McSpatial (version 1.1.1)

fourier: Fourier expansion smoothing

Description

Estimates a model of the form y=f(z)+XB+u using a fourier expansion for z. The variable z is first transformed to $z = 2\pi\frac{(z-min(z))}{max(z)-min(z))}$. The fourier model is $y = \alpha_1 z + \alpha_2 z^2 + \sum(\lambda_q sin(qz) + \delta_q cos(qz)) + X \beta +u$. Estimation can be carried out for a fixed value of Q or for a range of Q. In the latter case, the function indicates the value of Q that produces the lowest value of one of the following criteria: the AIC, the Schwarz information criterion, or the gcv.

Usage

fourier(form,q=1,minq=0,maxq=0,crit="gcv",data=NULL)

Arguments

form

Model formula. The expansion is applied to the first explanatory variable.

q

If q is specified and minq=maxq, fits a fourier expansion with Q set to q. Default is q=1, which implies a model with z, z^2, sin(z), cos(z) and X as explanatory variables.

minq

The lower bound to search for the value of Q that minimizes crit. minq can take any value greater than zero. Default: not used.

maxq

The upper bound to search for the value of Q that minimizes crit. maxq must be great than or equal to minq. Default: not used.

crit

The selection criterion. Must be in quotes. The default is the generalized cross-validation criterion, or "gcv". Options include the Akaike information criterion, "aic", and the Schwarz criterion, "sc". Let nreg be the number of explanatory variables

data

A data frame containing the data. Default: use data in the current working directory

Value

• yhat
  The predicted values of the dependent variable at the original data points
• rss
  The residual sum of squares
• sig2
  The estimated error variance
• aic
  The value for AIC
• sc
  The value for sc
• gcv
  The value for gcv
• coef
  The estimated coefficient vector, B
• fourierhat
  The predicted values for z alone, normalized to have the same mean as the dependent variable. If no X variables are included in the regression, fourierhat = yhat.
• q
  The value of Q used in the final estimated model.

References

Gallant, Ronald, "On the Bias in Flexible functional Forms and an Essentially Unbiased Form: The Fourier Flexible Form," Journal of Econometrics 15 (1981), 211-245. Gallant, Ronald, "Unbiased Determination of Production Technologies," Journal of Econometrics 20 (1982), 285-323. McMillen, Daniel P. and Jonathan Domborw, "A Flexible Fourier Approach to Repeat Sales Price Indexes," Real Estate Economics 29 (2001), 207-225. McMillen, Daniel P., "Neighborhood Price Indexes in Chicago: A Fourier Repeat Sales Approach," Journal of Economic Geography 3 (2003), 57-73. McMillen, Daniel P., "Issues in Spatial Data Analysis," Journal of Regional Science 50 (2010), 119-141.

See Also

cparlwr cubespline lwr lwrgrid semip

Examples

set.seed(23849103)
n = 1000
x <- runif(n,0,2*pi)
x <- sort(x)
ybase <- x - .1*(x^2) + sin(x) - cos(x) -.5*sin(2*x) + .5*cos(2*x)
sig = sd(ybase)/2
y <- ybase + rnorm(n,0,sig)

par(ask=TRUE)
plot(x,y)
lines(x,ybase,col="red")

fit <- fourier(y~x,minq=1,maxq=10)
plot(x,ybase,type="l",xlab="x",ylab="y")
lines(x,fit$yhat,col="red")
legend("topright",c("Base","Fourier"),col=c("black","red"),lwd=1)