$a$ = AgeGroups = 5yr groups
$b$ = BroadAgeGroups = 3 age groups, i.e. LT15, 15-64, GE65
$s$ = Sex, i.e. female or male
$sa$ = SexAgeGroup = combination of AgeGroups en Sex
$i$ = grid cell id
$r$ = regio id, i.e. LAU-regios
$P_i$ = Populatie per 1km gridcell i, obv ARDECO dataset.
$E_{is}$ = Populatie per 1 km gridcell i, per sex, obv ESTAT dataset
$E_{ib}$ = Populatie per 1 km gridcell i, per BroadAgeGroup, obv ESTAT dataset
$Q_{asr}$ = Populatie per LAU regio, uitsplitst per SexAgeGroup
$A_i$ = balancing factor, totale populatie in cell i
$B_{asr}$ = balancing factor SexAgeGroup in region r
$X_{asi}$ = resulterende populatie per SexAgeGroup in cell i
$I_i^r$= the amount of population in gridcell $i$ that is related to region $r$
The population of each gridcell $i$ relates to one region:
$$\forall i \sum_r I_i^r == 1$$
incidence of age group $a$ into BroadAgeGroup $b$:
$$\forall a \sum_b I_a^b == 1$$
Target population per $as$ in cell $i$;
$$x_{asi} := E_{is} \cdot I_a^b \cdot E_{ib} \cdot A_i \cdot I_i^r \cdot B_{asr}$$
Balancing factor $A_i$ halen we uit:
$$\sum_{asr}[x_{asi}] = P_i \quad \forall i$$
From which follows:
$$A_i := \frac{P_i}{\sum_{as}[E_{is} \cdot I_a^b \cdot E_{ib} \cdot I_i^r \cdot B_{asr}]}$$
Balancing factor $B_{asr}$ halen we uit:
$$\sum_i [x_{asi} \cdot I_i^r] = Q_{asr} \quad \forall a,s,r$$
From which follows:
$$B_{asr} := \frac{Q_{asr}}{\sum_i[E_{is} \cdot I_i^b \cdot E_{ib} \cdot A_i \cdot I_i^r \cdot I_i^r]]}$$
