Restricted Cubic Spline

Fits the so called restricted cubic spline via least squares and obtains 95% bootstrap based CIs.

%Fits the so called restricted cubic spline via least squares (see Harrell
%(2001)). The obtained spline is linear beyond the first and the last
%knot. The truncated power basis representation is used. That is, the
%fitted spline is of the form:
%f(x)=b0+b1*x+b2*(x-t1)^3*(x>t1)+b3*(x-t2)^3*(x>t2)+...
%where t1 t2,... are the desired knots.
%95% confidence intervals are provided based on the bootstrap procedure.

%For more information see also:
%Frank E Harrell Jr, Regression Modelling Strategies (With application to
%linear models, logistic regression and survival analysis), 2001,
%Springer Series in Statistics, pages 20-21.
%
%INPUT ARGUMENTS:
%x: A vector containing the covariate values x.
%y: A vector of length(x) that contains the response values y.
%knots: A vector of points at which the knots are to be placed.
% Alternatively, it can be set as 'prc3', 'prc4', ..., 'prc8' and 3
% or 4 or...8 knots placed at equally spaced percentiles will be used.
% It can also be set to 'eq3', 'eq4', ...,'eq8' to use 3 or 4 or ...
% or 8 equally spaced knots. There is a difference in using one of
% these strings to define the knots instead of passing them directly
% as a vector of numbers and the difference involves only the
% bootstrap option and not the fit itself. When the bootstrap is used
% and the knots are passed in as numbers, then the knot sequence will
% be considered fixed as provided by the user for each bootstrap
% iteration. If a string as the ones mentioned above is used, then
% the knot sequence is re-evaluated for each bootstrap sample
% based on this choice.
%
%OPTIONAL INPUT ARGUMENTS: (These can be not reached at all or set as [] to
%proceed to the next optional input argument):
%
%bootsams: The number of bootstrap samples if the user wants to derive 95% CIs.
%atwhich: a vector of x values at which the CIs of the f(x) are to be evaluated.
%plots: If set to 1, it returns a plot of the spline and the data.
% Otherwise it is ignored. This input argument can also not be reached
% at all. (It also plots the CIs provided that they are requested).
%
%OUTPUT ARGUMENTS:
%bhat: the estimated spline coefficients.
%f: a function handle from which you can evaluate the spline value at a
% given x (which can be a scalar or a vector). For example ff(2) will
% yield the spline value for x=2. You can use a vector (grid) of x values to
% plot the f(x) by requesting plot(x,f(x)).
%sse: equals to sum((y-ff(x)).^2)
%knots: the knots used for fitting the spline.
%CI : 95% bootstrap based confidence intervals.
% Obtained only if the bootstrap is requested and and only fot the
% points at which the CIs were requested. Hence, CI is a three column
% matrix with its first column be the spline value at the points
% supplied by the user, and the second and third column are
% respectively the lower and upper CI limits for that points.
%
%References: Frank E. Harrell, Jr. Regression Modeling Strategies (With
%applications to linear models, logistic regression, and survival
%analysis). Springer 2001.
%
%
%Code author: Leonidas E. Bantis,
%Dept. of Statistics & Actuarial-Financial Mathematics, School of Sciences
%University of the Aegean, Samos Island.
%
%E-mail: [e-mail]
%Date: January 14th, 2013.
%Version: 1.