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-What is a Spline Function
-How do I choose the order m of a spline?




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The Characteristics of Spline Functions
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Expertise: Intermediate

What is a spline function?
We need to know what the essential characteristics of splines are before we consider how to construct a basis system for them.

Spline functions are formed by joining polynomials together at fixed points called knots. That is, we divide the interval extending from lower limit tL to upper limit tU over which we wish to approximate a curve into L+1 sub-intervals separated by L interior boundaries ξl called knots, or sometimes breakpoints.) There is a distinction between these two terms, but we will come to this later.

Consider the simplest case in which a single breakpoint divides interval [tL; tU] into two subintervals. The spline function is, within each interval, a polynomial of specified degree (the highest power defining the polynomial) or order (the number of coefficients defining the polynomial, which is one more than its degree). Let's use m to designate the order of the polynomial, so that the degree is m - 1:

At the interior breakpoint ξ1, the two polynomials are required to join smoothly. In the most common case, this means that the derivatives match up to the order one less than the degree. In fact, if they matched up to the derivative whose order equaled the degree, they would be the same polynomial. Thus, a spline function defined in this was has one extra degree of freedom than a polynomial extending over the entire interval.

For example, let each polynomial be a straight line segment, and therefore of degree one. In this, they join at the breakpoint with matching derivatives up to degree 0; in short, they simply join and having identical values at the break point. Since the first polynomial has two degree of freedom (slope and intercept), and the second, having its value already defined at the break point, is left with only one degree of freedom (slope), the total polygonal line has three degrees of freedom.

Correspondingly, if both polynomials are quadratics, then the match both in terms of values and in terms of slope of first derivative at ξ1. The first polynomial has three degrees of freedom, but the second loses two because if the constraint on its value and slope at ξi, and thus retains only one. This leaves a total of four degrees of freedom for the spline function formed in this way, as opposed to three for a quadratic polynomial over the entire interval [tL; tU]. Figure 1 shows these linear and quadratic cases with a single breakpoint.

What are some examples of commonly used basis functions?
We can now generalize the situation to L interior breakpoints, and a spline function being of order m or degree m - 1 over each sub-interval. The first polynomial segment has a full complement of m degrees of freedom, but each subsequent segment has only one degree of freedom because of the m - 1 constraints on its behavior. This gives a total of L + m degrees of freedom, or number of interior breakpoints plus the order of the polynomial segments.

Thus, spline functions are essentially generalizations of the notion of polygonal lines. They gain their flexibility in two ways: First, by the order of the polynomials from which they are built, and second, by the number of breakpoints used. We usually elect to keep the order fixed, and add breakpoints as needed to get the required flexibility.

Figure 1: The left figure shows an order two spline function, which is piecewise linear. The right figure is an order 3 spline, which is piecewise quadratic. The breakpoint at 0.5 is indicated by the red vertical dotted line.

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