lsConstrain.fit: Minimize Inequality Constrained Mahalanobis Distance

In dawai: Discriminant Analysis with Additional Information

Minimize Inequality Constrained Mahalanobis Distance

Description

Find the vector z that solves:

min{ (x - z)'inv(S)(x - z); Az <= b },

where x is an input vector, S its covariance matrix, A is a matrix of known contrasts, and b is a vector of known constraint constants.

Usage

lsConstrain.fit(x, b, s, a, iflag, itmax=4000, eps=1e-06, eps2=1e-06)

Arguments

x	vector of length n
b	vector of length k, containing constraint constants
s	matrix of dim n x n, the covariance matrix for vector x
a	matrix of dim k x n, for the contraints
iflag	vector of length k; an item = 0 if inequality constraint, 1 if equality constraint
itmax	scalar for number of max interations
eps	scalar of accuracy for convergence
eps2	scalar to determine close to zero

Value

List with the following components:

itmax: (defined above)

eps: (defined above)

eps2: (defined above)

iflag: (defined above)

xkt: vector of length k, for the Kuhn-Tucker coefficients.

iter: number of completed iterations.

supdif: greatest difference between estimates across a full cycle

ifault: error indicator: 0 = no error 1 = itmax exceeded 3 = invalid constraint function for some row ASA'=0.

a: (defined above)

call: function call

x.init: input vector x.

x.final: the vector "z" that solves the equation (see z in description).

s: (defind above)

min.dist: the minimum value of the function (see description).

References

Wollan PC, Dykstra RL. Minimizing inequality constrained mahalanobis distances. Applied Statistics Algorithm AS 225 (1987).

Examples

# An simulation example with linear regression with 3 beta's, 
# where we have the contraints:
#
# b[1] > 0
# b[2] - b[1] < 0
# b[3] < 0


set.seed(111)

n <- 100
x <- rep(1:3,rep(n,3))
x <- 1*outer(x,1:3,"==")

beta <- c(2,1,1)

y <- x%*%beta + rnorm(nrow(x))

fit <- lm(y ~-1 + x)

s <- solve( t(x) %*% x )

bhat <- fit$coef


a <-  rbind(c(-1, 0, 0),
            c(-1, 1, 0),
            c( 0, 0, 1))

# View expected constraints (3rd not met):

a %*% bhat
#            [,1] 
# [1,] -1.8506811
# [2,] -0.9543320
# [3,]  0.8590827

b <- rep(0, nrow(a))
iflag <- rep(0,length(b))

save <- lsConstrain.fit(x=bhat,b=b, s=s, a=a, iflag=iflag, itmax=500, 
                        eps=1e-6, eps2=1e-6)

save

dawai documentation built on Oct. 15, 2024, 5:06 p.m.