B-Spline Bases and Local Support

Why Use B-Spline Bases?

A cubic regression spline can be represented with a truncated power basis, as in the earlier material on that basis. That representation is useful for seeing how polynomial pieces are joined, but its columns can be strongly related to one another, and changes to a coefficient may affect a broad part of the curve. A B-spline basis provides another way to represent the same kind of spline: it builds the curve from basis functions that are nonzero only over a limited range of the predictor.

This limited range is called local support. It makes the representation practical: at a given predictor value, only a few B-spline basis functions contribute to the fitted value. The curve is still a weighted sum, with coefficients estimated from data, but its basis functions are organized around the knot sequence.

Definition by Recursion

Let the knots be \(t_0,t_1,\ldots,t_m\), in nondecreasing order. To keep the calculations transparent here, assume the knots are distinct. Knot multiplicity and its effect on continuity are a separate issue. Define the degree-zero functions as indicators of consecutive half-open intervals:

For degrees \(p\ge 1\), define the functions recursively:

A term with a zero denominator is defined to be zero. With distinct knots, the denominators in these expressions are positive. Each new degree combines two functions from the preceding degree, multiplying them by weights that vary linearly with \(x\). The resulting function is piecewise polynomial of degree at most \(p\). In regression splines, degree \(p=3\) gives cubic pieces.

The indexing above starts a degree-zero function on \([t_{i-1},t_i)\). State the indexing convention when calculating values: formulas written with other index conventions can shift the labels even though they describe the same shapes.

Worked Example: Applying the Recursion at a Knot

Local Support and Which Coefficients Matter

The support statement follows directly from the recursion. A degree-zero function \(B_{i,0}\) is zero outside \([t_{i-1},t_i)\). Suppose each degree-\((p-1)\) function is zero outside its stated support. In the degree-\(p\) recursion, the first term can be nonzero only where \(B_{i,p-1}\) can be nonzero, and the second only where \(B_{i+1,p-1}\) can be nonzero. Those ranges are \([t_{i-1},t_{i+p-1})\) and \([t_i,t_{i+p})\). Their union is contained in \([t_{i-1},t_{i+p})\). This proves the support result by induction on \(p\).

For a cubic basis function, the support is \([t_{i-1},t_{i+3})\). If knots are distinct, this includes four consecutive knot intervals. So a change to one cubic coefficient affects the curve only across that function's support. At an \(x\) lying between two consecutive knots, at most four cubic B-spline basis functions can be nonzero. This is the local character of the basis.

A Fitted Spline Is a Weighted Sum

For observations with predictor values \(x_1,\ldots,x_n\) and responses \(y_1,\ldots,y_n\), a regression spline can be written as a sum of B-spline basis functions. For a cubic spline with \(K\) basis functions, a common model is

Here \(c_i\) is the coefficient multiplying basis function \(B_{i,3}\), and \(\varepsilon_j\) is the error for observation \(j\). For the usual linear regression interpretation, assume the errors have mean zero and constant variance and are independent across observations. Normal errors are additionally needed for exact small-sample \(t\)- and \(F\)-inference, but not for evaluating a fitted curve. Once the coefficients have been estimated, the fitted value at any \(x\) is

The estimate \(\hat{y}(x)\) is a weighted sum: the basis values at \(x\) are the weights, and the estimated coefficients are the quantities being weighted. For cubic B-splines, only the functions whose supports contain \(x\) contribute. If the basis is constructed over a suitable knot sequence and evaluated in its spline domain, the B-spline basis functions are nonnegative and sum to one at each point in that domain. Consequently, the fitted value is a weighted average of the contributing coefficients there. This does not mean that a coefficient is generally the fitted response at a particular observation.

How B-Spline Bases Relate to Regression

Once the basis functions and knots are fixed, the model is linear in its coefficients even though the fitted curve can bend. For each observation, evaluate each basis function at its predictor value. These evaluations form a design matrix: rows correspond to observations, and columns correspond to basis functions. Estimating the spline is then a linear regression using those columns. The nonlinearity is in the relationship between \(x\) and the fitted response, not in the way the coefficients enter the model.

B-spline and truncated power bases can represent the same spline space when constructed for the same degree, knots, and boundary setup. They express that space in different coordinates. B-splines are especially useful computationally because local support makes the design matrix sparse: most basis columns are zero for any one observation. This can reduce computation and makes the local effect of a basis coefficient easier to track.

The coefficient values themselves should not be read as segment slopes or as response values at knots. A coefficient scales a basis function; a fitted response is obtained by adding the contributions from all basis functions at the chosen \(x\). The curve, rather than an isolated coefficient, is usually the direct object of interpretation.

Common Mistakes

• Using the wrong lower-degree values. The recursion is only as accurate as its inputs. Calculate the degree-zero and intermediate terms at the stated \(x\) before substituting into the next degree.
• Ignoring endpoint conventions. With half-open degree-zero intervals, a knot belongs to the interval beginning at that knot, not the interval ending there. Use the convention consistently in recursive calculations.
• Assuming a coefficient affects the whole fitted curve. A B-spline coefficient changes the fitted curve only over the support of its basis function.
• Treating a coefficient as a fitted response. Fitted values are sums of coefficients multiplied by basis values. They are not generally equal to a single coefficient.
• Confusing basis degree with curve flexibility alone. Degree sets the maximum polynomial degree on each piece; the number and placement of knots also shape the spline space.