R code for Shape-Restricted Regression

SHAPE-RESTRICTED LEAST-SQUARES REGRESSION

"Constraints-only" -- no smoothing.

PART I: Unsmoothed monotone (isotonic) regression

Given a data vector y in &realⁿ, minimize ||y &minus &phi||² , over &phi₁ &le &phi₂ &le ... &le &phi_n

The following code has argument y, and returns the least-squares nondecreasing solution.

R code for monotone regression

PART II: Unsmoothed convex regression

Given a scatterplot of data vector (x,y) where x,y in &realⁿ, minimize ||y &minus &phi||² , over A&phi&ge 0, where the non-zero elements of the (n-2) by n constraint matrix A are

A_i,i=x_i+2 &minus x_i+1; A_i,i+1=x_i &minus x_i+2; and A_i,i+2=x_i+1 &minus x_i, for i=1,...n-2.

The following code has arguments x and y and returns the solution.

R code for convex regression

PART III: Unsmoothed monotone increasing and convex regression

Given a scatterplot of data vector (x,y) where x,y in &realⁿ, minimize ||y &minus &phi||² , over A&phi&ge 0, where the non-zero elements of the (n-1) by n constraint matrix A are

A_i,i=x_i+2 &minus x_i+1; A_i,i+1=x_i &minus x_i+2; A_i,i+2=x_i+1 &minus x_i for i=1,...n-2, and A_n-1,1=-1; A_n-1,2=1.

The following code has arguments x and y and returns the solution.

R code for increasing monotone convex regression

PART IV: Unsmoothed monotone decreasing and convex regression

Given a scatterplot of data vector (x,y) where x,y in &realⁿ, minimize ||y &minus &phi||² , over A&phi&ge 0, where the non-zero elements of the (n-1) by n constraint matrix A are

A_i,i=x_i+2 &minus x_i+1; A_i,i+1=x_i &minus x_i+2; A_i,i+2=x_i+1 &minus x_i for i=1,...n-2, and A_n-1,n-1=1; A_n-1,n=-1.

The following code has arguments x and y and returns the solution.

R code for decreasing monotone convex regression

meyer@stat.colostate.edu