At each age, tables such as table 1, provide 4 independent ratios, , i,j=1,2. If we assume that the 4 unknown forces are constant during , we may estimate these forces. T is the interval between two rounds, i.e. about 2 years.
Matrix from equation 1 can be diagonalized; the eigenvalues are the roots of equation:
, this equation leads to 2 distinct real and negative roots, and , r>s, where:
Therefore, , or the probability for an individual in state j at time 0 to be in state k at time t, can be written:
Constants are determined by initial conditions at time 0:
and by conditions which derivates at time 0 must satisfy:
This leads to:
Since , the expression can also be written:
If corresponds to the () matrix with element , it can be written:
As is known from the LSOA survey for each single age, this leads to a system of four equations with four unknowns, , , and .
For the numeric calculations we used IMSL
As T is about 2 years, we can either use single age or double age
(x, x+2) estimates.