# cvx.lse.reg: Convex Least Squares Regression.

### Description

This function provides an estimate of the non-parametric regression function with a shape constraint of convexity and no smoothness constraint. Note that convexity by itself provides some implicit smoothness.

### Usage

 1 2 3 4 5 6 7 8 9 cvx.lse.reg(t, z, w = NULL,...) ## Default S3 method: cvx.lse.reg(t, z, w = NULL, ...) ## S3 method for class 'cvx.lse.reg' plot(x,...) ## S3 method for class 'cvx.lse.reg' print(x,...) ## S3 method for class 'cvx.lse.reg' predict(object, newdata = NULL, deriv = 0, ...) 

### Arguments

 t a numeric vector giving the values of the predictor variable. z a numeric vector giving the values of the response variable. w an optional numeric vector of the same length as t; Defaults to all elements 1/n. ... additional arguments. x An object of class ‘cvx.lse.reg’. This is for plot and print function. object An object of class ‘cvx.lse.reg’. newdata a matrix of new data points in the predict function. deriv a numeric either 0 or 1 representing which derivative to evaluate.

### Details

The function minimizes

∑_{i=1}^n w_i(y_i - θ_i)^2

subject to

\frac{θ_2 - θ_1}{x_2 - x_1}≤\cdots≤\frac{θ_n - θ_{n-1}}{x_n - x_{n-1}}

for sorted x values and y reorganized such that y_i corresponds to the new sorted x_i. This function previously used the coneA function from the coneproj package to perform the constrained minimization of least squares. Currently, the code makes use of the nnls function from nnls package for the same purpose. plot function provides the scatterplot along with fitted curve; it also includes some diagnostic plots for residuals. Predict function now allows computation of the first derivative.

### Value

An object of class ‘cvx.lse.reg’, basically a list including the elements

 x.values sorted ‘x’ values provided as input. y.values corresponding ‘y’ values in input. fit.values corresponding fit values of same length as that of ‘x.values’. deriv corresponding values of the derivative of same length as that of ‘x.values’. iter number of steps taken to complete the iterations. residuals residuals obtained from the fit. minvalue minimum value of the objective function attained. convergence a numeric indicating the convergence of the code.

### Author(s)

Arun Kumar Kuchibhotla, arunku@wharton.upenn.edu

### Source

Lawson, C. L and Hanson, R. J. (1995). Solving Least Squares Problems. SIAM.

### References

Chen, D. and Plemmons, R. J. (2009). Non-negativity Constraints in Numerical Analysis. Symposium on the Birth of Numerical Analysis.

Liao, X. and Meyer, M. C. (2014). coneproj: An R package for the primal or dual cone projections with routines for constrained regression. Journal of Statistical Software 61(12), 1 – 22.

### Examples

 1 2 3 4 5 6 7 args(cvx.lse.reg) x <- runif(50,-1,1) y <- x^2 + rnorm(50,0,0.3) tmp <- cvx.lse.reg(x, y) print(tmp) plot(tmp) predict(tmp, newdata = rnorm(10,0,0.1)) 

Questions? Problems? Suggestions? or email at ian@mutexlabs.com.

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