Example: birthdays and birthdates#

Part 5 · Stage 15 · 🌊 Basis Functions & Gaussian Processes · Lesson 130 of 144 · advanced

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Decomposing a time series with kernels#

The number of babies born in the United States each day, over two decades, is a deceptively rich time series — and the showcase, on the cover of Bayesian Data Analysis, for what additive Gaussian-process kernels can do. The daily count carries several overlapping patterns at different timescales, and the GP recovers them by adding kernels, one per component.

The components#

Births vary on at least four rhythms, each captured by a kernel matched to its shape:

a long-term trend over the years — a slow squared-exponential (large length-scale);
a seasonal pattern within the year (more births in late summer) — a periodic kernel with a one-year period;
a day-of-week effect (far fewer births on weekends, when scheduled deliveries pause) — a periodic kernel with a one-week period;
special days — sharp dips on holidays, a spike or dip on dates like Valentine’s Day and a notable avoidance of the 13th — short-scale deviations.

The additive model#

Because a sum of independent GPs is a GP whose kernel is the sum of the components’ kernels, the whole decomposition is a single Gaussian process:

\[f(t) = f_{\text{trend}}(t) + f_{\text{year}}(t) + f_{\text{week}}(t) + f_{\text{special}}(t), \qquad k = k_{\text{trend}} + k_{\text{year}} + k_{\text{week}} + k_{\text{special}} .\]

Fitting the sum and then examining each term’s posterior separates the rhythms — the model discovers the trend, the seasonal curve and the weekly dip individually, each with uncertainty.

import pymc as pm
with pm.Model():
    # additive kernel: long trend + yearly + weekly periodic components
    k_trend = pm.HalfNormal("a1", 1)**2 * pm.gp.cov.ExpQuad(1, ls=pm.Gamma("l1", 4, 0.1))
    k_year  = pm.HalfNormal("a2", 1)**2 * pm.gp.cov.Periodic(1, period=365.25,
                                                             ls=pm.Gamma("l2", 2, 1))
    k_week  = pm.HalfNormal("a3", 1)**2 * pm.gp.cov.Periodic(1, period=7,
                                                             ls=pm.Gamma("l3", 2, 1))
    gp = pm.gp.Marginal(cov_func=k_trend + k_year + k_week)     # sum of GPs is a GP
    gp.marginal_likelihood("y", X=t[:, None], y=births, sigma=pm.HalfNormal("s", 1))

What it demonstrates#

Two lessons close out the GP method. Kernel composition is a modelling language — adding and multiplying kernels expresses structural beliefs (a trend plus a cycle times a slow modulation) directly in the covariance, and the posterior teases the pieces apart. And a Gaussian process turns time-series decomposition into inference: trend, seasonality and calendar effects that classical methods extract by separate procedures fall out of one coherent model, with propagated uncertainty on every component. The birthday series is the vivid proof that flexible, structured, interpretable models need not be built from a fixed basis.

Hint