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# 4. Examples

Three examples with generated data sets demonstrate the use of the algorithm. The first two examples deal with one-dimensional Gaussian distributions. The third example handles two-dimensional Gaussian distributions as overlapping main data set and several blocks as underlying substructures. All data sets are adapted with a Poisson-distributed error.

In order to be able to present the principle of the algorithm only examples using artificial data sets will be presented. The detection of structures in color-color diagrams by means of the algorithm will be presented in a further paper.

## 4.1. A small subcomponent

Two Gaussian distributions build the data set I. Table 1 (click here) gives the parameters. The last column shows the intensity of each component (). The intensity of I equals to 4693 (, the adapted error does not change the sum of the intensities).

 h Int a1 90 15 120 4512 a2 99 4 18 181
Table 1: The parameters of the building components of I

As an estimation of the superimposing main structure, a Gaussian distribution f0 is generated with the same values as a1. The intensity of f0 is set to 1. Figure 1 (click here) shows the sets I, a1 and a2. The algorithm has as input values the sets I and f0. Set I has to be smoothed in contrast to the generated set f0.

The algorithm runs 4 times and calculates as intensity of the main structure (4% less than the real intensity of a1). The intensity of the remaining residuals R4 reaches 361.

The result concerning the whole data set, F4 + R4, demonstrates that the algorithm does not change the intensity during the calculation: the algorithm is intensity-invariant.

Figure 1: The main data set I together with the building components a1 and a2 (broken lines)

The big difference between the real and the estimated substructure () arises due to the fact that the intensity of the main component is 25 times(!) the intensity of the substructure. If the algorithm reaches the correct intensity of the main component up to 1% and due to the invariance of the algorithm, the difference has to be exactly 25% between the original and the calculated substructures.

After the iteration, a probability function p is calculated in order to be able to distinguish between an error and the real substructure. Actually, this probability function only takes into account a ratio between an estimated maximal error (), the real data set I and the found substructure. Figure 2 (click here) displays the found substructure R4 and the function p (which is multiplied with the factor 10 in order to be able to compare it with R4). p has its highest values in the regions of the real substructure ai and at the edges of the figure.

Figure 2: The residual R4 together with the probability function p

## 4.2. Two clusters of similar size

The second example deals with the problem often found in astronomy: Two clustars with about the same size and a non empty intersection. Again two gaussian distributions were used.

 h Int a1 80 20 40 2005 a2 120 20 40 2005
Table 2: The parameters of the building components of I

Figure 3: Top: The data set I using two clusters of similar size together with its building components a1 and a2 (broken lines). The left component a1 is used as "main structure". Bottom: The residual R4 using two clusters of similar size (solid line), the building function a2 (thick broken line) and the propability function (thin broken line)

The residuum represents extremely well the building component a2. One should remember, that the algorithm does not know anything about the form of this function a2.

## 4.3. The covered E

The second example illustrates further capabilities of the algorithm. On the one hand, different forms build the set. The superimposing main structure is built with two Gaussian distributions, whereas the covered structures are built with several blocks. On the other hand this example demonstrates the use of the algorithm in the two dimensional case.

A so-called block is a kind of a two dimensional trapecium with the following parameters:

Figure 4: The defining parameters a, b () and h (in one dimension)

The original set is shown in Fig. 5 (click here). The intensity-ratio between the main component and the subcomponent is about 90 : 1. The substructure has a constant height of 8 units (cf. Fig. 6 (click here)), the maximum height of the real main component is at 409.6 units and consists of two Gaussian distributions, each of them having a height of 240.0 units. The total intensity of the main component has 3114430 units.

Figure 5: The real structure I (the main component together with the substructure and the Poisson noise). A visual inspection does not show any sign of the hidden E

After 2 iterations, the algorithm achieves the following result: the found height of the main component has 410.4 units (), the corresponding intensity has 3135670 units (). The intensity of the achieved residuals is at 23464.5 units (cf. Figs. 7 (click here) and 8 (click here)), while the intensity of the real substructures is at 34272 units.

Figure 6: The substructure built with 4 blocks

Figure 7: The estimated residuals including the remaining noise

Figure 8: The contour-plot of the estimated residuals. The substructure is very well recognizable

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