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\title{A Primer on Regression Splines}
\author{Jeffrey S.~Racine}
\date{\today}
\thanks{These notes are culled from a variety of sources. I am solely
  responsible for all errors. Suggestions are welcomed (racinej@mcmaster.ca).} 

\begin{document}

<<eval=TRUE,echo=FALSE,keep.source=TRUE,results=hide>>=
library(crs)
options(prompt = "R> ", crs.messages = FALSE, digits = 3)
@ 

\setkeys{Gin}{width=0.45\textwidth}

\maketitle

\onehalfspacing

\section{Overview}

B-splines constitute an appealing method for the nonparametric
estimation of a range of statistical objects of interest. In this
primer we focus our attention on the estimation of a conditional mean,
i.e.\ the `regression function'.

A `spline' is a function that is constructed piece-wise from polynomial
functions. The term comes from the tool used by shipbuilders and
drafters to construct smooth shapes having desired
properties. Drafters have long made use of a bendable strip fixed in
position at a number of points that relaxes to form a smooth curve
passing through those points. The malleability of the spline material
combined with the constraint of the control points would cause the
strip to take the shape that minimized the energy required for bending
it between the fixed points, this being the smoothest possible
shape. We shall rely on a class of splines called `B-splines'
(`basis-splines'). A B-spline function is the maximally differentiable
interpolative basis function. The B-spline is a generalization of the
B\'ezier curve (a B-spline with no `interior knots' is a B\'ezier
curve). B-splines are defined by their `order' $m$ and number of
interior `knots' $N$ (there are two `endpoints' which are themselves
knots so the total number of knots will be $N+2$). The degree of the
B-spline polynomial will be the spline order $m$ minus one (degree =
$m-1$).

To best appreciate the nature of B-splines, we shall first consider a
simple type of spline, the B\'ezier function, and then move on to the
more flexible and powerful generalization, the B-spline itself. We
begin with the univariate case in Section \ref{sec:bezier} where we
consider the univariate B\'ezier function. In Section
\ref{sec:b-spline} we turn to the univariate B-spline function, and
then in Section \ref{sec:mv spline} we turn to the multivariate case
where we also briefly mention how one could handle the presence of
categorical predictors.

We presume that interest lies in `regression spline' methodology which
differs in a number of ways from `smoothing splines', both of which
are popular in applied settings. The fundamental difference between
the two approaches is that smoothing splines explicitly penalize
roughness and use the data points themselves as potential knots
whereas regression splines place knots at equidistant/equiquantile
points. We direct the interested reader to \citet{WAHBA:1990} for
a treatment of smoothing splines.

\section{\label{sec:bezier}B\'ezier curves}

We present an overview of B\'ezier curves which form the basis for the
B-splines that follow. We begin with a simple illustration, that of a
quadratic B\'ezier curve.

\begin{example}{A quadratic B\'ezier curve.}

  A quadratic B\'ezier curve is the path traced by the function
  $B(x)$, given points $\beta_0$, $\beta_1$, and $\beta_2$, where
  \begin{align*}
    B(x)&=\beta_0(1-x)^2+2\beta_1(1-x)x+\beta_2x^2\cr
    &=\sum_{i=0}^2 \beta_iB_i(x),\quad x\in [0,1].
  \end{align*}
  The terms $B_0(x)=(1-x)^2$, $B_1(x)=2(1-x)x$, and $B_2(x)=x^2$ are
  the `bases' which is this case turn out to be `Bernstein
  polynomials' \citep{BERNSTEIN:1912}.  For our purposes the
  `control points' $\beta_i$, $i=0,1,2$, will be parameters that could
  be selected by least squares fitting in a regression setting, but
  more on that later. Consider the following simple example where we
  plot a quadratic B\'ezier curve with arbitrary control points:
  
\begin{center}
<<fig=TRUE,echo=FALSE>>=
B <- function(x,b0,b1,b2) { (1-x)^2*b0+(1-x)*x*b1+x^2*b2 }
x <- seq(0,1,length=1000)
b0 <- 1
b1 <- -1
b2 <- 2
plot(x,B(x,b0,b1,b2),ylab="B(x)",type="l",lwd=2,cex.lab=1.25)
@ 
\end{center}

For this simple illustration we set $\beta_0=\Sexpr{b0}$,
$\beta_1=\Sexpr{b1}$, $\beta_2=\Sexpr{b2}$.
  
Note that the derivative of this curve is
  \begin{equation*}
    B'(x)=2(1-x)(\beta_1-\beta_0)+2x(\beta_2-\beta_1),
  \end{equation*}
  which is a polynomial of degree one. 
  
  This example of a B\'ezier curve will also be seen to be a
  `second-degree B-spline with no interior knots' or, equivalently, `a
  third-order B-spline with no interior knots'. Using the terminology
  of B-splines, in this example we have a third-order B-spline ($m=3$)
  which is of polynomial degree two ($m-1=2$) having highest
  derivative of polynomial degree one ($m-2=1$).
  
\end{example}

\subsection{The B\'ezier curve defined}

More generally, a B\'ezier curve of degree $n$ (order $m$) is composed
of $m=n+1$ terms and is given by
\begin{align}
  \label{eq:bezier}
  B(x)&=\sum_{i=0}^n\beta_i{n\choose i}(1-x)^{n-i}x^i\cr
  &=\sum_{i=0}^n\beta_iB_{i,n}(x),
\end{align}
where ${n\choose i}=\frac{n!}{(n-i)!i!}$, which can be expressed
recursively as
\begin{equation*}
  B(x)=(1-x)\left(\sum_{i=0}^{n-1}\beta_iB_{i,n-1}(x)\right)+x\left(\sum_{i=1}^{n}\beta_iB_{i-1,n-1}(x)\right),
\end{equation*}
so a degree $n$ B\'ezier curve is a linear interpolation between two
degree $n-1$ B\'ezier curves.

%% B_{i,n-1} above changed to B_{i-1,n-1} 20/1/15: "Dear Dr. Racine, I
%% have spotted a small typo in your otherwise excellent Primer on
%% Regression Splines
%% (http://cran.r-project.org/web/packages/crs/vignettes/spline_primer.pdf
%% ): on page 3, in the unnumbered equation below Eq (3) the
%% parameters of the Bernstein polynomial in the second term should be
%% B_{i-1,n-1} I believe (not B_{i,n-1}). Tamas
%% (ferenci.tamas@nik.uni-obuda.hu)"

\begin{example}{A quadratic B\'ezier curve as a linear interpolation
    between two linear B\'ezier curves.}

  The linear B\'ezier curve is given by $\beta_0(1-x)+\beta_1 x$, and
  above we showed that the quadratic B\'ezier curve is given by
  $\beta_0(1-x)^2+2\beta_1(1-x)x+\beta_2x^2$. So, when $n=2$
  (quadratic), we have
\begin{align*}
B(x)
&=(1-x)(\beta_0(1-x)+\beta_1 x)
+x(\beta_1(1-x)+\beta_2 x)\cr
&=\beta_0(1-x)^2+2\beta_1(1-x)x+\beta_2x^2.
\end{align*}

\end{example}

This is essentially a modified version of the idea of taking linear
interpolations of linear interpolations of linear interpolations and
so on. Note that the polynomials
\begin{equation*}
  B_{i,n}(x)={n\choose i}(1-x)^{n-i}x^i
\end{equation*}
are called `Bernstein basis polynomials of degree $n$' and are such
that $\sum_{i=0}^nB_{i,n}(x)=1$, unlike raw
polynomials.\footnote{Naturally we define $x^0=(1-x)^0=1$, and by
  `raw' polynomials we simply mean $x^j$, $j=0,\dots,n$.} 

The $m=n+1$ control points $\beta_i$, $i=0,\dots,n$, are somewhat
ancillary to the discussion here, but will figure prominently when we
turn to regression as in a regression setting they will be the
coefficients of the regression model.

\begin{example}{The quadratic B\'ezier curve basis functions.}

  The figure below presents the bases $B_{i,n}(x)$ underlying a B\'ezier
  curve for $i=0,\dots,2$ and $n=2$.
  
\begin{center}
<<fig=TRUE,echo=FALSE>>=
Bernstein <- function(n,i,x) { factorial(n)/(factorial(n-i)*factorial(i))*(1-x)^{n-i}*x^i }
x <- seq(0,1,length=100)
degree <- 2
plot(x,Bernstein(degree,0,x),type="l",lwd=2,ylab="B(x)",col=1)
for(i in 1:degree) lines(x,Bernstein(degree,i,x),lty=i+1,lwd=2,col=i+1)
@ 
\end{center}

These bases are $B_{0,2}(x)=(1-x)^2$, $B_{1,2}(x)=2(1-x)x$, and
$B_{2,2}(x)=x^2$ and illustrate the foundation upon which the B\'ezier
curves are built.

\end{example}

\subsection{Derivatives of spline functions}

From \citet{DE_BOOR:2001} we know that the derivatives of spline
functions can be simply expressed in terms of lower order spline
functions. In particular, for the B\'ezier curve we have
\begin{equation*}
  B^{(l)}(x)=\sum_{i=0}^{n-l}\beta^{(l)}_iB_{i,n-l}(x),
\end{equation*}
where $\beta^{(0)}_i=\beta_i$, $0\le i\le n$, and
\begin{equation*}
  \beta^{(l)}_i=(n-l)\left(\beta^{(l-1)}_{i+1}-\beta^{(l-1)}_{i}\right)/(t_i-t_{i-n+l}),\quad
  0\le i\le n-l.
\end{equation*}
See \citet{ZHOU_WOLFE:2000} for details.

We now turn our attention to the B-spline function. This can be
thought of as a generalization of the B\'ezier curve where we now
allow for there to be additional breakpoints called `interior knots'.

\section{\label{sec:b-spline}B-splines}

\subsection{B-spline knots}

B-spline curves are composed from many polynomial pieces and are
therefore more versatile than B\'ezier curves.  Consider $N+2$ real
values $t_i$, called `knots' ($N\ge 0$ are called `interior knots' and
there are always two endpoints, $t_0$ and $t_{N+1}$), with
\begin{equation*}
t_0 \le t_1 \le \cdots \le t_{N+1}.
\end{equation*}
When the knots are equidistant they are said to be `uniform',
otherwise they are said to be `non-uniform'. One popular type of knot
is the `quantile' knot sequence where the interior knots are the
quantiles from the empirical distribution of the underlying
variable. Quantile knots guarantee that an equal number of sample
observations lie in each interval while the intervals will have
different lengths (as opposed to different numbers of points lying in
equal length intervals).

B\'ezier curves possess two endpoint knots, $t_0$ and $t_1$, and no
interior knots hence are a limiting case, i.e.\ a B-spline for which
$N=0$.

\subsection{The B-spline basis function}

Let $\boldsymbol{t}=\lbrace t_i\mid i\in \mathbb{Z}\rbrace$ be a
sequence of non-decreasing real numbers ($t_i\le t_{i+1}$) such
that\footnote{This description is based upon the discussion found at
  http://planetmath.org/encyclopedia/BSpline.html.}
\begin{equation*}
t_0 \le t_1 \le \cdots \le t_{N+1}.
\end{equation*}
Define the augmented knot set 
\begin{equation*} 
  t_{-\left( m-1\right)}=\cdots
  =t_{0}\le t_{1}\le\cdots \le t_{N}\le t_{N+1}=\cdots=t_{N+m},
\end{equation*}
%% changed $t_{1}$ $n=m-1$ times to $t_{N+1}$...
where we have appended the lower and upper boundary knots $t_0$ and
$t_{N+1}$ $n=m-1$ times (this is needed due to the recursive nature of the
B-spline). If we wanted we could then reset the index for the first
element of the augmented knot set (i.e.\ $t_{-(m-1)}$) so that the
$N+2m$ augmented knots $t_i$ are now indexed by $i=0,\ldots,N+2m-1$
(see the example below for an illustration).

For each of the augmented knots $t_i$, $i=0,\ldots,N+2m-1$, we
recursively define a set of real-valued functions $B_{i,j}$ (for
$j=0,1,\ldots,n$, $n$ being the degree of the B-spline basis) as
follows:
\begin{align*}
  B_{i,0}(x)&=\left\{ \begin{array}{ll} 1 & \mbox{ if } t_i\leq
      x<t_{i+1}\\ 0 & \mbox{ otherwise. } \end{array}\right.\\ \\
  B_{i,j+1}(x)&=\alpha_{i,j+1}(x) B_{i,j}(x) +
  [1-\alpha_{i+1,j+1}(x)] B_{i+1,j}(x) ,
\end{align*}
where
\begin{align*} \alpha_{i,j}(x) &= \left\{ \begin{array}{ll}
      \displaystyle{\frac{x - t_i}{t_{i+j} - t_i}} & \mbox{ if }
      t_{i+j}\ne t_i \\ 0 & \mbox{ otherwise. } \end{array}\right.
\end{align*}
For the above computation we define 0/0 as 0.

\textbf{Definitions}.

Using the notation above:

\begin{enumerate}

\item the sequence $\boldsymbol{t}$ is known as a \emph{knot
    sequence}, and the individual term in the sequence is a
  \emph{knot}.

%% changed order to degree aug 26 2021

\item the functions $B_{i,j}$ are called the \emph{$i$-th B-spline
    basis functions of degree $j$}, and the recurrence relation is
  called the \emph{de Boor recurrence relation}, after its discoverer
  Carl de Boor \citep{DE_BOOR:2001}.

%% changed order to degree aug 26 2021

\item given any non-negative integer $j$, the vector space
  $V_j(\boldsymbol{t})$ over $\mathbb{R}$, generated by the set of all
  B-spline basis functions of degree $j$ is called the \emph{B-spline
    of degree $j$}. In other words, the B-spline
  $V_j(\boldsymbol{t})=\operatorname{span}\lbrace B_{i,j}(x)\mid
  i=0,1,\ldots \rbrace$ over $\mathbb{R}$.

\item Any element of $V_j(\boldsymbol{t})$ is a \emph{B-spline
    function} of degree $j$.

\end{enumerate}

The first term $B_{0,n}$ is often referred to as the `intercept'. In
typical spline implementations the option \verb+intercept=FALSE+
denotes dropping this term while \verb+intercept=TRUE+ denotes keeping
it (recall that $\sum_{i=0}^nB_{i,n}(x)=1$ which can lead to perfect
multicollinearity in a regression setting; also see
\citet{ZHOU_WOLFE:2000} who instead apply shrinkage methods).

\begin{example}{A fourth-order B-spline basis function with three
    interior knots and its first derivative function.}

  Suppose there are $N=3$ interior knots given by $(0.25,0.5,0.75)$,
  the boundary knots are $(0,1)$, and the degree of the spline is
  $n=3$ hence the order is $m=4$. The set of all knot points needed to
  construct the B-spline is
  \begin{equation*}
    (0, 0, 0, 0, 0.25, 0.5, 0.75, 1,1,1,1)
  \end{equation*}
  and the number of basis functions is $K=N+m=7$. The seven cubic
  spline basis functions will be denoted $B_{0,3},\dots,B_{6,3}$.

  The figure below presents this example of a third degree B-spline
  with three interior knots along with its first derivative (the
  spline derivatives would be required in order to compute derivatives
  from the spline regression model).

\begin{center}
<<fig=TRUE,echo=FALSE,keep.source=TRUE>>=
degree <- 3
m <- degree+1
## nbreak is the total number of knots (2 indicates 2 endpoints, 0 interior)
nbreak <- 2+3
## N is the number of interior knots
N <- nbreak-2
x <- seq(0,1,length=1000)
B <- gsl.bs(x,degree=degree,nbreak=nbreak,intercept=TRUE)
matplot(x,B,type="l",lwd=2)
@ 
<<fig=TRUE,echo=FALSE,keep.source=TRUE>>=
deriv <- 1
B.deriv <- gsl.bs(x,degree=degree,nbreak=nbreak,deriv=deriv,intercept=TRUE)
matplot(x,B.deriv,type="l",lwd=2)
@ 
\end{center}

To summarize, in this illustration we have an order $m=\Sexpr{m}$
(degree = \Sexpr{degree}) B-spline (left) with \Sexpr{nbreak-1}
sub-intervals (segments) using uniform knots ($N=\Sexpr{N}$ interior
knots, $\Sexpr{N+2}$ knots in total (2 endpoint knots)) and its
\Sexpr{deriv}st-order derivative (right). The dimension of $B(x)$ is
$K=N+m=\Sexpr{ncol(B)}$.

\end{example}

See the appendix for R code \citep{R} that implements the
B-spline basis function.

\subsection{The B-spline function}

A B-spline of degree $n$ (of spline order $m=n+1$) is a parametric
curve composed of a linear combination of basis B-splines $B_{i,n}(x)$
of degree $n$ given by
\begin{equation}
  \label{eq:bspline}
  B(x)= \sum_{i=0}^{N+n} \beta_{i}B_{i,n}(x),\quad x \in [t_0,t_{N+1}].
\end{equation}
The $\beta_i$ are called `control points' or `de Boor points'.  For an
order $m$ B-spline having $N$ interior knots there are $K=N+m=N+n+1$
control points (one when $j=0$).

The B-spline order $m$ must be at least 2 (hence at least linear,
i.e.\ degree $n$ is at least 1) and the number of interior knots must
be non-negative ($N\ge0$).

See the appendix for R code \citep{R} that implements the
B-spline function.

\section{\label{sec:mv spline}Multivariate B-spline regression}

The functional form of parametric regression models must naturally be
specified by the user. Typically practitioners rely on raw polynomials
and also often choose the form of the regression function (i.e.\ the
order of the polynomial for each predictor) in an ad-hoc
manner. However, raw polynomials are not sufficiently flexible for our
purposes, particularly because they possess no interior knots which is
where B-splines derive their unique properties. Furthermore, in a
regression setting we typically encounter multiple predictors which
can be continuous or categorical in nature, and traditional splines
are for continuous predictors. Below we briefly describe a
multivariate kernel weighted tensor product B-spline regression method
(kernel weighting is used to handle the presence of the categorical
predictors). This method is implemented in the R package `crs'
\citep{crs}.

\subsection{Multivariate knots, intervals, and spline bases}

In general we will have $q$ predictors,
$\mathbf{X}=(X_1,\dots,X_q)^T$. We assume that each $X_{l}$, $1\leq
l\leq q$, is distributed on a compact interval $\left[
  a_{l},b_{l}\right] $, and without loss of generality, we take all
intervals $ \left[ a_l,b_l\right] =\left[ 0,1\right] $. Let
$G_{l}=G_{l}^{\left( m_{l}-2\right) }$ be the space of polynomial
splines of order $m_{l}$. We note that $G_{l}$ consists of functions
$\varpi $ satisfying (i) $\varpi $ is a polynomial of degree $m_{l}-1$
on each of the sub-intervals $I_{j_{l},l},j_{l}=0,\ldots ,N_{l} $; (ii)
for $m_{l}\geq 2$, $\varpi $ is $m_{l}-2$ times continuously
differentiable on $[0,1]$.

Pre-select an integer $N_{l}=N_{n,l}$. Divide $\left[
  a_{l},b_{l}\right] =\left[ 0,1\right] $ into $\left( N_{l}+1\right)
$ sub-intervals $I_{j_{l},l}=\left[ t_{j_{l},l},t_{j_{l}+1,l}\right) $,
$ j_{l}=0,\ldots ,N_{l}-1$, $I_{N_{l},l}=\left[ t_{N_{l},l},1\right]
$, where $ \left\{ t_{j_{l},l}\right\} _{j_{l}=1}^{N_{l}}$ \ is a
sequence of equally-spaced points, called interior knots, given
as
\begin{equation*} 
  t_{-\left( m_{l}-1\right),l }=\cdots
  =t_{0,l}=0<t_{1,l}<\cdots <t_{N_{l},l}<1=t_{N_{l}+1,l}=\cdots
  =t_{N_{l}+m_{l},l}, 
\end{equation*} 
in which $t_{j_{l},l}=j_{l}h_{l}$, $j_{l}=0,1\ldots ,N_{l}+1$,
$h_{l}=1/\left( N_{l}+1\right) $ is the distance between neighboring
knots.

Let $K_{l}=K_{n,l}=N_{l}+m_{l}$, where $N_{l}$ is the number of
interior knots and $m_{l}$ is the spline order, and let $B_{l}\left(
  x_{l}\right) =\left\{ B_{j_{l},l}\left( x_{l}\right) :1-m_{l}\leq
  j_{l}\leq N_{l}\right\} ^{\mathrm{T}}$ be a basis system of the
space $G_{l}$.

We define the space of tensor-product polynomial splines by
$\mathcal{G} =\otimes _{l=1}^{q}G_{l}$. It is clear that $\mathcal{G}$
is a linear space of dimension
$\mathbf{K}_{n}=\prod\nolimits_{l=1}^{q}K_{l}$. Then\footnote{The
  notation here may throw off those used to sums of the form
  $\sum_{i=1}^n$, $n>0$ (i.e.\ sum indices that are positive
  integers), so consider a simple illustration that may defuse this
  issue. Suppose there are no interior knots ($N=0$) and we consider a
  quadratic (degree $n$ equal to two hence the `spline order' is
  three). Then $\sum_{i=1-m}^{N}$ contains three terms having indices
  $i=-2,-1,0$. In general the number of terms is the number the number
  of interior knots $N$ plus the spline order $m$, which we denote
  $K=N+m$.  We could alternatively sum from $1$ to $N+m$, or from $0$
  to $N+m-1$ of from $0$ to $N+n$ (the latter being consistent with
  the B\'ezier curve definition in \eqref{eq:bezier} and the B-spline
  definition in \eqref{eq:bspline}).}
\begin{equation*}
  \mathcal{B}\left( \mathbf{x}\right) =\left[ \left\{ \mathcal{B}
      _{j_{1},\ldots ,j_{q}}\left( \mathbf{x}\right) \right\}
    _{j_{1}=1-m_{1},\ldots ,j_{q}=1-m_{q}}^{N_{1},\ldots ,N_{q}}\right] _{
    \mathbf{K}_{n}\times 1}=B_{1}\left( x_{1}\right) \otimes \cdots \otimes
  B_{q}\left( x_{q}\right)
\end{equation*}
is a basis system of the space $\mathcal{G}$, where $\mathbf{x=}\left(
  x_{l}\right) _{l=1}^{q}$. Let $\mathbf{B=}\left[ \left\{
    \mathcal{B}\left( \mathbf{X}_{1}\right) ,\ldots ,\mathcal{B}\left(
      \mathbf{X}_{n}\right) \right\} ^{\mathrm{T}}\right] _{n\times
  \mathbf{K}_{n}}$.

\subsection{Spline regression}

In what follows we presume that the reader is interested in the
unknown conditional mean in the following location-scale model,
\begin{equation}
  Y=g\left( \mathbf{X},\mathbf{Z}\right) +\sigma \left( \mathbf{X},\mathbf{Z}
  \right) \varepsilon ,  \label{MOD:csm}
\end{equation}
where $g(\cdot )$ is an unknown function, $\mathbf{X=}\left(
  X_{1},\ldots ,X_{q}\right) ^{\mathrm{T}}$ is a $q$-dimensional
vector of continuous predictors, and $\mathbf{Z}=\left( Z_{1},\ldots
  ,Z_{r}\right) ^{\mathrm{T}}$ is an $r$-dimensional vector of
categorical predictors. Letting $\mathbf{z} =\left( z_{s}\right)
_{s=1}^{r}$, we assume that $z_{s}$ takes $c_{s}$ different values in
$D_{s}\equiv \left\{ 0,1,\ldots ,c_{s}-1\right\} $, $ s=1,\ldots ,r$,
and let $c_{s}$ be a finite positive constant. Let $\left(
  Y_{i},\mathbf{X}_{i}^{\mathrm{T}},\mathbf{Z}_{i}^{\mathrm{T}}\right)
_{i=1}^{n}$ be an i.i.d copy of $\left(
  Y,\mathbf{X}^{\mathrm{T}},\mathbf{Z}^{\mathrm{T}}\right) $. Assume
for $1\leq l\leq q$, each $X_{l}$ is distributed on a compact interval
$\left[ a_{l},b_{l}\right] $, and without loss of generality, we take
all intervals $ \left[ a_l,b_l\right] =\left[ 0,1\right] $.

In order to handle the presence of categorical predictors, we define
\begin{align}
  l\left( Z_{s},z_{s},\lambda _{s}\right) & =\left\{
    \begin{array}{c}
      1\text{,when~}Z_{s}=z_{s} \\
      \lambda _{s}\text{, otherwise.\ \ }
    \end{array}
  \right. ,  \notag \\
  L\left( \mathbf{Z},\mathbf{z},\mathbf{\lambda }\right) &
  =\prod\limits_{s=1}^{r}l\left( Z_{s},z_{s},\lambda _{s}\right)
  =\prod\limits_{s=1}^{r}\lambda _{s}^{1\left( Z_{s}\neq z_{s}\right)
  },
  \label{DEF:Lzs}
\end{align}
where $l(\cdot )$ is a variant of Aitchison and Aitken's
univariate categorical kernel function \citep{AITCHISON_AITKEN:1976}, $L(\cdot )$ is a product categorical kernel function, and
$\mathbf{\lambda }=(\lambda _{1},\lambda _{2},\dots ,\lambda
_{r})^{\mathrm{T}}$ is the vector of bandwidths for each of the
categorical predictors. See \citet{MA_RACINE_YANG:2011} and
\citet{MA_RACINE:2013} for further details.

We estimate $\beta \left( \mathbf{z}\right) $ by minimizing the
following weighted least squares criterion,
\begin{equation*}
  \widehat{\beta }\left( \mathbf{z}\right) =\arg \min_{\beta \in \mathbb{R}^{\mathbf{K}_{n}}}\sum\limits_{i=1}^{n}\left\{ Y_{i}-
    \mathcal{B}\left( \mathbf{X}_{i}\right) ^{\mathrm{T}}\beta \right\} ^{2}L\left( \mathbf{Z}_{i},\mathbf{z},\mathbf{\lambda }
  \right) .
\end{equation*}

Let $\mathcal{L}_{z}=\text{diag}\left\{ L\left(
    \mathbf{Z}_{1},\mathbf{z}, \mathbf{\lambda }\right) ,\ldots
  ,L\left( \mathbf{Z}_{n},\mathbf{z},\mathbf{ \lambda }\right)
\right\} $ be a diagonal matrix with $L\left( \mathbf{Z}
  _{i},\mathbf{z},\mathbf{\lambda }\right) $, $1\leq i\leq n$ as the
diagonal entries. Then $\widehat{\beta }\left( \mathbf{z}\right) $ can
be written as
\begin{equation}
  \widehat{\beta }\left( \mathbf{z}\right) =\left( n^{-1}\mathbf{B}^{\mathrm{T}
    }\mathcal{L}_{z}\mathbf{B}\right) ^{-1}\left( n^{-1}\mathbf{B}^{\mathrm{T}}
    \mathcal{L}_{z}\mathbf{Y}\right) ,  \label{EQ:betahat}
\end{equation}
where $\mathbf{Y=}\left( Y_{1},\ldots ,Y_{n}\right)
^{\mathrm{T}}$. $g\left( \mathbf{x},\mathbf{z}\right) $ is estimated
by $\widehat{g}\left( \mathbf{x}, \mathbf{z}\right) =\mathcal{B}\left(
  \mathbf{x}\right) ^{\mathrm{T}}\widehat{ \beta }\left(
  \mathbf{z}\right) $.

See the appendix for R code \citep{R} that implements the
B-spline basis function and then uses least squares to construct the
regression model for a simulated data generating process.

\singlespacing

\bibliographystyle{agsm}
\bibliography{spline_primer}

\clearpage
\appendix

\onehalfspacing

\section{Sample R code for constructing B-splines}

The following code uses recursion to compute the B-spline basis and
B-spline function. Then a simple illustration demonstrates how one
could immediately compute a least-squares fit using the B-spline. In
the spirit of recursion, it has been said that ``To iterate is human;
to recurse divine.''  (L.~Peter Deutsch).

\subsection*{R Code for Implementing B-spline basis functions and the
  B-spline itself}

\singlespacing
{\scriptsize
\begin{verbatim}
## $Id: spline_primer.Rnw,v 1.30 2015/01/20 10:53:40 jracine Exp jracine $

## April 23 2011. The code below is based upon an illustration that
## can be found in http://www.stat.tamu.edu/~sinha/research/note1.pdf
## by Dr. Samiran Sinha (Department of Statistics, Texas A&M). I am
## solely to blame for any errors and can be contacted at
## racinej@mcmaster.ca (Jeffrey S. Racine).

## This function is a (simplified) R implementation of the bs()
## function in the splines library and illustrates how the Cox-de Boor
## recursion formula is used to construct B-splines.

basis <- function(x, degree, i, knots) {
  if(degree == 0){
    B <- ifelse((x >= knots[i]) & (x < knots[i+1]), 1, 0)
  } else {
    if((knots[degree+i] - knots[i]) == 0) {
      alpha1 <- 0
    } else {
      alpha1 <- (x - knots[i])/(knots[degree+i] - knots[i])
    }
    if((knots[i+degree+1] - knots[i+1]) == 0) {
      alpha2 <- 0
    } else {
      alpha2 <- (knots[i+degree+1] - x)/(knots[i+degree+1] - knots[i+1])
    }
    B <- alpha1*basis(x, (degree-1), i, knots) + alpha2*basis(x, (degree-1), (i+1), knots)
  }
  return(B)
}

bs <- function(x, degree=3, interior.knots=NULL, intercept=FALSE, Boundary.knots = c(0,1)) {
  if(missing(x)) stop("You must provide x")
  if(degree < 1) stop("The spline degree must be at least 1")
  Boundary.knots <- sort(Boundary.knots)
  interior.knots.sorted <- NULL
  if(!is.null(interior.knots)) interior.knots.sorted <- sort(interior.knots)
  knots <- c(rep(Boundary.knots[1], (degree+1)), interior.knots.sorted, rep(Boundary.knots[2], (degree+1)))
  K <- length(interior.knots) + degree + 1
  B.mat <- matrix(0,length(x),K)
  for(j in 1:K) B.mat[,j] <- basis(x, degree, j, knots)
  if(any(x == Boundary.knots[2])) B.mat[x == Boundary.knots[2], K] <- 1
  if(intercept == FALSE) {
    return(B.mat[,-1])
  } else {
    return(B.mat)
  }
}

## A simple illustration that computes and plots the B-spline bases.

par(mfrow = c(2,1))

n <- 1000
x <- seq(0, 1, length=n)
B <- bs(x, degree=5, intercept = TRUE, Boundary.knots=c(0, 1))
matplot(x, B, type="l", lwd=2)

## Next, simulate data then construct a regression spline with a
## prespecified degree (in applied settings we would want to choose
## the degree/knot vector using a sound statistical approach).

dgp <- sin(2*pi*x)
y <- dgp + rnorm(n, sd=.1)

model <- lm(y~B-1)

plot(x, y, cex=.25, col="grey")
lines(x, fitted(model), lwd=2)
lines(x, dgp, col="red", lty=2)
\end{verbatim}
}

\end{document}

