Multivariate Analysis using Biplots

Description

Classical PCA biplot with aditional features as non-standard data transformations, scales for the variables, together with many graphical aids as sizes or colors of the points according to their qualities of representation or predictiveness. The package includes also Alternating Least Squares (ALS) or Criss-Cross procedures for the calculation of the reduced rank approximation that can deal with missing data, differencial weights for each element of the data matrix or even ronust versions of the procedure.

This is part of a bigger project called MULTBIPLOT that contains many other biplot techniques and is a translation to R of the package MULBIPLOT programmed in MATLAB. A GUI for the package is also in preparation.

Details

Author(s)

Jose Luis Vicente Villardon Maintainer: Jose Luis Vicente Villardon <villardon@usal.es>

References

Vicente-Villardon, J.L. (2010). MULTBIPLOT: A package for Multivariate Analysis using Biplots. Departamento de Estadistica. Universidad de Salamanca. (http://biplot.usal.es/ClassicalBiplot/index.html).

Vicente-Villardon, J. L. (1992). Una alternativa a las técnicas factoriales clasicas basada en una generalización de los metodos Biplot (Doctoral dissertation, Tesis. Universidad de Salamanca. España. 248 pp.[Links]).

Gabriel KR (1971) The biplot graphic display of matrices with application to principal component analysis. Biometrika 58(3):453-467

Gabriel KR (1998) Generalised bilinear regresion, J. L. (1998). Use of biplots to diagnose independence models in three-way contingency tables. Visualization of Categorical Data. Academic Press. London.

Gabriel, K. R. (2002). Le biplot-outil d'exploration de donnes multidimensionnelles. Journal de la Societe francaise de statistique, 143(3-4).

Gabriel KR, Zamir S (1979) Lower rank approximation of matrices by least squares with any choice of weights. Technometrics 21(4):489-498.

Gower J, Hand D (1996) Biplots. Monographs on statistics and applied probability. 54. London: Chapman and Hall., 277 pp.

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Qüestiió. 1986, vol. 10, núm. 1.

Demey J, Vicente-Villardon JL, Galindo MP, Zambrano A (2008) Identifying molecular markers associated with classification of genotypes using external logistic biplots. Bioinformatics 24(24):2832-2838.

Vicente-Villardon JL, Galindo MP, Blazquez-Zaballos A (2006) Logistic biplots. Multiple Correspondence Analysis and related methods pp 491-509.

Santos, C., Munoz, S. S., Gutierrez, Y., Hebrero, E., Vicente, J. L., Galindo, P., Rivas, J. C. (1991). Characterization of young red wines by application of HJ biplot analysis to anthocyanin profiles. Journal of Agricultural and food chemistry, 39(6), 1086-1090.

Rivas-Gonzalo, J. C., Gutierrez, Y., Polanco, A. M., Hebrero, E., Vicente, J. L., Galindo, P., Santos-Buelga, C. (1993). Biplot analysis applied to enological parameters in the geographical classification of young red wines. American journal of enology and viticulture, 44(3), 302-308.

Examples

data(iris)
bip=PCA.Biplot(iris[,1:4])
plot(bip)

Add suplementary binary variables to a biplot

Description

Add suplementary binary variables to a biplot of any kind

Usage

AddBinVars2Biplot(bip, Y, IncludeConst = TRUE, penalization = 0.2, 
freq = NULL, tolerance = 1e-05, maxiter = 100)

Arguments

Details

Fits binary variables to an existing biplot using penalized logistic regression.

Value

The biplot object with supplementary binary variables added.

Author(s)

Jose Luis Vicente Villardon

References

Vicente-Villardón, J. L., & Hernández-Sánchez, J. C. (2020). External Logistic Biplots for Mixed Types of Data. In Advanced Studies in Classification and Data Science (pp. 169-183). Springer, Singapore.

Examples

## No examples yet

Add clusters to a biplot object

Description

The function add clusters to a biplot object to be represented on the biplot. The clusters can be defined by a nominal variable provided by the user, obtained from the hclust function of the base package or from the kmeans function

Usage

AddCluster2Biplot(Bip, NGroups=3, ClusterType="hi", Groups=NULL, 
                  Original=FALSE, ClusterColors=NULL, ...)

Arguments

Details

One of the main shortcomings of cluster analysis is that it is not easy to search for the variables associated to the obtained classification; representing the clusters on the biplot can help to perform that interpretation. If you consider the technique for dimension reduction as a way to separate the signal from the noise, clusters should be constructed using the dimensions retained in the biplot, otherwise the complete original data matrix can be used. The colors used by each cluster should match the color used in the Dendrogram. User defined clusters can also be plotted, for example, to investigate the relation of the biplot solution to an external nominal variable.

Value

The function returns the biplot object with the information about the clusters added in new fields

Author(s)

Jose Luis Vicente Villardon

References

Demey, J. R., Vicente-Villardon, J. L., Galindo-Villardon, M. P., & Zambrano, A. Y. (2008). Identifying molecular markers associated with classification of genotypes by External Logistic Biplots. Bioinformatics, 24(24), 2832-2838.

Gallego-Alvarez, I., & Vicente-Villardon, J. L. (2012). Analysis of environmental indicators in international companies by applying the logistic biplot. Ecological Indicators, 23, 250-261.

Galindo, P. V., Vaz, T. D. N., & Nijkamp, P. (2011). Institutional capacity to dynamically innovate: an application to the Portuguese case. Technological Forecasting and Social Change, 78(1), 3-12.

Vazquez-de-Aldana, B. R., Garcia-Criado, B., Vicente-Tavera, S., & Zabalgogeazcoa, I. (2013). Fungal Endophyte (Epichloë festucae) Alters the Nutrient Content of Festuca rubra Regardless of Water Availability. PloS one, 8(12), e84539.

See Also

For clusters not provided by the user the function uses the standard procedures in hclust and kmeans.

Examples

data(Protein)
bip=PCA.Biplot(Protein[,3:11])
plot(bip)
# Add user defined clusters containing the region (North, South, Center)
bip=AddCluster2Biplot(bip, ClusterType="us", Groups=Protein$Region)
plot(bip, mode="a", margin=0.1, PlotClus=TRUE)

# Hierarchical clustering on the biplot coordinates using the Ward method
bip=AddCluster2Biplot(bip, ClusterType="hi", method="ward.D")
op <- par(mfrow=c(1,2))
plot(bip, mode="s", margin=0.1, PlotClus=TRUE)
plot(bip$Dendrogram)
par(op)
# K-means cluster on the biplot coordinates using the Ward method
bip=AddCluster2Biplot(bip, ClusterType="hi", method="ward.D")
op <- par(mfrow=c(1,2))
plot(bip, mode="s", margin=0.1, PlotClus=TRUE)
plot(bip$Dendrogram)
par(op)

Adds supplementary continuous variables to a biplot object

Description

Adds supplementary continuous variables to a biplot object

Usage

AddContVars2Biplot(bip, X, dims = NULL, Scaling = 5, Fit = NULL)

Arguments

Details

More types of fit will be added in the future

Value

A biplot object with the coordinates for the supplementary variables added.

Author(s)

Jose Luis Vicente Villardon

See Also

AddSupVars2Biplot

Examples

# Not yet

Adds supplementary ordinal variables to an existing biplot objects.

Description

Adds supplementary ordinal variables to an existing biplot objects.

Usage

AddOrdVars2Biplot(bip, Y, tol = 1e-06, maxiterlogist = 100, 
penalization = 0.2, showiter = TRUE, show = FALSE)

Arguments

Details

Adds supplementary ordinal variables to an existing biplot objects.

Value

An object with the information of the fits

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardon, J. L., & Hernandez-Sanchez, J. C. (2020). External Logistic Biplots for Mixed Types of Data. In Advanced Studies in Classification and Data Science (pp. 169-183). Springer, Singapore.

Examples

# not yet

Adds supplementary variables to a biplot object

Description

Adds supplementary bariables to a biplot object constructed with any of the biplot methods of the package. The new variables are fitted using the coordinates for the rows. Each variable is fitted using the adequate procedure for its type.

Usage

AddSupVars2Biplot(bip, X)

Arguments

Details

Binary, nominal or ordinal variables are fitted using logistic biplots. Continuous variables are fitted with linear regression.

Value

A biplot object with the coordinates for the supplementary variables added.

Author(s)

Jose Luis Vicente Villardon

See Also

AddContVars2Biplot

Examples

# Not yet

Bartlett tests

Description

Bartlett tests foor the columns of a matrix and a grouping variable

Usage

Bartlett.Tests(X, groups = NULL)

Arguments

Details

Bartlett tests foor the columns of a matrix and a grouping variable

Value

A matrix with the tests for each column

Author(s)

Jose Luis Vicente Villardon

References

Bartlett, M. S. (1937). "Properties of sufficiency and statistical tests". Proceedings of the Royal Statistical Society, Series A 160, 268-282 JSTOR 96803

Examples

data(wine)
Bartlett.Tests(wine[,4:8], groups = wine$Origin)

Basic descriptive sataistics

Description

Basic descriptive sataistics of several variables by the categories of a factor.

Usage

BasicDescription(X, groups = NULL, SortByGroups = FALSE, na.rm = FALSE, Intervals = TRUE)

Arguments

Details

Basic descriptive sataistics of several variables by the categories of a factor.

Value

A list with the description of each variable.

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
BasicDescription(wine[,4:8], groups = wine$Origin)

Binary Distances

Description

Calculates distances among rows of a binary data matrix or among the rows of two binary matrices. The end user will use BinaryProximities rather than this function. Input must be a matrix with 0 or 1 values.

Usage

BinaryDistances(x, y = NULL, coefficient= "Simple_Matching", transformation="sqrt(1-S)")

Arguments

Details

The following coefficients are calculated

1.- Kulezynski = a/(b + c)

2.- Russell_and_Rao = a/(a + b + c+d)

3.- Jaccard = a/(a + b + c)

4.- Simple_Matching = (a + d)/(a + b + c + d)

5.- Anderberg = a/(a + 2 * (b + c))

6.- Rogers_and_Tanimoto = (a + d)/(a + 2 * (b + c) + d)

7.- Sorensen_Dice_and_Czekanowski = a/(a + 0.5 * (b + c))

8.- Sneath_and_Sokal = (a + d)/(a + 0.5 * (b + c) + d)

9.- Hamman = (a - (b + c) + d)/(a + b + c + d)

10.- Kulezynski = 0.5 * ((a/(a + b)) + (a/(a + c)))

11.- Anderberg2 = 0.25 * (a/(a + b) + a/(a + c) + d/(c + d) + d/(b + d))

12.- Ochiai = a/sqrt((a + b) * (a + c))

13.- S13 = (a * d)/sqrt((a + b) * (a + c) * (d + b) * (d + c))

14.- Pearson_phi = (a * d - b * c)/sqrt((a + b) * (a + c) * (d + b) * (d + c))

15.- Yule = (a * d - b * c)/(a * d + b * c)

The following transformations of the similarity3 are calculated

1.- 'Identity' dis=sim

2.- '1-S' dis=1-sim

3.- 'sqrt(1-S)' dis = sqrt(1 - sim)

4.- '-log(s)' dis=-1*log(sim)

5.- '1/S-1' dis=1/sim -1

6.- 'sqrt(2(1-S))' dis== sqrt(2*(1 - sim))

7.- '1-(S+1)/2' dis=1-(sim+1)/2

8.- '1-abs(S)' dis=1-abs(sim)

9.- '1/(S+1)' dis=1/(sim)+1

Value

An object of class proximities.This has components:

Author(s)

Jose Luis Vicente-Villardon

References

Gower, J. C. (2006) Similarity dissimilarity and Distance, measures of. Encyclopedia of Statistical Sciences. 2nd. ed. Volume 12. Wiley

See Also

PrincipalCoordinates

Examples

data(spiders)

Binary logistic biplot with the EM algorithm.

Description

Binary logistic biplot with the EM algorithm

Usage

BinaryLogBiplotEM(x, freq = matrix(1, nrow(x), 1), aini = NULL,
dimens = 2, nnodos = 15, tol = 1e-04, maxiter = 100, penalization = 0.2)

Arguments

Details

Binary logistic biplot with the EM algorithm based on marginal maximum likelihood.

Value

A logistic biplot object.

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardón, J. L., Galindo-Villardón, M. P., & Blázquez-Zaballos, A. (2006). Logistic biplots. Multiple correspondence analysis and related methods. London: Chapman & Hall, 503-521.

Examples

# Not yet

Binary Logistic Biplot with Gradient Descent Estimation

Description

Binary Logistic Biplot with Gradient Descent Estimation. An external optimization function is used to calculate the parameters.

Usage

BinaryLogBiplotGD(X, freq = matrix(1, nrow(X), 1), dim = 2, tolerance =
                   1e-07, penalization = 0.01, num_max_iters = 100,
                   RotVarimax = FALSE, seed = 0, OptimMethod = "CG",
                   Initial = "random", Orthogonalize = FALSE, Algorithm =
                   "Joint", ...)

Arguments

Details

Fits a binary logistic biplot using gradient descent. The general function optim is used to optimize the loss function. Conjugate gradien is used as a default although other alternatives can be USED.

Value

An object of class "Binary.Logistic.Biplot".

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez, A. (2006) Logistic Biplots. In Multiple Correspondence Análisis And Related Methods. Grenacre, M & Blasius, J, Eds, Chapman and Hall, Boca Raton.

Demey, J., Vicente-Villardon, J. L., Galindo, M.P. AND Zambrano, A. (2008) Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics, 24(24): 2832-2838.

Examples

data(spiders)
X=Dataframe2BinaryMatrix(spiders)

logbip=BinaryLogBiplotGD(X,penalization=0.1)
plot(logbip, Mode="a")
summary(logbip)

Binary Logistic Biplot with Recursive Gradient Descent Estimation

Description

Binary Logistic Biplot with Recursive Gradient Descent Estimation. An external optimization function is used to calculate the parameters.

Usage

BinaryLogBiplotGDRecursive(X, freq = matrix(1, nrow(X), 1), dim = 2, tolerance = 1e-04, 
                          penalization = 0.2, num_max_iters = 100, 
                          RotVarimax = FALSE, OptimMethod = "CG", 
                          Initial = "random", ...)

Arguments

Details

Fits a binary logistic biplot using recursive gradient descent. The general function optim is used to optimize the loss function. Conjugate gradien is used as a default although other alternatives can be USED. It can be considered as a generalization of the NIPALS algorithm for a matrix of binary data.

Value

An object of class "Binary.Logistic.Biplot".

Author(s)

José Luis Vicente Villardon

References

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez, A. (2006) Logistic Biplots. In Multiple Correspondence Análisis And Related Methods. Grenacre, M & Blasius, J, Eds, Chapman and Hall, Boca Raton.

Demey, J., Vicente-Villardon, J. L., Galindo, M.P. AND Zambrano, A. (2008) Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics, 24(24): 2832-2838.

Examples

data(spiders)
X=Dataframe2BinaryMatrix(spiders)
logbip=BinaryLogBiplotGDRecursive(X,penalization=0.1)
plot(logbip, Mode="a")
summary(logbip)

Binary logistic biplot with a gradient descent algorithm.

Description

Binary logistic biplot with a gradient descent algorithm.

Usage

BinaryLogBiplotJoint(x, freq = matrix(1, nrow(x), 1), dim = 2, 
ainit = NULL, tolerance = 1e-04, maxiter = 30, penalization = 0.2, 
maxcond = 7, RotVarimax = FALSE, lambda = 0.1, ...)

Arguments

Details

Binary logistic biplot with a gradient descent algorithm. Estimates row and column parameters at the same time.

Value

A logistic biplot object.

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardón, J. L., Galindo-Villardón, M. P., & Blázquez-Zaballos, A. (2006). Logistic biplots. Multiple correspondence analysis and related methods. London: Chapman & Hall, 503-521.

Vicente-Villardon, J. L., & Vicente-Gonzalez, L. Redundancy Analysis for Binary Data Based on Logistic Responses in Data Analysis and Rationality in a Complex World. Springer.

Examples

# not yet

Binary logistic biplot with Item Response Theory.

Description

Binary logistic biplot with Item Response Theory.

Usage

BinaryLogBiplotMirt(x, dimens = 2, tolerance = 1e-04, 
maxiter = 30, penalization = 0.2, Rotation = "varimax", ...)

Arguments

Details

Binary logistic biplot with Item Response Theory.

Value

A logistic biplot object.

Author(s)

Jose Luis Vicente Villardon

References

Vicente-Villardón, J. L., Galindo-Villardón, M. P., & Blázquez-Zaballos, A. (2006). Logistic biplots. Multiple correspondence analysis and related methods. London: Chapman & Hall, 503-521.

Examples

# Not yet

Binary Logistic Biplot

Description

Fits a binary lo gistic biplot to a binary data matrix.

Usage

BinaryLogisticBiplot(x, dim = 2, compress = FALSE, init = "mca", 
method = "EM", rotation = "none", tol = 1e-04, 
maxiter = 100, penalization = 0.2, similarity = "Simple_Matching", ...)

Arguments

Details

Fits a binary lo gistic biplot to a binary data matrix.

Different Initial configurations can be selected:

1.- random : Random coordinates for each point.

2.- mirt: scores of the procedure mirt (Multidimensional Item Response Theory)

3.- PCoA: Principal Coordinates Analysis

4.- mca: Multiple Correspondence Analysis

We can use also different methods for the estimation

1.- Joint: Joint estimation of the row and column parameters. The Initial alorithm.

2.- EM: Marginal Maximum Likelihood

3.- mirt: Similar to the previous but fitted using the package mirt.

4.- JointGD: Joint estimation of the row and column methods using the gradient descent method.

5.- AlternatedGD: Alternated estimation of the row and column methods using the gradient descent method.

6.- External: Logistic fits on the Principal Coordinates Analysis.

7.- Recursive: Recursive (one axis at a time) estimation of the row and column methods using the gradient descent method. This is similar to the NIPALS algorithm for PCA

Value

A Logistic Biplot object.

Author(s)

Jose Luis Vicente Villardon

References

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez, A. (2006) Logistic Biplots. In Multiple Correspondence Análisis And Related Methods. Grenacre, M & Blasius, J, Eds, Chapman and Hall, Boca Raton.

Demey, J., Vicente-Villardon, J. L., Galindo, M.P. AND Zambrano, A. (2008) Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics, 24(24): 2832-2838.

See Also

BinaryLogBiplotJoint, BinaryLogBiplotEM, BinaryLogBiplotGD, BinaryLogBiplotMirt,

Examples

# data(spiders)
# X=Dataframe2BinaryMatrix(spiders)

# logbip=BinaryLogBiplotGD(X,penalization=0.1)
# plot(logbip, Mode="a")
# summary(logbip)

Binary PLS Regression.

Description

Fits Binary PLS regression.

Usage

BinaryPLSFit(Y, X, S = 2, tolerance = 5e-06, maxiter = 100, show = FALSE, 
          penalization = 0.1, OptimMethod = "CG", seed = 0) 

Arguments

Details

Fits Binary PLS Regression. It is used for a higher level function.

Value

The PLS fit used by the BinaryPLSR function.

Author(s)

Jose Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Vicente-Gonzalez, L., & Vicente-Villardon, J. L. (2022). Partial Least Squares Regression for Binary Responses and Its Associated Biplot Representation. Mathematics, 10(15), 2580.

Examples

## Not yet

Partial Least Squares Regression with Binary Data

Description

Fits Partial Least Squares Regression with Binary Data

Usage

BinaryPLSR(Y, X, S = 2, tolerance = 5e-05, maxiter = 100, show = FALSE,
                   penalization = 0.1, OptimMethod = "CG", seed = 0)

Arguments

Details

The function fits the PLSR method for the case when there are two sets of binary variables, using logistic rather than linear fits to take into account the nature of responses. We term the method BPLSR (Binary Partial Least Squares Regression). This can be considered as a generalization of the NIPALS algorithm when the data are all binary.

Value

Author(s)

José Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Vicente-Gonzalez, L., & Vicente-Villardon, J. L. (2022). Partial Least Squares Regression for Binary Responses and Its Associated Biplot Representation. Mathematics, 10(15), 2580.

Examples

X=as.matrix(wine[,4:21])
Y=cbind(Factor2Binary(wine[,1])[,1], Factor2Binary(wine[,2])[,1])
rownames(Y)=wine[,3]
colnames(Y)=c("Year", "Origin")
pls=PLSRBin(Y,X, penalization=0.1, show=TRUE, S=2)

Proximity Measures for Binary Data

Description

Calculation of proxymities among rows or columns of a binary data matrix or a data frame that will be converted into a binary data matrix.

Usage

BinaryProximities(x, y = NULL, coefficient = "Jaccard", transformation =
                 NULL, transpose = FALSE, ...)

Arguments

Details

A binary data matrix is a matrix with values 0 or 1 coding the absence or presence of several binary characters. When a data frame is provided, every variable in the data frame is converted to a binary variable using the function Dataframe2BinaryMatrix. Factors with two levels are converted directly to binary variables, factors with more than two levels are converted to a matrix with as meny columns as levels and numerical variables are converted to binary variables using a cut point that can be the median, the mean or a value provided by the user.

The following coefficients are calculated

1.- Kulezynski = a/(b + c)

2.- Russell_and_Rao = a/(a + b + c+d)

3.- Jaccard = a/(a + b + c)

4.- Simple_Matching = (a + d)/(a + b + c + d)

5.- Anderberg = a/(a + 2 * (b + c))

6.- Rogers_and_Tanimoto = (a + d)/(a + 2 * (b + c) + d)

7.- Sorensen_Dice_and_Czekanowski = a/(a + 0.5 * (b + c))

8.- Sneath_and_Sokal = (a + d)/(a + 0.5 * (b + c) + d)

9.- Hamman = (a - (b + c) + d)/(a + b + c + d)

10.- Kulezynski = 0.5 * ((a/(a + b)) + (a/(a + c)))

11.- Anderberg2 = 0.25 * (a/(a + b) + a/(a + c) + d/(c + d) + d/(b + d))

12.- Ochiai = a/sqrt((a + b) * (a + c))

13.- S13 = (a * d)/sqrt((a + b) * (a + c) * (d + b) * (d + c))

14.- Pearson_phi = (a * d - b * c)/sqrt((a + b) * (a + c) * (d + b) * (d + c))

15.- Yule = (a * d - b * c)/(a * d + b * c)

The following transformations of the similarity3 are calculated

1.- 'Identity' dis=sim

2.- '1-S' dis=1-sim

3.- 'sqrt(1-S)' dis = sqrt(1 - sim)

4.- '-log(s)' dis=-1*log(sim)

5.- '1/S-1' dis=1/sim -1

6.- 'sqrt(2(1-S))' dis== sqrt(2*(1 - sim))

7.- '1-(S+1)/2' dis=1-(sim+1)/2

8.- '1-abs(S)' dis=1-abs(sim)

9.- '1/(S+1)' dis=1/(sim)+1

Note that, after transformation the similarities are converted to distances except for "Identity". Not all the transformations are suitable for all the coefficients. Use them at your own risk. The default values are admissible combinations.

Value

An object of class proximities.This has components:

Author(s)

Jose Luis Vicente-Villardon

References

Gower, J. C. (2006) Similarity dissimilarity and Distance, measures of. Encyclopedia of Statistical Sciences. 2nd. ed. Volume 12. Wiley

See Also

BinaryDistances, Dataframe2BinaryMatrix

Examples

data(spiders)
D=BinaryProximities(spiders, coefficient="Jaccard", transformation="sqrt(1-S)")
D2=BinaryProximities(spiders, coefficient=3, transformation=3)

Biplot for a PLSR model with binary data

Description

Builds a Biplot for a PLSR model with binary data

Usage

Biplot.BinaryPLSR(plsr, BinBiplotType=1)

Arguments

Details

Builds a Biplot for a PLSR model with binary data. The result is a biplot for the matrix with the binary predictors (X) adding the binary responses as suplementary variables. There are two possible types, 1 for the biplot directly obtained in the fit (the default) and 2 for the biplot obtaines after refitting the binary variables using Ridge Logistic Regression.

Value

An object of class Binary.Logistic.Biplot

Author(s)

Jose Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Vicente-Gonzalez, L., & Vicente-Villardon, J. L. (2022). Partial Least Squares Regression for Binary Responses and Its Associated Biplot Representation. Mathematics, 10(15), 2580.

Examples

X=as.matrix(wine[,4:21])
Y=cbind(Factor2Binary(wine[,1])[,1], Factor2Binary(wine[,2])[,1])
rownames(Y)=wine[,3]
colnames(Y)=c("Year", "Origin")
pls=PLSRBin(Y,X, penalization=0.1, show=TRUE, S=2)
plsbip=Biplot.PLSRBIN(pls, BinBiplotType=1)
plsbip=AddCluster2Biplot(plsbip, ClusterType = "us", 
       Groups = wine$Group)
plot(plsbip, margin=0.05, mode="s", PlotClus = TRUE, 
    ModeSupBinVars = "s", ShowAxis = FALSE, 
    ColorSupBinVars = "blue",     CexInd=0.5, 
    ClustCenters = TRUE, LabelInd = FALSE, ShowBox = TRUE)

Partial Least Squares Biplot

Description

Adds a Biplot to a Partial Lest Squares (plsr) object.

Usage

Biplot.PLSR(plsr)

Arguments

Details

Adds a Biplot to a Partial Lest Squares (plsr) object. The biplot is constructed with the matrix of predictors, the dependent variable is projected onto the biplot as a continuous supplementary variable.

Value

An object of class ContinuousBiplot with the dependent variables as supplemntary.

Author(s)

Jose Luis Vicente Villardon

References

Oyedele, O. F., & Lubbe, S. (2015). The construction of a partial least-squares biplot. Journal of Applied Statistics, 42(11), 2449-2460.

See Also

PLSR

Examples

X=as.matrix(wine[,4:21])
y=as.numeric(wine[,2])-1
mifit=PLSR(y,X, Validation="None")
mibip=Biplot.PLSR(mifit)
plot(mibip, PlotVars=TRUE, IndLabels = y, ColorInd=y+1)

Biplot for a PLSR model with a binary response

Description

Biplot for a PLSR model with a binary response

Usage

Biplot.PLSR1BIN(plsr)

Arguments

Details

Biplot for a PLSR model with a binary response

Value

The biplot for the independent variables with the response as supplementary binary variable.

Author(s)

Jose Luis Vicente Villardon

References

Ugarte-Fajardo, J., Bayona-Andrade, O., Criollo-Bonilla, R., Cevallos-Cevallos, J., Mariduena-Zavala, M., Ochoa-Donoso, D., & Vicente-Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

See Also

PLSR1Bin

Examples

# Not Yet

Biplot for a PLSR model with binary responses

Description

Builds a Biplot for a PLSR model with binary responses

Usage

Biplot.PLSRBIN(plsr, BinBiplotType = 1)

Arguments

Details

Builds a Biplot for a PLSR model with binary responses. The result is a biplot for the matrix with the predictors (X) adding the binary responses as suplementary variables. There are two possible types, 1 for the biplot directly obtained in the fit ( the default) and 2 for the biplot obtaines after refitting the binary variables using Ridge Logistic Regression.

Value

An object of class ContinuousBiplot

Author(s)

Jose Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Examples

X=as.matrix(wine[,4:21])
Y=cbind(Factor2Binary(wine[,1])[,1], Factor2Binary(wine[,2])[,1])
rownames(Y)=wine[,3]
colnames(Y)=c("Year", "Origin")
pls=PLSRBin(Y,X, penalization=0.1, show=TRUE, S=2)
plsbip=Biplot.PLSRBIN(pls, BinBiplotType=1)
plsbip=AddCluster2Biplot(plsbip, ClusterType = "us", 
       Groups = wine$Group)
plot(plsbip, margin=0.05, mode="s", PlotClus = TRUE, 
    ModeSupBinVars = "s", ShowAxis = FALSE, 
    ColorSupBinVars = "blue",     CexInd=0.5, 
    ClustCenters = TRUE, LabelInd = FALSE, ShowBox = TRUE)

External Biplot for functional data from a functional PCA object.

Description

The function calculates a biplot from a functional PCA object and the data used tocalculate it.

Usage

BiplotFPCA(FPCA, X)

Arguments

Details

The function calculates a biplot from a functional PCA object and the data used tocalculate it. At this moment the function calculates only an external biplot by regressing X o the funcional components. Furure versions will include the internal biplot.

Value

A Continuous biplot object

Author(s)

José Luis Vicente Villardón

Examples

# not yet

Bootstrap on the distance matrices used for Principal Coordinates Analysis (PCoA)

Description

Obtains bootstrap replicates of a distance matrix using ramdom samples or permuatations of the residual matrix from a Principal Coordinates (Components) Analysis. The object is to estimate the sampling variability of absorbed variances, coordinates and qualities of representation in a PCoA.

Usage

BootstrapDistance(D, W=diag(nrow(D)), nB=200, dimsol=2, 
                  ProcrustesRot=TRUE, method=c("Sampling", "Permutation"))

Arguments

Details

The function calculates bootstrap confidence intervals for the inertia, coordinates and qualties of representation of a Principal Coordinates Analysis using a distance matrix as a basis. The funcion uses random sampling or permutations of the residuals to obtain the bootstrap replications. The procedure preserves the length of the points in the multidimensional space perturbating only the angles among the vectors. It is done so to preserve the property of positiveness of the diagonal elements of the scalar product matrices. The procedure may result into a scalar product that does not have an euclidean configuration and then has some negative eigenvalues; to avoid this problem the negative eigenvalues are removed to approximate the perturbated matrix by the closest with the required properties.

It is well known that the eigenvectors of a matrix are unique except for reflections, that is, if we change the sign of each component of the eigenvector we have the same solution. If that happens, an unwanted increase in the variability due to this artifact may invalidate the results. To avoid this we can calculate the scalar product of each eigenvector of the initial matrix with the corresponding eigenvector of the bootstrap replicate and change the signs of the later if the result is negative.

Another artifact of the procedure may arise when the dimension of the solution is higher than 1 because the eigenvectors of a replicate may generate the same subspace although are not in the same directions, i. e., the subspace is referred to a different system. That also may produce an unwanted increase of the variability that invalidates the results. To avoid this, every replicate may be rotated to match as much as possible the subspace generated by the eigenvectors of the initial matrix. This is done by Procrustes Analysis, taking the rotated matrix as solution. The solution to this problem is also a sulution to the reflection, then only this problem is considered.

Value

Returns an object of class "PCoABootstrap" with the information for each bootstrap replication.

Author(s)

Jose L. Vicente-Villardon

References

Efron, B.; Tibshirani, RJ. (1993). An introduction to the bootstrap. New York: Chapman and Hall. 436p.

Ringrose, T. J. (1992). Bootstrapping and correspondence analysis in archaeology. Journal of Archaeological Science, 19(6), 615-629.

MILAN, L., & WHITTAKER, J. (1995). Application of the parametric bootstrap to models that incorporate a singular value decomposition. Applied statistics, 44(1), 31-49.

See Also

BootstrapScalar, ~~~

Examples

data(spiders)
D=BinaryProximities(spiders, coefficient="Jaccard", transformation="sqrt(1-S)")
DB=BootstrapDistance(D$Proximities)

Bootstrap on the scalar product matrices used for Principal Coordinates Analysis (PCoA)

Description

Obtains bootstrap replicates of a scalar products matrix using ramdom samples or permuatations of the residual matrix from a Principal Coordinates (Components) Analysis. The object is to estimate the sampling variability of absorbed variances, coordinates and qualities of representation in a PCoA.

Usage

BootstrapScalar(B, W=diag(nrow(B)), nB=200, dimsol=2, 
                ProcrustesRot=TRUE, method=c("Sampling", "Permutation"))

Arguments

Details

The function calculates bootstrap confidence intervals for the inertia, coordinates and qualties of representation of a Principal Coordinates Analysis using a distance matrix as a basis. The funcion uses random sampling or permutations of the residuals to obtain the bootstrap replications. The procedure preserves the length of the points in the multidimensional space perturbating only the angles among the vectors. It is done so to preserve the property of positiveness of the diagonal elements of the scalar product matrices. The procedure may result into a scalar product that does not have an euclidean configuration and then has some negative eigenvalues; to avoid this problem the negative eigenvalues are removed to approximate the perturbated matrix by the closest with the required properties.

It is well known that the eigenvectors of a matrix are unique except for reflections, that is, if we change the sign of each component of the eigenvector we have the same solution. If that happens, an unwanted increase in the variability due to this artifact may invalidate the results. To avoid this we can calculate the scalar product of each eigenvector of the initial matrix with the corresponding eigenvector of the bootstrap replicate and change the signs of the later if the result is negative.

Another artifact of the procedure may arise when the dimension of the solution is higher than 1 because the eigenvectors of a replicate may generate the same subspace although are not in the same directions, i. e., the subspace is referred to a different system. That also may produce an unwanted increase of the variability that invalidates the results. To avoid this, every replicate may be rotated to match as much as possible the subspace generated by the eigenvectors of the initial matrix. This is done by Procrustes Analysis, taking the rotated matrix as solution. The solution to this problem is also a sulution to the reflection, then only this problem is considered.

Value

Returns an object of class "PCoABootstrap" with the information for each bootstrap replication.

Author(s)

Jose L. Vicente-Villardon

References

Efron, B.; Tibshirani, RJ. (1993). An introduction to the bootstrap. New York: Chapman and Hall. 436p.

Ringrose, T. J. (1992). Bootstrapping and correspondence analysis in archaeology. Journal of Archaeological Science, 19(6), 615-629.

Milan, L., & Whittaker, J. (1995). Application of the parametric bootstrap to models that incorporate a singular value decomposition. Applied statistics, 44(1), 31-49.

See Also

BootstrapScalar

Examples

## Not yet

Bootstrap on the distance matrices used for MDS with Smacof

Description

Obtains bootstrap replicates of a distance matrix using ramdom samples or permuatations of a distance matrix. The object is to estimate the sampling variability of the results of the Smacof algorithm.

Usage

BootstrapSmacof(D, W=NULL, Model=c("Identity", "Ratio", "Interval", "Ordinal"), 
                dimsol=2, maxiter=100, maxerror=0.000001, StandardizeDisparities=TRUE,
                ShowIter=TRUE, nB=200, ProcrustesRot=TRUE, 
                method=c("Sampling", "Permutation"))

Arguments

Details

The function calculates bootstrap confidence intervals for coordinates and different stress measures using a distance matrix as a basis. The funcion uses random sampling or permutations of the residuals to obtain the bootstrap replications. The procedure preserves the length of the points in the multidimensional space perturbating only the angles among the vectors. It is done so to preserve the property of positiveness of the diagonal elements of the scalar product matrices. The procedure may result into a scalar product that does not have an euclidean configuration and then has some negative eigenvalues; to avoid this problem the negative eigenvalues are removed to approximate the perturbated matrix by the closest with the required properties.

It is well known that the eigenvectors of a matrix are unique except for reflections, that is, if we change the sign of each component of the eigenvector we have the same solution. If that happens, an unwanted increase in the variability due to this artifact may invalidate the results. To avoid this we can calculate the scalar product of each eigenvector of the initial matrix with the corresponding eigenvector of the bootstrap replicate and change the signs of the later if the result is negative.

Another artifact of the procedure may arise when the dimension of the solution is higher than 1 because the eigenvectors of a replicate may generate the same subspace although are not in the same directions, i. e., the subspace is referred to a different system. That also may produce an unwanted increase of the variability that invalidates the results. To avoid this, every replicate may be rotated to match as much as possible the subspace generated by the eigenvectors of the initial matrix. This is done by Procrustes Analysis, taking the rotated matrix as solution. The solution to this problem is also a sulution to the reflection, then only this problem is considered.

Value

Returns an object of class "PCoABootstrap" with the information for each bootstrap replication.

Author(s)

Jose L. Vicente-Villardon

References

Efron, B.; Tibshirani, RJ. (1993). An introduction to the bootstrap. New York: Chapman and Hall. 436p.

Ringrose, T. J. (1992). Bootstrapping and correspondence analysis in archaeology. Journal of Archaeological Science, 19(6), 615-629.

MILAN, L., & WHITTAKER, J. (1995). Application of the parametric bootstrap to models that incorporate a singular value decomposition. Applied statistics, 44(1), 31-49.

Jacoby, W. G., & Armstrong, D. A. (2014). Bootstrap Confidence Regions for Multidimensional Scaling Solutions. American Journal of Political Science, 58(1), 264-278.

See Also

BootstrapScalar

Examples

data(spiders)
D=BinaryProximities(spiders, coefficient="Jaccard", transformation="sqrt(1-S)")
DB=BootstrapDistance(D$Proximities)

Panel of box plots

Description

Panel of box plots for a set of numerical variables and a grouping factor.

Usage

BoxPlotPanel(X, groups = NULL, nrows = NULL, panel = TRUE, 
notch = FALSE, GroupsTogether = TRUE, ...)

Arguments

Details

Panel of box plots for a set of numerical variables and a grouping factor.

Value

The box plot panel

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
BoxPlotPanel(wine[,4:7], groups = wine$Origin, nrows = 2, ylab="")

Correspondence Analysis

Description

Correspondence Analysis for a frequency or abundace data matrix.

Usage

CA(x, dim = 2, alpha = 1)

Arguments

Details

Calculates Correspondence Analysis for a tww-way frequency or abundance table

Value

Correspondence analysis solution

Author(s)

Jose Luis Vicente Villardon

References

Benzécri, J. P. (1992). Correspondence analysis handbook. New York: Marcel Dekker.

Greenacre, M. J. (1984). Theory and applications of correspondence analysis. Academic Press.

Examples

data(SpidersSp)
cabip=CA(SpidersSp)
plot(cabip)

Canonical Correspondence Analysis

Description

Calculates the solution of a Canonical Correspondence Analysis Biplot

Usage

CCA(P, Z, alpha = 1, dimens = 4)

Arguments

Details

A pair of ecological tables, made of a species abundance matrix and an environmental variables matrix measured at the same sampling sites, is usually analyzed by Canonical Correspondence Analysis (CCA) (Ter BRAAK, 1986). CCA can be considered as a Correspondence Analysis (CA) in which the ordination axis are constrained to be linear combinations of the environmental variables. Recently the procedure has been extended to other disciplines as Sociology or Psichology and it is potentially useful in many other fields.

Value

A CCA solution object

Author(s)

Jose Luis vicente Villardon

References

Ter Braak, C. J. (1986). Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology, 67(5), 1167-1179.

Johnson, K. W., & Altman, N. S. (1999). Canonical correspondence analysis as an approximation to Gaussian ordination. Environmetrics, 10(1), 39-52.

Graffelman, J. (2001). Quality statistics in canonical correspondence analysis. Environmetrics, 12(5), 485-497.

Graffelman, J., & Tuft, R. (2004). Site scores and conditional biplots in canonical correspondence analysis. Environmetrics, 15(1), 67-80.

Greenacre, M. (2010). Canonical correspondence analysis in social science research (pp. 279-286). Springer Berlin Heidelberg.

Examples

data(riano)
Sp=riano[,3:15]
Env=riano[,16:25]
ccabip=CCA(Sp, Env)
plot(ccabip)

Biplot representation of a Canonical Variate Analysis or a Manova (Canonical-Biplot or MANOVA-Biplot)

Description

Calculates a canonical biplot with confidence regions for the means.

Usage

Canonical.Variate.Analysis(X, group, InitialTransform = 5)

Arguments

Details

The Biplot method (Gabriel, 1971; Galindo, 1986; Gower and Hand, 1996) is becoming one of the most popular techniques for analysing multivariate data. Biplot methods are techniques for simultaneous representation of the n rows and n columns of a data matrix \bf{X}, in reduced dimensions, where the rows represent individuals, objects or samples and the columns the variables measured on them. Classical Biplot methods are a graphical representation of a Principal Components Analysis (PCA) that it is used to obtain linear combinations that successively maximize the total variability. PCA is not considered an appropriate approach where there is known a priori group structure in the data. The most general methodology for discrimination among groups, using multiple observed variables, is Canonical Variate Analysis (CVA). CVA allows us to derive linear combinations that successively maximize the ratio of "between-groups"" to "pooled within-group" sample variance. Several authors propose a Biplot representation for CVA called Canonical Biplot (CB) (Vicente-Villardon, 1992 and Gower & Hand, 1996) when it is oriented to the discrimination between groups or MANOVA-Biplot Gabriel (1972, 1995) when the aim is to study the variables responsible for the discrimination. The main advantage of the Biplot version of the technique is that it is possible not only to establish the differences between groups but also to characterise the variables responsible for them. The methodology is not yet widely used mainly because it is still not available in the major statistical packages. Amaro, Vicente-Villardon & Galindo (2004) extend the methodology for two-way designs and propose confidence circles based on univariate and multivariate tests to perform post-hoc analysis of each variable.

Value

An object of class "Canonical.Biplot"

Author(s)

Jose Luis Vicente Villardon

References

Amaro, I. R., Vicente-Villardon, J. L., & Galindo-Villardon, M. P. (2004). Manova Biplot para arreglos de tratamientos con dos factores basado en modelos lineales generales multivariantes. Interciencia, 29(1), 26-32.

Vicente-Villardón, J. L. (1992). Una alternativa a las técnicas factoriales clásicas basada en una generalización de los métodos Biplot (Doctoral dissertation, Tesis. Universidad de Salamanca. España. 248 pp.[Links]).

Gabriel KR (1971) The biplot graphic display of matrices with application to principal component analysis. Biometrika 58(3):453-467.

Gabriel, K. R. (1995). MANOVA biplots for two-way contingency tables. WJ Krzanowski (Ed.), Recent advances in descriptive multivariate analysis, Oxford University Press, Toronto. 227-268.

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Qüestiió. 1986, vol. 10, núm. 1.

Gower y Hand (1996): Biplots. Chapman & Hall.

Varas, M. J., Vicente-Tavera, S., Molina, E., & Vicente-Villardon, J. L. (2005). Role of canonical biplot method in the study of building stones: an example from Spanish monumental heritage. Environmetrics, 16(4), 405-419.

Santana, M. A., Romay, G., Matehus, J., Villardon, J. L., & Demey, J. R. (2009). simple and low-cost strategy for micropropagation of cassava (Manihot esculenta Crantz). African Journal of Biotechnology, 8(16).

Examples

data(wine)
X=wine[,4:21]
canbip=CanonicalBiplot(X, group=wine$Group)
plot(canbip, mode="s")

Biplot representation of a Canonical Variate Analysis or a Manova (Canonical-Biplot or MANOVA-Biplot)

Description

Calculates a canonical biplot with confidence regions for the means.

Usage

CanonicalBiplot(X, group, SUP = NULL, InitialTransform = 5, LDA=FALSE, MANOVA = FALSE)

Arguments

Details

The Biplot method (Gabriel, 1971; Galindo, 1986; Gower and Hand, 1996) is becoming one of the most popular techniques for analysing multivariate data. Biplot methods are techniques for simultaneous representation of the n rows and n columns of a data matrix \bf{X}, in reduced dimensions, where the rows represent individuals, objects or samples and the columns the variables measured on them. Classical Biplot methods are a graphical representation of a Principal Components Analysis (PCA) that it is used to obtain linear combinations that successively maximize the total variability. PCA is not considered an appropriate approach where there is known a priori group structure in the data. The most general methodology for discrimination among groups, using multiple observed variables, is Canonical Variate Analysis (CVA). CVA allows us to derive linear combinations that successively maximize the ratio of "between-groups"" to "pooled within-group" sample variance. Several authors propose a Biplot representation for CVA called Canonical Biplot (CB) (Vicente-Villardon, 1992 and Gower & Hand, 1996) when it is oriented to the discrimination between groups or MANOVA-Biplot Gabriel (1972, 1995) when the aim is to study the variables responsible for the discrimination. The main advantage of the Biplot version of the technique is that it is possible not only to establish the differences between groups but also to characterise the variables responsible for them. The methodology is not yet widely used mainly because it is still not available in the major statistical packages. Amaro, Vicente-Villardon & Galindo (2004) extend the methodology for two-way designs and propose confidence circles based on univariate and multivariate tests to perform post-hoc analysis of each variable.

Value

An object of class "Canonical.Biplot"

Author(s)

Jose Luis Vicente Villardon

References

Amaro, I. R., Vicente-Villardon, J. L., & Galindo-Villardon, M. P. (2004). Manova Biplot para arreglos de tratamientos con dos factores basado en modelos lineales generales multivariantes. Interciencia, 29(1), 26-32.

Vicente-Villardón, J. L. (1992). Una alternativa a las técnicas factoriales clásicas basada en una generalización de los métodos Biplot (Doctoral dissertation, Tesis. Universidad de Salamanca. España. 248 pp.[Links]).

Gabriel KR (1971) The biplot graphic display of matrices with application to principal component analysis. Biometrika 58(3):453-467.

Gabriel, K. R. (1995). MANOVA biplots for two-way contingency tables. WJ Krzanowski (Ed.), Recent advances in descriptive multivariate analysis, Oxford University Press, Toronto. 227-268.

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Qüestiió. 1986, vol. 10, núm. 1.

Gower y Hand (1996): Biplots. Chapman & Hall.

Varas, M. J., Vicente-Tavera, S., Molina, E., & Vicente-Villardon, J. L. (2005). Role of canonical biplot method in the study of building stones: an example from Spanish monumental heritage. Environmetrics, 16(4), 405-419.

Santana, M. A., Romay, G., Matehus, J., Villardon, J. L., & Demey, J. R. (2009). simple and low-cost strategy for micropropagation of cassava (Manihot esculenta Crantz). African Journal of Biotechnology, 8(16).

Examples

data(wine)
X=wine[,4:21]
canbip=CanonicalBiplot(X, group=wine$Group)
plot(canbip, mode="s")

MANOVA and Canonical Analysis of Distances

Description

Performs a MANOVA and a Canonical Analysis based on of Distance Matrices (usally for continuous data)

Usage

CanonicalDistanceAnalysis(Prox, group, dimens = 2, Nsamples = 1000, 
PCoA = "Standard", ProjectInd = TRUE)

Arguments

Details

Performs a MANOVA and a Canonical Analysis based on of Distance Matrices (usally for continuous data). The MANOVA statistics is calculated from a decomposition of the distance matrix based on a factor of a external classification. The significance of the test is calculated using a premutation test. The approach depens only on the distances and then is useful with any kind of data.

The Canonical Representation is calculated from a Principal Coordinates Analysis od the distance matrix among the means. Thus, it is only possible for continuous data. The PCoA representation can be "Standard" using the means directly, "Weighted" weighting each group with its sample size or using weighted Princiopal Components Analysis of the matrix of means.

A measure of the quality of representation of the groups is provided. When possible, the measure is also provided for the individual points.

Soon, a biplot representation will also be developed.

Value

An object of class "CanonicalDistanceAnalysis" with:

Note

The MANOVA and the representation of the means can be applied to any Distance althoug the projection of the individuals can be made only for continuous data.

Author(s)

Jose Luis Vicente Villardon

References

Gower, J. C., & Krzanowski, W. J. (1999). Analysis of distance for structured multivariate data and extensions to multivariate analysis of variance. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(4), 505-519.

Krzanowski, W. J. (2004). Biplots for multifactorial analysis of distance. Biometrics, 60(2), 517-524.

Examples

data(iris)
group=iris[,5]
X=as.matrix(iris[1:4])
D=ContinuousProximities(X,  coef = 1)
CDA=CanonicalDistanceAnalysis(D, group, dimens=2)
summary(CDA)
plot(CDA)

CANONICAL STATIS-ACT for multiple tables with common rows and its associated Biplot

Description

The procedure performs STATIS-ACT methodology for multiple tables with common rows and its associated biplot

Usage

CanonicalStatisBiplot(X, Groups, InitTransform = "Standardize columns", dimens = 2,
                 SameVar = FALSE)

Arguments

Details

The procedure performs Canonical STATIS-ACT methodology for multiple tables with common rows and its associated biplot. When the variables are the same for all occasions trajectories for the variables can also be plotted.

Value

An object of class StatisBiplot

Author(s)

Jose Luis Vicente Villardon

References

Vallejo-Arboleda, A., Vicente-Villardon, J. L., & Galindo-Villardon, M. P. (2007). Canonical STATIS: Biplot analysis of multi-table group structured data based on STATIS-ACT methodology. Computational statistics & data analysis, 51(9), 4193-4205.

Abdi, H., Williams, L.J., Valentin, D., & Bennani-Dosse, M. (2012). STATIS and DISTATIS: optimum multitable principal component analysis and three way metric multidimensional scaling. WIREs Comput Stat, 4, 124-167.

Efron, B.,Tibshirani, RJ. (1993). An introduction to the bootstrap. New York: Chapman and Hall. 436p.

Escoufier, Y. (1976). Operateur associe a un tableau de donnees. Annales de laInsee, 22-23, 165-178.

Escoufier, Y. (1987). The duality diagram: a means for better practical applications. En P. Legendre & L. Legendre (Eds.), Developments in Numerical Ecology, pp. 139-156, NATO Advanced Institute, Serie G. Berlin: Springer.

L'Hermier des Plantes, H. (1976). Structuration des Tableaux a Trois Indices de la Statistique. [These de Troisieme Cycle]. University of Montpellier, France.

Ringrose, T.J. (1992). Bootstrapping and Correspondence Analysis in Archaeology. Journal of Archaeological Science. 19:615-629.

Examples

data(Chemical)
x= Chemical[37:144,5:9]
weeks=as.factor(as.numeric(Chemical$WEEKS[37:144]))
levels(weeks)=c("W2" , "W3", "W4")
X=Convert2ThreeWay(x,weeks, columns=FALSE)
Groups=Chemical$Treatment[1:36]
canstbip=CanonicalStatisBiplot(X, Groups, SameVar = TRUE)
plot(canstbip, mode="s", PlotVars=TRUE, ShowBox=TRUE)

Distances among individuals using nominal variables.

Description

Distances among individuals using nominal variables.

Usage

CategoricalDistances(x, y = NULL, coefficient = "GOW", transformation = "sqrt(1-S)")

Arguments

Details

The function calculates similarities and dissimilarities among a set ob ogjects characterized by a set of nominal variables. The function uses similarities and converts into dissimilarities using a variety of transformations controled by the user.

Value

A matrix with distances among the rows of x and y. If y is NULL the interdistances among the rows of x are calculated.

Author(s)

Jose Luis Vicente Villardon

References

dos Santos, T. R., & Zarate, L. E. (2015). Categorical data clustering: What similarity measure to recommend?. Expert Systems with Applications, 42(3), 1247-1260.

Boriah, S., Chandola, V., & Kumar, V. (2008). Similarity measures for categorical data: A comparative evaluation. red, 30(2), 3.

Examples

##---- Should be DIRECTLY executable !! ----

Proximities among individuals using nominal variables.

Description

Proximities among individuals using nominal variables.

Usage

CategoricalProximities(Data, SUP = NULL, coefficient = "GOW", transformation = 3, ...)

Arguments

Details

The function calculates similarities and dissimilarities among a set ob ogjects characterized by a set of nominal variables. The function uses similarities and converts into dissimilarities using a variety of transformations controled by the user.

Value

A list of Values

Author(s)

Jose Luis Vicente Villardon

References

dos Santos, T. R., & Zarate, L. E. (2015). Categorical data clustering: What similarity measure to recommend?. Expert Systems with Applications, 42(3), 1247-1260.

Boriah, S., Chandola, V., & Kumar, V. (2008). Similarity measures for categorical data: A comparative evaluation. red, 30(2), 3.

Examples

data(Doctors)
Dis=CategoricalProximities(Doctors, SUP=NULL, coefficient="GOW" , transformation=3)
pco=PrincipalCoordinates(Dis)
plot(pco, RowCex=0.7, RowColors=as.integer(Doctors[[1]]), RowLabels=as.character(Doctors[[1]]))

Checks if a data matrix is binary

Description

Checks if a data matrix is binary

Usage

CheckBinaryMatrix(x)

Arguments

Details

Checks if all the entries of the matix are either 0 or 1.

Value

TRUE if the matrix is binary.

Author(s)

Jose Luis Vicente-Villardon

Examples

data(spiders)
sp=Dataframe2BinaryMatrix(spiders)
CheckBinaryMatrix(sp)

Checks if a vector is binary

Description

Checks if all the entries of a vector are 0 or 1

Usage

CheckBinaryVector(x)

Arguments

Value

The logical result

Author(s)

Jose luis Vicente Villardon

Examples

x=c(0, 0, 0, 0,  1, 1, 1, 2)
CheckBinaryVector(x)

Chemical data

Description

Ecological data

Usage

data("Chemical")

Format

A data frame with 324 observations on the following 16 variables.

Treatment

a factor with levels F0N0 F0N1 F0N2 F0N3 F1N0 F1N1 F1N2 F1N3 F2N0 F2N1 F2N2 F2N3

FISH

a factor with levels F0 F1 F2

NUTRIENTS

a factor with levels N0 N1 N2 N3

WEEKS

a factor with levels W1 W2 W3 W4 W5 W6 W7 W8 W9

TEMPERATURE

a numeric vector

pH

a numeric vector

ALKALINITYmeql

a numeric vector

CO2free

a numeric vector

NNH4mgl

a numeric vector

NNO3mgl

a numeric vector

SRPmglP

a numeric vector

TPmgl

a numeric vector

TSSmgl

a numeric vector

CONDUCTIVITYmScm

a numeric vector

TSPmglP

a numeric vector

Chlorophyllamgl

a numeric vector

Details

Chemical Data

Source

Department of Ecology. University of Leon. (Spain)

References

To add

Examples

data(Chemical)
## maybe str(Chemical) ; plot(Chemical) ...

Draws a circle

Description

Draws a circle for a given radius at the specified center with the given color

Usage

Circle(radius = 1, origin = c(0, 0), col = 1, ...)

Arguments

Details

Draws a circle for a given radius at the specified center with the given color

Value

No value is returned

Author(s)

Jose Luis Vicente Villardon

Examples

plot(0,0)
Circle(1,c(0,0))

Coinertia Analysis.

Description

Calculates a Coinertia Analysis for two matrices of continuous data

Usage

Coinertia(X, Y, ScalingX = 5, ScalingY = 5, dimsol = 3)

Arguments

Details

Coinertia analysis for two continuous data matrices.

Value

An object of class Coinertia.SOL

Author(s)

Jose Luis Vicente Villardon

References

Doledec, S., & Chessel, D. (1994). Co-inertia analysis: an alternative method for studying species-environment relationships. Freshwater biology, 31(3), 277-294.

Examples

SSI$Year == "a2006"
SSI2D=SSI[SSI$Year == "a2006",3:23]
rownames(SSI2D)=as.character(SSI$Country[SSI$Year == "a2006"])
SSIHuman2D=SSI2D[,1:9]
SSIEnvir2D=SSI2D[,10:16]
SSIEcon2D=SSI2D[,17:21]
Coin=Coinertia(SSIHuman2D, SSIEnvir2D)

Plots the contributios of a biplot

Description

Plots the contributios of a biplot

Usage

ColContributionPlot(bip, A1 = 1, A2 = 2, Colors = NULL, Labs = NULL, 
MinQuality = 0, CorrelationScale = FALSE, ContributionScale = TRUE, 
AddSigns2Labs = TRUE, ...)

Arguments

Details

Plots the contributions on a plot that sows also the sum for both axes-

Value

The contribution plot

Author(s)

Jose Luis Vicente Villardon

Examples

## Simple Biplot with arrows
data(Protein)
bip=PCA.Biplot(Protein[,3:11])

# Plot of the Variable Contributions
ColContributionPlot(bip, cex=1)

Concentration ellipse for a se of two-dimensional points

Description

The function calculates a non-parametric concentration ellipse for a set of two-dimensional points.

Usage

ConcEllipse(data, confidence=1, npoints=100)

Arguments

Details

The procedre uses the Mahalanobis distances to determine the points that will be used for the calculations.

Value

A list with the following fields

Author(s)

Jose Luis Vicente Villardon

References

Meulman, J. J., & Heiser, W. J. (1983). The display of bootstrap solutions in multidimensional scaling. Murray Hill, NJ: Bell Laboratories.

Linting, M., Meulman, J. J., Groenen, P. J., & Van der Kooij, A. J. (2007). Stability of nonlinear principal components analysis: An empirical study using the balanced bootstrap. Psychological Methods, 12(3), 359.

Examples

data(iris)
dat=as.matrix(iris[1:50,1:2])
plot(iris[,1], iris[,2],col=iris[,5], asp=1)
E=ConcEllipse(dat, 0.95)
plot(E)

Confidence Interval for the mean

Description

Calculates Confidence Interval for the mean of a Numerical Variable.

Usage

ConfidenceInterval(x, Desv = NULL, df = NULL, Confidence = 0.95)

Arguments

Details

Calculates Confidence Interval for the mean of a Numerical Variable.

Value

The confidence Interval for the mean

Author(s)

Jose Luis Vicente Villardon

Examples

# Not yet

Constrained Binary Logistic Biplot

Description

Constrained Binary Logistic Biplot or Redundancy Analysis for Binary Data based on logistic responses

Usage

ConstrainedLogisticBiplot(Y, X, dim = 2, Scaling = 5, tolerance = 1e-05, 
maxiter = 100, penalization = 0.1)

Arguments

Details

Constrained Binary Logistic Biplot or Redundancy Analysis for Binary Data based on logistic responses.

Value

A logistic Biplot with the reponse and the predictive variables projected onto it.

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardon, J. L., & Vicente-Gonzalez, L. Redundancy Analysis for Binary Data Based on Logistic Responses in Data Analysis and Rationality in a Complex World. Springer.

Examples

# not yet

Constrained Ordinal Logistic Biplot

Description

Constrained Ordinal Logistic Biplot or Redundancy Analysis for Ordinal Data based on logistic responses

Usage

ConstrainedOrdinalLogisticBiplot(Y, X, dim = 2, Scaling = 5, 
tolerance = 1e-05, maxiter = 100, penalization = 0.1, show = FALSE)

Arguments

Details

Constrained Ordinal Logistic Biplot or Redundancy Analysis for Ordinal Data based on logistic responses.

Value

An ordinal logistic Biplot with the reponse and the predictive variables projected onto it.

Author(s)

Jose Luis Vicente-Villardon

References

Vicente-Villardon, J. L., & Vicente-Gonzalez, L. Redundancy Analysis for Binary Data Based on Logistic Responses in Data Analysis and Rationality in a Complex World. Springer.

Examples

# not yet

Distances for Continuous Data

Description

Calculates distances among rows of a continuous data matrix or among the rows of two continuous matrices.

Usage

  ContinuousDistances(x, y = NULL, coef = "Pythagorean", r = 1)

Arguments

Details

The following coefficients are calculated

1.- Pythagorean = sqrt(sum((y[i, ] - x[j, ])^2)/p)

2.- Taxonomic = sqrt(sum(((y[i,]-x[j,])^2)/r^2)/p)

3.- City = sum(abs(y[i,]-x[j,])/r)/p

4.- Minkowski = (sum((abs(y[i,]-x[j,])/r)^t)/p)^(1/t)

5.- Divergence = sqrt(sum((y[i,]-x[j,])^2/(y[i,]+x[j,])^2)/p)

6.- dif_sum = sum(abs(y[i,]-x[j,])/abs(y[i,]+x[j,]))/p

7.- Camberra = sum(abs(y[i,]-x[j,])/(abs(y[i,])+abs(x[j,])))

8.- Bray_Curtis = sum(abs(y[i,]-x[j,]))/sum(y[i,]+x[j,])

9.- Soergel = sum(abs(y[i,]-x[j,]))/sum(apply(rbind(y[i,],x[j,]),2,max))

10.- Ware_hedges = sum(abs(y[i,]-x[j,]))/sum(apply(rbind(y[i,],x[j,]),2,max))

Value

A list with:

Author(s)

Jose Luis Vicente-Villardon

References

Gower, J. C. (2006) Similarity dissimilarity and Distance, measures of. Encyclopedia of Statistical Sciences. 2nd. ed. Volume 12. Wiley

See Also

PrincipalCoordinates

Examples

data(wine)
dis=ContinuousDistances(wine[,4:21])

Proximities for Continuous Data

Description

Calculates proximities among rows of a continuous data matrix or among the rows of two continuous matrices.

Usage

ContinuousProximities(x, y = NULL, ysup = FALSE, 
transpose = FALSE, coef = "Pythagorean", r = 1)

Arguments

Details

The following coefficients are calculated

1.- Pythagorean = sqrt(sum((y[i, ] - x[j, ])^2)/p)

2.- Taxonomic = sqrt(sum(((y[i,]-x[j,])^2)/r^2)/p)

3.- City = sum(abs(y[i,]-x[j,])/r)/p

4.- Minkowski = (sum((abs(y[i,]-x[j,])/r)^t)/p)^(1/t)

5.- Divergence = sqrt(sum((y[i,]-x[j,])^2/(y[i,]+x[j,])^2)/p)

6.- dif_sum = sum(abs(y[i,]-x[j,])/abs(y[i,]+x[j,]))/p

7.- Camberra = sum(abs(y[i,]-x[j,])/(abs(y[i,])+abs(x[j,])))

8.- Bray_Curtis = sum(abs(y[i,]-x[j,]))/sum(y[i,]+x[j,])

9.- Soergel = sum(abs(y[i,]-x[j,]))/sum(apply(rbind(y[i,],x[j,]),2,max))

10.- Ware_hedges = sum(abs(y[i,]-x[j,]))/sum(apply(rbind(y[i,],x[j,]),2,max))

Value

Author(s)

Jose Luis Vicente-Villardon

References

Gower, J. C. (2006) Similarity dissimilarity and Distance, measures of. Encyclopedia of Statistical Sciences. 2nd. ed. Volume 12. Wiley

Examples

data(wine)
dis=ContinuousProximities(wine[,4:21])

Three way array from a two way matrix

Description

Converts a two-dimensional matrix into a list where each cell is the two dimensional data matrix for an occasion or group.

Usage

Convert2ThreeWay(x, groups, columns = FALSE, RowNames = NULL)

Arguments

Details

Converts a two dimensional matrix into a multitable list according to the groups provided by the user. Each field of the list has the name of the corresponding group.

Value

A Multitable list. Ech filed is the data matrix for a group.

Author(s)

Jose Luis Vicente Villardon

Examples

data(Chemical)
x= Chemical[,5:16]
X=Convert2ThreeWay(x,Chemical$WEEKS, columns=FALSE)

Converts a three way array into a list

Description

Converts a three way array into a list

Usage

Convert3wArray2List(X)

Arguments

Details

Converts a three way array into a list

Value

A list

Author(s)

Jose Luis Vicente-Villardon

Examples

#No examples yet

Convert a factor to integer numbers

Description

Convert a factor to integer numbers

Usage

ConvertFactors2Integers(x)

Arguments

Details

Convert a factor to integer numbers

Value

a vector with the converted values

Author(s)

Jose Luis Vicente Villardon

Examples

##---- Should be DIRECTLY executable !! ----

Converts a list of matrices into a three way array

Description

Converts a list of matrices into a three way array. All the matrices in the list must have the same size.

Usage

ConvertList23wArray(X)

Arguments

Details

Converts a list of matrices into a three way array. All the matrices in the list must have the same size.

Value

A three-way array

Author(s)

Jose Luis Vicente-Villardon

Examples

# No examples yet

Circle of correlations

Description

Circle of correlations among the manifiest variables and the principal comonents (or dimensions of any biplot).

Usage

CorrelationCircle(bip, A1 = 1, A2 = 2, Colors = NULL, Labs = NULL, ...)

Arguments

Details

Circle of correlations among the manifiest variables and the principal comonents (or dimensions of any biplot).

Value

The plot of the circle of correlations

Author(s)

Jose Luis Vicente Villardon

Examples

bip=PCA.Biplot(wine[,4:21])
CorrelationCircle(bip)

Alternated Least Squares Biplot

Description

Alternated Least Squares Biplot with any choice of weigths for each element of the data matrix

Usage

CrissCross(x, w = matrix(1, dim(x)[1], dim(x)[2]), dimens = 2, a0 = NULL, 
b0 = NULL, maxiter = 100, tol = 1e-04, addsvd = TRUE, lambda = 0)

Arguments

Details

The function calculates Alternated Least Squares Biplot with any choice of weigths for each element of the data matrix. The function is useful when we want a low rank approximation of a data matrix in which each element of the matrix has a different weight, for example, all the weights are 1 except for the missing elements that are 0, or to model the logarithms of a frequency table using the frequencies as weights.

Value

An object of class .Biplot" with the following components:

Author(s)

Jose Luis Vicente Villardon

References

GABRIEL, K.R. and ZAMIR, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21: 489-498.

See Also

LogFrequencyBiplot

Examples

data(Protein)
X=as.matrix(Protein[,3:11])
X = InitialTransform(X, transform=5)$X
bip=CrissCross(X)

Cummulative sums

Description

Cummulative sums

Usage

CumSum(X, dimens = 1)

Arguments

Details

Cummulative sums within rows (dimens=1) or columns (dimens=2) of a data matrix

Value

A matrix of the same size as X with cummulative sums within each row or each column

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
X=wine[,4:21]
CumSum(X,1)
CumSum(X,2)

Prepares a matrix for regression from a data frame

Description

Prepares a matrix for regression from a data frame

Usage

DataFrame2Matrix4Regression(X, last = TRUE, Intercept = FALSE)

Arguments

Details

Nominal variables are converted to a matrix of dummy variables for regression.

Value

A matrix ready to use as independent variables in a regression

Author(s)

Jose Luis Vicente Vilardon

Examples

##---- Should be DIRECTLY executable !! ----

Converts a Data Frame into a Binary Data Matrix

Description

Converts a Data Frame into a Binary Data Matrix

Usage

Dataframe2BinaryMatrix(dataf, cuttype = "Median", cut = NULL, BinFact = TRUE)

Arguments

Details

The function converts a data frame into a Binary Data Matrix (A matrix with entries either 0 or 1).

Factors with two levels are directly transformed into a column of 0/1 entries.

Factors with more than two levels are converted into a binary submatrix with as many rows as x and as many columns as levels or categories. (Indicator matrix)

Integer Variables are treated as factors

Continuous Variables are converted into binary variables using a cut point that can be the median, the mean or a value provided by the user.

Value

A Binary Data Matrix.

Author(s)

Jose Luis Vicente Villardon

Examples

data(spiders)
Dataframe2BinaryMatrix(spiders)

Adds Non-parametric densities to a biplot. Separated densities are calculated for different clusters

Description

Adds Non-parametric densities to a biplot. Separated densities are calculated for different clusters

Usage

DensityBiplot(X, y = NULL, grouplabels = NULL, ncontours = 6, 
groupcolors = NULL, ncolors=20, ColorType=4)

Arguments

Details

Non parametric densities are used to investigate the concentration of row points on different areas of the biplot representation. The densities can be calculated for different groups or clusters in order to investigate if individuals with differnt characteristics are concentrated on particular areas of the biplot. The procedure is particularly useful with a high number of individuals.

Value

No value returned. It has effect on the graph.

Author(s)

Jose Luis Vicente Villardon

References

Gower, J. C., Lubbe, S. G., & Le Roux, N. J. (2011). Understanding biplots. John Wiley & Sons.

Examples

bip=PCA.Biplot(iris[,1:4])
plot(bip, mode="s", CexInd=0.1)

Calculation of Disparities

Description

Calculation of Disparities for a MDS model

Usage

Dhats(P, D, W, Model = c("Identity", "Ratio", "Interval", "Ordinal"), Standardize = TRUE)

Arguments

Details

Calculation of disparities using standard or monotone regression depending on the MDS model.

Value

Returns the proximities.

Author(s)

Jose L. Vicente Villardon

References

Borg, I., & Groenen, P. J. (2005). Modern multidimensional scaling: Theory and applications. Springer.

Examples

## Function is used inside MDS or smacof

Labels for the selected dimensions in a biplot

Description

Creates a character vector with labels for the dimensions of the biplot

Usage

DimensionLabels(dimens, Root = "Dim")

Arguments

Details

An auxiliary function to cretae labels for the dimensions. Useful to label the matrices of results

Value

Returns a vector of labels

Author(s)

Jose Luis Vicente Villardon

Examples

DimensionLabels(dimens=3, Root = "Dim")

Data set extracted from the Careers of doctorate holders survey carried out by Spanish Statistical Office in 2008.

Description

The sample data, as part of a large survey, corresponds to 100 people who have the PhD degree and it shows the level of satisfaction of the doctorate holders about some issues.

Usage

data(Doctors)

Format

This data frame contains 100 observation for the following 5 ordinal variables, with four categories each: (1= "Very Satisfied", 2= "Somewhat Satisfied",3="Somewhat dissatisfied", 4="Very dissatisfied")

Salary

Benefits

Job Security

Job Location

Working conditions

Source

Spanish Statistical Institute. Survey of PDH holders, 2006. URL: http://www.ine.es.

Examples

data(Doctors)
## maybe str(Doctors) ; plot(Doctors) ...

Plots a panel of error bars

Description

Plots a panel of error bars to compare the means of several variables in the levels of a factor using confidence intervals.

Usage

ErrorBarPlotPanel(X, groups = NULL, nrows = NULL, panel = TRUE, 
GroupsTogether = TRUE, Confidence = 0.95, p.adjust.method = "None", 
UseANOVA = FALSE, Colors = "blue", Title = "Error Bar Plot", 
sort = TRUE, ...)

Arguments

Details

The funtion plots a panel of error bars plots to compare several groups for several variables.

Value

A panel of error bars plots.

Author(s)

Jose Luis Vicente Villardon

Examples

ErrorBarPlotPanel(wine[4:9], wine$Group, UseANOVA=TRUE, Title="", sort=FALSE)

Classical Euclidean Distance (Pythagorean Distance)

Description

Calculates the eucliden distances among the rows of an euclidean configurations in any dimensions

Usage

EuclideanDistance(x)

Arguments

Details

eucliden distances among the rows of an euclidean configurations in any dimensions

Value

Returns the distance matrix

Author(s)

Jose Luis Vicente Villardon

Examples

x=matrix(runif(20),10,2)
D=EuclideanDistance(x)

Expands a compressed table of patterns and frequencies

Description

Expands a compressed table of patterns and frequencies

Usage

ExpandTable(table)

Arguments

Details

To simplify the calculations of some of the algorithms we compress the tables by searching for the distinct patterns and its frequencies. This function recovers the original data. It serves also to assign the corrdinates on the biplot to the original individuals.

Value

A matrix with the original data

Author(s)

Jose Luis Vicente Villardon

Examples

##---- Should be DIRECTLY executable !! ----

External Logistic Biplot for binary Data

Description

Fits an External Logistic Biplot to the results of a Principal Coordinates Analysis obtained from binary data.

Usage

ExternalBinaryLogisticBiplot(Pco, IncludeConst=TRUE,  penalization=0.2, freq=NULL, 
tolerance = 1e-05, maxiter = 100)

Arguments

Details

Let {\bf{X}} be the matrix of binary data scored as present or absent (1 or 0), in which the rows correspond to n individuals or entries (for example, genotypes) and the columns to p binary characters (for example alleles or bands), let {\bf{S}} = ({s_{ij}}) be a matrix containing the similarities among rows, obtained from the binary data matrix , and let \Delta = ({\delta _{ij}}) be the corresponding dissimilarity/distance matrix, taking for example {\delta _{ij}} = \sqrt {1 - {s_{ij}}}. Despite the fact that, in Cluster Analysis and Principal Coordinates Analysis, interpretation of the variables responsible for grouping or ordination is not straightforward, those methods are normally used to classify individual in which binary variables have been measured. we use a combination of Principal Coordinates Analysis (PCoA), Cluster Analysis (CA) and External Logistic Regression (ELB), as a better way to interpret the binary variables associated to the classification of genotypes. The combination of three standard techniques with some new ideas about the geometry of the procedures, allows to construct a External Logistic Regression (ELB), that helps the interpretation of the variables responsible for the classification or ordination. Suppose we have obtained an euclidean configuration {\bf{Y}} obtained from the Principal Coordinates (PCoA) of the similarity matrix. To search for the variables associated to the ordination obtained in PCoA, we can look for the directions in the ordination diagram that better predict the probability of presence of each allele. More formally, if we defined {\pi _{ij}} = E({x_{ij}})= {\textstyle{1 \over {1 + \exp ( - ({b_{j0}} + \sum\limits_{s = 1}^k {{b_{js}}{y_{is}}} ))}}} as the expected probability that the allele j be present at genotype for a genotype with coordinates y_{is} (i=1, ...,n; s=1, ..., k) on the ordination diagram, as where bjs ( j=1,..., p) are the logistic regression coefficients that correspond to the jth variable (alleles or bands) in the sth dimension. The model is a generalized linear model having the logit as a link function. where and , y's and b's define a biplot in logit scale. This is called External Logistic Biplot because the coordinates of the genotypes are calculated in an external procedure (PCoA). Given that the y's are known from PCoA, obtaining the b´s is equivalent to performing a logistic regression using the j-th column of X as a response variable and the columns of y as regressors.

Value

An object of class External.Binary.Logistic.Biplot with the fields of the Principal.Coordinates object with the following fields added.

Author(s)

Jose Luis Vicente Villardon

References

Demey, J., Vicente-Villardon, J. L., Galindo, M.P. AND Zambrano, A. (2008) Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics, 24(24): 2832-2838.

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez, A. (2006) Logistic Biplots. In Multiple Correspondence Análisis And Related Methods. Grenacre, M & Blasius, J, Eds, Chapman and Hall, Boca Raton.

Examples

data(spiders)
x2=Dataframe2BinaryMatrix(spiders)
colnames(x2)=colnames(spiders)
dist=BinaryProximities(x2)
pco=PrincipalCoordinates(dist)
pcobip=ExternalBinaryLogisticBiplot(pco)

Extracts unique patterns and its frequencies for a discrete data matrix (numeric)

Description

Extracts the patterns and the frequencies of a discrete data matrix reducing the size of the data matrix in order to accelerate calculations in some techniques.

Usage

ExtractTable(x)

Arguments

Details

For any numerical matrix, calculates the different patterns and the frequencies associated to each pattern The result contains the pattern matrix, a vector with the frequencies, a list with rows sharing the same pattern. The final pattern matrix has different ordering than the original matrix.

Value

Author(s)

Jose Luis Vicente-Villardon

Examples

data(spiders)
spidersbin=Dataframe2BinaryMatrix(spiders)
spiderstable=ExtractTable(spidersbin)

Biplot for Factor Analysis.

Description

Biplot used as a graphical representation of Factor Analysis.

Usage

FA.Biplot(X, dimension = 3, Extraction="PC", Rotation="varimax", 
         InitComunal="A1", normalize=FALSE, Scores= "Regression",  
         MethodArgs=NULL, sup.rows = NULL, sup.cols = NULL, ...)

Arguments

Details

Biplots represent the rows and columns of a data matrix in reduced dimensions. Usually rows represent individuals, objects or samples and columns are variables measured on them. The most classical versions can be thought as visualizations associated to Principal Components Analysis (PCA) or Factor Analysis (FA) obtained from a Singular Value Decomposition or a related method. From another point of view, Classical Biplots could be obtained from regressions and calibrations that are essentially an alternated least squares algorithm equivalent to an EM-algorithm when data are normal This routine Calculates a biplot as a graphical representation of a classical Factor Analysis alowing for different extraction methods and different rotations.

Value

An object of class "ContinuousBiplot" with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Gabriel, K.R.(1971): The biplot graphic display of matrices with applications to principal component analysis. Biometrika, 58, 453-467.

Gabriel, K. R. AND Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21(21):489–498, 1979.

Gabriel, K.R.(1998): Generalised Bilinear Regression. Biometrika, 85, 3, 689-700.

Gower y Hand (1996): Biplots. Chapman & Hall.

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez-Zaballos, A. (2006). Logistic Biplots. Multiple Correspondence Analysis and related methods 491-509.

See Also

InitialTransform

Examples

data(Protein)
X=Protein[,3:11]
bip=FA.Biplot(X, Extraction="ML", Rotation="oblimin")
plot(bip, mode="s", margin=0.05, AddArrow=TRUE)

Converts a Factor into its indicator matrix

Description

Converts a factor into a binary matrix with as many columns as categories of the factor

Usage

Factor2Binary(y, Name = NULL)

Arguments

Value

An indicator binary matrix

Author(s)

Jose Luis Vicente Villardon

Examples

y=factor(c(1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1))
Factor2Binary(y)

Selection of a fraction of the data

Description

Selects a percentage of the data eliminating the observations with higher Mahalanobis distances to the center.

Usage

Fraction(data, confidence = 1)

Arguments

Details

The function is used to select a fraction of the data to be plotted for example when clusters are used. The function eliminates the extreme values.

Value

An object of class fraction with the following fields

Author(s)

Jose Luis Vicente Villardon

References

Meulman, J. J., & Heiser, W. J. (1983). The display of bootstrap solutions in multidimensional scaling. Murray Hill, NJ: Bell Laboratories.

Linting, M., Meulman, J. J., Groenen, P. J., & Van der Kooij, A. J. (2007). Stability of nonlinear principal components analysis: An empirical study using the balanced bootstrap. Psychological Methods, 12(3), 359.

See Also

ConcEllipse, AddCluster2Biplot

Examples

a=matrix(runif(50), 25,2)
a2=Fraction(a, 0.7)

Biplot for continuous data based on gradient descent methods

Description

Biplot for continuous data based on gradient descent methods.

Usage

GD.Biplot(X, dimension = 2, Scaling = 5, 
         lambda = 0.01, OptimMethod = "CG", 
         Orthogonalize = FALSE, Algorithm = "Alternated", 
         sup.rows = NULL, sup.cols = NULL,
         grouping = NULL, tolerance = 1e-04, 
         num_max_iters = 300, Initial = "random")

Arguments

Details

The function calculates a bilot using gradient descent methods. The function optim is used to optimize the loss function. By default CG (Conjugate Gradient) method is used althoug other possibilities can be used.

Value

An object of class "ContinuousBiplot" is returned.

Author(s)

Jose Luis Vicente Villardon

Examples

data("Protein")
X=Protein[,3:11]
gdbip=GD.Biplot(X, dimension=2, Algorithm="Joint", 
Orthogonalize=FALSE, lambda=0.3, Initial="random")
plot(gdbip)
summary(gdbip)

Games-Howell post-hoc tests for Welch's one-way analysis

Description

This function produces results from Games-Howell post-hoc tests for Welch's one-way analysis of variance (ANOVA) for a matrix of numeric data and a grouping variable.

Usage

Games_Howell(data, group)

Arguments

Details

This function produces results from Games-Howell post-hoc tests for Welch's one-way analysis of variance (ANOVA) for a matrix of numeric data and a grouping variable.

Value

The tests for each column of the data matrix

Author(s)

Jose Luis Vicente Villardon

References

Ruxton, G. D., & Beauchamp, G. (2008). Time for some a priori thinking about post hoc testing. Behavioral ecology, 19(3), 690-693.

Examples

# Not yet

Generalized Procrustes Analysis

Description

Generalized Procrustes Analysis

Usage

  GeneralizedProcrustes(x, tolerance = 1e-05, maxiter = 100, Plot = FALSE)

Arguments

Details

Generalized Procrustes Analysis for several configurations contained in a three-dimensional array.

Value

An object of class GenProcustes.This has components:

Author(s)

Jose Luis Vicente-Villardon

References

Gower, J.C., (1975). Generalised Procrustes analysis. Psychometrika 40, 33-51.

Ingwer Borg, I. & Groenen, P. J.F. (2005). Modern Multidimensional Scaling. Theory and Applications. Second Edition. Springer

See Also

PrincipalCoordinates

Examples

data(spiders)
n=dim(spiders)[1]
p=dim(spiders)[2]
prox=array(0,c(n,2,4))

p1=BinaryProximities(spiders,coefficient=5)
prox[,,1]=PrincipalCoordinates(p1)$RowCoordinates
p2=BinaryProximities(spiders,coefficient=2)
prox[,,2]=PrincipalCoordinates(p2)$RowCoordinates
p3=BinaryProximities(spiders,coefficient=3)
prox[,,3]=PrincipalCoordinates(p3)$RowCoordinates
p4=BinaryProximities(spiders,coefficient=4)
prox[,,4]=PrincipalCoordinates(p4)$RowCoordinates
GeneralizedProcrustes(prox)

Calculates the scales for the variables on a linear biplot

Description

Calculates the scales for the variables on a linear prediction biplot There are several types of scales and values that can be shown on the graphical representation. See details.

Usage

GetBiplotScales(Biplot, nticks = 3, TypeScale = "Complete", ValuesScale = "Original")

Arguments

Details

The function calculates the points on the biplot axes where the scales should be placed.

There are three types of scales when the transformations of the raw data are made by columns:

"Complete": Covers the whole range of the variable using the number of ticks specified in "nticks". A smaller number of points could be shown if some fall outsite the range of the scatter.

"StdDev": The mean +/- 1, 2 and 3 times the standard deviation.A smaller number of points could be shown if some fall outsite the range of the scatter.

"BoxPlot": Median, 25, 75 percentiles maximum and minimum values are shown. The extremes of the interquartile range are connected with a thicker line. A smaller number of points could be shown if some fall outsite the range of the scatter.

There are two kinds of values that can be shown on the biplot axis:

"Original": The values before transformation. Only makes sense when the transformations are for each column.

"Transformed": The values after transformation, for example, after standardization.

Although the function is public, the end used will not normally use it.

Value

A list with the following components:

Author(s)

Jose Luis Vicente Villardon

Examples

data(iris)
bip=PCA.Biplot(iris[,1:4])
GetBiplotScales(bip)

Calculates scales for plotting the environmental variables in a Canonical Correspondence Analysis

Description

Calculates scales for plotting the environmental variables in a Canonical Correspondence Analysis

Usage

GetCCAScales(CCA, nticks = 7, TypeScale = "Complete", ValuesScale = "Original")

Arguments

Details

Calculates scales for plotting the environmental variables in a Canonical Correspondence Analysis

Value

Returns the points and the labels for each biplot axis

Author(s)

Jose Luis Vicente Villardon

References

Gower, J. C., & Hand, D. J. (1995). Biplots (Vol. 54). CRC Press.

Gower, J. C., Lubbe, S. G., & Le Roux, N. J. (2011). Understanding biplots. John Wiley & Sons.

Vicente-Villardón, J. L., Galindo Villardón, M. P., & Blázquez Zaballos, A. (2006). Logistic biplots. Multiple correspondence analysis and related methods. London: Chapman & Hall, 503-521.

Examples

# No examples yet

Gower Dissimilarities for mixed types of data

Description

Gower Dissimilarities for mixed types of data

Usage

  GowerProximities(x, y = NULL, Binary = NULL, Classes = NULL,
                   transformation = 3, IntegerAsOrdinal = FALSE, BinCoef
                   = "Simple_Matching", ContCoef = "Gower", NomCoef =
                   "GOW", OrdCoef = "GOW")

Arguments

Details

The transformation sqrt(1-S) is applied to the similarity.

Value

An object of class proximities.This has components:

Author(s)

Jose Luis Vicente-Villardon

References

J. C. Gower. (1971) A General Coefficient of Similarity and Some of its Properties. Biometrics, Vol. 27, No. 4, pp. 857-871.

Examples

data(spiders)

Gower Dissimilarities for mixed types of data

Description

Gower Dissimilarities for mixed types of data

Usage

  GowerSimilarities(x, y = NULL, Classes = NULL, transformation =
                   "sqrt(1-S)", BinCoef = "Simple_Matching", ContCoef =
                   "Gower", NomCoef = "GOW", OrdCoef = "GOW")

Arguments

Details

Gower Dissimilarities for mixed types of data. The transformation sqrt(1-S) is applied to the similarity by default.

Value

An object of class proximities.This has components:

Author(s)

Jose Luis Vicente-Villardon

References

J. C. Gower. (1971) A General Coefficient of Similarity and Some of its Properties. Biometrics, Vol. 27, No. 4, pp. 857-871.

Examples

data(spiders)

HJ Biplot with added features.

Description

HJ Biplot with added features.

Usage

HJ.Biplot(X, dimension = 3, Scaling = 5, sup.rows = NULL, 
         sup.cols = NULL, grouping = NULL)

Arguments

Details

Biplots represent the rows and columns of a data matrix in reduced dimensions. Usually rows represent individuals, objects or samples and columns are variables measured on them. The most classical versions can be thought as visualizations associated to Principal Components Analysis (PCA) or Factor Analysis (FA) obtained from a Singular Value Decomposition or a related method. From another point of view, Classical Biplots could be obtained from regressions and calibrations that are essentially an alternated least squares algorithm equivalent to an EM-algorithm when data are normal.

Value

An object of class ContinuousBiplot with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Questiio. 1986, vol. 10, núm. 1.

See Also

InitialTransform

Examples

## Simple Biplot with arrows
data(Protein)
bip=HJ.Biplot(Protein[,3:11])
plot(bip)

Gauss-Hermite quadrature

Description

Find the Gauss-Hermite abscissae and weights.

Usage

Hermquad(N)

Arguments

Details

Find the Gauss-Hermite abscissae and weights.

Value

Author(s)

Jose Luis Vicente Villardon (translated from a Matlab function by Greg von Winckel) )

References

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical Recipes in C: The Art of Scientific Computing (New York. Cambridge University Press, 636-9.

http://www.mathworks.com/matlabcentral/fileexchange/8836-hermite-quadrature/content/hermquad.m

Examples

Hermquad(10)

Panel of histograms

Description

Panel of histograms for a set of numerical variables.

Usage

HistogramPanel(X, nrows = NULL, separated = FALSE, ...)

Arguments

Details

Jose Luis Vicente Villardon

Value

The histogram panel.

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
HistogramPanel(wine[,4:7], nrows = 2, xlab="")

Checks if a point is inside a box.

Description

Checks if a point is inside a box. The point is specified bi its x and y coordinates and the bom with the minimum and maximum values on both coordinate axis: xmin, xmax, ymin, ymax. The vertices of the box are then (xmin, ymin), (xmax, ymin), (xmax, ymax) and (xmin, ymax)

Usage

InBox(x, y, xmin, xmax, ymin, ymax)

Arguments

Value

Returns a logical value : TRUE if the point is inside the box and FALSE otherwise.

Author(s)

Jose Luis Vicente Villardon

Examples

InBox(0, 0, -1, 1, -1, 1) 

Initial transformation of data

Description

Initial transformation of data before the construction of a biplot. (or any other technique)

Usage

InitialTransform(X, sup.rows = NULL, sup.cols = NULL, 
InitTransform = "None", transform = "Standardize columns", 
grouping = NULL)

Arguments

Details

Possible Transformations are:

1.- "Raw Data": When no transformation is required.

2.- "Substract the global mean": Eliminate an eefect common to all the observations

3.- "Double centering" : Interaction residuals. When all the elements of the table are comparable. Useful for AMMI models.

4.- "Column centering": Remove the column means.

5.- "Standardize columns": Remove the column means and divide by its standard deviation.

6.- "Row centering": Remove the row means.

7.- "Standardize rows": Divide each row by its standard deviation.

8.- "Divide by the column means and center": The resulting dispersion is the coefficient of variation.

9.- "Normalized residuals from independence" for a contingency table.

The transformation can be provided to the function by using the string beetwen the quotes or just the associated number.

The supplementary rows and columns are not used to calculate the parameters (means, standard deviations, etc). Some of the transformations are not compatible with supplementary data.

Value

A list with the following components

Author(s)

Jose Luis Vicente Villardon

References

M. J. Baxter (1995) Standardization and Transformation in Principal Component Analysis, with Applications to Archaeometry. Journal of the Royal Statistical Society. Series C (Applied Statistics). Vol. 44, No. 4 (1995) , pp. 513-527

Kroonenberg, P. M. (1983). Three-mode principal component analysis: Theory and applications (Vol. 2). DSWO press. (Chapter 6)

Examples

data(iris)
x=as.matrix(iris[,1:4])
x=InitialTransform(x, transform=4)
x

Transforms an Integer Variable into a Binary Variable

Description

Transforms an Integer Variable into a Binary Variable

Usage

  Integer2Binary(y, name = "My_Factor")

Arguments

Details

Transforms an Integer vector into a Binary Indicator Matrix

Value

A Binary Data Matrix

Author(s)

Jose Luis Vicente-Villardon

Examples

dat=c(1, 2, 2, 4, 1, 1, 4, 2, 4)
Integer2Binary(dat,"Myfactor")

Kruskal Wallis Tests

Description

Kruskal Wallis Tests for a matrix of continuous variables and a grouping factor.

Usage

Kruskal.Wallis.Tests(X, groups, posthoc = "none", alternative = "two.sided", digits = 4)

Arguments

Details

Kruskal Wallis Tests for a matrix of continuous variables and a grouping factor, including the Dunn test for multiple comparisons.

Value

the organized output.

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
Kruskal.Wallis.Tests(wine[,4:7], wine$Group, posthoc = "bonferroni")

Levene Tests

Description

Levene Tests for a matrix of continuous variables and a grouping factor.

Usage

Levene.Tests(X, groups = NULL)

Arguments

Details

Levene Tests for a matrix of continuous variables and a grouping factor.

Value

The organized output

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
Levene.Tests(wine[,4:7], wine$Group)

Weighted Biplot for a table of frequencies

Description

Biplot for the logarithms of the frequencies of a contingency table using the frequencies as weights.

Usage

LogFrequencyBiplot(x, Scaling = 2, logoffset = 1, freqoffset = logoffset, ...)

Arguments

Details

Biplot for the logarithms of the frequencies of a contingency table using the frequencies as weigths.

Value

An object of class .Biplot" with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Gabriel, K. R., Galindo, M. P. & Vicente-Villardon, J. L. (1995) Use of Biplots to Diagnose Independence Models in Three-Way Contingency Tables. in: M. Greenacre & J. Blasius. eds. Visualization of Categorical Data. Academis Press. London.

GABRIEL, K.R. and ZAMIR, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21: 489-498.

See Also

CrissCross, ~~~

Examples

data(smoking)
logbip=LogFrequencyBiplot(smoking, Scaling=1, logoffset=0, freqoffset=0)

Multidimensional Scaling

Description

Multidimensional Scaling using SMACOF algorithm and Bootstraping the coordinates.

Usage

MDS(Proximities, W = NULL, Model = c("Identity", "Ratio", "Interval", "Ordinal"), 
dimsol = 2, maxiter = 100, maxerror = 1e-06, Bootstrap = FALSE, nB = 200, 
ProcrustesRot = TRUE, BootstrapMethod = c("Sampling", "Permutation"), 
StandardizeDisparities = FALSE, ShowIter = FALSE)

Arguments

Details

Multidimensional Scaling using SMACOF algorithm and Bootstraping the coordinates. MDS performs multidimensional scaling of proximity data to find a least- squares representation of the objects in a low-dimensional space. A majorization algorithm guarantees monotone convergence for optionally transformed, metric and nonmetric data under a variety of models.

Value

An object of class Principal.Coordinates and MDS. The function adds the information of the MDS to the object of class proximities. Together with the information about the proximities the object has:

Author(s)

Jose Luis Vicente Villardon

References

Commandeur, J. J. F. and Heiser, W. J. (1993). Mathematical derivations in the proximity scaling (PROXSCAL) of symmetric data matrices (Tech. Rep. No. RR- 93-03). Leiden, The Netherlands: Department of Data Theory, Leiden University.

Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 28-42.

De Leeuw, J. & Mair, P. (2009). Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software, 31(3), 1-30, http://www.jstatsoft.org/v31/i03/

Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling (2nd ed.). Springer.

Borg, I., Groenen, P. J. F., & Mair, P. (2013). Applied Multidimensional Scaling. Springer.

Groenen, P. J. F., Heiser, W. J. and Meulman, J. J. (1999). Global optimization in least squares multidimensional scaling by distance smoothing. Journal of Classification, 16, 225-254.

Groenen, P. J. F., van Os, B. and Meulman, J. J. (2000). Optimal scaling by alternating length-constained nonnegative least squares, with application to distance-based analysis. Psychometrika, 65, 511-524.

See Also

BootstrapSmacof

Examples

data(spiders)
Dis=BinaryProximities(spiders)
MDSSol=MDS(Dis, Bootstrap=FALSE)
plot(MDSSol)

Mixture Gaussian Clustering

Description

Model based clustering using mixtures of gaussian distriutions.

Usage

MGC(x, NG = 2, init = "km", RemoveOutliers=FALSE, ConfidOutliers=0.995, 
tolerance = 1e-07, maxiter = 100, show=TRUE, ...)

Arguments

Details

A basic algorithm for clustering with mixtures of gaussians with no restrictions on the covariance matrices

Value

Clusters

Author(s)

Jose Luis Vicente Villardon

References

Me falta

Examples

X=as.matrix(iris[,1:4])
mod1=MGC(X,NG=3)
plot(iris[,1:4], col=mod1$Classification)
table(iris[,5],mod1$Classification)

Matrix to Proximities

Description

Converts a matrix of proximities into a Proximities object as used in Principal Coordinates or MDS

Usage

Matrix2Proximities(x, TypeData = "User Provided", 
Type = c("dissimilarity", "similarity", "products"), 
Coefficient = "None", Transformation = "None", Data = NULL)

Arguments

Details

Converts a matrix of proximities into a Proximities object as used in Principal Coordinates or MDS aading some extra information about the procedure used to obtain the proximities. Is mainly used when the proximities matrix has been provided by the user and not calculated from raw data using BinaryProximities, ContinuousDistances or any other function.

Value

An object of class Proximities containing the proximities matrix and some extra information about it.

Author(s)

Jose Luis Vicente Villardon

Weighted Isotonic Regression (Weighted Monotone Regression)

Description

Performs weighted isotonic (monotone) regression using the non-negative weights in w. The function is a direct translation of the matlab function lsqisotonic.

Usage

MonotoneRegression(x, y, w = NULL)

Arguments

Details

YHAT = MonotoneRegression(X,Y) returns a vector of values that minimize the sum of squares (Y - YHAT).^2 under the monotonicity constraint that X(I) > X(J) => YHAT(I) >= YHAT(J), i.e., the values in YHAT are monotonically non-decreasing with respect to X (sometimes referred to as "weak monotonicity"). LSQISOTONIC uses the "pool adjacent violators" algorithm.

If X(I) == X(J), then YHAT(I) may be <, ==, or > YHAT(J) (sometimes referred to as the "primary approach"). If ties do occur in X, a plot of YHAT vs. X may appear to be non-monotonic at those points. In fact, the above monotonicity constraint is not violated, and a reordering within each group of ties, by ascending YHAT, will produce the desired appearance in the plot.

Value

The fitted values after the monotone regression

Note

The function is a direct translation of the matlab function lsqisotonic.

Author(s)

Jose L. Vicente Villardon (from a matlab functiom)

References

Kruskal, J.B. (1964) "Nonmetric multidimensional scaling: a numerical method", Psychometrika 29:115-129.

Cox, R.F. and Cox, M.A.A. (1994) Multidimensional Scaling, Chapman&Hall.

Examples

## Used inside MDS

Statistics for multiple tables

Description

Statistics for multiple tables

Usage

MultiTableStatistics(X, dual = FALSE)

Arguments

Details

Statistics for multiple tables

Value

A list with vectors of statistics for each table

Author(s)

Jose Luis Vicente Villardon

Examples

##---- Should be DIRECTLY executable !! ----

Initial Transformation of a multi table object

Description

Initial Transformation of a multi table object

Usage

MultiTableTransform(X, InitTransform = "Standardize columns", dual = FALSE,
CommonSD = TRUE)

Arguments

Details

Initial Transformation of a multi table object

Value

he table transformed

Author(s)

Jose Luis Vicente Villardon

Multidimensional Gauss-Hermite quadrature

Description

Multidimensional Gauss-Hermite quadrature

Usage

Multiquad(nnodes, dims)

Arguments

Details

Multidimensional Gauss-Hermite quadrature

Value

Multidimensional Gauss-Hermite quadrature

Author(s)

Jose Luis Vicente Villardon

References

Jackel, P. (2005). A note on multivariate Gauss-Hermite quadrature. http://www.awdz65.dsl.pipex.com/ANoteOnMultivariateGaussHermiteQuadrature.pdf

Examples

Multiquad(5, 3)

Biplot using the NIPALS algorithm

Description

Biplot using the NIPALS algorithm including a truncated and a sparse version.

Usage

NIPALS.Biplot(X, alpha = 1, dimension = 3, Scaling = 5, 
Type = "Regular", grouping = NULL, ...)

Arguments

Details

Biplot using the NIPALS algorithm including a truncated and a sparse version.

Value

An object of class ContinuousBiplot with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Wold, H. (1966). Estimation of principal components and related models by iterative least squares. Multivariate analysis. ACEDEMIC PRESS. 391-420.

Examples

bip1=NIPALS.Biplot(wine[,4:21], Type="Sparse", lambda=0.15)
plot(bip1)

NIPALS algorithm for PCA

Description

Classical NIPALS algorithm for PCA and Biplot.

Usage

NIPALSPCA(X, dimens = 2, tol = 1e-06, maxiter = 1000)

Arguments

Details

Classical NIPALS algorithm for the singular value decomposition that allows for the construction of PCA and Biplot.

Value

The singular value decomposition

Author(s)

Jose Luis Vicente Villardon

References

Wold, H. (1966). Estimation of principal components and related models by iterative least squares. Multivariate analysis. ACEDEMIC PRESS. 391-420.

Examples

# Not yet

Nice numbers: simple decimal numbers

Description

Calculates a close nice number, i. e. a number with simple decimals.

Usage

NiceNumber(x = 6, round = TRUE)

Arguments

Details

Calculates a close nice number, i. e. a number with simple decimals.

Value

A number with simple decimals

Author(s)

Jose Luis Vicente Villardon

References

Heckbert, P. S. (1990). Nice numbers for graph labels. In Graphics Gems (pp. 61-63). Academic Press Professional, Inc..

See Also

PrettyTicks

Examples

NiceNumber(0.892345)

Distances among individuals with nominal variables

Description

This function computes several measures of distance (or similarity) among individuals from a nominal data matrix.

Usage

NominalDistances(X, method = 1, diag = FALSE, upper = FALSE, similarity = TRUE)

Arguments

Details

Let be the table of nominal data. All these distances are of type d=\sqrt{1-s} with s a similarity coefficient.

1 = Overlap method

The overlap measure simply counts the number of attributes that match in the two data instances.

2 = Eskin

Eskin et al. proposed a normalization kernel for record-based network intrusion detection data. The original measure is distance-based and assigns a weight of \frac{2}{n_{k}^{2}} for mismatches; when adapted to similarity, this becomes a weight of \frac{n_{k}^{2}}{n_{k}^{2}+2}.This measure gives more weight to mismatches that occur on attributes that take many values.

3=IOF (Inverse Occurrence Frequency .)

This measure assigns lower similarity to mismatches on more frequent values. The IOF measure is related to the concept of inverse document frequency which comes from information retrieval, where it is used to signify the relative number of documents that contain a spe- cific word.

4 = OF (Ocurrence Frequency)

This measure gives the opposite weighting of the IOF measure for mismatches, i.e., mismatches on less frequent values are assigned lower similarity and mismatches on more frequent values are assigned higher similarity

5 = Goodall3

This measure assigns a high similarity if the matching values are infrequent regardless of the frequencies of the other values.

6 = Lin

This measure gives higher weight to matches on frequent values, and lower weight to mismatches on infrequent values.

Value

An object of class distance

Author(s)

Jose L. Vicente-Villardon

References

Boriah, S., Chandola, V. & Kumar,V.(2008). Similarity measures for categorical data: A comparative evaluation. In proceedings of the eight SIAM International Conference on Data Mining, pp 243–254.

See Also

BinaryDistances,ContinuousDistances

Examples

## Not run: 
data(Env)
Distance<-NominalDistances(Env,upper=TRUE,diag=TRUE,similarity=FALSE,method=1)

## End(Not run)

Normality tests

Description

Normality tests foor the columns of a matrix and a grouping variable.

Usage

NormalityTests(X, groups = NULL, plot = FALSE, SortByGroups = FALSE)

Arguments

Details

Normality tests foor the columns of a matrix and a grouping variable.

Value

The normality tests and the plots

Author(s)

Jose Luis Vicente Villardon

Examples

data(wine)
NormalityTests(wine[,4:6], groups = wine$Origin, plot=TRUE)

Converts a numeric variable into a binary one

Description

Converts a numeric variable into a binary one using a cut point

Usage

Numeric2Binary(y, name= "MyVar", cut = NULL)

Arguments

Details

Converts a numeric variable into a binary one using a cut point. If the cut is NULL the median is used.

Value

A binary Variable

Author(s)

Jose Luis Vicente-Villardon

See Also

Dataframe2BinaryMatrix

Examples

y=c(1, 1.2, 3.2, 2.4, 1.7, 2.2, 2.7, 3.1)
Numeric2Binary(y)

Alternated EM algorithm for Ordinal Logistic Biplots

Description

This function computes, with an alternated algorithm, the row and column parameters of an Ordinal Logistic Biplot for ordered polytomous data. The row coordinates (E-step) are computed using multidimensional Gauss-Hermite quadratures and Expected a posteriori (EAP) scores and parameters for each variable or items (M-step) using Ridge Ordinal Logistic Regression to solve the separation problem present when the points for different categories of a variable are completely separated on the representation plane and the usual fitting methods do not converge. The separation problem is present in almost avery data set for which the goodness of fit is high.

Usage

OrdLogBipEM(Data, freq=NULL, dim = 2, nnodes = 15, 
tol = 0.0001, maxiter = 100, maxiterlogist = 100, 
penalization = 0.2, show = FALSE, initial = 1, alfa = 1, 
Orthogonalize=TRUE, Varimax=TRUE, ...) 

Arguments

Value

An object of class "Ordinal.Logistic.Biplot".This has components:

Author(s)

Jose Luis Vicente-Villardon

References

Bock,R. & Aitkin,M. (1981),Marginal maximum likelihood estimation of item parameters: Aplication of an EM algorithm, Phychometrika 46(4), 443-459.

Examples

## Not run: 
    data(Doctors)
    olb = OrdLogBipEM(Doctors,dim = 2, nnodes = 10, initial=4,
    tol = 0.001, maxiter = 100, penalization = 0.1, show=TRUE)
    olb
    summary(olb)
    PlotOrdinalResponses(olb)
    
## End(Not run)

Plots an ordinal variable on the biplot

Description

Plots an ordinal variable on the biplot from its fitted parameters

Usage

OrdVarBiplot(bi1, bi2, threshold, xmin = -3, xmax = 3, ymin = -3, 
ymax = 3, label = "Point", mode = "a", CexPoint = 0.8,
PchPoint = 1, Color = "green", tl = 0.03, textpos = 1, CexScale= 0.5, ...)

Arguments

Details

Plots an ordinal variable on the biplot from its fitted parameters. The plot uses the same parameters as any other biplot.

Value

Returns a graphical representation of the ordinal variable on the current plot

Author(s)

Jose Luis Vicente Villardon

References

Vicente-Villardon, J. L., & Sanchez, J. C. H. (2014). Logistic Biplots for Ordinal Data with an Application to Job Satisfaction of Doctorate Degree Holders in Spain. arXiv preprint arXiv:1405.0294.

Examples

##---- Should be DIRECTLY executable !! ----

Coordinates of an ordinal variable on the biplot.

Description

Coordinates of an ordinal variable on the biplot.

Usage

OrdVarCoordinates(tr, b = c(1, 1), inf = -12, sup = 12, step = 0.01,
                 plotresponse = FALSE, label = "Item", labx = "z", laby
                 = "Probability", catnames = NULL, Legend = TRUE,
                 LegendPos = 1)

Arguments

Details

The function calculates the coordinates of the points that define the separation among the categories of an ordinal variable projected onto an ordinal logistic biplot.

Value

An object of class OrdVarCoord

Author(s)

Jose Luis Vicente Villardon

References

Vicente-Villardon, J. L., & Sanchez, J. C. H. (2014). Logistic Biplots for Ordinal Data with an Application to Job Satisfaction of Doctorate Degree Holders in Spain. arXiv preprint arXiv:1405.0294.

Examples

# No examples

Fits an ordinal logistic regression with ridge penalization

Description

This function fits a logistic regression between a dependent ordinal variable y and some independent variables x, and solves the separation problem using ridge penalization.

Usage

OrdinalLogisticFit(y, x, penalization = 0.1, tol = 1e-04, maxiter = 200, show = FALSE)

Arguments

Details

The problem of the existence of the estimators in logistic regression can be seen in Albert (1984); a solution for the binary case, based on the Firth's method, Firth (1993) is proposed by Heinze(2002). All the procedures were initially developed to remove the bias but work well to avoid the problem of separation. Here we have chosen a simpler solution based on ridge estimators for logistic regression Cessie(1992).

Rather than maximizing {L_j}(\left. {\bf{G}} \right|{{\bf{b}}_{j0}},{{\bf{B}}_j}) we maximize

{{L_j}(\left. {\bf{G}} \right|{{\bf{b}}_{j0}},{{\bf{B}}_j})} - \lambda \left( {\left\| {{{\bf{b}}_{j0}}} \right\| + \left\| {{{\bf{B}}_j}} \right\|} \right)

Changing the values of \lambda we obtain slightly different solutions not affected by the separation problem.

Value

An object of class "pordlogist". This has components:

Author(s)

Jose Luis Vicente-Villardon

References

Albert,A. & Anderson,J.A. (1984),On the existence of maximum likelihood estimates in logistic regression models, Biometrika 71(1), 1–10.

Bull, S.B., Mak, C. & Greenwood, C.M. (2002), A modified score function for multinomial logistic regression, Computational Statistics and dada Analysis 39, 57–74.

Firth, D.(1993), Bias reduction of maximum likelihood estimates, Biometrika 80(1), 27–38

Heinze, G. & Schemper, M. (2002), A solution to the problem of separation in logistic regression, Statistics in Medicine 21, 2109–2419

Le Cessie, S. & Van Houwelingen, J. (1992), Ridge estimators in logistic regression, Applied Statistics 41(1), 191–201.

Examples

# No examples yet

Orthogonalize a set of Scores calculated by other procedure

Description

Orthogonalize a set of Scores calculated by other procedure

Usage

OrthogonalizeScores(scores)

Arguments

Details

Orthogonalize a set of Scores calculated by other procedure proyecting onto the dimensions defined by the eigenvectors of the covariance matrix

Value

The orthogonalised scores.

Author(s)

Jose Luis Vicente Villardon

Examples

##---- Should be DIRECTLY executable !! ----

Classical PCA Biplot with added features.

Description

Classical PCA Biplot with added features.

Usage

PCA.Analysis(X, dimension = 3, Scaling = 5, ...)

Arguments

Details

Biplots represent the rows and columns of a data matrix in reduced dimensions. Usually rows represent individuals, objects or samples and columns are variables measured on them. The most classical versions can be thought as visualizations associated to Principal Components Analysis (PCA) or Factor Analysis (FA) obtained from a Singular Value Decomposition or a related method. From another point of view, Classical Biplots could be obtained from regressions and calibrations that are essentially an alternated least squares algorithm equivalent to an EM-algorithm when data are normal.

Value

An object of class ContinuousBiplot with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Gabriel, K.R.(1971): The biplot graphic display of matrices with applications to principal component analysis. Biometrika, 58, 453-467.

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Questiio. 1986, vol. 10, núm. 1.

Gabriel, K. R. AND Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21(21):489–498, 1979.

Gabriel, K.R.(1998): Generalised Bilinear Regression. Biometrika, 85, 3, 689-700.

Gower y Hand (1996): Biplots. Chapman & Hall.

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez-Zaballos, A. (2006). Logistic Biplots. Multiple Correspondence Analysis and related methods 491-509.

Demey, J., Vicente-Villardon, J. L., Galindo, M. P. and Zambrano, A. (2008). Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics 24 2832-2838.

See Also

InitialTransform

Examples

## Simple Biplot with arrows
data(Protein)
bip=PCA.Biplot(Protein[,3:11])
plot(bip)

## Biplot with scales on the variables
plot(bip, mode="s", margin=0.2)

# Structure plot (Correlations)
CorrelationCircle(bip)

# Plot of the Variable Contributions
ColContributionPlot(bip, cex=1)

Classical PCA Biplot with added features.

Description

Classical PCA Biplot with added features.

Usage

PCA.Biplot(X, alpha = 1, dimension = 2, Scaling = 5, sup.rows = NULL, 
          sup.cols = NULL, grouping = NULL)

Arguments

Details

Biplots represent the rows and columns of a data matrix in reduced dimensions. Usually rows represent individuals, objects or samples and columns are variables measured on them. The most classical versions can be thought as visualizations associated to Principal Components Analysis (PCA) or Factor Analysis (FA) obtained from a Singular Value Decomposition or a related method. From another point of view, Classical Biplots could be obtained from regressions and calibrations that are essentially an alternated least squares algorithm equivalent to an EM-algorithm when data are normal.

Value

An object of class ContinuousBiplot with the following components:

Author(s)

Jose Luis Vicente Villardon

References

Gabriel, K.R.(1971): The biplot graphic display of matrices with applications to principal component analysis. Biometrika, 58, 453-467.

Galindo Villardon, M. (1986). Una alternativa de representacion simultanea: HJ-Biplot. Questiio. 1986, vol. 10, núm. 1.

Gabriel, K. R. AND Zamir, S. (1979). Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21(21):489–498, 1979.

Gabriel, K.R.(1998): Generalised Bilinear Regression. Biometrika, 85, 3, 689-700.

Gower y Hand (1996): Biplots. Chapman & Hall.

Vicente-Villardon, J. L., Galindo, M. P. and Blazquez-Zaballos, A. (2006). Logistic Biplots. Multiple Correspondence Analysis and related methods 491-509.

Demey, J., Vicente-Villardon, J. L., Galindo, M. P. and Zambrano, A. (2008). Identifying Molecular Markers Associated With Classification Of Genotypes Using External Logistic Biplots. Bioinformatics 24 2832-2838.

See Also

InitialTransform

Examples

## Simple Biplot with arrows
data(Protein)
bip=PCA.Biplot(Protein[,3:11])
plot(bip)

## Biplot with scales on the variables
plot(bip, mode="s", margin=0.2)

# Structure plot (Correlations)
CorrelationCircle(bip)

# Plot of the Variable Contributions
ColContributionPlot(bip, cex=1)

Principal Components Analysis with bootstrap confidence intervals.

Description

Calculates a Principal Components Analysis with bootstrap confidence intervals for its parameters

Usage

PCA.Bootstrap(X, dimens = 2, Scaling = "Standardize columns", B = 1000, type = "np")

Arguments

Details

The types of bootstrap used are:

"np : "

Non Parametric

"pa : "

parametric (data is obtained from a Multivariate Normal Distribution)

"spper : "

Semi-parametric Residuals are permutated

"spres : "

Semi-parametric Residuals are resampled

For the moment, only the non-parametric bootstrap is implemented.

The Principal Components (eigenvectors) are obtained using bootstrap samples.

The Row scotes are obtained projecting the completen data matrix into the bootstrap Principal Components. In this way all the individulas have the same number of replications.

Value

Author(s)

Jose Luis Vicente Villardon

References

Daudin, J. J., Duby, C., & Trecourt, P. (1988). Stability of principal component analysis studied by the bootstrap method. Statistics: A journal of theoretical and applied statistics, 19(2), 241-258.

Chateau, F., & Lebart, L. (1996). Assessing sample variability in the visualization techniques related to principal component analysis: bootstrap and alternative simulation methods. COMPSTAT, Physica-Verlag, 205-210.

Babamoradi, H., van den Berg, F., & Rinnan, Å. (2013). Bootstrap based confidence limits in principal component analysis—A case study. Chemometrics and Intelligent Laboratory Systems, 120, 97-105.

Fisher, A., Caffo, B., Schwartz, B., & Zipunnikov, V. (2016). Fast, exact bootstrap principal component analysis for p> 1 million. Journal of the American Statistical Association, 111(514), 846-860.

See Also

PCA.Biplot

Examples

## Not run: X=wine[,4:21]
grupo=wine$Group
rownames(X)=paste(1:45, grupo, sep="-")
pcaboot=PCA.Bootstrap(X, dimens=2, Scaling = "Standardize columns", B=1000)
plot(pcaboot, ColorInd=as.numeric(grupo))
summary(pcaboot)

## End(Not run)

Partial Least Squares Regression

Description

Partial Least Squares Regression for numerical variables.

Usage

PLSR(Y, X, S = 2, InitTransform = 5, grouping = NULL, 
centerY = TRUE, scaleY = TRUE, tolerance = 5e-06, 
maxiter = 100, show = FALSE, Validation = NULL, nB = 500)

Arguments

Details

Partial Least Squares Regression for numerical variables.

Value

An object of class plsr with fiends

Author(s)

Jose Luis Vicente Villardon

References

H. Abdi, Partial least squares regression and projection on latent structure regression (PLS regression), WIREs Comput. Stat. 2 (2010), pp. 97-106.

See Also

Biplot.PLSR

Examples

X=as.matrix(wine[,4:21])
y=as.numeric(wine[,2])-1
mifit=PLSR(y,X, Validation="None")

Partial Least Squares Regression with Binary Response

Description

Fits Partial Least Squares Regression with Binary Response

Usage

PLSR1Bin(Y, X, S = 2, InitTransform = 5, grouping = NULL, 
tolerance = 5e-06, maxiter = 100, show = FALSE, penalization = 0, 
cte = TRUE, Algorithm = 1, OptimMethod = "CG")

Arguments

Details

The procedure uses the algorithm proposed by Bastien et al () to fit a Partial Lest Squares Regression when the response is Binary. The procedure will be later converted into a Biplot to visulize the results.

Value

Still to be finished

Author(s)

Jose Luis Vicente Villardon

Examples

# No examples yet

Partial Least Squares Regression with several Binary Responses

Description

Fits Partial Least Squares Regression with several Binary Responses

Usage

PLSRBin(Y, X, S = 2, InitTransform = 5, grouping = NULL, 
tolerance = 5e-05, maxiter = 100, show = FALSE, penalization = 0.1, 
cte = TRUE, OptimMethod = "CG", Multiple = FALSE)

Arguments

Details

The function fits the PLSR method for the case when there is a set binary dependent variables, using logistic rather than linear fits to take into account the nature of responses. We term the method PLS-BLR (Partial Least Squares Binary Logistic Regression). This can be considered as a generalization of the NIPALS algorithm when the responses are all binary.

Value

Author(s)

José Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Examples

X=as.matrix(wine[,4:21])
Y=cbind(Factor2Binary(wine[,1])[,1], Factor2Binary(wine[,2])[,1])
rownames(Y)=wine[,3]
colnames(Y)=c("Year", "Origin")
pls=PLSRBin(Y,X, penalization=0.1, show=TRUE, S=2)

PLS binary regression.

Description

Fits PLS binary regression.

Usage

PLSRBinFit(Y, X, S = 2, tolerance = 5e-06, maxiter = 100, 
show = FALSE, penalization = 0.1, cte = TRUE, OptimMethod = "CG")

Arguments

Details

Fits PLS binary regression. It is used for a higher level function.

Value

The PLS fit used by the PLSRBin function.

Author(s)

Jose Luis Vicente Villardon

References

Ugarte Fajardo, J., Bayona Andrade, O., Criollo Bonilla, R., Cevallos‐Cevallos, J., Mariduena‐Zavala, M., Ochoa Donoso, D., & Vicente Villardon, J. L. (2020). Early detection of black Sigatoka in banana leaves using hyperspectral images. Applications in plant sciences, 8(8), e11383.

Examples

## Not yet

Partial Least Squares Regression (PLSR)

Description

Fits a Partial Least Squares Regression (PLSR) to two continuous data matrices

Usage

PLSRfit(Y, X, S = 2, tolerance = 5e-06,
maxiter = 100, show = FALSE)

Arguments

Details

Fits a Partial Least Squares Regression (PLSR) to a set of two continuous data matrices

Value

An object of class "PLSR"