,
Felix Pape [ctb, ctr],
Tim Schreier [ctb, ctr],
Luis Winckelman [ctb, ctr],
Quirin Stier [ctb, ctr],
Alfred Ultsch [ths]| Title: | Credible Visualization for Two-Dimensional Projections of Data |
|---|---|
| Description: | Projections are common dimensionality reduction methods, which represent high-dimensional data in a two-dimensional space. However, when restricting the output space to two dimensions, which results in a two dimensional scatter plot (projection) of the data, low dimensional similarities do not represent high dimensional distances coercively [Thrun, 2018] <DOI: 10.1007/978-3-658-20540-9>. This could lead to a misleading interpretation of the underlying structures [Thrun, 2018]. By means of the 3D topographic map the generalized Umatrix is able to depict errors of these two-dimensional scatter plots. The package is derived from the book of Thrun, M.C.: "Projection Based Clustering through Self-Organization and Swarm Intelligence" (2018) <DOI:10.1007/978-3-658-20540-9> and the main algorithm called simplified self-organizing map for dimensionality reduction methods is published in <DOI: 10.1016/j.mex.2020.101093>. |
| Authors: | Michael Thrun [aut, cre, cph] ,
Felix Pape [ctb, ctr],
Tim Schreier [ctb, ctr],
Luis Winckelman [ctb, ctr],
Quirin Stier [ctb, ctr],
Alfred Ultsch [ths] |
| Maintainer: | Michael Thrun <[email protected]> |
| License: | GPL-3 |
| Version: | 1.2.6 |
| Built: | 2024-08-09 02:47:12 UTC |
| Source: | https://github.com/mthrun/generalizedumatrix |
Projections are common dimensionality reduction methods, which represent high-dimensional data in a two-dimensional space. However, when restricting the output space to two dimensions, which results in a two dimensional scatter plot (projection) of the data, low dimensional similarities do not represent high dimensional distances coercively [Thrun, 2018] <DOI: 10.1007/978-3-658-20540-9>. This could lead to a misleading interpretation of the underlying structures [Thrun, 2018]. By means of the 3D topographic map the generalized Umatrix is able to depict errors of these two-dimensional scatter plots. The package is derived from the book of Thrun, M.C.: "Projection Based Clustering through Self-Organization and Swarm Intelligence" (2018) <DOI:10.1007/978-3-658-20540-9> and the main algorithm called simplified self-organizing map for dimensionality reduction methods is published in <DOI: 10.1016/j.mex.2020.101093>.
For a brief introduction to GeneralizedUmatrix please see the vignette Introduction of the Generalized Umatrix Package.
For further details regarding the generalized Umatrix see [Thrun, 2018], chapter 4-5, or [Thrun/Ultsch, 2020].
If you want to verifiy your clustering result externally, you can use Heatmap or SilhouettePlot of the CRAN package DataVisualizations.
Index of help topics:
CalcUstarmatrix Calculate the U*matrix for a given Umatrix and
Pmatrix.
Chainlink Chainlink is part of the Fundamental Clustering
Problem Suit (FCPS) [Thrun/Ultsch, 2020].
DefaultColorSequence Default color sequence for plots
Delta3DWeightsC intern function
EsomNeuronsAsList Converts wts data (EsomNeurons) into the list
form
ExtendToroidalUmatrix Extend Toroidal Umatrix
GeneralizedUmatrix Generalized U-Matrix for Projection Methods
published in [Thrun/Ultsch, 2020]
GeneralizedUmatrix-package
Credible Visualization for Two-Dimensional
Projections of Data
GeneratePmatrix Generates the P-matrix
ListAsEsomNeurons Converts List to WTS
LowLand LowLand
NormalizeUmatrix Normalize Umatrix
ReduceToLowLand ReduceToLowLand
TopviewTopographicMap Top view of the topographic map in 2D
Uheights4Data Uheights4Data
UmatrixColormap U-Matrix colors
UniqueBestMatchingUnits
UniqueBestMatchingUnits
XYcoords2LinesColumns XYcoords2LinesColumns(X,Y) Converts points
given as x(i),y(i) coordinates to integer
coordinates Columns(i),Lines(i)
addRowWiseC intern function
plotTopographicMap Visualizes the generalized U-matrix in 3D
sESOM4BMUs simplified ESOM
setdiffMatrix setdiffMatrix shortens Matrix2Curt by those
rows that are in both matrices.
trainstepC internal function for s-esom
upscaleUmatrix Upscale a Umatrix grid
Michal Thrun
Maintainer: Michael Thrun <[email protected]>
[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Uncovering High-Dimensional Structures of Projections from Dimensionality Reduction Methods, MethodsX, Vol. 7, pp. 101093, DOI doi:10.1016/j.mex.2020.101093, 2020.
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
[Ultsch/Thrun, 2017] Ultsch, A., & Thrun, M. C.: Credible Visualizations for Planar Projections, in Cottrell, M. (Ed.), 12th International Workshop on Self-Organizing Maps and Learning Vector Quantization, Clustering and Data Visualization (WSOM), IEEE Xplore, France, 2017.
data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods #see DatabionicSwarm for projection method without parameters or objective function # ProjectedPoints=DatabionicSwarm::Pswarm(Data)$ProjectedPoints resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestmatches,Cls) ##Interactive Island Generation ## from a tiled Umatrix (toroidal assumption) ## Not run: Imx = ProjectionBasedClustering::interactiveGeneralizedUmatrixIsland(resUmatrix$Umatrix, resUmatrix$Bestmatches) plotTopographicMap(resUmatrix$Umatrix, resUmatrix$Bestmatches, Imx = Imx) ## End(Not run) #External Verification ## Not run: DataVisualizations::Heatmap(Data,Cls) #if spherical cluster strcuture DataVisualizations::SilhouettePlot(Data,Cls) ## End(Not run)data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods #see DatabionicSwarm for projection method without parameters or objective function # ProjectedPoints=DatabionicSwarm::Pswarm(Data)$ProjectedPoints resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestmatches,Cls) ##Interactive Island Generation ## from a tiled Umatrix (toroidal assumption) ## Not run: Imx = ProjectionBasedClustering::interactiveGeneralizedUmatrixIsland(resUmatrix$Umatrix, resUmatrix$Bestmatches) plotTopographicMap(resUmatrix$Umatrix, resUmatrix$Bestmatches, Imx = Imx) ## End(Not run) #External Verification ## Not run: DataVisualizations::Heatmap(Data,Cls) #if spherical cluster strcuture DataVisualizations::SilhouettePlot(Data,Cls) ## End(Not run)
Adds the Vector DataPoint to every row of the matrix WeightVectors
addRowWiseC(WeightVectors,DataPoint)addRowWiseC(WeightVectors,DataPoint)
WeightVectors |
WeightVectors. n weights with m components each |
DataPoint |
Vector with m components |
WeightVectors |
[1:m,1:n] |
Calculate the U*matrix for a given Umatrix and Pmatrix.
Umatrix |
[1:Lines,1:Column] Local averages of distances at each point of the trainedGridWts[1:Lines,1:Column,1:variables] of ESOM or other SOM of same format |
Pmatrix |
[1:Lines,1:Column] Local densities at each point of the trainedGridWts[1:Lines,1:Column,1:variables] of ESOM or other SOM of same format. |
UStarMatrix |
[1:Lines,1:Column] |
Michael Thrun
Ultsch, A. U* C: Self-organized Clustering with Emergent Feature Maps. in Lernen, Wissensentdeckung und Adaptivitaet (LWA). 2005. Saarbruecken, Germany.
linear not separable dataset of two interwined chains.
data("Chainlink")data("Chainlink")
Size 1000, Dimensions 3, stored in Chainlink$Data
Teo clusters, stored in Chainlink$Cls
Published in [Ultsch et al.,1994] in German and [Ultsch 1995] in English.
[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief,Vol. 30(C), pp. 105501, DOI 10.1016/j.dib.2020.105501 , 2020.
[Ultsch 1995] Ultsch, A.: Self organizing neural networks perform different from statistical k-means clustering, Proc. Society for Information and Classification (GFKL), Vol. 1995, Basel 8th-10th March, 1995.
[Ultsch et al.,1994] Ultsch, A., Guimaraes, G., Korus, D., & Li, H.: Knowledge extraction from artificial neural networks and applications, Parallele Datenverarbeitung mit dem Transputer, pp. 148-16Chainlink, Springer, 1994.
data(Chainlink) str(Chainlink) ## Not run: require(DataVisualizations) DataVisualizations::Plot3D(Chainlink$Data,Chainlink$Cls) ## End(Not run)data(Chainlink) str(Chainlink) ## Not run: require(DataVisualizations) DataVisualizations::Plot3D(Chainlink$Data,Chainlink$Cls) ## End(Not run)
Defines the default color sequence for plots made within the Projections package.
data("DefaultColorSequence")data("DefaultColorSequence")
A vector with 562 different strings describing colors for plots.
Thr implementation of the main formula of SOM, ESOM, sESOM algorithms.
Delta3DWeightsC(vx,Datasample)Delta3DWeightsC(vx,Datasample)
vx |
Numeric array of weights [1:Lines,1:Columns,1:Weights] |
Datasample |
Numeric vector of one datapoint[1:n] |
intern function in case of ComputeInR==FALSE in GeneralizedUmatrix
modified array of weights [1:Lines,1:Columns,1:Weights]
Michael Thrun
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
Converts wts data into the list form
EsomNeurons |
[1:Lines, 1:Columns, 1:Variables] high dimensional array with grid positions in the first two dimensions. |
One could describe this function as a transformation or a special case
of wide to long format, see also ListAsEsomNeurons
TrainedNeurons |
[1:(Lines*Columns),1:Variables] List of Weights as a
matrix (not |
Michael Thrun, Florian Lerch
Ultsch, A. Maps for the visualization of high-dimensional data spaces. in Proc. Workshop on Self organizing Maps. 2003.
Extends Umatrix by toroidal continuation of the given Umatrix defined by
ExtendBorders in all four directions.
ExtendToroidalUmatrix(Umatrix, Bestmatches, ExtendBorders)ExtendToroidalUmatrix(Umatrix, Bestmatches, ExtendBorders)
Umatrix |
[1:Lines,1:Columns] Matrix of Umatrix Heights |
Bestmatches |
[1:n, 1:2] Matrix with positions of Bestmatches for n
datapoints, first columns is the position in |
ExtendBorders |
number of lines and columns the umatrix should be extended with |
Function assumes that U-matrix is not planaer (has no borders), i.e. is toroidal, and not tiled. Bestmatches are moved to new positions accordingly. Example is shown in conference talk of [Thrun et al., 2020].
Umatrix |
[1:Lines+2*ExtendBorders,1:Columns+2*ExtendBorders] Matrix of U-Heights |
Bestmatches |
Array with positions of Bestmatches |
Currently can be only used if untiled U-Matrix (the default) is presented, but 4-tiled U-matrix does not work.
Michael Thrun
[Thrun et al., 2020] Thrun, M. C., Pape, F., & Ultsch, A.: Interactive Machine Learning Tool for Clustering in Visual Analytics, 7th IEEE International Conference on Data Science and Advanced Analytics (DSAA 2020), Vol. accepted, pp. 1-9, IEEE, Sydney, Australia, 2020.
#ToDO#ToDO
Generalized U-Matrix visualizes high-dimensional distance and density based structurs in two-dimensional scatter plots of projectios methods like CCA, MDS, PCA or NeRV [Ultsch/Thrun, 2017] with the help of a topographic map with hypsometrioc tints [Thrun et al. 2016] using a simplified emergent SOM published in [Thrun/Ultsch, 2020].
GeneralizedUmatrix(Data,ProjectedPoints, PlotIt=FALSE,Cls=NULL,Toroid=TRUE, Tiled=FALSE, ComputeInR=FALSE,Parallel=TRUE,DataPerEpoch=1,...)GeneralizedUmatrix(Data,ProjectedPoints, PlotIt=FALSE,Cls=NULL,Toroid=TRUE, Tiled=FALSE, ComputeInR=FALSE,Parallel=TRUE,DataPerEpoch=1,...)
Data |
[1:n,1:d] array of data: n cases in rows, d variables in columns |
ProjectedPoints |
[1:n,2] matrix containing coordinates of the Projection: A matrix of the fitted configuration. |
PlotIt |
Optional,bool, defaut=FALSE, if =TRUE: U-Marix of every current Position of Databots will be shown
However, the amount of details shown will be less than in |
Cls |
Optional, For plotting, see |
Toroid |
Optional, Default=TRUE, ==FALSE planar computation with borders defined by projection method ==TRUE: toroid borderless (toroidal) computation, the four borders defined by projection method are ignored. |
Tiled |
Optional,For plotting see |
ComputeInR |
Optional, =T: Rcode, =F Cpp Code |
Parallel |
Optional, =TRUE: compute parallel Cpp Code, =FALSE do not compute parallel Cpp Code |
DataPerEpoch |
Optional, scalar, value above zero and below 1 starts sampling and defines percentage of data points sampled in each epoch during the learning phase. Beware: Experimental! |
... |
Further parameters. |
Introduced first in the PhD thesis in [Thrun, 2018, p.46]. Furthermore the two parts of the work were peer-reviewed and published in [Ultsch/Thrun, 2017, Thrun/Ultsch, 2020].
List with
Umatrix |
[1:Lines,1:Columns] Umatrix to be plotted, numerical matrix storing the U-heights, see [Thrun, 2018] for definition. |
EsomNeurons |
[1:Lines,1:Columns,1:weights] 3-dimensional numeric array (wide format), not wts (long format). |
Bestmatches |
[1:n,1:2] Positions of GridConverted Projected Points on the Umatrix to the predefined Grid by Lines and Columns, First Columns has the content of the Line No and second Column of the Column number. |
sESOMparamaters |
internals for debugging |
Lines |
Number of Lines |
Columns |
Number of Columns |
gplotres |
output of ggplot2 |
Michael Thrun
[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, http://wscg.zcu.cz/wscg2016/short/A43-full.pdf, 2016.
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
[Ultsch/Thrun, 2017] Ultsch, A., & Thrun, M. C.: Credible Visualizations for Planar Projections, in Cottrell, M. (Ed.), 12th International Workshop on Self-Organizing Maps and Learning Vector Quantization, Clustering and Data Visualization (WSOM), IEEE Xplore, France, 2017.
[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Uncovering High-Dimensional Structures of Projections from Dimensionality Reduction Methods, MethodsX, Vol. 7, pp. 101093, DOI doi:10.1016/j.mex.2020.101093, 2020.
data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) ## Not run: Stress = ProjectionBasedClustering::KruskalStress(InputDistances, as.matrix(dist(ProjectedPoints))) ## End(Not run) resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestmatches,Cls)data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) ## Not run: Stress = ProjectionBasedClustering::KruskalStress(InputDistances, as.matrix(dist(ProjectedPoints))) ## End(Not run) resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) plotTopographicMap(resUmatrix$Umatrix,resUmatrix$Bestmatches,Cls)
Generates a P-matrix too visualize only density based structures of high-dimensional data.
Data |
[1:n,1:d], A |
EsomNeurons |
[1:Lines,Columns,1:Weights] 3D array of weights given by ESOM or sESOM algorithm. |
Radius |
The radius for measuring the density within the hypersphere. |
PlotIt |
If set the Pmatrix will also be plotted |
... |
If set the Pmatrix will also be plotted |
To set the Radius the ABCanalysis of high-dimensional distances can be used [Ultsch/Lötsch, 2015]. For a deteailed definition and equation of automated density estimation (Radius) see Thrun et al. 2016.
PMatrix [1:Lines,1:Columns]
Michael Thrun
Ultsch, A.: Maps for the visualization of high-dimensional data spaces, Proc. Workshop on Self organizing Maps (WSOM), pp. 225-230, Kyushu, Japan, 2003.
Ultsch, A., Loetsch, J.: Computed ABC Analysis for Rational Selection of Most Informative Variables in Multivariate Data, PloS one, Vol. 10(6), pp. e0129767. doi 10.1371/journal.pone.0129767, 2015.
Thrun, M. C., Lerch, F., Loetsch, J., Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision,Plzen, 2016.
Converts wts data in list form into a 3 dimensional array
wts_list |
[1:(Lines*Columns),1:Variables] Matrix with weights in the 2nd dimension(not list() like in R) |
Lines |
Lines/Height of the desired grid |
Columns |
Columns/Width of the desired grid |
One could describe this function as a transformation or a special case
of long to wide format, see also EsomNeuronsAsList
EsomNeurons |
[1:Lines, 1:Columns, 1:Variables] 3 dimensional array containing the weights of the neural grid. For a more general explanation see reference |
Michael Thrun, Florian Lerch
Ultsch, A.: Maps for the visualization of high-dimensional data spaces, Proc. Workshop on Self organizing Maps (WSOM), pp. 225-230, Kyushu, Japan, 2003.
LowLand
LowLand(BestMatchingUnits, GeneralizedUmatrix, Data, Cls, Key, LowLimit)LowLand(BestMatchingUnits, GeneralizedUmatrix, Data, Cls, Key, LowLimit)
BestMatchingUnits |
[1:n,1:n,1:n] BestMatchingUnits =[BMkey, BMLineCoords, BMColCoords] |
GeneralizedUmatrix |
[1:l,1:c] U-Matrix heights in Matrix form |
Data |
[1:n,1:d] data cases in lines, variables in Columns or [] or 0 |
Cls |
[1:n] a possible classification of the data or [] or 0 |
Key |
[1:n] the keys of the data or [] or 0 |
LowLimit |
GeneralizedUmatrix heights up to this are considered to lie in the low lands default: LowLimit = prctile(Uheights,80) nur die 80# tiefsten |
LowLandBM |
the unique BestMatchingUnits in the low lands of an u-Matrix |
LowLandInd |
index such that UniqueBM = BestMatchingUnits(UniqueInd,] |
LowLandData |
Data reduced to LowLand: LowLandData = Data(LowLandInd,] |
LowLandCls |
Cls reduced to LowLand: LowLandCls = Cls(LowLandInd) |
LowLandKey |
Key reduced to LowLand: LowLandKey = Key(LowLandInd) |
ALU 2021 in matlab, MCT reimplemented in R
Normalizing the U-matrix using the abstact U-Matrix concept [Loetsch/Ultsch, 2014].
NormalizeUmatrix(Data, Umatrix, BestMatches)NormalizeUmatrix(Data, Umatrix, BestMatches)
Data |
[1:n,1:d] numerical matrix of data with n cases and d variables |
Umatrix |
[1:lines,1:Columns] matrix of U-heights |
BestMatches |
[1:n,1:2] Bestmatching units. |
see publication [Loetsch/Ultsch, 2014]..
Normalized Umatrix[1:lines,1:Columns] using the abstact U-Matrix concept.
Felix Pape, Michael Thrun
Loetsch, J., Ultsch, A.: Exploiting the structures of the U-matrix, in Villmann, T., Schleif, F.-M., Kaden, M. & Lange, M. (eds.), Proc. Advances in Self-Organizing Maps and Learning Vector Quantization, pp. 249-257, Springer International Publishing, Mittweida, Germany, 2014.
data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## Normalization normalizedUmatrix=NormalizeUmatrix(Data,resUmatrix$Umatrix,resUmatrix$Bestmatches) ## visualization TopviewTopographicMap(GeneralizedUmatrix = normalizedUmatrix,resUmatrix$Bestmatches)data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## Normalization normalizedUmatrix=NormalizeUmatrix(Data,resUmatrix$Umatrix,resUmatrix$Bestmatches) ## visualization TopviewTopographicMap(GeneralizedUmatrix = normalizedUmatrix,resUmatrix$Bestmatches)
The generalized U-matrix is visualized as the topographic map with hypsometric tints. The topographic map represents high-dimensional distance and density-based structurs in form of a 3D landscape.
plotTopographicMap(GeneralizedUmatrix, BestMatchingUnits, Cls=NULL,ClsColors=NULL,Imx=NULL,Names=NULL, BmSize=0.5,RenderingContourLines=TRUE,...)plotTopographicMap(GeneralizedUmatrix, BestMatchingUnits, Cls=NULL,ClsColors=NULL,Imx=NULL,Names=NULL, BmSize=0.5,RenderingContourLines=TRUE,...)
GeneralizedUmatrix |
[1:Lines,1:Columns] U-matrix to be plotted, numerical matrix storing the U-heights, see [Thrun, 2018] for definition. |
BestMatchingUnits |
[1:n,1:2], Positions of bestmatches to be plotted as spheres onto the topographic map |
Cls |
[1:n], numerical vector of classification of |
ClsColors |
Vector of colors that will be used to colorize the different clusters, default is GeneralizedUmatrix::DefaultColorSequence |
Imx |
a mask (Imx) that will be used to cut out the U-matrix |
Names |
If set: [1:k] character vector naming the k clusters for the
legend. In this case, further parameters with the possibility to adjust are:
|
BmSize |
size(diameter) of the points in the visualizations. The points represent the BestMatchingUnits |
RenderingContourLines |
FALSE: disables plotting of contour lines resulting in a much faster plot. |
... |
Besides the legend/names parameter the list of further parameters, use only of you know what you are doing:
|
The visualization of this function is a topographic map with hypsometric tints (Thrun, Lerch, L?tsch, & Ultsch, 2016). "Hypsometric tints are surface colors that represent ranges of elevation (Patterson and Kelso 2004). Here, contour lines are combined with a specific color scale. The color scale is chosen to display various valleys, ridges, and basins: blue colors indicate small distances (sea level), green and brown colors indicate middle distances (low hills), and white colors indicate vast distances (high mountains covered with snow and ice). Valleys and basins represent clusters, and the watersheds of hills and mountains represent the borders between clusters. In this 3D landscape, the borders of the visualization are cyclically connected with a periodicity (L,C). The number of clusters can be estimated by the number of valleys of the visualization. The clustering is valid if mountains do not partition clusters indicated by colored points of the same color and colored regions of points (see examples in section 4.1 and 4.2)."[Thrun/Ultsch, 2020].
A central problem in clustering is the correct estimation of the number of clusters. This is addressed by the topographic map which allows assessing the number of clusters as the number of valleys (Thrun et al., 2016). Please see chapter 5 of [Thrun, 2018] for further details.
An object of class "htmlwidget" in mode invisible, please rglwidget for details.
First version of algorithm was partly based on the U-matrix package.
Michael Thrun
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, http://wscg.zcu.cz/wscg2016/short/A43-full.pdf, 2016.
[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A. : Using Projection based Clustering to Find Distance and Density based Clusters in High-Dimensional Data, Journal of Classification, DOI 10.1007/s00357-020-09373-2, in press, Springer, 2020.
data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## visualization plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) ## Open window in specific resolution #relevant if Names given library(rgl) r3dDefaults$windowRect = c(0,0,1200,1200) plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) ## Not run: ## To save as STL for 3D printing rgl::writeSTL("GenerelizedUmatrix_3d_model.stl") ## Save the visualization as a picture with library(rgl) rgl.snapshot('test.png') ## End(Not run) ## Save interactive html file ## Not run: widgets=plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) if(requireNamespace("htmlwidgets")) htmlwidgets::saveWidget(widgets,file = "interactiveTopographicMap.html") ## End(Not run)data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## visualization plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) ## Open window in specific resolution #relevant if Names given library(rgl) r3dDefaults$windowRect = c(0,0,1200,1200) plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) ## Not run: ## To save as STL for 3D printing rgl::writeSTL("GenerelizedUmatrix_3d_model.stl") ## Save the visualization as a picture with library(rgl) rgl.snapshot('test.png') ## End(Not run) ## Save interactive html file ## Not run: widgets=plotTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches) if(requireNamespace("htmlwidgets")) htmlwidgets::saveWidget(widgets,file = "interactiveTopographicMap.html") ## End(Not run)
ReduceToLowLand
ReduceToLowLand(BestMatchingUnits, GeneralizedUmatrix, Data = NULL, Cls = NULL, Key = NULL, LowLimit,Force=FALSE)ReduceToLowLand(BestMatchingUnits, GeneralizedUmatrix, Data = NULL, Cls = NULL, Key = NULL, LowLimit,Force=FALSE)
BestMatchingUnits |
[1:n,1:n,1:n] BestMatchingUnits =[BMkey, BMLineCoords, BMColCoords] |
GeneralizedUmatrix |
[1:l,1:c] U-Matrix heights in Matrix form |
Data |
[1:n,1:d] data cases in lines, variables in Columns or [] or 0 |
Cls |
[1:n] a possible classif( ication of the data or [] or 0 |
Key |
[1:n] the keys of the data or [] or 0 |
LowLimit |
GeneralizedUmatrix heights up to this are considered to lie in the low lands default: LowLimit = prctile(Uheights,80) nur die 80# tiefsten |
Force |
==TRUE: Always perform reduction |
LowLandBM |
the unique BestMatchingUnits in the low lands of an u-Matrix |
LowLandInd |
index such that UniqueBM = BestMatchingUnits(UniqueInd,] |
LowLandData |
Data reduced to LowLand: LowLandData = Data(LowLandInd,] |
LowLandCls |
Cls reduced to LowLand: LowLandCls = Cls(LowLandInd) |
LowLandKey |
Key reduced to LowLand: LowLandKey = Key(LowLandInd) |
ALU 2021 in matlab, MCT reimplemented in R
internfunction for the simplified ESOM Algorithmus [Thrun/Ultsch, 2020] for fixed BestMatchingUnits
sESOM4BMUs(BMUs,Data, esom, toroid, CurrentRadius,ComputeInR=FALSE,Parallel=TRUE)sESOM4BMUs(BMUs,Data, esom, toroid, CurrentRadius,ComputeInR=FALSE,Parallel=TRUE)
BMUs |
[1:Lines,1:Columns], BestMAtchingUnits generated by ProjectedPoints2Grid() |
Data |
[1:n,1:d] array of data: n cases in rows, d variables in columns |
esom |
[1:Lines,1:Columns,1:weights] array of NeuronWeights, see ListAsEsomNeurons() |
toroid |
TRUE/FALSE - topology of points |
CurrentRadius |
number betweeen 1 to x |
ComputeInR |
=T: Rcode, =F Cpp Code |
Parallel |
=T: Rcode, =F Cpp Code |
Algorithm is described in [Thrun, 2018, p. 48, Listing 5.1].
esom |
array [1:Lines,1:Columns,1:d], d is the dimension of the weights, the same as in the ESOM algorithm. modified esomneuros regarding a predefined neighborhood defined by a radius |
Usually not for seperated usage!
Michael Thrun
[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Uncovering High-Dimensional Structures of Projections from Dimensionality Reduction Methods, MethodsX, Vol. in press, pp. 101093. doi 10.1016/j.mex.2020.101093, 2020.
setdiffMatrix shortens Matrix2Curt by those rows that are in both matrices.
Matrix2Curt |
[n,k] matrix, which will be shortened by x rows |
Matrix2compare |
[m,k] matrix whose rows will be compared to those of Matrix2Curt x rows in Matrix2compare equal rows of Matrix2Curt (order of rows is irrelevant). Has the same number of columns as Matrix2Curt. |
V$CurtedMatrix |
[n-x,k] Shortened Matrix2Curt |
Michael Thrun with the help of Catharina Lippmann
Fast visualization of the generalized U-matrix in 2D which visualizes high-dimensional distance and density based structurs of the combination two-dimensional scatter plots (projections) with high-dimensional data.
TopviewTopographicMap(GeneralizedUmatrix, BestMatchingUnits, Cls, ClsColors = NULL, Imx = NULL, ClsNames = NULL, BmSize = 6, DotLineWidth = 2, alpha = 1, ...)TopviewTopographicMap(GeneralizedUmatrix, BestMatchingUnits, Cls, ClsColors = NULL, Imx = NULL, ClsNames = NULL, BmSize = 6, DotLineWidth = 2, alpha = 1, ...)
GeneralizedUmatrix |
[1:Lines,1:Columns] U-matrix to be plotted, numerical matrix storing the U-heights, see [Thrun, 2018] for definition. |
BestMatchingUnits |
[1:n,1:2], Positions of bestmatches to be plotted onto the U-matrix |
Cls |
[1:n], numerical vector of classification of |
ClsColors |
Vector of colors that will be used to colorize the different classes |
Imx |
a mask (Imx) that will be used to cut out the U-matrix |
ClsNames |
If set: [1:k] character vector naming the k classes for the
legend. In this case, further parameters with the possibility to adjust are:
|
BmSize |
size(diameter) of the points in the visualizations. The points represent the BestMatchingUnits |
DotLineWidth |
... |
alpha |
... |
... |
|
Please see plotTopographicMap. This function is currently still experimental because not all functionallity is fully tested yet.
plotly handler
Names are currently under development, Imx in testing phase.
Tim Schreier, Luis Winckelmann, Michael Thrun
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, http://wscg.zcu.cz/wscg2016/short/A43-full.pdf, 2016.
data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## visualization TopviewTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches)data("Chainlink") Data=Chainlink$Data Cls=Chainlink$Cls InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) #see also ProjectionBasedClustering package for other common projection methods resUmatrix=GeneralizedUmatrix(Data,ProjectedPoints) ## visualization TopviewTopographicMap(GeneralizedUmatrix = resUmatrix$Umatrix,resUmatrix$Bestmatches)
Does the training for fixed bestmatches in one epoch of the sESOM.
trainstepC(vx,vy, DataSampled,BMUsampled,Lines,Columns, Radius, toroid)trainstepC(vx,vy, DataSampled,BMUsampled,Lines,Columns, Radius, toroid)
vx |
array [1:Lines,1:Columns,1:Weights], WeightVectors that will be trained, internally transformed von NumericVector to cube |
vy |
array [1:Lines,1:Columns,1:2], meshgrid for output distance computation |
DataSampled |
NumericMatrix, n cases shuffled Dataset[1:n,1:d] by |
BMUsampled |
NumericMatrix, n cases shuffled BestMatches[1:n,1:2] by |
Lines |
double, Height of the grid |
Columns |
double, Width of the grid |
Radius |
double, The current Radius that should be used to define neighbours to the bm |
toroid |
bool, Should the grid be considered with cyclically connected borders? |
Algorithm is described in [Thrun, 2018, p. 48, Listing 5.1].
WeightVectors, array[1:Lines,1:Columns,1:weights] with the adjusted Weights
Usually not for seperated usage!
Michael Thrun
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
Uheights4Data
Uheights4Data(BestMatchingUnits, GeneralizedUmatrix)Uheights4Data(BestMatchingUnits, GeneralizedUmatrix)
BestMatchingUnits |
[1:n,1:d] BMKey = BestMatchingUnits[,1) |
GeneralizedUmatrix |
[1:Lines,1:Columns] a GeneralizedUmatrix |
Uheights |
Uheights |
BMLineCoords |
BMLineCoords |
BMColCoords |
BMColCoords |
ALU 2021 in matlab, MCT reimplemented in
Defines the default color sequence for plots made for Umatrix
data("UmatrixColormap")data("UmatrixColormap")
Returns the vectors for a (heat) colormap.
UniqueBestMatchingUnits
UniqueBestMatchingUnits(NonUniqueBestMatchingUnits)UniqueBestMatchingUnits(NonUniqueBestMatchingUnits)
NonUniqueBestMatchingUnits |
[1:n,1:n,1:n] UniqueBestMatchingUnits =[BMkey, BMLineCoords, BMColCoords] |
UniqueBM |
[1:u,1:u,1:u] UniqueBM =[UBMkey, UBMLineCoords, UBMColCoords] |
UniqueInd |
Index such that UniqueBM = UniqueBestMatchingUnits(UniqeInd,:) |
Uniq2AllInd |
Index such that UniqueBestMatchingUnits = UniqueBM(Uniq2AllInd,:) |
ALU 2021 in matlab, MCT reimplemented in R
Use linear interpolation to increase the size of a umatrix. This can be used to produce nicer ggplot plots in plotTopographicMap and is going to be used for further normalization of the umatrix.
upscaleUmatrix(Umatrix, Factor = 2,BestMatches, Imx)upscaleUmatrix(Umatrix, Factor = 2,BestMatches, Imx)
Umatrix |
The umatrix which should be upscaled |
BestMatches |
The BestMatches which should be upscaled |
Factor |
Optional: The factor by which the axes will be scaled. Be aware that the size of the matrix will grow by Factor squared. Default: 2 |
Imx |
Optional: Island cutout of the umatrix. Should also be scaled to the new size of the umatrix. |
A List consisting of:
Umatrix |
A matrix representing the upscaled umatrix. |
BestMatches |
If BestMatches was given as parameter: The rescaled
BestMatches for an island cutout. Otherwise: |
Imx |
If Imx was given as parameter: The rescaled matrix for an island
cutout. Otherwise: |
Felix Pape
XYcoords2LinesColumns(X,Y) Converts points given as x(i),y(i) coordinates to integer coordinates Columns(i),Lines(i)
X |
[1:n] first coordinate: x(i), y(i) is the i-th point on a plane |
Y |
[1:n] second coordinate: x(i), y(i) is the i-th point on a plane |
minNeurons |
minimal size of the corresponding grid i.e max(Lines)*max(Columns)>=MinGridSize , default MinGridSize = 4096 defined by the numer of neurons |
MaxDifferentPoints |
TRUE: the discretization error is minimal FALSE: number of Lines and Columns is minimal |
PlotIt |
Plots the result |
na.rm |
if non finite values should be disregarded in the computation then set to TRUE |
Non finite values are not filtered out even if na.rm=TRUE, only ignored. Details are written down in [Thrun, 2018, p. 47].
GridConvertedPoints[1:Columns,1:Lines,2] IntegerPositions on a grid corresponding to x,y
Michael Thrun
[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.
data("Chainlink") Data=Chainlink$Data InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) GridConvertedPoints=XYcoords2LinesColumns(ProjectedPoints[,1],ProjectedPoints[,2],PlotIt=FALSE)data("Chainlink") Data=Chainlink$Data InputDistances=as.matrix(dist(Data)) res=cmdscale(d=InputDistances, k = 2, eig = TRUE, add = FALSE, x.ret = FALSE) ProjectedPoints=as.matrix(res$points) GridConvertedPoints=XYcoords2LinesColumns(ProjectedPoints[,1],ProjectedPoints[,2],PlotIt=FALSE)