How to interpret model summary of multiple covariates in a single tensor?

Modeling
1

Hello,

I am using brms with cmdstanr backend to fit a non-linear bayesian regression model with a univariate smooth term and a tensor term with 4 covariates. My model is formulated with the following code:

priors <- c(set_prior("normal(-0.1, 0.15)", class = "Intercept"),  # Weak normal prior for intercept
            set_prior("normal(0.01, 0.05)", lb = 0, class = "sds"),
            set_prior("normal(0,0.05)", class = "b"),
            set_prior("normal(0,1)", lb=0, class = "sigma"))

# Specify the model formula
formula <- bf(gpp_stdzed ~ s(doy, bs="cc") + t2(NDVI_stdzed, vh_stdzed, vv_stdzed, lstC_stdzed))

# Fit the model using well-informed priors based on our prior-only predictions on the real data
b1_sol_ten_informed <- brm(
  formula,
  data = training_data_train,
  prior = priors,
  family = skew_normal(),  # Specify the appropriate family
  chains = 10,
  warmup = 5000,
  iter = 10000, # seems large but needed for reasonable ESS 
  seed = 123, 
  backend = "cmdstanr",
  cores = 10,
  thin = 10,
  control = list(adapt_delta = 0.99, max_treedepth=15),
  normalize = FALSE,
  silent=0)

The output of my model summary looks like this:

Family: skew_normal
Links: mu = identity; sigma = identity; alpha = identity
Formula: gpp_stdzed ~ s(doy, bs = “cc”) + t2(NDVI_stdzed, vh_stdzed, vv_stdzed, lstC_stdzed)
Data: training_data_train (Number of observations: 14485)
Draws: 10 chains, each with iter = 4000; warmup = 2000; thin = 10;
total post-warmup draws = 2000

Smoothing Spline Hyperparameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sds(sdoy_1) 0.08 0.02 0.05 0.12 1.00 1942 1917
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_1) 0.13 0.08 0.01 0.29 1.00 2103 1949
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_2) 0.05 0.04 0.00 0.14 1.00 1933 1864
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_3) 0.08 0.06 0.00 0.21 1.00 2059 2013
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_4) 0.05 0.04 0.00 0.13 1.00 1954 1991
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_5) 0.06 0.04 0.00 0.16 1.00 2037 1955
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_6) 0.05 0.04 0.00 0.14 1.00 2072 1962
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_7) 0.05 0.04 0.00 0.15 1.00 2102 1989
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_8) 0.05 0.03 0.00 0.13 1.00 1993 1856
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_9) 0.05 0.03 0.00 0.13 1.00 2233 1994
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_10) 0.04 0.03 0.00 0.12 1.00 2022 2038
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_11) 0.04 0.03 0.00 0.12 1.00 1946 1955
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_12) 0.04 0.03 0.00 0.11 1.00 2079 2032
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_13) 0.05 0.03 0.00 0.12 1.00 1914 2083
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_14) 0.05 0.03 0.00 0.12 1.00 2188 1994
sds(t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_15) 0.05 0.03 0.00 0.13 1.00 1972 2083

Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.11 0.00 -0.12 -0.10 1.00 2045 1993
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_1 -0.08 0.01 -0.10 -0.07 1.00 2043 1910
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_2 0.08 0.01 0.07 0.10 1.00 2070 1833
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_3 -0.10 0.01 -0.12 -0.08 1.00 2017 1877
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_4 0.17 0.01 0.16 0.19 1.00 1933 2039
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_5 -0.12 0.01 -0.14 -0.10 1.00 2117 2037
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_6 0.14 0.01 0.12 0.15 1.00 1989 1861
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_7 -0.01 0.01 -0.04 0.01 1.00 1935 1842
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_8 0.46 0.01 0.44 0.47 1.00 2038 1840
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_9 -0.38 0.01 -0.39 -0.36 1.00 2125 2024
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_10 0.31 0.01 0.28 0.33 1.00 1864 2048
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_11 -0.27 0.02 -0.30 -0.24 1.00 2166 2076
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_12 0.43 0.01 0.41 0.46 1.00 2030 1858
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_13 -0.32 0.01 -0.35 -0.29 1.00 1900 1684
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_14 0.18 0.01 0.16 0.20 1.00 2156 1891
t2NDVI_stdzedvh_stdzedvv_stdzedlstC_stdzed_15 -0.06 0.02 -0.09 -0.02 1.00 2019 1781

Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.48 0.00 0.47 0.48 1.00 1911 1990
alpha 1.65 0.07 1.50 1.79 1.00 2093 2086

Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

I am having a hard time interpreting the t2 regression coefficient terms. Specifically, I don’t understand which term corresponds to which marginal effect and which term corresponds to a respective interactive effect. I think if I could disentangle this information, then interpretation might be a bit easier. I understand if this is not possible, as I have not found anything in the literature on how to parse these unlabeled parameters.

Thanks very much!