The Truncated Power Basis for Cubic Splines

Why Use a Truncated Power Basis?

A single cubic polynomial can bend smoothly, but its curvature is governed by one polynomial over the entire predictor range. A cubic spline allows the curve’s cubic behavior to change at chosen locations called knots, while keeping the fitted curve smooth where the pieces meet. The truncated power basis expresses that curve as a sum of familiar predictor powers and terms that become active only after their knots.

This representation is useful because it turns a piecewise curve into a regression model that is linear in its coefficients. Once the knots are fixed, each basis function can be calculated for every observation and entered as a predictor in a regression.

The Basis and Its Regression Model

Suppose there are \(K\) fixed interior knots, ordered as \(\xi_1<\xi_2<\cdots<\xi_K\). The truncated power basis for a cubic spline consists of the constant function, \(x\), \(x^2\), \(x^3\), and one truncated cubic function for each knot. The resulting mean curve is

Here, \(\beta_0,\beta_1,\beta_2,\beta_3\), and \(\gamma_1,\ldots,\gamma_K\) are coefficients. In a regression setting, the model for observation \(i\) is \(Y_i=f(x_i)+\varepsilon_i\). The conditional mean assumption is \(E(\varepsilon_i\mid x_i)=0\). For ordinary least squares, the coefficients are chosen to minimize the sum of squared differences between the observed responses and the fitted values. A unique least-squares coefficient estimate also requires the basis columns in the data to be linearly independent. Usual standard-error calculations require additional error assumptions; they are not needed just to define or evaluate the basis.

The first four terms describe a cubic polynomial that applies everywhere. Each truncated term adds a cubic adjustment only to the right of its knot. With \(K\) interior knots, the expression has \(K+4\) coefficients, including the intercept.

Why the Curve Is Smooth at a Knot

A cubic spline in this setting is a piecewise cubic curve whose value, first derivative, and second derivative agree on both sides of each interior knot. The truncated power form enforces these continuity conditions by construction. To see why, consider the contribution from one knot:

For \(x<\xi\), \(g(x)=0\). For \(x>\xi\), \(g(x)=\gamma(x-\xi)^3\). On the right side, its first and second derivatives are \(3\gamma(x-\xi)^2\) and \(6\gamma(x-\xi)\). At \(x=\xi\), the function and both derivatives approach zero from either side. Thus this term changes the cubic behavior after the knot without introducing a jump in the curve, slope, or second derivative.

A short derivation also explains why these terms represent the general cubic spline with these continuity conditions. Start with a cubic polynomial on the interval before the first knot. At a knot \(\xi_j\), the difference between the cubic polynomial to the right and the one to the left must have value, first derivative, and second derivative equal to zero at \(\xi_j\). A cubic polynomial with these three zeros at the same point must be a constant multiple of \((x-\xi_j)^3\): it has a root of multiplicity at least three, and its degree is at most three. Therefore, crossing each knot adds a term \(\gamma_j(x-\xi_j)^3\) on the right and zero on the left. Applying this reasoning in order at all knots gives the stated truncated power representation.

How the Coefficients Shape the Curve

Before the first knot, all truncated terms are zero, so the curve is the ordinary cubic polynomial \(\beta_0+\beta_1x+\beta_2x^2+\beta_3x^3\). On an interval after some knots, the active truncated terms add to that polynomial. For example, after the first \(m\) knots and before the next one, expanding the active terms gives

Thus the coefficients \(\gamma_j\) are not, in general, the slopes or intercepts of separate segments. They modify the local polynomial after their knots. More specifically, away from the knots the third derivative is \(6\beta_3+6\sum_{j:x>\xi_j}\gamma_j\). Crossing knot \(\xi_j\) changes the third derivative by \(6\gamma_j\). The curve, its slope, and its second derivative remain continuous.

Using the Basis Carefully

Once the knots and basis functions are set, fitting the model amounts to estimating coefficients for the basis columns. The selected knot locations determine where the curve can change its cubic behavior; they do not by themselves determine how much it changes. That depends on the estimated \(\gamma_j\) values along with the polynomial coefficients.

• Keep the predictor units in mind. Powers and truncated powers have units: if \(x\) is measured in months, a cubic term is measured in months cubed. Coefficients therefore depend on the scale used for \(x\).
• Check which terms are active. At \(x=\xi_j\), the \(j\)th truncated term is zero. Only values strictly greater than that knot produce a nonzero contribution.
• Do not treat \(\gamma_j\) as a slope change. A truncated cubic term has zero first derivative at its knot. Its coefficient controls a change in third derivative, not an immediate jump in slope.
• Do not confuse fitted means with predictions for individual responses. The basis defines the fitted mean curve; individual observations can also differ from that curve because of their errors.

The truncated power basis is a global algebraic representation: a coefficient associated with an early knot affects the curve everywhere to the right of that knot. This is useful for deriving and interpreting the curve in terms of polynomial pieces. The practical task is to calculate the basis values consistently and read coefficients in relation to the whole expression, rather than in isolation.