The smaller the target region is, the harder we gain time series data for the systems analysis (for example, econometric approaches). Therefore, methods using few sort of time series data are needed for regional planning, especially forecasting.
The purpose of this paper is to give a simple estimate of the transition probability matrix of a finite Markov process and to use this matrix for a regional analysis. Consequently, the procedure of estimating this matrix is a generalized least squares method with linear constraints. We make the following brief statement. Where A is an n-dimensional matrix of which each element is a transition probability air (from i-th region to r-th region), xt is a column vector of n elements xit which are the share of i-th region, et is a column vector of n errors eit of t-th period.
We will obtain the least squares estimate of this matrix A, when the following quadratic form is a minimum. uhk and λ are Lagrange multipliers which satisfy the following constraints. Where l is a column vector of 1 and D is an appropriate symmetric and positive definite matrix. And we finally get which is the same form as that of Theil-Rey [8].
We used this method for a forecast of office demand in Tokyo's 23 wards. We divided Tokyo's 23 wards into four districts, which are the Central district (3 wards), the sub central district (3 wards), the surrounding district (8 wards) and others (9 wards).
