# locally.weighted.polynomial: Locally-Weighted Polynomial Regression Smoother

### Description

Smoothes the given bivariate data using kernel regression.

### Usage

 1 2 locally.weighted.polynomial(x, y, h = NA, x.grid = NA, degree = 1, kernel.type = "Normal") 

### Arguments

 x Vector of data for the independent variable y Vector of data for the dependent variable h The bandwidth for the kernel x.grid What x-values should the value of the smoother be calculated at. degree The degree of the polynomial to be fit at each x-value. The default is to fit a linear regression, ie degree=1. kernel.type What kernel to use. Valid choices are 'Normal', 'Epanechnikov', 'biweight', and 'triweight'

### Details

The confidence intervals are created using the row-wise method of Hannig and Marron (2006).

Notice that the derivative to be estimated must be less than or equal to the degree of the polynomial initially fit to the data.

If the bandwidth is not given, the Sheather-Jones bandwidth selection method is used.

### Value

Returns a LocallyWeightedPolynomial object that has the following elements:

 data A structure of the data used to generate the smoothing curve h The bandwidth used to generate the smoothing curve. x.grid The grid of x-values that we have estimated function value and derivative(s) for. degrees.freedom The effective sample size at each grid point Beta A matrix of estimated beta values. The number of rows is degrees+1, while the number of columns is the same as the length of x.grid. Notice that \hat{f}(x_i) = β[1,i] \hat{f'}(x_i) = β[2,i]*1! \hat{f''}(x_i) = β[3,i]*2! and so on... Beta.var Matrix of estimated variances for Beta. Same structure as Beta.

### Author(s)

Derek Sonderegger

### References

Chaudhuri, P., and J. S. Marron. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94 807-823.

Hannig, J., and J. S. Marron. 2006. Advanced distribution theory for SiZer. Journal of the American Statistical Association 101 484-499.

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195

SiZer, plot.LocallyWeightedPolynomial, spm in package 'SemiPar', loess, smooth.spline, interpSpline
  1 2 3 4 5 6 7 8 9 10 11 12 13 data(Arkansas) x <- Arkansas$year y <- Arkansas$sqrt.mayflies layout(cbind(1,2,3)) model <- locally.weighted.polynomial(x,y) plot(model, main='Smoothed Function', xlab='Year', ylab='Sqrt.Mayflies') model2 <- locally.weighted.polynomial(x,y,h=.5) plot(model2, main='Smoothed Function', xlab='Year', ylab='Sqrt.Mayflies') model3 <- locally.weighted.polynomial(x,y, degree=1) plot(model3, derv=1, main='First Derivative', xlab='Year', ylab='1st Derivative')