Hierarchical model with grouped, ragged data

Modeling
1

I am trying to fit a hierarchical, exponential decay model with n phases to grouped data. That is, if an individual belongs to group k, then their expected value is
V_k(t) = \sum_{i=1}^n \lambda_{ik} exp((-1/r_i) t),
with the constraint \sum_{i=1}^n \lambda_i = 1.

Effectively, my model is representing fixed decay rates per exponential decay phase, with the only differences between groups being the proportion of the individual in each phase.

I am assuming that my data is normally distributed with rate above, and fixed variance \sigma, and that r_j is drawn from some population level distribution. That is, individual j, in group k, and considering just exponential phase l gives
r_{jkl} \sim \mathcal{N}(\hat{r}_l + \epsilon_{ik}).

While highly structured, this is fairly simple to model. The challenge comes from the structure of my data. Inside each group is a varying number of individuals.

This means that when I set up my square data structures, I have to pad parameters (the \epsilon's here). This runs fine but I am concerned it is affecting the sampling.

Here is some R code to generate some synthetic data that has the same structure I am on about:

r_hyper <- 20

groups <- 1:2

num_groups <- length(groups)
num_individuals <- c(7, 12)
obs_times <- seq(0, 100, length.out=10)
num_obs <- length(obs_times)

rates <- rnorm(n=sum(num_individuals), mean = r_hyper, sd = 0.1) + rnorm(n=sum(num_individuals), mean = 0, sd = 5)

d <- lapply(1:length(rates), function(i) {
  value <- exp(-1/rates[i] * obs_times)
  if (i <= 7)
    group <- 1
  else
    group <- 2
  
  data.frame(t=obs_times, group=group, individual = i, value=value)
}) %>% do.call(rbind, .)

setupStanData <- function(data, id = individual, time = t, group = group, value = value) {
  
  group <- enquo(group)
  id <- enquo(id)
  time <- enquo(time)
  value <- enquo(value)
  
  groups <- data %>% select(!!group) %>% unique() %>% .[,1]
  num_groups <- length(groups)
  
  num_individuals <- sapply(groups, function(g) {
    df <- filter(data, (!!group) == g)
    
    length(unique(df[,quo_name(id)]))
  })
  
  num_obs <- lapply(groups, function(g) {
    df <- dplyr::filter(data, (!!group) == g)
    
    monkeys <- unique(df[, quo_name(id)])
    
    num_obs <- by(df, df[, quo_name(id)], function(m) length(m[,quo_name(id)])) %>% as.numeric()
    num_obs <- c(num_obs, rep(-1, times=max(num_individuals)-length(num_obs)))
  }) %>% do.call(rbind, .)
  
  max_obs <- max(num_obs)
  
  
  times <- lapply(groups, function(g) {
    
    df <- filter(data, (!!group) == g)
    monkeys <- unique(df[, quo_name(id)])
    
    t <- lapply(monkeys, function(m) {
      df2 <- filter(df, (!!id) == m)
      obs <- df2[, quo_name(time)]
      
      obs <- c(obs, rep(-1, times=max(max_obs)-length(obs)))
    }) %>% do.call(rbind, .)
  })
  
  max_monkeys <- sapply(times, nrow) %>% max()
  for (i in 1:length(times)) {
    padding <- matrix(-1, nrow = max_monkeys - nrow(times[[i]]), ncol = max_obs)
    times[[i]] <- rbind(times[[i]], padding)
  }
  
  a <- array(NA, dim=c(num_groups, max_monkeys, max_obs))
  for (i in 1:length(times)) {
    a[i,,] <- times[[i]]
  }
  
  times <- a
  
  y <- lapply(groups, function(g) {
    df <- filter(data, (!!group) == g)
    monkeys <- unique(df[, quo_name(id)])
    
    v <- lapply(monkeys, function(m) {
      df2 <- filter(df, (!!id) == m)
      obs <- df2[, quo_name(value)]
      
      obs <- c(obs, rep(-1, times=max(max_obs)-length(obs)))
    }) %>% do.call(rbind, .)
  })
  for (i in 1:length(y)) {
    padding <- matrix(-1, nrow = max_monkeys - nrow(y[[i]]), ncol = max_obs)
    y[[i]] <- rbind(y[[i]], padding)
  }
  
  a <- array(NA, dim=c(num_groups, max_monkeys, max_obs))
  for (i in 1:length(y)) {
    a[i,,] <- y[[i]]
  }
  
  y <- a
  
  
  
  return (list(num_obs=num_obs, times=times, y=y, num_groups=num_groups, max_obs=max_obs, num_individuals=as.array(num_individuals)))
}

s <- setupStanData(d)

(change s$num_phases to be how many decay phases you would like)
and the corresponding stan file:

data {
  int num_groups;
  int num_phases;
  int num_individuals[num_groups];
  int max_obs;

  int num_obs[num_groups,max(num_individuals)];
  real times[num_groups,max(num_individuals), max_obs];
  real y[num_groups, max(num_individuals), max_obs];
}

parameters {  
  real<lower=0> r_hyper[num_phases];
  real error[num_groups, max(num_individuals)];
  real<lower=0> sigma;
  simplex[num_phases] lambda[num_groups];
}

transformed parameters {
  real<lower=0>r[num_groups, num_phases, max(num_individuals)];
  
  for (g in 1:num_groups) {
    for (p in 1:num_phases) {
       for (i in 1:max(num_individuals)) {
         if (i > num_individuals[g]) {
           r[g,p,i] = 0;
         } else {
            r[g,p,i] = r_hyper[p] + error[g,i];
         }
        }
    }
  }
}

model {
  r_hyper ~ normal(5, 20);
  for (g in 1:num_groups) {
    for (n in 1:num_individuals[g]) {
      error[g,n] ~ normal(0, 0.1);
    }
  }
  
  for (g in 1:num_groups) {
    for (j in 1:num_individuals[g]) {
      for (t in 1:num_obs[g][j]) {
        real rate = 0;
        for (k in 1:num_phases) {
          rate += lambda[g][k]*exp(-1/r[g,k,j] * times[g,j,t]);  
        }
          y[g,j,t] ~ normal(rate, sigma);
      }
    }
  }
  
}

I have made heavy use of 3D arrays to structure the data in some way that made sense to me.

Here, I get estimates for error[1,8] through error[1,12], which are ridiculous and unnecessary - group 1 only has 7 individuals.

Is there any way I can get stan to not estimate these parameters, but still estimate error[2,8] through error[2,12], and still continue only requiring a single stan file?

2

Update: I have re-structured my data to be long-form instead. This has fixed the issues of the padded parameters at the expense of slightly more challenging interpretation, and probably a speed cost.

New stan code below for those interested:

data {
  int<lower=0> N;
  int<lower=0> num_groups;
  int group_sizes[num_groups];
  int<lower=1, upper=num_groups> group_index[N];
  int<lower=1, upper=sum(group_sizes)> individual_index[N];
  real y[N]; //observations
  real times[N];
  
  int num_phases;
}

transformed data {
  int<lower=1> num_individuals;
  num_individuals = sum(group_sizes);
}

parameters {  
  real<lower=0> r_hyper[num_phases];
  real error[num_individuals];
  real<lower=0> sigma;

  simplex[num_phases] lambda[num_groups];
}

transformed parameters {
  real<lower=0> r[num_individuals, num_phases];
  
  for (n in 1:num_individuals) {
    for (p in 1:num_phases) {
      r[n,p] = r_hyper[p] + error[n];
    }
  }
}

model {
  r_hyper ~ normal(5, 20);
  for (n in 1:num_individuals) {
    error ~ normal(0, 20);
  }
 
  for (n in 1:N) {
    real rate = 0;
    for (p in 1:num_phases) {
      rate += lambda[group_index[n]][p] * exp(-1/r[individual_index[n], p] * times[n]);
    }
    
    y[n] ~ normal(rate, sigma);
    
  }
}
3

You don’t need to put

error ~ normal(0, 20);

in the loop.

Also if you define vector[N] rate at the start of your model block. You can use

y ~ normal(rate, sigma);

outside of the loop. I think both changes might give you a decent speedup.

4

There’s a chapter in the manual on how to code ragged structures more efficiently than in long table form. We’re working on building ragged structures in directly, but that’s probably a year or more off at this point. You need to apply that technique to both the data and the parameters.

The way you’re doing it in the original post would work, but many of those error parameter entries are just going to have the prior as their posterior as nothing else touches them.

You need to not put that in the loop—it’s not just a speedup, it’s a different model. The code

for (n in 1:num_individuals)
  error ~ normal(0, 20);

is equivalent to

for (n in 1:num_individuals)
  for (k in 1:num_individuals)
    error[n] ~ normal(0, 20);

which has the effect of introducing a prior

p(\mathrm{err}_n) = \prod_{n = 1}^{\mathrm{num\_individuals}} \, \mathsf{normal}(\mathrm{err}_n \mid 0,\, 20).

because each iteration.

@stijn’s suggestion to vectorize is always good for speeding things up. It’s even worth constructing arguments in loops if you can use a vectorized log density—it’s the gradient calls, not the loops that are slow in Stan.

5

Thanks stijn and Bob.
I’ve replaced the model block with

model {
  vector[N] rate;
  r_hyper ~ normal(5, 20);
  error ~ normal(0, 5);

 
  for (n in 1:N) {
    rate[n] = 0;
    for (p in 1:num_phases) {
      rate[n] += lambda[group_index[n]][p] * exp(-1/r[individual_index[n], p] * times[n]);
    }
  }
  y ~ normal(rate, sigma);
  
}

Taken the error out of the loop (that it was mistakenly in in the first place) and vectorised as suggested. I also reduced the variance in the error to a more appropriate scale, as I was getting a lot of rejected samples (r < 0).

My data is quite small, and so the process is quite fast anyways, but always nice for some more speedup.