Algorithm

Abstract

This chapter contains the description of the developed algorithm, applying the different aspects considered in the earlier chapters.

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1 Data

The developed algorithm will first be illustrated using the spatial disaggregation problem. The reason is that the problem of regridding can be considered a special case of a spatial disaggregation problem when introducing the segment grid as introduced in Sect. 3.1.2. The problem of data fusion is slightly different, and will only partly be considered.

The algorithm for spatial disaggregation starts with identifying suitable parameters. This is done by exhaustively calculating the value and an associated range for each of the output cells (Sects. 5.1 and 5.2). The range is used to rescale the values, which are assigned to a new grid that has the same layout as the output grid. The similarity of this grid to the output grid can be used to assess how good the parameter is: a good parameter will result in a grid that resembles the output grid. This is only the case for parameters with a positive correlation, however the rescaling operation can be done in such a way that the evaluation will hold also for parameters with a negative correlation. This selection procedure is described in Sect. 5.3. For this similarity, the method developed in Chap. 8. This process is also performed to determine which output parameter is the most suitable: this parameter and the most suitable range for it need to be calculated when the rulebase will be applied.

The best parameter/range combinations for input values will be used in the rulebase construction. The number of parameters is chosen beforehand; in addition a threshold can be imposed on the match value of the parameter to further limit the selection of less suitable parameters. For solving a regridding problem, the grid OUTPUT is replaced by the segment grid as defined in Sect. 3.1.2. For a data-fusion problem, there is no input grid, so only parameter and range definitions that make no use of an input grid can be considered.

2 Constructing the Rulebase

To construct the rulebase, a training set is needed. In the case of a disaggregation or a data fusion problem, this can be available and used. It is however unlikely that there is a training set for regridding, as the segment grid is very specific to the combination of input and output grid. From experiments it was observed that the relative layouts of the grids are enough to find suitable parameters; as such is it possible based on the input, output and proxy data to generate a training set. In Chap. 10, the method for generating trainingsets in the experiments is presented. A trainingset can also be generated for spatial disaggregation or datafusion. The rulebase construction makes use of the developed algorithm for constructing rulebases with variable spaces, as developed in Sect. 6.4. For the algorithm, it is assumed that we have a list of suitable parameters: this is a combination of a definition for a value and an associated range.

With a list of value-definitions, ranges and a training set, the algorithm to construct rules using variable spaces described in Sect. 6.4 can be applied.

3 Applying the Rulebase and Defuzzifying the Results

The application of the rulebase has to take into account the fact that the rulebase uses variable spaces. The modification to the standard rulebase application (Chap. 6) needs to calculate the range when it is needed, which is at the same time as when the value is calculated. This modification is listed in the pseudocode below.

The above pseudocode results in a fuzzy set for every output cell, for the defuzzification of the fuzzy sets the methodology developed in Chap. 7 will be applied:

The constraint of the input cell depends on the problem considered: for a property that is an absolute value, this constraint is that the sum of the defuzzified values has to add up to the value of the input cell. However, for other properties, this constraint can be different (average, weighted average, etc.). The defuzzification results in the final values. When the rulebase approach is used to solve a general regridding problem, this end result is the segment grid. To present it as the original output, the segments have to be combined to form the cells of the original output grid.

4 Complexity

The complexity of the algorithm mainly depends on the operators used to determine the parameters. The construction of the rulebase requires calculating all the parameters for each of the output cells. This is linear in the number of output cells, if the calculation of a parameter (the value or the most possible range) is not dependent on the number of output cells. The only situation in which this occurs is if the most possible range is a global range, but in this case the complexity can be controlled by performing this calculation only once and using this cached value whenever it is needed. As such, the complexity of the construction of the rulebase is linear.

The rulebase is evaluated once for every cell in the output grid. The size of the rulebase is not dependent on the output grid. Consequently, the complexity of its application also depends on the calculation of the parameters. The same remark as for the construction of the rulebase holds, making the evaluation of the rulebase of linear complexity in terms of the number of output cells.

As a last step, the defuzzification is necessary for all output cells, but the complexity of the defuzzification a single cell is not dependent on the size of the output grid. As such, the final defuzzification step is also of linear complexity.