In addition to fitting conventional DFA models with trends modeled as random walks (or ARMA processes), we can also construct models where underlying trends are treated as smooth trends (B-splines, P-splines, or Gaussian processes).

Let’s load the necessary packages:

Data simulation

The sim_dfa function normally simulates loadings \(\sim N(0,1)\), but here we will simulate time series that are more similar with loadings \(\sim N(1,0.1)\)

s = sim_dfa(num_trends = 1, num_years = 1000, num_ts = 4,
            loadings_matrix = matrix(nrow = 4, ncol = 1, rnorm(4 * 1,
    1, 0.1)), sigma=0.05)
matplot(t(s$y_sim), type="l")

Comparing approaches

All of the smooth trend methods are flexible and able to capture the wiggliness of latent trends. Based on our experience, the B-spline and P-spline models will generally fit faster than the Gaussian predicitve process models (because they omit a critical matrix inversion step). The full rank Gaussian process models tend to be faster than the predictive process models.