means, that
parameter a belongs to the set of integer numbers {ao, ao+1,
... , A}.Quality Assessment of Demographic Data in the WHO MONICA Project
In the WHO MONICA Project, annual data on population size were collected for a period of about ten years. The age range covered was 25-64 years or optionally 25-74 years, and the data were reported by 5-year age group. We describe here the method which was used for checking the consistency of the population dynamics as part of the data quality assessment in the MONICA Data Centre (MDC).
The approach is to estimate the migration pattern from the reported data on population size. If the migration pattern shows year specific or isolated jumps, very large annual migration, or other abnormalities, the data are considered to be inconsistent, and concern is aroused as to the quality of the data. Here we will use the term "migration" for the sum of all changes in the population size, including conventional migration and mortality. (Birth is not included because it is irrelevant in the age groups considered in MONICA.)
To facilitate the estimation of migration, we will need a model and a number of assumptions. The scheme for estimating migration is based on four elements:
These elements together with the reported data enable the estimation of one year population sizes and migration. The next three sections of this appendix give a mathematical description of the model and the three assumptions. The last two sections set up the optimisation problem, the solution of which yields the required estimate of the migration pattern, and introduces some computational aspects. The details of the optimisation algorithm and the computational scheme are beyond the scope of this appendix.
Matrices and vectors are denoted by bold-faced letters such as B and y, and scalars by unbold lower case letters.
Specific notation from set theory and theory of optimisation will be incorporated in
the text. We will use one- and two-dimensional scalar functions defined on integer-valued
domains. Notation [.,.] indicates a contiguous range of a function argument or index of
summation: the first dot indicates the position of the value or variable for the lower
limit of the range, and the second one the upper limit. A letter with subscript 0 denotes
the lower limit of a parameter range, while a capital letter denotes its upper
limit, unless otherwise specified. For example, means, that
parameter a belongs to the set of integer numbers {ao, ao+1,
... , A}.
Symbol 
means "is defined as". For example, expression 
means that vector po is defined as a set of the specified
values of function p.
In some cases where capital letters are used to denote the value of a parameter, the corresponding lower case letter denotes its logarithm. Such cases will be specified in the text.
A certain amount of linear algebra is assumed to be known by the reader.
Our model for the population dynamics, using integer valued calendar time and age, specifies the yearly changes in the population size of one year specific cohort groups:
 |
(1) |
where
By introducing
we may rewrite (1) in form:
Taking the logarithm of both sides of the equation, we get
 |
(2) |
where p = log(P) and c = log(C).
Here c(t,a) is actually the rate of change in the population size. For example c(t,a)=0.05 corresponds approximately to a 5% increase in the following year in the size of the population (t,a).
For the younger age groups the change reflects predominantly migration, and the contribution of mortality to the change increases with age. In the following analysis we call c the "migration rate" or just "migration", regardless of the fact that it also reflects mortality.
For simplicity, we set t=0 for the first calendar year and count the later years from the first one. We consider the "squared area”:
 |
(3) |
and assume that c(t,a) is defined everywhere in the area (3), except the last
year T. Furthermore, we assume that p(0,a) is given for all 
(initial condition), and p(t,ao) is given for all 
(boundary condition). Using (2), we can then recursively express p(t,a) in area
(3) for each t>0 and a>ao. For notational
convenience, we extend the domains of c and p beyond the area (3):
 |
(4) |
Based on (2) and (4) we can now express the logarithm of the population size p(t, a) in area (3) for calendar years t>0 using the population size of the first year and the annual migration rates:
 |
(5) |
We will base further consideration in this formulation of the model of demographic dynamics . It defines the logarithm of the size of any one year birth cohort in any calendar year as a linear function of the migration rates c and the logarithm of its population size in the first year.
Our model (5) specifies the population size in one year age groups, but the MONICA data
were reported by five year age groups, which cover the age range [ao,A]
for every calendar year . Let j be an index for the five year age
groups: j=1 denotes the youngest age group (i.e. 25-29) and J the oldest age
group (i.e. 60-64 or 70-74). By a(j) we denote the first year of age in age group
j (e.g. a(1)=25). By R(t,j) we denote the reported population
size of the five-year age group j in year t.
As model (5) deals with the logarithms of the population size, at first sight it seems that we should use r(t,j), defined as the logarithm of R(t,j). However, as model (5) uses the logarithms of the population sizes of the one year age groups, r(t,j) should be analogous to the sums of the logarithms of the five one year age group sizes. This is not the case with the logarithm of R(t,j), but using a simple approximation we may define:
 |
(6) |
From now on we refer to r(t,j) as the "reported data".
For the given reported data r(t,j), our aim is to find a vector
and a matrix
such that the the degree of variability in their elements is feasible and, when substituted in (5), they provide a satisfactory fit with the reported data. In order that we can set exact criteria for these conditions, we construct three indicators:
We construct indicator S1 as the mean squared relative deviation of the 1-year age group sizes in the first calendar year:
 |
(7) |
where aF = ao-T, and
Thus µ(a) is the moving average of the logarithms of the five one-year age group sizes.
We construct indicator S2 as the mean squared relative discrepancy between the population size estimates from the model and the reported data:
 |
(8) |
where 
and 
Before constructing indicator S3 we simplify our model by assuming that at any given year, matrix c is constant within the reported 5-year age group. Indicator S3 is then defined as:
 |
(9) |
This indicator shows how far matrix c is from the matrix with all elements equal to zero, which corresponds to the case of no migration.
For the further analysis it is convenient to present all indicators as functions in a vector space, using vectors:
x = po,
u = (c(0,a(1)),...,c(0,a(J)),...,c(T-1,a(1)),...,c(T-1,a(J)))T,
z = (xT,uT)T,
y2 =( r(0,1),...,r(0,J),...,r(T,1),...,r(T,J))T.
Let k=dim(x), m=dim(u), n=k+m=dim(z).
We can define matrices B1x, B2x, B2u and B3u such that
| S1(z) = S1(x,
u) = ||B1xx||2; S2(z) = S2(x, u) = ||B2xx+B2uu-y2||2; S3(z) = S3(x, u) = ||B3uu||2 = ||Iu||2, where I is the m×m identity matrix. |
(10) |
where ||.|| denotes the Euclidean norm and I is the m×m identity matrix. One can identify the elements of matrices B1x, B2x, B2u and B3u by inspecting equations (7), (8) and (9). Note that all indicators S1, S2 and S3 are quadratic forms in the finite dimensional vector space En = Ek×Em with elements z .
Our optimisation problem, which we call "Problem P1", is to:
| minimise |  |
| subject to |  |
where 
and 
are positive constants.
An analysis of Problem P1 has shown that there exists a unique solution for
practically any values of constants and , except when they
both are too small. After computational experimentation, the constants were fixed to =
0.0025 and = 0.01.
The theory of optimisation provides an iterative algorithm for obtaining a solution. At each step an auxiliary optimisation problem is solved.This is minimising Lagrangian function for the Problem P1:
with non-negative parameters 
and 
. Such an
optimisation can be performed iteratively as the by-product of the weighted least squares
estimation, where the matrix of observations is composed of matrices B1x,
B2x, B2u and B3u
and vector y2, and the weights are determined by
and of the previous round of iteration.
The computation in the MDC was done using SAS. The program prepared for this includes tools for controlling the convergence and a more advanced method for approximating the logarithmic analogue of the reported data than that described in (6).