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## Estimating forces from the LSOA survey

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:

Since:

, 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 `dneqnf` subroutine. As T is about 2 years, we can either use single age or double age (x, x+2) estimates.

Nicolas Brouard
Tue Jun 6 00:15:46 DF 1995