\( \newcommand{\pen}{\lambda} \newcommand{\minage}{\alpha} \newcommand{\maxage}{\omega} \newcommand{\knotspacing}{\Delta x} \newcommand{\nintern}{n} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\infotype}{I} \)

PSPLINE

The PSPLINE multi-transition build method smooths migration ratios using quadratic P-Splines (see Dyrting and Taylor 2021 and Dyrting and Taylor 2024) .

ExtraInfo

The user must specify the total number of movers in each age group using an extrainfo string of the form PSPLINE:MInfo[,ExtraInfo]. If MInfo is a number then the number of movers in each age interval of the migration ratios subtable is flat at this value. If it is a string then it specifies the name of a PCURVE object. ExtraInfo is an optional comma-separated list of quoted strings of the form "Parameter=Value" which can be used to change the method's default settings.

Degree

By default PSPLINE uses quadratic B-Splines. The degree of splines can be changed with the extrainfo string "Degree=m".

Order

By default PSPLINE uses a first order penalty. The order can be changed with the extrainfo string "Order=k".

Knots

There are two ways to set the B-Spline knots. The first way is to set the minimum and maximum ages using extrainfo strings "MinAge=\(\minage\)" and "MaxAge=\(\maxage\)" respectively. The internal points are then equally spaced upwards from \(\minage\) as follows \begin{eqnarray} x_i&=&\minage+\knotspacing\times i,\quad i=1,\ldots,\nintern\\ \end{eqnarray} where \begin{equation} \nintern=\ceil{\frac{\maxage-\minage}{\knotspacing}}-1 \end{equation} and the spacing \(\knotspacing\) is specified by the extrainfo string "KnotSpacing=\(\knotspacing\)". The default values are \(\minage=0\), \(\maxage=90\) and \(\knotspacing=3\). The second way is to set the knots explicitly using the extrainfo string "Knots=\(\minage\)|\(x_1\)|\(x_2\)|\(\ldots\)|\(x_\nintern\)|\(\maxage\)".

Penalty

The P-Spline penalty can be set specified explicitly using "Penalty=\(\pen\). The default value is "Penalty=-1" which signals that the penalty should be calculated automatically by minimizing the information criterion specified by InfoType. A separate penalty can be specified for each ToPop using the extrainfo string "Penalty=\(\pen_1\)|\(\pen_2\)|\(\ldots\)|\(\pen_d\)".

InfoType

This extrainfo parameter specifies the information criterion used to automatically calculate the penalty. The default method is the Akaike information criterion with corrections ("Penalty=AICC"). Alternatives are the Akaike information criterion ("Penalty=AIC") and the Bayesian information criterion ("Penalty=BIC"). A separate method can be specified for each ToPop using the extrainfo string "InfoType=\(\infotype_1\)|\(\infotype_2\)|\(\ldots\)|\(\infotype_d\)"