# fit.boundary: Multivariate smooth boundary fitting with additional... In snfa: Smooth Non-Parametric Frontier Analysis

## Description

Fits boundary of data with kernel smoothing, imposing monotonicity and/or concavity constraints.

## Usage

 1 2 3 fit.boundary(X.eval, y.eval, X.bounded, y.bounded, X.constrained = NA, X.fit = NA, y.fit.observed = NA, H.inv = NA, H.mult = 1, method = "u", scale.constraints = TRUE) 

## Arguments

 X.eval Matrix of inputs used for fitting y.eval Vector of outputs used for fitting X.bounded Matrix of inputs where bounding constraints apply y.bounded Vector of outputs where bounding constraints apply X.constrained Matrix of inputs where monotonicity/concavity constraints apply X.fit Matrix of inputs where curve is fit; defaults to X.constrained y.fit.observed Vector of outputs corresponding to observations in X.fit; used for efficiency calculation H.inv Inverse of the smoothing matrix (must be positive definite); defaults to rule of thumb H.mult Scaling factor for rule of thumb smoothing matrix method Constraints to apply; "u" for unconstrained, "m" for monotonically increasing, and "mc" for monotonically increasing and concave scale.constraints Boolean, whether to scale constraints by their average value, can help with convergence

## Details

This method fits a smooth boundary of the data (with all data points below the boundary) while imposing specified monotonicity and concavity constraints. The procedure is derived from Racine et al. (2009), which develops kernel smoothing methods with bounding, monotonicity and concavity constraints. Specifically, the smoothing procedure involves finding optimal weights for a Nadaraya-Watson estimator of the form

\hat{y} = m(x) = ∑_{i=1}^N p_i A(x, x_i) y_i,

where x are inputs, y are outputs, p are weights, subscripts index observations, and

A(x, x_i) = \frac{K(x, x_i)}{∑_{h=1}^N K(x, x_h)}

for a kernel K. This method uses a multivariate normal kernel of the form

K(x, x_h) = \exp≤ft(-\frac12 (x - x_h)'H^{-1}(x - x_h)\right),

where H is a bandwidth matrix. Bandwidth selection is performed via Silverman's (1986) rule-of-thumb, in the function H.inv.select.

Optimal weights \hat{p} are selected by solving the quadratic programming problem

\min_p \mbox{\ \ }-\mathbf{1}'p + \frac12 p'p.

This method always imposes bounding constraints as specified points, given by

m(x_i) - y_i = ∑_{h=1}^N p_h A(x_i, x_h) y_h - y_i ≥q 0 \mbox{\ \ \ \ }\forall i.

Additionally, monotonicity constraints of the following form can be imposed at specified points:

\frac{\partial m(x)}{\partial x^j} = ∑_{h=1}^N p_h \frac{\partial A(x, x_h)}{\partial x^j} y_h ≥q 0 \mbox{\ \ \ \ }\forall x, j,

where superscripts index inputs. Finally concavity constraints of the following form can also be imposed using Afriat's (1967) conditions:

m(x) - m(z) ≤q \nabla_x m(z) \cdot (x - z) \mbox{\ \ \ \ }\forall x, z.

The gradient of the frontier at a point x is given by

\nabla_x m(x) = ∑_{i=1}^N \hat{p}_i \nabla_x A(x, x_i) y_i,

where \hat{p}_i are estimated weights.

## Value

Returns a list with the following elements

 y.fit Estimated value of the frontier at X.fit gradient.fit Estimated gradient of the frontier at X.fit efficiency Estimated efficiencies of y.fit.observed solution Boolean; TRUE if frontier successfully estimated X.eval Matrix of inputs used for fitting X.constrained Matrix of inputs where monotonicity/concavity constraints apply X.fit Matrix of inputs where curve is fit H.inv Inverse smoothing matrix used in fitting method Method used to fit frontier scaling.factor Factor by which constraints are multiplied before quadratic programming

## References

\insertRef

ParmeterRacinesnfa

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 data(univariate) #Set up data for fitting X <- as.matrix(univariate$x) y <- univariate$y N.fit <- 100 X.fit <- as.matrix(seq(min(X), max(X), length.out = N.fit)) #Reflect data for fitting reflected.data <- reflect.data(X, y) X.eval <- reflected.data$X y.eval <- reflected.data$y #Fit frontiers frontier.u <- fit.boundary(X.eval, y.eval, X.bounded = X, y.bounded = y, X.constrained = X.fit, X.fit = X.fit, method = "u") frontier.m <- fit.boundary(X.eval, y.eval, X.bounded = X, y.bounded = y, X.constrained = X.fit, X.fit = X.fit, method = "m") frontier.mc <- fit.boundary(X.eval, y.eval, X.bounded = X, y.bounded = y, X.constrained = X.fit, X.fit = X.fit, method = "mc") #Plot frontier library(ggplot2) frontier.df <- data.frame(X = rep(X.fit, times = 3), y = c(frontier.u$y.fit, frontier.m$y.fit, frontier.mc$y.fit), model = rep(c("u", "m", "mc"), each = N.fit)) ggplot(univariate, aes(X, y)) + geom_point() + geom_line(data = frontier.df, aes(color = model)) #Plot slopes slope.df <- data.frame(X = rep(X.fit, times = 3), slope = c(frontier.u$gradient.fit, frontier.m$gradient.fit, frontier.mc$gradient.fit), model = rep(c("u", "m", "mc"), each = N.fit)) ggplot(slope.df, aes(X, slope)) + geom_line(aes(color = model)) 

### Example output  snfa documentation built on May 1, 2019, 9:16 p.m.