Seasonal and Periodic Temporal Covariance

Why Model Seasonal Dependence?

Temporal dependence need not fade steadily as observations get farther apart. Measurements taken in the same season in successive years may be more alike than measurements separated by only a few months. A covariance model that decreases monotonically with lag, such as the exponential covariance associated with a continuous-time Ornstein–Uhlenbeck process, cannot express this recurring pattern by itself.

In a GLMM, a temporal covariance function can describe dependence in a latent Gaussian effect included in the linear predictor. It describes how the latent effect varies over time; it does not, by itself, specify a seasonal pattern in the expected response. A deterministic seasonal pattern in the mean and recurring covariance in a latent effect are distinct modeling choices.

For a latent Gaussian effect in a GLMM, one common specification is

Here \(\eta_i\) is the linear predictor for observation \(i\), and \(\boldsymbol{\Sigma}\) must be a valid covariance matrix. The temporal effect is assumed to have mean zero, and the stationary covariance specification assumes the same dependence at any pair of times separated by the same lag. If observations are clustered in different subjects or locations, the intended scope of this temporal covariance must also be stated; this tutorial considers one temporal process, not a full space-time structure.

Exact Periodicity and Damped Recurrence

Let \(P>0\) denote a period, measured in the same time units as \(h\). Exact periodicity means that the covariance repeats without weakening as the lag increases by a whole period: \(C(h+P)=C(h)\). A simple valid covariance is

where \(\sigma_u^2>0\) is the latent-effect variance. This covariance is valid because, for any times \(t_i,t_j\), its correlation entry can be written as \(\cos(\omega(t_i-t_j))=\cos(\omega t_i)\cos(\omega t_j)+\sin(\omega t_i)\sin(\omega t_j)\), where \(\omega=2\pi/P\). Thus its covariance matrix is the sum of two positive semidefinite outer-product matrices, multiplied by \(\sigma_u^2\). It is nevertheless a restrictive model: the matrix has rank at most two, and observations exactly one period apart have correlation one. It represents a perfectly repeating latent trajectory, not a seasonal pattern that gradually loses strength.

A damped cosine allows recurrence to weaken over time:

The cosine produces the repeating peaks and troughs; the exponential envelope reduces their magnitude as \(|h|\) grows. The parameter \(\phi\) has inverse-time units. At a full-period lag \(P\), the correlation is \(\exp(-\phi P)\), not one. At a half-period lag \(P/2\), it is \(-\exp(-\phi P/2)\).

Another useful exact-periodic option is the periodic squared-exponential covariance:

The dimensionless parameter \(\ell\) controls how quickly correlation falls as times move apart in phase within a period. This is exactly periodic because \(\sin^2\{\pi(h+P)/P\}=\sin^2(\pi h/P)\). Its validity follows from the Fourier expansion of \(\exp\{\kappa\cos(\theta)\}\), whose cosine-series coefficients are nonnegative for \(\kappa>0\); setting \(\kappa=1/\ell^2\) gives a nonnegative weighted sum of valid cosine covariance functions. Like the pure cosine, this exact-periodic covariance gives correlation one at every whole-period lag.

Calculating Damped Seasonal Correlations

For the damped cosine, calculate the lag in the chosen time unit, evaluate the exponential envelope, and then multiply by the cosine. The sign matters: a negative correlation at some lags is a permitted feature of this model, not an invalid covariance. The decay parameter and period should be interpreted jointly, since the correlation at a seasonal recurrence is determined by both.

Assumptions, Diagnostics, and Alternatives

A periodic covariance is most defensible when the time scale is meaningful, a plausible period can be specified or estimated, and dependence is reasonably stable across the observation window. A stationary periodic covariance assumes the same covariance at the same lag regardless of calendar date. A damped model further assumes a particular oscillatory shape and exponential weakening; an exact-periodic model assumes no weakening at all. These are modeling assumptions, not consequences of observing a seasonal-looking series.

Diagnostics should ask whether the fitted dependence resembles the model's implications. Examine residual or simulation-based checks for recurring patterns at the proposed period and for remaining lag dependence, using methods appropriate to the GLMM and response distribution. A damped cosine predicts alternating positive and negative lobes as lag increases, with their absolute magnitudes bounded by the exponential envelope. A monotone exponential covariance predicts nonnegative correlations that decrease with lag. Clear unexplained oscillation, a persistent seasonal pattern where the model predicts strong damping, or dependence that changes across calendar time indicates possible lack of fit.

Period estimates can be weakly determined when the time series is short, observations are sparse, or only a small number of cycles are observed. Irregularly spaced times can be used by evaluating the covariance at actual time differences, but do not eliminate weak information about the period. In discrete-time data, a period that is not distinguishable at the observation schedule may be difficult to identify from other oscillations. Compare scientifically plausible periods and inspect uncertainty in the covariance parameters rather than treating a single fitted period as certain.

The alternatives answer different questions. A discrete-time AR(1) or continuous-time exponential covariance describes dependence that decays monotonically; it is simpler when recurrence is not supported. The exact periodic squared-exponential covariance allows a flexible, smooth pattern within a cycle but forces perfect recurrence after every whole period. The damped cosine allows weakening recurrence and negative correlations, but imposes a specific oscillatory shape. Model choice should reflect the timing and scientific process, not just which covariance can produce a visually appealing fit.

Periodic covariance functions are useful when repeated phases of a cycle plausibly share latent deviations, as with month-of-year or day-of-week patterns. Their limitations are equally important: a single stationary period may not capture changing amplitude, shifting phase, multiple cycles, or an isolated long-term change. Exact periodicity can be especially restrictive because observations one period apart have identical latent effects. A covariance model should therefore be interpreted as a claim about the latent dependence structure, and checked against the design and the observed time scale.