Page 42 - IJPS-11-2

International Journal of
Population Studies Satellite data analysis of South Africa population grid

denoted according to their coordinates as: The location and clustering of over and under-

Z-obs = f(x , y ) + μ (I) representative population point are evident for both
i i i i periods.
Given the particular value of the i-th mark (x ) of
i
observation Z-obs with coordinates x and y , that is, term The computed Akaike information criterion (AIC)
i
i
i
components (x , y ), with residual μ . Each point (x , y ) is scores for the first and second order models are displayed
i
i
i
i
i
unique in the sense that only one z value corresponds to it, in Appendix A3. The results for the selected years suggest
and the samples to be analyzed consist of (x, y, z) triplets— that the results of the second-order trend surface are
one triplet for each observation, one for each year of the superior to the first-order results.
study period. Using Equation I, the trend component of Third- and higher-degree polynomial plots can
any location on the map can be determined. represent surfaces with a maximum number of extrema
possible in each case being (n-f)², where n is the polynomial
The slanted flat plane surface (first order) equals z = a +
bx +cy (Equation II) while the parabolic or geodesic dome degree. The regression results of the third order trend
2
2
(second order) is given as z = a + bx + cy + dx + ey + fxy surface model over the period, in logarithmic format, can
be obtained using the Equation IV:
(Equation III), depending on the highest degrees of x and
y (Hota, 2014). In general, first-order effects suggest that z = a + bx + cy + d(x ) + e(y ) + f(yx) + g(x y) + h(xy ) +
2
2
2
2
3
3
the structure of the underlying area influences the location i(x ) + j(y ) (IV)
of the point. When some points do, however, guide the The AIC and residual results for the third-order trend
location of others, second- and higher-order effects are surface compared to the second-order one are illustrated in
possible. Appendix A4. The results for the selected years suggest the
To select the correct equation for the trend, the smallest third-order trend surface is superior to the second-order one.
variance should be determined by minimizing the squared One can test the significance of the trend surface
deviation. When the equations optimize the minimum by examining whether the distribution of “z” values is
square value simultaneously, trend surface expression can random and independent of “x” and “y.” Stated differently,
be derived. the “z” values should not be a function of “x and y.” The
Appendix A1 gives the results of the total population F-test is employed to evaluate the null hypothesis (H0) that
per point’s first-order trend surface, that is, Equation II, the coefficients in the trend surface expression are all zero,
in logarithmic format, which was obtained using the 2000 indicating no trend. Conversely, the alternative hypothesis
and 2020 GPW grid for South Africa. (H1) suggests that at least some of the coefficients in
the trend surface equation are not zero and should be
The first-order trend surface model generates
population predictions for each point based on a smooth accepted. Given the significance of the parameters, it can
be concluded that the trend surface analysis proposes that
surface that can be compared to the actual population a non-random spatial pattern of the population exists. The
for each location. The residuals (actual minus predicted population distribution is non-linear, that is, a higher-
population) for 2000 and 2020 are displayed below order distribution.
(Figure 6A and 6B). Locations, with positive residuals
(under-predictions of the population) are shown in green 3.3. Spatial interpolation
and locations with negative residuals (over-predictions of The Kriging method was applied in this section to examine
the population) in red. The location and clustering of an the spatial interpolation of the GPW for South Africa.
under-representative population point are evident for both Kriging method involves using known population values
periods. The presence of large residuals that tend to be near at certain points to estimate values at unknown points,
to each other, and vice versa, suggests a higher-order trend. building on the results of the trend surface analysis that
To test the hypothesis that improved results could utilizes higher-order surfaces. The method calculates
be achieved, we applied the second-order equation weights for linear and non-linear observed samples based
(Equation III) to the dome surface population per on correlations, which are then used to estimate values for
point in logarithmic format. The results are presented in unobserved regions. By considering the distances between
Appendix A2. Similar as per the first-order trend surface neighboring observations, Kriging calculates a locally
model, locations with positive residuals (under-predictions weighted average to estimate values at new points. Two
of the population) are shown in green and locations with spatial domains are needed, one having values associated
negative residuals (over-predictions of the population) in with the points and one for which estimates are needed.
red (Figure 6C and D). As the average weights of unknown regions are estimated

Volume 11 Issue 2 (2025) 36 https://doi.org/10.36922/ijps.3297