,,[ 1

)()()( Zx

)()( Cx )()( B

*** ,, xCZ

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FANOVA: goal and solution

Goal: estimate from

Solution:

B C

**1**^

)( CZZZB T

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FANOVA: significance testing

Locally:

Globally: ig

igig Zxerrordf

MSE 2^

)]()()([)(

1)(

)(/)(sup CMSEContrastM

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FANOVA: case study

Spectra of manure

4 types of animals: dairy, beef, calf, hog

3 ambient temperatures: 4°C, 12°C, 20°C

3 sample temperatures: 4°C, 12°C, 20°C

9 replicates

=> 324 samples

Model: )()()()()( ijklkjiijkl SATI

]9,1[],3,1[],3,1[],4,1[ lkji

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FANOVA: case study (ct'd)

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FANOVA: case study (ct'd)

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Conclusions

Splines are a good basis for fitting spectral

data

Using FDA, it is possible to include vicinity

constraint in prediction models in

chemometrics

FANOVA is a good tool to explore the

variance in spectral data