Towards an uncertainty reduction framework for landcover change prediction using possibility theory
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Abstract
This paper presents an approach for reducing uncertainty related to the process of landcover change (LCC) prediction. LCC prediction models have, almost, two sources of uncertainty which are the uncertainty related to model parameters and the uncertainty related to model structure. These uncertainties have a big impact on decisions of the prediction model. To deal with these problems, the proposed approach is divided into three main steps: (1) an uncertainty propagation step based on possibility theory is used as a tool to evaluate the performance of the model; (2) a sensitivity analysis step based on Hartleylike measure is then used to find the most important sources of uncertainty; and (3) a knowledge base based on machine learning algorithm is built to identify the reduction factors of all uncertainty sources of parameters and to reshape their values to reduce in a significant way the uncertainty about future changes of land cover. In this study, the present and future growths of two case studies were anticipated using multitemporal Spot4 and Landsat satellite images. These data are used for the preparation of prediction map of year 2025. The results show that our approach based on possibility theory has a potential for reducing uncertainty in LCC prediction modeling.
Keywords
LCC prediction Parameter uncertainty Structural uncertainty Possibility theory Sensitivity analysis1 Introduction
LCC is a central issue in the sustainability debate because of its wide range of environmental impacts. Models of LCC start with an initial landcover situation for a given case study area. Then, they use an inferred transition function, representing the processes of change, to simulate the expansion and contraction of a predefined set of landcover types over a given period. LCC models help to improve our understanding of the land system by establishing causeeffect relations and testing them on historic data. They help to identify the drivers of LCC and their relative importance. In addition, LCC models can be used to explore future landcover pathways for different scenarios. However, the performance of the LCC prediction models is affected by different types of uncertainties (i.e., aleatory or/and epistemic uncertainties). These uncertainties can be subdivided into two sources: parameter uncertainty (adequate values of model parameters) [1, 2] and structural uncertainty (ability of the model to describe the catchment’s response) [3]. These sources contribute with different levels to the uncertainty associated with the predictive model. It is important to quantify the uncertainty due to uncertain model parameter, but methods for quantifying uncertainty due to uncertainty in model structure are less well developed. For quantifying, probability theory is generally used. Moreover, numerous authors conclude that there are limitations in using probability theory in this context. So far, several alternative frameworks based on nonprobabilistic theories have been proposed in the literature. By no means do the promoters of theories pretend to replace probability theory; they just present different levels of expressiveness that leave room for properly representing the lack of background knowledge [4]. The most common theories that are used from these alternatives are imprecise probabilities [5], random sets [6], belief function theory [7], fuzzy sets [8], and possibility theory [9]. In our context of continuous measurements, the possibility theory is more adapted, because it generalises interval analysis and provides a bridge with probability theory by its ability to represent a family of probability distributions. In summary, the possibility distribution has the ability to handle both aleatory and epistemic uncertainty of pixel detection through a possibility and a necessity measures. In this framework, the possibility distributions of the model outputs are used to derive the prediction uncertainty bounds.
Understanding the impact of parameter and structural uncertainty on LCC prediction models outcomes is crucial to the successful use of these models. On the other hand, model optimization with multiple uncertainty sources is complex and very timeconsuming task. However, the sensitivity analysis has been proved to be efficient and robust to find the most important sources of uncertainty that have effect on LCC prediction models output [1, 9, 10]. Parameter sensitivity analysis allows to examine effects of model parameter on results, whereas structural sensitivity analysis allows to modify the structure of the model and to identify the possible structural factors that affect the robustness of the results (vary structure of model and see impact on results and tradeoffs between choices). Several sensitivity analysis methods exist, including screening method [11], differential analysis [12], variancebased methods [13], samplingbased methods [14], and a relative entropybased method [15]. However, all these require specific probability distribution in modeling both model parameters and model structure. In the literature, previous nonprobabilistic methods of sensitivity analysis are developed [16, 17]. Several studies have confirmed the robustness of use of Hartleylike measure to apply sensitivity analysis in fuzzy theory framework in numerous fields [33, 34, 35]. Minimum value to Hartleylike measure of the model output is considered to be the most sensitive source.
Based on possibilistic approach, this study proposes an approach for reducing parameter and structural uncertainty in LCC prediction modeling. The proposed approach is divided into three main steps: (1) an uncertainty propagation step based on possibility theory is used as a tool to evaluate the performance of the model; (2) a sensitivity analysis step based on Hartleylike measure is used to find the most important sources of uncertainty; and (3) a knowledge base based on machine learning algorithm is built to identify the reduction factors of all uncertainty sources of parameters. Then, values of these parameters are reshaped to improve decisions about future changes of land cover in SaintDenis city, Reunion Island and Cairo region, Egypt.
The rest of this paper is organized as follows: Sect. 2 presents a description of the proposed approach for reducing uncertainty throughout the model of LCC prediction. Results are given and described in Sect. 3. Finally, conclusion and future works are outlined in Sect. 4.
2 Proposed approach
In this paper, the proposed approach for reducing parameter and structural uncertainty is applied to model presented in [18] and it has the following steps (Fig. 1): (1) identifying uncertainty related to parameters and model structure; (2) propagating the uncertainty through the LCC prediction model using the possibility theory; (3) performing a sensitivity analysis using the Hartleylike measure; and (4) constructing knowledge base using machine learning algorithm to improve parameters’ quality.
2.1 Step 1: identifying parameters and structure of LCC prediction model
2.1.1 Choice of parameters
Input parameters of LCC prediction model describe the objects’ features extracted from satellite images which are the subject of studying changes. In this study, we consider 26 features: ten spectral, five texture, seven shape, one vegetation, and three climate features. Spectral features are: mean values and standard deviation values of green (MG, SDG), red (MR, SDR), NIR (MN, SDN), SWIR (MS, SDS), and monospectral (MM, SDM) bands for each image object. Texture features are: homogeneity (Hom), contrast (Ctr), entropy (Ent), standard deviation (SD), and correlation (Cor) generated from graylevel cooccurrence matrix (GLCM). Shape and spatial relationship features are: area (A), length/width (LW), shape index (SI), roundness (R), density (D), metric relations (MR), and direction relations (DR). Vegetation feature is: Normalized Difference Vegetation Index (NDVI) that is the ratio of the difference between NIR and red reflectance. Finally, climate features are: temperature (Tem), humidity (Hum), and pressure (Pre). These features are selected based on previous results, as reported in [18], and are considered as input parameters to the LCC model.

Uncertainty sources of spectral parameters Several studies investigated effects of spectral parameters [28]. Among these effects, we list: spectral reflectance of the surface (S1), sensor calibration (S2), effect of mixed pixels (S3), effect of a shift in the channel location (S4), pixel registration between several spectral channels (S5), atmospheric temperature and moisture profile (S6), effect of haze particles (S7), instrument’s operation conditions (S8), atmospheric conditions (S9), as well as by the stability of the instrument itself characteristics (S10).

Uncertainty sources of texture parameters Among these sources, we list: the spatial interaction between the size of the object in the scene and the spatial resolution of the sensor (S11), a border effect (S12), and ambiguity in the object/background distinction (S13).

Uncertainty sources of shape parameters Uncertainty related to shape parameters can rely to the following factors [28]: accounting for the seasonal position of the sun with respect to the Earth (S14), conditions in which the image was acquired changes in the scene’s illumination (S15), atmospheric conditions (S16), and observation geometry (S17).

Uncertainty sources of NDVI Among factors that affect NDVI, we can list: variation in the brightness of soil background (S18), red and NIR bands (S19), atmospheric perturbations (S20), and variability in the subpixel structure (S21).

Uncertainty sources of climate parameters According to [29], uncertainty sources related to climate parameters can be: atmospheric correction (S22), noise of the sensor (S23), land surface emissivity (S24), aerosols and other gaseous absorbers (S25), angular effects (S26), wavelength uncertainty (S27), fullwidth halfmaximum of the sensor (S28), and bandpass effects (S29).
2.1.2 Description of model structure

Similarity measure step: Distance between states (\(d(S_{t},S_{t_1})\ge 0.9\) indicates a higher similarity between the query and the retrieved states). In addition, similarity measure between states is based on time assumption.

Spatiotemporal change tree building step: The aim of this step is to determine the confidence degrees and the percentage of changes of the model between two dates and for different landcover types. The confidence degree of changes is achieved by a fuzzy decision tree (fuzzy ID3). This method is based on a number of assumptions such as: the proportion of a data set of landcover type, the size of a data set, etc. The percentage of changes is achieved by computing the distances between two states and the centroid of the classes.
2.2 Step 2: propagating the uncertainty
In this step, we focus on how to propagate parameter and structural uncertainty through the LCC prediction model described in [18] via the possibility theory.
2.2.1 Basics of possibility theory
2.2.2 Propagation of parameter uncertainty
 1.
Set \(\alpha = 0\).
 2.
Select the \(\alpha \) cuts \(A^{X1}_\alpha , A^{X2}_\alpha , , A^{Xj}_\alpha , \ldots , A^{Xn}_\alpha \) of the possibility distributions \(\pi _{X_1}(x_1), \pi _{X_2}(x_2), \ldots , \pi _{X_j}(x_j), \ldots , \pi _{X_n}(x_n)\) of the possibilistic parameters \(X_j\), \(j = 1, 2, \ldots , n,\) as intervals of possible values \(\lfloor \underline{x}_{j,\alpha }, \overline{x}_{j,\alpha } \rfloor \) \(j = 1, 2, \ldots , n\).
 3.
Calculate the smallest and largest values of Y, denoted by \(\underline{y}_\alpha \) and \(\overline{y}_\alpha \), respectively, letting variables \(X_j\) range within the intervals \(\lfloor \underline{x}_{j,\alpha }, \overline{x}_{j,\alpha } \rfloor \) \(j = 1, 2, \ldots , n;\) in particular, \(\underline{y}_\alpha =\mathrm{inf}_{j,X_j\in [\underline{x}_{j,\alpha }, \overline{x}_{l,\alpha }]} f(X_1, X_2, \ldots , X_j, \ldots , X_n)\) and \(\overline{y}_\alpha = \mathrm{sup}_{j,X_j\in [\underline{x}_{j,\alpha }, \overline{x}_{l,\alpha }]} f (X_1, X_2, \ldots , X_j, \ldots , X_n)\).
 4.
Take the values \(\underline{y}_\alpha \) and \(\overline{y}_\alpha \) found in step 3 as the lower and upper limits of the \(\alpha \) cut \(A^{Y}_{\alpha }\) of Y;
 5.
If \(\alpha < 1\), then set \(\alpha = \alpha + \triangle \alpha \) and return to step 2; otherwise, stop the algorithm. The possibility distribution \(\pi _Y(y)\) of \(Y = f (X_1, X_2, \ldots , X_n)\) is constructed as the collection of the values \(\underline{y}_\alpha \) and \(\overline{y}_\alpha \) for each \(\alpha \) cut.
2.2.3 Propagation of structural uncertainty
The propagation of structural uncertainty is implemented in combination with the propagation of parameter uncertainty. In this section, as parameter uncertainty is modeled by possibility theory, we use this method in this framework.
2.3 Step 3: performing the sensitivity analysis
2.4 Step 4: constructing the knowledge base
3 Experimental results
The aim of this section is to validate and to evaluate the performance of the proposed approach through two case studies for reducing parameter and structural uncertainty in LCC prediction modeling.
3.1 Case study 1
3.1.1 Description of the study area and data
Reunion Island is a French territory of 2500 \(\mathrm{km}^{2}\) located in the Indian Ocean, 200 km SouthWest of Mauritius and 700 km to the East of Madagascar (Fig. 3). Mean annual temperatures decrease from 24 \(^{\circ } \mathrm{C}\) in the lowlands to 12 \(^{\circ } \mathrm{C}\) 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 1 m along the south western coast. Vegetation is most clearly structured along gradients of altitude and rainfall [27].
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 [24]. It has been significant changes, putting pressure on agricultural and natural areas. The urban areas expanded by 189 % over the period from 1989 to 2002 [25] and available land became a rare and coveted resource. The landscapes are now expected to fulfil multiple functions, i.e., urbanization, agriculture production, and ecosystem conservation, and this causes conflicts among stakeholders about their planning and management [26].
3.1.2 Results of uncertainty propagation
As mentioned perviously, the model parameter and model structure of LCC prediction are marred by uncertainty. Ignoring each of these sources can affect the results of uncertainty propagation. To illustrate the importance of propagating uncertainty related to model parameter and model structure through the LCC prediction model, the analysis with pure parameter uncertainty assumption is conducted. In this case, the possibility distribution of output representing only parameter uncertainty is obtained via possibility theory. Figure 5 shows this distribution based on 10,000 samples. With uncertainty in model parameter, there is uncertainty in model structure. Therefore, it is also import to illustrate the importance of structural uncertainty in LCC prediction modeling by the proposed approach. This is the reason behind using the LCC prediction model described in [18] with three different structures. Then, we obtain three different models \((M_1,\) \(M_2,\), and \(M_3)\) with different assumptions. To take into account structural uncertainty in the final result, uncertainty related to parameters is first propagated and this for each prediction model.
Figure 5 shows the possibility distribution of the LCC prediction model output, where only parameter uncertainty is propagated.
Percentages of LCC of the actual and simulated LCC
Water (%)  Urban (%)  Forest (%)  Bare soil (%)  Vegetation (%)  

Predicted changes in 2025  1.9  37.4  39.31  26.95  26.7 
Output of proposed model  1.5  23.18  35.97  22.87  20.08 
Real changes in 2011  1.7  21.4  36.1  24.1  16.7 
Figure 7 shows possibility distribution representing integrated parameter and structural uncertainty through the LCC prediction modeling. Note that combining parameter and structural uncertainty can be crucially important to enhance the accuracy of the LCC prediction model.
3.1.3 Results of sensitivity analysis
In this paper, the sensitivity analysis based on Hartleylike measure is implemented to estimate the effect of 26 uncertain parameters through three different LCC prediction model structures. Results of the sensitivity analysis are shown in Fig. 8.
3.1.4 Results of LCC prediction maps
LCC prediction maps are validated based on temporal series of multispectral SPOT images. First, the 2011 LCC was simulated using the 2006 data sets. Then, the simulated changes are compared with the real LCC in 2011 to evaluate the accuracy and the performance of the proposed approach. Second, the process of LCC is conducted to predict landcover distributions for forthcoming dates.
Table 1 illustrates a comparison between the actual and simulated percentages occupied by the different landcover types (water, urban, forest, bare soil, and vegetation) between 2006 and 2011. It shows that the modeled changes generally matched that of the actual changes. These results confirm that the proposed approach can simulate the prediction of LCC with an acceptable accuracy.
After the validation, the next step is to simulate the LCC in 2025, assuming the changes between 2006 and 2011. In this simulation, the LCC and the parameters acquired in 2011 are used as input to simulate the LCC in 2025.
Table 1 shows the simulated changes between 2006 and 2025. Urban expansion is the dominant change process. This can be attributed to the increase in population by increased demands for residential land. There have been significant LCC, where urban land covered 21.4 % of simulated changes in 2011 and 37.4 % in 2025. From these results, it can be found the replacing of the land natural cover (forest and vegetation lands) in the study area by residential land (urban land).
Figure 9 depicts the simulated future changes compared with landcover maps for the years 2006 and 2011.
3.1.5 Evaluation of the proposed approach
To evaluate the proposed approach in improving LCC prediction, we apply the proposed uncertainty propagation approach on the LCC model described by Qiang and Lam in [40] to the SaintDenis city, Reunion Island. The LCC prediction model proposed in [40] uses the Artificial Neural Network (ANN) to derive the LCC rules and then applies the Cellular Automate (CA) model to simulate future scenarios.
Comparison between real changes, predicted changes of the proposed approach, and changes made by the proposed approach applied to model described in [40]
Water (%)  Urban (%)  Forest (%)  Bare soil (%)  Vegetation (%)  

Proposed approach  1.5  23.18  35.97  22.87  20.08 
Approach applied to model in [40]  1.5  25.32  34.98  20.03  16.24 
Real changes in 2011  1.7  21.4  36.1  24.1  16.7 
3.2 Case study 2
3.2.1 Description of the study area and data
Cairo, the capital of Egypt, is one of the most crowded cities in Egypt (Fig. 10) and is considered as a world megacity. Mapping LCC is important to understand and analyze the relationships between the geomorphology (highlands and deserts), natural resources (agricultural lands and the Nile River), and human activities. Agricultural lands around Cairo have witnessed severe encroachment practices due to the accelerated population growth. However, adjacent desert plains have also witnessed urbanization practices to encompass the intensive population growth. Different studies have previously been carried out for LCC detection and modeling in the Cairo Region [36, 37, 38, 39]. Population of Cairo (Cairo city and Giza) increased from about 6.4 millions in 1976 [36] to about 12.5 million in 2006 according to the Egyptian Central Agency for Public Mobilization and Statistics. The importance of Cairo arises from its location in the midway between the Nile Valley and the delta. Main government facilities and services occur at Cairo.
In this case, two Landsat TM5 satellite images are obtained from the United States Geological Survey (USGS) database online resources. These two images acquired in 6 April 1987 and 15 March 2014, respectively, are classified into four landcover types which are urban, agriculture, desert, and water to produce LCC maps (Fig. 11). During this time period, Cairo population has increased from an estimated 7 million in 1987 to over 15 million in 2014. The recent population growth has caused the city and its associated urban areas to expand into the surrounding desert, as seen in the right image in Fig. 11. Within the main Nile River Valley, these two images also show an overall increase in developed urban area (red) versus agricultural land (green). As new urban and agricultural areas are being developed in the desert, they require diversion of water supplies from the main Nile River Valley.
3.2.2 Results of uncertainty propagation
Output of the proposed LCC prediction model in comparison with real changes between 1987 and 2014 in Cairo region
Urban (%)  Agriculture (%)  Water (%)  Desert (%)  

Output of proposed model  15.63  13.80  0.01  4.03 
Real changes in 2014  17.32  13.00  0.02  5.00 
Output of the proposed LCC prediction model of the predicted LCC between 2014 and 2025 in Cairo region
Urban (%)  Agriculture (%)  Water (%)  Desert (%)  

Predicted changes in 2025  20.16  14.72  0.03  6.11 
Real changes in 2014  17.32  13.00  0.02  5.00 
3.2.3 Results of sensitivity analysis
In this case study, we have also used Hartleylike measure to estimate the 26 uncertain parameters through three different LCC prediction model structures. The main objective is to test the impact of parameter and structural uncertainties. Results of the sensitivity analysis are shown in Fig. 15. In this case, parameters in \(M_3\) are highly sensitive compared to \(M_1\) and \(M_2\). On the other hand, as in the first case study, the overall contribution of spectral, shape, and NDVI parameters to the LCC prediction model is the highest and represents the most sensitive parameters for the three model structures.
3.2.4 Results of LCC prediction maps
The validation of LCC prediction maps consists of two phases. First, the 2014 LCC is simulated using the 1987 data sets, which is then compared with the real LCC in 2014 to evaluate the accuracy and the performance of the proposed approach. Second, future changes are simulated using the real 2014 data sets.
To check the accuracy of our approach, Table 3 compares actual and simulated percentages occupied by the different landcover types (urban, agriculture, water, and desert) between 1987 and 2014. According to the proposed model output, the most significant changes in this period are the transitions from agriculture and desert to urban areas (Fig. 11). Over 27 years, from 1987 to 2014, agriculture lost 12 % to urban areas. In addition, 4 % of desert areas became urban between 1987 and 2014, which is equivalent to 24,687 hectares. This percentage results from the application of desert reconstruction strategies to build new communities outside the Nile Valley. The obtained results depict that the proposed approach gives an accurate prediction with about 3.96 % of error through a comparison with the real changes in Cairo region. These results confirm that the proposed LCC prediction model is able to describe the LCC. The proposed approach can simulate the prediction of LCC with an accepTable accuracy. After the validation of the proposed approach, the next step is to simulate the LCC in 2025, assuming that the changes between 1987 and 2014 will continue during the next 11 years. In this simulation, the LCC and the input parameters acquired in 2014 are used as input to simulate the LCC in 2025.
Comparaison between real changes and changes prediction for the proposed approach and the proposed approach applied to model described in [40]
Urban (%)  Agriculture (%)  Water (%)  Desert (%)  

Proposed approach  15.63  13.80  0.01  4.03 
Approach applied to model in [40]  14.93  13.61  0.01  5.92 
Real changes in 2014  17.32  13.00  0.02  5.00 
Figure 16 depicts the simulated future changes compared with landcover maps for the years 1987 and 2014. These results indicate usefulness and applicability of the proposed approach in predicting the LCC.
3.2.5 Evaluation of the proposed approach
In this case study, we also apply the proposed uncertainty propagation approach on the LCC model described by Qiang and Lam in [40] to the Cairo region, Egypt.
Table 5 depicts the percentages of change of the four landcover types (urban, agriculture, water, and desert). This table shows the difference between real changes and changes prediction for the proposed approach and the proposed approach applied to model described in [40].
4 Conclusion
This study has proposed an approach for reducing parameter and structural uncertainty in LCC prediction modeling. The proposed approach herein quantifies uncertainty based on possibility theory. Subsequently, the Hartleylike measure is used to perform the sensitivity of the LCC prediction model parameter and structure. Using the sensitivity analysis, we are able to quantify precisely the effect of each LCC prediction model parameter, and also the effect of model structure. This analysis yields that the spectral, shape, and vegetation parameters are the most sensitive parameters in three different model structures.
To validate the proposed approach, we choose two case studies which are: SaintDenis city, Reunion Island, and Cairo region, Egypt. We study spectral parameters, texture parameters, shape parameters, vegetation parameter, and climate parameters for three different model structures to simulate forthcoming LCC. Results show that the urban expansion in the two case studies is rapid and should be monitored in the future.
As future work, we propose to put online a tool for uncertainty propagation and sensitivity analysis based on possibility theory. This tool will help researchers to improve the performance of their models. It has also as input parameters of a considered model and as output which of these input parameters that most influence the model output.
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