The Egyptian Journal of Remote Sensing and Space Sciences (2016) xxx, xxx-xxx

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The Egyptian Journal of Remote Sensing and Space

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RESEARCH PAPER

Statistical independence test and validation of CA Markov land use land cover (LULC) prediction results

Md. Surabuddin Mondala *, Nayan Sharmab, P.K. Gargc, Martin Kappas a

aDept. of Cartography, GIS & Remote Sensing, Institute of Geography, Georg-August University of Gottingen, Gottingen, 37077, Germany

b Dept. of W R D & M, Indian Institute of Technology, Roorkee, 247667, India c Dept. of Civil (Geomatic) Engineering, Indian Institute of Technology, Roorkee, 247667, India

Received 19 August 2015; revised 20 July 2016; accepted 1 August 2016

KEYWORDS

CA Markov model; Predictions;

Statistical independence test;

Validation;

Kappa index

Abstract Statistical independence test and validity of the CA (Cellular Automata) Markov process for projecting future land use and land cover (LULC) changes were carried out in this study. Predicting quantity and location changes have been analyzed, and statistically evaluated. Validity of the CA Markov process has been examined using various Kappa Index of Agreement (KIA or Kstandard) and related statistical variations on the KIA. Statistical test of independence (K2) was performed and markovian suitability has been checked using hypothesis of goodness of fit (Xc2). Hypothesis of statistical independence was rejected, which proved that land use land cover change trends are similar like previous development of land. With acceptance of the hypothesis of goodness of fit (Xc2) proved that actual transition probability of matrix is fitted with expected transition probability prepared using Markov chain method. Statistics indicates Kno, Klocation, Klocation Strata and Kstandard are 0.8347, 0.859, 0.8591 and 0.7928, respectively.

© 2016 National Authority for Remote Sensing and Space Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/

by-nc-nd/4.0/).

1. Introduction

* Corresponding author.

E-mail addresses: msk.iit@gmail.com (M.S. Mondal), nayanfwt@ iitr.ac.in (N. Sharma), gargpfce@iitr.ac.in (P.K. Garg), mkappas@ uni-goettingen.de (M. Kappas).

Peer review under responsibility of National Authority for Remote Sensing and Space Sciences.

Land use/land cover changes (LULCC) are continuous process and have to be understood from more dynamics information. Traditionally change detection methods can only provide a static diagnosis of the land use/land cover change for the fixed beginning and end dates. Land use/land cover change process model aims at predicting the spatial distribution of the specific

http://dx.doi.org/10.1016/j.ejrs.2016.08.001

1110-9823 © 2016 National Authority for Remote Sensing and Space Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Figure 1a Location map.

land cover and land use classes in a later year utilizing the knowledge gained from previous year. Modeling of land use/land cover change (LULCC) has been a topic of research since over a decade and there are several methods and models existing for the same. Baker (1989), followed by Lambin (1994) reviewed some initial LULCC models. Agarwal et al. (2002) provide details of LULCC models and recently, Mondal et al. (2012) provides an updated detail of LULCC models. The Markov model is very good and useful to understand the stochastic nature and the stability of the land use/land cover (LULC). The Markov model has become more popular due to advancement of Remote Sensing and GIS technology. The Markov model is frequently used to simulate landscape change (Baker, 1989; Muller and Middleton, 1994), analyze land use types, trends and dimension of changes (Weng, 2002; Huang et al., 2008). Two representative models are the Markov chain model (Muller and Middleton, 1994) and the CA (Cellular Automata) Markov model (Clarke, 1997). The Markov chain model treats as a stochastic process; the later state (land cover type) of a pixel is only related to its immediate preceding state, but not to any other previous states. A transition probability is the direct outcome from the Markov chain model. The CA Markov model, on the other hand, achieved a significant improvement in incorporating the spatial contingency information when making predictions. As a step forward, research has been made with the Markov chain model to achieve better accuracy. Pontius and Malanson (2005) reported their success in applying spatial contiguity in a combined CA Markov model when predicting land cover changes in Central Massachusetts.

The CA Markov model combines both the concept of a CA filter and Markov chain procedure. Markov chain and CA both is the discrete dynamic model in time and state. The transition probabilities may be accurate on per category basis, but there is no knowledge of the spatial distribution of occurrences within each LULC category. CA will add spatial character to the model. CA is discrete dynamic systems in which the state of each cell at time t + 1 is determined by the stated of its neighboring cells at time according the pre-defined transition rules. CA as a method with temporal-spatial dynamics can simulate the evolution of things in two dimensions. Using the outputs from the Markov chain analysis, the transition matrix, CA Markov will apply a contiguity filter to 'grow out' LULC from the time two to later time periods. CA Markov will use the transition areas tables and the conditional probability images to predict land use and land cover changes over the periods specified in Markov chain analysis. CA Markov will produce much better results geographically using the contiguity filter; those areas likely to change will do so closer to the existing LULC classes.

It is also important to validate the model output in an intelligent manner because a negative interpretation of the accuracy can give extremely misleading results. There are various methods of estimating the accuracy of prediction. Pontius et al. (2003) suggested use of kappa for location statistics in order to estimate the pixel level accuracy of a model as it extrapolates backwards in time for several land categories. Several studies estimated the accuracy using kappa for location statistics (et al.). Statistical test of independence (K2) can be also used to understand whether the changes in LULC are dependent

Figure 1b Satellite images of study area.

Table 1 Details of satellite data used in the study.

Satellite Sensor Path/row Data Spatial resolution Spectral band Data sources

acquired (m)

LANDSAT-5 TM 136/042 26-12-1987 30 (120 m - B 1 (blue): 0.45-0.52 im GLCF*-Earth

(WRS-2 thermal (B 6)) B 2 (green): 0.52-0.60 im Science Data

footprints) B 3 (red): 0.63-0.69 im Interface

B 4 (NIR): 0.76-0.90 im

B 5 (SWIR): 1.55-1.75 im

B 6 (thermal IR):10.4-12.5 im

B 7 (Mid-Infrared): 2.08-2.35 im

IRS-1C LISS-III 110/53 05-03-1997 23.5 (70 m - B 2 (green): 0.52-0.59 im NRSC

B5 (SWIR)) B 3 (red): 0.62-0.68 im

B 4 (NIR): 0.77-0.86 im

B 5 (SWIR): 1.55-1.70 im

IRS-P6 LISS-III 110/53 14-12-2007 23.5 B 2 (green): 0.52-0.59 im NRSC

(Resourcesat-1) B 3 (red): 0.62-0.68 im

B 4 (NIR): 0.77-0.86 im

B 5 (SWIR): 1.55-1.70 im

* The Global Land Cover Facility (GLCF) is a NASA-funded member of the Earth Science Information Partnership at the University of

Maryland, providing free satellite images to users all over world.

or not. The Markovian suitability can be checked using the hypothesis of goodness of fit (Xc2) which is availed to test that the land use/land cover change trends are dependent or not dependent on previous development of land. Using the hypothesis of goodness of fit (Xc2) it will check that actual transition probability of matrix of land use/land cover is fitted or not fitted with expected transition probability prepared using Markov chain method.

The prediction results in this study are tested and validated using traditional kappa for location statistics. Statistical test of independence (K2) was also performed, the Markovian suitability has been checked using hypothesis of goodness of fit (Xc2) and tested that the land use/land cover change trends are dependent or not depended on previous development of land. Using the hypothesis of goodness of fit (Xc2) it has been also checked that actual transition probability of matrix of

Table 2 Levels and LULC (land use land cover) classes

considered for classification.

Level I Level II

1. Built up land 1.1. Built up land

2. Agricultural land 2.1. Agricultural crop land

2.2. Agricultural fallow land

2.3. Plantations

3. Forest 3.1. Dense forest

3.2. Degraded forest

4. Waste land 4.1. Land with or without scrub

4.2. Marshy/swampy

4.3. Waterlogged area

4.4. Sandy area (river bed)

5. Water bodies 5.1. River/stream

5.2. Lake/reservoir/pond/tank

6. Others 6.1. Open land

6.2. Aquatic vegetation

land use/land cover is fitted or not fitted with expected transition probability prepared using Markov chain method.

2. Data used & methods for LULC prediction

In this study, the spatio-temporal CA (Cellular Automata) Markov model of landscape change using multi-temporal satellite imagery has been used which enabled us to predict spatial pattern of future land use/land cover for the study area - Kamrup Metropolitan district of Assam state in India (Fig. 1a). For this purpose, land use/land cover maps of the study area have been extracted from multi temporal satellite images. LANDSAT-5 TM image acquired on December 26, 1987, IRS-1C LISS III image acquired on March 5, 1997, IRS-P6 LISS III image acquired on 14th December of 2007 digitally classified for land use/land cover mapping (Fig. 1b and Table 1). Land use/land cover (LULC) maps derived from satellite images of 1987 and 1997 were used to predict future land use/land cover of 2007. The CA Markov model is simulated for a especial study area which covered a large proportion by urban landscape with or surrounding by other 14 classes of LULC. The CA model, coupled with the Markov transition probability, has indicated the capability of trend

Figure 2 Classified land use land cover map of 1987, 1997 and 2007.

projection for landscape change. This spatio-temporal model provided not only the quantitative description of change in the past but also the direction and magnitude of change in the future.

2.1. Preparation of LULC maps

The image dataset used in this study consists of LANDSAT-5 TM images of December 1987, IRS-1C images of March 1997 and IRS-P6 images of December 2007. Only images acquired in December and March months (winter season) were considered. The available images were selected based on the absence of cloud cover. When multi-data images from different sources are used, different atmospheric and terrain conditions may cause variations in data. Therefore, radiometric corrections including atmospheric correction - Top-of-Atmosphere (TOA) reflectance calibration were applied in this study. After radiometric correction, geometric correction was applied to the images. For accurate change detection, an accurate geometric registration is needed. The 1987 Landsat image from Global Land Cover Facility (GLCF) was chosen which has been orthorectified by the United States Geological Survey (USGS). Then, IRS-1C images of 1997 and IRS-P6 images of 2007 were rectified (geometrically corrected) with reference to the orthorectified Landsat satellite image of 1987 with two-order polynomial transformation and more than 14 ground control points (GCPs—mainly road junctions) to further improve the georeferencing accuracy. All images were resampled using Nearest Neighbor resampling method with a root mean square error of less than ±0.5 pixels per image to a 23.5 m resolution with the UTM coordinate system (zone 46, WGS 84 datum system).

For this study, supervised maximum likelihood classifier is used to classify all satellite images. Modified (modified from NRSA classification system for India and classification scheme adopted for European Commission sponsored Brahmatwin projects) classification scheme (level II) is adopted for different

categories of LULC (Table 2). 14 LULC classes i.e., built up land, agricultural crop land, agricultural fallow land, plantation, dense forest land, degraded forest land, land with or without scrub, marshy/swampy land, waterlogged area, sandy area, river, lakes/reservoirs/ponds, open land, aquatic vegetation area derived from satellite images. As supervised classification technique has been used for this study, it requires a priori knowledge of the number of classes, as well as knowledge concerning statistical aspects of the classes. Areas of visually homogeneous spectral response were chosen (10-12 training set for per class) well distributed all over images as AOI (area of interest) and added to the spectral signature editor. Limited pre-classification ground truth (using GPS) helped to select the training samples. The pre-classification ground truth was conducted on 14 December 2007, the same date when satellite collected the images for the study area. In the classification, the signature separability functions were used to examine the quality of training sites and class signature, before performing the classification. The land use and land cover types derived from digital image classification validate with data obtained from limited post-classification ground verification and using highresolution Google earth images. Land use/land cover (LULC) maps derived from satellite images of 1987, 1997 and 2007 are shows in Fig. 2 & area statistics are shown in Table 3.

2.2. CA Markov model

CA Markov model is a combination of the concept of a CA filter and Markov chain procedure. The CA model can be expressed as follows:

S(t, t +1) =f(S(t), N

where, S is the set of limited and discrete cellular states, N is the Cellular field, t and t + 1 indicate the different times, and f is the transformation rule of cellular states in local space.

The Markov model is a theory based on the process of the formation of Markov random process systems for the predic-

Table 3 Area statistics of LULC (land use land cover).

Sl. No. Class name 1987 1997 2007

Area (km2) % of area Area (km2) % of area Area (km2) % of area

1. Built up land 60.54 14.63 102.4 24.73 141.35 34.14

2. Agricultural crop land 25.91 6.26 5.99 1.45 7.17 1.73

3. Agricultural fallow land 48.27 11.66 34.08 8.23 25.12 6.07

4. Plantations 1.38 0.33 3.68 0.89 3.35 0.81

5. Dense forest 86.26 20.84 80.56 19.46 74.84 18.08

6. Degraded forest 83.48 20.17 76.95 18.59 60.31 14.57

7. Land with or without scrub 9.48 2.29 24.82 6 23.78 5.74

8. Marshy/swampy 13.42 3.24 10.26 2.48 6.82 1.65

9. Water logged area 3.57 0.86 1.86 0.45 1.52 0.37

10. Sandy area (river bed) 14.83 3.58 16.08 3.88 15.92 3.85

11. River/stream 37.27 9 32.51 7.85 33.42 8.07

12. Lake/reservoir/pond/tank 7.99 1.93 6.05 1.46 6.59 1.59

13. Open land 13.8 3.33 7.28 1.76 6.97 1.68

14. Aquatic vegetation 7.78 1.88 11.46 2.77 6.82 1.65

Total 413.98 100.00 413.98 100.00 413.98 100.00

tion and optimal control theory method. Based on the conditional probability formula—Bayes, the prediction of land use changes is calculated by the following equation:

S(t + 1) = Pj x S(t) (1)

where, S(t), S(t + 1) are the system status at the time of t or t +1; Ptj is the transition probability matrix in a state which is calculated as follows:

P = (Pj) =

Pn1 Pn

P1n P2n

Tpj -1

where, P is the Markov transition matrix P,

i, j is the land use land cover type of the first and second time period,

and Pij is the probability from land use and land cover type i to land type j.

In this expression, n is the number of land use and land cover types in the target area, and "Pj' is the probability of transition of type i into that of type j from the initiation to the end. In the transition matrix, it requests that each rate is a non-negative quantity, and each line factor 0 to 1. The estimate of Markov chain is the relative frequency of transitions observed over the entire time period. The result of the estimation can be used for prediction.

2.2.1. Markov chain - transition probability matrix

The transition probability matrix has been calculated for the time period of 1987-1997 & 1997-2007 for the prediction of LULC of 2007. The transition probability matrix for the time period of 1987-1997 & 1997-2007 displayed in Tables 4 and 5, respectively. The expected probability of transition of LULC category is displayed in Table 6. The transition probability matrix is the cross tabulation of the two images (1987 and 1997 & 1997 and 2007), that each LULC category will change to every other category. The transition probability areas matrix records the number of pixels that are expected to change over the specified time (1987-2007).

2.2.2. Preparation of suitability map (evidence likelihood map) and calibration of the CA Markov model

According to the underlying land use and land cover change dynamics between years 1987 and 1997, a series of suitability maps (evidence likelihood map) consisting of built up land suitability, agricultural crop land suitability, agricultural fallow land suitability, plantation suitability, dense forest land suitability, degraded forest land suitability, land with or without scrub suitability, marshy/swampy land suitability, waterlogged area suitability, sandy area suitability, river suitability, lakes/reservoirs/ponds suitability, open land suitability, aquatic vegetation land suitability were prepared (Fig. 3). The number thus expresses the likelihood of finding

Figure 3 Suitability (evidence likelihood) map used to predict future LULC.

Figure 4 Predicted LULC of 2007 using 1987 & 1997 LULC image.

the LULC at the pixel in question, if this lies in transition area. These images (evidence likelihood maps) are calculated as projections from the later date image (1997) of two input LULC images (before image 1987 and later image 1997). The output images are the conditional probability images. This conditional probability images report the probability that each LULC type would be found at each pixel in future after the specified time. The procedure looks at the relative frequency of pixels belonging to the different categories of that variable within areas of change. In effect, it asks the question of each category of the variable, ''How likely is it that you would have a value like this if you was an area that would experience change?" (Eastman et al., 2009). To project land use and land cover change for next 10 years using known LULC of 1987 and 1997, probability statistics for land use and land cover change for 2007 has been generated through cross tabulation of two LULC maps. Thus, the CA Markov model combines both the concepts of Markov chain procedure and CA filters, after getting Markov transition probability, CA Markov used the transition probability matrix and probability images (here, suitability/evidence likelihood map) to predict the LULC over a 10 years period i.e., 2007. The total numbers of iterations are based on the number of time steps, for 10 years model will choose to complete run in 10 iterations. The predicted locations of LULC are shows in Fig. 4. The quantitative results of predicted LULC are shows in Table 7.

3. Statistical independence test for Markov chain transition probability

The Markov model considers that LULC as stochastic process, and different categories of LULC as the states of chain. A chain is defined as stochastic process having the conditional probability distribution of the process at time n + 1, Xn + 1 depends upon only value of Xn, and is not dependent on all other previous value Xn _ 1, Xn _ 2,..., X0. It can be explained as:

P[Xn+1 — Xn+1jXn — Xn] X0 — x0

P[Xn+1 — xn+1jXn — xn] (1-1)

This can also be expressed as Pij — P[X„+1 — jjXn — i] (1.2)

ij — 0; 1; 2;

Here, Pij is transition probability of one step, which can be analyzed as the conditional probability at time n when the process in state 1 and at time n +1 the process is in state j. Two step transition probabilities are defined with generalization of Chapman-Kolmogorov equation.

Pj — P[X„+2 — j|Xn — i] —Y, P[Xn+2 — j|X„+,

— k]P[Xn — k|Xn — i] (1.3)

This is equivalent to(P)m + n — (P)n * (P)m (1.4)

Table 4 Transition matrix of 1987-1997.

LULC classes Built Agricultural Agricultural up crop land fallow land

Plantations Dense Degraded Land with or Marshy/ forest forest without scrub swampy

Water Sandy area River/ Lake/ Open Aquatic

logged (river bed) stream reservoir / land vegetation area pond/tank

Built up land Agricultural crop land Agricultural fallow land Plantations Dense forest Degraded forest

Land with or without scrub Marshy/ swampy Water logged area

Sandy area (river bed) River/stream Lake/

reservoir/pond /tank Open land Aquatic vegetation

0.0001 0

0.083 0

0.0001 0

0.0001 0.0002

0.0001

0.0001 0

0.0007 0

0.0004 0

0.0002 0 0 0

0.0001

0.0003 0.0001

0.0101

0.0027

0.0041

0.0005 0.0004 0.0096

0.0057

0.0012

0.0001

0.0002

0.0001 0.0002

0.0007 0.0006

0.0133 0.0023

0.0302

0.0002 0.0003 0.0075

0.0054

0.0023

0.0006

0.0001

0.0004

0.0041 0.0007

0.0002 0.0003 00

0.0009 0.0001 0.0002 0.1025 0.0004 0.0163

0.0001 0.0011

0.0001

00 0.0001 0

00 0.0001 0

0.0212 0.0004

0.0088

0.0134

0.0026

0.0004

0.0001

0.0006

0.0009 0.0014

0.0038 0

0.0011

0.0001 0.0002 0.0032

0.0035

0.0003

0.0001

0.0003 0.0005

0.0028 0.0009

0.0021

0.0003 0.0002 0.0028

0.0006

0.0007

0.0006

0.0017 0.0012

0.0009 0.001

0.0006 0

0.0008

0.0001 0

0.0005

0.0002

0.0001

0.0006

0.0001

0.0001 0.0005

0.0007 0.0007

0.0007 0.0005 0.0019 0

0.0008 0

0.0003 0.0003 0.0001 0 0.0026 0.0007 00

0.0113

0.0041 0.0394 0.0001 0

0.0001 0 00

0.0005 0

0.0057 0.0001

0.0001 0.0035

0.0001 0.0007

0.0003

0.0001

0.0037

0.0001 0.0001 0.0038

0.0028

0.0007

0.0001

0.0002

0.0002 0.0017 0.0054 0.0006

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Table 6 Transition probability of prepared LULC data for 1987-2007.

LULC classes Built Agricultural Agricultural Plantations Dense Degraded Land with or Marshy/ Water Sandy area River/ Lake/ Open Aquatic

up crop land fallow land forest forest without scrub swampy logged (river bed) stream reservoir/ land vegetation

land area pond/tank

Built up land 0.4190 0.0001 0.0001 0.0002 0 0.0004 0.0003 0 0 0 0 0 0 0

Agricultural 0.0002 0.0821 0.0178 0.0264 0.0006 0.0011 0.0415 0.0066 0.0052 0.0013 0.0009 0.0009 0.0011 0.0105

crop land

Agricultural 0 0 0.0024 0.0016 0 0.0004 0.0021 0.0001 0.0009 0.0002 0.0017 0 0.0001 0.0002

fallow land

Plantations 0.0001 0.0001 0.0026 0.0223 0 0.0001 0.0033 0.0009 0.0011 0.0005 0.001 0 0.0001 0.002

Dense forest 0 0.0001 0.0005 0.0003 0.0009 0.0005 0.0016 0.0002 0.0002 0.0001 0 0 0.0001 0.0001

Degraded 0.0001 0 0 0.0001 0.0001 0.1005 0.0039 0 0.0001 0 0 0 0 0

forest

Land with or 0.0002 0.0005 0.0076 0.0048 0.0002 0.0157 0.0481 0.0019 0.0018 0.0003 0.0001 0 0.0004 0.0023

without scrub

Marshy/ 0.0001 0.0003 0.0031 0.007 0 0.0012 0.0114 0.0022 0.0024 0.0016 0.0002 0 0.0005 0.0021

swampy

Water logged 0 0.0002 0.0004 0.001 0 0.0006 0.0018 0.0002 0.0019 0.0001 0.0017 0.001 0.0001 0.0001

area Sandy area 0 0 0.0001 0.0002 0 0.0002 0.0003 0.0003 0.0002 0.0001 0.0002 0.0002 0.0001 0.0001

(river bed)

River/stream 0.0001 0 0.0003 0.0001 0 0 0 0 0.0016 0.0001 0.0105 0.0094 0 0

Lake/ 0.0001 0 0.0001 0 0 0 0 0 0.0016 0 0.0043 0.0406 0 0

reservoir/

pond/tank

Open land 0 0 0.0001 0.0003 0.0001 0 0.0004 0.0001 0.0002 0.0001 0 0 0.006 0.0001

Aquatic 0 0.0014 0.001 0.0026 0 0.0001 0.0015 0.0003 0.0009 0.0001 0.0001 0 0.0001 0.0013

vegetation

Table 7 Area statistics of predicted land use land cover (LULC) of 2007 using 1987 & 1997 LULC (land use land cover) image and

LULC (land use land cover) derived from LISS III image of 2007.

LULC class Area (in Km2)

Predicted LULC 2007 LULC 2007 Differences

(Using 1987 & 1997 (Derived from

LULC Image) LISS III

Imageof 2007)

Built up land 125.09 141.35 -16.26

Agricultural crop land 4.32 7.17 -2.85

Agricultural fallow land 23.62 25.12 -1.50

Plantation 10.57 3.35 + 7.22

Dense forest 66.26 74.84 -8.58

Degraded forest 76.19 60.31 + 15.88

Land with or without scrub 24.95 23.78 + 1.17

Marshy/swampy 10.91 6.82 + 4.09

Waterlogged 1.46 1.52 -0.06

Sandy area 17.39 15.92 + 1.47

River 25.72 33.42 -7.70

Lakes/reservoirs/ponds 6.31 6.59 -0.28

Open land 8.67 6.97 + 1.70

Aquatic vegetation 12.52 6.82 + 5.70

Total 413.98 413.98

3.1. Hypothesis test for statistical independence

To follow the hypothesis of statistical independence involves a process of comparing the actual data with expected data of land use adopting following formula:

K2 - Elkf /Ek (1.5)

where,

Eik = expected value under Markov hypothesis

Ak = actual value of data from category in I to category in

If the value of K2 is greater than the tabulated value on the critical region 0.05 with degree of freedom (D.F. — 1)2 the hypothesis will be rejected. Expected value calculated with the use of Chapman-Kolmogorov equation following the Markov method. For calculation of transition probability matrix for the period 1987-2007 (Table 8) can be obtained by multiplying the 1987-1997 matrices (Table 4) and 19972007 matrices (Table 5). The expected value is calculated by following formula:

Ek = E(Eij)(Ejk)/Ej (1.6)

where,

Eij = the number of transition from category i to j during the period 1987-1997,

Ejk = the number of transition from category j to k during the period 1997-2007,

Ej = the number of cells in category j in 1987.

3.2. Test of goodness of fit

Chi square test of goodness of fit is used to test order Marko-vian suitability with the data. This test analyzes whether the

particular distribution is adequately described or not. By making comparison between actual observed probability and expected probability.

Xc2 = ^ I] (Ok — Eik)2/Eik (1.7)

where,

Oik = observed transition probability data from 1987 to

Eik = expected data of transition probability from 1987 to

If the Xc2 is less than the value of X 1-a on the 0.05 critical regions then the hypothesis is accepted.

3.3. Output of statistical independence test

The transition probability matrix has been calculated between 1987-1997 & 1997-2007 for prediction of LULC for 2007. The excepted probability of transition of LULC category is displayed in Table 8. The transition probability matrix is the cross tabulation of the two images (images of 1987 and 1997).

The statistical test (Table 9) of independence is used to understand whether the changes in LULC are dependent or not. For this statistical test of independence, (K2) is performed on LULC data. The results of K2 is 497.12 which is more than the significance 201.1 on critical region 0.05 with degree of freedom (14 — 1)2. So the hypothesis of statistical independence is rejected. Therefore, the changes in LULC are dependent. One can say that the land use and land cover change trends are dependent on previous development of land use/land cover or in another language one can say the land use/land cover change trends are likely to similar kinds of previous trends of land use/land cover change.

Table 8 Transition probability of LULC from 1987 to 2007 under Markov Hypothesis.

. £ 0 a

1n "" p.

LULC classes Built Agricultural Agricultural Plantations Dense Degraded Land with or Marshy/ Water Sandy area River/ Lake/ Open Aquatic

up crop land fallow land forest forest without scrub swampy logged (river bed) stream reservoir/ land vegetation

land area pond/tank

Built up land 0.9799 0 0 0.0078 0 0.0047 0 0.0019 0 0 0.0003 0.0012 0.0037 0.0014

Agricultural 0.2812 0.0736 0.1141 0.0149 0.0098 0.2637 0.1569 0.0342 0.0016 0.006 0.0033 0.005 0.019 0.0172

crop land

Agricultural 0.1998 0.0343 0.4459 0.0031 0.0045 0.1109 0.0801 0.0343 0.0084 0.0016 0.0007 0.0061 0.0602 0.0103

fallow land

Plantations 0.0821 0.0056 0.0111 0.4736 0.0984 0.1829 0.044 0.002 0.0004 0 0 0.0595 0.0052 0.0353

Dense forest 0.0055 0 0.0002 0.0004 0.8486 0.1349 0.0093 0.0005 0 0 0.0001 0.0001 0.0003 0

Degraded 0.1827 0.0032 0.0343 0.017 0.0751 0.5218 0.1143 0.0221 0.0038 0.0005 0.0001 0.0049 0.0078 0.0121

forest

Land with or 0.2883 0.0015 0.0837 0.0112 0.0137 0.243 0.2666 0.0215 0.0019 0.0001 0 0.0054 0.0227 0.0406

without scrub

Marshy/ 0.1491 0.0472 0.1102 0.0135 0.0083 0.1487 0.0336 0.1599 0.0369 0.0316 0.0904 0.066 0.0499 0.0546

swampy

Water logged 0.1301 0.0008 0.1624 0.0159 0.0017 0.0946 0.036 0.0259 0.1196 0.0259 0.0102 0.1072 0.1379 0.132

area Sandy area 0.0334 0.0928 0.039 0.002 0.0001 0.0132 0.0063 0.1235 0 0.4823 0.1981 0.0026 0.0067 0

(river bed)

River/stream 0.0092 0.0009 0.0003 0.0003 0 0.005 0 0.0137 0.0001 0.2156 0.7537 0.0002 0.0009 0

Lake/ 0.0442 0.0018 0.0133 0.0026 0.0067 0.0585 0.0287 0.0097 0.0028 0.0008 0.0014 0.3293 0.0158 0.4845

reservoir/

pond/tank

Open land 0.2967 0.0032 0.1786 0.0056 0.0067 0.1966 0.1448 0.0369 0.0032 0.0023 0.0007 0.0102 0.0861 0.0293

Aquatic 0.0639 0 0.0774 0.0077 0.005 0.0872 0.1318 0.0400 0.0091 0.0001 0 0.1151 0.0155 0.4473

vegetation

The Markovian suitability has been checked using hypothesis of goodness of fit. In this test, actual LULC from 1987 to 2007 has been compared with expected data (LULC) which were calculated using the Markov model. This hypothesis is accepted for these data. The calculated value of Xc2 is 0.52 and it is very much less than the significance of 22.4 on the critical region of 0.05 with 13 degrees of freedom (Table 9). With acceptance of the hypothesis one can say that actual transition probability of matrix from 1987 to 2007 is fitted with expected transition probability prepared using Markov method. Actual transition probability of matrix from 1987 to 2007 is similar to expected transition probability prepared using Markov method.

4. Validation of CA Markov prediction - kappa indices of agreement and disagreement

The international scientific community has called for research into land cover change, specifically models that predict spatial patterns of future change (Turner et al., 1995; Lambin et al., 2003). Modelers are satisfying this need with a variety of approaches (Baker, 1989; Pontius et al., 2004; Hall et al., 1995; Veldkamp and Fresco, 1996; Geoghegan et al., 1997; Mertens and Lambin, 1997; Liverman et al., 1998; Wu and Webster, 1998). In most cases, the models are connected to a raster-based GIS. Scientists are required to necessarily develop statistical methods to validate such models, because it is essential to know its prediction accuracy (Pontius and Schneider, 2001). Pontius (2002) have suggested the use of Kappa

statistics for testing accuracy in terms of location (Kappa for location) and quantity of correct cells (Kappa for quantity). Therefore, land use and land cover change data derived from satellite images for describing and projecting land use and cover changes establishes the validity of the predicted results of the CA Markov process in this study. For validation, a map of simulated future change is compared to a map of recent real land cover change. For appropriate validation, the map of reality used for validation should not be used in calibration (Pontius and Schneider, 2001). Here, LULC of 2007 is predicted using LULC maps of 1987 and 1997, derived from Landsat and IRS-P6 satellite images, respectively. This provides a method to measure agreement between two categorical images, a "comparison" map (here the predicted LULC of 2007 - Fig. 4) and a "reference" map (LULC map derived from IRS-P6 LISS III image of 2007 - Fig. 2c). The comparison map is the result of CA Markov model simulation results, whose validity is to be assessed against a reference map that depicts reality.

The statistical methods separate error and agreement by components due to specification of quantity and location. The simulated map of 2007 is compared to the reference map of 2007, a Kappa for quantity and location statistic is derived (Table 10). The statistics for location showing Kno is 0.8347, Klocation is 0.859, Klocation Strata is 0.8591 and Kstandard is 0.7928 (Table 11). The results indicate that CA Markov model's ability to specify grid cell level location of future change is nearly perfect (here Klocation value is 0.859, where Klocation value of 1 is perfect).

Table 11 Kappa Index of Agreement to ability to specify accurately quantity and location to predict 2007 LULC.

Statistics Index

Kno 0.8347

Klocation 0.8591

Klocation Strata 0.8591

Kstandard 0.7928

Table 10 Agreement/disagreement according to ability to specify accurately quantity and location to predict 2007 LULC.

Sl. No. Information of location Information of quality

No [n] Medium [m] Perfect [p]

1. Perfect [P(x)] P(n) = 0.4592 P(m) = 0.9478 P(P) = 1.0000

2. Perfect Stratum [K(x)] K(n) = 0.4592 K(m) = 0.9478 K(p) = 1.0000

3. Medium Grid [M(x)] M(n) = 0.4398 M(m) = 0.8550 M(p) = 0.8856

4. Medium Stratum [H(x)] H(n) = 0.1522 H(m) = 0.3235 H(p) = 0.3261

5. No [N(x)] N(n) = 0.1522 N(m) = 0.3235 N(p) = 0.3261

Agreement chance 0.1522

Agreement quantity 0.1713

Agreement strata 0.0000

Agreement grid cell 0.5315

Disagree grid cell 0.0928

Disagree strata 0.0000

Disagree quantity 0.0522

Table 9 Statistical results of data.

Test perform Calculated Chi sq. table value

value on .05 critical

region

Statistical independence 497.12 201.1

test (K2)

Goodness of fit test (Xc2) 0.52 22.4

5. Conclusions

Currently, land-change modelers are not being held accountable for their prediction of future landscapes. Most land-change modelers fail to validate models and fail to state the uncertainty in future prediction. Consequently, policy makers and the general public develop opinions based on misleading research that fails to give them the appropriate interpretations required to make informed decisions. Validation efforts to a known point in time are necessary to make an estimate of the uncertainty for the extrapolation to an unknown point in time. CA Markov LULCC Model prediction results were tested and validated in this study using traditional kappa for location statistics. Statistical test of independence (K2) was performed; the Markovian suitability has been checked using hypothesis of goodness of fit (Xc2) and proved that the land use/land cover change trends are dependent on previous development of land. The calculated value of Xc2 is 0.52 and it is very less than significance 22.4 on critical region 0.05 with 13 degree of freedom. With acceptance of the hypothesis established that actual transition probability of matrix from 1987 to 2007 is fitted with expected transition probability prepared using Markov method. Hypothesis of goodness of fit (Xc2) value established that the actual transition probability of matrix of land use/land cover is similar to expected transition probability prepared using the Markov chain method. The validation for CA Markov model land use/land cover prediction results calculated using various Kappa Index of Agreement (KIA or Kstandard) and related statistical variations on the KIA. The simulated map of 2007 was compared to the reference map of 2007, Kappa for quantity and location statistic was derived and statistics for location showing Kno is 0.8347, Klocation is 0.859, Klocation Strata is 0.8591 and Kstandard is 0.7928 and this results indicated that CA Markov model's ability to specify grid cell level location of future change is nearly perfect. This study concludes that use of statistical independence test, Kappa indices are potentially useful techniques for purposes of validation of CA Markov model land use/land cover (LULC) prediction results.

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