Derivation of a yearly transition probability matrix for land-use dynamics and its applications
Authors
- First Online:
- Received:
- Accepted:
DOI: 10.1007/s10980-009-9433-x
- Cite this article as:
- Takada, T., Miyamoto, A. & Hasegawa, S.F. Landscape Ecol (2010) 25: 561. doi:10.1007/s10980-009-9433-x
- 21 Citations
- 503 Views
Abstract
Transition matrices have often been used in landscape ecology and GIS studies of land-use to quantitatively estimate the rate of change. When transition matrices for different observation periods are compared, the observation intervals often differ because satellite images or photographs of the research site taken at constant time intervals may not be available. If the observation intervals differ, the transition probabilities cannot be compared without calculating a transition matrix with the normalized observation interval. For such calculation, several previous studies have utilized a linear algebra formula of the power root of matrices. However, three difficulties may arise when applying this formula to a practical dataset from photographs of a research site. We examined the first difficulty, namely that plural solutions could exist for a yearly transition matrix, which implies that there could be multiple scenarios for the same transition in land-use change. Using data for the Abukuma Mountains in Japan and the Selva el Ocote Biosphere Reserve in Mexico, we then looked at the second difficulty, in which we may obtain no positive Markovian matrix and only a matrix partially consisting of negative numbers. We propose a way to calibrate a matrix with some negative transition elements and to estimate the prediction error. Finally, we discuss the third difficulty that arises when a new land-use category appears at the end of the observation period and how to solve it. We developed a computer program to calculate and calibrate the yearly matrices and to estimate the prediction error.
Keywords
Abukuma Mountains (Japan)Computer programMultiple scenariosn-th power roots of matricesObservation intervalIntroduction
In the late 1990s, a number of international research projects such as the Land Use and Cover Change project (Messerli 1997) began to examine the intensity of land-use change and the implications for global environmental change (Lambin and Geist 2006; Turner et al. 2007). These projects examined relatively large areas, including suburbs and cities, and focused on land-use change induced by human activities. The results indicated the necessity for intensive studies of land-use changes to determine the rate of changes and the associated driving forces. To quantitatively estimate the rate of land-use changes, satellite images, aerial photographs, and geographic information systems (GIS) have been widely used to identify and examine land-use and land cover change (Ehlers et al. 1990; Meyer and Turner 1991; Hathout 2002; Braimoh and Vlek 2004). The type, amount, and location of land-use changes can now be quantified, and some GIS software now provides a flexible environment for displaying, storing, and analyzing the digital data necessary to detect such changes. The software includes a procedure to classify the patterns of land use and land cover and to calculate transitions in areas of these classifications of land use. Area-based tables can be constructed using these procedures, allowing users to conveniently grasp the transition at a glance.
Probability-based transition tables, such as Markovian models or cross-tabulation matrices, are often obtained from area-based transition as a theoretical tool of landscape ecology. These tables provide a simple method for comparing the dynamics among research sites of different sizes and have been the focus of extensive theoretical studies (Usher 1981; Kachi et al. 1986; Gardner et al. 1987; Baker 1989a, b; Gustafson and Parker 1992; Lewis and Brown 2001; Pontius 2002; Pontius et al. 2004; Pontius and Cheuk 2006). Therefore, since the 1990s, many researchers have used Markovian models or cross-tabulation matrices (Meyer and Turner 1991; Mertens and Lambin 2000; Hathout 2002; Braimoh and Vlek 2004; Mundia and Aniya 2005; Braimoh 2006; Flamenco-Sandoval et al. 2007) to grasp dynamical characteristics of land use such as the diversity, driving forces, or scale dependence of land use (Turner et al. 1989; Turner 1990; Lo and Yang 2002).
Several authors (Mertens and Lambin 2000; Petit et al. 2001; Flamenco-Sandoval et al. 2007) have tried to construct yearly transition matrices using mathematical formulae from stochastic process theory (Cinlar 1975; Lipschutz 1979). Mertens and Lambin (2000) used four satellite images of East Province in Cameroon that were taken in 1973, 1986, 1991, and 1996 (at one 13-year and two 5-year intervals). They constructed 2 × 2 transition matrices with forest and non-forest land-use categories, obtained the yearly transition matrices, and compared them to detect the annual rate of changes in land cover. Flamenco-Sandoval et al. (2007) also conducted a similar analysis with 7 × 7 transition matrices in 1986, 1995, and 2000. Obtaining yearly transition matrices from the original transition matrices is becoming increasingly popular in land-use analysis. The above two papers established the mathematical formulae for obtaining the yearly transition matrix. However, it is not well known that several practical difficulties arise in the general way of obtaining the yearly matrix, although Flamenco-Sandoval et al. (2007) experienced one of the difficulties.
In the present paper, we clarify the three practical difficulties and the reasons they occur. The first difficulty is that the yearly transition matrix basically has plural solutions, which implies that multiple scenarios may exist for the same transition in land-use change. The second is that the yearly transition matrix could have some negative elements. We show two examples of land-use change, one from the Abukuma Mountains of central Japan and the other from Flamenco-Sandoval et al.’s (2007) study. The third is that a new land-use category may appear at the end of the observation period. A new land-use category could appear when the land-use change is very large. In this case, the transition matrix is not a square matrix, and we cannot apply the established formula. Finally, we propose ways of solving these problems and construct an algorithm to obtain the yearly transition matrix, presented by Mathematica and C++ programs.
Methods
Therefore, the n-by-n yearly matrix, B, is the c-th power root of an original transition matrix, A.
We developed computer programs to solve Eq. (4). Here, we describe the calculation of several examples and clarify the three practical difficulties in obtaining the yearly transition matrices in the “Result” section.
Result
The first practical difficulty
In calculating the yearly transition matrix, we may encounter the difficulty of obtaining more than one yearly matrix. The number of c-th power roots of the matrix is easily obtained from Eq. (4), in which λ^{1/c} represents the c-th power root of the scalar λ. Because λ could be a complex number, we can set \( \lambda = {\text{re}}^{i\theta } = r(\cos \theta + i\sin \theta ) \)\( (r > 0\,{\text{and}}\,0 \le \theta < 2\pi ) \), using polar coordinates. Therefore, \( \lambda^{1/c} = r^{1/c} \left( {\cos {\frac{\theta + 2\pi k}{c}} + i\sin {\frac{\theta + 2\pi k}{c}}} \right) \) for k = 0, 1,…, c–1 generally has c solutions, including complex numbers, as long as λ is not equal to zero. Therefore, the number of combinations for n λ_{i}s is c^{n}, and the number of whole solutions is c^{n}.
The second practical difficulty
- (1)
The Abukuma Mountains
Area-based transition tables among land-use categories in the Abukuma Mountains
(a) From 1947 to 1962 | 1962 | Total in 1947 | ||||
---|---|---|---|---|---|---|
1947 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other | |
Coniferous planted forest | 1,642.4 | 143.8 | 0.0 | 19.6 | 1.3 | 1,807.1 |
Secondary forest | 852.6 | 4,940.0 | 0.0 | 87.0 | 67.5 | 5,947.0 |
Old forest | 211.7 | 164.7 | 437.4 | 0.6 | 2.4 | 816.7 |
Grassland | 96.1 | 483.4 | 0.0 | 85.6 | 26.1 | 691.2 |
Other | 1.1 | 7.3 | 0.0 | 0.9 | 741.5 | 750.9 |
Total | 2,803.9 | 5,739.1 | 437.4 | 193.8 | 838.7 | 10,012.9 |
(b) From 1962 to 1975 | 1975 | Total in 1962 | ||||
---|---|---|---|---|---|---|
1962 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other | |
Coniferous planted forest | 2,656.5 | 77.6 | 0.0 | 55.2 | 14.6 | 2,803.9 |
Secondary forest | 1,805.1 | 3,433.0 | 2.5 | 391.3 | 107.3 | 5,739.1 |
Old forest | 49.3 | 65.5 | 228.8 | 93.8 | 0.0 | 437.4 |
Grassland | 92.0 | 24.3 | 0.0 | 77.4 | 0.0 | 193.8 |
Other | 33.0 | 5.5 | 0.0 | 11.2 | 789.1 | 838.7 |
Total | 4,635.9 | 3,606.0 | 231.3 | 626.8 | 911.0 | 10,012.9 |
(c) From 1975 to 1997 | 1997 | Total in 1975 | ||||
---|---|---|---|---|---|---|
1975 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other | |
Coniferous planted forest | 3,249.8 | 1,265.7 | 18.2 | 29.7 | 70.3 | 4,633.7 |
Secondary forest | 1,110.8 | 2,158.6 | 31.2 | 190.3 | 115.4 | 3,606.4 |
Old forest | 45.9 | 39.7 | 145.3 | 0.0 | 0.9 | 231.9 |
Grassland | 308.6 | 165.3 | 3.8 | 124.6 | 26.1 | 628.5 |
Other | 127.6 | 159.3 | 3.3 | 76.5 | 545.6 | 912.4 |
Total | 4,842.8 | 3,788.7 | 201.9 | 421.2 | 758.4 | 10,012.9 |
Transition matrices among land-use categories in the Abukuma Mountains
(a) From 1947 to 1962 | 1962 | ||||
---|---|---|---|---|---|
1947 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.909 | 0.080 | 0 | 0.011 | 0.001 |
Secondary forest | 0.143 | 0.831 | 0 | 0.015 | 0.011 |
Old forest | 0.259 | 0.202 | 0.536 | 0.001 | 0.003 |
Grassland | 0.139 | 0.699 | 0 | 0.124 | 0.038 |
Other | 0.002 | 0.010 | 0 | 0.001 | 0.988 |
(b) From 1962 to 1975 | 1975 | ||||
---|---|---|---|---|---|
1962 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.947 | 0.028 | 0 | 0.020 | 0.005 |
Secondary forest | 0.315 | 0.598 | 0 | 0.068 | 0.019 |
Old forest | 0.113 | 0.150 | 0.523 | 0.214 | 0 |
Grassland | 0.475 | 0.125 | 0 | 0.400 | 0 |
Other | 0.039 | 0.007 | 0 | 0.013 | 0.941 |
(c) From 1975 to 1997 | 1997 | ||||
---|---|---|---|---|---|
1975 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.701 | 0.273 | 0.004 | 0.006 | 0.015 |
Secondary forest | 0.308 | 0.599 | 0.0009 | 0.053 | 0.032 |
Old forest | 0.198 | 0.171 | 0.627 | 0 | 0.004 |
Grassland | 0.491 | 0.263 | 0.006 | 0.198 | 0.042 |
Other | 0.140 | 0.175 | 0.004 | 0.084 | 0.598 |
We obtained the yearly transition matrices in the Abukuma Mountains using our computer program. The matrix between 1947 and 1962 has 15^{5} solutions, elements of which could include negative and complex numbers, as explained previously. We omitted solutions with negative or complex numbers after obtaining the whole solutions and could not obtain any positive solutions. The computer program was then modified to detect solutions with real elements ≥ −0.1, taking into account approximately positive solutions.
Yearly and calibrated transition matrices in the Abukuma Mountains
(a) From 1947 to 1962 | 1962 | ||||
---|---|---|---|---|---|
1947 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.9931 | 0.0052 | 0 | 0.0016 | 0.0000 |
Secondary forest | 0.0109 | 0.9860 | 0 | 0.0023 | 0.0008 |
Old forest | 0.0225 (0.0224) | 0.0189 (0.0189) | 0.9592 (0.9585) | −0.0007 (0.0000) | 0.0002 (0.0002) |
Grassland | 0.0105 | 0.1193 | 0 | 0.8652 | 0.0002 |
Other | 0.0000 | 0.0008 | 0 | 0.0002 | 0.9992 |
(b) From 1962 to 1975 | 1975 | ||||
---|---|---|---|---|---|
1962 | Coniferous planted forest | Secondary Forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.9949 | 0.0026 | 0 | 0.0022 | 0.0004 |
Secondary forest | 0.0287 | 0.9597 | 0 | 0.097 | 0.0018 |
Old forest | −0.0005 (0.0000) | 0.0163 (0.0163) | 0.9514 (0.9507) | 0.0330 (0.0329) | −0.0002 (0.0000) |
Grassland | 0.0530 (0.0530) | 0.0178 (0.0178) | 0.0 (0.0) | 0.9296 (0.9293) | −0.0004 (0.0000) |
Other | 0.0027 | 0.0005 | 0 | 0.0015 | 0.9953 |
(c) From 1975 to 1997 | 1997 | ||||
---|---|---|---|---|---|
1975 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.9800 (0.9790) | 0.0202 (0.0202) | 0.0002 (0.0002) | −0.0010 (0.0000) | 0.0006 (0.0006) |
Secondary forest | 0.0199 | 0.9708 | 0.0006 | 0.0066 | 0.0021 |
Old forest | 0.0116 (0.0116) | 0.0104 (0.0104) | 0.9789 (0.9780) | −0.0008 (0.0000) | 0.0000 (0.0000) |
Grassland | 0.0504 | 0.0190 | 0.0004 | 0.9263 | 0.0039 |
Other | 0.0036 | 0.0106 | 0.0001 | 0.0094 | 0.9763 |
Error estimation between the observed area and estimated area using a calibrated transition matrix
Area vector (ha) | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
---|---|---|---|---|---|
(a) From 1947 to 1962 in the Abukuma Mountains; c = 15 | |||||
At time t | 1,807.1 | 5,947.0 | 816.7 | 691.2 | 750.9 |
At time t + 15 | 2,803.9 | 5,739.1 | 437.4 | 193.8 | 838.7 |
Estimated area vector | 2,802.8 | 5,741.6 | 432.6 | 196.5 | 839.5 |
Error (%) | −0.01 | 0.02 | −0.05 | 0.03 | 0.01 |
(b) From 1962 to 1975 in the Abukuma Mountains; c = 13 | |||||
At time t | 2,803.9 | 5,739.1 | 437.4 | 193.8 | 838.7 |
At time t + 13 | 4,635.9 | 3,606.0 | 231.3 | 628.8 | 911.0 |
Estimated area vector | 4,637.3 | 3,605.6 | 229.4 | 627.0 | 913.6 |
Error (%) | 0.01 | 0.00 | −0.02 | −0.02 | 0.03 |
(c) From 1975 to 1997 in the Abukuma Mountains; c = 22 | |||||
At time t | 4,633.7 | 3,606.4 | 231.9 | 628.5 | 912.4 |
At time t + 22 | 4,842.8 | 3,788.7 | 201.9 | 421.2 | 758.4 |
Estimated area vector | 4,792.3 | 3,786.9 | 198.8 | 474.5 | 760.4 |
Error (%) | −0.50 | −0.02 | −0.03 | 0.53 | 0.02 |
- (2)
The Selva el Ocote Biosphere Reserve in Mexico studied by Flamenco-Sandoval et al. (2007)
They examined land-use shift in the Selva el Ocote Biosphere Reserve in Mexico with 7 × 7 transition matrices in 1986, 1995, and 2000. Their seven categories of land use were agriculture and pasture (A/P), temperate forest (TemF), tropical forest (TroF), shrub and savanna (S/S), second-growth temperate forest (SGTemF), second-growth tropical forest (SGTtroF), and second-growth forest with slash and burn agriculture (SGF + SBA). They constructed two transition matrices, from 1986 to 1995 and from 1995 to 2000, and obtained the yearly matrices to discern whether they were significantly different by a log-linear statistical test.
Yearly transition matrices recalculated from Flamenco-Sandoval et al. (2007)
(a) From 1986 to 1995 | 1995 | ||||||
---|---|---|---|---|---|---|---|
1986 | A/P | TemF | TroF | S/S | SGTemF | SGTtroF | SGF + SBA |
A/P | 0.995 | 0.000 | 0.000 | 0.000 | 0.000 | 0.005 | 0.000 |
TemF | 0.001 | 0.997 | 0.000 | 0.000 | 0.002 | 0.000 | 0.000 |
TroF | 0.001 | 0.000 | 0.988 | 0.000 | 0.000 | 0.009 | 0.002 |
S/S | 0.006 | 0.000 | 0.000 | 0.994 | 0.000 | 0.000 | 0.000 |
SGTemF | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 |
SGTtroF | 0.006 | 0.000 | 0.000 | 0.000 | 0.000 | 0.994 | 0.000 |
SGF + SBA | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.006 | 0.993 |
(b) From 1995 to 2000 | 2000 | ||||||
---|---|---|---|---|---|---|---|
1995 | A/P | TemF | TroF | S/S | SGTemF | SGTtroF | SGF + SBA |
A/P | 0.959 | 0.000 | 0.000 | 0.003 | 0.001 | 0.036 | 0.000 |
TemF | 0.010 | 0.971 | 0.000 | −0.000 | 0.020 | −0.001 | 0.000 |
TroF | 0.013 | −0.000 | 0.913 | −0.000 | −0.001 | 0.057 | 0.019 |
S/S | 0.008 | −0.000 | −0.000 | 0.993 | −0.000 | −0.001 | −0.000 |
SGTemF | 0.037 | −0.000 | −0.000 | −0.000 | 0.966 | −0.003 | −0.000 |
SGTtroF | 0.069 | 0.000 | −0.000 | 0.002 | −0.000 | 0.929 | −0.000 |
SGF + SBA | 0.059 | −0.000 | 0.000 | −0.001 | 0.028 | 0.127 | 0.787 |
(c) Calibrated matrix from 1995 to 2000 | 2000 | ||||||
---|---|---|---|---|---|---|---|
1995 | A/P | TemF | TroF | S/S | SGTemF | SGTtroF | SGF + SBA |
A/P | 0.959 | 0.000 | 0.000 | 0.003 | 0.001 | 0.036 | 0.000 |
TemF | 0.010 | 0.970 | 0.000 | 0.000 | 0.020 | 0.000 | 0.000 |
TroF | 0.013 | 0.000 | 0.911 | 0.000 | 0.000 | 0.057 | 0.019 |
S/S | 0.008 | 0.000 | 0.000 | 0.992 | 0.000 | 0.000 | 0.000 |
SGTemF | 0.037 | 0.000 | 0.000 | 0.000 | 0.963 | 0.000 | 0.000 |
SGTtroF | 0.069 | 0.000 | 0.000 | 0.002 | 0.000 | 0.929 | 0.000 |
SGF + SBA | 0.059 | 0.000 | 0.000 | 0.000 | 0.028 | 0.127 | 0.786 |
Error (%) | 0.00 | −0.40 | −0.62 | 1.11 | −0.07 | 0.18 | −0.52 |
Table 5c shows the calibrated matrix of the second period, including the error estimation (that of the first period is the same in the range of three decimal digits because of extremely small negative elements; see Table 5a). The result of error estimation is very low, at most 1.1%, implying that the calibrated matrix is a reasonably good estimator of the area vectors in the final year.
The third practical difficulty
Here, we are confronted with the second difficulty again and could obtain the calibrated yearly Markovian matrix using Eq. (8). If an obtained matrix includes many and large negative elements, the result of error estimation would become large. Then, we should not adopt the solution because the assumption of setting the fourth row in Eq. (9) to (0, 0, 0, 1) does not hold.
Discussion
Ten-year transition matrices in the Abukuma Mountains
(a) From 1947 to 1962 | 1962 | ||||
---|---|---|---|---|---|
1947 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.936 | 0.054 | 0 | 0.009 | 0.000 |
Secondary forest | 0.100 | 0.879 | 0 | 0.013 | 0.008 |
Old forest | 0.189 | 0.152 | 0.655 | 0.002 | 0.002 |
Grassland | 0.097 | 0.631 | 0 | 0.241 | 0.001 |
Other | 0.001 | 0.006 | 0 | 0.001 | 0.992 |
(b) From 1962 to 1975 | 1975 | ||||
---|---|---|---|---|---|
1962 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.957 | 0.022 | 0 | 0.016 | 0.004 |
Secondary forest | 0.253 | 0.671 | 0.000 | 0.061 | 0.015 |
Old forest | 0.075 | 0.126 | 0.603 | 0.195 | 0.001 |
Grassland | 0.398 | 0.112 | 0.000 | 0.489 | 0.002 |
Other | 0.030 | 0.005 | 0 | 0.011 | 0.954 |
(c) From 1975 to 1997 | 1997 | ||||
---|---|---|---|---|---|
1975 | Coniferous planted forest | Secondary forest | Old forest | Grassland | Other |
Coniferous planted forest | 0.825 | 0.162 | 0.002 | 0.005 | 0.007 |
Secondary forest | 0.171 | 0.764 | 0.005 | 0.043 | 0.018 |
Old forest | 0.104 | 0.091 | 0.801 | 0.002 | 0.001 |
Grassland | 0.344 | 0.154 | 0.003 | 0.470 | 0.028 |
Other | 0.054 | 0.093 | 0.002 | 0.063 | 0.789 |
The present paper has described three practical difficulties in obtaining the yearly transition matrix. One is that the number of appropriate solutions for yearly matrices could be >1, as in Eq. (7). This implies that there could be multiple scenarios that lead to the same aerial photographs in the final year of the observation period, which could be caused by different driving forces among the scenarios, i.e., political reasons (e.g., adoption of new ordinances), economic reasons (e.g., price reductions in the timber market), or environmental reasons (e.g., soil erosion). Mathematically, there would be no way to identify which matrix (or which driving force) is correct. To determine the correct transition matrix among plural solutions, extra aerial photographs are required for a middle year (m) during the observation period (c years; c > m). Using the observed area distribution in the initial year, x_{0}, the discrepancy between the observed area distribution from the extra photograph (x_{m}) and the expected area distribution \( {\bf{x}}_{0} {\bf{B}}_{i}^{m} \) can be calculated for each scenario (B_{i}). The most appropriate scenario can then be chosen such that the norm of \( \left\| {\bf{x}}_{0} {\bf{B}}_{i}^{m} - {\bf{x}}_{m} \right\| \) is minimized.
The second difficulty is that only a yearly transition matrix with negative elements close to zero may be obtained, rather than transition matrices with positive elements. Previous studies did not refer explicitly to these two points. Mertens and Lambin (2000) calculated the yearly transition matrices of 2 × 2 matrices and obtained a positive matrix. Using our computer program, we confirmed only one positive solution existed in their case. Similarly, Flamenco-Sandoval et al. (2007) obtained two 7 × 7 yearly matrices, one of which included a few small negative elements, as shown in the present paper. This could occur because of the non-stationarity of the Markov process in land-use change or errors such as mistaken image analysis in land-use classifications. If the transition among land-use categories is not stationary during the observation period, the possibility of not obtaining positive yearly transition matrices would increase. The percentage error between the observed and estimated area vectors in Table 4 would express the index of the non-stationarity of the Markovian process. Furthermore, an improbable transition can be picked up from photographs because of the precision of GIS software. To avoid mistakes in classification, a technique to compute the transition probabilities for soft-classified pixels would be useful, as proposed by Pontius and Cheuk (2006). From our experience, small negative elements in yearly transition matrices are likely to occur when many zero or small elements are included in the original matrix. For example, in the second period in the Abukuma Mountains (Table 2b), there are six zero elements among the 5 × 5 elements because transitions among categories are usually slow in forest ecosystems.
There is a kind of trade-off between the first and second problems. Matrix A (Eq. (6), original matrix) includes sufficiently large positive elements, indicating that there are large transitions among the land-use categories and increasing the possibility of obtaining plural solutions. This is likely to occur for land-use changes in cities or agricultural areas, where the effect of agricultural innovation or human activity is large. In contrast, in natural forests, transitions are usually slow, and the original matrices include many small elements. In this case, a yearly transition matrix with negative elements close to zero is sometimes obtained.
To solve the above difficulties, an adequate procedure is to find all the solutions and then select the appropriate solutions, including those with a few small negative elements. If a solution is unique, that is the solution we want to obtain. If a solution includes negative elements, we should check whether it can be calibrated and the extent to which the calibrated matrix causes the percentage error between the observed and estimated area vectors (Table 4). We developed a computer program using Mathematica (Wolfrum Research, Inc.) and C++, which can be accessed on the website http://hosho.ees.hokudai.ac.jp/~takada/enews.html. The procedure used in our computer program could be easily incorporated into popular GIS software as a standard subprogram. The subprogram can be used for the comparison of yearly transition matrices among different observation periods when temporal change of exogenous driving factors occurs, and for the analysis of spatial heterogeneity in yearly transition matrices.
Acknowledgments
We express our sincere thanks to Masahiro Ichikawa, Takashi Kohyama, Toru Nakashizuka, and Ken-Ichi Akao for their helpful suggestions. Prof. Ichikawa encouraged us to continue this study. Profs. Kohyama and Nakashizuka provided the opportunity to solve the mechanism of the dynamics of land use. Prof. Akao provided mathematical advice at an early stage of our study. This research was funded in part by Grants-in-Aid from the Japanese Society for the Promotion of Science (JSPS) for Scientific Research (nos. A-21247006, B-20370006 and B-21310152) and project 2–2 “Sustainability and Biodiversity Assessment on Forest Utilization Options” and D-04 of the Research Institute for Humanity and Nature.