Continuity Constraints for Spline Trajectories

Why Coefficients Must Match at Knots

A piecewise cubic trajectory can adapt its shape over time by using a separate cubic polynomial on each interval between knots. But fitting a polynomial on each interval does not, by itself, ensure a coherent trajectory: the pieces could disagree at a knot, or meet while having sharply different slopes or curvature. In longitudinal models, such behavior can create an abrupt change in the modeled biomarker mean that comes only from the chosen parameterization.

The remedy is to impose equations on the polynomial coefficients. These equations express exactly what it means for adjacent pieces to agree in value, first derivative, and second derivative. The resulting constraints are algebraic and apply whether the trajectory is used as a mean or as a component of a longitudinal model’s linear predictor. They do not, on their own, make assumptions about the residual distribution or the event-time submodel.

Set Up the Local Polynomials

Let the ordered knots be \(\kappa_0<\kappa_1<\cdots<\kappa_{K+1}\), with \(\kappa_1,\ldots,\kappa_K\) interior knots. On interval \(j\), from \(\kappa_{j-1}\) to \(\kappa_j\), write the trajectory using time measured from that interval’s left endpoint:

The local coordinate makes the matching equations especially transparent. At the interior knot \(\kappa_j\), the left-hand polynomial has coordinate \(h_j=\kappa_j-\kappa_{j-1}\), while the right-hand polynomial has coordinate zero. The interval widths \(h_j\) are positive because the knots are strictly ordered. We assume each piece is an ordinary cubic polynomial on its interval and that the same trajectory is being joined on both sides of each knot.

If \(m_j(t)\) is measured in marker units and time is measured in years, then \(a_{j0}\) has units of marker units, \(a_{j1}\) has units of marker units per year, \(a_{j2}\) has units of marker units per year squared, and \(a_{j3}\) has units of marker units per year cubed. Each term in the polynomial therefore has marker units. The constraints below also match quantities with the same units on both sides.

Deriving the Matching Constraints

Differentiate the polynomial on interval \(j\), using \(u=t-\kappa_{j-1}\):

At \(\kappa_j\), the left piece has \(u=h_j\). The right piece, \(m_{j+1}\), starts at its local coordinate zero. Equating the corresponding quantities gives the following conditions.

The \(C^0\) equation is the condition for continuity alone. Requiring \(C^1\) means imposing both the \(C^0\) and first-derivative equations. Requiring \(C^2\) means imposing all three. There is no matching condition on third derivatives for a \(C^2\) cubic spline: a jump in the third derivative is allowed.

These equations can also be used constructively. Given the coefficients on the left piece, choose the next piece’s constant, linear, and quadratic coefficients to satisfy the three equations. Its cubic coefficient remains free. Solving for the right-hand coefficients gives:

This is a short derivation of the constraint form: evaluate the left polynomial and its first two derivatives at the knot, evaluate the right polynomial and its derivatives at local coordinate zero, and equate each pair. The matching equations are linear in the coefficients when the knot locations are fixed.

How Many Coefficients Remain?

With \(K\) interior knots, there are \(K+1\) intervals and \(4(K+1)\) local polynomial coefficients before matching. At each interior knot, a \(C^2\) join imposes three scalar equations. For distinct knots, these equations are independent: starting with the first interval’s four coefficients and adding one new interval at a time, each new interval has three coefficients fixed by matching and one cubic coefficient still free. Consequently, the dimension is \(4+K=K+4\).

This count explains the equivalence with a familiar cubic-spline representation from the earlier material on truncated power spline bases. A \(C^2\) piecewise cubic with \(K\) interior knots can be written using four global cubic terms and one truncated cubic term per knot:

Each added term is zero to the left of its knot and is a cubic to the right. Its value, first derivative, and second derivative are zero at the knot, so it changes the cubic piece to the right without disrupting \(C^2\) matching. The representation has \(4+K\) coefficients, agreeing with the constraint count. The local-coefficient equations and this basis representation describe the same class of trajectories, though their coefficients have different meanings.

Equivalent Constraints in Global Coordinates

Some software or derivations express every piece in powers of \(t\), rather than powers of time since the interval began. If the left piece is \(A_0+A_1t+A_2t^2+A_3t^3\) and the right piece is \(B_0+B_1t+B_2t^2+B_3t^3\), matching at \(\kappa\) requires:

These are the same value, slope, and curvature conditions, evaluated in a different coordinate system. Local coordinates often simplify interpretation because a segment’s value at its left endpoint is exactly its constant coefficient. Global coordinates can be convenient for algebra, but it is easy to confuse coefficients from the two coordinate systems. Translating a polynomial between coordinates before comparing coefficients avoids that mistake.

Diagnostics, Alternatives, and Limitations

A practical check should evaluate each adjacent pair at every interior knot. Compare the two fitted values for \(C^0\), the two first derivatives for \(C^1\), and the two second derivatives for \(C^2\). A nonzero discrepancy indicates either a violated constraint, a coding error, or a different smoothness requirement than intended. If coefficients are estimated numerically, tiny discrepancies may be due to rounding; assess them at the precision of the computation rather than treating every floating-point difference as a substantive jump.

There are several common failure modes:

• Matching values but not slopes. This gives a continuous trajectory, but a visible corner may remain. Continuity alone is not differentiability.
• Matching slopes while forgetting values. Derivative equality does not force the two pieces to meet at the same height; all lower-order conditions required for the target smoothness must be imposed.
• Using the wrong interval width. In local coordinates, the left piece is evaluated at \(h_j\), not at \(\kappa_j\). Substituting the absolute knot time in place of \(h_j\) generally produces incorrect constraints.
• Mixing coefficient units. The coefficient of a squared-time term has marker units per year squared, not merely per year squared. Unit checks can reveal a missing time scale or a mismatched derivative.
• Constraining only a population-average trajectory. If subject-specific trajectories are also intended to be \(C^2\), their coefficients must satisfy the matching conditions too. Constraining fixed effects alone does not generally guarantee smooth individual trajectories.

The main alternative is to parameterize the trajectory directly in a basis that enforces the desired smoothness, such as the truncated-cubic form above, rather than fit separate local polynomials and impose explicit equalities. B-spline bases provide another basis representation with local support, introduced in the earlier material on B-spline bases. Explicit local constraints are useful when segment-specific coefficients are needed or when the smoothness order must be tailored. A basis that builds continuity in can be more compact and reduces the risk of accidentally fitting unconstrained jumps.

Neither approach guarantees that the chosen trajectory is scientifically adequate. Smoothness is a structural restriction, not evidence that the mean path is correctly specified. A \(C^2\) spline can still be too rigid, too wiggly for the data, or poorly supported near the ends of follow-up. The present coefficient constraints also do not choose knot locations or determine how much flexibility is appropriate; those are separate modeling decisions.