# B3spline

## Contents

### Purpose

Univariate spline fit and prediction.

### Synopsis

modl = b3spline(x,y,t,options);
pred = b3spline(x,modl,options);
valid = b3spline(x,y,modl,options);

### Description

Curve fitting using second order splines where

yi = f(xi) for i=1,...,M.

#### Inputs

• x = Mx1 vector of independent variable values.
• y = Mx1 vector of corresponding dependendent variable values.
• t = defines the number of knots or knot positions.
• = 1x1 scalar integer defining the number of uniformly distributed INTERIOR knots. There will be t+2 knots positioned at: modl.t = linspace(min(x),max(x),t+2)';
• = Kx1 vector defining manually placed knot positions, where: modl.t = sort(t);

Note that knot positions need not be uniform, and that t(1) can be <min(x) and t(K) can be >max(x).

Note that knot positions must be such that there are at least 3 unique data points between each knot: tk, tk+1 for k = 1,...,K.

#### Outputs

• pred = structure array with predictions.
• valid = structure array with predictions.

### Options

options = a structure array with the following fields:

• display: [ {'on'} | 'off' ] level of display to command window.
• plots: [ {'final'} | 'none' ] governs level of plotting. If 'final' and calibrating a model, the plot shows plot(xi,yi) and plot(xi,f(xi),'-') with knots.
• algorithm: [ {'b3spline'} | 'b3_0' | 'b3_01' ] fitting algorithm
• 'b3_0': is the same as 'b3spline' but also constrains the ends to 0: f{1,2}(t1) = 0 and f{K-1,K}(tK) = 0.
• 'b3_01': is 'b3_0' but also constrains the derivatives at the ends to 0: f'{1,2}(t1) = 0 and f'{K-1,K}(tK) = 0.
• 'b3spline: fits quadradic polynomials f{k,k+1} to the data between knots tk, k=1,...,K, subject to:
f{k,k+1}(tk+1) = f{k+1,k+2}(tk+1) and
f'{k,k+1}(tk+1) = f'{k+1,k+2}(tk+1) for k=1,...,K-1.

The default options can be retreived using: options = baseline('options');.