Land cover classification with Support Vector Machine applied to MODIS imagery

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Oct 16, 2013 (4 years and 23 days ago)

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Land cover classification with Support Vector Machine applied to MODIS imagery

Paulo Gonçalves

INRIA Rhône
-
Alpes, 655 Av. de l’Europe, 38334 St. Ismier, France


paulo.goncalves@inria.fr


Hugo Carrão, Andre

Pinheiro and Mário Caetano

Portuguese Geographic Institute, Av. Artilharia Um, 1099
-
052 Lisbon, Portugal


hugo.carrao@igeo.pt
;
mario.caetano@igeo.pt
;


Abstract

The
reported study is the first part of an on
-
going work
which objective is to

produc
e

a land cover
classification of continental Portugal from multi
-
spectral and multi
-
temporal MODIS satellite images
acquired at a 500m nominal resolution. Our goal is to achie
ve an automatic pixel level classification
using a Support Vector Machine (SVM) learning approach. More precisely, we use the time evolution
o
f reflectances measured in several

spectral bands from weekly composited images acquired during a
complete year pe
riod. As temporal profiles are relevant fingerprints of local phenologies, we believe
time series
offer

great
potential
to
improve
discrimination among the different land cover types.

In
order to reduce the input

space dimensionality, we will ende
avor at i
dentifying a parsimonious set of
fitting parameters
that
adequately model the time series.
Eventually
, o
ur model parameters
will be
used as

inputs of a supervised SVM classifier.
P
erformances

will be exhaustively compared to th
ose

obtained when the same cl
assifier is direc
tly applied to
a
single date
multi
-
spectral
reflectance
data
. It
is this preparator
y study
demonstrating

the importance of
the
chosen
analyzing

date that is at t
he core
of the present article. In particular, w
e show that global misclassifi
cation error

reaches a

minimum a
t a
specific period of the year. However,
when
each class

is
taken

s
eparately,
this
date

does

not
necessarily
correspond to
the
most favorable time

for
its identification.

1.

INTRODUCTION

Land cover mapping is an essential
task

for

many applications,
such as

landscape management,
biodiversity assessment, and,
more generally

to support environmental, social, and economic policies.

As
landscape is persistently changing under the influence of several factors (e.g.,
forest fires, cl
ear
cuts, urban fabric growth
), land cover
maps become rapidly out of date, and the need for
updated land
cover maps
is constantly
increasing

to help

near real
-
time spatial decision
-
making.
While different in
g
oals and motivations, it is
also

evident

that
most organizations action is

now

implemented at

regional,
national, or

even continental scales. I
t

is

then expected that

in order to satisfy most users’ actual
need
s,

assessment of

continuous

landscape changes should

more and more

rely on land cover mappin
g at
regional scales
,

based on medium resolution satellite information.

Until recently, revisit frequency and
spatial coverage matching

were two severe obstacles

that
penalized the use of

satellite

imagery

to
produce

land cover mapping at high temporal sam
pling rate
s
.

Recent design

of
Moderate Resolution Imaging Spectroradiometers
(
MODIS
)

sensor
s

opened up

a
wide range of new possibilities for high periodic land cover chara
cterization at regional scale.
MODIS
sensors are mounted on two satellites from Natio
nal Aeronautics and Space Administration (NASA):
TERRA and AQUA launched on December 18, 1999 and on May 4, 2002, respectively. MODIS
images correspond to high radiometric sensitivity (radiance) measured in 36 spectral bands that span a
wavelength range fr
om 0.4 µm to 14.4 µm. The two first bands (620
-
670 nm and 841
-
876 nm,
primarily used for land/cloud/aerosols boundaries) are imaged at a nominal spatial resolution of 250 m
at nadir. The next five bands (459
-
479 nm, 545
-
565 nm, 1230
-
1250 nm, 1628
-
1652 nm a
nd 2105
-
2155 nm primarily used for land/cloud/aerosols properties) correspond to a 500 m pixel resolution.
The remaining 29 bands are acquired at a coarser spatial resolution of 1 km.
In addition, a
ll produced
images undergo

a first level processing for ge
ographic calibration and atmospheric surface reflectance
correction.

A
utomatic classification from

multi
-
spectral satellite imagery is a
n

important stake for

the remote
sensing community, and

i
t is widely reckoned

that
it should

provide with
accurate land
cover mapping
and change detection in
near real time.

S
till
, several pitfalls

remain
: (1
) different land covers share

similar spectral signatures;

(2) conventional image classification
technique
s
fail at separating a large
number of
land cover classes with

acceptable misclassification rates.

On the other hand, several studies have proved the advantages of multi
-
temporal images analysis for
vegetation characterization, environmental monitoring and land cover mapping. As there is not one
workbench methodology

for exploring satellite images time series, several different approaches have
been proposed in the literature (e.g. DeFries et al., 1997, Roberts et al., 1999, Boles et al. 2004
,
Oliveira et al., 2005,
to cite but a few).

Our objective is to exploit withi
n the same framework

of land cover classification
,

complementary
information stemming from multi
-
spectral

imagery and multi
-
temporal
data
, two components

that are
easily accessible in MODIS products.

Our approach is based on the
observation

that
the differ
ent land
cover classes exhibit specific spectral reflectances as functions of time, which should significantly
improve classification scores obtained from single date measurements.
To this end, the present article
i
s a systematic study to determine the cla
ssifying capacity of each available date,
so that we can

use
the best score as a benchmark in our future developments.

This work should be understood as a
milestone towards our final objective that is
to fit a simple parametric model
to

the time series, an
d use
these as the inputs of
a
Support Vector Machine classifier, still
of
confidential use in the remote
sensing community (Pal and Mather, 2005).

2.

STUDY AREA AND DATA

The study area is the entire Portuguese continental territory. Our study relies on the M
OD09A1
product, an 8 days composite of surface reflectance images, freely available from MODIS Data
Product web site (http://modis.gsfc.nasa.gov). We considered a full year observation period from
February 2000 to January 2001 (43 images). Surface reflecta
nce measured within seven disjoint
spectral bands (VIS+SWIR+MIR) were used and imaged at a nominal spatial resolution of 500 meters.
Moreover, two vegetation indices (
Normalized Difference Vegetation Index


NDVI;
Enhanced
Vegetation Index

-

EVI) were als
o calculated for each date and used as input information for land
cover classification.

CORINE Land Cover 2000 cartography (Instituto do Ambiente, 2005), derived from visual
interpretation of LANDSAT ETM+ data, was used as auxiliary data to select both the

training and the
validating sets. We assigned to each land cover class, a set of samples, each one representing a vector
in [0,1]

7


[
-
1,1]

2
associated to one pixel of the reflectance composite images and vegetation indices,
i.e. a 500m
-
by
-
500m square a
rea. Then, to ensure intra
-
class homogeneity, a dispersion criterion
(based on minimum integrated squared distance between individuals and the mean population) was
applied to perform a data clustering so as to remove possible outliers. The retained samples

are finally
randomly split into five subsets of the same size that were used for a cross validation purpose: each
time one subset is used for the training task, and the remnants for classification.

Nine Land Cover classes were defined according to the CLC
2000 nomenclature. These classes
received the following designations:
(i)

Water areas,
(ii)

Urban areas,
(iii)

Bare soils,
(iv)

Natural
grasslands,
(v)

Shrublands,
(vi)

Needle leaf forests,
(vii)

Broad leaf forests,
(viii)

Non irrigated lands
and
(ix)

Irri
gated lands. For each of these classes a set of samples we collected, 15, 30, 15, 30, 30, 30,
30, 31, and 30, respectively, and uniformly distributed all over the mainland territory.


3.

METHODOLOGY

3.1.

Data pre
-
processing



Compositing process aims at selecting
for each pixel
-
size area of an image, the “best” pixel among
those available within a given period of time (in most cases, optimality is meant for “cloud
-
free”
situations). Clearly, this selection implicitly assumes land cover to be stationary in time, whi
ch in turn
restricts the time window extent to a sensibly small period. Our study is particularly sensitive to this
limit since it precisely relies on the specific non
-
stationarity (cyclo
-
stationarity) of the land cover
classes. Therefore, we chose a seven
-
day compositing period that is sufficiently short not to smooth
out the reflectance changes due to vegetation phenology. On the other hand, such window size is not
large enough to remove all clouds effect, and locally the censored reflectances take on abe
rrant values
(see Fig. 1).



Figure
1
.
Mean reflectances in the red, green and blue wavebands (spectral bands 1, 4 and 3
respectively) as functions of time. Each displayed curve is obtained by averaging together the time
profiles

of all samples belonging to the same land cover class: water (squares □), natural grasslands
(

thick dotted line), broad leaf forests (Δ up
-
triangles), barren lands (thick line), shrub lands (+
crosses), Needle leaf forests (
-

-

-
dashed line), irrigated l
ands (thin line), non irrigated lands (circle o),
urban zone (thick dots ●). Clouds presence results in local bursts of extraordinarily high amplitude that
equally distort the classes’ responses.

These artificial bursts, not only pollute the time series, b
ut also, as they equally affect the mean class
profiles, attenuate possible dissimilarities between them. Table 1 gives the inter
-
class distances:




(1)

where

i

designa
tes the
i
-
th class center of mass in the 9
-
dimensional observation space (7 spectral
bands + 2 vegetation indices), and

i

the
i
-
th intra
-
class samples dispersion around the mean.


W

NG

BLF

BS

SL

NLF

IL

NIL

U

W

0

0,234

0,44

0,188

0,208

0,293

0,379

0,351

0,101

NG

0,234

0

0,332

0,249

0,045

0,135

0,621

0,409

0,346

BLF

0,44

0,332

0

0,752

0,4

0,185

0,596

1,222

1,575

BS

0,188

0,249

0,752

0

0,212

0,352

0,792

1,356

0,399

SL

0,208

0,045

0,4

0,212

0

0,113

0,676

0,576

0,276

NLF

0,293

0,135

0,185

0,352

0,113

0

0
,477

1,091

0,768

IL

0,379

0,621

0,596

0,792

0,676

0,477

0

2,561

1,66

NIL

0,351

0,409

1,222

1,356

0,576

1,091

2,561

0

1,711

U

0,101

0,346

1,575

0,399

0,276

0,768

1,66

1,711

0

Table
1
.

Euclidean distances between lands cover class
es (Eq. 1).

According to this table, the most distant classes are “Irrigated lands” and “Non irrigated lands”
(d
78
=2.561), whereas the closest two are “Natural grasslands” and “Shrublands” (d
25
=0.045). Aside the
possibility that those latter are indeed ver
y similar in kind, it deemed important to remove the clouds
shot noise effect and measure its influence on the inter
-
class distances.

We use a time sliding windowed median filter that replaces each sample
of a time series by the
Q
-
t
h

percentile of the windowed series

centered on the sliding index
k
. The same int
er
-
class distances (ref. Eq. 1
) computed on the filtered time series show that distinctness between any two
classes systematically increases, except f
or the “Barren lands”


“Irrigated lands” pair that slightly got
closer. The “Natural grasslands”


“Shrublands” duo remains very close (d
25

= 0.057), supporting the
fact that these two classes are effectively much alike in kind. This is corroborated in Fi
gure 2, where
the filtered time series indubitably evidence disparities and similarities between land cover classes.


W

NG

BLF

BS

SL

NLF

IL

NIL

U

W

0

4,352

9,368

3,879

4,15

6,883

5,576

4,556

1,939

NG

4,352

0

2,054

0,43

0,057

0,657

1,041

0,878

1,84

BLF

9,368

2,054

0

4,133

3,056

1,405

2,853

7,765

12,175

BS

3,879

0,43

4,133

0

0,221

1,376

0,764

2,785

1,343

SL

4,15

0,057

3,056

0,221

0

0,935

1,071

1,375

1,595

NLF

6,883

0,657

1,405

1,376

0,935

0

1,124

5,109

6,847

IL

5,576

1,041

2,853

0,764

1,071

1,124

0

4,
803

3,947

NIL

4,556

0,878

7,765

2,785

1,375

5,109

4,803

0

3,614

U

1,939

1,84

12,175

1,343

1,595

6,847

3,947

3,614

0

Table
2
.

Normalized Euclidean distances between lands cover classes (Eq. 1). To remove cloud effects,
reflectance

and vegetation indices time series have been filtered using a sliding time windowed median filter (
Q
-
th percentile is fixed to 40% and window width corresponds to a 2 months period).































Figure
2
.
Mean
reflectances and vegetation indices as functions of time. To remove cloud effects visible in
Figure 1, a sliding time windowed median filter has been applied to the pixel time series (percentile is fixed to
40%, window width corresponds to a 2 months perio
d and threshold is set to 0.5). Each displayed curve is
obtained by averaging together filtered time profiles of all samples belonging to the same land cover class (see
fig. 1 caption for legend).

3.2

Classification with Support Vector Machine


Support Vector
Machines (SVM) are a new generation of supervised learning systems based on recent
advances in statistical learning theory (Cristianini and Shawe
-
Taylor, 2000). Pioneered by the work on
learning strategy by Vapnik and collaborators (Boser
et al.
, 1992; Vap
nik, 1998), they have rapidly
and successfully been applied to numerous real
-
world classification problems.

Conceptually, SVM rationale is that a classification problem that does not have a satisfying solution in
its own observation space may have one sim
ple and efficient in a more complicated representative
system. As so, SVM use a hypothesis space of linear indicator functions to draw classification in a
high dimensional (possibly infinite) feature space, image of the observation space by a non
-
linear
ma
pping



C潮獩d敲⁡扩n慲y⁣a獳ifi捡i潮
-
l敡r湩湧⁴a獫
1

and the following data training set:

,

(2)

with samples

drawn i.i.d. according to some unknown but fixed probability di
stribution
. The
non
-
linear mapping


r慮af潲m猠桥h
-
dimensional
input space

into the feature
space
, and let the set of hypotheses be linear hypothesis functions

of the type:

.

(3)

One remarkable fact about this hypothesis function (written here in its dual form) is that it implies the
data only through their inner products in the feature spac
e. Therefore, if


i猠pr潰敲ly捨潳敮e獯s桡
i敹猠s桥h獯
-
捡lle搠
kernel

condition,


(4)

we do not even need to know the underlying feature map


漠扥⁡bl攠漠l敡r渠n渠nh
攠ee慴r攠獰sc攮e

A獳畭i湧 獯Ⱐ桥h灲潢l敭 i猠桥h o 摥敲mi湥n桥捯cffi捩e湴s
桡h mi湩miz攠桥
捬慳獩fi捡i潮error潮畮e敮es慭灬敳⸠I摥dllyⰠ桥h扥獴捬a獳ifi敲獨s畬dmi湩miz攠桥h數灥捴敤ev慬攠
潦⁴h攠
loss

or

risk
:



(5)

where the loss function
, penalizes the deviations. In practice however, the joint
probability function
is unknown, and the expected loss (also called
gener
alization loss
) is
approximated by the
empirical classification error

based on the available information (i.e. the training
set). Then, the empirical risk functional reads:

.

(6)

O
ften though, finding the hypothesis
that minimizes this empirical risk leads to an ill
-
posed
problem that results in the well
-
known phenomenon described as
over
-
fitting
in the literature (i.e. the
selected hypothesis is too complex).

One way to avoid over
-
fitting tolerating noise and outliers, is to
restrict the complexity of the hypothesis function, by introducing a regularization term ([
66
] Vapnik,
1982). This regularization term is closely related to the notion of margin, another i
mportant concept
for SVM that reflects the sensitivity and tolerance of the classifier to the samples

that
stand “close” to the (non
-
linear) separator. Those points are called support vectors, and the solution
that minimizes the reg
ularized empirical risk function is referred to as a soft margin SVM (in contrast
to a hard margin SVM that systematically zeroes the empirical loss). Solution to this quadratic
optimization problem is classically obtained by Lagrangian theory and comes ou
t to find the
Lagrangian multipliers

*

that maximize the following quantity:




1

For the sake of simplicity we restrict this introductory study to the case of a two
-
classes classification.
Generalization to multiple classes problems is straightforward using a one
-
versus
-
the
-
rest strategy.


subject to

and
,
i
=1,…N.

(7)

C is the regularization paramet
er that bounds the Lagrangian multipliers (i.e. the weights associated to
the support vectors) controlling this way the capacity of the hypothesis function class
2
.

Finally, a Radially Basis Function (RBF) kernel,
, with user pre
-
defi
ned sample
variance

2
, is chosen because it maps the input space into an infinite dimensional feature space and
often yields good results for nonlinear regression (Suykens
et al.
, 2002; Seeger, 2004).


4.

RESULTS

All experimental results we are reporting in
this section have been obtained using a cross validation
technique that permits evaluating classifiers’ efficiency (in terms of generalized bias) using only the
training set. We fixed to five the number of cross validation folds.

Moreover, to appraise lan
d cover classification performance of SVM applied to MODIS reflectance
data, a standard multi
-
class K
-
Nearest Neighbor (K
-
NN) classifier was trained on the same samples
set and results used as nominal values.

In the course of our study, we were led to ques
tion our training set capacity to deal with the input
space dimensionality (recall we are working with 9
-
dimensional data points corresponding to 7
spectral reflectances and 2 vegetation indices). We then investigated several strategies to perform a
dimens
ion reduction: combination of a reduced number of spectral reflectances and indices, Principal
Component Analysis (PCA) with and without components chop off.
3

It appeared that quite
systematically the best classification rates were obtained with the full r
ank output of PCA applied to
the 9
-
dimensional training set. For that reason, we retain this data configuration in the remainder of
our analysis.

Figure 3 presents the expected losses obtained for all available dates, after optimizing the classifiers’
para
meters (i.e. SVM and K
-
NN). In both cases, lowest misclassification rates are attained in summer
times. This is actually consistent with the class profiles of figure 2 that show maximum discrepancy
during this same period. In terms of relative performance,

SVM outperforms K
-
NN classifier at most
discriminating dates, and conversely at winter season. Nonetheless, as SVM achieves the best global
classification rate in July (expected loss is then equal to 22.02 %), this reasonably good result will
serve us as
benchmark in our ongoing work on multi
-
temporal data classification.

Let us now split the

global misclassification rate into generalization losses per class (figure 4). All
curves, except those for “Natural Grasslands” and “Non Irrigated lands”, display a
minimum value
during summer time. Again, this is fully consistent with NDVI and EVI time series of figure 2. For
similar reasons, “Natural Grasslands” and “Non Irrigated lands” succeed better classification rates in
April and October respectively, in perf
ect conformity with their respective phenologies. Furthermore,
a closer view at the omission and commission errors (Table 3) confirms that “Natural Grasslands” is
definitely the less distinctive land cover class, frequently confounded with “Shrub lands” an
d “Non
Irrigated lands”. On the contrary, pixels of “Bare Soils” or “Irrigated Lands”, two land cover classes
that were very close according to Euclidean distances of Table 2, surprisingly never mix up.




2

There exists an alternative way for contr
olling the hypothesis function capacity, which amounts to fix the
proportion of training samples that will lie between the boundary and the margin hyperplane. In SVM literature,
this regularization strategy is often referred to as the

-
parameterization, a
nd we will use this in our study.

3

Strikingly, a PCA performed at each available date reveals that in all cases, only the first two principal
components suffice to explain more than 90% of data variability. Yet, chopping the less explanatory components
of
f always degrades the classification performances.


Date

SVM (

=0.08)

KNN





Mean Loss

K

Mean Loss

0
0.02.26

1.5

36.85 %

9

31.15 %

00.03.31

1.2

40.30 %

5

31.96 %

00.04.30

1.4

33.65 %

3

34.25 %

00.05.24

1.3

35.68 %

5

33.21 %

00.06.17

0.8

27.39 %

7

31.98 %

00.07.11

1

22.02 %

5

28.86 %

00.08.28

1.9

25.33 %

7

27.62 %

00.09.29

2.1

30.29 %

7

27.20 %

00.
10.23

2

30.31 %

3

30.53 %

00.11.08

2.6

32.77 %

7

30.93 %

00.12.10

1.3

37.37 %

3

34.44 %

01.01.25

0.7

39.06 %

9

30.31 %











Classification

Reference Data



W

NG

BLF

BS

SL

NLF

IL

NIL

U

User's Accuracy (%)

W

13

1

0

0

0

0

0

0

0

92.9

NG

0

18

1

2

5

3

0

5

0

52.9

BLF

0

1

25

1

0

2

0

0

0

86.2

BS

0

1

1

10

0

0

0

0

1

76.9

SL

2

5

2

2

24

0

3

1

0

61.5

NLF

0

1

1

0

0

25

0

0

0

92.6

IL

0

0

0

0

0

0

25

1

1

92.6

NIL

0

3

0

0

1

0

2

22

2

73.3

U

0

0

0

0

0

0

0

2

26

92.9

Producer's Ac
curacy (%)

86.7

60.0

83.3

66.7

80.0

83.3

83.3

71.0

86.7



Table
3
.

Omission and commission errors per class. This repartition corresponds to a SVM classification based
on reflectance
s

and vegetation indices measured at a single dat
e (July 11, 2000).

Figure
3
.

Generalization loss as a function of time. For each available date, the “minimum” expected classification
error corresponding to the nine land cover classes
, is obtained optimizing the classifier parameters set.

Full line: using SVM classification. Dashed line: using a K
-
Nearest Neighbor classification. Table on the right
indicates for both classifiers, and for a monthly date, the corresponding optimized par
ameter values.

2

designates
the variance of the SVM
-
RBF kernel; K is the number of Nearest Neighbors classifier.

Figure 4.

Generalization loss per class as a function of time. Left plot: experimental curves. Right plot: 2
nd

order
polynomial fit. See caption of figure 1 for legends.

5.

CONCLUSIONS


The
results we

obtained are promising for they show a significant in
fluence of the year date on
classification performance
s
.
Our forthcoming work on time series modeling will be
motivated by the
necessity of data fusion and

dimensionality reduction. Indeed, pre
liminary tests showed that
embedding

information available for all dates simultaneously had to face the problem of
insufficient
training data, which

sensibly degraded
the
classification scores
.

From a methodological vi
ewpoint, SVM
are powerful

and easy to tune

learning system
s that should
rapidly enter the standard
classifiers’

toolbox used in remote sensing
applications
.


6.

ACKOWLEDGEMENTS


We are
very grateful to Pedro Oliveira (ISEGI, Portugal) who
collected the sampl
es used in this
study

a
nd to Frédéric Desobry (Univ. of Cambridge, UK) for fruitful discussion on SVM theory.


7.

REFERENCES


B
oles
,

S.

H.,

X
iao
,

X.,

L
iu
,

J.,

Z
hang
,

Q.,

M
unkhtuya
,

S.,

C
hen
,

S.,

O
jima, D., 2004
.

L
and cover
characterization of Temperate East A
sia using multi
-
temporal VEGETATION sensor data.
Remote
Sensing of Environment

90: 47
-
489.

Boser, B. E., Guyon, I. M., Vapnik, V. N., 1992. A training algorithm for optimal margin classifiers.
In D. Haussler, editor,
Proceedings of the 5
th

Annual ACM Works
hop on Computational Learning
Theory
, pages 144
-
152. ACM Press.

Cristianini, N., Shawe
-
Taylor, J., 2000.
An Introduction to Support Vector Machines and Other
Kernel
-
based Learning Methods
. Cambridge University Press.

De
F
ries
,

R.,
H
ansen
,

M.,

S
teininger
,

M.
,

D
ubayah
,

R.,

S
ohlberg
,

R.,

T
ownshend
,

J.,

1997.

S
ubpixel

F
orest Cover in Central Africa from Multisensor, Multitemporal Data.
Remote Sensing of
Environment
60: 228
-
246.

Instituto do Ambiente, 2005. CLC2000 Portugal. Technical Report.

Oliveira, P., Gonçal
ves, P. And Caetano, M., 2005.
Land cover time profiles from linear mixture
models applied to MODIS images,
Proceedings of the 31st International Symposium on Remote
Sensing of Environment
, June 20
-

24, Saint Petersburg, Russian Federation.

Pal, M., Mathe
r, P. M., 2005. Support vector machines for classification in remote sensing.
International Journal of Remote Sensing

26(5):1007
-
1011.

R
oberts
,

D.

A.,

B
atista
,

G.

T.,

P
ereira
,

J.

L.

G.,

W
aler
,

E.

K.,

N
elson
,

B.

W.,

1999.

Change identification
using multit
emporal spectral mixture analysis: applications in eastern Amazonia.
Remote Sensing
Change Detection


Environmental Monitoring Methods and Applications
. R. S.
L
unetta
,

and
C.

D.

E
lvidge (Eds), Taylor & Francis Editions, 137
-
159.

Scholkopf, B., Smola, A.,
2002.
Learning with Kernels: Support Vector Machines, Regularization,
Optimization, and Beyond
. MIT Press.

Seeger, M., 2004. Gaussian processes for machine learning.
Interrnational Journal of Neural Systems

14 (2):1
-
38.

Suykens, J., Gestel, T., de Brabante
r, J., de Moor, B., Vandewalle, J., 2002. Le
ast Squares Support
Vector Machines.

World Scientific.

Vapnik, V., 1982.
Estimation of dependencies based on empirical data
. Springer Verlag, New York.

Vapnik, V., 1998.
Statistical Learning Theory.
Wiley, 1998.