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Parameters

• $a$ = 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 and Sex
• $i$ = grid cell id
• $r$ = region id, i.e. LAU-regios
• $P_i$ = Population per 1km gridcell i, based on ARDECO dataset.
• $E_{is}$ = Population per 1 km gridcell i, per sex, based on ESTAT dataset
• $E_{ib}$ = Population per 1 km gridcell i, per BroadAgeGroup, based on ESTAT dataset
• $Q_{asr}$ = Population per LAU region, split per SexAgeGroup
• $A_i$ = balancing factor, total population in cell i
• $B_{asr}$ = balancing factor SexAgeGroup in region r
• $X_{asi}$ = resulting population per SexAgeGroup in cell i
• $I_i^r$= the amount of population in gridcell $i$ that is related to region $r$

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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 $b$:

$$\forall a \sum_b I_a^b == 1$$

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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:

$$B_{asr} := \frac{Q_{asr}}{\sum_i[E_{is} \cdot I_i^b \cdot E_{ib} \cdot A_i \cdot I_i^r \cdot I_i^r]]}$$