# Sparse and Ragged Data Structures

Stan does not directly support either sparse or ragged data structures, though both can be accommodated with some programming effort. The sparse matrices chapter introduces a special-purpose sparse matrix times dense vector multiplication, which should be used where applicable; this chapter covers more general data structures.

## Sparse data structures

Coding sparse data structures is as easy as moving from a matrix-like data structure to a database-like data structure. For example, consider the coding of sparse data for the IRT models discussed in the item-response model section. There are \(J\) students and \(K\) questions, and if every student answers every question, then it is practical to declare the data as a \(J \times K\) array of answers.

```
data {
int<lower=1> J;
int<lower=1> K;
array[J, K] int<lower=0, upper=1> y;
// ...
model {
for (j in 1:J) {
for (k in 1:K) {
y[j, k] ~ bernoulli_logit(delta[k] * (alpha[j] - beta[k]));
}
}// ...
}
```

When not every student is given every question, the dense array coding will no longer work, because Stan does not support undefined values.

The following missing data example shows an example with \(J=3\) and \(K=4\), with missing responses shown as NA, as in R.

\[\begin{equation*} y = \left[ \begin{array}{cccc} 0 & 1 & \mbox{NA} & 1 \\ 0 & \mbox{NA} & \mbox{NA} & 1 \\ \mbox{NA} & 0 & \mbox{NA} & \mbox{NA} \end{array} \right] \end{equation*}\]

There is no support within Stan for R’s NA values, so this data structure cannot be used directly. Instead, it must be converted to a “long form” as in a database, with columns indicating the indices along with the value. With columns \(jj\) and \(kk\) used for the indexes (following Gelman and Hill (2007)), the 2-D array \(y\) is recoded as a table. The number of rows in the table equals the number of defined array elements, here \(y_{1,1} = 0\), \(y_{1,2} = 1\), up to \(y_{3,2} = 1\). As the array becomes larger and sparser, the long form becomes the more economical encoding.

jj |
kk |
y |
---|---|---|

1 | 1 | 0 |

1 | 2 | 1 |

1 | 4 | 1 |

2 | 1 | 0 |

2 | 4 | 1 |

3 | 2 | 0 |

Letting \(N\) be the number of \(y\) that are defined, here \(N=6\), the data and model can be formulated as follows.

```
data {
// ...
int<lower=1> N;
array[N] int<lower=1, upper=J> jj;
array[N] int<lower=1, upper=K> kk;
array[N] int<lower=0, upper=1> y;
// ...
}model {
for (n in 1:N) {
y[n] ~ bernoulli_logit(delta[kk[n]]
* (alpha[jj[n]] - beta[kk[n]]));
}// ...
}
```

In the situation where there are no missing values, the two model formulations produce exactly the same log posterior density.

## Ragged data structures

Ragged arrays are arrays that are not rectangular, but have different sized entries. This kind of structure crops up when there are different numbers of observations per entry.

A general approach to dealing with ragged structure is to move to a full database-like data structure as discussed in the previous section. A more compact approach is possible with some indexing into a linear array.

For example, consider a data structure for three groups, each of which has a different number of observations.

\(y_1 = \left[1.3 \ \ 2.4 \ \ 0.9\right]\\\) \(y_2 = \left[-1.8 \ \ -0.1\right]\\\) \(y_3 = \left[12.9 \ \ 18.7 \ \ 42.9 \ \ 4.7\right]\)

\(z = [1.3 \ \ 2.4 \ \ 0.9 \ \ -1.8 \ \ -0.1 \ \ 12.9 \ \ 18.7 \ \ 42.9 \ \ 4.7]\\\) \(s = \{ 3 \ \ 2 \ \ 4 \}\)

On the left is the definition of a ragged data structure \(y\) with three rows of different sizes (\(y_1\) is size 3, \(y_2\) size 2, and \(y_3\) size 4). On the right is an example of how to code the data in Stan, using a single vector \(z\) to hold all the values and a separate array of integers \(s\) to hold the group row sizes. In this example, \(y_1 = z_{1:3}\), \(y_2 = z_{4:5}\), and \(y_3 = z_{6:9}\).

Suppose the model is a simple varying intercept model, which, using vectorized notation, would yield a log-likelihood \[\begin{equation*} \sum_{n=1}^3 \log \textsf{normal}(y_n \mid \mu_n, \sigma). \end{equation*}\] There’s no direct way to encode this in Stan.

A full database type structure could be used, as in the sparse example, but this is inefficient, wasting space for unnecessary indices and not allowing vector-based density operations. A better way to code this data is as a single list of values, with a separate data structure indicating the sizes of each subarray. This is indicated on the right of the example. This coding uses a single array for the values and a separate array for the sizes of each row.

The model can then be coded up using slicing operations as follows.

```
data {
int<lower=0> N; // # observations
int<lower=0> K; // # of groups
vector[N] y; // observations
array[K] int s; // group sizes
// ...
}model {
int pos;
1;
pos = for (k in 1:K) {
segment(y, pos, s[k]) ~ normal(mu[k], sigma);
pos = pos + s[k]; }
```

This coding allows for efficient vectorization, which is worth the copy cost entailed by the `segment()`

vector slicing operation.

## References

*Data Analysis Using Regression and Multilevel-Hierarchical Models*. Cambridge, United Kingdom: Cambridge University Press.