| Title: | Multivariate Statistical Analysis in Chemometrics |
|---|---|
| Description: | R companion to the book "Introduction to Multivariate Statistical Analysis in Chemometrics" written by K. Varmuza and P. Filzmoser (2009). |
| Authors: | Peter Filzmoser [aut, cre, cph] |
| Maintainer: | Peter Filzmoser <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 1.4.4 |
| Built: | 2024-12-17 06:50:46 UTC |
| Source: | CRAN |
Included are functions for multivariate statistical methods, tools for diagnostics, multivariate calibration, cross validation and bootstrap, clustering, etc.
The package can be used to verify the examples in the book. It can also be used to analyze own data.
P. Filzmoser <[email protected]
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
A data transformation according to the additive logratio transformation is done.
alr(X, divisorvar)alr(X, divisorvar)
X |
numeric data frame or matrix |
divisorvar |
number of the column of X for the variable to divide with |
The alr transformation is one possibility to transform compositional data to a real space. Afterwards, the transformed data can be analyzed in the usual way.
Returns the transformed data matrix with one variable (divisor variable) less.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) glass_alr <- alr(glass,1)data(glass) glass_alr <- alr(glass,1)
Data from 99 ash samples originating from different biomass, measured on 9 variables; 8 log-transformed variables are added.
data(ash)data(ash)
A data frame with 99 observations on the following 17 variables.
SOTa numeric vector
P2O5a numeric vector
SiO2a numeric vector
Fe2O3a numeric vector
Al2O3a numeric vector
CaOa numeric vector
MgOa numeric vector
Na2Oa numeric vector
K2Oa numeric vector
log(P2O5)a numeric vector
log(SiO2)a numeric vector
log(Fe2O3)a numeric vector
log(Al2O3)a numeric vector
log(CaO)a numeric vector
log(MgO)a numeric vector
log(Na2O)a numeric vector
log(K2O)a numeric vector
The dependent variable Softening Temperature (SOT) of ash should be modeled by the elemental composition of the ash data. Data from 99 ash samples - originating from different biomass - comprise the experimental SOT (630-1410 centigrades), and the experimentally determined eight mass concentrations the listed elements. Since the distribution of the elements is skweed, the log-transformed variables have been added.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(ash) str(ash)data(ash) str(ash)
For 15 cereals an X and Y data set, measured on the same objects, is available. The X data are 145 infrared spectra, and the Y data are 6 chemical/technical properties (Heating value, C, H, N, Starch, Ash). Also the scaled Y data are included (mean 0, variance 1 for each column). The cereals come from 5 groups B=Barley, M=Maize, R=Rye, T=Triticale, W=Wheat.
data(cereal)data(cereal)
A data frame with 15 objects and 3 list elements:
Xmatrix with 15 rows and 145 columns
Ymatrix with 15 rows and 6 columns
Yscmatrix with 15 rows and 6 columns
The data set can be used for PLS2.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(cereal) names(cereal)data(cereal) names(cereal)
A data transformation according to the centered logratio transformation is done.
clr(X)clr(X)
X |
numeric data frame or matrix |
The clr transformation is one possibility to transform compositional data to a real space. Afterwards, the transformed data can be analyzed in the usual way.
Returns the transformed data matrix with the same dimension as X.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) glass_clr <- clr(glass)data(glass) glass_clr <- clr(glass)
A cluster validity measure based on within- and between-sum-of-squares is computed and plotted for the methods k-means, fuzzy c-means, and model-based clustering.
clvalidity(x, clnumb = c(2:10))clvalidity(x, clnumb = c(2:10))
x |
input data matrix |
clnumb |
range for the desired number of clusters |
The validity measure for a number k of clusters is
divided by with
is the sum of squared distances of the objects in each cluster
cluster to its center, and is the squared distance between
the cluster centers of cluster j and l.
validity |
vector with validity measure for the desired numbers of clusters |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) require(robustbase) res <- pcaCV(glass,segments=4,repl=100,cex.lab=1.2,ylim=c(0,1),las=1)data(glass) require(robustbase) res <- pcaCV(glass,segments=4,repl=100,cex.lab=1.2,ylim=c(0,1),las=1)
A utility function to delete any intercept column from a model matrix, and adjust the assign attribute correspondingly.
delintercept(mm)delintercept(mm)
mm |
Model matrix |
A model matrix without intercept column.
B.-H. Mevik and Ron Wehrens
For 2-dimensional data a scatterplot is made. Additionally, ellipses corresponding to certain Mahalanobis distances and quantiles of the data are drawn.
drawMahal(x, center, covariance, quantile = c(0.975, 0.75, 0.5, 0.25), m = 1000, lwdcrit = 1, ...)drawMahal(x, center, covariance, quantile = c(0.975, 0.75, 0.5, 0.25), m = 1000, lwdcrit = 1, ...)
x |
numeric data frame or matrix with 2 columns |
center |
vector of length 2 with multivariate center of x |
covariance |
2 by 2 covariance matrix of x |
quantile |
vector of quantiles for the Mahalanobis distance |
m |
number of points where the ellipses should pass through |
lwdcrit |
line width of the ellipses |
... |
additional graphics parameters, see |
For multivariate normally distributed data, a fraction of 1-quantile of data should be outside the ellipses. For center and covariance also robust estimators, e.g. from the MCD estimator, can be supplied.
A scatterplot with the ellipses is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) data(glass.grp) x=glass[,c(2,7)] require(robustbase) x.mcd=covMcd(x) drawMahal(x,center=x.mcd$center,covariance=x.mcd$cov,quantile=0.975,pch=glass.grp)data(glass) data(glass.grp) x=glass[,c(2,7)] require(robustbase) x.mcd=covMcd(x) drawMahal(x,center=x.mcd$center,covariance=x.mcd$cov,quantile=0.975,pch=glass.grp)
13 different measurements for 180 archaeological glass vessels from different groups are included.
data(glass)data(glass)
A data matrix with 180 objects and 13 variables.
This is a matrix with 180 objects and 13 columns.
Janssen, K.H.A., De Raedt, I., Schalm, O., Veeckman, J.: Microchim. Acta 15 (suppl.) (1998) 253-267. Compositions of 15th - 17th century archaeological glass vessels excavated in Antwerp.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) str(glass)data(glass) str(glass)
13 different measurements for 180 archaeological glass vessels from different groups are included. These groups are certain types of glasses.
data(glass.grp)data(glass.grp)
The format is: num [1:180] 1 1 1 1 1 1 1 1 1 1 ...
This is a vector with 180 elements referring to the groups.
Janssen, K.H.A., De Raedt, I., Schalm, O., Veeckman, J.: Microchim. Acta 15 (suppl.) (1998) 253-267. Compositions of 15th - 17th century archaeological glass vessels excavated in Antwerp.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass.grp) str(glass.grp)data(glass.grp) str(glass.grp)
30 objects (Wild growing, flowering Hyptis suaveolens) and 7 variables (chemotypes), and 2 variables that explain the grouping (4 groups).
data(hyptis)data(hyptis)
A data frame with 30 observations on the following 9 variables.
Sabinenea numeric vector
Pinenea numeric vector
Cineolea numeric vector
Terpinenea numeric vector
Fenchonea numeric vector
Terpinolenea numeric vector
Fenchola numeric vector
Locationa factor with levels East-high East-low
North South
Groupa numeric vector with the group information
This data set can be used for cluster analysis.
P. Grassi, M.J. Nunez, K. Varmuza, and C. Franz: Chemical polymorphism of essential oils of Hyptis suaveolens from El Salvador. Flavour and Fragrance, 20, 131-135, 2005. K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009
data(hyptis) str(hyptis)data(hyptis) str(hyptis)
A data transformation according to the isometric logratio transformation is done.
ilr(X)ilr(X)
X |
numeric data frame or matrix |
The ilr transformation is one possibility to transform compositional data to a real space. Afterwards, the transformed data can be analyzed in the usual way.
Returns the transformed data matrix with one dimension less than X.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) glass_ilr <- ilr(glass)data(glass) glass_ilr <- ilr(glass)
Evaluation for k-Nearest-Neighbors (kNN) classification by cross-validation
knnEval(X, grp, train, kfold = 10, knnvec = seq(2, 20, by = 2), plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)knnEval(X, grp, train, kfold = 10, knnvec = seq(2, 20, by = 2), plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)
X |
standardized complete X data matrix (training and test data) |
grp |
factor with groups for complete data (training and test data) |
train |
row indices of X indicating training data objects |
kfold |
number of folds for cross-validation |
knnvec |
range for k for the evaluation of kNN |
plotit |
if TRUE a plot will be generated |
legend |
if TRUE a legend will be added to the plot |
legpos |
positioning of the legend in the plot |
... |
additional plot arguments |
The data are split into a calibration and a test data set (provided by "train"). Within the calibration set "kfold"-fold CV is performed by applying the classification method to "kfold"-1 parts and evaluation for the last part. The misclassification error is then computed for the training data, for the CV test data (CV error) and for the test data.
trainerr |
training error rate |
testerr |
test error rate |
cvMean |
mean of CV errors |
cvSe |
standard error of CV errors |
cverr |
all errors from CV |
knnvec |
range for k for the evaluation of kNN, taken from input |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(class) set.seed(123) train=sample(1:n,ntrain) resknn=knnEval(X,grp,train,knnvec=seq(1,30,by=1),legpos="bottomright") title("kNN classification")data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(class) set.seed(123) train=sample(1:n,ntrain) resknn=knnEval(X,grp,train,knnvec=seq(1,30,by=1),legpos="bottomright") title("kNN classification")
Plots the coefficients of Lasso regression
lassocoef(formula, data, sopt, plot.opt = TRUE, ...)lassocoef(formula, data, sopt, plot.opt = TRUE, ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data frame to be analyzed |
sopt |
optimal fraction from Lasso regression, see details |
plot.opt |
if TRUE a plot will be generated |
... |
additional plot arguments |
Using the function lassoCV for cross-validation, the optimal
fraction sopt can be determined. Besides a plot for the Lasso coefficients
for all values of fraction, the optimal fraction is taken to compute the
number of coefficients that are exactly zero.
coefficients |
regression coefficients for the optimal Lasso parameter |
sopt |
optimal value for fraction |
numb.zero |
number of zero coefficients for optimal fraction |
numb.nonzero |
number of nonzero coefficients for optimal fraction |
ind |
index of fraction with optimal choice for fraction |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) res=lassocoef(y~X,data=PAC,sopt=0.3)data(PAC) res=lassocoef(y~X,data=PAC,sopt=0.3)
Performs cross-validation (CV) for Lasso regression and plots the results in order to select the optimal Lasso parameter.
lassoCV(formula, data, K = 10, fraction = seq(0, 1, by = 0.05), trace = FALSE, plot.opt = TRUE, sdfact = 2, legpos = "topright", ...)lassoCV(formula, data, K = 10, fraction = seq(0, 1, by = 0.05), trace = FALSE, plot.opt = TRUE, sdfact = 2, legpos = "topright", ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data frame to be analyzed |
K |
the number of segments to use for CV |
fraction |
fraction for Lasso parameters to be used for evaluation, see details |
trace |
if 'TRUE', intermediate results are printed |
plot.opt |
if TRUE a plot will be generated that shows optimal choice for "fraction" |
sdfact |
factor for the standard error for selection of the optimal parameter, see details |
legpos |
position of the legend in the plot |
... |
additional plot arguments |
The parameter "fraction" is the sum of absolute values of the regression coefficients
for a particular Lasso parameter on the sum of absolute values of the regression
coefficients for the maximal possible value of the Lasso parameter (unconstrained
case), see also lars.
The optimal fraction is chosen according to the following criterion:
Within the CV scheme, the mean of the SEPs is computed, as well as their standard
errors. Then one searches for the minimum of the mean SEPs and adds
sdfact*standarderror. The optimal fraction is the smallest fraction with an MSEP
below this bound.
cv |
MSEP values at each value of fraction |
cv.error |
standard errors for each value of fraction |
SEP |
SEP value for each value of fraction |
ind |
index of fraction with optimal choice for fraction |
sopt |
optimal value for fraction |
fraction |
all values considered for fraction |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) # takes some time: # res <- lassoCV(y~X,data=PAC,K=5,fraction=seq(0.1,0.5,by=0.1))data(PAC) # takes some time: # res <- lassoCV(y~X,data=PAC,K=5,fraction=seq(0.1,0.5,by=0.1))
Repeated Cross Validation for multiple linear regression: a cross-validation is performed repeatedly, and standard evaluation measures are returned.
lmCV(formula, data, repl = 100, segments = 4, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, ...)lmCV(formula, data, repl = 100, segments = 4, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data set including y and X |
repl |
number of replication for Cross Validation |
segments |
number of segments used for splitting into training and test data |
segment.type |
"random", "consecutive", "interleaved" splitting into training and test data |
length.seg |
number of parts for training and test data, overwrites segments |
trace |
if TRUE intermediate results are reported |
... |
additional plotting arguments |
Repeating the cross-validation with allow for a more careful evaluation.
residuals |
matrix of size length(y) x repl with residuals |
predicted |
matrix of size length(y) x repl with predicted values |
SEP |
Standard Error of Prediction computed for each column of "residuals" |
SEPm |
mean SEP value |
RMSEP |
Root MSEP value computed for each column of "residuals" |
RMSEPm |
mean RMSEP value |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(ash) set.seed(100) res=lmCV(SOT~.,data=ash,repl=10) hist(res$SEP)data(ash) set.seed(100) res=lmCV(SOT~.,data=ash,repl=10) hist(res$SEP)
For multivariate outlier detection the Mahalanobis distance can be used. Here a plot of the classical and the robust (based on the MCD) Mahalanobis distance is drawn.
Moutlier(X, quantile = 0.975, plot = TRUE, ...)Moutlier(X, quantile = 0.975, plot = TRUE, ...)
X |
numeric data frame or matrix |
quantile |
cut-off value (quantile) for the Mahalanobis distance |
plot |
if TRUE a plot is generated |
... |
additional graphics parameters, see |
For multivariate normally distributed data, a fraction of 1-quantile of data can be declared as potential multivariate outliers. These would be identified with the Mahalanobis distance based on classical mean and covariance. For deviations from multivariate normality center and covariance have to be estimated in a robust way, e.g. by the MCD estimator. The resulting robust Mahalanobis distance is suitable for outlier detection. Two plots are generated, showing classical and robust Mahalanobis distance versus the observation numbers.
md |
Values of the classical Mahalanobis distance |
rd |
Values of the robust Mahalanobis distance |
cutoff |
Value with the outlier cut-off |
...
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) data(glass.grp) x=glass[,c(2,7)] require(robustbase) res <- Moutlier(glass,quantile=0.975,pch=glass.grp)data(glass) data(glass.grp) x=glass[,c(2,7)] require(robustbase) res <- Moutlier(glass,quantile=0.975,pch=glass.grp)
Performs a careful evaluation by repeated double-CV for multivariate regression methods, like PLS and PCR.
mvr_dcv(formula, ncomp, data, subset, na.action, method = c("kernelpls", "widekernelpls", "simpls", "oscorespls", "svdpc"), scale = FALSE, repl = 100, sdfact = 2, segments0 = 4, segment0.type = c("random", "consecutive", "interleaved"), length.seg0, segments = 10, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = FALSE, selstrat = "hastie", ...)mvr_dcv(formula, ncomp, data, subset, na.action, method = c("kernelpls", "widekernelpls", "simpls", "oscorespls", "svdpc"), scale = FALSE, repl = 100, sdfact = 2, segments0 = 4, segment0.type = c("random", "consecutive", "interleaved"), length.seg0, segments = 10, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = FALSE, selstrat = "hastie", ...)
formula |
formula, like y~X, i.e., dependent~response variables |
ncomp |
number of PLS components |
data |
data frame to be analyzed |
subset |
optional vector to define a subset |
na.action |
a function which indicates what should happen when the data contain missing values |
method |
the multivariate regression method to be used, see
|
scale |
numeric vector, or logical. If numeric vector, X is scaled by dividing each variable with the corresponding element of 'scale'. If 'scale' is 'TRUE', X is scaled by dividing each variable by its sample standard deviation. If cross-validation is selected, scaling by the standard deviation is done for every segment. |
repl |
Number of replicattion for the double-CV |
sdfact |
factor for the multiplication of the standard deviation for the determination of the optimal number of components |
segments0 |
the number of segments to use for splitting into training and test
data, or a list with segments (see |
segment0.type |
the type of segments to use. Ignored if 'segments0' is a list |
length.seg0 |
Positive integer. The length of the segments to use. If specified, it overrides 'segments' unless 'segments0' is a list |
segments |
the number of segments to use for selecting the optimal number if
components, or a list with segments (see |
segment.type |
the type of segments to use. Ignored if 'segments' is a list |
length.seg |
Positive integer. The length of the segments to use. If specified, it overrides 'segments' unless 'segments' is a list |
trace |
logical; if 'TRUE', the segment number is printed for each segment |
plot.opt |
if TRUE a plot will be generated that shows the selection of the optimal number of components for each step of the CV |
selstrat |
method that defines how the optimal number of components is selected, should be one of "diffnext", "hastie", "relchange"; see details |
... |
additional parameters |
In this cross-validation (CV) scheme, the optimal number of components is determined by an additional CV in the training set, and applied to the test set. The procedure is repeated repl times. There are different strategies for determining the optimal number of components (parameter selstrat): "diffnext" compares MSE+sdfact*sd(MSE) among the neighbors, and if the MSE falls outside this bound, this is the optimal number. "hastie" searches for the number of components with the minimum of the mean MSE's. The optimal number of components is the model with the smallest number of components which is still in the range of the MSE+sdfact*sd(MSE), where MSE and sd are taken from the minimum. "relchange" is a strategy where the relative change is combined with "hastie": First the minimum of the mean MSE's is searched, and MSE's of larger components are omitted. For this selection, the relative change in MSE compared to the min, and relative to the max, is computed. If this change is very small (e.g. smaller than 0.005), these components are omitted. Then the "hastie" strategy is applied for the remaining MSE's.
resopt |
array [nrow(Y) x ncol(Y) x repl] with residuals using optimum number of components |
predopt |
array [nrow(Y) x ncol(Y) x repl] with predicted Y using optimum number of components |
optcomp |
matrix [segments0 x repl] optimum number of components for each training set |
pred |
array [nrow(Y) x ncol(Y) x ncomp x repl] with predicted Y for all numbers of components |
SEPopt |
SEP over all residuals using optimal number of components |
sIQRopt |
spread of inner half of residuals as alternative robust spread measure to the SEPopt |
sMADopt |
MAD of residuals as alternative robust spread measure to the SEPopt |
MSEPopt |
MSEP over all residuals using optimal number of components |
afinal |
final optimal number of components |
SEPfinal |
vector of length ncomp with final SEP values; use the element afinal for the optimal SEP |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10)
NIPALS is an algorithm for computing PCA scores and loadings.
nipals(X, a, it = 10, tol = 1e-04)nipals(X, a, it = 10, tol = 1e-04)
X |
numeric data frame or matrix |
a |
maximum number of principal components to be computed |
it |
maximum number of iterations |
tol |
tolerance limit for convergence of the algorithm |
The NIPALS algorithm is well-known in chemometrics. It is an algorithm for computing PCA scores and loadings. The advantage is that the components are computed one after the other, and one could stop at a desired number of components.
T |
matrix with the PCA scores |
P |
matrix with the PCA loadings |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) res <- nipals(glass,a=2)data(glass) res <- nipals(glass,a=2)
For 166 alcoholic fermentation mashes of different feedstock (rye, wheat and corn) we have 235 variables (X) containing the first derivatives of near infrared spectroscopy (NIR) absorbance values at 1115-2285 nm, and two variables (Y) containing the concentration of glucose and ethanol (in g/L).
data(NIR)data(NIR)
A data frame with 166 objects and 2 list elements:
xNIRdata frame with 166 rows and 235 columns
yGlcEtOHdata frame with 166 rows and 2 columns
The data can be used for linear and non-linear models.
B. Liebmann, A. Friedl, and K. Varmuza. Determination of glucose and ethanol in bioethanol production by near infrared spectroscopy and chemometrics. Anal. Chim. Acta, 642:171-178, 2009.
B. Liebmann, A. Friedl, and K. Varmuza. Determination of glucose and ethanol in bioethanol production by near infrared spectroscopy and chemometrics. Anal. Chim. Acta, 642:171-178, 2009.
data(NIR) str(NIR)data(NIR) str(NIR)
Evaluation for Artificial Neural Network (ANN) classification by cross-validation
nnetEval(X, grp, train, kfold = 10, decay = seq(0, 10, by = 1), size = 30, maxit = 100, plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)nnetEval(X, grp, train, kfold = 10, decay = seq(0, 10, by = 1), size = 30, maxit = 100, plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)
X |
standardized complete X data matrix (training and test data) |
grp |
factor with groups for complete data (training and test data) |
train |
row indices of X indicating training data objects |
kfold |
number of folds for cross-validation |
decay |
weight decay, see |
size |
number of hidden units, see |
maxit |
maximal number of iterations for ANN, see |
plotit |
if TRUE a plot will be generated |
legend |
if TRUE a legend will be added to the plot |
legpos |
positioning of the legend in the plot |
... |
additional plot arguments |
The data are split into a calibration and a test data set (provided by "train"). Within the calibration set "kfold"-fold CV is performed by applying the classification method to "kfold"-1 parts and evaluation for the last part. The misclassification error is then computed for the training data, for the CV test data (CV error) and for the test data.
trainerr |
training error rate |
testerr |
test error rate |
cvMean |
mean of CV errors |
cvSe |
standard error of CV errors |
cverr |
all errors from CV |
decay |
value(s) for weight decay, taken from input |
size |
value(s) for number of hidden units, taken from input |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(nnet) set.seed(123) train=sample(1:n,ntrain) resnnet=nnetEval(X,grp,train,decay=c(0,0.01,0.1,0.15,0.2,0.3,0.5,1), size=20,maxit=20)data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(nnet) set.seed(123) train=sample(1:n,ntrain) resnnet=nnetEval(X,grp,train,decay=c(0,0.01,0.1,0.15,0.2,0.3,0.5,1), size=20,maxit=20)
For 209 objects an X-data set (467 variables) and a y-data set (1 variable) is available. The data describe GC-retention indices of polycyclic aromatic compounds (y) which have been modeled by molecular descriptors (X).
data(PAC)data(PAC)
A data frame with 209 objects and 2 list elements:
ynumeric vector with length 209
Xmatrix with 209 rows and 467 columns
The data can be used for linear and non-linear models.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) names(PAC)data(PAC) names(PAC)
By splitting data into training and test data repeatedly the number of principal components can be determined by inspecting the distribution of the explained variances.
pcaCV(X, amax, center = TRUE, scale = TRUE, repl = 50, segments = 4, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = TRUE, ...)pcaCV(X, amax, center = TRUE, scale = TRUE, repl = 50, segments = 4, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = TRUE, ...)
X |
numeric data frame or matrix |
amax |
maximum number of components for evaluation |
center |
should the data be centered? TRUE or FALSE |
scale |
should the data be scaled? TRUE or FALSE |
repl |
number of replications of the CV procedure |
segments |
number of segments for CV |
segment.type |
"random", "consecutive", "interleaved" splitting into training and test data |
length.seg |
number of parts for training and test data, overwrites segments |
trace |
if TRUE intermediate results are reported |
plot.opt |
if TRUE the results are shown by boxplots |
... |
additional graphics parameters, see |
For cross validation the data are split into a number of segments, PCA is computed (using 1 to amax components) for all but one segment, and the scores of the segment left out are calculated. This is done in turn, by omitting each segment one time. Thus, a complete score matrix results for each desired number of components, and the error martrices of fit can be computed. A measure of fit is the explained variance, which is computed for each number of components. Then the whole procedure is repeated (repl times), which results in repl numbers of explained variance for 1 to amax components, i.e. a matrix. The matrix is presented by boxplots, where each boxplot summarized the explained variance for a certain number of principal components.
ExplVar |
matrix with explained variances, repl rows, and amax columns |
MSEP |
matrix with MSEP values, repl rows, and amax columns |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) x.sc <- scale(glass) resv <- clvalidity(x.sc,clnumb=c(2:5))data(glass) x.sc <- scale(glass) resv <- clvalidity(x.sc,clnumb=c(2:5))
Score distances and orthogonal distances are computed and plotted.
pcaDiagplot(X, X.pca, a = 2, quantile = 0.975, scale = TRUE, plot = TRUE, ...)pcaDiagplot(X, X.pca, a = 2, quantile = 0.975, scale = TRUE, plot = TRUE, ...)
X |
numeric data frame or matrix |
X.pca |
PCA object resulting e.g. from |
a |
number of principal components |
quantile |
quantile for the critical cut-off values |
scale |
if TRUE then X will be scaled - and X.pca should be from scaled data too |
plot |
if TRUE a plot is generated |
... |
additional graphics parameters, see |
The score distance measures the outlyingness of the onjects within the PCA space using Mahalanobis distances. The orthogonal distance measures the distance of the objects orthogonal to the PCA space. Cut-off values for both distance measures help to distinguish between outliers and regular observations.
SDist |
Score distances |
ODist |
Orthogonal distances |
critSD |
critical cut-off value for the score distances |
critOD |
critical cut-off value for the orthogonal distances |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) require(robustbase) glass.mcd <- covMcd(glass) rpca <- princomp(glass,covmat=glass.mcd) res <- pcaDiagplot(glass,rpca,a=2)data(glass) require(robustbase) glass.mcd <- covMcd(glass) rpca <- princomp(glass,covmat=glass.mcd) res <- pcaDiagplot(glass,rpca,a=2)
Diagnostics of PCA to see the explained variance for each variable.
pcaVarexpl(X, a, center = TRUE, scale = TRUE, plot = TRUE, ...)pcaVarexpl(X, a, center = TRUE, scale = TRUE, plot = TRUE, ...)
X |
numeric data frame or matrix |
a |
number of principal components |
center |
centring of X (FALSE or TRUE) |
scale |
scaling of X (FALSE or TRUE) |
plot |
if TRUE make plot with explained variance |
... |
additional graphics parameters, see |
For a desired number of principal components the percentage of explained variance is computed for each variable and plotted.
ExplVar |
explained variance for each variable |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) res <- pcaVarexpl(glass,a=2)data(glass) res <- pcaVarexpl(glass,a=2)
The data consist of mass spectra from 600 chemical compounds, where 300 contain a phenyl substructure (group 1) and 300 compounds do not contain this substructure (group 2). The mass spectra have been transformed to 658 variables, containing the mass spectral features. The 2 groups are coded as -1 (group 1) and +1 (group 2), and is provided as first last variable.
data(Phenyl)data(Phenyl)
A data frame with 600 observations on the following 659 variables.
grpa numeric vector
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spec.V658a numeric vector
The data set can be used for classification in high dimensions.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(Phenyl) str(Phenyl)data(Phenyl) str(Phenyl)
Generate plot showing optimal number of components for Repeated Double Cross-Validation
plotcompmvr(mvrdcvobj, ...)plotcompmvr(mvrdcvobj, ...)
mvrdcvobj |
object from repeated double-CV, see |
... |
additional plot arguments |
After running repeated double-CV, this plot helps to decide on the final number of components.
optcomp |
optimal number of components |
compdistrib |
frequencies for the optimal number of components |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot2 <- plotcompmvr(res)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot2 <- plotcompmvr(res)
Generate plot showing optimal number of components for Repeated Double Cross-Validation of Partial Robust M-regression
plotcompprm(prmdcvobj, ...)plotcompprm(prmdcvobj, ...)
prmdcvobj |
object from repeated double-CV of PRM, see |
... |
additional plot arguments |
After running repeated double-CV for PRM, this plot helps to decide on the final number of components.
optcomp |
optimal number of components |
compdistrib |
frequencies for the optimal number of components |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot2 <- plotcompprm(res)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot2 <- plotcompprm(res)
Generate plot showing predicted values for Repeated Double Cross Validation
plotpredmvr(mvrdcvobj, optcomp, y, X, method = "simpls", ...)plotpredmvr(mvrdcvobj, optcomp, y, X, method = "simpls", ...)
mvrdcvobj |
object from repeated double-CV, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
method |
the multivariate regression method to be used, see
|
... |
additional plot arguments |
After running repeated double-CV, this plot visualizes the predicted values.
A plot is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot3 <- plotpredmvr(res,opt=7,y,X,method="simpls")data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot3 <- plotpredmvr(res,opt=7,y,X,method="simpls")
Generate plot showing predicted values for Repeated Double Cross Validation of Partial Robust M-regression
plotpredprm(prmdcvobj, optcomp, y, X, ...)plotpredprm(prmdcvobj, optcomp, y, X, ...)
prmdcvobj |
object from repeated double-CV of PRM, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
... |
additional plot arguments |
After running repeated double-CV for PRM, this plot visualizes the predicted values. The result is compared with predicted values obtained via usual CV of PRM.
A plot is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot3 <- plotpredprm(res,opt=res$afinal,y,X)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot3 <- plotpredprm(res,opt=res$afinal,y,X)
The predicted values and the residuals are shown for robust PLS using the optimal number of components.
plotprm(prmobj, y, ...)plotprm(prmobj, y, ...)
prmobj |
resulting object from CV of robust PLS, see |
y |
vector with values of response variable |
... |
additional plot arguments |
Robust PLS based on partial robust M-regression is available at prm.
Here the function prm_cv has to be used first, applying cross-validation
with robust PLS. Then the result is taken by this routine and two plots are generated
for the optimal number of PLS components: The measured versus the predicted y, and
the predicted y versus the residuals.
A plot is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(cereal) set.seed(123) res <- prm_cv(cereal$X,cereal$Y[,1],a=5,segments=4,plot.opt=FALSE) plotprm(res,cereal$Y[,1])data(cereal) set.seed(123) res <- prm_cv(cereal$X,cereal$Y[,1],a=5,segments=4,plot.opt=FALSE) plotprm(res,cereal$Y[,1])
Generate plot showing residuals for Repeated Double Cross Validation
plotresmvr(mvrdcvobj, optcomp, y, X, method = "simpls", ...)plotresmvr(mvrdcvobj, optcomp, y, X, method = "simpls", ...)
mvrdcvobj |
object from repeated double-CV, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
method |
the multivariate regression method to be used, see
|
... |
additional plot arguments |
After running repeated double-CV, this plot visualizes the residuals.
A plot is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot4 <- plotresmvr(res,opt=7,y,X,method="simpls")data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot4 <- plotresmvr(res,opt=7,y,X,method="simpls")
Generate plot showing residuals for Repeated Double Cross Validation for Partial Robust M-regression
plotresprm(prmdcvobj, optcomp, y, X, ...)plotresprm(prmdcvobj, optcomp, y, X, ...)
prmdcvobj |
object from repeated double-CV of PRM, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
... |
additional plot arguments |
After running repeated double-CV for PRM, this plot visualizes the residuals. The result is compared with predicted values obtained via usual CV of PRM.
A plot is generated.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot4 <- plotresprm(res,opt=res$afinal,y,X)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot4 <- plotresprm(res,opt=res$afinal,y,X)
Two plots from Ridge regression are generated: The MSE resulting from Generalized Cross Validation (GCV) versus the Ridge parameter lambda, and the regression coefficients versus lambda. The optimal choice for lambda is indicated.
plotRidge(formula, data, lambda = seq(0.5, 50, by = 0.05), ...)plotRidge(formula, data, lambda = seq(0.5, 50, by = 0.05), ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data frame to be analyzed |
lambda |
possible values for the Ridge parameter to evaluate |
... |
additional plot arguments |
For all values provided in lambda the results for Ridge regression are computed.
The function lm.ridge is used for cross-validation and
Ridge regression.
predicted |
predicted values for the optimal lambda |
lambdaopt |
optimal Ridge parameter lambda from GCV |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) res=plotRidge(y~X,data=PAC,lambda=seq(1,20,by=0.5))data(PAC) res=plotRidge(y~X,data=PAC,lambda=seq(1,20,by=0.5))
Generate plot showing SEP values for Repeated Double Cross Validation
plotSEPmvr(mvrdcvobj, optcomp, y, X, method = "simpls", complete = TRUE, ...)plotSEPmvr(mvrdcvobj, optcomp, y, X, method = "simpls", complete = TRUE, ...)
mvrdcvobj |
object from repeated double-CV, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
method |
the multivariate regression method to be used, see
|
complete |
if TRUE the SEPcv values are drawn and computed for the same range of components as included in the mvrdcvobj object; if FALSE only optcomp components are computed and their results are displayed |
... |
additional plot arguments |
After running repeated double-CV, this plot visualizes the distribution of the SEP values.
SEPdcv |
all SEP values from repeated double-CV |
SEPcv |
SEP values from classical CV |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot1 <- plotSEPmvr(res,opt=7,y,X,method="simpls")data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- mvr_dcv(y~.,data=NIR.Glc,ncomp=10,method="simpls",repl=10) plot1 <- plotSEPmvr(res,opt=7,y,X,method="simpls")
Generate plot showing trimmed SEP values for Repeated Double Cross Validation for Partial RObust M-Regression (PRM)
plotSEPprm(prmdcvobj, optcomp, y, X, complete = TRUE, ...)plotSEPprm(prmdcvobj, optcomp, y, X, complete = TRUE, ...)
prmdcvobj |
object from repeated double-CV of PRM, see |
optcomp |
optimal number of components |
y |
data from response variable |
X |
data with explanatory variables |
complete |
if TRUE the trimmed SEPcv values are drawn and computed from
|
... |
additional arguments ofr |
After running repeated double-CV for PRM, this plot visualizes the distribution of the SEP values. While the gray lines represent the resulting trimmed SEP values from repreated double CV, the black line is the result for standard CV with PRM, and it is usually too optimistic.
SEPdcv |
all trimmed SEP values from repeated double-CV |
SEPcv |
trimmed SEP values from usual CV |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot1 <- plotSEPprm(res,opt=res$afinal,y,X)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=4,repl=2) plot1 <- plotSEPprm(res,opt=res$afinal,y,X)
Plot results of Self Organizing Maps (SOM).
plotsom(obj, grp, type = c("num", "bar"), margins = c(3,2,2,2), ...)plotsom(obj, grp, type = c("num", "bar"), margins = c(3,2,2,2), ...)
obj |
result object from |
grp |
numeric vector or factor with group information |
type |
type of presentation for output, see details |
margins |
plot margins for output, see |
... |
additional graphics parameters, see |
The results of Self Organizing Maps (SOM) are plotted either in a table with numbers (type="num") or with barplots (type="bar"). There is a limitation to at most 9 groups. A summary table is returned.
sumtab |
Summary table |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(glass) require(som) Xs <- scale(glass) Xn <- Xs/sqrt(apply(Xs^2,1,sum)) X_SOM <- som(Xn,xdim=4,ydim=4) # 4x4 fields data(glass.grp) res <- plotsom(X_SOM,glass.grp,type="bar")data(glass) require(som) Xs <- scale(glass) Xn <- Xs/sqrt(apply(Xs^2,1,sum)) X_SOM <- som(Xn,xdim=4,ydim=4) # 4x4 fields data(glass.grp) res <- plotsom(X_SOM,glass.grp,type="bar")
Computes the PLS solution by eigenvector decompositions.
pls_eigen(X, Y, a)pls_eigen(X, Y, a)
X |
X input data, centered (and scaled) |
Y |
Y input data, centered (and scaled) |
a |
number of PLS components |
The X loadings (P) and scores (T) are found by the eigendecomposition of X'YY'X. The Y loadings (Q) and scores (U) come from the eigendecomposition of Y'XX'Y. The resulting P and Q are orthogonal. The first score vectors are the same as for standard PLS, subsequent score vectors different.
P |
matrix with loadings for X |
T |
matrix with scores for X |
Q |
matrix with loadings for Y |
U |
matrix with scores for Y |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(cereal) res <- pls_eigen(cereal$X,cereal$Y,a=5)data(cereal) res <- pls_eigen(cereal$X,cereal$Y,a=5)
NIPALS algorithm for PLS1 regression (y is univariate)
pls1_nipals(X, y, a, it = 50, tol = 1e-08, scale = FALSE)pls1_nipals(X, y, a, it = 50, tol = 1e-08, scale = FALSE)
X |
original X data matrix |
y |
original y-data |
a |
number of PLS components |
it |
number of iterations |
tol |
tolerance for convergence |
scale |
if TRUE the X and y data will be scaled in addition to centering, if FALSE only mean centering is performed |
The NIPALS algorithm is the originally proposed algorithm for PLS. Here, the y-data are only allowed to be univariate. This simplifies the algorithm.
P |
matrix with loadings for X |
T |
matrix with scores for X |
W |
weights for X |
C |
weights for Y |
b |
final regression coefficients |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) res <- pls1_nipals(PAC$X,PAC$y,a=5)data(PAC) res <- pls1_nipals(PAC$X,PAC$y,a=5)
NIPALS algorithm for PLS2 regression (y is multivariate)
pls2_nipals(X, Y, a, it = 50, tol = 1e-08, scale = FALSE)pls2_nipals(X, Y, a, it = 50, tol = 1e-08, scale = FALSE)
X |
original X data matrix |
Y |
original Y-data matrix |
a |
number of PLS components |
it |
number of iterations |
tol |
tolerance for convergence |
scale |
if TRUE the X and y data will be scaled in addition to centering, if FALSE only mean centering is performed |
The NIPALS algorithm is the originally proposed algorithm for PLS. Here, the Y-data matrix is multivariate.
P |
matrix with loadings for X |
T |
matrix with scores for X |
Q |
matrix with loadings for Y |
U |
matrix with scores for Y |
D |
D-matrix within the algorithm |
W |
weights for X |
C |
weights for Y |
B |
final regression coefficients |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(cereal) res <- pls2_nipals(cereal$X,cereal$Y,a=5)data(cereal) res <- pls2_nipals(cereal$X,cereal$Y,a=5)
Robust PLS by partial robust M-regression.
prm(X, y, a, fairct = 4, opt = "l1m",usesvd=FALSE)prm(X, y, a, fairct = 4, opt = "l1m",usesvd=FALSE)
X |
predictor matrix |
y |
response variable |
a |
number of PLS components |
fairct |
tuning constant, by default fairct=4 |
opt |
if "l1m" the mean centering is done by the l1-median, otherwise if "median" the coordinate-wise median is taken |
usesvd |
if TRUE, SVD will be used if X has more columns than rows |
M-regression is used to robustify PLS, with initial weights based on the FAIR weight function.
coef |
vector with regression coefficients |
intercept |
coefficient for intercept |
wy |
vector of length(y) with residual weights |
wt |
vector of length(y) with weights for leverage |
w |
overall weights |
scores |
matrix with PLS X-scores |
loadings |
matrix with PLS X-loadings |
fitted.values |
vector with fitted y-values |
mx |
column means of X |
my |
mean of y |
Peter Filzmoser <[email protected]>
S. Serneels, C. Croux, P. Filzmoser, and P.J. Van Espen. Partial robust M-regression. Chemometrics and Intelligent Laboratory Systems, Vol. 79(1-2), pp. 55-64, 2005.
data(PAC) res <- prm(PAC$X,PAC$y,a=5)data(PAC) res <- prm(PAC$X,PAC$y,a=5)
Cross-validation (CV) is carried out with robust PLS based on partial robust M-regression. A plot with the choice for the optimal number of components is generated. This only works for univariate y-data.
prm_cv(X, y, a, fairct = 4, opt = "median", subset = NULL, segments = 10, segment.type = "random", trim = 0.2, sdfact = 2, plot.opt = TRUE)prm_cv(X, y, a, fairct = 4, opt = "median", subset = NULL, segments = 10, segment.type = "random", trim = 0.2, sdfact = 2, plot.opt = TRUE)
X |
predictor matrix |
y |
response variable |
a |
number of PLS components |
fairct |
tuning constant, by default fairct=4 |
opt |
if "l1m" the mean centering is done by the l1-median, otherwise by the coordinate-wise median |
subset |
optional vector defining a subset of objects |
segments |
the number of segments to use or a list with segments (see
|
segment.type |
the type of segments to use. Ignored if 'segments' is a list |
trim |
trimming percentage for the computation of the SEP |
sdfact |
factor for the multiplication of the standard deviation for
the determination of the optimal number of components, see
|
plot.opt |
if TRUE a plot will be generated that shows the selection of the
optimal number of components for each step of the CV, see
|
A function for robust PLS based on partial robust M-regression is available at
prm. The optimal number of robust PLS components is chosen according
to the following criterion: Within the CV scheme, the mean of the trimmed SEPs
SEPtrimave is computed for each number of components, as well as their standard
errors SEPtrimse. Then one searches for the minimum of the SEPtrimave values and
adds sdfact*SEPtrimse. The optimal number of components is the most parsimonious
model that is below this bound.
predicted |
matrix with length(y) rows and a columns with predicted values |
SEPall |
vector of length a with SEP values for each number of components |
SEPtrim |
vector of length a with trimmed SEP values for each number of components |
SEPj |
matrix with segments rows and a columns with SEP values within the CV for each number of components |
SEPtrimj |
matrix with segments rows and a columns with trimmed SEP values within the CV for each number of components |
optcomp |
final optimal number of PLS components |
SEPopt |
trimmed SEP value for final optimal number of PLS components |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(cereal) set.seed(123) res <- prm_cv(cereal$X,cereal$Y[,1],a=5,segments=4,plot.opt=TRUE)data(cereal) set.seed(123) res <- prm_cv(cereal$X,cereal$Y[,1],a=5,segments=4,plot.opt=TRUE)
Performs a careful evaluation by repeated double-CV for robust PLS, called PRM (partial robust M-estimation).
prm_dcv(X,Y,a=10,repl=10,segments0=4,segments=7,segment0.type="random", segment.type="random",sdfact=2,fairct=4,trim=0.2,opt="median",plot.opt=FALSE, ...)prm_dcv(X,Y,a=10,repl=10,segments0=4,segments=7,segment0.type="random", segment.type="random",sdfact=2,fairct=4,trim=0.2,opt="median",plot.opt=FALSE, ...)
X |
predictor matrix |
Y |
response variable |
a |
number of PLS components |
repl |
Number of replicattion for the double-CV |
segments0 |
the number of segments to use for splitting into training and
test data, or a list with segments (see |
segments |
the number of segments to use for selecting the optimal number if
components, or a list with segments (see |
segment0.type |
the type of segments to use. Ignored if 'segments0' is a list |
segment.type |
the type of segments to use. Ignored if 'segments' is a list |
sdfact |
factor for the multiplication of the standard deviation for
the determination of the optimal number of components, see
|
fairct |
tuning constant, by default fairct=4 |
trim |
trimming percentage for the computation of the SEP |
opt |
if "l1m" the mean centering is done by the l1-median, otherwise if "median", by the coordinate-wise median |
plot.opt |
if TRUE a plot will be generated that shows the selection of the optimal number of components for each step of the CV |
... |
additional parameters |
In this cross-validation (CV) scheme, the optimal number of components is determined by an additional CV in the training set, and applied to the test set. The procedure is repeated repl times. The optimal number of components is the model with the smallest number of components which is still in the range of the MSE+sdfact*sd(MSE), where MSE and sd are taken from the minimum.
b |
estimated regression coefficients |
intercept |
estimated regression intercept |
resopt |
array [nrow(Y) x ncol(Y) x repl] with residuals using optimum number of components |
predopt |
array [nrow(Y) x ncol(Y) x repl] with predicted Y using optimum number of components |
optcomp |
matrix [segments0 x repl] optimum number of components for each training set |
residcomp |
array [nrow(Y) x ncomp x repl] with residuals using optimum number of components |
pred |
array [nrow(Y) x ncol(Y) x ncomp x repl] with predicted Y for all numbers of components |
SEPall |
matrix [ncomp x repl] with SEP values |
SEPtrim |
matrix [ncomp x repl] with trimmed SEP values |
SEPcomp |
vector of length ncomp with trimmed SEP values; use the element afinal for the optimal trimmed SEP |
afinal |
final optimal number of components |
SEPopt |
trimmed SEP over all residuals using optimal number of components |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=3,repl=2)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- prm_dcv(X,y,a=3,repl=2)
Performs repeated cross-validation (CV) to evaluate the result of Ridge regression where the optimal Ridge parameter lambda was chosen on a fast evaluation scheme.
ridgeCV(formula, data, lambdaopt, repl = 5, segments = 10, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = TRUE, ...)ridgeCV(formula, data, lambdaopt, repl = 5, segments = 10, segment.type = c("random", "consecutive", "interleaved"), length.seg, trace = FALSE, plot.opt = TRUE, ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data frame to be analyzed |
lambdaopt |
optimal Ridge parameter lambda |
repl |
number of replications for the CV |
segments |
the number of segments to use for CV,
or a list with segments (see |
segment.type |
the type of segments to use. Ignored if 'segments' is a list |
length.seg |
Positive integer. The length of the segments to use. If specified, it overrides 'segments' unless 'segments' is a list |
trace |
logical; if 'TRUE', the segment number is printed for each segment |
plot.opt |
if TRUE a plot will be generated that shows the predicted versus the observed y-values |
... |
additional plot arguments |
Generalized Cross Validation (GCV) is used by the function
lm.ridge to get a quick answer for the optimal Ridge parameter.
This function should make a careful evaluation once the optimal parameter lambda has
been selected. Measures for the prediction quality are computed and optionally plots
are shown.
residuals |
matrix of size length(y) x repl with residuals |
predicted |
matrix of size length(y) x repl with predicted values |
SEP |
Standard Error of Prediction computed for each column of "residuals" |
SEPm |
mean SEP value |
sMAD |
MAD of Prediction computed for each column of "residuals" |
sMADm |
mean of MAD values |
RMSEP |
Root MSEP value computed for each column of "residuals" |
RMSEPm |
mean RMSEP value |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(PAC) res=ridgeCV(y~X,data=PAC,lambdaopt=4.3,repl=5,segments=5)data(PAC) res=ridgeCV(y~X,data=PAC,lambdaopt=4.3,repl=5,segments=5)
A matrix with pandom projection (RP) directions (columns) is generated according to a chosen distributions; optionally the random vectors are orthogonalized.
RPvectors(a, m, ortho = "none", distr = "uniform", par_unif = c(-1, 1), par_norm = c(0, 1), par_eq = c(-1, 0, 1), par_uneq = c(-sqrt(3), 0, sqrt(3)), par_uneqprob = c(1/6, 2/3, 1/6))RPvectors(a, m, ortho = "none", distr = "uniform", par_unif = c(-1, 1), par_norm = c(0, 1), par_eq = c(-1, 0, 1), par_uneq = c(-sqrt(3), 0, sqrt(3)), par_uneqprob = c(1/6, 2/3, 1/6))
a |
number of generated vectors (>=1) |
m |
dimension of generated vectors (>=2) |
ortho |
orthogonalization of vectors: "none" ... no orthogonalization (default); "onfly" ... orthogonalization on the fly after each generated vector; "end" ... orthogonalization at the end, after the whole random matrix was generated |
distr |
distribution of generated random vector components: "uniform" ... uniformly distributed in range par_unif (see below); default U[-1, +1]; "normal" ... normally distributed with parameters par_norm (see below); typical N(0, 1); "randeq" ... random selection of values par_eq (see below) with equal probabilities; typically -1, 0, +1; "randuneq" ... random selection of values par_uneq (see below) with probabilties par_uneqprob (see below); typical -(3)^0.5 with probability 1/6; 0 with probability 2/3; +(3)^0.5 with probability 1/6 |
par_unif |
parameters for range for distr=="uniform"; default to c(-1,1) |
par_norm |
parameters for mean and sdev for distr=="normal"; default to c(0,1) |
par_eq |
values for distr=="randeq" which are replicated; default to c(-1,0,1) |
par_uneq |
values for distr=="randuneq" which are replicated with probabilties par_uneqprob; default to c(-sqrt(3),0,sqrt(3)) |
par_uneqprob |
probabilities for distr=="randuneq" to replicate values par_uneq; default to c(1/6,2/3,1/6) |
The generated random projections can be used for dimension reduction of multivariate data. Suppose we have a data matrix X with n rows and m columns. Then the call B <- RPvectors(a,m) will produce a matrix B with the random directions in its columns. The matrix product X times t(B) results in a matrix of lower dimension a. There are several options to generate the projection directions, like orthogonal directions, and different distributions with different parameters to generate the random numbers. Random Projection (RP) can have comparable performance for dimension reduction like PCA, but gives a big advantage in terms of computation time.
The value returned is the matrix B with a columns of length m, representing the random vectors
Peter Filzmoser <[email protected]>
K. Varmuza, P. Filzmoser, and B. Liebmann. Random projection experiments with chemometric data. Journal of Chemometrics. To appear.
B <- RPvectors(a=5,m=10) res <- t(B)B <- RPvectors(a=5,m=10) res <- t(B)
The trimmed standard deviation as a robust estimator of scale is computed.
sd_trim(x,trim=0.2,const=TRUE)sd_trim(x,trim=0.2,const=TRUE)
x |
numeric vector, data frame or matrix |
trim |
trimming proportion; should be between 0 and 0.5 |
const |
if TRUE, the appropriate consistency correction is done |
The trimmed standard deviation is defined as the average trimmed sum of squared deviations around the trimmed mean. A consistency factor for normal distribution is included. However, this factor is only available now for trim equal to 0.1 or 0.2. For different trimming percentages the appropriate constant needs to be used. If the input is a data matrix, the trimmed standard deviation of the columns is computed.
Returns the trimmed standard deviations of the vector x, or in case of a matrix, of the columns of x.
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
x <- c(rnorm(100),100) # outlier 100 is included sd(x) # classical standard deviation sd_trim(x) # trimmed standard deviationx <- c(rnorm(100),100) # outlier 100 is included sd(x) # classical standard deviation sd_trim(x) # trimmed standard deviation
Stepwise regression, starting from the empty model, with scope to the full model
stepwise(formula, data, k, startM, maxTime = 1800, direction = "both", writeFile = FALSE, maxsteps = 500, ...)stepwise(formula, data, k, startM, maxTime = 1800, direction = "both", writeFile = FALSE, maxsteps = 500, ...)
formula |
formula, like y~X, i.e., dependent~response variables |
data |
data frame to be analyzed |
k |
sensible values are log(nrow(x)) for BIC or 2 for AIC; if not provided -> BIC |
startM |
optional, the starting model; provide a binary vector |
maxTime |
maximal time to be used for algorithm |
direction |
either "forward" or "backward" or "both" |
writeFile |
if TRUE results are shown on the screen |
maxsteps |
maximum number of steps |
... |
additional plot arguments |
This function is similar to the function step for stepwise
regression. It is especially designed for cases where the number of regressor
variables is much higher than the number of objects. The formula for the full model
(scope) is automatically generated.
usedTime |
time that has been used for algorithm |
bic |
BIC values for different models |
models |
matrix with no. of models rows and no. of variables columns, and 0/1 entries defining the models |
Leonhard Seyfang and (marginally) Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- stepwise(y~.,data=NIR.Glc,maxsteps=2)data(NIR) X <- NIR$xNIR[1:30,] # first 30 observations - for illustration y <- NIR$yGlcEtOH[1:30,1] # only variable Glucose NIR.Glc <- data.frame(X=X, y=y) res <- stepwise(y~.,data=NIR.Glc,maxsteps=2)
Evaluation for Support Vector Machines (SVM) by cross-validation
svmEval(X, grp, train, kfold = 10, gamvec = seq(0, 10, by = 1), kernel = "radial", degree = 3, plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)svmEval(X, grp, train, kfold = 10, gamvec = seq(0, 10, by = 1), kernel = "radial", degree = 3, plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)
X |
standardized complete X data matrix (training and test data) |
grp |
factor with groups for complete data (training and test data) |
train |
row indices of X indicating training data objects |
kfold |
number of folds for cross-validation |
gamvec |
range for gamma-values, see |
kernel |
kernel to be used for SVM, should be one of "radial", "linear",
"polynomial", "sigmoid", default to "radial", see |
degree |
degree of polynome if kernel is "polynomial", default to 3, see
|
plotit |
if TRUE a plot will be generated |
legend |
if TRUE a legend will be added to the plot |
legpos |
positioning of the legend in the plot |
... |
additional plot arguments |
The data are split into a calibration and a test data set (provided by "train"). Within the calibration set "kfold"-fold CV is performed by applying the classification method to "kfold"-1 parts and evaluation for the last part. The misclassification error is then computed for the training data, for the CV test data (CV error) and for the test data.
trainerr |
training error rate |
testerr |
test error rate |
cvMean |
mean of CV errors |
cvSe |
standard error of CV errors |
cverr |
all errors from CV |
gamvec |
range for gamma-values, taken from input |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(e1071) set.seed(143) train=sample(1:n,ntrain) ressvm=svmEval(X,grp,train,gamvec=c(0,0.05,0.1,0.2,0.3,0.5,1,2,5), legpos="topright") title("Support vector machines")data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(e1071) set.seed(143) train=sample(1:n,ntrain) ressvm=svmEval(X,grp,train,gamvec=c(0,0.05,0.1,0.2,0.3,0.5,1,2,5), legpos="topright") title("Support vector machines")
Evaluation for classification trees by cross-validation
treeEval(X, grp, train, kfold = 10, cp = seq(0.01, 0.1, by = 0.01), plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)treeEval(X, grp, train, kfold = 10, cp = seq(0.01, 0.1, by = 0.01), plotit = TRUE, legend = TRUE, legpos = "bottomright", ...)
X |
standardized complete X data matrix (training and test data) |
grp |
factor with groups for complete data (training and test data) |
train |
row indices of X indicating training data objects |
kfold |
number of folds for cross-validation |
cp |
range for tree complexity parameter, see |
plotit |
if TRUE a plot will be generated |
legend |
if TRUE a legend will be added to the plot |
legpos |
positioning of the legend in the plot |
... |
additional plot arguments |
The data are split into a calibration and a test data set (provided by "train"). Within the calibration set "kfold"-fold CV is performed by applying the classification method to "kfold"-1 parts and evaluation for the last part. The misclassification error is then computed for the training data, for the CV test data (CV error) and for the test data.
trainerr |
training error rate |
testerr |
test error rate |
cvMean |
mean of CV errors |
cvSe |
standard error of CV errors |
cverr |
all errors from CV |
cp |
range for tree complexity parameter, taken from input |
Peter Filzmoser <[email protected]>
K. Varmuza and P. Filzmoser: Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press, Boca Raton, FL, 2009.
data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(rpart) set.seed(123) train=sample(1:n,ntrain) par(mar=c(4,4,3,1)) restree=treeEval(X,grp,train,cp=c(0.01,0.02:0.05,0.1,0.15,0.2:0.5,1)) title("Classification trees")data(fgl,package="MASS") grp=fgl$type X=scale(fgl[,1:9]) k=length(unique(grp)) dat=data.frame(grp,X) n=nrow(X) ntrain=round(n*2/3) require(rpart) set.seed(123) train=sample(1:n,ntrain) par(mar=c(4,4,3,1)) restree=treeEval(X,grp,train,cp=c(0.01,0.02:0.05,0.1,0.15,0.2:0.5,1)) title("Classification trees")