Global Shape Constraints

Enforce positivity and monotonicity over an entire one-dimensional domain.

Source: Examples/Tutorials/GlobalShapeConstraints.m

Enforce positivity and monotonicity across the whole domain

GlobalConstraint objects do not act at a few selected points. Instead they describe the shape of a ConstrainedSpline fit everywhere on the domain:

\[f(t)\ge 0,\qquad f'(t)\ge 0 \quad \text{for all } t.\]

rng(4)
t = linspace(0, 1, 45)';
xTrue = 0.15 + 0.85*(1 - exp(-4*t));
xObs = xTrue + 0.05*randn(size(t));
tq = linspace(t(1), t(end), 400)';

noiseModel = NormalDistribution(sigma=0.05);
freeFit = ConstrainedSpline.fromData(t, xObs, S=3, splineDOF=12, distribution=noiseModel);

shapeConstraints = [ ...
    GlobalConstraint.positive()
    GlobalConstraint.monotonicIncreasing()];

shapeFit = ConstrainedSpline.fromData(t, xObs, S=3, splineDOF=12, distribution=noiseModel, constraints=shapeConstraints);

Plot the unconstrained and globally constrained fits together.

figure(Position=[100 100 820 320])
plot(tq, 0.15 + 0.85*(1 - exp(-4*tq)), "k--", LineWidth=1.5), hold on
plot(tq, freeFit(tq), LineWidth=2)
plot(tq, shapeFit(tq), LineWidth=2)
scatter(t, xObs, 28, "filled", MarkerFaceAlpha=0.65)
xlabel("t")
ylabel("x(t)")
legend("Underlying signal", "Unconstrained fit", "Positive monotone fit", "Observations", Location="southoutside")
grid on

GlobalConstraint objects can enforce positivity and monotonic increase over the full domain in one fit.

Check the constrained derivative

The first derivative is the clearest way to see the monotonicity condition. Once the global monotone-increasing constraint is active, the fitted derivative stays nonnegative throughout the interval.

dFree = freeFit.valueAtPoints(tq, D=1);
dShape = shapeFit.valueAtPoints(tq, D=1);

Plot the first derivative to check the monotonicity condition directly.

figure(Position=[100 100 780 300])
plot(tq, dFree, LineWidth=1.6), hold on
plot(tq, dShape, LineWidth=2)
yline(0, "k--")
xlabel("t")
ylabel("dx/dt")
legend("Unconstrained", "Positive monotone fit", Location="southoutside")
grid on

The constrained fit keeps the first derivative nonnegative across the whole domain.