Action: BF_CUBIC_B_SPLINES

Module	ves
Description	Usage
Cubic B spline basis functions.

Details and examples

Cubic B spline basis functions.

These basis functions do not form orthogonal bases. We recommend using wavelets (BF_WAVELETS) instead that do for orthogonal bases.

A basis using cubic B spline functions according to the first paper cited below. See the second paper cited below for full details.

The mathematical expression of the individual splines is given by $\begin{aligned} h\left(x\right) = \begin{cases} \left(2 - \lvert x \rvert\right)^3, & 1 \leq \lvert x \rvert \leq 2\\ 4 - 6\lvert x \rvert^2 + 3 \lvert x \rvert^3,\qquad & \lvert x \rvert \leq 1\\ 0, & \text{elsewhere}. \end{cases} \end{aligned}$

The full basis consists of equidistant splines at positions $\mu_i$ which are optimized in their height: $f_i\left(x\right) = h\left(\frac{x-\mu_i}{\sigma}\right)$

Note that the distance between individual splines cannot be chosen freely but is equal to the width: $\mu_{i+1} = \mu_{i} + \sigma$ .

The ORDER keyword of the basis set determines the number of equally sized sub-intervalls to be used. On the borders of each of these sub-intervalls the mean $\mu_i$ of a spline function is placed.

The total number of basis functions is $\text{ORDER}+4$ as the constant $f_{0}(x)=1$ , as well as the two splines with means just outside the interval are also included.

As an example two adjacent basis functions can be seen below. The full basis consists of shifted splines in the full specified interval.

A graph illustrating the cubic B spline basisfunctions

When the splines are used for a periodic CV (with the PERIODIC keyword), the sub-intervals are chosen in the same way, but only $\text{ORDER}+1$ functions are required to fill it (the ones at the boundary coincide and the ones outside can be omitted).

To avoid 'blind' optimization of the basis functions outside the currently sampled area, it is often beneficial to use the OPTIMIZATION_THRESHOLD keyword of the VES_LINEAR_EXPANSION (set it to a small value, e.g. 1e-6)

Examples

The bias is expanded with cubic B splines in the intervall from 0.0 to 10.0 specifying an order of 20. This results in 24 basis functions.

Click on the labels of the actions for more information on what each action computes

bf: BF_CUBIC_B_SPLINESCubic B spline basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=0.0 MAXIMUMThe maximum of the interval on which the basis functions are defined=10.0 ORDERThe order of the basis function expansion=20
The BF_CUBIC_B_SPLINES action with label bf calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
ORDER	compulsory	none	The order of the basis function expansion
MINIMUM	compulsory	none	The minimum of the interval on which the basis functions are defined
MAXIMUM	compulsory	none	The maximum of the interval on which the basis functions are defined
DEBUG_INFOThis keyword do not have examples	optional	false	Print out more detailed information about the basis set
NORMALIZATIONThis keyword do not have examples	optional	not used	the normalization factor that is used to normalize the basis functions by dividing the values
PERIODICThis keyword do not have examples	optional	false	Use periodic version of basis set

References

More information about how this action can be used is available in the following articles: