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.
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 |
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
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 |
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.
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