# Minimize Inequality Constrained Mahalanobis Distance 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. ```r 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 ## Returns 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 ```r # 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 ```

lsConstrain.fit function