cvpvs.wnn {pvclass}

Cross-Validated P-Values (Weighted Nearest Neighbors)

Description

Computes cross-validated nonparametric p-values for the potential class memberships of the training data. The p-values are based on 'weighted nearest-neighbors'.

Usage

cvpvs.wnn(X, Y, wtype = c('linear', 'exponential'), W = NULL,
          tau = 0.3, distance = c('euclidean', 'ddeuclidean',
          'mahalanobis'), cova = c('standard', 'M', 'sym'))

Arguments

X

matrix containing training observations, where each observation is a row vector.

Y

vector indicating the classes which the training observations belong to.

wtype

type of the weight function (see section 'Details' below).

W

vector of the (decreasing) weights (see section 'Details' below).

tau

parameter of the weight function. If tau is a vector or tau = NULL, the program searches for the best tau. For more information see section 'Details'.

distance

the distance measure:
"euclidean": fixed Euclidean distance,
"ddeuclidean": data driven Euclidean distance (component-wise standardization),
"mahalanobis": Mahalanobis distance.

cova

estimator for the covariance matrix:
'standard': standard estimator,
'M': M-estimator,
'sym': symmetrized M-estimator.

Details

Computes cross-validated nonparametric p-values for the potential class memberships of the training data. Precisely, for each feature vector X[i,] and each class b the number PV[i,b] is a p-value for the null hypothesis that Y[i] equals b.
This p-value is based on a permutation test applied to an estimated Bayesian likelihood ratio, using 'weighted nearest neighbors' with estimated prior probabilities N(b)/n. Here N(b) is the number of observations of class b and n is the total number of observations.
The (decreasing) weights for the observations can be either indicated with a n dimensional vector W or (if W = NULL) one of the following weight functions can be used:
linear:

W_i = \max(1-\frac{i}{n}/\tau,0),

exponential:

W_i = (1-\frac{i}{n})^\tau.

If tau is a vector, the program searches for the best tau. To determine the best tau for the p-value PV[i,b], the class label of the training observation X[i,] is set temporarily to b and then for all training observations with Y[j] != b the sum of the weights of the observations belonging to class b is computed. Then the tau which minimizes the sum of these values is chosen.
If W = NULL and tau = NULL, tau is set to seq(0.1,0.9,0.1) if wtype = "l" and to c(1,5,10,20) if wtype = "e".

Value

PV is a matrix containing the cross-validated p-values. Precisely, for each feature vector X[i,] and each class b the number PV[i,b] is a p-value for the null hypothesis that Y[i] = b.
If tau is a vector or NULL (and W = NULL), PV has an attribute "opt.tau", which is a matrix and opt.tau[i,b] is the best tau for observation X[i,] and class b (see section 'Details'). "opt.tau" is used to compute the p-values.

Author(s)

Niki Zumbrunnen [e-mail]
Lutz Dümbgen [e-mail]
www.imsv.unibe.ch/duembgen/index_ger.html

References

Zumbrunnen N. and Dümbgen L. (2017) pvclass: An R Package for p Values for Classification. Journal of Statistical Software 78(4), 1–19. doi:10.18637/jss.v078.i04

Dümbgen L., Igl B.-W. and Munk A. (2008) P-Values for Classification. Electronic Journal of Statistics 2, 468–493, available at http://dx.doi.org/10.1214/08-EJS245.

Zumbrunnen N. (2014) P-Values for Classification – Computational Aspects and Asymptotics. Ph.D. thesis, University of Bern, available at http://boris.unibe.ch/id/eprint/53585.

See Also

cvpvs, cvpvs.gaussian, cvpvs.knn, cvpvs.logreg

Examples

X <- iris[, 1:4]
Y <- iris[, 5]

cvpvs.wnn(X, Y, wtype = 'l', tau = 0.5)