\( \newcommand{\kernelregression}{\texttt{KERNEL\_REGRESSION}} \newcommand{\objfunc}{S} \newcommand{\minage}{\alpha} \newcommand{\maxage}{\beta} \newcommand{\migprob}{q} \newcommand{\smigprob}{\tilde{\migprob}} \newcommand{\tofunc}{f} \newcommand{\fromfunc}{\tofunc^{-1}} \newcommand{\degree}{p} \newcommand{\xinfovar}[1]{\textit{#1}} \newcommand{\xinfoval}[1]{\texttt{#1}} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\esm}{\hat{m}} \newcommand{\kernel}{K} \)

Kernel Regression

This build method smooths a migration curve using local polynomial regression. Given migration probabilities \(\migprob_x\) for \(x=0,1,2,\ldots,\omega\), the smoothed migration probability \(\smigprob_x\) is given by \begin{equation} \smigprob_x=\left\{\begin{array}{ll} \fromfunc\left(\esm(x)\right) & x\in [\minage,\maxage]\\ \migprob_x & \mathrm{otherwise} \end{array}\right. \end{equation} where \(\tofunc(\migprob)\) is a transformation function and \(\esm(x)\) is a local polynomial regression fit to the observations \((X_i,Y_i)\) where \(X_i=i\) and \(Y_i=\tofunc(\migprob_i)\).

Extra Information

The default behaviour can be modified by appending an extrainfo string to the buildmethod, eg KERNEL_REGRESSION:ExtraInfo. Here ExtraInfo is a comma-separated list of quoted strings of the form "Parameter=Value".

ModelType
The transformation function \(\tofunc(\migprob)\) is controlled by the extrainfo string "ModelType=type" where \begin{equation} \tofunc(q) = \left\{\begin{array}{ll} q, & \xinfovar{type}=\xinfoval{LINEAR}\\ \log q, & \xinfovar{type}=\xinfoval{LOG} \end{array}\right. \end{equation} The default value is ModelType=LINEAR.

Degree
The degree \(\degree\) of the local polynomial regression can be set with the extrainfo string "Degree=\(\degree\)". The default value is \(\degree=1\).

Bandwidth
The bandwidth of the local polynomial regression can be set with the extrainfo string "Bandwidth=value" where value is either a number specifying the value or a string specifying the bandwidth selector. The default value is Bandwidth=ROT, the modified Ruppert-Sheather-Wand rule-of-thumb bandwidth selector.

Kernel
The local polynomial regression kernel function can be set with the extrainfo string "Kernel=type" where type is a kernel function code. The default value is Kernel=GAUSSIAN.

Age Range
The minimum and maximum ages for smoothing can be set by the extrainfo strings "MinAge=\(\minage\)" and "MaxAge=\(\maxage\)" respectively. The default values are \(\minage=0\) and \(\maxage=90\).