Using Evidence Theory in Land Cover Change Prediction to Model

Imperfection Propagation with Correlated Inputs Parameters

Ahlem Ferchichi, Wadii Boulila and Imed Riadh Farah

Ecole Nationale des Sciences de l’Informatique, Laboratoire RIADI, Manouba, Tunisia

Keywords:

LCC Prediction, Imperfection Propagation, Parameter and Model Imperfection, Aleatory and Epistemic

Imperfection, Correlated Parameters, Evidence Theory.

Abstract:

The identiﬁcation and the propagation of imperfection are important. In general, imperfection in land cover

change (LCC) prediction process can be categorized as both aleatory and epistemic. This imperfection, which

can be subdivided into parameter and structural model imperfection, is recognized to have an important im-

pact on results. On the other hand, correlation of input system parameters is often neglected when modeling

this system. However, correlation of parameters often blurs the model imperfection and makes it difﬁcult to

determine parameter imperfection. Several studies in literature depicts that evidence theory can be applied

to model aleatory and epistemic imperfection and to solve multidimensional problems, with consideration of

the correlation among parameters. The effective contribution of this paper is to propagate the imperfection

associated with both correlated input parameters and LCC prediction model itself using the evidence theory.

The proposed approach is divided into two main steps: 1) imperfection identiﬁcation step is used to identify

the types of imperfection (aleatory and/or epistemic), the sources of imperfections, and the correlations of the

uncertain input parameters and the used LCC prediction model, and 2) imperfection propagation step is used

to propagate aleatory and epistemic imperfection of correlated input parameters and model structure using the

evidence theory. The results show the importance to propagate both parameter and model structure imperfec-

tion and to consider correlation among input parameters in LCC prediction model. In this study, the changes

prediction of land cover in Saint-Denis City, Reunion Island of next 5 years (2016) was anticipated using

multi-temporal Spot-4 satellite images in 2006 and 2011. Results show good performances of the proposed

approach in improving prediction of the LCC of the Saint-Denis City on Reunion Island.

1 INTRODUCTION

Propagation of the imperfections existing in a system

and the corresponding reliability analysis have be-

come an essential task for current product design pro-

cesses, especially, in LCC prediction process (Boulila

et al., 2014). Generally, imperfection in a system

can be classiﬁed into aleatory and epistemic (Hoff-

man and Hammonds, 1994). Aleatory imperfection

results from randomness (e.g. uncertainty). How-

ever, epistemic imperfection stems from the lack of

knowledge (e.g. imprecision, conﬂict). Imperfection

types can be roughly split into two categories: param-

eter and structural model (Duy et al., 2013)(Droguett.

and Mosleh, 2012). Imperfection in model param-

eter is due to natural variability, measurement inac-

curacy, and errors in handling and processing data

(Duy et al., 2013). Model structure shows imper-

fection from model assumptions/approximations, hy-

potheses, and scale effects. On the other hand, Corre-

lated parameters are often neglected when modeling a

natural system. However, parameters correlation of-

ten blurs the model imperfection and makes it difﬁ-

cult to determine parameter imperfection. In review

of literature, model parameter imperfection has re-

ceived the most attention in previous studies (Boulila

et al., 2014)(Eckhardt et al., 2003). (Boulila et al.,

2014) focuses on the imperfection in LCC predic-

tion model parameters, and explores how those im-

perfection propagate through to model responses us-

ing probabilistic method to model only random im-

perfections, and have not taken into account the cor-

relation among parameters. The focus here partly

reﬂects a consensus that model parameter is an im-

portant source of imperfection in simulation predic-

tions. However, it also reﬂects the relative ease with

which imperfection in model parameter can be quan-

tiﬁed in comparison to assessments of structural im-

Ferchichi, A., Boulila, W. and Farah, I..

Using Evidence Theory in Land Cover Change Prediction to Model Imperfection Propagation with Correlated Inputs Parameters.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 2: FCTA, pages 47-56

ISBN: 978-989-758-157-1

perfection in models. In practice, structural imper-

fection may be more important than parameter imper-

fection in evaluating model performance (Gong et al.,

2013)(Rohmer et al., 2014). For example, (Gong

et al., 2013) show the importance of using the im-

perfection of hydrological model structure in the im-

perfection processing. In LCC prediction process,

the correlated parameters and structural model im-

perfections are generally treated by probability the-

ory. Moreover, numerous authors conclude that there

are limitations in using probability theory in this con-

text (Droguett. and Mosleh, 2012). So far, sev-

eral non-probabilistic methods have been developed,

which include evidence theory (Shafer, 1976), possi-

bility theory (Zadeh, 1965), etc. As a generalization

of all the above probability and non-probability im-

perfection models, evidence theory combines random

and epistemic imperfections in a very straightforward

way, and is capable to solve multidimensional prob-

lems, with consideration of the correlation among pa-

rameters (Jiang et al., 2013). Based on the ﬂexibility

of evidence theory, the main aim of our work is to

propagate the imperfection associated with both cor-

related input parameters and LCC prediction model

itself using the evidence theory. The proposed ap-

proach is divided into two main steps: 1) imperfec-

tion identiﬁcation step is used to identify the types of

imperfection (aleatory and/or epistemic), the sources

of imperfections, and the correlations of the uncertain

input parameters and the used LCC prediction model,

and 2) imperfection propagation step is used to prop-

agate aleatory and epistemic imperfection of corre-

lated input parameters and model structure using the

evidence theory. The remainder of this paper is orga-

nized as follows. A brief introduction of evidence the-

ory is given in Section 2. The proposed framework is

presented in section 3. Section 4 presents our results.

Finally, section 5 concludes the paper with summary

and future works.

2 EVIDENCE THEORY

Evidence theory, also called as Dempster-Shafer the-

ory, was initially developed by (Klir, 1994) and for-

malized by (Shafer, 1976). The evidence theory has

the potential to quantify aleatory and epistemic imper-

fections. This theory has also become the accepted

method for propagating correlated and uncorrelated

input parameters imperfection through LCC predic-

tion models. In this section, the basic notations of the

evidence theory are introduced.

Frame of Discernment (FD). The FD is deﬁned by

the ﬁnest possible subdivisions of the sets, and the

ﬁnest possible subdivision is called the elementary

proposition.

Basic Probability Assignment (BPA). Let Θ be a ﬁ-

nite set of mutually exclusive and exhaustive hypothe-

ses, and 2

be the power set of Θ. The fundamen-

tal concept for representing imperfection is the BPA,

which deﬁnes a mapping function (m) of 2

to the

interval between 0 and 1. The measure m, BPA func-

tion, must satisfy the following axioms:

m(A) ≥ 0, ∀A ⊆ Θ. (1)

0) = 0 (2)

∑

m(A) = 1. ∀A ⊆ Θ. (3)

Belief and Plausibility Functions. The measures of

imperfection provided by evidence theory are known

as belief (Bel) and plausibility (Pl), which also lie in

the interval [0,1]. Given a body of evidence, the (Bel)

and (Pl) can be derived from the BPA by

Bel(B) =

∑

A⊆B

m(A). (4)

Pl(B) =

∑

B∩A6=

m(A). (5)

The formulas make it easy to see that the belief func-

tion, (Bel), is calculated by summing the BPAs that

totally agree with the event B, while the plausibility

function, (Pl), is calculated by summing BPAs that

agree with the event B totally and partially. These

two functions can be derived from each other. For

example, the belief function can be derived from the

plausibility function in the following way:

Bel(B) = 1 − Pl(B) (6)

The relationship between belief and plausibility func-

tions is

Bel(B) ≤ Pl(B) (7)

which shows that as a measure of event B is true, if

P(B) is the true value of the measure of set B is true,

then Pl(B) is the upper bound of P(B),and Bel(B) is

the lower bound, so

Bel(A) ≤ P(A) ≤ Pl(A) (8)

Dempster’s Rule of Combining. The Dempster’s

rule of combination is an operation that plays a cen-

tral role in the evidence theory. The BPAs induced by

several sources are aggregated using this rule in order

to yield a global BPA that synthesizes the knowledge

of the different sources. Take two BPA structures, m

and m

, for instance, the combined structure m

calculated in the following manner:

(A) =

∑

B∩C=A

(B)m

(C)

1 − K

when A 6=

0 (9)

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

(

0) = 0, when K =

∑

B∩C6=

(B)m

The coefﬁcient K represents the mass that the combi-

nation assigns to

0 and reﬂects the conﬂict among the

sources. The denominator in Dempster’s rule, 1 − K,

is a normalization factor, which throws out the opin-

ion of those experts who assert that the object under

consideration does not exist.

3 METHODS AND DATA

3.1 Study Area and Data

Reunion Island is a French territory of 2500 km

located in the Indian Ocean, 200 km South-West of

Mauritius and 700 km to the East of Madagascar

(Fig. 1). Mean annual temperatures decrease from 24

degC in the lowlands to 12 degC at ca 2000 m. Mean

annual precipitation ranges from 3 m on the eastern

windward coast, up to 8 m in the mountains and

down to 1m along the south western coast. Vegetation

is most clearly structured along gradients of altitude

and rainfall (Cadet, 1980). Reunion Island has a

strong growth in a limited area with an estimated

population of 833,000 in 2010 that will probably

be more than 1 million in 2030 (Reunion, 2011).

It have been a signiﬁcant changes, putting pressure

on agricultural and natural areas. The urban areas

expanded by 189% over the period from 1989 to 2002

(Durieux et al., 2008) and available land became a

rare and coveted resource. The landscapes are now

expected to fulﬁl multiple functions i.e. urbanisation,

agriculture production and ecosystem conservation

and this causes conﬂicts among stakeholders about

their planning and management (van der Valk, 2002).

Saint−Denis is the capital of Reunion Island, and

the city with the most inhabitants on the island (Fig.

1). It hosts all the important administrative ofﬁces,

and is also a cultural center with numerous museums.

Saint-Denis is also the largest city in all of the French

Overseas Departments.

Figure 1: Location of the study area.

Available remote sensing data for this research in-

clude classiﬁed images of land over of Saint−Denis

from SPOT-4 images for the years 2006 and 2011

(Fig. 2). Selecting these images beneﬁts from ad-

vantages such as a broad and integrated view, multi-

spectral images and replicated coverage in different

time periods. For this study, satellite data were clas-

siﬁed after initial corrections and processing in order

to prepare the data for extracting useful information.

Spectral, geometric, and atmospheric corrections of

images were conducted to make features manifest, in-

crease the quality of images and to eliminate the ad-

verse effects of light and atmosphere. According to

the study objective and different land cover of the

area, ﬁve categories including water, urban, forest,

bare soil, and vegetation were identiﬁed and classi-

ﬁed.

Figure 2: Satellite images.

3.2 Methodology

In this section, we propose the application of the ev-

idence theory for propagating imperfections of cor-

related parameters and imperfections of structural

model in the context of LCC prediction. The pro-

posed approach is presented in the ﬁgure 3.

Figure 3: General modelling proposed framework.

The ﬁrst step is to map sources and types of imper-

fection of parameter and model structure (Ferchichi

et al., 2014) and to study correlation among parame-

ters. Finally, the propagation of all sources simultane-

ously provides an analysis of their effect on the model

response.

Using Evidence Theory in Land Cover Change Prediction to Model Imperfection Propagation with Correlated Inputs Parameters

3.2.1 Processing and Modeling of Parameter

and Model Structure Imperfection

The objective of this step is to identify imperfections

related to both correlated input parameters of LCC

prediction model in one hand, and imperfections

related to structure of this model in the other hand.

a) Parameter Imperfection

These imperfections are related to input data of

LCC prediction model.

X Sources of parameter imperfection: To

better provide decisions about LCC, prediction model

include several input parameters. These parameters

describe objects extracted from satellite images and

which are subject of studying changes. In this study,

objects are extracted from images coming from

SPOT-4 image, including four multispectral bands

and one monospectral band:

• 10 spectral parameters: Mean values and standard

deviation values of green, red, NIR, SWIR and

monospectral bands for each image object;

• 6 texture parameters: These parameters generated

from GLCM (Gray Level Co-occurrence Matrix)

include homogeneity, contrast, dissimilarity, en-

tropy, standard deviation and correlation. Their

deﬁnitions as following:

Hom =

∑

P(i− j)

1+(i− j)

Ctr =

∑

P(i − j)

Ent =

∑

P(i − j)[−logP(i − j)]

SD =

∑

P(i −

∑

i • P(i − j))

Cor =

∑

(i−

∑

i•P(i− j))( j−

∑

i•P(i− j))

where i and j are two different grey levels of the

image, P is the number of the co-appearance of

grey levels i and j.

• 7 shape and spatial relationships parameters: area,

length/width, shape index, roundness, density,

metric relations and direction relations;

• 1 vegetation parameter: The NDVI (Normalized

Difference Vegetation Index) is the ratio of the

difference between NIR and red reﬂectance.

• 3 climate parameters: temperature, humidity and

pressure.

The inclusion of these 27 uncertain parameters

was based on previous studies (Boulila et al.,

2014)(Boulila et al., 2011).

− Spectral parameters imperfection sources: Sev-

eral studies have presented effects of spectral param-

eters on prediction decisions (Atanassov et al., 2013).

Among these effects we list: spectral reﬂectance of

the surface, sensor calibration, effect of mixed pixels,

effect of a shift in the channel location, pixel registra-

tion between several spectral channels, atmospheric

temperature and moisture proﬁle. To reduce the spec-

tral imperfections, (Atanassov et al., 2013) have pro-

posed some solutions e.g. strict requirements for the

instrument’s design, envisaging of appropriate proce-

dures for on-board calibration, choosing appropriate

algorithms for radiometric and atmospheric correc-

tion, reducing the wavelength range of the irradiance

or spectral response measurement, reducing the cloud

shadows and cloud contamination effects and reduc-

ing errors of sensor system itself.

− Texture parameters imperfection sources: Those

factors may produce a textural effect derived by the

spatial interaction between the size of the object in the

scene and the spatial resolution of the sensor, a border

effect, and ambiguity in the object/background dis-

tinction. (Peters, 1995) (Pesaresi and Benediktsson,

2001) have proposed some solutions for reduction of

texture data imperfection e.g. using high spatial reso-

lution and choosing appropriate methods for segmen-

tation.

− Shape parameters imperfection sources: Imper-

fection related to shape parameters can rely to the fol-

lowing factors (Atanassov et al., 2013): accounting

for the seasonal position of the sun with respect to

the Earth, conditions in which the image was acquired

changes in the scene’s illumination, atmospheric con-

ditions and observation geometry. (Atanassov et al.,

2013) have proposed some solutions for reduction of

shape data imperfection e.g. improving the platforms’

stability and the carrier’s velocity, technological en-

hancement of the sensors themselves, reducing the ef-

fects of atmospheric conditions, improving the overall

segmentation quality, reducing of the number of bad

pixels and the size of bad areas, and improvement of

the imperfection of pixels’ response.

− NDVI imperfection sources: Among factors that

affect NDVI, we can list: variation in the brightness

of soil background, red and NIR bands, atmospheric

perturbations as a function of the state of the atmo-

sphere and the soil surface at the time of the two ac-

quisitions and variability in the structure of sub-pixel

of the vegetation, and variability in the structure of

sub-pixel. To reduce the NDVI imperfections, (Miura

et al., 2000) (Hird and McDermid, 2009) have pro-

posed some solutions e.g. choosing appropriate al-

gorithms for atmospheric correction, reducing errors

in surface measurements for the NIR and red bands,

reducing the temporal variations effects in the solar

zenith and azimuth angles, and reducing the sun an-

gle effects and noise contamination.

− Climate parameters imperfection sources: Sev-

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

eral studies have investigated the inﬂuence of several

factors on the climate parameters accuracy (Jimenez-

Munoz and Sobrino, 2006). These factors can be: at-

mospheric correction, noise of the sensor, land sur-

face emissivity, aerosols and other gaseous absorbers,

angular effects, wavelength imperfection, full-width

half maximum of the sensor and band-pass effects.

To reduce the climate data imperfections, (Jimenez-

Munoz and Sobrino, 2006) (Hulley et al., 2012) have

proposed some solutions e.g. choosing appropriate al-

gorithms for atmospheric correction, reducing errors

of sensor system itself, reducing the emissivity varia-

tions, reducing the sun angle effects and solar heating,

reducing the errors of radiometer calibration and the

errors of radiation, and reducing errors of spatial and

temporal variability of clouds.

X Types of parameter imperfection: Imper-

fection related to input parameters to the LCC predic-

tio model can be of two types: epistemic and random.

The imperfection type for each parameter depends on

sources of their imperfection. For example, for the

NDVI parameter, imperfection sources are the bright-

ness of pixels, the red and NIR bands, etc. Thus, each

imperfection source provides an imperfection type for

NDVI parameter. For example, the brightness of pix-

els on darker pixels leads to imprecise information in

a vegetation zone. In this case, the NDVI parameter

is marred by an epistemic imperfection.

X Modeling of parameter imperfection: The

goal of this step is to determine the distribution of

each uncertain parameter. For random parameters,

various distributions, such as uniform, triangular, nor-

mal, or lognormal distributions, can be used in the

model. For epistemic parameters, there are also many

distributions such as: belief and plausibity distribu-

tions, possibility distribution, etc. Herein, each im-

perfection propagation method has its own speciﬁc

distributions.

X Analysis of parameters correlation: Its

objective is to determine the relationships between

parameters. In general, input parameters in remote

sensing systems are not independent of each other.

The value of one parameter can affect the value of

another. A LCC prediction model usually contains a

large number of correlated parameters leading to non-

identiﬁability problems. For example, the distribution

and the production of vegetation are strongly inﬂu-

enced by temperature. Then, temperature and NDVI

parameters are correlated.

b) Model Structure Imperfection

These are imperfections related to LCC predic-

tion model structure itself. The proposed model

for LCC prediction is presented in previous work

(Boulila et al., 2011). In this work, we presented a

model to predict spatiotemporal changes in satellite

image databases. The proposed model exploits data

mining concepts to build predictions and decisions

for several remote sensing ﬁelds. Imperfections

related to the structure of LCC prediction model can

be very numerous and affect different parts of the

model. Therefore necessary to identify imperfections

sources that should be considered for processing.

Then, imperfection related to model structure are

resulting from conceptualization, simpliﬁcations, and

hypotheses related to a lack of knowledge about a

system, a structure, and the behavior of components

in varied conditions during development. In majority,

the imperfection of structural representation is the

most inﬂuential imperfection on LCC prediction

model. The development of LCC prediction model

is based on a number of hypotheses which are

decisions or judgments considered by analysts. For

example, when two hypotheses H

, H

are given

by two different experts, then we have two differ-

ent structural models M

and M

. In most cases,

imperfection about LCC prediction model structure

is a form of epistemic imperfection because we are

unsure whether their constructions are reasonable

and complete. It would be aleatory imperfection

only if the structure of the governing model were

itself to change over time, across space, or among

components in some population.

3.2.2 Propagation of Imperfection using

Evidence Theory

The objective of this step is to propagate the imper-

fection of correlated parameters and the imperefction

of model structure using evidence theory.

a) Parameters Imperfection Propagation

using Evidence Theory

Using the evidence theory to model imperfec-

tions has been studied extensively in several ﬁelds

(Duy et al., 2013) (Abdallah et al., 2013). In this

section, the procedures of propagating the uniﬁed

structures dealing with both random and epistemic

imperfection and with considering correlation among

parameters will be addressed. For the proposed

model, ﬁrst we should identify which type of imper-

fection each parameter. To illustrate the proposed

method, we use a simple transfer function which has

two uncertain parameters

Y = f (E,A). (11)

where E represents the epistemic imperfection param-

eter, A represents the random imperfection parameter

Using Evidence Theory in Land Cover Change Prediction to Model Imperfection Propagation with Correlated Inputs Parameters

and Y is the model response of the LCC. For E, the

epistemic imperfection is generally expressed by a se-

ries of subsets of the universal set associated with a

BPA structure just as {[E

]/m(1),[E

]/m(2)

,...,[E

]/m(k),...|k ∈ (1,2,...,M)}. Where M is

the total number of subintervals of E and m(k) repre-

sents the BPA value associated with the kth subinter-

val [E

]. When there are different BPA structures,

we can use combining rule to integrate them into a

combined BPA structure as E

/m(E

)( j ∈ [1,2,, M])

ultimately, where E

is also an interval as [E

]

and m(E

) is the BPA value associated with the in-

terval E

. For A, assuming A is normal distribution

A ∼ (µ,σ), the distribution scope can be truncated to

[µ − ξσ,µ + ξσ] approximately and then we can dis-

cretize the approximate interval into N subintervals

], i ∈ [1,2, , N], and for each subinterval the ba-

sic probability value is deﬁned:

m(A

) =

f (x)dx,i ∈ [1,2,...,N]. (12)

where A

is deﬁned as {A

|x ∈ [A

]} and f (x) is

the probability density distribution function (pdf) of

x. Obviously for the random parameter, the equiva-

lent BPA values within speciﬁed intervals are equal to

the area under the pdf. After obtaining the BPA struc-

tures of all the uncertain parameters, we can integrate

them into a joint structure. The joint BPA structure

is deﬁned by the Cartesian product, which is synthe-

sized as:

C = A × E = {c

i j

= A

× E

} (13)

where C denotes the Cartesian set of all the uncertain

parameters and c

i j

is the element of C.

− When the uncertain parameters, E and A, are inde-

pendent, the joint BPA for c

i j

is deﬁned by multiply-

ing the BPA of A

to the BPA of E

m(c

i j

) = m(A

) × m(E

) (14)

The focal element c

i j

is included by the joint FD, and

its BPA is just equal to the multiplication of the cor-

responding marginal BPAs.

− When the uncertain parameters, E and A, are

correlated, we will develop a new evidence the-

ory model which takes into account the correlation

among parameters based on ellipsoidal model (Luo

et al., 2008). Then, the ellipsoidal model is originally

proposed for non-probabilistic imperfection analysis.

Here the ellipsoidal model is extended to deal with the

correlated evidence parameters. For this purpose, a

multidimensional ellipsoid is constructed by making

all possible realizations of the N-dimensional inter-

correlated evidence parameters fall into a joint FD:

Ω = {X|(X − X

)

G(X − X

) ≤ 1} (15)

where the ellipsoidal center X

is obtained through

the marginal FDs:

+ X

,m = 1,2,...,N (16)

where X

∈ c

i j

is the evidence parameters (random

and epistemic parameters).

The symmetric positive-deﬁnite characteristic matrix

G determines the size and orientation of the ellip-

soid, reﬂecting the degree and the manner of corre-

lation between the evidence parameters. Obviously,

one should assign the belief probabilities only to the

elements c

i j

that are partially or totally falling into the

ellipsoid model. Thus, a joint BPA is formulated as:

m(c

i j

∩ Ω) =

m(A

) × m(E

)

i j

∩ Ω 6= 0 (17)

where S is a normalization factor to make the total

BPAs of m equal to 1.0, which is given by

S =

∑

i j

∩Ω6=0

m(c

i j

) (18)

Then, get the upper and lower CDFs of system re-

sponse y via evidence reasoning.

Let Θ

= {d

i j

: d

i j

= f (c

i j

),c

i j

⊂ Θ

} denote the

frame of discernment of Y, where d

i j

is its focal el-

ement, f is the LCC prediction model in (11), and Θ

is the frame of discernment of X. After determining

the sets, c

i j

and d

i j

, the belief and plausibility func-

tions are evaluated by checking all propositions of the

joint BPA structure, as given in the following equa-

tions (Joslyn and Helton, 2002).

Bel

i j

) = Bel

[ f

−1

i j

)] =

∑

i j

⊂ f

−1

i j

)

i j

)

(19)

i j

) = Pl

[ f

−1

i j

)] =

∑

i j

∩ f

−1

i j

)6=

i j

)

(20)

Then

Bel

(y < v) = Bel

[ f

−1

)] =

∑

i j

⊂ f

−1

)

i j

)

(21)

(y < v) = Pl

[ f

−1

)] =

∑

i j

∩ f

−1

)6=

i j

)

(22)

= {y : y < v,y ∈ Θ

} (23)

From (8),

Bel

(y < v) ≤ P(y < v) ≤ Pl

(y < v) (24)

Obviously, Bel

is the lower CDF of the LCC predic-

tion system response Y , and Pl

is the upper CDF.

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

• Algorithm of the ellipsoidal model construction:

Assuming that there are t experimental samples

(r)

, r = 1,2, ...,t for the N evidence parameters

and each sample is an N-dimensional vector, the

ellipsoidal model can be established as follows:

1. Take a pair of evidence parameters X

and X

(m 6= n) at a time from the uncertain parameter

set.

2. Extract the values of X

and X

from the t

experimental samples and construct a corre-

sponding bivariant sample set (X

(r)

), r =

1,2,...,t.

3. Create a minimum ellipse enveloping the ob-

tained bivariant samples and obtain the corre-

sponding rotation angle θ as shown in Fig. 5.

Figure 4: Ellipsoidal-model-based joint FD.

4. Compute the covariance (Cov) and correlation

coefﬁcient (ρ) of the two uncertain parameters

and X

based on the value of θ:

– Cov(X

) =

tan(θ)

1−tan

(θ)

((X

)

− (X

)

where X

and X

represent the radii of X

and X

, respectively.

– ρ

Cov(X

)

, −1 ≤ ρ

≤ 1.

5. Repeat the above process for all pairs of uncer-

tain parameters, and obtain a total of N(N −

1)/2 covariances and correlation coefﬁcients

for all the parameters.

6. Create a covariance matrix C based on the cal-

culated covariances.

7. Finally, an ellipsoidal model can be obtained:

Ω = {X|(X − X

)

−1

(X − X

) ≤ 1} (25)

b) Model Structure Imperfection Propagation

using Evidence Theory

The model structure imperfection propagation is

implemented in combination with the propagation

of the parametric imperfection. In this section, as

parametric imperfections are modeled by evidence

theory, we use this technique in this framework (Fer-

chichi et al., 2014). Therefore, the ﬁnal imperfection

representation of output variable Y can be obtained

by the following formulas.

Bel

∗

(Y ) = min(Bel

(Y ),Bel

(Y ),...,Bel

(Y )) (26)

∗

(Y ) = max(Pl

(Y ),Pl

(Y ),...,Pl

(Y )) (27)

The belief and plausibility functions

[Bel

∗

(Y ),Pl

∗

(Y )] take into account both param-

eter and structural model imperfection in the ﬁnal

output result.

4 EXPERIMENTAL RESULTS

The aim of this section is to validate and to evaluate

the performance of the proposed approach in propa-

gating imperfection related to correlated parameters

and structural model. The proposed approach was

used that subdivided the database into training (60%),

validation set (10%), and test set (30%). The train-

ing data set is used for building the LCC prediction

model. A validation set was used to stop the train-

ing procedure, and a test set was used to validate the

performance of the LCC prediction model.

4.1 Validation of the Proposed

Approach

4.1.1 Validation of the Imperfection

Propagation of LCC Prediction Model

In proposed changes prediction model, the process-

ing and propagating imperfections associated with 27

input parameters in the framework of evidence the-

ory are realized. To validate the proposed approach,

consider that the all parameters are independent. The

cumulative distribution function (CDF) of output rep-

resenting only the imperfection in parameters is ob-

tained via evidence theory. In fact, Figure 6 shows

this distribution based on 10,000 samples. Now, con-

sider that the parameters are correlated. The CDF of

output representing only the imperfection in param-

eters is obtained in ﬁgure 6. Here, the difference in

both distributions representation presents the effect

and the impact of parameters correlation.

Also, Figure 7 shows the belief (Bel*) and plau-

sibility (Pl*) functions representing the integrated pa-

rameter and structural model imperfections about the

LCCs.

Note that the combined effect of structural model

and parameter imperfection lead to a wider imperfec-

tion bound of the LCC when compared against the

parameter imperfection case.

Using Evidence Theory in Land Cover Change Prediction to Model Imperfection Propagation with Correlated Inputs Parameters

Figure 5: Distributions of model output (LCCs) of only pa-

rameter imperfection with (in)dependence parameters.

Figure 6: Belief and plausibility functions of the combined

parameter and structural model imperfections for LCCs.

4.1.2 Validation of LCC Prediction Maps

The validation of LCC prediction maps consisted of

two phases. First, the 2011 LCC was simulated using

the 2006 datasets, which was then compared with the

actual LCC in 2011 to evaluate the accuracy and the

performance of the proposed approach. Second, the

future changes was simulated using the actual 2011

datasets. Figure 8 compares the actual and the simu-

lated percentages occupied by the different land cover

types (water, urban, forest, bare soil, and vegetation)

between 2006 and 2011; it shows that the simulated

changes generally matched that of the actual changes.

These results conﬁrms that the LCC prediction model

were reasonable to describe the LCC and the pro-

posed approach can simulate the prediction of LCC

with an acceptable accuracy.

After the validation of the proposed approach, the

next step was to simulate the LCC in 2016, assum-

ing the changes between 2006 and 2011 will con-

tinue during the next time interval. In this simula-

tion, the LCC and input parameters acquired in 2011

were used as input to simulate the LCC in 2016. Fig-

ure 9 shows that the simulated changes between 2006

and 2016. There have been signiﬁcant LCC where ur-

Figure 7: Categorical distribution of the actual and simu-

lated LCC between 2006 and 2011.

ban land covered 26,5% of simulated changes in 2011

and 37,4% in 2016. This could be attributed to the

increase in population by increased demands for res-

idential land. The resulting effect was the decrease

in forest land from 27,8% of simulated changes in

2011 and to 20,1% in 2016. From these results, it

can be found the replacing the land natural cover (for-

est land) in the study area by residential land (urban

land).

Figure 8: Categorical distribution of the simulated LCC be-

tween 2011 and 2016.

Figure 11 maps the simulated future changes com-

pared with land cover maps for the years 2006 and

2011. These results indicate usefulness and applica-

bility of the proposed approach in predicting the LCC.

4.2 Evaluation of the Proposed

Approach

In order to evaluate the proposed approach in LCC

prediction, we apply the proposed method and the

methods presented by (Boulila et al., 2014) [1] and

(Gong et al., 2013) [5]. Then, we compare the pro-

posed prediction changes to these two methods.

Table 1 depicts the error calculated between real

LCCs, proposed method and methods presented in [1]

and [5] between dates 2007 and 2011. As we note,

the proposed approach provides a better results than

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

Figure 9: Comparison between the land cover maps for

years 2006 and 2011 and the predicted land cover map for

2016.

Table 1: Error for the prediction of LCCs between dates

2007 and 2011.

Approach predicting LCCs (%)

Approach in [1] 0.18

Approach in [5] 0.23

Proposed approach 0.09

the methods presented in [1] and [5] in predicting

LCCs. This shows the effectiveness of the proposed

approach in reducing imperfection related to the pre-

diction process. Indeed, ﬁgure 10 shows the differ-

ence between real changes and changes prediction for

the three approaches: proposed approach, approach in

[1] and approach in [5]. For example, as we note, the

proposed approach predicts a change of 72.15% for

forest object, while the real image shows a change of

72.21%.

Figure 10: Comparaison between real changes and changes

prediction for the three approaches: proposed approach, ap-

proach proposed in [1] and approach proposed in [5] .

This provides a difference in the order of 0.06%.

This result conﬁrms the effectiveness of the proposed

approach in improving LCC prediction. This is made

by reducing the effect of imperfection related to pa-

rameters and model structure and the effect of their

propagation on LCC prediction model.

5 CONCLUSION

This article has expanded the evidence theory frame-

work for propagating correlated parameters and struc-

tural model imperfections through a LCC prediction

model. We used the ellipsoidal model to analyze cor-

related parameters. The proposed approach has also

modeled both imperfections types (random and epis-

temic) that are associated with input parameters and

model itself.

The results show the importance to propagate both

aleatory and epistemic imperfection and to consider

correlation among input parameters in LCC predic-

tion model. Proposed approach studied the changes

prediction of land cover in Saint-Denis City, Reunion

Island of next 5 years (2016) using multi-temporal

Spot-4 satellite images in 2006 and 2011. Results

indicated that the urban land covered 26,5% of sim-

ulated changes in 2011 and 37,4% in 2016 and the

forest land covered 27,8% of simulated changes in

2011 and 20,1% in 2016. From these results, it can

be found the replacing the land natural cover (for-

est land) in the study area by residential land (urban

land).

Proposed approach was compared on error predic-

tion to existing propagation methods. Results show

good performance of the proposed approach in im-

proving prediction of the LCC. More work, however,

is needed to understand how to reduce the computa-

tional cost in our proposed approach. LCC prediction

model with a large number of uncertain input param-

eters is more complex. To optimize, it is important to

study the sensitivity of input parameters and also the

sensitivity of model structure.

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