| Type: | Package |
| Title: | SOM Algorithm for the Analysis of Multivariate Environmental Data |
| Version: | 1.1.2 |
| Maintainer: | Sabina Licen <slicen@units.it> |
| Description: | Analysis of multivariate environmental high frequency data by Self-Organizing Map and k-means clustering algorithms. By means of the graphical user interface it provides a comfortable way to elaborate by self-organizing map algorithm rather big datasets (txt files up to 100 MB ) obtained by environmental high-frequency monitoring by sensors/instruments. The functions present in the package are based on 'kohonen' and 'openair' packages implemented by functions embedding Vesanto et al. (2001) http://www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf heuristic rules for map initialization parameters, k-means clustering algorithm and map features visualization. Cluster profiles visualization as well as graphs dedicated to the visualization of time-dependent variables Licen et al. (2020) <doi:10.4209/aaqr.2019.08.0414> are provided. |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| Depends: | R (≥ 3.6.0) |
| RoxygenNote: | 7.1.1 |
| Imports: | rlist, kohonen, shiny, dplyr, plyr, openair, colourpicker, shinycssloaders, shinycustomloader |
| URL: | https://github.com/SomEnv/somenv |
| BugReports: | https://github.com/SomEnv/somenv/issues |
| Author: | Sabina Licen [aut, cre], Marco Franzon [aut], Tommaso Rodani [aut], Pierluigi Barbieri [aut] |
| NeedsCompilation: | no |
| Packaged: | 2021-07-26 09:43:08 UTC; rstudio-user |
| Repository: | CRAN |
| Date/Publication: | 2021-07-26 13:30:02 UTC |
BMUs of the cluster centroids
Description
The function finds the Best Matching Units of the cluster centroids
Usage
BmusCentr(centroids, som_model, k)
Arguments
centroids |
Centroids array (output of kmeans_clustersR function) |
som_model |
An object of class kohonen |
k |
Number of clusters |
Value
An array containing the BMU for each centroid
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Cluster assignment for the experimental data
Description
Generate a vector containing the cluster assignment to experimental data
Usage
BmusClus(Bmus, Cluster)
Arguments
Bmus |
Best Matching Unit assignment to the experimental data |
Cluster |
Vector containing cluster number assignment for prototypes |
Value
A vector containing the cluster assignment to experimental data
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Boxplot of prototype variables split by cluster and variable
Description
Boxplot function is used, box whiskers are omitted
Usage
BoxClus(Dms, codebook, Cluster, Centroids)
Arguments
Dms |
A vector of length 2, where the first argument specifies the number of rows and the second the number of columns of plots (see mfrow in par) |
codebook |
De-normalized prototype codebook |
Cluster |
Vector containing cluster number assignment for prototypes |
Centroids |
Centroids matrix |
Value
Boxplot of prototype variables split by cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot, par
Boxplot of prototype variables split by cluster
Description
Boxplot function is used, box whiskers are omitted
Usage
BoxUnits(codebook, Cluster, Centroids, Ylim = NA, pitch = NA, xdim = 0.75)
Arguments
codebook |
Prototype codebook normalized by variable |
Cluster |
Vector containing cluster number assignment for prototypes |
Centroids |
Centroids matrix |
Ylim |
Vector of length 2 for y-axis limits |
pitch |
Vector containing the position of horizontal grid lines |
xdim |
x axes label dimensions |
Value
Boxplot of prototype variables split by cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot
Custom color sequence for clusters
Description
Generate the sequence of colors to plot the SOM map according to clusters
Usage
ClusCol(Centroids, Cluster, colSeq = rainbow(nrow(data.frame(Centroids))))
Arguments
Centroids |
Centroids matrix |
Cluster |
Vector containing cluster number assignment for prototypes |
colSeq |
Color sequence for the clusters |
Value
A vector of colors with length equal to Cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Prototype coordinates for graph
Description
Generate X and Y coordinates for plotting a SOM map shaped according to Vesanto visualization fashion
Usage
CodeCoord(Row, Col)
Arguments
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
Value
This function returns a data.frame including columns:
X
Y
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf; Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Examples
Coord<-CodeCoord(10,5)
Plot of daily percentages for each cluster
Description
The function produces a plot representing the the daily percentage for each cluster
Usage
DailyBar(
experimental,
TrainClus,
Centroids,
colSeq = rainbow(nrow(data.frame(Centroids))),
Total = 1440,
xdim = 0.7,
ydim = 0.8
)
Arguments
experimental |
Experimental data (must contain variable "date") |
TrainClus |
Vector containing cluster number assignment for experimental data |
Centroids |
Centroids matrix |
colSeq |
Color sequence for the clusters |
Total |
Number of observations per day |
xdim |
x axes label dimensions |
ydim |
y axes label dimensions |
Value
Plot of daily percentages for each cluster, the latter element in the legend represents percentage of not determined data
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Percentage frequency for each cluster
Description
Percentage frequency for each cluster
Usage
Freq(Cluster, Centroids)
Arguments
Cluster |
Vector containing cluster number assignment for experimental data |
Centroids |
Centroids matrix |
Value
A data frame containing the percentage frequency of each cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Daily percentage frequency for each cluster
Description
Daily percentage frequency for each cluster
Usage
FreqD(Date, Cluster, Centroids, Total = 1440)
Arguments
Date |
Vector containing date/time variable for experimental data |
Cluster |
Vector containing cluster number assignment for experimental data |
Centroids |
Centroids matrix |
Total |
Number of observations per day |
Value
A data frame containing the daily percentage frequency of each cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Monthly percentage frequency for each cluster
Description
Monthly percentage frequency for each cluster
Usage
FreqM(Date, Cluster, Centroids)
Arguments
Date |
Vector containing date/time variable for experimental data |
Cluster |
Vector containing cluster number assignment for experimental data |
Centroids |
Centroids matrix |
Value
A data frame containing the monthly percentage frequency of each cluster
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Function to draw an hexagon around a point
Description
Draws an hexagon around a point of x and y coordinates
Usage
Hexa(x, y, color = NA, border = "gray", unitcell = 1)
Arguments
x |
X-coordinate of the hexagon center |
y |
Y-coordinate of the hexagon center |
color |
Filling color of the hexagon (default NA) |
border |
Border color of the hexagon (default "gray") |
unitcell |
The distance side to side between two parallel sides of the hexagon (default 1) |
Value
This function draws an hexagon on a plot
Author(s)
Sabina Licen
Hits distribution on the SOM map
Description
Plot a SOM map with filled hexagons according to the number of hits
Usage
HexaHits(hits, Coord, Row, Col, color = "black")
Arguments
hits |
Vector with number of hits for each prototype |
Coord |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
color |
color filling of the hexagons |
Value
Plot a SOM map with filled hexagons according to the number of hits
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Hits distribution on the SOM map
Description
Plot a SOM map with hits plotted as grayscale according to quartiles
Usage
HexaHitsQuant(hits, Coord, Row, Col)
Arguments
hits |
Vector with number of hits for each prototype |
Coord |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
Details
The function plots a SOM map with hits represented as grayscale according to quartiles, from white (lower outliers) followed by grayscale (quartiles) and black (upper outiliers). The prototype with the maximum number of hits is represented by a red hexagon. The outilers and quartiles are evaluated by boxplot function applying default parameters.
Value
Plot a SOM map with hits represented as grayscale according to quartiles
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot
Relative quantization error distribution on the SOM map
Description
Plot a SOM map with relative quantization error plotted as grayscale according to quartiles
Usage
HexaQerrs(bmus, qerrs, Coord, Row, Col, color = "black")
Arguments
bmus |
Vector with Best Matching Unit for each experimental sample |
qerrs |
Vector with quantization error for each experimental sample |
Coord |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
color |
color filling of the hexagonsType a message |
Details
The function evaluate the relative quantization error for each prototype dividing the sum of quantization errors for experimental samples represented by the single prototype by the number of hits of the same prototype, then plots a SOM map with filled hexagons according to the relative quantization error
Value
Plot a SOM map with filled hexagons according to the relative quantization error
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Realtive quantization error distribution on the SOM map
Description
Plot a SOM map with realtive quantization error plotted as grayscale according to quartiles
Usage
HexaQerrsQuant(bmus, qerrs, Coord, Row, Col)
Arguments
bmus |
Vector with Best Matching Unit for each experimental sample |
qerrs |
Vector with quantization error for each experimental sample |
Coord |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
Details
The function evaluate the relative quantization error for each prototype dividing the sum of quantization errors for experimental sample represented by the single prototype by the number of hits of the same prototype, then plots a SOM map with the realtive quantization error represented as grayscale according to quartiles, from white (lower outliers) followed by grayscale (quartiles) and black (upper outiliers). The outilers and quartiles are evaluated by boxplot function applying default parameters.
Value
Plot a SOM map with realtive quantization error represented as grayscale according to quartiles
Author(s)
S. Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot
Function to draw an hexagonal SOM map
Description
Draws an hexagonal SOM map using x, y coordinates for the hexagon centers
Usage
Hexagons(Coords, Row, Col, color = NA, border = "gray", unitcell = 1)
Arguments
Coords |
matrix containing the x and y coordinates of the hexagon centers |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
color |
Filling color of the hexagons (default NA) |
border |
Border color of the hexagons (default "gray") |
unitcell |
The distance side to side between two parallel sides of the hexagon (default 1) |
Value
A hexagonal SOM map
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Examples
Coord<-CodeCoord(10,5)
Hexagons(Coord,10,5)
SOM map with clusters
Description
Generates a SOM map colored according to cluster splitting
Usage
HexagonsClus(
Centroids,
Cluster,
BCentr,
Coord,
Row,
Col,
colSeq = rainbow(nrow(Centroids))
)
Arguments
Centroids |
Centroids matrix |
Cluster |
Vector containing cluster number assignment for prototypes |
BCentr |
Best Matching Unit of the cluster centroids |
Coord |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
colSeq |
Color sequence for the clusters |
Value
A SOM map colored according to cluster splitting
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
Heatmaps
Description
Multiple plots that show the distribution of the modeled variables on the SOM map
Usage
HexagonsVar(Dms, codebook, Coords, Row, Col)
Arguments
Dms |
A vector of length 2, where the first argument specifies the number of rows and the second the number of columns of plots (see mfrow in par) |
codebook |
SOM codebook |
Coords |
Prototype coordinates for plotting the map |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
Details
The function plots a SOM map for the values of each modeled variable using a grayscale according to quartiles, from white (lower outliers), followed by grayscale (quartiles) and black (upper outiliers). The outilers and quartiles are evaluated by boxplot function applying default parameters.
Value
SOM map plots for the values of each modeled variable using a grayscale according to quartiles
Author(s)
Sabina Licen
References
Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot, par
Custom number sequence for clusters
Description
Changes the input vector according the custom number sequence for clusters
Usage
NClusChange(Vector, Centroids, NCh)
Arguments
Vector |
Vector containing cluster number assignment for prototypes or experimental data |
Centroids |
Centroids matrix |
NCh |
Vector with custom sequence of numbers for clusters |
Value
A vector of same length as input vector with cluster numbers changed according to custom input
Author(s)
Sabina Licen
Topographical error for the SOM map
Description
Calculate topographical error for the SOM map
Usage
SOMtopol(dataset, codebook, grid)
Arguments
dataset |
Experimental data used for training the map |
codebook |
SOM codebook |
grid |
SOM grid expressed as a matrix of x and y coordinates of the map units |
Value
This function returns the topographical error
Author(s)
Sabina Licen
References
Clark, S., Sisson, S.A., Sharma, A. (2020) Adv. Water Resour. 143, art. no. 103676 DOI: 10.1016/j.advwatres.2020.103676
The function starts the SOMEnv GUI
Description
The function starts the SOMEnv GUI
Usage
SomEnvGUI()
Value
This function starts the graphical user interface with the default system browser. The main help suggestion for using the tool are embedded in the GUI
Author(s)
Sabina Licen, Marco Franzon, Tommaso Rodani
References
Winston Chang, Joe Cheng, JJ Allaire, Yihui Xie and Jonathan McPherson (2019). shiny: Web Application Framework for R. R package version 1.4.0. https://CRAN.R-project.org/package=shiny
seealso shiny
Examples
## Not run:
SomEnvGUI()
## End(Not run)
U-matrix plot
Description
Plot of Unified Distance Matrix using a colored scale according to quartiles
Usage
UmatGraph(umat, Row, Col, colorscale = c("bw", "gs"))
Arguments
umat |
Unified Distance Matrix |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
colorscale |
Either "bw" for grayscale or "gs" for green-white scale |
Details
The function plots a U-matrix map for the values of each modeled variable using a grayscale according to quartiles, from darker color (lower distances) to lighter color (higher distances). The quartiles are evaluated by boxplot function applying default parameters.
Value
Plot of Unified Distance Matrix using a grayscale or (green-white scale) according to quartiles
Author(s)
Sabina Licen
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf; Licen, S., Cozzutto, S., Barbieri, P. (2020) Aerosol Air Qual. Res., 20 (4), pp. 800-809. DOI: 10.4209/aaqr.2019.08.0414
See Also
boxplot, som_umatR
Evaluate Davis-Bouldin index for the cluster split of data input
Description
The function has been coded in R code starting from db_index.m script present in somtoolbox for Matlab by Vesanto and adapted for the use in the shiny app
Usage
db_indexR(codebook, k_best, c_best)
Arguments
codebook |
SOM codebook |
k_best |
Vector with cluster number assignment for each sample |
c_best |
Matrix with cluster centroids |
Value
The mean DB-index for the clustering
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
See Also
som_mdistR, kmeans_clustersRProg
K-means algorithm applied for different values of clusters
Description
The som_kmeansR function with 100 epochs training is run for a custom number of times for each k value of clusters and the best of these is selected based on sum of squared errors (err). The Davies-Bouldin index is calculated for each k-clustering. The function has been coded in R code starting from kmeans_clusters.m script present in somtoolbox for Matlab by Vesanto and adapted to show a progress bar when working embedded in the shiny app.
Usage
kmeans_clustersRProg(codebook, k = 5, times = 5, seed = NULL)
Arguments
codebook |
SOM codebook |
k |
Maximum number of clusters (the function will be run from 2 to k clusters) |
times |
Number of times the som_kmeansR function is iterated |
seed |
Number for set.seed function |
Value
This function returns a list containing the cluster number assignment for each sample, the cluster centroids, the total quantization error, the DB-index for each number of clusters, and the random seed number used
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
See Also
som_mdistR, som_kmeansRProg, db_indexR
Basic statistics of values present in the input vector
Description
Generate basic statistics for the input vector
Usage
paramQuant(param)
Arguments
param |
Numeric vector |
Details
The outilers and quartiles are evaluated by boxplot function applying default parameters.
Value
A table which contains basic statistics for the input vector
Author(s)
Sabina Licen
See Also
boxplot
Examples
library(datasets)
paramQuant(iris[,1])
Calculate map dimensions
Description
Generate SOM map dimensions according to Vesanto heuristic rules based on the first two eigenvalues of the experimental data and their related eigenvectors The function has been coded in R code starting from som_dim.m script present in somtoolbox for Matlab by Vesanto and adapted for the use in the shiny app
Usage
som_dimR(dataset, type = "regular")
Arguments
dataset |
Experimental data |
type |
Either "regular", "small" or "big" map (default ="regular") |
Value
This function returns a list containing the number of rows, columns and overall map units
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
See Also
eigen, cor
Examples
library(datasets)
som_dimR(iris[,1:4], type="small")
Calculate initialization matrix for SOM training
Description
Generate SOM map initialization matrix according to Vesanto heuristic rules related to map dimensions, the first two eigenvalues of the experimental data and their related eigenvectors The function has been coded in R code starting from som_init.m script present in somtoolbox for Matlab by Vesanto and adapted for the use in the shiny app
Usage
som_initR(dataset, Row, Col, munits)
Arguments
dataset |
Experimental data |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
munits |
Number of SOM map units (Row*Col) |
Value
This function returns an initialization matrix for SOM training
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
Examples
SOMdim<-som_dimR(iris[,1:4], type="small")
SOMinit<-som_initR(iris[,1:4],SOMdim$Row,SOMdim$Col,SOMdim$munits)
K-means algorithm applied for a specific number of clusters
Description
The training is run for a custom number of epochs for k number of clusters
Usage
som_kmeansRProg(codebook, k, epochs, seed = NULL)
Arguments
codebook |
SOM codebook |
k |
Number of clusters |
epochs |
Number of training epochs |
seed |
Number for set.seed function |
Details
The function has been coded in R code starting from som_kmeans.m script present in somtoolbox for Matlab by Vesanto and adapted to show a progress bar when working embedded in the shiny app.
Value
This function returns a list containing the cluster number assignment for each sample, the cluster centroids, the total quantization error, and the random seed number used
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
See Also
set.seed
Evaluate pairwise distance matrix for the given codebook
Description
The function has been coded in R code starting from som_mdist.m script present in somtoolbox for Matlab by Vesanto and adapted for the use in the shiny app
Usage
som_mdistR(codebook)
Arguments
codebook |
SOM codebook |
Value
Distance matrix
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf
See Also
db_indexR
Unified distance matrix for the SOM map
Description
The function has been coded in R code starting from som_umat.m script present in somtoolbox for Matlab by Vesanto and adapted for the use in the shiny app
Usage
som_umatR(codebook, Row, Col)
Arguments
codebook |
SOM codebook |
Row |
Number of SOM map rows |
Col |
Number of SOM map columns |
Value
The unified distance matrix for the SOM map
Author(s)
Sabina Licen, Pierluigi Barbieri
References
J. Vesanto, J. Himberg, E. Alhoniemi, J. Parhankagas, SOM Toolbox for Matlab 5, Report A57, 2000, Available at: www.cis.hut.fi/projects/somtoolbox/package/papers/techrep.pdf; A. Ultsch, H.P. Siemon, Proceedings of International Neural Network Conference (INNC?90), Kluwer academic Publishers, Dordrecht, 1990, pp. 305?308.