B-splines

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B-Splines

Lagrange interpolation

Consider 2 points and


is a random point between and :

where and are base functions.

Consider 3 points , and


Knowing that these 3 points will get a certain quadratic function. To get this function, for every point, there is ,

The result for interpolation is:

To fit that ,

The sum for these 3 points:

Largragne interpolation


For points, , .

  • , therefore, it contains all the points
  • is the base function.

Bezier curve

Given 3 points , assume in the line :



Hence, .


Suppose there are points, the Bezier curve is:

The is determained by the former points’ ’ Bezier curve and later points :

B-Splines

Basic concepts:

  • Control points: control the shape of curves. Suppose there are control points: .
  • Knot: affect for the weight, suppose knots, the curve is devided into pieces.
  • Degree & Order: order = degree + 1, degree is usually denoted as .


where represents the -th points’ weight function.


How to get (de Boor):

B-splines regression

For multilinear regression, , where .\
For splines estimates, assume:

where is the base spline function.


For more details, see here for reference.

Reference

  1. Introduction of Lagrange interpolation.

  2. 陈广雷, 王兆军. 多元部分线性模型的B-样条估计[J]. 应用概率统计, 2010, 26(2): 138-150.