An R package for denoising censored, Gaussian means with empirical Bayes \(g\)-modeling. The general model is as follows:

\[ \theta_i \sim_{iid} g \quad (\subseteq \mathbb{R}^p) \]

\[ X_{ij} \mid \theta_{ij} \sim_{indep.} N(\theta_{ij}, \sigma^2) \]

\[ L_{ij} \leq X_{ij} \leq R_{ij} \]

The data is represented with matrices:

\[ \theta = \begin{bmatrix} \theta_{11} & \dots & \theta_{1p} \\ \theta_{21} & \dots & \theta_{2p} \\ \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{np} \\ \end{bmatrix} \qquad X = \begin{bmatrix} X_{11} & \dots & X_{1p} \\ X_{21} & \dots & X_{2p} \\ \vdots & \ddots & \vdots \\ X_{n1} & \dots & X_{np} \\ \end{bmatrix} \]

\[ L = \begin{bmatrix} L_{11} & \dots & L_{1p} \\ L_{21} & \dots & L_{2p} \\ \vdots & \ddots & \vdots \\ L_{n1} & \dots & L_{np} \\ \end{bmatrix} \qquad R = \begin{bmatrix} R_{11} & \dots & R_{1p} \\ R_{21} & \dots & R_{2p} \\ \vdots & \ddots & \vdots \\ R_{n1} & \dots & R_{np} \\ \end{bmatrix} \]

The bounds \(L_{ij}\) and \(R_{ij}\) are assumed to be known. When \(L_{ij} = R_{ij}\) there is a direct (noisy) measurement of \(\theta_{ij}\), if \(L_{ij} < R_{ij}\) then there is a censored measurement of \(\theta_{ij}\). This structure is commonly referred to as partially interval censored data and it allows for any combination of observed measurements and left-, right-, and interval-censored measurements.

We use a Tobit likelihood for each measurement:

\[ P(L, R \mid \theta) = \begin{cases} \phi_{\sigma} ( L - \theta ) & L = R \\ \Phi_{\sigma} ( R - \theta ) - \Phi_{\sigma} ( L - \theta ) & L < R \end{cases} \]

where the standard Gaussian likelihood is used when there is a direct Gaussian measurement (ie \(L = X = R\)) and a Gaussian probability is used when there is a censored Gaussian measurement (ie \(L < R\)).

This package provides an object `ebTobit`

(Empirical Bayes model with Tobit likelihood) that estimates the prior, \(g\) over a user-specified grid `gr`

and then computes the posterior mean or \(\ell_1\) mediod as estimates for \(\theta\). In one dimension, the \(\ell_1\) mediod is the median. By default `gr`

is set using the exemplar method so the grid is the maximum likelihood estimate for each \(\theta_{ij}\). When the censoring interval is finite, the maximum likelihood estimate for each \(\theta_{ij}\) is \(0.5 ( L_{ij} + R_{ij} )\)

Suppose \(p = 1\) and there is no censoring, then the basic utility is:

```
library(ebTobit)
# create noisy measurements
n <- 100
t <- sample(c(0, 5), size = n, replace = TRUE, prob = c(0.8, 0.2))
x <- t + stats::rnorm(n)
# fit g-model with default prior grid
res1 <- ebTobit(x)
# measure performance of estimated posterior mean
mean((t - fitted(res1))^2)
```

Next we can look at a more complicated example with \(p = 10\):

```
library(ebTobit)
# create noisy measurements (low rank structure)
n <- 1000; p <- 10
t <- matrix(stats::rgamma(n*p, shape = 5, rate = 1), n, p)
x <- t + matrix(stats::rnorm(n*p), n, p)
# assume we can't accurately measure x < 1 but we know theta > 0
L <- ifelse(x < 1, 0, x)
R <- ifelse(x < 1, 1, x)
# fit g-model with default prior grid
res2 <- ebTobit(x)
res3 <- ebTobit(L, R)
# oberve that the censoring affects the fitted range
range(fitted(res2))
range(fitted(res3))
# fit censored data with a different grid (large and random not MLE)
res4 <- ebTobit(
L = L,
R = R,
gr = sapply(1:p, function(j) stats::runif(1e+4, min = min(L[,j]), max = max(R[,j]))),
algorithm = "EM"
)
# compute posterior mean and L1mediod given new data
# we can also predict based on partially interval-censored observations
y <- matrix(stats::rexp(5*p, rate = 0.5), 5, p)
predict(res4, y) # posterior mean
predict(res4, y, method = "L1mediod") # posterior L1-mediod
```

This package is available on CRAN. It can also be installed directly from GitHub:

This R package also includes a real bile acid `data.frame`

taken directly from Lei et al. (2018) (https://doi.org/10.1096/fj.201700055R) via https://github.com/WandeRum/GSimp (https://doi.org/10.1371/journal.pcbi.1005973). The bile acid data contains measurements of 34 bile acids for 198 patients; no missing values are present in the data. In our modeling, we assume the bile acid values are independent log-normal measurements.

Alton Barbehenn and Sihai Dave Zhao

GPL (>= 3)