Results and discussion

This section tested the dasymetric techniques using a single study. The source spatial units with known population data are the SLAs and the target spatial unit where the data are received are the UCLs. The errors for the spatial disaggregation were evaluated using the actual values from the independent data sources at the target zones (UCLs). They contained the true population data for a smaller spatial unit than the SLAs and can be used to check the performance of the different disaggregation methods. Because the population count for the UCLs are reported for two types of urban areas — urban centres and urban localities — the errors in each of the spatial disaggregation techniques are only identified for the urban area and the error assessment on the non-urban area is not included.

In Figure 4.5 (a—h), the absolute errors of the disaggregated value for each technique are visualised at the target zones. The examination of the visualised error has been an efficient method for analysing the results of the spatial disaggregation techniques (Eicher and Brewer, 2001; Reibel, 2005). A visual presentation of the results provides a comparison against our knowledge of the existing development or patterns of settlement change (Langford et al, 1991). The absolute errors for the eight spatial disaggregation

Land

distinguish the degree of errors between each technique for the same target locations. By comparing the error maps, the binary dasymetric mapping produced the highest error especially in the large urban centre for Brisbane. The 3-class dasymetric method and the locally-fitted regression (3 classes) approach gave a similar distribution of error. The dasymetric methods incorporating the multiple land classes present a lower overall degree of error across the region. This suggests that multiple-class dasymetric methods are more accurate than the simple approaches to solve the spatial heterogeneity issue in the population disaggregation for SEQ. However, owing to the small number of target zones used, the comparative accuracy between the dasymetric methods is not distinguishable by using a simple error visualisation.

(a) Regression (b) 2-class

(c) 3-class (d) 4-class

Figure 4.5 (a)–(d): The visualised absolute errors of eight techniques that

(e) 5-class (f) 6-class

(g) 7-class (h) 8-class

Figure 4.5 (e)–(h): The visualised absolute errors of eight techniques that disaggregate population from SLAs to urban footprint polygons

Figure 4.5: The visualised absolute errors of eight techniques that disaggregate population from SLAs to urban footprint polygons

Table 4.2 summaries the outputs and errors for each technique for two types of urban areas, the urban centre and the urban locality. The italicised figures in the table show the value of the overestimation and the value of the underestimation.

The overall errors of the results are measured by root mean square errors (RMSE):

(

)

1 



 



 −

=

∑

t

t t

RMS Y Y

E t _(4.6)

where ^ERMSis the RMSE and t is the number of target zones (UCLs); Ytis the actual value at target area t; Yˆtis the estimated value at target area t. RMSE is a rigorous measure of overall error which is more sensitive to the outliers in the error distribution (Gregory, 2002).

Table 4.2: Absolute error of spatial disaggregation techniques at Urban Centres and Urban

22,661 2,137,642 211,801 21,60,303 (Total UCL

(The italicised figures represent the value of overestimation or underestimation)

From Table 4.4, we can see that all the techniques gave an overestimate of the population for the urban areas, and an underestimate of the population for the non-urban areas. The binary dasymetric mapping technique was found the least accurate for interpolating the population from the SLAs onto the urban areas. This is evidenced by the high overestimation for both the urban centres (+146,214) and the urban localities (+62,293);

and there is no population left on the non-urban land. The main problem would be the limitation of the data for the test of the binary dasymetric mapping, because target zones are congruent with the control zones. Based upon the principle of binary dasymetric mapping, there was no population in the populated area that could be allocated to the non-urban area. Therefore, from this case, we can see that the application of he binary dasymetric mapping will be constrained when the binary land classifications are defined at the target zones.

Next, the locally-fitted regression model using three density classes presents a better accuracy than the binary dasymetric method in each category. This is acceptable, because the technique takes advantages of the locally-fitting approach and the more refined population density classification. In addition, calibrating the population density at each source zone level was necessary when dealing with large areas with complex population distributions.

The 3-class dasymetric mapping produced lower errors in each density category than the binary dasymetric approach and the regression model. In comparison to the 3-class dasymetric method tested by Langford (2006) using constant density fractions, this study applied a variable density fraction that is more appropriate for resolving the spatial heterogeneity for each land class over space. However, we can see the 3-class dasymetric mapping produced a greater error in the urban centres than in the urban localities. This is possibly caused by a relatively coarse land classification in comparison to a high complexity of the study area. Some errors in the coarse land classes might mistakenly inform the disaggregation process and the potential improvement can be made by a refinement of the land classification.

showed that the technique based on the complex density assumption tends to be more accurate.

Unlike the great improvement made by the 4-class method, the 5-class dasymetric method did not show a significantly better result for the urban centres (+70,247), but returned a higher overestimate of the population for urban localities (+19,989). This indicates a further density classification does not always yield a significantly improved disaggregation accuracy.

When the density division moves into six classes, the dasymetric method presents a notably better estimate for both the urban categories. However, from the six classes, the seven and eight class dasymetric methods showed very limited accuracy improvements for each urban category.

To compare the overall error for each spatial disaggregation solutions, I graphed the RMSE against the total number of density classes incorporated by the method. Figure 4.6 presents a very clear non-linear correlative relationship between the RMSE and the total number of the density classes used for the dasymetric method.

2,000

2-Class 3-Class 4-Class 5-Class 6-Class 7-Class 8-Class

Method

RMSE

Figure 4.6: RMSE and the number of classes for the dasymetric method

The binary dasymetric mapping shows the overall poorer error (6,545 by the RMSE) among all techniques. Comparably, the regression model using variable and locally-fitted

densities presents a better accuracy (5,700 by the RMSE). However, as a simple 3-class regression approach, its accuracy did not seem to be competitive with the 3-class dasymetric mapping (5,079 smaller than 5,699 by RMSE). The 3-class dasymetric mapping produced a lower error than the simpler techniques because, (a) SEQ is more complex than the previous study areas and, (b) the dasymetric method uses a more complex local-fitting approach to account for the spatial heterogeneity in the population density. These results consolidated the previous finding in that a 3-class dasymetric technique is expected to be more accurate than the simpler methods (Fisher and Langford, 1995; Langford, 2006).

Nevertheless, the Figure 4.6 further illustrates that a 3-class dasymetric method was not satisfactory for explaining the complex heterogeneity in our dataset. The 4-class dasymetric method based on a more relaxed density assumption significantly reduced the RMSE from 5,079 to 3,801 (which is about 12% of the average population in the target zones).

However, the RMSE does not continuously present an expected decrease when the density class increases from four to five classes. Following that, the dasymetric models using six and seven classes return slightly better results, but the improvement between the six and seven classes was very limited (3,428 and 3,399 by RMSE). Finally, when the density class is increased to eight classes, the RMSE stop decreasing and the method starts to show a poorer accuracy (3,533 greater than 3,399 by RMSE). Although the 8-class gives a good overall estimation for the urban centres and the localities, it does not show a good internal variation of the data in the target areas. A possible reason for this is that the regression model for the 8-class densities involves a higher standard error in the density estimation for highest density areas (class eight) that is derived from a ‘natural breaks’ classification.

In this study, we can find a relationship between the model complexity and the error of the disaggregation that has not been revealed in the previous research. Figure 4.6 indicates a trend that the error of the spatial disaggregation decreases when the density class increases. However, such a relationship tends to be unstable and insignificant as excessive numbers of land classes are imposed. The traditional 3-class dasymetric method can be further improved by a more refined density division. But, a large number of land classes is

be made by modelling spatial heterogeneity, for instance, using smaller source zones.

However, the capacity for improvement that can be made by the more refined density classes is very limited for going beyond the existing 4-class configuration.

Overall, the findings from this research are considered very topical and more conclusive than the previous study in the spatial disaggregation method (see Langford, 2006). Even though an independent ancillary dataset (land use data) was not employed to disaggregate the population data, the simulated land classes using the natural breaks classification of the data at the CCD level is considered as efficient to test the performance of the dasymetric method with a different total number of land classes.

The goal of these tests is to select one model from a set of competing models that best capture of spatial structure of the study area. In this study, the idea of choosing an appropriate technique uses the concept of standard model selection methods. These methods penalize goodness of fit of the models by incorporating the model complexity, for example, Akaike information criterion (AIC) (Akaike, 1973), cross-validation criterion (CV) (Browne, 2000), and the Bayes information criterion (BIC) (Albert and Chib, 1997).

Doing so to ensure that the method selection is not solely based on the fit to the data but also taking into account of the model complexity. Model selection solely based on the fit to a particular set of data will result in the choice of an unnecessarily complex method that overfits the data and thus generalized poorly to other data (for example, the future data or other study areas) (Myung, 2000).

The number of classes is the only dimension of complexity that these spatial disaggregation methods consider. Thus, the method selection is carried out by trading off the accuracy (RMSE) against complexity. Based on the assessment of the results, the 4-class dasymetric method is the most effective technique to model the spatial structure of the study area with a relatively low complexity. More complex models with more than 4 density classes, having a large value in the complexity term, are not selected because their contribution to the model fit are not good enough to justify the extra complexity.

Although choosing the 4-class method appears to be subjective and a statistical model should be used to determine a reasonable number of classes from the data, the idea of method selection in this research is similar to the concept of standard model selection methods (e.g. AIC). Based on its good accuracy and generalizability, I nominate the 4-class

dasymetric method as the most effective technique to spatially disaggregate the population forecasts for SEQ.