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--- determine_transition_matrix [2015/10/11 23:25]
admin
+++ determine_transition_matrix [2026/08/28 03:17] (current)
hermann Sync from local documentation review
@@ Line 1: / Line 1: @@
 ====== Determine Transition Matrix ======
 ===== Description =====
-This functor determines a matrix of transition rates between two time-series maps.
+Determines a matrix of transition rates between two time-series maps: a single-step transition matrix for the entire period between the initial and final landscape maps, and a multi-step transition matrix for a given number of time steps into which that period is divided.
 ===== Inputs =====
-^ Name              ^ Type                                       ^ Description                                ^
+^ Name  ^ Type  ^ Description  ^
-| Initial Landscape | [[Categorical Map Type]]  | Initial map of land use and cover classes. |
+| Initial Landscape  | [[Categorical Map Type]]  | Initial map of land use and cover classes.  |
-| Final Landscape   | [[Categorical Map Type]]  | Final map of land use and cover classes.   |
+| Final Landscape  | [[Categorical Map Type]]  | Final map of land use and cover classes.  |
-| Time Steps        | [[Positive Int Type]]  | Number of time steps between initial and final landscape maps. Step can be any time unit, such as year, month, etc.  |
+| Time Steps  | [[Positive Integer Value Type]]  | Number of time steps between the initial and final landscape maps. A step can be any unit of time, such as a year or a month, since Dinamica EGO uses it only as an external reference parameter.  |
-===== Output =====
+===== Optional Inputs =====
-^ Name                   ^ Type                                           ^ Description                               ^
+None.
-| Single Step Matrix     | [[Transition Matrix Type]]  | Transition matrix for the entire period.  |
-| Single Step Matrix     | [[Transition Matrix Type]]  | Transition matrix for the time step specified by the number of units that the time period is divided.  |
+===== Outputs =====
+^ Name  ^ Type  ^ Description  ^
+| Single Step Matrix  | [[Transition Matrix Type]]  | Transition matrix for the entire period between the initial and final landscape maps.  |
+| Multi Step Matrix  | [[Transition Matrix Type]]  | Transition matrix for a single time step, derived by dividing the period between the initial and final landscape maps by Time Steps.  |
 ===== Group =====
@@ Line 24: / Line 28: @@
 ===== Notes =====
-To analyze a historical context, the initial map should be considered the older map of the time series.
+Only the classes present in the categorization of the initial or final map are used when determining the transitions that occurred between the two maps; any other values are ignored.
-Multi-step transition matrix only applies to an ergodic matrix, i.e. a matrix that possesses eigenvalues and vectors.
+A cell is ignored in the calculation if its value is null in either the initial or the final map.
-The transition matrix describes a system that changes over discrete time increments, in which the value of any variable in a given time period is the sum of fixed percentages of the value of the variables in the previous time period. The sum of fractions along the column of the transition matrix is equal to one. The diagonal Line of the transition matrix needs not to be filled in since it models the percentage of unchangeable cells. The transition rates are passed on to the model as a fixed parameter within a given phase. For Dinamica, time step can comprise any span of time, since the time unit is only a reference parameter externally set.
+To analyze a historical context, Initial Landscape should be the older map of the time series.
-<m>
+Deriving the multi-step matrix from the single-step matrix requires an eigendecomposition of the single-step matrix, which is only possible when that matrix is ergodic, that is, when it has well-defined eigenvalues and eigenvectors. See http://mathworld.wolfram.com/EigenDecompositionTheorem.html for details on this decomposition and on why it is not always possible.
-delim{[}
-{matrix{5}{1}
-   {
-     vdots
-     j
-   }
-}
-{]}_{t=v}
-=
-delim{[}
-{matrix{5}{5}
-   {
-     P_{11} P_{21} P_{31} cdots P_{i1}
-     P_{12} P_{22} P_{32} cdots P_{i2}
-     P_{13} P_{23} P_{33} cdots P_{i3}
-     vdots vdots vdots ddots vdots
-     P_{1j} P_{2j} P_{3j} cdots P_{ij}
-   }
-}
-{]}
-*
-delim{[}
-{matrix{5}{1}
-   {
-     vdots
-     j
-   }
-}
-{]}_{t=0}
-</m>
-<m>
+The transition matrix describes a system that changes over discrete time increments, in which the value of any variable in a given period is the sum of fixed percentages of the values of the variables in the previous period. The fractions in each column of the transition matrix sum to one; the diagonal need not be filled in, since it represents the percentage of unchanged cells. The transition rates are passed to the model as a fixed parameter within a given phase; the time step itself can span any amount of time, since Dinamica treats the time unit only as an externally set reference parameter.
-sum{i=1}{n}{
-  P_{ij}
-}
-</m>, j = 1, 2 ... n
-An estimation of <m>P_{ij}</m> is given below, where n is the number of states
+===== Internal Name =====
-...
+DetermineTransitionMatrix
-The transition matrix is calculated for a time period. Dinamica can also be run in multiple time steps. It is necessary for this purpose to derive the multiple time step transition matrix, as this is equivalent to the number of time steps in which the time period is divided.
+===== Usage examples =====
-<m>
+See practical examples of this functor in [[lesson_18|Lesson 18: Building a land-use and land-cover change simulation model]]
-P^t=H*V^t*H^{-1}
-</m>
-H and V are Eigen values and Eigen vector matrices.
-===== Internal Name =====
-DetermineTransitionMatrix