The smoothing spline is a method of smoothing, or fitting a smooth curve to a set of noisy observations.

Definition

Let $(x\_i,Y\_i);\; i=1,dots,n$ be a sequence of observations, modeled by the relation $E(Y\_i)\; =\; mu(x\_i)$. The smoothing spline estimate $hatmu$ of the function $mu$ is defined to be the minimizer (over the class of twice differentiable functions) of

$$

sum_{i=1}^n (Y_i - hatmu(x_i))^2 + lambda int hatmu(x)^2 ,dx.

Remarks:

1. $lambda\; ge\; 0$ is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate.
2. The integral is evaluated over the range of the $x\_i$.
3. As $lambdato\; 0$ (no smoothing), the smoothing spline converges to the interpolating spline.
4. As $lambdatoinfty$ (infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a linear least-squares estimate.
5. The roughness penalty based on the second derivative is the most common in modern statistics literature, although the method can easily be adapted to penalties based on other derivatives.
6. In early literature, with equally-spaced $x\_i$, second or third-order differences were used in the penalty, rather than derivatives.
7. When the sum-of-squares term is replaced by a log-likelihood, the resulting estimate is termed penalized likelihood. The smoothing spline is the special case of penalized likelihood resulting from a Gaussian likelihood.

Derivation of the smoothing spline

It is useful to think of fitting a smoothing spline in two steps:

1. First, derive the values $hatmu(x\_i);i=1,ldots,n$.
2. From these values, derive $hatmu(x)$ for all x.

Now, treat the second step first.

Given the vector $hat\{m\}\; =\; (hatmu(x\_1),ldots,hatmu(x\_n))^T$ of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize $int\; hatmu(x)^2\; ,\; dx$, and the minimizer is a natural cubic spline that interpolates the points $(x\_i,hatmu(x\_i))$. This interpolating spline is a linear operator, and can be written in the form

$$

hatmu(x) = sum_{i=1}^n hatmu(x_i) f_i(x) where $f\_i(x)$ are a set of spline basis functions. As a result, the roughness penalty has the form

$$

int hatmu''(x)^2 dx = hat{m}^T A hat{m}. where the elements of A are $int\; f\_i(x)\; f\_j(x)dx$. The basis functions, and hence the matrix A, depend on the configuration of the predictor variables $x\_i$, but not on the responses $Y\_i$ or $hat\; m$.

Now back the first step. The penalized sum-of-squares can be written as

$$

|Y - hat m|^2 + lambda hat{m}^T A hat m, where $Y=(Y\_1,ldots,Y\_n)^T$. Minimizing over $hat\; m$ gives

$$

hat m = (I + lambda A)^{-1} Y.

Related methods

Smoothing splines are related to, but distinct from:

• Regression splines. In this method, the data is fitted to a set of spline basis functions with a reduced set of knots, typically by least squares. No roughness penalty is used.
• Penalized Splines. This combines the reduced knots of regression splines, with the roughness penalty of smoothing splines.

Further reading

• Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
• Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models. CRC Press.

References