Splines, knots, and penalties
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Truncated power function bases with equally spaced knots. (a) Linear, (b) cubic.
Interpolation and extrapolation with a combination of penalties of first and second order:
(a) Impulse response of a P‐spline smoother with a only a first or second‐order difference penalty; (b) impulse response of a P‐spline smoother with second and first‐order penalties:
Smoothing of simulated data (dots) with and without exponentially varying weights on the differences in the penalty. (a) Uniform weights; (b) varying weights. Parameters optimized with grid search and leave‐one‐out cross validation. Full line: fitted curve (100 cubic B‐splines, second‐order penalty); broken line: true curve.
A standard B‐spline basis (top) and the corresponding wrapped basis.
Automatic choice of the smoothing parameter with the hybrid algorithm for variance estimation. Simulated data. Broken line: true curve; full line: automatically estimated smooth.
Condition numbers of three types of bases: truncated power functions (crosses), B‐splines (squares) and Z‐matrices for mixed models (diamonds). The sample sizes are 100, 200, 500, or 1000, but this is difficult to see, because the lines and symbols overlap strongly.
An illustration of optimal smoothing and interpolation with many B‐splines and a large gap. The upper left panel shows the numerical condition, and the lower left panel the leave‐one‐out cross‐validation profile. The panels show results of smoothing for the approximately optimal λ and for small λ, the latter showing overshoot. The number of cubic B‐splines is 53 and the order of the penalty is 2.
Smoothing and interpolation of simulated data with a basis of linear truncated power functions, with knots at unique values of x. Ridge penalty with κ = 0.1.
Smoothing and interpolation of simulated data with a basis of linear truncated power functions, with 100 equally spaced knots. Ridge penalty with κ = 0.1.
Smoothing and interpolation of simulated data with a large basis of cubic B‐splines and a second‐order penalty (λ = 10). The scaled B‐splines are shown on the bottom of the graph. Their sum gives the full line, which is the fitted curve. The encircled dots represent the value of the B‐spline coefficients.
Components of a fit with 15 (κ = 0.1) linear truncated power functions to simulated data.
Components of a fit with 18 cubic B‐splines and a second‐order penalty to simulated data (squares). The encircled dots show the coefficients of the B‐splines. (a) λ = 0.01, (b) λ = 10.
Logarithm of the absolute value of one cubic B‐spline, computed from the fourth difference of cubic truncated polynomials. The B‐splines has been scaled to a maximum of 1.
B‐spline bases with equally spaced knots. (a) Linear, (b) quadratic.
