Training sites are the areas defined for each land cover type within the image. The chosen color composite is used for digitizing polygons around each training site for similar land cover. Then a unique identifier is assigned to each known land cover type [12]. At the beginning, the training sites have been developed based on the collected reference data and ancillary information.

This is the stage of creating the spectral signature for each type of land cover. This is done by analyzing the pixels of the training sites. When the digitization of training sites is finished, the statistical characterizations of each land cover class are needed. These are called signatures [12].

After developing signature files for all the land cover types the images have been classified using a hard classifier called “Fisher Classifier”. Fisher classifier uses the concept of the linear discrimination analysis [12]. Fisher Classifier performs well when there are very few areas of unknown classes and when the training sites are representative of their informational classes [12]. This is why fisher classifier is appropriate for this particular research, because most areas for the classes are known.

After image classification, sometimes many isolated pixels may be found [12]. These isolated pixels may belong to one or more classes that differ from surrounding pixels. Therefore it is necessary to generalize the image and remove the isolated pixels. Filtering is the solution for this type of problem. Therefore a 3 × 3 mode filter has been applied to generalize the fisher classified land cover images. This post-processing operation replaces the isolated pixels to the most common neighboring class. Finally the generalized images are reclassified to produce the final version of land cover maps for different years (Figure 2). It is important to remember that the supervised classification in every year is independent of the classification of the other years, because this method can lead to substantial overestimates of land change in the post-classification change analysis.

User’s accuracy for category K is the percent of category K in the reference information, given that the map shows category K. Producer’s accuracy for category K is the percent of category K in the map, given that the reference information shows category K [14]. The producer’s and user’s accuracy for all the years are found ranging approximately from 71% to 100% (Table 3,Table 4andTable 5).

The overall accuracy represents the percentage of correctly classified pixels [14]. It is achieved by dividing the number of correct observations by the number of actual observations. The overall accuracies for 1989, 1999 and 2009 are found 85.20%, 86.80% and 91.60% respectively.

There is 14.8%, 13.2% and 8.4% reported map error in the maps of 1989, 1999 and 2009 respectively. These errors in the maps are clearly not negligible. It is a very common problem in land change science that the amount of error in the maps is nearly as large as the amount of change on the ground. Thus it is necessary to compare the error in the maps with how much difference there is during 1989–1999 and 1999–2009.

The amount of land change ranges from approximately 2% to 20% for 1989–1999, while the figure is approximately 1% to 17% for 1999–2009 (Table 6 and Table 7). Therefore, on an average, it can be stated that the errors in the maps are not much larger than the amount of land change between the two points in time (1989–1999 and 1999–2009).

Finally it can be stated that few misclassifications have been observed in the classified land cover maps of Dhaka city. The reasons may be as follows:

The overall accuracies of the base maps can be achieved better if the above mentioned limitations can be reduced.

NP equals the number of patches of the corresponding patch type (class). Number of patches of a particular class or land cover type is a simple measure of the extent of subdivision or fragmentation of the class [16]. The numbers of patches (urban blocks in this case) of built-up area and fallow land types have increased over time. Again the decrease of water body and vegetation is prominent (Figure 3). This reveals the gradual development of urban infrastructures converting water bodies, vegetation and low lands.

In case of built-up area, the core southern part of Dhaka city has remained the same. While the north-east and south-west parts have converted to built-up areas. The northern part of Dhaka city has gained water body followed by a massive decrease in the south-east and south-west parts (Figure 4).

No particular pattern on gains or losses is found for vegetation. In cases of low land the changes are evident in eastern and western parts. Fallow land has decreased markedly and the losses are clear in north-western and mid parts (Figure 4).

In a Markov chain the probability of the next state is only dependent upon the current state. This is called Markov property and stated as [19]:

(1)

The probability of a Markov chain ξ₁, ξ₂,........can be calculated as [19]:

(2)

The conditional probabilities:

(3)

These are called the “Transition Probabilities” of the Markov chain [19].

Let’s consider a Markov chain with n states s₁, s₂,.......,s_n. Let p_ij denote the transition probability from state s_i to state s_j, i.e.,

(4)

The transition matrix (n × n) of this Markov process is then defined as [19]:

(5)

Predictions of the future state probabilities can be calculated by solving the matrix equation [19]:

p(t) = p(t − 1)∙P (6)

With increasing time steps, a Markov chain may approach to a constant state probability vector, which is called limiting distribution [19]:

p(∞) = = ∙P^t (7)

At the beginning, Markov chain produces a transition matrix (Table 8), a transition areas matrix (Table 9) and a set of conditional probability images by analyzing two qualitative land cover images (Figure 5) from two different dates 1989 and 1999 [12].

The matrix of transition probabilities (Table 8) shows the probability that each land cover category will change to other categories in 2009. Table 9 represents the number of cells/pixels (30 m × 30 m) that will be transformed over time from one land cover type to other types. Markov Chain Analysis also produces related conditional probability images (Figure 5) with the help of transition probability matrices. Each conditional probability image shows the possibility of transitioning to another land cover class.

After analyzing Figure 5, it is clear that most areas will convert into built-up areas. The Markovian conditional probability of being built-up area ranges up to 0.66 which is the highest among all other land cover types. This probabilistic prediction is dependent upon the past trend of the last ten years (1989–1999).

The next step is to make one single land cover map for future prediction aggregating all the Markovian conditional probability images. This prediction is performed by a stochastic choice decision model. Stochastic choice creates a stochastic land cover map by evaluating and aggregating the conditional probabilities in which each land cover can exist at each pixel location against a rectilinear random distribution of probabilities [12].

Stochastic Choice creates a stochastic land cover map by evaluating and aggregating the conditional probabilities in which each land cover can exist at each pixel location against a rectilinear random distribution of probabilities [12]. The Stochastic Markov (St_Markov) predicted land cover map of 2009 is shown in Figure 6.

The components that comprise an elementary cellular automaton are as follows [23]:

A CA model represents discrete dynamic system consisting of four elements [20]:

(8)

where, L = the discrete lattice or the physical environment; ∑ = the set of possible states, where each i^th cell of the lattice at time step t has a state σ_i(t) ∑; = the neighborhood of a cell automaton, which is defined as all cells that fall within a radius r around the actual cell. It denotes the idea of neighborhood template [24]:

where ; = the relative index of all neighbors of a particular cell; δ = the local transition rule, which is denoted as follows [20]:

(9)

It shows that the state of the i^th cell at the next time step t+1 is computed by δ based on the states of all the cells in its neighborhood at the current time step t.

Here, _i(t) = the associated neighborhood with i^th cell at time t; || = the number of cells in the neighborhood.

The local transition rule is given by a rule table where given the sizes of ∑ and , the total number of possible rules equals [20]:

(10)

where each of the |∑^N| possible configurations of a cell’s neighborhood is mapped to the number of possible states a cell can be in. Now considering the ordered set of all the states of all cells collectively at time step t, a CA’s global configuration can be denoted as follows [20]:

(11)

Now applying the local transition rule to all the cells in the CA’s lattice, the next configuration of the CA can be computed by its induced global map [20]:

(12)

In brief, the standard CA can be generalized as follows [24]:

(13)

where, S = the set of all possible states of the cellular automata; N = a neighborhood of all cells providing input values for the function f and f = a transition function that defines the change of the state from t to t+1.

CA_Markov is useful for modeling the state of several categories of a cell based on a matrix of Markov transition areas; transitional suitability images and a user defined contiguity filter. A Markov model applies contiguity rule like a pixel near to an urban area is most likely to be changed into urban area [12]. In this research, a 3 × 3 mean contiguity filter has been used (Figure 7) for modeling purpose. This kind of filtering is applied on suitability images for each land cover class. This is a defined neighborhood. The suitability of a pixel is determined by the pixel values within this defined filtering kernel. The more pixels of the same category of land cover exist in the neighborhood, the more the suitability value for that particular land cover type increases. Otherwise the pixel value remains the same [12].

The suitability maps determine which pixels will change as per the highest suitability of each land cover type. The higher the suitability of a pixel, the possibility of the neighboring pixels to change into that particular class is higher.

Preparing a suitability map for each land cover type is difficult in terms of data and information availability. It is not possible to incorporate all types of factors or constraints that exist within the study area. Therefore a simple assumption has been assumed for fuzzy factor standardization.

The basic assumption for preparing suitability images is the pixel closer to an existing land cover type has the higher suitability. It means a pixel that is completely within vegetation has the highest suitability value (255) and pixels far from existing vegetation pixels will have less suitability values. The farthest pixels from vegetation will show the lowest suitability values. Here the suitability decreases with distance. Though this idea is not perfect always, like in case of water body and vegetation the scenario can be different. In reality, a pixel near to forest is more likely to convert into built-up area rather in forest. But this can be considered to be the research limitation.

Therefore a simple linear distance decay function is appropriate for this basic assumption. It serves the basic idea of contiguity. The land cover maps have been standardized (Figure 8) to the same continuous suitability scale (0–255) using fuzzy set membership analysis process [12].

At the end, the Markov transition area matrix (Table 9), all the suitability images (Figure 8), the 3 × 3 CA contiguity filter (Figure 7) and the base map of Dhaka city (1999) have been used to predict the land cover map of 2009. The CA_Markov predicted final land cover image (2009) of Dhaka city is illustrated in Figure 9.

MLP neural network uses the back propagation (BP) algorithm. The calculation is based on information from training sites [12]. Back propagation involves two major steps, forward and backward propagation. The input that a single node receives is weighted as:

(14)

where, w_ij = the weights between node i and node j; O_i = the output from the node i

The output from a given node j is computed as [26]:

(15)

f = a non-linear sigmoid function that is applied to the weighted sum of inputs before the signal passes to the next layer

This is known as “Forward Propagation”. Once it is finished, the activities of the output nodes are compared with their expected activities. In normal circumstances, the network output differs from the desired output (a set of training data, e.g., known classes). The difference is termed as the error in the network [26]. The error is then back-propagated through the network. Now the weights of the connections are corrected as follows [12]:

(16)

η = the learning rate parameter; δ_j = an index of the rate of change of the error; α = the momentum parameter.

The process of the forward and backward propagation is repeated iteratively, until the errors of the network minimized or reaches an acceptable magnitude [26]. The purpose of training the network is to get proper weights both for the connection between the input and hidden layer, and between the hidden and the output layer for the classification of unknown pixels [12]. Several factors affect the capabilities of the neural network to generalize [26]. These include:

In general, the larger the number of nodes in the hidden layer, the better the neural network represents the training data [26]. The number of hidden layer nodes is estimated by the following equation [12]:

(17)

where, N_h = the number of hidden nodes; N_i = the number of input nodes; N_o = the number of output nodes

The number of training sample also affects the training accuracy. Too few samples may not represent the pattern of each category while too many samples may cause overlap. Again too many iterations can cause over training that may cause poor generalization of the network [12]. Over training can be prevented by early stopping of training [25]. The acceptable error rate is evaluated based on the Root Mean Square (RMS) Error [25]:

(18)

where, N = the number of elements; i = the index for elements; e_i = the error of the ith element; t_i = the target value (measured) for i^th element; a_i = the calculated value for the i^th element.

The basic concept of modeling with MLP neural network adopted in this research is to consider the change in built-up area over the years. In general, it means other land cover types are primarily contributing to increase the built-up area. At this stage, the issue of which variables affect the change to built-up area (1989–1999) has been considered. Therefore, only the transitions from “water body to built-up area”, “vegetation to built-up area”, “low land to built-up area” and “fallow land to built-up area” have been considered for model simulation. These four transitions have been termed as “All” here. Figure 10 exhibits the transition from all to built-up area.

It is logical that new areas will be converted to built-up area where there are existing built-up areas. Therefore six driver variables have been selected for MLP_Markov modeling. These are (1989–1999): Distance from all to built-up area, distance from water body, distance from vegetation, distance from low land, distance from fallow land and empirical likelihood image).

The empirical likelihood transformation is an effective means of incorporating categorical variables into the analysis (Figure 11). It has been produced by determining the relative frequency of different land cover types occurred within areas of transition (1989 to 1999). The numbers (legend) indicate the likelihood of changing into built-up area. The higher the value the likeliness of the pixel to change into the built-up cover type is more.

Now it is important to test the potential explanatory power of each variable. The quantitative measures of the variables have been tested through Cramer’s V [27]. It is suggested that the variables that have a Cramer’s V of about 0.15 or higher are useful while those with values of 0.4 or higher are good [12].

After getting satisfactory Cramer’s V values for all the driving variables, now the turn is to run MLP neural network model. For this purpose, 10,000 iterations have been chosen. The minimum number of cells that transitioned from 1989 to 1999 is 4,794. Therefore, the maximum sample size has been chosen as 4,794. For each principal transition particular weights have to be obtained. The RMS error curve has been found smooth and descent after running MLP neural network. After all these combinations, the MLP running statistics gives a very high accuracy rate of 91.36% (this accuracy is a measure of calibration, not validation). Based on these running statistics the transition potential maps have been produced (Figure 12). These maps depict, for each location, the potential it has for each of the modeled transitions [12]. These are not the probability maps where the sum of values for a particular pixel location will not be 1. The reason behind this is because the MLP neural network outputs are obtained by applying fuzzy set to the signals into values from 0 to 1 with activation function (sigmoid). Here the higher values represent a higher degree of membership for that corresponding land cover type [12].

Using this kind of MLP neural network analysis it is possible to determine the weights of the transitions that will be included in the matrix of probabilities of Markov Chain for future prediction. The transition probabilities are shown in Table 10. Based on all these information from MLP neural network, the final land cover map of 2009 (Figure 13) has been simulated through Markov chain analysis. The whole procedure, for predicting the land cover map by this way, has been termed as “MLP_Markov” model.

The three maps comparison method consists of two components of agreement and three components of disagreement. The components of agreement are persistence simulated correctly and change simulated correctly [30]. The components of disagreement are change simulated as persistence (the entries where reference t1 matches simulation t2 but does not match reference t2), persistence simulated as change (the entries where reference t1 matches reference t2 but does not match simulation t2) and change simulated as change to wrong category (the entries where all three maps disagree) [30].

Among the three models, the percentages of disagreement components are lowest (28.066%) while the percentages of agreement components (71.934%) are found highest for MLP_Markov (Table 11).

This is how, implementing the three maps comparison method, it is found that the simulated map of “MLP_Markov 2009” is showing the best results in terms of the percentages of disagreement and agreement components. This is why; MLP_Markov is giving the best outputs among the three modeling techniques.

Figure 14 shows the results from an overlay of the three maps (the base map of 1999, the base map of 2009 and the St_Markov/CA_Markov/MLP_Markov simulated maps of time 2009). From this figure it is possible to get a clear idea about the components of agreement and disagreement.

Moreover, if we look at Table 12, then it is clear that MLP_Markov model comprises with all the basic elements of modeling. But St_Markov model lacks most of the elements, while CA_Markov is in a moderately better position.

Therefore, at the end, the MLP_Markov model has been selected for predicting the land cover map of Dhaka City for the year of 2019.