ANALYSIS OF VARIANCE WITH FUNCTIONAL DATA TO DETECT

COLOR CHANGES IN GRANITE

T. Rivas, J. Taboada, C. Ord´o˜nez

Dpto. de Recursos Naturales, Universidad de Vigo, 36200, Vigo, Spain

J. M. Mat´ıas

Dpto. de Estad´ıstica, Universidad de Vigo, 36200, Vigo, Spain

Keywords:

Functional data, Anova, Spectral reﬂectance curve, Granite, Protective treatment.

Abstract:

Analysis of spectral reﬂectance curves is useful in many application ﬁelds. Despite the functional nature of

these curves, statistical methods used to date for analysing these curves (classiﬁcation, analysis of variance,

etc.) have tended to be scalar and do not fully take advantage of the information they contain as functional

objects. In this article we applied functional analysis of variance to spectral reﬂectance curves in order to

evaluate the impact of protective treatments on granite colour.

The use of raw information on the spectrum means that signiﬁcant changes are detected that might go unno-

ticed in models that use scalar values to measure colour. The application of the funcional approach enables

information to be obtained on changes whose intensity are statistically signiﬁcant at each point of the spectrum

without the need to perform different analyses for each area of the spectrum. Furthermore, the computational

load is no greater than for classical multivariate models.

1 INTRODUCTION

Analysis of spectral reﬂectance curves is useful in

several application ﬁelds, such as astronomy, agricul-

ture, engineering, meteorology, environment, mining,

geographical information systems, etc. (see for exam-

ple, (Cho and Skidmore, 2006), (Stevens et al., 2006),

(Ayala-Silva and Beyl, 2005), (Henry et al., 2004),

(I˜nigo et al., 2004), (Lacar et al., 2001), (Richardson

et al., 2001), (Lebow et al., 1996) and references cited

therein).

However, statistical methods currently used for

this kind of analysis (classiﬁcation, discrimination,

regression, etc) has a basically scalar or, at best, multi-

variate, nature and so curves are considered merely as

sets of observed points. When curves are considered

under a functional view, the goal is not to implement

a statistical analysis that handles them as such func-

tional objects, but to implement a preliminary step to

graphic or analytical studies (such as, for example, a

functional approximation, the calculation of deriva-

tives, or interpolation or extrapolation at some point

of the spectrum, etc.), or statistical analysis of scalar

indicators of some of their overall analytical proper-

ties (see the references given above).

In this article we describe a novel application

of analysis of variance (e.g. (Montgomery, 1997))

with a functional approach (functional ANOVA or

FANOVA) (Ramsay and Silverman, 1997), (Cuevas

et al., 2004) to an analysis of spectral reﬂectance

curves with a view to evaluating the impact of several

protective treatments on granite rock colour.

In the architectural heritage preservation ﬁeld, the

colour of stone surfaces constitutes an element of the

historical and artistic value of a monument, and, as

such, should not undergo changes following cleaning,

protection or consolidation processes (Lazzarini and

Tabasso, 1986). It is therefore important to be aware

of the potential impact of such treatments on orna-

mental rock so as to be able to avoid treatments that

cause colour changes and choose treatments that are

more suitable for preservation purposes.

Colour is quantitatively expressed using spec-

trophotometers which, based on measuring re-

ﬂectance throughout the visible spectrum, deﬁne

colour using colorimetric coordinates in the stimu-

lus space of a standard observer. Coordinate sys-

tems widely used in this context include, for ex-

540

Rivas T., Taboada J., Ordóñez C. and M. Matías J. (2009).

ANALYSIS OF VARIANCE WITH FUNCTIONAL DATA TO DETECT COLOR CHANGES IN GRANITE.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 540-546

DOI: 10.5220/0002314405400546

 SciTePress

ample, CIE L

∗

and CIE L

∗

systems

(L’ECLAIRAGE-CIE, 2007), which are based on

combining the spectral properties of the light source,

the reﬂectance curve of an object and the sensitiv-

ity of a standard observer. In this research, we fo-

cused particularly on the changes produced in the ob-

ject itself, in an endeavour to determine, directly from

changes in the spectral reﬂectance curve, the mean-

ing of changes in rock colour produced by protective

treatment. Another article currently in preparation

(Rivas et al., 2009) will focus on analysing changes

produced at the level of the stimulus space for a stan-

dard observer.

The application of a functional approach is partic-

ularly signiﬁcant for the problem at hand, given the

great heterogeneity of colour in the rocks that are typ-

ically used in historical buildings. This is the case of

igneous rocks (like granite), where the overall colour

is the result of the colour contributions of different

minerals: the whites and greys of quartz and mus-

covite, the pinks and greys of the feldspars and the

browns and blacks of iron-rich minerals. The rela-

tive quantities for each mineral (mineralogy), grain

size and the distribution of mineral grains relative to

each other (texture) are features that have a bearing

on colours and colour measure repeatability. Changes

brought about by a treatment can easily be masked or

erroneously evaluated as a consequence of the innate

variability of the colour of a rock ((Rivas et al., 1998),

(Berns, 2000)).

The article is structured as follows: ﬁrst we brieﬂy

describe the FANOVA methodology; next we de-

scribe the application problem, specifying the meth-

ods used and the results obtained; and ﬁnally we de-

scribe our conclusions.

2 FUNCTIONAL ANOVA

2.1 Introduction to Functional Linear

Models

The linear regression model for a single input variable

X and a response variable Y takes the form:

E(Y|X) = α+ βX

where α, β are the regression coefﬁcients. This model

is easily generlaized to d regression variables X =

, ..., X

)

by means of

E(Y|X) = µ+

∑

i=1

= α+ hX, βi (1)

where α and β = (β

, ..., β

)

are now the regression

coefﬁcients.

A functional linear regression model (Ramsay and

Silverman, 1997) is an extension of the multivari-

ate linear regression model to the case of inﬁnite-

dimensional or functional data. For each t ∈ T :

E(Y(t)|X) = α(t) + hX, β(·, t)i

= α(t) +

X(s)β(s, t)ds (2)

with a parameter function β : T × T → R and a over-

all mean response function α : T → R. The data are

a sample of pairs of random functions (X, Y), with X

the predictor and Y the response functions.

The above functional model can also be extended

to the case of several functional predictors as follows:

E(Y(t)|X) = α(t) + hX, β(·, t)i

= α(t) +

∑

j=1

(s)β

(s, t)ds (3)

where X = (X

, ..., X

) are the functional predictors

and Y is the response function.

Intermediate models between classical (1) and

general (3) are models that produce a scalar response

with functional predictors and models (like those used

in this research) that produce a functional response

with scalar predictors. The general model is as fol-

lows:

E(Y(t)|X) = α(t) + hX, βi

= α(t) + Xβ(t)

= α(t) +

∑

j=1

(t) (4)

where X is the matrix n × d of the design with el-

ements (X)

= X

j,i

(the i-th observation of the j-th

scalar covariable), α is the overall functional mean

and β is the functional vector of the d functional co-

efﬁcients β

, with i = 1, ..., n, j = 1, ..., d.

2.2 Function Smoothing

In the functional model, we do not see the sample

functions (in our case, response functions) y

, i =

1, ..., n in the majority of applications, but only their

values y

) at a set of n

points t

∈ R, j = 1, ..., n

For the sake of simplicity, we shall assume these to

be common to all the functions y

, i = 1, ..., n. These

observations may, moreover, be subject to noise, in

which case they take the form: z

= y

)+ε

, where

we assume that ε

is random noise with zero mean,

i = 1, ..., n, j = 1, ..., n

Therefore, the functional approach ﬁrst requires

estimating each sample function y

∈ F , i =

1, ..., n. One way to do so is to assume that F =

ANALYSIS OF VARIANCE WITH FUNCTIONAL DATA TO DETECT COLOR CHANGES IN GRANITE

541

span{φ

, ..., φ

} with {φ

} sets of basis functions

(Ramsay and Silverman, 1997). For our research, we

chose a family of B-splines as the set of basis func-

tions, given their good local behaviour. If, for the sake

of simplicity, we represent as y any of the functions y

i = 1, ..., n in the sample, we have:

y(t) =

∑

k=1

(t) = c

Φ(t) (5)

where Φ(t) = (φ

(t), ..., φ

(t))

Hence, the smoothing problem consists in deter-

mining the solution y to the following regularization

problem:

min

x∈F

∑

j=1



− y(t

)



+ λΓ(y) (6)

where z

= y(t

)+ε

is the result of observing y at the

point t

, Γ is an operator that penalizes the complex-

ity of the solution, and λ is a regularization parameter

that regulates the intensity of this penalization. In our

case, we used the operator Γ(y) =



y(t)



dt,

where T =[t

min

, t

max

] and D

is the second-order dif-

ferential operator.

Bearing in mind the expansion (5), the above

problem (6) may be written as:

min



(z− Φc)

(z− Φc) + λc



where z= (z

, ..., z

)

, c = (c

, ..., c

)

, Φ is the

×n

matrix with elements Φ

= φ

) and R is the

×n

matrix with elements R



, D



(T )

(t)D

(t)dt.

The solution to this problem is given by

c =

(Φ

Φ + λR)

−1

z, such that the estimated values

of the true function y at the observation points are

obtained by means of ˆy = Sz, where S = Φ(Φ

Φ +

λR)

−1

and ˆy = ( ˆy(t

), ..., ˆy





)

The selection of the λ forms part of the model se-

lection problem and is usually performed using cross

validation.

2.3 Fitting the Functional Response –

Scalar Predictor Linear Model

Below we focus on the functional response model

with scalar covariables (4) as used in this research.

So as to avoid the independent term α(t), below, we

propose assuming that the vector of covariables in-

cludes a ﬁrst constant covariable equal to 1 whose

functional coefﬁcient is β

= α. We can thus refor-

mulate the model (4) as:

E(Y(t)|X) = Xβ(t) =

∑

j=1

(t) (7)

Within this framework, we assume that the above

smoothing process on the example functions produces

the expansions y

(t) = c

Φ(t), i = 1, ..., n which may

be jointly written by means of y(t) = Cφ(t) where C

is a matrix n× m with rows c

i = 1, ..., n.

We likewise assume that each of the functional co-

efﬁcient β

admits an expansion in terms of the afore-

mentioned bases:

(t) =

∑

k=1

(t) = b

η(t)

where b

= (b

, ..., b

)

. This may be written in

functional form as: β

= b

Φ. Using this expression

for all the coefﬁcients β

jointly, we obtain:

β = (β

, ..., β

)

= Bη

where B is a matrix d×m

with rows b

, j = 1, ..., d.

From all the above, for each observation i =

1, ..., n, the model (7) becomes: ˆy

(t) = x

β(t) =

Bη(t) or equivalently, in compact functional form:

ˆy

= x

Bη. Applying the above to all the example

observations, we obtain the following functional-type

vectorial expression:

ˆy = XBη

The ﬁt is carried out using the criterion of regular-

ized minimum squared errors:

∑

i=1

− ˆy

(T )

+ λΓ(β)

(y(t) − ˆy(t))

(y(t) − ˆy(t))dt + λΓ(β)

(Cφ(t) − XBη(t))

(Cφ(t) − XBη(t))dt

+ λ

[Lβ(t)]

[Lβ(t)]dt

= trace



φφ



+ trace



XBJ

ηη



− 2trace



ηφ



+ λtrace



BRB



(8)

where J

ηη

η(t)η(t)

dt, J

φφ

Φ(t)φ(t)

dt,

ηφ

Φ(t)η(t)

dt and Γ is a regularization oper-

ator that penalize the complexity of β as a function of

Γ(β) =

[Lβ(t)]

[Lβ(t)] dt

[LBη(t)]

[LBη(t)]dt

= trace



BRB



where R is an m

× m

matrix with (R)

k,l

hLη

, Lη

(T )

Deriving the expression (8) with respect to B, the

following normal equations are obtained (Ramsay and

Silverman, 1997):

XBJ

ηη

+ λBR = X

φη

IJCCI 2009 - International Joint Conference on Computational Intelligence

542

and also the following expression of the solution in

terms of the Kronecker product ⊗:

vec(B) =



ηη

+ λR)⊗





−1

vec



φη



where vec(B) is the column vector obtained by stack-

ing the columns of the matrix B on top of one another

and where λ can be selected using, for example, cross-

validation.

2.4 ANOVA Table and F Test

The signiﬁcance of the model can be evaluated us-

ing one of two possible approaches: by evaluating

the signiﬁcance of the functional model as a whole

(e.g. (Cuevas et al., 2004), (Ramsay and Silverman,

1997)), or by performing an analysis of signiﬁcance

at each point t ∈ T . This latter approach is of partic-

ular interest in our research as we wish to identify the

wavelengths where signiﬁcant changes occur. Signif-

icance, in any case, for each t ∈ T logically provides

overall signiﬁcance for the functional model.

The signiﬁcance of the model for each point

can be determined using a functional version of

Snedecor’s F distribution, in which the statistic is

also a function deﬁned in the support T for the F

functions (Ramsay and Silverman, 1997). This pro-

duces an F distribution for each point of this support,

enabling detection of possibly statistically signiﬁcant

changes in areas where changes occur. Given that the

following expressions are functions of t ∈ T :

SSE(t) =

∑

i=1

(t) − ˆy

(t))

SSY(t) =

∑

i=1

−

α(t))

SSM(t) = SSY(t) − SSE(t)

the statistics of the ANOVA table also are functions

of t:

Source of

Variation

d f MS(t) =

SS(t)

d f

F(t)

Model d − 1 MS

(t)

Error n− d MS

(t)

Total n− 1

Therefore, an F test can be performed for each

t ∈ T .

3 DETECTING CHANGES IN

REFLECTANCE CURVES

USING FUNCTIONAL ANOVA

3.1 Data and Experiment Design

Used in the experiment was a medium-grained pre-

Hercynian granite composed of quartz, mica (mus-

covite and biotite) and feldspar. This granite,

brownish-yellow in colour, is widely used in archi-

tectural monuments in northwest Spain.

The rock was treated with water repellent prod-

ucts intended to prevent or slow down water entry in

the rock and consequent deterioration. The products

applied were BS29 (supplied by Wacker Chimie) and

Tegosivin HL100 (supplied by Evonik). BS29 is a

water thinnable mixture of silane, siloxane and syn-

thetic resins and Tegosivin HL100 is an solvent or-

ganic methylethoxy polysiloxane. For each product,

ﬁve prisms (5x5x1 cm) were selected. Colour was

measured at eight points chosen at random prior to

the application of each product. A 5% (w/w) solu-

tion of each product (using white spirit as a solvent in

the case of Tegosivin HL100 and distilled water in the

case of BS29) was applied to the samples via partial

immersion for one minute. After 48 hours colour was

measured at another eight points of the treated surface

of the prism chosen at random.

Colour was measured using a Minolta CM710

spectrophotometer in SCI mode, using D65 as a light

source, an 8 mm target area and an observer angle

of 10

◦

. Obtained for each point were values of the

reﬂectance spectrum, expressing the percentage of re-

ﬂectance of the object for every 10 nm of wavelength

in the range between 400 and 720 nm (support for the

functions was thus the wavelength interval T = [400,

720] nm).

Observations for each product totalled 40,

whether as colorimetric coordinates and attributes

or as spectral reﬂectance curves. The curves were

smoothed and assumed to belong to a functionalspace

F = h{φ

, ..., φ

}i generated by a set of order 6 B-

splines with 12 knots, resulting in a total of d = 16 ba-

sis functions. (This number of basis functions, chosen

in view of the complexity of the observed functions,

does not produce signiﬁcant changes in the results for

a wide range of values). Hence, all the spectral func-

tions are unambiguously determined by their coefﬁ-

cients in terms of the expression:

y =

∑

i=1

Changes in colour were evaluated by means of

the application of FANOVA to the reﬂectance curves

ANALYSIS OF VARIANCE WITH FUNCTIONAL DATA TO DETECT COLOR CHANGES IN GRANITE

543

obtained. For each protective treatment used (BS29

or HL100), a full factorial design for treatment fac-

tors (no/yes), prism (1, 2, 3, 4 and 5) and interaction

was postulated, with the following functional linear

model:

ijk

= µ+ α

+ β

+ (αβ)

+ ε

ijk

i = 1, ..., p, j = 1, ..., q, k = 1, ..., r,

with p = 2 levels of treatment (treated or untreated),

q = 5 different test prisms and r = 8 replications, sub-

ject to restrictions as follows:

∑

i=1

= 0;

∑

j=1

= 0;

∑

i=1

(αβ)

= 0;

∑

(αβ)

= 0

(9)

where y

ijk

, ∈ F was the spectral reﬂectance curve ob-

tained from the spectrophotometer for the observa-

tion ijk, µ ∈ F was the global mean for these curves,

∈ F was the main effect corresponding to level i of

the treatment factor (treated or untreated), β

∈ F was

the main effect corresponding to level j of the prism

factor (prism j), (αβ)

∈ F was the interaction effect

between level i and prism j and where ε

ijk

∈ F was a

residual function accounting for the unexplained vari-

ation speciﬁc to the k-th observation for prism j with

treatment i.

Note that the restrictions (9) imply sum zero in

such a way that each main effect represents the mean

alteration of the global mean for each treatment, and

each interaction effect indicates the mean alteration

for each main effect caused by each combination of

levels of the factors.

3.2 Results

Figure 1 shows for BS29 the overall mean and the

effect of no treatment and the effect of treatment. As

can be observed, the effect of treatment was negative,

indicating a reduction in the intensity of the spectral

reﬂectance curves which should indicate that colour

of the rock becames darker.

The signiﬁcance of these effects are analysed in

Figure 2 (BS29) and 3. The last one shows (left to

right) the curves for the FANOVA table and the func-

tional F statistics corresponding to the treatment fac-

tor, the prism and the crossed treatment × prism fac-

tor. In these ﬁgures, the broken line indicates the

value of F that corresponds to 5% signiﬁcance for

each t ∈ T ; values below this level should be consid-

ered less signiﬁcant or non-signiﬁcant. The content of

each FANOVA table responds to Table 1, with p = 2

(levels of treatment), q = 5 (prisms), r = 8 (replica-

tions) and n = pqr = 80 observations, and where (e.g.

(Montgomery, 1997)):

∑

i=1

∑

j=1

∑

k=1

ijk

−

···

kpr

∑

i=1

i··

−

···

kpr

∑

j=1

· j·

−

···

kpr

∑

i=1

∑

j=1

ij·

−

···

kpr

− SS

= SS

− SS

are the different ANOVA decomposition functions.

400 600

wave−length (nm)

effect

Grand Mean

400 600

0.95

1.05

1.1

1.15

1.2

1.25

wave−length (nm)

BS29: No

400 600

−1.25

−1.2

−1.15

−1.1

−1.05

−1

−0.95

wave−length (nm)

BS29: Yes

Figure 1: For BS29, overall mean, effect of level 1 treatment

(No) and level 2 treatment (Yes).

Table 1: ANOVA table corresponding to the full factorial

design used.

Variation d f MS(t) =

SS(t)

d f

F(t)

Treat.

p− 1 MS

(t)

Prism

q− 1 MS

(t)

Treat.×

Prism

(p− 1)×

(q− 1)

(t)

Error

pq(r− 1) MS

(t)

Total

n− 1

As can be observed in Figures 2 and 3, the ap-

plication of BS29 produces signiﬁcant changes in the

reﬂectance curve as a consequence of the three fac-

tors, namely, application of the treatment, prism and

treatment×prism. For the last two of these factors,

this change is signiﬁcant for the entire spectrum con-

sidered, whereas in regard to the treatment factor, the

change is not signiﬁcant above 600nm (orange-red re-

ﬂectance).

Combining these results with the sign of the ef-

fect (Figure 1) it can be concluded that treatment with

IJCCI 2009 - International Joint Conference on Computational Intelligence

544

350 400 450 500 550 600 650 700 750

100

120

140

160

BS29 − Functional ANOVA MS Table

wave−length

Concentr

Test−prism

Conc*Test−prism

Error

Figure 2: Components of the FANOVA table for BS29.

400 500 600 700

wave−length (nm)

F concentration

400 500 600 700

wave−length (nm)

F test−prism

400 500 600 700

wave−length (nm)

F conc*test−prism

Figure 3: F tests for the product factor, prism factor and

treatment×prism factor for BS29. The horizontal broken

line indicates a 5% signiﬁcance level.

BS29 produces a reduction in colour reﬂectance in the

entire spectrum except for the orange-red part.

As for application of THL100, the FANOVA

results (not shown) indicate that the prism and

treatment×prism factors are equally signiﬁcant in the

entire spectrum considered, whereas the treatment

factor is only signiﬁcant for smaller wavelengths, cor-

responding to reﬂectance in the lower green part and

in all the blue and purple parts. It can be concluded

that treatment with THL100 also produces a reduc-

tion in colour reﬂectance, but only conclusively be-

low 500nm (the lower green part and all the blue and

purple parts). This reduction is also more intense for

smaller wavelengths.

Finally, it should be noted that for both products

there was high standard deviation in the upper part of

the spectrum, from the wavelength value above which

there was no signiﬁcance for the treatment factor for

THL100 and low signiﬁcance for the treatment factor

for BS29. This would appear to indicate that some

factor exists, not associated with the presence of the

treatments, whose dispersion in a part of the spectrum

prevents the model from capturing the variability as-

sociated with the presence of the product.

The indications are that this factor arises in the

chromatic variability existing in the rock, with stan-

dard deviation higher in the range of the spectrum

covering the colours yellow, orange and red. The

colour of the rock is yellow-reddish, resulting, on

the one hand, from cream-yellow feldspars (less in-

tense colours for albite-anortite compositions and yel-

lower for potassium compositions) and, on the other

hand, from brownish-reddish iron oxyhydroxide pati-

nas distributed unevenly throughout the rock.

Furthermore, since the rock is ﬁne-grained hetero-

granular granite, colour is strongly determined by tex-

tural unevenness in the rock. This is manifested in the

great dispersion in the reﬂectance spectra in the range

corresponding to the rock colour range, i.e., mainly

above 600 nm (orange-red). This variability prevents

the model from detecting, in this particular range, the

effects of the treatments, although it does not prevent

it doing so in the range where dispersion is less—that

is, below green and towards the blues. The signiﬁ-

cance in the entire spectrum for the prism and treat-

ment prism factors is also a consequence of this chro-

matic variability and textural unevenness in the rock.

4 CONCLUSIONS

We described how we used FANOVA to evaluate

changes in spectral reﬂectance curves resulting from

the application of two commercial protective treat-

ments to granite rock (BS29 and Tegosivin HL100).

Methods used to date are scalar and have to be ap-

plied repeatedly at points or areas of the spectrum in

order to determine the statistical signiﬁcance of wave-

length changes. The results obtained in our research

demonstrate the usefulness of the functional approach

when it is necessary to analyse changes in functional

objects, such as reﬂectance curves. The application of

the funcional approach enables information to be ob-

tained on changes whose intensity are statistically sig-

niﬁcant at each point of the spectrum (without having

to perform analyses for each point of the spectrum)

and changes that might go unnoticed in models that

use scalar values to measure colour. Furthermore, the

computational load is no greater than that required for

multivariate ANOVA (except for the effort required to

smooth the functions in the sample).

The method was applied to evaluating the changes

in colour reﬂectance for granite rock following the ap-

plication of two water repellent products (BS29 and

ANALYSIS OF VARIANCE WITH FUNCTIONAL DATA TO DETECT COLOR CHANGES IN GRANITE

545

HL100). The results showed a reduction in colour re-

ﬂectance in the rock, although, for the BS29, the re-

duction affected wavelengths below 650 nm (yellow)

and for the HL100, the reduction was only signiﬁcant

in wavelengths below 500 nm.

For both products, a reduction in the luminosity

of the rock can be anticipated, greater in the case of

BS29. In upcoming research, we will analyse how de-

tected reﬂectance changes inﬂuence changes in colour

as perceived by a standard observer. In other words,

we will apply this functional approach to the colour

curves resulting from superimposing the source spec-

tral curve, the object reﬂectance curve and the colour

matching curves of the standard observer, so as to be

able to measure the colour in the stimulus space. The

results obtained will be compared with those that clas-

sical statistical methods (e.g. multivariate ANOVA)

obtain in any normal space for measuring colour (e.g.

CIE L

∗

Finally, the signiﬁcance of the prism effect and of

the interaction between the prism factor and the treat-

ment factors suggest a design that reduces the vari-

ability produced by the experimental unit (prism) as

far as possible. In regard to FANOVA, the lesser sig-

niﬁcance in the upper part of the spectrum may be a

result of the greater variance in the curves in this part

of the spectrum as a consequence of the granite’s own

colour characteristics. One of our new lines of re-

search is the design of new experiments that deal with

this variability (e.g. paired designs or designs with

additional random blocking factors).

ACKNOWLEDGEMENTS

We wish to express our gratitude to Professor J. O.

Ramsay and his team for the functional data analy-

sis software for Matlab that served as the nucleus for

the developments that were necessary to carry out this

study. The authors also thank to Evonik Industries

and Wacker Chemie for supplying Tegosivin HL100

and BS29 respectively. J.M. Mat´ıas’s research is sup-

ported by the Spanish Ministry of Education and Sci-

ence, Grant No. MTM2008-03010.

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