How do I make a basis function matrix?

Modeling
1

I am trying to fit a model with splines, and I’m struggling to understand what is happening. In brms, I have this model for some dummy data.

library(brms)
df <- data.frame(y = rnorm(10), x = 1:10)
fit_spline <- brm(y ~ s(x), df)

stancode(fit_spline)
standata(fit_spline)


// generated with brms 2.16.3

functions {
}

data {
  int<lower=1> N;  // total number of observations
  vector[N] Y;  // response variable

  // data for splines
  int Ks;  // number of linear effects
  matrix[N, Ks] Xs;  // design matrix for the linear effects

  // data for spline s(x)
  int nb_1;  // number of bases
  int knots_1[nb_1];  // number of knots

  // basis function matrices
  matrix[N, knots_1[1]] Zs_1_1;
  int prior_only;  // should the likelihood be ignored?
}
transformed data {
}
parameters {
  real Intercept;  // temporary intercept for centered predictors
  vector[Ks] bs;  // spline coefficients

  // parameters for spline s(x)
  // standarized spline coefficients
  vector[knots_1[1]] zs_1_1;
  real<lower=0> sds_1_1;  // standard deviations of spline coefficients
  real<lower=0> sigma;  // dispersion parameter
}
transformed parameters {
  // actual spline coefficients
  vector[knots_1[1]] s_1_1;

  // compute actual spline coefficients
  s_1_1 = sds_1_1 * zs_1_1;
}
model {
  // likelihood including constants
  if (!prior_only) {
    // initialize linear predictor term
    vector[N] mu = Intercept + rep_vector(0.0, N) + Xs * bs + Zs_1_1 * s_1_1;
    target += normal_lpdf(Y | mu, sigma);
  }
  // priors including constants
  target += student_t_lpdf(Intercept | 3, 0.3, 2.5);
  target += student_t_lpdf(sds_1_1 | 3, 0, 2.5)
    - 1 * student_t_lccdf(0 | 3, 0, 2.5);
  target += std_normal_lpdf(zs_1_1);
  target += student_t_lpdf(sigma | 3, 0, 2.5)
    - 1 * student_t_lccdf(0 | 3, 0, 2.5);
}
generated quantities {
  // actual population-level intercept
  real b_Intercept = Intercept;
}

From the data section,
Xs is a vector of length N ranging from -0.126 to +0.126
Zs_1_1 is then a knots_1 x N (8x10) matrix of numbers in the range (-1, 1)

How has brms has done to calculate these, and what do they represent?

In standata(fit_splines) I also have a X variable, which isn’t in the Stan code. Is this being created extra by brms?

2

Xs are a centered and scaled version of the covariate values over which you are going to smooth. I’m not sure how the choice of the scaling factor is determined.

Zs_1_1 contains the basis functions evaluated over your covariate values.