Should you scale your time series for nonlinear analysis?

Benjamin Rosenbaum — Wed, 28 Oct 2026 23:00:00 GMT

Linear and nonlinear analysis

Linear time series analysis is popular with observed time series of community abundances from the field. Classically, a multivariate time series of species is log-transformed , before a multivariate autoregressive model MAR(1) is fitted With matrix algebra, this can be written as where is an interaction matrix, and contains the effects of environmental variables . This model is purely linear. [Something about SSM, MARSS package, etc]

This MAR(1) model actually corresponds to an ecological community model. If we transform densities back to the original scale, we get the discrete-time, multi-species Gompertz model.

More common in modeling, however, is the generalized Lotka-Volterra model. In its discrete-time version, the Ricker model, it reads On logscale, it becomes an additive model just as the Gompertz, although it is not linear in anymore. However, it is still linear in its parameters, and by defining new predictors , we could still push it through lm(...) to estimate all parameters [link to blog post].

Scaling and reparametrization

For the linear MAR(1) / Gompertz model, it makes sense to scale log-transformed observations species-wise, i.e., . This effectively is a reparametrization of the model, with parameter estimates describing relative interaction strength. Setting up the MAR(1) equations for the scaled variables, we get Now multiplying both sides by and rearranging coefficients, we get the MAR(1) equations for the original log-abundances .

This only worked because the Gompertz model is, on logscale, linear both in the parameters as well as the state variables.

It does not work the same way for the Ricker model. Here, reparameterization is only feasible for mean-centering .

However, scaling we get stuck.

The nonlinear transformation of the Ricker-predictors (raised to the power ), does not allow us to transform these equations back to the original Ricker model by reparameterization of the model coefficients.

Conclusions

When fitting an linear MAR(1) on log-scaled population abundances we can indeed mean-center and sd-scale the data. The result is still a Gompertz community model, and parameters for the non-scaled abundances can be obtained by simple transformations.

The same holds for mean-centered data in the Ricker model. However, when also sd-scaling the data, we are not fitting a Ricker model anymore. Estimated parameters cannot be transformed back to the coefficients of generalized Lotka-Volterra model.