snfa (version 0.0.1)

fit.boundary: Multivariate smooth boundary fitting with additional constraints

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

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

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

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

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

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

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

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

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

$$m(x_i) - y_i = \sum_{h=1}^N p_h A(x_i, x_h) y_h - y_i \geq 0 \mbox{\ \ \ \ }\forall i.$$

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

$$\frac{\partial m(x)}{\partial x^j} = \sum_{h=1}^N p_h \frac{\partial A(x, x_h)}{\partial x^j} y_h \geq 0 \mbox{\ \ \ \ }\forall x, j,$$

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

$$m(x) - m(z) \leq \nabla_x m(z) \cdot (x - z) \mbox{\ \ \ \ }\forall x, z.$$

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

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

where \(\hat{p}_i\) are estimated weights.

References

ParmeterRacinesnfa

Examples

# NOT RUN {
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))
  
# }