cvx.pen.reg: Penalized Smooth Convex Regression.
In simest: Constrained Single Index Model Estimation

Description Usage Arguments Details Value Author(s) Source Examples

This function provides an estimate of the non-parametric regression function with a shape constraint of convexity and smoothness constraint provided through square integral of second derivative.

cvx.pen.reg(x, y, lambda, w = NULL, tol = 1e-05, maxit = 1000,...)
## Default S3 method:
cvx.pen.reg(x, y, lambda, w = NULL, tol = 1e-05, maxit = 1000,...)
## S3 method for class 'cvx.pen.reg'
plot(x,...)
## S3 method for class 'cvx.pen.reg'
print(x,...)
## S3 method for class 'cvx.pen.reg'
predict(object, newdata = NULL,...)

`x`	a numeric vector giving the values of the predictor variable. For `plot` and `print` functions, `x` represents an object of class `cvx.pen.reg`.
`y`	a numeric vector giving the values of the response variable.
`lambda`	a numeric value giving the penalty value.
`w`	an optional numeric vector of the same length as x; Defaults to all 1.
`maxit`	an integer giving the maxmimum number of steps taken by the algorithm; defaults to 1000.
`tol`	a numeric providing the tolerance level for convergence.
`...`	any additional arguments.
`object`	An object of class ‘cvx.pen.reg’. This is for predict function.
`newdata`	a vector of new data points to be used in the predict function.

The function minimizes

∑_{i=1}^n w_i(y_i - f(x_i))^2 + λ\int\{f''(x)\}^2dx

subject to convexity constraint on f. plot function provides the scatterplot along with fitted curve; it also includes some diagnostic plots for residuals. Predict function returns a matrix containing the inputted newdata along with the function values, derivatives and second derivatives.

An object of class ‘cvx.pen.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.
`alpha`	a numeric vector of length 2 less than ‘x’. This represents the coefficients of the B-splines in the second derivative of the estimator.
`AlphaMVal`	a numeric vector needed for predict function.
`lower`	a numeric vector needed for predict function.
`upper`	a numeric vector needed for predict function.

Arun Kumar Kuchibhotla, arunku@wharton.upenn.edu, Rohit Kumar Patra, rohit@stat.columbia.edu.

Elfving, T. and Andersson, L. (1988). An Algorithm for Computing Constrained Smoothing Spline Functions. Numer. Math., 52(5):583–595.

Dontchev, A. L., Qi, H. and Qi, L. (2003). Quadratic Convergence of Newton's Method for Convex Interpolation and Smoothing. Constructive Approximation, 19(1):123-143.

1 2 3 4 5 6 7	args(cvx.pen.reg) x <- runif(50,-1,1) y <- x^2 + rnorm(50,0,0.3) tmp <- cvx.pen.reg(x, y, lambda = 0.01) print(tmp) plot(tmp) predict(tmp, newdata = rnorm(10,0,0.1))

Loading required package: nnls
Loading required package: cobs
function (x, y, lambda, w = NULL, tol = 1e-05, maxit = 1000, 
    ...) 
NULL
Call:
cvx.pen.reg.default(x = x, y = y, lambda = 0.01)
Minimum Criterion Value Obtained:
[1] 3.078794
Number of Iterations:
[1] 7
Convergence Status:
[1] 0
          newdata           fun       1der      2der
 [1,]  0.02235507 -0.0661781642  0.3750100 2.0956590
 [2,]  0.08450423 -0.0401507128  0.4521583 0.9613187
 [3,]  0.06902296 -0.0470335122  0.4368915 1.0109729
 [4,] -0.18927517 -0.0845818087 -0.2017507 2.6889231
 [5,]  0.08158771 -0.0414653402  0.4493410 0.9706731
 [6,]  0.16644492 -0.0001673682  0.5201618 0.6985043
 [7,]  0.09518042 -0.0352692462  0.4622387 0.9270761
 [8,] -0.02449673 -0.0812901791  0.2648712 2.8752968
 [9,]  0.10501686 -0.0306781225  0.4712027 0.8955270
[10,]  0.22266225  0.0300888114  0.5545830 0.4971582