# Regression Model with B-Splines not converge

Hi, I’m new to Stan and my model can not converge.

Here’s a brief introduction to my model.

I need to estimate the mortality rate for 21 countys in 21 years (1995~2015). Thus the data is one 21x21 matrix for births x and one for deaths y.

So denote underlying mortality rate as Alpha, this is a binomial distribution with y \sim binomial(x, Alpha). And it is obvious that the mortality rate should be quite small, say 96 deaths out of 22626.

For Alpha, I use a bayesian penalized spline regression model. The original model is shown in this picture (cut from the paper I read).

I did a little change to this original model.

1. \Psi in this original model is the log transformation of true mortality rate. Yet I found that when calculating true rate, \exp(\Psi) are not constrained in [0,1]. So I modeled \Psi as the logit transformation of true rate.

2. The original model has different sources of data, and my data is quite simple, so I simplified the data model part to
\delta_c \sim N(0, \nu_c), \nu_c \sim U(1, 10), suggesting that different county has different variation.

Here’s my stan code.

data {
int TT; // number of years, 21
int K; // number of splines, 13
int C; // number of countys, 21
real I; // between-knot intervals, set to 2.5 years

int x[C,TT]; // births
int y[C,TT]; // deaths
matrix[K,K-2] D_Kc; // transformation of difference matrix
matrix[TT,K] B; // calculated cubic spline matrix
}
parameters {
real chi;
real<lower=0> phi;
real sigma_lg[C];
real<lower=0> lambda_0_exp[C];
vector[C] lambda_1_unit;
real<lower=0> nu[C];
matrix[C,K-2] epsilon;
row_vector[TT] delta[C];
}
transformed parameters {
real<lower=0> sigma[C];
real lambda_0[C];
vector[C] lambda_1;

row_vector[K] alpha[C]; // spline coefficients

sigma = exp(sigma_lg);
lambda_0 = log(lambda_0_exp);
lambda_1 = lambda_1_unit * I;

for (c in 1:C){
for (k in 1:K)
alpha[c,k] = lambda_0[c] + lambda_1[c] * (k - K * 1.0 / 2) + D_Kc[k] * epsilon[c]';
}
}
model {
chi ~ normal(-3, sqrt(10));
phi ~ uniform(0,5);
sigma_lg ~ normal(chi, phi);
lambda_0_exp ~ uniform(1,1000);
lambda_1_unit ~ uniform(-0.25, 0.2);
nu ~ uniform(1, 10);

for (c in 1:C){
delta[c] ~ normal(0, nu[c]);
epsilon[c] ~ normal(0, sigma[c]);
target += binomial_logit_lpmf( y[c] | x[c] , exp(alpha[c] * B' + delta[c]) );
}
}
generated quantities {
row_vector<lower=0, upper=1>[TT] Alpha[C]; // estimated true mortality rate

for (c in 1:C)
Alpha[c] = inv_logit(alpha[c] * B' + delta[c]);
}


Here’s my R code.

library(bdefdata)
library(demest)
library(dplyr)
library(purrr)
library(tidyr)
library(boot)
library(latticeExtra)
library(rstan)
library(splines)

options(mc.cores = parallel::detectCores())
rstan_options(auto_write = TRUE)

y <- bdefdata::sweden_deaths
x <- bdefdata::sweden_births
C <- dim(x)[1]
TT <- dim(x)[2]
tpower <- function(x, t, p){
# Truncated p-th power function
(x - t) ^ p * (x > t)
}
bbase <- function(x, xl, xr, dx, deg){
# Construct a B-spline basis of degree ’deg’
knots <- seq(xl - deg * dx, xr + deg * dx, by = dx)
P <- outer(x, knots, tpower, deg)
n <- dim(P)[2]
D <- diff(diag(n), diff = deg + 1) / (gamma(deg + 1) * dx ^ deg)
B <- (-1) ^ (deg + 1) * P %*% t(D)
B
}
B <- round(bbase(1995:2015, 1995, 2015, 2.5, 3), 8)

K <- dim(B)[2]
I <- 2.5
D <- matrix(rep(0, (K-2)*K), nrow=K-2)
for (i in 1:K-2){
for (j in 1:K){
if (i == j | j == i + 2) D[i,j] = 1
if (j == i + 1) D[i,j] = -2
}
}
D_Kc <- t(D) %*% solve(D %*% t(D))
d <- list(C=C,
TT=TT,
K=K,
I=I,
x=x,
y=y,
D_Kc=D_Kc,
B=B)

set.seed(12)
initf <- function(chain_id){
list(lambda_0_exp = runif(C, 1, 1000),
lambda_1_unit = runif(C, -0.25, 0.2),
nu = runif(C, 1, 10),
epsilon = matrix(rnorm(C*(K-2), 0, 1), nrow=C)
)
}

n_chains <- 6
a <- lapply(1:n_chains, function(id) initf(chain_id = id))
f <- stan(file = 'mr.stan', data = d, chains = n_chains,
iter = 50000, warmup = 10000, thin = 10,
init = a)


And the 6 chains do not converge. Warning message suggests “divergent transitions after warmup”, which is almost all of them.

Here’s the traceplot of several Alpha.

Here’s another traceplot of another run with only one chain.

> sessionInfo()
R version 3.5.3 (2019-03-11)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)

Matrix products: default

locale:
[1] LC_COLLATE=Chinese (Simplified)_China.936
[2] LC_CTYPE=Chinese (Simplified)_China.936
[3] LC_MONETARY=Chinese (Simplified)_China.936
[4] LC_NUMERIC=C
[5] LC_TIME=Chinese (Simplified)_China.936

attached base packages:
[1] splines   stats     graphics  grDevices utils     datasets  methods
[8] base

other attached packages:
[4] latticeExtra_0.6-28 RColorBrewer_1.1-2  lattice_0.20-38
[7] boot_1.3-23         tidyr_1.0.0         purrr_0.3.2
[10] dplyr_0.8.3         demest_0.0.0.202    dembase_0.0.0.108
[13] bdefdata_0.0.0.9002


I really need some suggestions. And if there’s naive mistakes in my code, please point them out. Thanks for your time : )

Sorry it took so long to reply. Let us first try the simplest solution. @paul.buerkner, is it possible to fit this model using brms?

You can fit splines with brms using the s() function. If that exact spline can be fitted, I don’t know.

1 Like

Hi Paul,
I spent some time reading documents about the s() function that you mentioned in package mgcv. But I’m not sure how to define the spline I need.

The splines I need are these ones in the bottom of the picture. Each one of them are non-zero for 10 years and all splines have the same shape.

I reformat my data to fit the s() function.

I found that s() function offers an option bs=‘ps’ for penalized splines. By specifying m = c(3, 2), I meant that I want cubic splines with penalty on the second derivative.

So I tested

make_stancode(formula =
deaths | trials(births) ~ s(year, bs = "ps", m = c(3, 2)),
data = sweden, family = binomial("logit")
)


and got the following error message:

Error in Summary.factor(c(1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L,  :
‘min’ not meaningful for factors


I’m not sure which variables should be offered to s() function to construct the splines.

> sessionInfo()
R version 3.5.3 (2019-03-11)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)

Matrix products: default

locale:
[1] LC_COLLATE=Chinese (Simplified)_China.936
[2] LC_CTYPE=Chinese (Simplified)_China.936
[3] LC_MONETARY=Chinese (Simplified)_China.936
[4] LC_NUMERIC=C
[5] LC_TIME=Chinese (Simplified)_China.936

attached base packages:
[1] splines   stats     graphics  grDevices utils     datasets  methods
[8] base

other attached packages: