STAR Model Overview

STAR models build upon continuous data models to provide a valid count-valued data-generating process. An example STAR model for linear regression is as follows: \[\begin{align*} y_i &= \mbox{floor}(y_i^*) \\ z_i^* &= \log(y_i^*) \\ z_i^* &= x_i'\beta + \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim}N(0, \sigma^2) \end{align*}\] The latent data \(y_i^*\) act as a continuous proxy for the count data \(y_i\), which is easier to model yet has a simple mapping via the floor function to the observed data. The latent data \(y_i^*\) are transformed to \(z_i^*\), as in common practice, and modeled using Gaussian linear regression. This model inherits the same structure as before, but the data-generating process is now count-valued.

More generally, STAR models are defined via a rounding operator \(h\), a (known or unknown) transformation \(g\), and a continuous data model \(\Pi_\theta\) with unknown parameters \(\theta\): \[\begin{align*} y &= h(y^*) \quad \mbox{(rounding)}\\ z^* &= g(y^*) \quad \mbox{(transformation)}\\ z^* & \sim \Pi_\theta \quad \mbox{(model)}\\ \end{align*}\] Importantly, STAR models are highly flexible count-valued processes, and provide the capability to model (i) discrete data, (ii) zero-inflation, (iii) over- or under-dispersion, and (iv) bounded or censored data.

We focus on conditionally Gaussian models of the form \[ z^*(x) = \mu_\theta(x) + \epsilon(x), \quad \epsilon(x) \stackrel{iid}{\sim}N(0, \sigma^2) \] where \(\mu_\theta(x)\) is the conditional expectation of the transformed latent data with unknown parameters \(\theta\). Examples include linear, additive, and tree-based regression models.

The Rounding Operator

The rounding operator \(h\) is a many-to-one function that sets \(y = j\) whenever \(y^*\in \mathcal{A}_j\) or equivalently when \(z^*=g(y^*) \in g(\mathcal{A}_j)\) . It could take many different forms, but generally the floor function, i.e where \(\mathcal{A}_j := [j, j+1)\) works well as a default, with the modification \(g(\mathcal{A}_0) := (-\infty, 0)\) so that \(y = 0\) whenever \(z^* < 0\). This latter modification ensures that much of the latent space is mapped to zero, and therefore STAR models can easily account for zero-inflation. Furthermore, when there is a known upper bound y_max\(=K\) for the data, there is an additional change to incorporate this structure, namely we let \(g(\mathcal{A}_K) := [g(a_K), \infty)\).

The rounding operator and its inverse are implemented in countSTAR with the functions a_j() and round_fun(), although these rarely need to be employed by the end user. Instead, the only thing that might need to be specified by the user would be whether an upper bound for the data exists, in which case there is a simple option of setting y_max in each of the modeling functions that will then be passed to the rounding function.

The Transformation Function

There are a variety of options for the transformation function \(g\), ranging from fixed functions to a-priori data-driven transformations to transformations learned along with the rest of the model. All models in countSTAR support three common fixed transformations: log, square root (‘sqrt’), and the identity transformation (essentially a rounding-only model). Furthermore, all functions support a set of transformations which are learned by matching marginal moments of the data \(y\) to the latent \(z\):

transformation='pois' uses a moment-matched marginal Poisson CDF
transformation='neg-bin' uses a moment-matched marginal Negative Binomial CDF
transformation='np' is a nonparametric transformation estimated from the empirical CDF of \(y\).

Details on the estimation of these transformations can be found in Kowal and Wu (2021). In particular the “np” transformation has proven very effective, especially for heaped data, and is the default across all functions.

Some STAR methods also support ways of learning the transformation alongside the model:

transformation='box-cox': the transformation is assumed to belong to the Box-Cox family; the sampler samples the \(\lambda\) parameter
transformation='ispline': the transformation is modeled as an unknown, monotone function using I-splines. The Robust Adaptive Metropolis (RAM) sampler is used for drawing the parameter of the transformation function.
transformation='bnp': the transformation is modeled using the Bayesian bootstrap, a Bayesian nonparametric model that incorporates the uncertainty about the transformation into posterior and predictive inference.

These learned transformations are not always available in every function; check the appropriate help page to see what options are supported.

Count-Valued Data: The Roaches Dataset

As an example of complex count-valued data, consider the roaches data from Gelman and Hill (2006). The response variable, \(y_i\), is the number of roaches caught in traps in apartment \(i\), with \(i=1,\ldots, n = 262\).

data(roaches, package="countSTAR") 

# Roaches:
y = roaches$y

# Function to plot the point mass function:
stickplot = function(y, ...){
  js = 0:max(y); 
  plot(js, 
       sapply(js, function(js) mean(js == y)), 
       type='h', lwd=2, ...)
}
stickplot(y, main = 'PMF: Roaches Data',
          xlab = 'Roaches', ylab = 'Probability mass')

There are several notable features in the data:

Zero-inflation: 36% of the observations are zeros.
(Right-) Skewness, which is clear from the histogram and common for (zero-inflated) count data.
Overdispersion: the sample mean is 26 and the sample variance is 2585.

A pest management treatment was applied to a subset of 158 apartments, with the remaining 104 apartments receiving a control. Additional data are available on the pre-treatment number of roaches, whether the apartment building is restricted to elderly residents, and the number of days for which the traps were exposed. We are interested in modeling how the roach incidence varies with these predictors.

Frequentist inference for STAR models

Frequentist (or classical) estimation and inference for STAR models is provided by an EM algorithm. Sufficient for estimation is an estimator function which solves the least squares (or Gaussian maximum likelihood) problem associated with \(\mu_\theta\)—or in other words, the estimator that would be used for Gaussian or continuous data. Specifically, estimator inputs data and outputs a list with two elements: the estimated coefficients \(\hat \theta\) and the fitted.values \(\hat \mu_\theta(x_i) = \mu_{\hat \theta}(x_i)\).

The Classical Linear Model

For many applications, the STAR linear model is often the first method to try. In countSTAR, the linear model is implemented with the lm_star function, which aims to mimic the functionality of lm by allowing users to input a formula. Standard functions like coef and fitted can be used on the output to extract coefficients and fitted values, respectively.

library(countSTAR)

# Select a transformation:
transformation = 'np' # Estimated transformation using empirical CDF

# EM algorithm for STAR (using the log-link)
fit_em = lm_star(y ~ roach1 + treatment + senior + log(exposure2),
                 data = roaches, transformation = transformation)


# Dimensions:
n = nrow(fit_em$X); p = ncol(fit_em$X)

# Fitted coefficients:
round(coef(fit_em), 3)
#>    (Intercept)         roach1      treatment         senior log(exposure2) 
#>          0.035          0.006         -0.285         -0.321          0.216

Here the np transformation was used, but other options are available; see ?lm_star for details.

Based on the fitted STAR linear model, we may further obtain confidence intervals for the estimated coefficients using confint:

# Confidence interval for all coefficients
confint(fit_em)
#>                      2.5 %       97.5 %
#> (Intercept)    -0.14010165  0.201243821
#> roach1          0.00490082  0.007461323
#> treatment      -0.48749734 -0.086246324
#> senior         -0.54512436 -0.101437843
#> log(exposure2) -0.19884729  0.637022686

Similarly, p-values are available using likelihood ratio tests, which can be applied for individual coefficients,

\[ H_0: \beta_j= 0 \quad \mbox{vs} \quad H_1: \beta_j \ne 0 \]

or for joint sets of variables, analogous to a (partial) F-test:

\[ H_0: \beta_1=\ldots=\beta_p = 0, \quad \mbox{vs.} \quad H_1: \beta_j \ne 0 \mbox{ for some } j=1,\ldots,p \] P-values for all individual coefficients as well as the p-value for any effects are computed with the pvals function.

# P-values:
print(pvals(fit_em))
#>        (Intercept)             roach1          treatment             senior 
#>       6.973323e-01       1.887411e-18       5.600904e-03       4.862508e-03 
#>     log(exposure2) Any linear effects 
#>       3.072370e-01       9.271136e-20

Finally, we can get predictions at new data points (or the training data) using predict, which actually outputs samples from the . Optionally, prediction intervals can be estimated using the (plug-in) predictive distribution at the MLEs (see ?predict.lmstar for details). Note that the “plug-in” predictive distribution is a crude approximation, and better approaches for uncertainty quantification are available using the Bayesian models.

#Compute the predictive draws (just using observed points here)
y_pred = predict(fit_em)

Machine Learning Models

In addition to the linear model, countSTAR also has implementations for STAR models paired with more flexible regression methods, in particular random forests (randomForest_star()) and generalized boosted machines (gbm_star()). In these functions, the user directly inputs the set of predictors \(X\) alongside any test points in \(X_{test}\), as can be seen in the example below

# Select a transformation:
transformation = 'np' # Estimated transformation using empirical CDF

# Construct data matrix
y = roaches$y
X = roaches[, c("roach1", "treatment", "senior", "exposure2")]

#Fit STAR with random forests
fit_rf = randomForest_star(y, X, transformation = transformation)

#Fit STAR with GBM
fit_gbm = gbm_star(y, X, transformation = transformation)

For all frequentist models, the functions output log-likelihood values at the MLEs, which allows for a quick comparison of model fit.

#Look at -2*log-likelihood
print(-2*c(fit_rf$logLik, fit_gbm$logLik))
#> [1] 1667.530 1594.125

In this case, it seems the GBM model has better fit to the data, although this could be further backed up by performing an out-of-sample comparison of the two models.

Bayesian inference for STAR models

For a Bayesian model, STAR requires only an algorithm for initializing and sampling from the posterior distribution under a continuous data model. More specifically, posterior inference under STAR is based on a Gibbs sampler, which augments the aforementioned continuous sampler with a draw from \([z^* | y, \theta]\). When \(\Pi_\theta\) is conditionally Gaussian, \([z^* | y, \theta]\) is a truncated Gaussian distribution.

Linear Model

As an illustration, consider the Bayesian linear regression model \[\begin{align*} z_i^* &= x_i'\beta + \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim}N(0, \sigma^2) \\ \sigma^2 &\sim \mbox{Gamma}(\alpha=0.001, \beta=0.001) \;. \end{align*}\] The model is completed by assigning a prior structure for \(\beta\). The default in countSTAR is Zellner’s g-prior, which has the most functionality, but other options are available (namely horseshoe and ridge priors). The model is estimated using blm_star(). Note that for the Bayesian models in countSTAR, the user must supply the design matrix \(X\) (and if predictions are desired, a matrix of predictors at the test points). We apply this to the roaches data, now using the default nonparametric transformation.

X = model.matrix(y ~ roach1 + treatment + senior + log(exposure2),
                 data = roaches)

# Dimensions:
n = nrow(X); p = ncol(X)

fit_blm = blm_star(y = y, X=X, transformation = 'np')

By default, the function uses a Gibbs sampler to draw from the posterior, but in some settings exact inference can be performed (see Kowal and Wu (2022) for details). To enable this, one can simply set use_MCMC=FALSE. In either case, the output of the function should be much the same, with the exception that \(\sigma\) is estimated a priori and thus has no posterior draws.

Posterior expectations and posterior credible intervals from the model are available as follows:

# Posterior mean of each coefficient:
round(coef(fit_blm),3)
#>    (Intercept)         roach1      treatment         senior log(exposure2) 
#>          0.029          0.006         -0.288         -0.323          0.218

# Credible intervals for regression coefficients
ci_all_bayes = apply(fit_blm$post.beta,
      2, function(x) quantile(x, c(.025, .975)))

# Rename and print:
rownames(ci_all_bayes) = c('Lower', 'Upper')
print(t(round(ci_all_bayes, 3)))
#>                 Lower  Upper
#> (Intercept)    -0.149  0.209
#> roach1          0.005  0.008
#> treatment      -0.497 -0.086
#> senior         -0.562 -0.093
#> log(exposure2) -0.202  0.659

We may further evaluate the model based on posterior diagnostics and posterior predictive checks on the simulated versus observed proportion of zeros. Posterior predictive checks are easily visualized using the bayesplot package.

# MCMC diagnostics for posterior draws of the regression coefficients
plot(as.ts(fit_blm$post.beta), main = 'Trace plots', cex.lab = .75)


# (Summary of) effective sample sizes across coefficients:
getEffSize(fit_blm$post.beta)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>    3389    3583    3643    3797    3868    4501

# Posterior predictive check using bayesplot
suppressMessages(library(bayesplot))
prop_zero <- function(y) mean(y == 0)
(ppc_stat(y=roaches$y, yrep=fit_blm$post.pred, stat = "prop_zero"))

BART STAR

One of the most flexible model options is to use Bayesian Additive Regression Trees (BART; Chipman, George, and McCulloch (2012)) as the latent regression model:

#Get the model matrix of predictors (no intercept necessary)
X = model.matrix(y ~ -1 + roach1 + treatment + senior + exposure2,
                 data = roaches)

fit_bart = bart_star(y = y, X=X, transformation = 'np')
#> [1] "Burn-In Period"
#> [1] "7.12 seconds remaining"
#> [1] "1.74 seconds remaining"
#> [1] "Starting sampling"
#> [1] "33.32 seconds remaining"
#> [1] "25.82 seconds remaining"
#> [1] "18.64 seconds remaining"
#> [1] "8.53 seconds remaining"
#> [1] "Total time:  52 seconds"

Once again, we can perform posterior predictive checks. This time we plot the densities.

ppc_dens_overlay(y=roaches$y, yrep=fit_bart$post.pred[1:50,])

With all Bayesian models, pointwise log-likelihoods and WAIC values are outputted for model comparison. Using this information, we can see the BART STAR model seems to have a better fit than the linear model.

waic <- c(fit_blm$WAIC, fit_bart$WAIC)
names(waic) <- c("STAR w/ Linear Model", "STAR w/ BART")
print(waic)
#> STAR w/ Linear Model         STAR w/ BART 
#>             1687.829             1639.028

Other Models

countSTAR also implements Bayesian additive models. With the function bam_star(), we can specify covariates to be modeled linearly and others to be modeled non-linearly using spline bases. As a word of warning, the additive models do take significantly longer to fit. For exploring the relationship between one predictor and count outcome \(y\), one can also use spline regression as implemented in spline_star(). See the appropriate help pages for examples on how to run the models.

Count Time Series Modeling: warpDLM

Up to this point, all of the attention has focused on static regression where the data does not depend on time. However, the ideas of simultaneous transformation and rounding were extended to the time series domain in King and Kowal (2021). In that work, we proposed linking time-dependent count data \(y_t\) to a powerful time series framework known as Dynamic Linear Models (DLMs). A DLM is defined by two equations: (i) the observation equation, which specifies how the observations are related to the latent state vector and (ii) the state evolution equation, which describes how the states are updated in a Markovian fashion. More concretely, we represent it in the following form: \[\begin{align*} z_t &= F_t \theta_t + v_t, \quad v_t \sim N_n(0, V_t) \\ \theta_t &= G_t \theta_{t-1} + w_t, \quad w_t \sim N_p( 0, W_t) \end{align*}\] for \(t=1,\ldots, T\), where \(\{ v_t, w_t\}_{t=1}^T\) are mutually independent and \(\theta_0 \sim N_p(a_0, R_0)\).

Of course, given the Gaussian assumptions of the model, a DLM alone is not appropriate for count data. Thus, a warping operation (the simultaneous transformation and rounding) is applied to the DLM, resulting in the count time series framework known as a warped DLM (warpDLM). More explicitly, it can be written as: \[\begin{align*} y_t &= h \circ g^{-1}(z_t) \\ \{z_t\}_{t=1}^T &\sim \text{DLM} \end{align*}\]

The DLM form shown earlier is very general. To keep things simple, countSTAR implements only a few options for the DLM, namely the local level model and the local linear trend model. A local level model (also known as a random walk with noise model) has a univariate state \(\theta_t:=\mu_t\), and the DLM equations are simply \[\begin{align*} z_t &= \mu_t + v_t, \quad v_t \sim N(0, V) \\ \mu_t &= \mu_{t-1} + w_t, \quad w_t \sim N(0, W) \end{align*}\] The local linear trend model extends the local level model, incorporating a time varying drift \(\nu_t\) in the dynamics. It is often described in the following three equation format: \[\begin{align*} z_t &= \mu_t + v_t, \quad v_t \sim N(0, V) \\ \mu_t &= \mu_{t-1} + \nu_{t-1} + w_{\mu,t}, \quad w_{\mu,t} \sim N( 0, W_ \mu) \\ \nu_t &= \nu_{t-1} + w_{\nu,t}, \quad w_{\nu,t} \sim N( 0, W_\nu) \end{align*}\] This can in turn be recast into the general two-equation DLM form. Namely, if we let \(\theta_t:=(\mu_t, \nu_t)\), the local linear trend is written as

\[\begin{align*} z_t & = \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} \mu_t \\ \nu_t \end{pmatrix} + v_t, \quad v_t \sim N(0, V) \\ \begin{pmatrix} \mu_t \\ \nu_t \end{pmatrix} & = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{pmatrix} \mu_{t-1} \\ \nu_{t-1} \end{pmatrix} + \boldsymbol{w_t}, \quad \boldsymbol{w_t} \sim N\begin{pmatrix} \boldsymbol{0}, \begin{bmatrix} W_ \mu & 0 \\ 0 & W_\nu \end{bmatrix} \end{pmatrix} \end{align*}\]

These two common forms have a long history and are also referred to as structural time series models (implemented in base R via StructTS()). With countSTAR, warpDLM time series modeling is accomplished via the warpDLM() function. In the below, we apply the model to a time series dataset included in base R concerning yearly numbers of important discoveries from 1860 to 1959 (?discoveries for more information).

#Visualize the data
plot(discoveries)


#Fit the model
warpfit <- warpDLM(y=discoveries, type="trend")
#> [1] "Time taken:  65.56  seconds"

Once again, we can check fit using posterior predictive checks. The median of the posterior predictive draws can act as a sort of count-valued smoother of the time series.

ppc_ribbon(y=as.vector(discoveries), yrep=warpfit$post_pred)

Getting Started with countSTAR

Introduction