| Title: | Multivariate Outlier Detection Based on Robust Methods |
|---|---|
| Description: | Various methods for multivariate outlier detection: arw, a Mahalanobis-type method with an adaptive outlier cutoff value; locout, a method incorporating local neighborhood; pcout, a method for high-dimensional data; mvoutlier.CoDa, a method for compositional data. References are provided in the corresponding help files. |
| Authors: | Peter Filzmoser <[email protected]> and Moritz Gschwandtner <[email protected]> |
| Maintainer: | P. Filzmoser <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 2.1.1 |
| Built: | 2024-11-03 06:26:22 UTC |
| Source: | CRAN |
The function aq.plot plots the ordered squared robust Mahalanobis distances of the observations against the empirical distribution function of the $MD^2_i$. In addition the distribution function of $chisq_p$ is plotted as well as two vertical lines corresponding to the chisq-quantile specified in the argument list (default is 0.975) and the so-called adjusted quantile. Three additional graphics are created (the first showing the data, the second showing the outliers detected by the specified quantile of the $chisq_p$ distribution and the third showing these detected outliers by the adjusted quantile).
aq.plot(x, delta=qchisq(0.975, df=ncol(x)), quan=1/2, alpha=0.05)aq.plot(x, delta=qchisq(0.975, df=ncol(x)), quan=1/2, alpha=0.05)
x |
matrix or data.frame containing the data; has to be at least two-dimensional |
delta |
quantile of the chi-squared distribution with ncol(x) degrees of freedom. This quantile appears as cyan-colored vertical line in the plot. |
quan |
proportion of observations which are used for mcd estimations; has to be between 0.5 and 1, default ist 0.5 |
alpha |
Maximum thresholding proportion (optional scalar, default: alpha = 0.05) |
The function aq.plot plots the ordered squared robust Mahalanobis distances of the observations against the empirical distribution function of the $MD^2_i$. The distance calculations are based on the MCD estimator.
For outlier detection two different methods are used. The first one marks observations as outliers if they exceed a certain quantile of the chi-squared distribution. The second is an adaptive procedure searching for outliers specifically in the tails of the distribution, beginning at a certain chisq-quantile (see Filzmoser et al., 2005).
The function behaves differently depending on the dimension of the data. If the data is more than two-dimensional the data are projected on the first two robust principal components.
outliers |
boolean vector of outliers |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
# create data: set.seed(134) x <- cbind(rnorm(80), rnorm(80), rnorm(80)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: aq.plot(z, alpha=0.1)# create data: set.seed(134) x <- cbind(rnorm(80), rnorm(80), rnorm(80)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: aq.plot(z, alpha=0.1)
Adaptive reweighted estimator for multivariate location and scatter with hard-rejection weights. The multivariate outliers are defined according to the supremum of the difference between the empirical distribution function of the robust Mahalanobis distance and the theoretical distribution function.
arw(x, m0, c0, alpha, pcrit)arw(x, m0, c0, alpha, pcrit)
x |
Dataset (n x p) |
m0 |
Initial location estimator (1 x p) |
c0 |
Initial scatter estimator (p x p) |
alpha |
Maximum thresholding proportion (optional scalar, default: alpha = 0.025) |
pcrit |
Critical value obtained by simulations (optional scalar, default value obtained from simulations) |
At the basis of initial estimators of location and scatter, the function arw performs a reweighting step to adjust the threshold for outlier rejection. The critical value pcrit was obtained by simulations using the MCD estimator as initial robust covariance estimator. If a different estimator is used, pcrit should be changed and computed by simulations for the specific dimensions of the data x.
m |
Adaptive location estimator (p x 1) |
c |
Adaptive scatter estimator (p x p) |
cn |
Adaptive threshold ("adjusted quantile") |
w |
Weight vector (n x 1) |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
x <- cbind(rnorm(100), rnorm(100)) arw(x, apply(x,2,mean), cov(x))x <- cbind(rnorm(100), rnorm(100)) arw(x, apply(x,2,mean), cov(x))
The Kola data were collected in the Kola Project (1993-1998, Geological Surveys of Finland (GTK) and Norway (NGU) and Central Kola Expedition (CKE), Russia). More than 600 samples in five different layers were analysed, this dataset contains the B-horizon.
data(bhorizon)data(bhorizon)
A data frame with 609 observations on the following 48 variables.
IDa numeric vector
XCOOa numeric vector
YCOOa numeric vector
Aga numeric vector
Ala numeric vector
Al_XRFa numeric vector
Asa numeric vector
Baa numeric vector
Bea numeric vector
Bia numeric vector
Caa numeric vector
Ca_XRFa numeric vector
Cda numeric vector
Coa numeric vector
Cra numeric vector
Cua numeric vector
ECa numeric vector
Fea numeric vector
Fe_XRFa numeric vector
Ka numeric vector
K_XRFa numeric vector
LOIa numeric vector
Laa numeric vector
Lia numeric vector
Mga numeric vector
Mg_XRFa numeric vector
Mna numeric vector
Mn_XRFa numeric vector
Moa numeric vector
Naa numeric vector
Na_XRFa numeric vector
Nia numeric vector
Pa numeric vector
P_XRFa numeric vector
Pba numeric vector
Sa numeric vector
Sca numeric vector
Sea numeric vector
Sia numeric vector
Si_XRFa numeric vector
Sra numeric vector
Tea numeric vector
Tha numeric vector
Tia numeric vector
Ti_XRFa numeric vector
Va numeric vector
Ya numeric vector
Zna numeric vector
Kola Project (1993-1998)
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
data(bhorizon) # classical versus robust correlation corr.plot(log(bhorizon[,"Al"]), log(bhorizon[,"Na"]))data(bhorizon) # classical versus robust correlation corr.plot(log(bhorizon[,"Al"]), log(bhorizon[,"Na"]))
Coordinates of the BSS data background map
data(bss.background)data(bss.background)
A data frame with 6093 observations on the following 2 variables.
V1a numeric vector with the x-coordinates
V2a numeric vector with the y-coordinates
Is used by pbb()
BSS project
Reimann C, Siewers U, Tarvainen T, Bityukova L, Eriksson J, Gilucis A, Gregorauskiene V, Lukashev VK, Matinian NN, Pasieczna A. Agricultural Soils in Northern Europe: A Geochemical Atlas. Geologisches Jahrbuch, Sonderhefte, Reihe D, Heft SD 5, Schweizerbart'sche Verlagsbuchhandlung, Stuttgart, 2003.
data(bss.background) pbb()data(bss.background) pbb()
The BSS data were collected in agrigultural soils from Northern Europe. from an area of about 1,800,000 km2. 769 samples on an iregular grid were taken in two different layers, the top layer (0-20cm) and the bottom layer. This dataset contains the bottom layer of the BSS data. It has 46 variables, including x and y coordinates.
data(bssbot)data(bssbot)
A data frame with 768 observations on the following 46 variables.
a numeric vector
a numeric vector
x coordinates: a numeric vector
y coordinates: a numeric vector
a numeric vector
a numeric vector
a numeric vector
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a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
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a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
BSS Project in Northern Europe
Reimann C, Siewers U, Tarvainen T, Bityukova L, Eriksson J, Gilucis A, Gregorauskiene V, Lukashev VK, Matinian NN, Pasieczna A. Agricultural Soils in Northern Europe: A Geochemical Atlas. Geologisches Jahrbuch, Sonderhefte, Reihe D, Heft SD 5, Schweizerbart'sche Verlagsbuchhandlung, Stuttgart, 2003.
data(bssbot) # classical versus robust correlation corr.plot(log(bssbot[, "Al2O3_B"]), log(bssbot[, "Na2O_B"]))data(bssbot) # classical versus robust correlation corr.plot(log(bssbot[, "Al2O3_B"]), log(bssbot[, "Na2O_B"]))
The BSS data were collected in agrigultural soils from Northern Europe. from an area of about 1,800,000 km2. 769 samples on an iregular grid were taken in two different layers, the top layer (0-20cm) and the bottom layer. This dataset contains the top layer of the BSS data. It has 46 variables, including x and y coordinates.
data(bsstop)data(bsstop)
A data frame with 768 observations on the following 46 variables.
a numeric vector
a numeric vector
x coordinates: a numeric vector
y coordinates: a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
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a numeric vector
a numeric vector
a numeric vector
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a numeric vector
a numeric vector
a numeric vector
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a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
a numeric vector
BSS Project in Northern Europe
Reimann C, Siewers U, Tarvainen T, Bityukova L, Eriksson J, Gilucis A, Gregorauskiene V, Lukashev VK, Matinian NN, Pasieczna A. Agricultural Soils in Northern Europe: A Geochemical Atlas. Geologisches Jahrbuch, Sonderhefte, Reihe D, Heft SD 5, Schweizerbart'sche Verlagsbuchhandlung, Stuttgart, 2003.
data(bsstop) # classical versus robust correlation corr.plot(log(bsstop[, "Al2O3_T"]), log(bsstop[, "Na2O_T"]))data(bsstop) # classical versus robust correlation corr.plot(log(bsstop[, "Al2O3_T"]), log(bsstop[, "Na2O_T"]))
The function chisq.plot plots the ordered robust mahalanobis distances of the data against the quantiles of the Chi-squared distribution. By user interaction this plotting is iterated each time leaving out the observation with the greatest distance.
chisq.plot(x, quan=1/2, ask=TRUE, ...)chisq.plot(x, quan=1/2, ask=TRUE, ...)
x |
matrix or data.frame containing the data |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
ask |
logical. specifies whether user interacton is allowed or not. default is TRUE |
... |
additional graphical parameters |
The function chisq.plot plots the ordered robust mahalanobis distances of the data against the quantiles of the Chi-squared distribution. If the data is normal distributed these values should approximately correspond to each other, so outliers can be detected visually. By user interaction this procedure is repeated, each time leaving out the observation with the greatest distance (the number of the observation is printed on the console). This method can be seen as an iterative deletion of outliers until a straight line appears.
outliers |
indices of the outliers that are removed by left-click on the plotting device. |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
R.G. Garrett (1989). The chi-square plot: a tools for multivariate outlier recognition. Journal of Geochemical Exploration, 32 (1/3), 319-341.
data(humus) res <-chisq.plot(log(humus[,c("Co","Cu","Ni")])) res$outliers # these are the potential outliersdata(humus) res <-chisq.plot(log(humus[,c("Co","Cu","Ni")])) res$outliers # these are the potential outliers
The Kola Data were collected in the Kola Project (1993-1998, Geological Surveys of Finland (GTK) and Norway (NGU) and Central Kola Expedition (CKE), Russia). More than 600 samples in five different layers were analysed, this dataset contains the C-horizon.
data(chorizon)data(chorizon)
A data frame with 606 observations on the following 110 variables.
IDa numeric vector
XCOOa numeric vector
YCOOa numeric vector
Aga numeric vector
Ag_INAAa numeric vector
Ala numeric vector
Al2O3a numeric vector
Asa numeric vector
As_INAAa numeric vector
Au_INAAa numeric vector
Ba numeric vector
Baa numeric vector
Ba_INAAa numeric vector
Bea numeric vector
Bia numeric vector
Br_ICa numeric vector
Br_INAAa numeric vector
Caa numeric vector
Ca_INAAa numeric vector
CaOa numeric vector
Cda numeric vector
Ce_INAAa numeric vector
Cl_ICa numeric vector
Coa numeric vector
Co_INAAa numeric vector
ECa numeric vector
Cra numeric vector
Cr_INAAa numeric vector
Cs_INAAa numeric vector
Cua numeric vector
Eu_INAAa numeric vector
F_ICa numeric vector
Fea numeric vector
Fe_INAAa numeric vector
Fe2O3a numeric vector
Hf_INAAa numeric vector
Hga numeric vector
Hg_INAAa numeric vector
Ir_INAAa numeric vector
Ka numeric vector
K2Oa numeric vector
Laa numeric vector
La_INAAa numeric vector
Lia numeric vector
LOIa numeric vector
Lu_INAAa numeric vector
wt_INAAa numeric vector
Mga numeric vector
MgOa numeric vector
Mna numeric vector
MnOa numeric vector
Moa numeric vector
Mo_INAAa numeric vector
Naa numeric vector
Na_INAAa numeric vector
Na2Oa numeric vector
Nd_INAAa numeric vector
Nia numeric vector
Ni_INAAa numeric vector
NO3_ICa numeric vector
Pa numeric vector
P2O5a numeric vector
Pba numeric vector
pHa numeric vector
PO4_ICa numeric vector
Rba numeric vector
Sa numeric vector
Sba numeric vector
Sb_INAAa numeric vector
Sca numeric vector
Sc_INAAa numeric vector
Sea numeric vector
Se_INAAa numeric vector
Sia numeric vector
SiO2a numeric vector
Sm_INAAa numeric vector
Sn_INAAa numeric vector
SO4_ICa numeric vector
Sra numeric vector
Sr_INAAa numeric vector
SUM_XRFa numeric vector
Ta_INAAa numeric vector
Tb_INAAa numeric vector
Tea numeric vector
Tha numeric vector
Th_INAAa numeric vector
Tia numeric vector
TiO2a numeric vector
U_INAAa numeric vector
Va numeric vector
W_INAAa numeric vector
Ya numeric vector
Yb_INAAa numeric vector
Zna numeric vector
Zn_INAAa numeric vector
ELEVa numeric vector
COUNa numeric vector
ASPa numeric vector
TOPCa numeric vector
LITOa numeric vector
Al_XRFa numeric vector
Ca_XRFa numeric vector
Fe_XRFa numeric vector
K_XRFa numeric vector
Mg_XRFa numeric vector
Mn_XRFa numeric vector
Na_XRFa numeric vector
P_XRFa numeric vector
Si_XRFa numeric vector
Ti_XRFa numeric vector
Kola Project (1993-1998)
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
data(chorizon) # classical versus robust correlation corr.plot(log(chorizon[,"Al"]), log(chorizon[,"Na"]))data(chorizon) # classical versus robust correlation corr.plot(log(chorizon[,"Al"]), log(chorizon[,"Na"]))
The function color.plot plots the (two-dimensional) data using different symbols according to the robust mahalanobis distance based on the mcd estimator with adjustment and using different colors according to the euclidean distances of the observations.
color.plot(x, quan=1/2, alpha=0.025, ...)color.plot(x, quan=1/2, alpha=0.025, ...)
x |
two dimensional matrix or data.frame containing the data. |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
amount of observations used for calculating the adjusted quantile (see function arw). |
... |
additional graphical parameters |
The function color.plot plots the (two-dimensional) data using different symbols (see function symbol.plot) according to the robust mahalanobis distance based on the mcd estimator with adjustment and using different colors according to the euclidean distances of the observations. Blue is typical for a little distance, whereas red is the opposite. In addition four ellipsoids are drawn, on which mahalanobis distances are constant. These constant values correspond to the 25%, 50%, 75% and adjusted quantiles (see function arw) of the chi-square distribution (see Filzmoser et al., 2005).
outliers |
boolean vector of outliers |
md |
robust mahalanobis distances of the data |
euclidean |
euclidean distances of the observations according to the minimum of the data. |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: color.plot(z, quan=0.75)# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: color.plot(z, quan=0.75)
The function corr.plot plots the (two-dimensional) data and adds two correlation ellipsoids, based on classical and robust estimation of location and scatter. Robust estimation can be thought of as estimating the mean and covariance of the 'good' part of the data.
corr.plot(x, y, quan=1/2, alpha=0.025, ...)corr.plot(x, y, quan=1/2, alpha=0.025, ...)
x |
vector to be plotted against y and of which the correlation with y is calculated. |
y |
vector to be plotted against x and of which the correlation with x is calculated. |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
Determines the size of the ellipsoids. An observation will be outside of the ellipsoid if its mahalanobis distance exceeds the 1-alpha quantile of the chi-squared distribution. |
... |
additional graphical parameters |
cor.cla |
correlation between x and y based on classical estimation of location and scatter |
cor.rob |
correlation between x and y based on robust estimation of location and scatter |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 3, 1), rnorm(10, 3, 1)) z <- rbind(x,y) # execute: corr.plot(z[,1], z[,2])# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 3, 1), rnorm(10, 3, 1)) z <- rbind(x,y) # execute: corr.plot(z[,1], z[,2])
Illustrative data example with 100 observations in two dimensions.
data(dat)data(dat)
The format is: num [1:100, 1:2] 3.39 4.08 4.35 4.89 4.55 ...
Data are constructed to contain global as well as local outliers.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
data(dat) plot(dat)data(dat) plot(dat)
The function dd.plot plots the classical mahalanobis distance of the data against the robust mahalanobis distance based on the mcd estimator. Different symbols (see function symbol.plot) and colours (see function color.plot) are used depending on the mahalanobis and euclidean distance of the observations (see Filzmoser et al., 2005).
dd.plot(x, quan=1/2, alpha=0.025, ...)dd.plot(x, quan=1/2, alpha=0.025, ...)
x |
matrix or data frame containing the data |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
amount of observations used for calculating the adjusted quantile (see function arw). |
... |
additional graphical parameters |
outliers |
boolean vector of outliers |
md.cla |
mahalanobis distances of the observations based on classical estimators of location and scatter. |
md.rob |
mahalanobis distances of the observations based on robust estimators of location and scatter (mcd). |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
symbol.plot, color.plot, arw, covPlot
# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 3, 1), rnorm(10, 3, 1)) z <- rbind(x,y) # execute: dd.plot(z) # # Identify multivariate outliers for Co-Cu-Ni in humus layer of Kola data: data(humus) dd.plot(log(humus[,c("Co","Cu","Ni")]))# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 3, 1), rnorm(10, 3, 1)) z <- rbind(x,y) # execute: dd.plot(z) # # Identify multivariate outliers for Co-Cu-Ni in humus layer of Kola data: data(humus) dd.plot(log(humus[,c("Co","Cu","Ni")]))
The Kola Data were collected in the Kola Project (1993-1998, Geological Surveys of Finland (GTK) and Norway (NGU) and Central Kola Expedition (CKE), Russia). More than 600 samples in five different layers were analysed, this dataset contains the humus layer.
data(humus)data(humus)
A data frame with 617 observations on the following 44 variables.
IDa numeric vector
XCOOa numeric vector
YCOOa numeric vector
Aga numeric vector
Ala numeric vector
Asa numeric vector
Ba numeric vector
Baa numeric vector
Bea numeric vector
Bia numeric vector
Caa numeric vector
Cda numeric vector
Coa numeric vector
Cra numeric vector
Cua numeric vector
Fea numeric vector
Hga numeric vector
Ka numeric vector
Laa numeric vector
Mga numeric vector
Mna numeric vector
Moa numeric vector
Naa numeric vector
Nia numeric vector
Pa numeric vector
Pba numeric vector
Rba numeric vector
Sa numeric vector
Sba numeric vector
Sca numeric vector
Sia numeric vector
Sra numeric vector
Tha numeric vector
Tla numeric vector
Ua numeric vector
Va numeric vector
Ya numeric vector
Zna numeric vector
Ca numeric vector
Ha numeric vector
Na numeric vector
LOIa numeric vector
pHa numeric vector
Conda numeric vector
Kola Project (1993-1998)
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
data(humus) # classical versus robust correlation: corr.plot(log(humus[,"Al"]), log(humus[,"Na"]))data(humus) # classical versus robust correlation: corr.plot(log(humus[,"Al"]), log(humus[,"Na"]))
Coordinates of the Kola background map
data(kola.background)data(kola.background)
The format is: List of 4 $ boundary:‘data.frame’: 50 obs. of 2 variables: ..$ V1: num [1:50] 388650 388160 386587 384035 383029 ... ..$ V2: num [1:50] 7892400 7881248 7847303 7790797 7769214 ... $ coast :‘data.frame’: 6259 obs. of 2 variables: ..$ V1: num [1:6259] 438431 439102 439102 439643 439643 ... ..$ V2: num [1:6259] 7895619 7896495 7896495 7895800 7895542 ... $ borders :‘data.frame’: 504 obs. of 2 variables: ..$ V1: num [1:504] 417575 417704 418890 420308 422731 ... ..$ V2: num [1:504] 7612984 7612984 7613293 7614530 7615972 ... $ lakes :‘data.frame’: 6003 obs. of 2 variables: ..$ V1: num [1:6003] 547972 546915 NA 547972 547172 ... ..$ V2: num [1:6003] 7815109 7815599 NA 7815109 7813873 ...
Is used by map.plot()
Kola Project (1993-1998)
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
example(map.plot)example(map.plot)
Computes global and pairwise Mahalanobis distances for visualizing global and local multivariate outliers. The size of the neighborhood (number of neighbors) is varying, but the fraction of neighbors is fixed.
locoutNeighbor(dat, X, Y, propneighb = 0.1, variant = c("dist", "knn"), usemax = 1/3, npoints = 50, chisqqu = 0.975, indices = NULL, xlab = NULL, ylab = NULL, colall = gray(0.7), colsel = 1, ...)locoutNeighbor(dat, X, Y, propneighb = 0.1, variant = c("dist", "knn"), usemax = 1/3, npoints = 50, chisqqu = 0.975, indices = NULL, xlab = NULL, ylab = NULL, colall = gray(0.7), colsel = 1, ...)
dat |
multivariate data set (without coordinates) |
X |
X coordinates of the data points |
Y |
Y coordinates of the data points |
propneighb |
proportion of neighbors to be included in tolerance ellipse |
variant |
either search for neighbors according to the Eucl.Distance, or according to kNN |
usemax |
for either variant: give fraction of points (max Dist) that is used for the plot |
npoints |
computation is done at most at npoints points |
chisqqu |
quantile of the chisquare distribution for splitting the plot |
indices |
if this is not NULL, these should be indices of observations to be highlighted |
xlab |
x-axis label for plot |
ylab |
y-axis label for plot |
colall |
color for lines if indices is NULL |
colsel |
color for lines if indices are selected |
... |
additional parameters for plotting |
For this diagnostic tool, the number of neighbors is varied up to a fraction of usemax observations. Then propneighb (called beta) is fixed, and for each observation we compute the degree of isolation from a fraction of 1-beta of its neighbors. Neighborhood can be defined either via the Euclidean distance or by k-Nearest-Neighbors. For computational reasons, all computations are done at most for npoints points. The critical value for outliers is the quantile chisqqu of the chisquare distribution. One can also provide indices of observations (for indices). Then the corresponding lines in the plots will be highlighted.
indices.reg |
indices of the (selected) observations being regular observations |
indices.out |
indices of the (selected) observations being golbal outliers |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
# use data from illustrative example in paper: data(X) data(Y) data(dat) res <- locoutNeighbor(dat,X,Y,variant="knn",usemax=1,chisqqu=0.975,indices=c(1,11,24,36), propneighb=0.1,npoints=100)# use data from illustrative example in paper: data(X) data(Y) data(dat) res <- locoutNeighbor(dat,X,Y,variant="knn",usemax=1,chisqqu=0.975,indices=c(1,11,24,36), propneighb=0.1,npoints=100)
Computes global and pairwise Mahalanobis distances for visualizing global and local multivariate outliers. The size of the neighborhood (number of neighbors) is fixed, but the fraction of neighbors is varying.
locoutPercent(dat, X, Y, dist = NULL, k = 10, chisqqu = 0.975, sortup = 10, sortlow = 90, nlinesup = 10, nlineslow = 10, indices = NULL, xlab = "(Sorted) Index", ylab = "Distance to neighbor", col = gray(0.7), ...)locoutPercent(dat, X, Y, dist = NULL, k = 10, chisqqu = 0.975, sortup = 10, sortlow = 90, nlinesup = 10, nlineslow = 10, indices = NULL, xlab = "(Sorted) Index", ylab = "Distance to neighbor", col = gray(0.7), ...)
dat |
multivariate data set (without coordinates) |
X |
X coordinates of the data points |
Y |
Y coordinates of the data points |
dist |
maximum distance to search for neighbors; if nothing is provided, k for kNN is used |
k |
number of nearest neighbors to search - not taken if a value for dist is provided |
chisqqu |
quantile of the chisquare distribution for splitting the plot |
sortup |
sort local outliers accorting to given percentage |
sortlow |
sort local inliers accorting to given percentage |
nlinesup |
number of lines to be plotted for upper part |
nlineslow |
number of lines to be plotted for lower part |
indices |
if this is not NULL, these should be indices of observations to be highlighted |
xlab |
x-axis label for plot |
ylab |
y-axis label for plot |
col |
color for lines |
... |
additional parameters for plotting |
For this diagnostic tool, the number of neighbors is fixed, but propneighb (called beta) is varied. For each observation we compute the degree of isolation from a fraction of 1-beta of its neighbors. Neighborhood can be defined either via the Euclidean distance or by k-Nearest-Neighbors. The critical value for outliers is the quantile chisqqu of the chisquare distribution. One can also provide indices of observations (for indices). Then the corresponding lines in the plots will be highlighted.
ret |
list containing indices of regular and outlying observations |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
# use data from illustrative example in paper: data(X) data(Y) data(dat) res <- locoutPercent(dat,X,Y,k=10,chisqqu=0.975, indices=c(1,11,24,36))# use data from illustrative example in paper: data(X) data(Y) data(dat) res <- locoutPercent(dat,X,Y,k=10,chisqqu=0.975, indices=c(1,11,24,36))
Computes global and pairwise Mahalanobis distances for visualizing global and local multivariate outliers. The plot is split into regular (left) and global (right) outliers, and points can be selected interactively. In a second plot, these points are shown by spatial coordinates.
locoutSort(dat, X, Y, distc = NULL, k = 10, propneighb = 0.1, chisqqu = 0.975, sel = NULL, ...)locoutSort(dat, X, Y, distc = NULL, k = 10, propneighb = 0.1, chisqqu = 0.975, sel = NULL, ...)
dat |
multivariate data set (without coordinates) |
X |
X coordinates of the data points |
Y |
Y coordinates of the data points |
distc |
maximum distance to search for neighbors; if nothing is provided, k for kNN is used |
k |
number of nearest neighbors to search - not taken if a value for dist is provided |
propneighb |
proportion of neighbors to be included in tolerance ellipse |
chisqqu |
quantile of the chisquare distribution for splitting the plot |
sel |
optional list with x and y, i.e. coordinates with selected polygon |
... |
additional parameters for plotting |
For this diagnostic tool, the number of neighbors is fixed, and propneighb (called beta) is also fixed. For each observation we compute the degree of isolation from a fraction of 1-beta of its neighbors. The observations are sorted according to this degree of isolation, and this sorted index forms the x-axis of the left plot. This plot is also split into regular (left) and global (right) outliers. Then one can select with the mouse a region in this plot, meaning an observation and (some of) its neighbors. Alternatively, this region can be supplied by sel. The selected observations are then shown in the right plot. Links to the neighbors are also shown.
list(sel=sel,index.regular=res$indices.regular,index.outliers=res$indices.outliers)
sel |
plot coordinates of the selected region |
indices.reg |
indices of the bservations being regular observations |
indices.out |
indices of the observations being golbal outliers |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
# use data from illustrative example in paper: data(X) data(Y) data(dat) sel <- locoutSort(dat,X,Y,k=10,propneighb=0.1,chisqqu=0.975, sel=list(x=c(87.5,87.5,89.3,89.3),y=c(4.3,0.7,0.7,4.3)))# use data from illustrative example in paper: data(X) data(Y) data(dat) sel <- locoutSort(dat,X,Y,k=10,propneighb=0.1,chisqqu=0.975, sel=list(x=c(87.5,87.5,89.3,89.3),y=c(4.3,0.7,0.7,4.3)))
The function map.plot creates a map using geographical (x,y)-coordinates. This is thought for spatially dependent data of which coordinates are available. Multivariate outliers are marked.
map.plot(coord, data, quan=1/2, alpha=0.025, symb=FALSE, plotmap=TRUE, map="kola.background",which.map=c(1,2,3,4),map.col=c(5,1,3,4), map.lwd=c(2,1,2,1), ... )map.plot(coord, data, quan=1/2, alpha=0.025, symb=FALSE, plotmap=TRUE, map="kola.background",which.map=c(1,2,3,4),map.col=c(5,1,3,4), map.lwd=c(2,1,2,1), ... )
coord |
(x,y)-coordinates of the data |
data |
matrix or data.frame containing the data. |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
amount of observations used for calculating the adjusted quantile (see function arw). |
symb |
logical for plotting special symbols (see details). |
plotmap |
logical for plotting the background map. |
map |
see plot.kola.background() |
which.map |
see plot.kola.background() |
map.col |
see plot.kola.background() |
map.lwd |
see plot.kola.background() |
... |
additional graphical parameters |
The function map.plot shows mutlivariate outliers in a map. If symb=FALSE (default), only two colors and no special symbols are used to mark multivariate outliers (the outliers are marked red). If symb=TRUE different symbols and colors are used. The symbols (cross means big value, circle means little value) are selected according to the robust mahalanobis distance based on the adjusted mcd estimator (see function symbol.plot) Different colors (red means big value, blue means little value) according to the euclidean distances of the observations (see function color.plot) are used. For details see Filzmoser et al. (2005).
outliers |
boolean vector of outliers |
md |
robust mahalanobis distances of the data |
euclidean |
(only if symb=TRUE) euclidean distances of the observations according to the minimum of the data. |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
data(humus) # Load humus data xy <- humus[,c("XCOO","YCOO")] # X and Y Coordinates myhumus <- log(humus[, c("As", "Cd", "Co", "Cu", "Mg", "Pb", "Zn")]) map.plot(xy, myhumus, symb=TRUE)data(humus) # Load humus data xy <- humus[,c("XCOO","YCOO")] # X and Y Coordinates myhumus <- log(humus[, c("As", "Cd", "Co", "Cu", "Mg", "Pb", "Zn")]) map.plot(xy, myhumus, symb=TRUE)
The Kola Data were collected in the Kola Project (1993-1998, Geological Surveys of Finland (GTK) and Norway (NGU) and Central Kola Expedition (CKE), Russia). More than 600 samples in five different layers were analysed, this dataset contains the moss layer.
data(moss)data(moss)
A data frame with 598 observations on the following 34 variables.
IDa numeric vector
XCOOa numeric vector
YCOOa numeric vector
Aga numeric vector
Ala numeric vector
Asa numeric vector
Ba numeric vector
Baa numeric vector
Bia numeric vector
Caa numeric vector
Cda numeric vector
Coa numeric vector
Cra numeric vector
Cua numeric vector
Fea numeric vector
Hga numeric vector
Ka numeric vector
Mga numeric vector
Mna numeric vector
Moa numeric vector
Naa numeric vector
Nia numeric vector
Pa numeric vector
Pba numeric vector
Rba numeric vector
Sa numeric vector
Sba numeric vector
Sia numeric vector
Sra numeric vector
Tha numeric vector
Tla numeric vector
Ua numeric vector
Va numeric vector
Zna numeric vector
Kola Project (1993-1998)
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
data(moss) # classical versus robust correlation: corr.plot(log(moss[,"Al"]), log(moss[,"Na"]))data(moss) # classical versus robust correlation: corr.plot(log(moss[,"Al"]), log(moss[,"Na"]))
Computes the basis information for plot functions supporting the interpretation of multivariate outliers in case of compositional data.
mvoutlier.CoDa(x, quan = 0.75, alpha = 0.025, col.quantile = c(0, 0.05, 0.1, 0.5, 0.9, 0.95, 1), symb.pch = c(3, 3, 16, 1, 1), symb.cex = c(1.5, 1, 0.5, 1, 1.5), adaptive = TRUE)mvoutlier.CoDa(x, quan = 0.75, alpha = 0.025, col.quantile = c(0, 0.05, 0.1, 0.5, 0.9, 0.95, 1), symb.pch = c(3, 3, 16, 1, 1), symb.cex = c(1.5, 1, 0.5, 1, 1.5), adaptive = TRUE)
x |
data set (matrix or data frame) containing the raw untransformed compositional data |
quan |
quantity of data used for robust estimation; between 0.5 and 1 |
alpha |
maximum threshold for adaptive outlier detection |
col.quantile |
quantiles of an average concentration defining the colors |
symb.pch |
plotting character for symbols |
symb.cex |
plotting size for symbols |
adaptive |
if TRUE then the adaptive method for the outlier threshold is used |
In a first step, the raw compositional data set in transformed by the isometric logratio (ilr) transformation to the usual Euclidean space. Then adaptive outlier detection is perfomed: Starting from a quantile 1-alpha of the chisquare distribution, one looks for the supremum of the differences between the chisquare distribution and the empirical distribution of the squared Mahalanobis distances. The latter are derived from the MCD estimator using the proportion quan of the data. The supremum is the outlier cutoff, and certain colors and symbols for the outliers are computed: The colors should reflect the magnitude of the median element concentration of the observations, which is done by computing for each observation along the single ilr variables the distances to the medians. The mediab of all distances determines the color (or grey scale): a high value, resulting in a red (or dark) symbol, means that most univariate parts have higher values than the average, and a low value (blue or light symbol) refers to an observation with mainly low values. The symbols are according to the cut-points from the quantiles 0.25, 0.5, 0.75, and the outlier cutoff of the squared Mahalanobis distances.
ilrvariables |
the ilr transformed data matrix |
outliers |
TRUE/FALSE vector; TRUE refers to outlier |
pcaobj |
object from PCA |
colcol |
vector with the colors |
colbw |
vector with the grey scales |
pchvec |
vector with plotting symbols |
cexvec |
vector with sizes of plot symbols |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, K. Hron, and C. Reimann. Interpretation of multivariate outliers for compositional data. Submitted to Computers and Geosciences.
plot.mvoutlierCoDa, arw, map.plot, uni.plot
data(humus) d <- humus[,c("As","Cd","Co","Cu","Mg","Pb","Zn")] res <- mvoutlier.CoDa(d) str(res)data(humus) d <- humus[,c("As","Cd","Co","Cu","Mg","Pb","Zn")] res <- mvoutlier.CoDa(d) str(res)
Plots the BSS background map
pbb(map = "bss.background", add.plot = FALSE, ...)pbb(map = "bss.background", add.plot = FALSE, ...)
map |
List of coordinates. For the correct format see also help(kola.background) |
add.plot |
logical. If true background is added to an existing plot |
... |
additional plot parameters, see help(par) |
The list of coordinates is plotted as a polygon line.
The plot is produced on the graphical device.
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
Reimann C, Siewers U, Tarvainen T, Bityukova L, Eriksson J, Gilucis A, Gregorauskiene V, Lukashev VK, Matinian NN, Pasieczna A. Agricultural Soils in Northern Europe: A Geochemical Atlas. Geologisches Jahrbuch, Sonderhefte, Reihe D, Heft SD 5, Schweizerbart'sche Verlagsbuchhandlung, Stuttgart, 2003.
See also pkb
data(bss.background) data(bsstop) plot(bsstop$XCOO,bsstop$YCOO,col="red",pch=3) pbb(add=TRUE)data(bss.background) data(bsstop) plot(bsstop$XCOO,bsstop$YCOO,col="red",pch=3) pbb(add=TRUE)
Fast algorithm for identifying multivariate outliers in high-dimensional and/or large datasets, using the algorithm of Filzmoser, Maronna, and Werner (CSDA, 2007).
pcout(x, makeplot = FALSE, explvar = 0.99, crit.M1 = 1/3, crit.c1 = 2.5, crit.M2 = 1/4, crit.c2 = 0.99, cs = 0.25, outbound = 0.25, ...)pcout(x, makeplot = FALSE, explvar = 0.99, crit.M1 = 1/3, crit.c1 = 2.5, crit.M2 = 1/4, crit.c2 = 0.99, cs = 0.25, outbound = 0.25, ...)
x |
a numeric matrix or data frame which provides the data for outlier detection |
makeplot |
a logical value indicating whether a diagnostic plot should be generated (default to FALSE) |
explvar |
a numeric value between 0 and 1 indicating how much variance should be covered by the robust PCs (default to 0.99) |
crit.M1 |
a numeric value between 0 and 1 indicating the quantile to be used as lower boundary for location outlier detection (default to 1/3) |
crit.c1 |
a positive numeric value used for determining the upper boundary for location outlier detection (default to 2.5) |
crit.M2 |
a numeric value between 0 and 1 indicating the quantile to be used as lower boundary for scatter outlier detection (default to 1/4) |
crit.c2 |
a numeric value between 0 and 1 indicating the quantile to be used as upper boundary for scatter outlier detection (default to 0.99) |
cs |
a numeric value indicating the scaling constant for combined location and scatter weights (default to 0.25) |
outbound |
a numeric value between 0 and 1 indicating the outlier boundary for defining values as final outliers (default to 0.25) |
... |
additional plot parameters, see help(par) |
Based on the robustly sphered data, semi-robust principal components are computed which are needed for determining distances for each observation. Separate weights for location and scatter outliers are computed based on these distances. The combined weights are used for outlier identification.
wfinal01 |
0/1 vector with final weights for each observation; weight 0 indicates potential multivariate outliers. |
wfinal |
numeric vector with final weights for each observation; small values indicate potential multivariate outliers. |
wloc |
numeric vector with weights for each observation; small values indicate potential location outliers. |
wscat |
numeric vector with weights for each observation; small values indicate potential scatter outliers. |
x.dist1 |
numeric vector with distances for location outlier detection. |
x.dist2 |
numeric vector with distances for scatter outlier detection. |
M1 |
upper boundary for assigning weight 1 in location outlier detection. |
const1 |
lower boundary for assigning weight 0 in location outlier detection. |
M2 |
upper boundary for assigning weight 1 in scatter outlier detection. |
const2 |
lower boundary for assigning weight 0 in scatter outlier detection. |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R. Maronna, M. Werner. Outlier identification in high dimensions, Computational Statistics and Data Analysis, 52, 1694-1711, 2008.
# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=pcout(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on pcout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=pcout(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on pcout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)
Plots the Kola background map
pkb(map = "kola.background", which.map = c(1, 2, 3, 4), map.col = c(5, 1, 3, 4), map.lwd = c(2, 1, 2, 1), add.plot = FALSE, ...)pkb(map = "kola.background", which.map = c(1, 2, 3, 4), map.col = c(5, 1, 3, 4), map.lwd = c(2, 1, 2, 1), add.plot = FALSE, ...)
map |
List of coordinates. For the correct format see also help(kola.background) |
which.map |
which==1 ... plot project boundary \# which==2 ... plot coast line \# which==3 ... plot country borders \# which==4 ... plot lakes and rivers |
map.col |
Map colors to be used |
map.lwd |
Defines linestyle of the background |
add.plot |
logical. if true background is added to an existing plot |
... |
additional plot parameters, see help(par) |
Is used by map.plot()
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
Reimann C, Äyräs M, Chekushin V, Bogatyrev I, Boyd R, Caritat P de, Dutter R, Finne TE, Halleraker JH, Jæger Ø, Kashulina G, Lehto O, Niskavaara H, Pavlov V, Räisänen ML, Strand T, Volden T. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway, 1998.
example(map.plot)example(map.plot)
Plots the computed information by mvoutlier.CoDa for supporting the interpretation
of multivariate outliers in case of compositional data.
## S3 method for class 'mvoutlierCoDa' plot(x, ..., which = c("biplot", "map", "uni", "parallel"), choice = 1:2, coord = NULL, map = NULL, onlyout = TRUE, bw = FALSE, symb = TRUE, symbtxt = FALSE, col = NULL, pch = NULL, obj.cex = NULL, transp = 1)## S3 method for class 'mvoutlierCoDa' plot(x, ..., which = c("biplot", "map", "uni", "parallel"), choice = 1:2, coord = NULL, map = NULL, onlyout = TRUE, bw = FALSE, symb = TRUE, symbtxt = FALSE, col = NULL, pch = NULL, obj.cex = NULL, transp = 1)
x |
resulting object from function |
... |
further plotting arguments |
which |
type of plot that should be made |
choice |
select the pair of PCs used for the biplot |
coord |
coordinates for the presentation in a map |
map |
coordinates for the background map; see details below |
onlyout |
if TRUE only the outliers are shown in the plot |
bw |
if TRUE symbold will be in grey scale rather than in color |
symb |
if TRUE special symbols are used according to outlyingness |
symbtxt |
if TRUE text labels are used for plotting |
col |
define colors to be used for outliers and non-outliers |
pch |
define plotting symbols to be used for outliers and non-outliers |
obj.cex |
define symbol size for outliers and non-outliers |
transp |
define transparancy for parallel coordinate plot |
The function mvoutlier.CoDa prepares the information needed for this plot function:
In a first step, the raw compositional data set in transformed by the isometric logratio
(ilr) transformation to the usual Euclidean space. Then adaptive outlier detection is
perfomed: Starting from a quantile 1-alpha of the chisquare distribution, one looks for the
supremum of the differences between the chisquare distribution and the empirical distribution
of the squared Mahalanobis distances. The latter are derived from the MCD estimator using
the proportion quan of the data. The supremum is the outlier cutoff, and certain colors
and symbols for the outliers are computed: The colors should reflect the magnitude of the
median element concentration of the observations, which is done by computing for each observation
along the single ilr variables the distances to the medians. The mediab of all distances
determines the color (or grey scale): a high value, resulting in a red (or dark) symbol,
means that most univariate parts have higher values than the average, and a low value (blue
or light symbol) refers to an observation with mainly low values. The symbols are according
to the cut-points from the quantiles 0.25, 0.5, 0.75, and the outlier cutoff of the
squared Mahalanobis distances. This plot function then allows to visualize the information.
The optional background map for the representation of the outliers in a map can be
included using the argument map. This should consist of one or more polygons
representing the geographical x- and y-coordinates of the background map. Of course,
this map should be represented in the same coordinate system as the coordinates for
the sample locations provided by coord. The structure of map is as follows:
It should consist of 2 columns, one for the x-, one for the y-coordinates. If a polygon
ends, a row with 2 entries NA should follow. At the end two NA rows are needed.
See also examples below.
A plot is drawn.
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, K. Hron, and C. Reimann. Interpretation of multivariate outliers for compositional data. Submitted to Computers and Geosciences.
mvoutlier.CoDa, arw, map.plot, uni.plot
data(humus) el=c("As","Cd","Co","Cu","Mg","Pb","Zn") dsel <- humus[,el] data(kola.background) # contains different information (coast, borders, etc.) coo <- rbind(kola.background$coast,kola.background$boundary,kola.background$borders) XY <- humus[,c("XCOO","YCOO")] set.seed(123) res <- mvoutlier.CoDa(dsel) par(ask=TRUE) ### Parallel coordinate plot: ## show for all obvervations (transp is only useful when generating e.g. a pdf): # plot(res,onlyout=FALSE,bw=TRUE,which="parallel",symb=FALSE,symbtxt=FALSE,transp=0.3) ## show only outliers with special colors and labels in the margins: plot(res,onlyout=TRUE,bw=FALSE,which="parallel",symb=TRUE,symbtxt=TRUE,transp=0.3) ### Biplot: ## show all data points, outliers are in different color and have different symbol: # plot(res,onlyout=FALSE,which="biplot",bw=FALSE,symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,onlyout=TRUE,which="biplot",bw=FALSE,symb=TRUE,symbtxt=TRUE) ### Map: ## show all data points, outliers are in different color and have different symbol: # plot(res,coord=XY,map=coo,onlyout=FALSE,which="map",bw=FALSE,symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,coord=XY,map=coo,onlyout=TRUE,which="map",bw=FALSE,symb=TRUE,symbtxt=TRUE) ### Univariate scatterplot: ## show all data points, outliers are in different color and have different symbol: # plot(res,onlyout=FALSE,which="uni",symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,onlyout=TRUE,which="uni",symb=TRUE,symbtxt=TRUE)data(humus) el=c("As","Cd","Co","Cu","Mg","Pb","Zn") dsel <- humus[,el] data(kola.background) # contains different information (coast, borders, etc.) coo <- rbind(kola.background$coast,kola.background$boundary,kola.background$borders) XY <- humus[,c("XCOO","YCOO")] set.seed(123) res <- mvoutlier.CoDa(dsel) par(ask=TRUE) ### Parallel coordinate plot: ## show for all obvervations (transp is only useful when generating e.g. a pdf): # plot(res,onlyout=FALSE,bw=TRUE,which="parallel",symb=FALSE,symbtxt=FALSE,transp=0.3) ## show only outliers with special colors and labels in the margins: plot(res,onlyout=TRUE,bw=FALSE,which="parallel",symb=TRUE,symbtxt=TRUE,transp=0.3) ### Biplot: ## show all data points, outliers are in different color and have different symbol: # plot(res,onlyout=FALSE,which="biplot",bw=FALSE,symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,onlyout=TRUE,which="biplot",bw=FALSE,symb=TRUE,symbtxt=TRUE) ### Map: ## show all data points, outliers are in different color and have different symbol: # plot(res,coord=XY,map=coo,onlyout=FALSE,which="map",bw=FALSE,symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,coord=XY,map=coo,onlyout=TRUE,which="map",bw=FALSE,symb=TRUE,symbtxt=TRUE) ### Univariate scatterplot: ## show all data points, outliers are in different color and have different symbol: # plot(res,onlyout=FALSE,which="uni",symb=FALSE,symbtxt=FALSE) ## show only the outliers with special symbols and colors: plot(res,onlyout=TRUE,which="uni",symb=TRUE,symbtxt=TRUE)
Fast algorithm for identifying multivariate outliers in high-dimensional and/or large datasets, using spatial signs, see Filzmoser, Maronna, and Werner (CSDA, 2007). The computation of the distances is based on Mahalanobis distances.
sign1(x, makeplot = FALSE, qcrit = 0.975, ...)sign1(x, makeplot = FALSE, qcrit = 0.975, ...)
x |
a numeric matrix or data frame which provides the data for outlier detection |
makeplot |
a logical value indicating whether a diagnostic plot should be generated (default to FALSE) |
qcrit |
a numeric value between 0 and 1 indicating the quantile to be used as critical value for outlier detection (default to 0.975) |
... |
additional plot parameters, see help(par) |
Based on the robustly sphered and normed data, robust principal components are computed. These are used for computing the covariance matrix which is the basis for Mahalanobis distances. A critical value from the chi-square distribution is then used as outlier cutoff.
wfinal01 |
0/1 vector with final weights for each observation; weight 0 indicates potential multivariate outliers. |
x.dist |
numeric vector with distances used for outlier detection. |
const |
outlier cutoff value. |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R. Maronna, M. Werner. Outlier identification in high dimensions, Computational Statistics and Data Analysis, 52, 1694-1711, 2008.
N. Locantore, J. Marron, D. Simpson, N. Tripoli, J. Zhang, and K. Cohen (1999). Robust principal components for functional data, Test 8, 1–73.
# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=sign1(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on signout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=sign1(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on signout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)
Fast algorithm for identifying multivariate outliers in high-dimensional and/or large datasets, using spatial signs, see Filzmoser, Maronna, and Werner (CSDA, 2007). The computation of the distances is based on principal components.
sign2(x, makeplot = FALSE, explvar = 0.99, qcrit = 0.975, ...)sign2(x, makeplot = FALSE, explvar = 0.99, qcrit = 0.975, ...)
x |
a numeric matrix or data frame which provides the data for outlier detection |
makeplot |
a logical value indicating whether a diagnostic plot should be generated (default to FALSE) |
explvar |
a numeric value between 0 and 1 indicating how much variance should be covered by the robust PCs (default to 0.99) |
qcrit |
a numeric value between 0 and 1 indicating the quantile to be used as critical value for outlier detection (default to 0.975) |
... |
additional plot parameters, see help(par) |
Based on the robustly sphered and normed data, robust principal components are computed which are needed for determining distances for each observation. The distances are transformed to approach chi-square distribution, and a critical value is then used as outlier cutoff.
wfinal01 |
0/1 vector with final weights for each observation; weight 0 indicates potential multivariate outliers. |
x.dist |
numeric vector with distances used for outlier detection. |
const |
outlier cutoff value. |
Peter Filzmoser <[email protected]> http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R. Maronna, M. Werner. Outlier identification in high dimensions, Computational Statistics and Data Analysis, 52, 1694–1711, 2008.
N. Locantore, J. Marron, D. Simpson, N. Tripoli, J. Zhang, and K. Cohen. Robust principal components for functional data, Test 8, 1-73, 1999.
# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=sign2(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on signout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)# geochemical data from northern Europe data(bsstop) x=bsstop[,5:14] # identify multivariate outliers x.out=sign2(x,makeplot=FALSE) # visualize multivariate outliers in the map op <- par(mfrow=c(1,2)) data(bss.background) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.out$wfinal01+2) title("Outlier detection based on signout") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) # compare with outlier detection based on MCD: x.mcd <- robustbase::covMcd(x) pbb(asp=1) points(bsstop$XCOO,bsstop$YCOO,pch=16,col=x.mcd$mcd.wt+2) title("Outlier detection based on MCD") legend("topleft",legend=c("potential outliers","regular observations"),pch=16,col=c(2,3)) par(op)
The function symbol.plot plots the (two-dimensional) data using different symbols according to the robust mahalanobis distance based on the mcd estimator with adjustment.
symbol.plot(x, quan=1/2, alpha=0.025, ...)symbol.plot(x, quan=1/2, alpha=0.025, ...)
x |
two dimensional matrix or data.frame containing the data. |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
amount of observations used for calculating the adjusted quantile (see function arw). |
... |
additional graphical parameters |
The function symbol.plot plots the (two-dimensional) data using different symbols. In addition a legend and four ellipsoids are drawn, on which mahalanobis distances are constant. As the legend shows, these constant values correspond to the 25%, 50%, 75% and adjusted (see function arw) quantiles of the chi-square distribution.
outliers |
boolean vector of outliers |
md |
robust mahalanobis distances of the data |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: symbol.plot(z, quan=0.75)# create data: x <- cbind(rnorm(100), rnorm(100)) y <- cbind(rnorm(10, 5, 1), rnorm(10, 5, 1)) z <- rbind(x,y) # execute: symbol.plot(z, quan=0.75)
The function uni.plot plots each variable of x parallel in a one-dimensional scatter plot and in addition marks multivariate outliers.
uni.plot(x, symb=FALSE, quan=1/2, alpha=0.025, ...)uni.plot(x, symb=FALSE, quan=1/2, alpha=0.025, ...)
x |
matrix or data.frame containing the data. |
symb |
logical. if FALSE, only two colors and no special symbols are used. outliers are marked red. if TRUE different symbols (cross means big value, circle means little value) according to the robust mahalanobis distance based on the mcd estimator and different colors (red means big value, blue means little value) according to the euclidean distances of the observations are used. |
quan |
amount of observations which are used for mcd estimations. has to be between 0.5 and 1, default ist 0.5 |
alpha |
amount of observations used for calculating the adjusted quantile (see function arw). |
... |
additional graphical parameters |
The function uni.plot shows the mutlivariate outliers in the single variables by one-dimensional scatter plots. If symb=FALSE (default), only two colors and no special symbols are used to mark multivariate outliers (the outliers are marked red). If symb=TRUE different symbols and colors are used. The symbols (cross means big value, circle means little value) are selected according to the robust mahalanobis distance based on the adjusted mcd estimator (see function symbol.plot) Different colors (red means big value, blue means little value) according to the euclidean distances of the observations (see function color.plot) are used. For details see Filzmoser et al. (2005).
outliers |
boolean vector of outliers |
md |
robust multivariate mahalanobis distances of the data |
euclidean |
(only if symb=TRUE) multivariate euclidean distances of the observations according to the minimum of the data. |
Moritz Gschwandtner <[email protected]>
Peter Filzmoser <[email protected]>
http://cstat.tuwien.ac.at/filz/
P. Filzmoser, R.G. Garrett, and C. Reimann. Multivariate outlier detection in exploration geochemistry. Computers & Geosciences, 31:579-587, 2005.
map.plot, symbol.plot, color.plot, arw
data(swiss) uni.plot(swiss) # # Geostatistical data: data(humus) # Load humus data uni.plot(log(humus[, c("As", "Cd", "Co", "Cu", "Mg", "Pb", "Zn")]),symb=TRUE)data(swiss) uni.plot(swiss) # # Geostatistical data: data(humus) # Load humus data uni.plot(log(humus[, c("As", "Cd", "Co", "Cu", "Mg", "Pb", "Zn")]),symb=TRUE)
Illustrative data example with 100 values for the X coordinate.
data(X)data(X)
The format is: num [1:100] 3.72 5.1 3.33 2.13 4.42 ...
Data are constructed to contain global as well as local outliers.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
data(X) data(Y) plot(X,Y)data(X) data(Y) plot(X,Y)
Illustrative data example with 100 values for the Y coordinate.
data(Y)data(Y)
The format is: num [1:100] 1.25 1.4 0.372 0.791 2.74 ...
Data are constructed to contain global as well as local outliers.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
P. Filzmoser, A. Ruiz-Gazen, and C. Thomas-Agnan: Identification of local multivariate outliers. Submitted for publication, 2012.
data(X) data(Y) plot(X,Y)data(X) data(Y) plot(X,Y)