2. Material and methods
The 'Liukuei ecosystem management area' is a part of the Liukuei experimental forest of the Taiwan Forestry Research Institute (TFRI). The study area encompasses seven forest compartments, and covers about 2500 ha. Elevation in the area ranges from 300m to 1800m, and the topography is up-sloped from southwest to northeast. The Shan ping post and several nurseries are also located inside the area and are connected by service roads stretching throughout the area. Natural forests dominate 78% of the landscape, and man-made stands occupy the remaining 22%. Most of the man-made stands are conifers. Natural forests consist of many broadleaf tree species, and some are mixed with conifers. There are some cut areas due to timber harvesting. In addition, there are some non-forested areas such as grasslands or bare lands resulted from landslides.

Aerial photographs taken in 1988 and 1996 were used to generate landscape maps of the study area. Positive films of the photographs were scanned into digital images. The images were subsequently rectified as orthogonal images with accurate positioning. Image pairs were then displayed as stereographs on computer screens, where features on the landscape were delineated and digitized. As a result, two sets of vector files were created. For each land cover patch, its land cover type, area, and perimeter were recorded. The two maps were then overlid together using a geographic information system to identify land cover changes during the period.

The analytical procedures of this study include three steps. In the first step, various landscape indices was used to measure the structures of the landscape under different conditions. The landscape was classified into natural forests, man-made stands, bare lands, streams, roads and other man-made structures (such as buildings and nurseries). The following landscape indices were used to measure the structural characteristics of the landscape: (1) number of patches, (2) mean patch size, (3) largest patch index, (4) total edge, and (5) mean shape index. The formula for calculating shape index (SI) is:


Where P is the perimeter and A is the area of a patch. The value of SI is equal is round-shaped, and the value increase if the shape of a patch becomes irregular.

In the second step. A first-order Markov model was used to examine the distributional changes of the landscape. The model can be expressed in matrix notation as:

N_t+1 = Pn_t (2)
Where n_t is a column vector whose elements are the fraction of landscape in each of the m land cover types at time t, and P is an m x m matrix whose elements, P_ij, represent the transition rate from one land cover type to each of the m land cover types during the time interval from to t+1. The transition probabilities were derived from the observed transitions occurred during the time interval. Maximum likelihood estimates of the transition probabilities were:


where nij are elements of the (m x m) 'transition matrix' representing the amount of transition from land cover type I to land cover type j. Several assumptions were made. First, the markov model was assumed to be a first-order process: that is, the state at time t+1 depends only on the state at time t and the transition probabilities, thus history previous to time t had no effect. Second, the transition probabilities were assumed to be stationary over time t had no effect. Second, the transition probabilities were assumed to be stationary over time. Then, Equation (2) was used project the subsequent landscape distributions. Under the assumption of stationary transition probabilities, iteration of equation (2) will ultimately culminate in an equilibrium distribution nt* which satisfies n*t+1 = Pn*t = n*t. The equilibrium distribution is the condition that the mangnitude of movements out of one land cover type is exactly equal to the movements into that land cover type. Elements of vector nt* represent the fraction of landscape in each of the m land cover types at the hypothetical equilibrium in the future.

While the Markov model predicts the distributional changes of the landscape in the future, it does not spatially predict where the changes might occur. Therefore, the third step of this study is to develop probabilistic models for predicting landscape changes spatially. To do so, multinomial models will be used to examine the factors contributing to landscape changes. Supposing landscape change from one land cover type to some other land cover type is dependent on Yi, which is a cumulative effect of several factors x1, x2, x3, … Yi can be written as:

Y_i = a_i + b_i1 x₁ + b_i2x₂ + .... + b_in X_n (i = 1..m) (4)
If there are m cover types, then the probability of changing into land cover type I is:

P = e^yj /(e_y1 + e_y2 ... +e_ym) (5)
Once the distributional changes predicted by the Markov model are determined, equation (5) can be used to predict where the changes might occur. The predicted landscape patterns can then be re-assessed by the landscape indices to evaluate the structural changes of the landscape.