====== Determine Transition Matrix ====== 

===== Description =====

This functor determines a matrix of transition rates between two time-series maps.

===== Inputs =====

^ Name              ^ Type                                       ^ Description                                ^
| 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.   |
| Time Steps        | [[Positive Integer Value Type]]  | Number of time steps between initial and final landscape maps. Step can be any time unit, such as year, month, etc.  |

===== Output =====

^ Name                   ^ Type                                           ^ Description                               ^
| 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.  |

===== Group =====

[[Functor List#Calibration | Calibration]]

===== Notes =====

To analyze a historical context, the initial map should be considered the older map of the time series.

Multi-step transition matrix only applies to an ergodic matrix, i.e. a matrix that possesses eigenvalues and vectors.

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.

<m>
delim{[}
{matrix{5}{1}
   {
     1
     2
     3
     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}
   {
     1
     2
     3
     vdots
     j
   }
}
{]}_{t=0}
</m>

<m>
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

...

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.

<m>
P^t=H*V^t*H^{-1}
</m>

H and V are Eigen values and Eigen vector matrices.

===== Internal Name =====

DetermineTransitionMatrix