fit.boundary: Multivariate smooth boundary fitting with additional...

In tkmckenzie/snfa: Smooth Non-Parametric Frontier Analysis

Description

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

Usage

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

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))
  

tkmckenzie/snfa documentation built on June 11, 2020, 4:34 a.m.