| Type: | Package |
| Title: | Quality Measurements for Dimensionality Reduction |
| Version: | 0.2.1 |
| Date: | 2023-10-10 |
| Maintainer: | Michael Thrun <m.thrun@gmx.net> |
| Description: | Several quality measurements for investigating the performance of dimensionality reduction methods are provided here. In addition a new quality measurement called Gabriel classification error is made accessible, which was published in Thrun, M. C., Märte, J., & Stier, Q: "Analyzing Quality Measurements for Dimensionality Reduction" (2023), Machine Learning and Knowledge Extraction (MAKE), <doi:10.3390/make5030056>. |
| License: | GPL-3 |
| Imports: | DatabionicSwarm |
| Suggests: | plotly, geometry, deldir, FCPS, ProjectionBasedClustering (≥ 1.2.1), DataVisualizations, FastKNN, ggplot2, pcaPP,pracma, spdep, grid, igraph, cccd, sf |
| Encoding: | UTF-8 |
| NeedsCompilation: | no |
| Packaged: | 2023-10-12 07:30:23 UTC; MCT |
| Author: | Quirin Stier [aut],
Florian Lerch [ctb],
Julian Märte [aut],
Hermann Tafo [ctb],
Laukert Schlichting [ctb],
Michael Thrun 
[aut, cph, cre] |
| Repository: | CRAN |
| Date/Publication: | 2023-10-12 09:40:15 UTC |
Classification Error (rate)
Description
Compares projected points to a given prior classification using knn classifier.
Usage
ClassificationError(OutputDistances,Cls,k=5)
Arguments
OutputDistances |
[1:n,1:n] numeric matrix with distance matrix of projected data. |
Cls |
[1:n] Numeric vector containing class information. |
k |
number of k nearest neighbors, in Venna 2010 set to 5 (here default) |
Details
Projected points are evaluated by k-nearest neighbor classification accuracy (with k = 5), that is, each sample in the visualization is classified by majority vote of its k nearest neighbors in the visualization, and the classification is compared to the ground truth label. [Venna 2010].
Value
List with three entries:
Error |
Classification Error: 1-Accuracy[1] |
Accuracy |
Accuracy |
KNNCls |
[1:n]] cls of knn classifier |
Note
Here, the Outputdistances of the Projected points are used.
Author(s)
Michael Thrun
References
Venna, J., Peltonen, J., Nybo, K., Aidos, H., and Kaski, S. Information retrieval perspective to nonlinear dimensionality reduction for data visualization. The Journal of Machine Learning Research, 11, 451-490. (2010)
Gracia, A., Gonzalez, S., Robles, V., and Menasalvas, E. A methodology to compare Dimensionality Reduction algorithms in terms of loss of quality. Information Sciences, 270, 1-27. (2014)
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
ClassificationError(as.matrix(dist(projection)),Hepta$Cls)
}
C-Measure subtypes
Description
Calculate the C-Measure subtypes of minimal path length and minimal wiring
Usage
Cmeasure(Data, Projection, k = 1)Arguments
Data |
[1:n,1:d] numerical matrix of points in input space. |
Projection |
[1:n,1:2] numerical matrix of points in output space. |
k |
Number of nearest neighbors, both measures set it always to k=1. |
Value
[1:2] Numerical vector of MinimalPathlength and MinimalWiring values.
Author(s)
Michael Thrun
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
Cmeasure(Hepta$Data,projection)
}
Gabriel Classification Error (GCE)
Description
GCE searches for the k-nearest neighbors of the first gabriel neighbors weighted by the Euclidean Distances of the Inputspace [Thrun et al, 2023]. GCE evaluates these neighbors in the Output space. A low value indicates a better two-dimensional projection of the high-dimensional Input space.
Usage
GabrielClassificationError(Data,ProjectedPoints,Cls,LC,
PlotIt=FALSE,Plotter = "native", Colors = NULL,LineColor= 'grey',
main = "Name of Projection", mainSize = 24,xlab = "X", ylab = "Y", xlim, ylim,
pch,lwd,Margin=list(t=50,r=0,l=0,b=0))
Arguments
Data |
[1:n,1:d] Numeric matrix with n cases and d variables |
ProjectedPoints |
[1:n,1:2] Numeric matrix with 2D points in cartesian coordinates |
Cls |
[1:n] Numeric vector with class labels |
LC |
Optional, Numeric vector of two values determining grid size of the underlying projection |
PlotIt |
Optional, Boolean: TRUE/FALSE => Plot/Do not plot (Default: FALSE) |
Plotter |
Optional, Character with plot technique (native or plotly) |
Colors |
Optional, Character vector of class colors for points |
LineColor |
Optional, Character of line color used for edges of graph |
main |
Optional, Character plot title |
mainSize |
Optional, Numeric size of plot title |
xlab |
Optional, Character name of x ax |
ylab |
Optional, Character name of y ax |
xlim |
Optional, Numeric vector with two values defining x ax range |
ylim |
Optional, Numeric vector with two values defining y ax range |
pch |
Optional, Numeric of point size (graphic parameter) |
lwd |
Optional, Numeric of linewidth (graphic parameter) |
Margin |
Optional, Margin of plotly plot |
Details
Gabriel Classification Error (GCE) makes an unbiased evaluation of distance- and density-based structures which might be even non-linear separable. First, GCE utilizes the information provided by a prior classification to assess projected structures. Second, GCE applies the insights drawn from graph theory. Details are described in [Thrun et al, 2023].
Value
list of several entries containing first the GCE itself as main result followed by further entries which contain potential important information
GCE |
Numeric: the 'Gabriel Classification Error' |
GCEperPoint |
[1:n] unnormalized GCE of each point: GCE = mean(GCEperPoint) |
nn |
the number of points in a relevant neghborhood: 0.5 * 85percentile(AnzNN) |
AnzNN |
[1:n] the number of points with a Gabriel graph neighborhood |
NNdists |
[1:n,1:nn] the distances within the relevant neighborhood, 1 for inter cluster distances and 0 for inner cluster distances |
HD |
[1:nn] HD = HarmonicDecay(nn) i.e weight function for the NNdists: GCEperPoint = HD*NNdists |
IsInterDistance |
Distances to the nn closest neighbors. |
GabrielDists |
Distance matrix implied by high dimensional distances and the underlying gabriel (Gabriel) graph |
ProjectionGraphError |
Plotly object in case, plotly is chosen. |
Author(s)
Michael Thrun, Quirin Stier, Julian Märte
References
[Thrun et al, 2023] Thrun, M.C, Märte, J., Stier, Q.: Analyzing Quality Measurements for Dimensionality Reduction, Machine Learning and Knowledge Extraction (MAKE), Vol 5., accepted, 2023.
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
GabrielClassificationError(Hepta$Data,projection,Hepta$Cls)$GCE
}
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
GabrielClassificationError(Hepta$Data,projection,Hepta$Cls)$GCE
}
Statistical correlation by Kendall
Description
Calculates the statistical correlation by Kendall. Basically a wrapper to pcaPP::cor.fk.
Usage
KendallsTau(InputDists, OutputDists)
Arguments
InputDists |
Matrix containing the distances of the first dataset. |
OutputDists |
Matrix containing the distances of the second dataset. |
Value
Equivalent to cor.fk
Author(s)
Michael Thrun
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
InputDist=dist(Hepta$Data)
projection=cmdscale(InputDist, k=2)
KendallsTau(as.matrix(InputDist),as.matrix(dist(projection)))
}
Trustworthiness and Discontinuity.
Description
In a trustworthy projection the visualized proximities hold in the original data as well, whereas a continuous projection visualizes all proximities of the original data.
Usage
MeasureTandD(Data, pData, NeighborhoodSize)Arguments
Data |
[1:n,1:d] points in input room with d attributes |
pData |
[1:n,1:2] projected points in output room, with index,x,y or index,line,column |
NeighborhoodSize |
Integer - sets the maximum number of neighbors to calculate trustworthiness and continuity for. |
Value
Numeric matrix [1:NeighborhoodSize,1:2] containing the trustworthiness values in the first column and the discontinuity values in the second column.
Author(s)
Julian Märte
References
Venna, J., & Kaski, S. (2005, September). Local multidimensional scaling with controlled tradeoff between trustworthiness and continuity. In Proceedings of 5th Workshop on Self-Organizing Maps (pp. 695-702).
Kaski, S., Nikkilä, J., Oja, M., Venna, J., Törönen, P., & Castrén, E. (2003). Trustworthiness and metrics in visualizing similarity of gene expression. BMC bioinformatics, 4(1), 1-13.
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
MeasureTandD(Hepta$Data,projection, 2)
}
Precision and Recall.
Description
Trade-off between missing similar points versus retrieving dissimilar points.
Usage
PrecisionAndRecall(Data, pData, NeighborhoodSize = 20)Arguments
Data |
[1:n,1:d] points in input room with d attributes |
pData |
[1:n,1:2] projected points in output room, with index,x,y or index,line,column |
NeighborhoodSize |
Sets the 'effective number of neighbors' used to control the width of the Gaussian, NeRV paper Seite 463 setzt Default auf 20 |
Value
Numeric matrix [1:NeighborhoodSize, 1:2] containing the precision values in the first column and the recall values in the second column of the matrix.
Author(s)
Felix Pape
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
PrecisionAndRecall(Hepta$Data,projection)
}
Rescaled average agreement rate
Description
Rescaled average agreement rate deduced by the co-ranking matrix from LCMC.
Usage
RAAR(Data, ProjectedPoints, kmax = nrow(Data) - 2, PlotIt = TRUE)
Arguments
Data |
Matrix containing n cases in rows, d variables in columns or a distance matrix which in this case has to be symmetric |
ProjectedPoints |
n by OutputDimension matrix containing coordinates of the Projection |
kmax |
maximum of intervall 1:kmax of k nearest neighbors |
PlotIt |
Optional: Should the output be plottet. Default: TRUE |
Value
A list containing:
Raar |
Rescaled average agreement rate |
Aar |
Average agreement rate |
Author(s)
Michael Thrun
References
Lee, J. A., Peluffo-Ordonez, D. H., & Verleysen, M. Multiscale stochastic neighbor embedding: Towards parameter-free dimensionality reduction. Paper presented at the Proceedings of 22st European Symposium on Artificial Neural Networks, Computational Intelligence And Machine Learning (ESANN) (2014).
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
RAAR(Hepta$Data,projection,kmax=nrow(Hepta$Data)-2,PlotIt=TRUE)
}
Calculates the error of a projection with spearman's rank correlation coefficient.
Description
Calculates the error of a projection with spearman's rank correlation coefficient.
Arguments
VectorOfInputDists(1:n2) |
dissimilarities in Input Space between the n data points in vector form as produced by squareform(Dists(1:n,1:n)) |
VectorOfOutputDists(1:n2) |
dissimilarities in Output Space between the n data points in vector form as produced by squareform(Dists(1:n,1:n)) |
Value
rho rank correlation coefficient
Author(s)
Florian Lerch
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
SpearmanError(as.matrix(dist(Hepta$Data)),as.matrix(dist(projection)))
}
Calculates the error of a projection with spearman's rank correlation coefficient
Description
Calculates the error of a projection with spearman's rank correlation coefficient
Usage
SpearmansRho(InputDists, OutputDists)Arguments
InputDists |
[1:d,1:d] numeric matrix with input distances |
OutputDists |
[1:d,1:d] numeric matrix with output distances |
Value
rho
Author(s)
Julian Märte
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
SpearmansRho(as.matrix(dist(Hepta$Data)),as.matrix(dist(projection)))
}
Topological Correlation
Description
Calculates the Topological Correlation
Usage
TopologicalCorrelation(Data,ProjectedPoints,type='norm',method,Kn=0)Arguments
Data |
[1:n, 1:d] a numeric matrix of the given n-dim. points: the rows represent the points and the columns represent the coordinates in the d-dim. space. |
ProjectedPoints |
[1:n, 1:2] numeric matrix of Projected Points, if missing, method should be set! |
method |
Determines whether the selected projections method for a given set of d-dim. points is a good choice. Therefor, a result of 1 means the selected projection method is good, and a result value of 0 means that the Visualization of the given Data in the two-dim. space doesnt fit the problem. |
type |
How the paths in the adjacency matrix should be weighted. 'norm' representes path lenthgs of 1 and eucldidean represents the distance in the euclidean metric. |
Kn |
k nearest neighbours in the graph. only needed in method is isomap and LocallyLinearEmbedding |
Value
TC value
Author(s)
Hermann Tafo, Laukert Schlichting 07/2015
Examples
#requires DatabionicSwarm v2.2.1
if(requireNamespace("FCPS")){
#data(Hepta,package="FCPS")
#projection=cmdscale(dist(Hepta$Data), k=2)
#TopologicalCorrelation(Hepta$Data,projection)
}
ZrehenMeasure4All
Description
A generalized version of the Zrehen-measure which defines the neighbourhood with a Gabriel Graph and is therefore not restricted to grid-based projections.
Usage
ZrehenMeasure4All(Data, Projection, width, height, isToroid = FALSE,
isGrid = TRUE, plotGabriel = FALSE)Arguments
Data |
[1:n,1:d] points in input room with d attributes |
Projection |
[1:n,1:2] projected points in output room, with index,x,y or index,line,column |
width |
Numeric: only necessary if toroid |
height |
Numeric: only necessary if toroid |
isToroid |
Boolean: are the points toroid? |
isGrid |
Boolean: is the grid a toroid? |
plotGabriel |
Boolean: plot the generated GabrielGraph (TRUE) or not (FALSE). Default: plotGabriel=FALSE. |
Value
List with
V$zrehen |
the raw zrehen measure |
V$normedzrehen |
the zrehen measure normed by the number of neighbours |
v$neighbourcounter |
the number of possible neighbours by which the zrehen measure is normed |
Author(s)
Florian Lerch 07/2015
Examples
if(requireNamespace("FCPS")){
data(Hepta,package="FCPS")
projection=cmdscale(dist(Hepta$Data), k=2)
ZrehenMeasure4All(Hepta$Data,projection)$zrehen
}
Computes Rescaled Average Agreement Rate
Description
Rescaled average agreement rate deduced by the co-ranking matrix from LCMC for various different sizes of the neighborhood.
Usage
plotMeasureRAAR(Raar, label = 'ProjectionMethod',
gPlotList = list(RAARplot = ggplot2::ggplot()), LineType="solid", Shape = 16,
PointsPerE = 10, fancy = FALSE)Arguments
Raar |
Output of RAAR() applied for a projection method. |
label |
Title of plot. |
gPlotList |
Settings for ggplot. |
LineType |
Character - graphic parameter: Line type of ggplot. |
Shape |
Integer: type of point |
PointsPerE |
Numeric graphic parameter: Distance between markers on plot line |
fancy |
Boolean graphic parameter: Some automatic settings for a more appealing plot. |
Value
ggplot object
Author(s)
Michael Thrun
Computes rank-based smoothed precision and recall
Description
Compares the projection in pData with the original data in Data and calculates trustworthiness and continuity of the projection for neighborhood sizes ranging from 1 to the size of the neighborhood.
Usage
plotMeasureTandD(TDmatrix, label = 'ProjectionMethod',
gPlotList = list(TW = ggplot2::ggplot(), DC = ggplot2::ggplot()), LineType = "solid",
Shape = 16, PointsPerE = 16)Arguments
TDmatrix |
Output of MeasureTundD() applied for a projection method. |
label |
Title of plot. |
gPlotList |
Settings for ggplot. |
LineType |
Character - graphic parameter: Line type of ggplot. |
Shape |
Integer: type of point |
PointsPerE |
Numeric graphic parameter: Distance between markers on plot line |
Value
ggplot object
Author(s)
Michael Thrun