%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: leobantis@gmail.com%Date: January 14th, 2013.%Version: 1.
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