knitr::opts_chunk$set( message = FALSE, warning = FALSE, error = FALSE, tidy = FALSE, cache = FALSE )
We start with a simulated Poisson example where the $\Theta_i$ are drawn from a chi-squared density with 10 degrees of freedom and the $X_i|\Theta_i$ are Poisson with expectation $\Theta_i:$
$$ \Theta_i \sim \chi^2_{10} \mbox{ and } X_i|\Theta_i \sim \mbox{Poisson}(\Theta_i) $$
The $\Theta_i$ for this setting, with N = 1000
observations can be
generated as follows.
set.seed(238923) ## for reproducibility N <- 1000 Theta <- rchisq(N, df = 10)
Next, the $X_i|\Theta_i$, for each of nSIM = 1000
simulations can
be generated as below.
nSIM <- 1000 data <- sapply(seq_len(nSIM), function(x) rpois(n = N, lambda = Theta))
We take the discrete set $\mathcal{T}=(1, 2, \ldots, 32)$ as the
$\Theta$-space and apply the deconv
function in the package
deconvolveR
to estimate $g(\theta).$
library(deconvolveR) tau <- seq(1, 32) results <- apply(data, 2, function(x) deconv(tau = tau, X = x, ignoreZero = FALSE, c0 = 1))
The default setting for deconv
uses the Poisson
family and a
natural cubic spline basis of degree 5 as $Q.$ The regularization
parameter for this example (c0
) is set to 1. The ignoreZero
parameter indicates that this dataset contains zero counts, i.e.,
there zeros have not been truncated. (In the Shakespeare example
below, the counts are of words seen in the canon, and so there is a
natural truncation at zero.)
Some warnings are emitted by the nlm
routine used for optimization,
but they are mostly inconsequential.
Since deconv
works on a sample at a time, the result above is a list
of lists from which various statistics can be extracted. Below, we
construct a table of values for various values of $\Theta$.
g <- sapply(results, function(x) x$stats[, "g"]) mean <- apply(g, 1, mean) SE.g <- sapply(results, function(x) x$stats[, "SE.g"]) sd <- apply(SE.g, 1, mean) Bias.g <- sapply(results, function(x) x$stats[, "Bias.g"]) bias <- apply(Bias.g, 1, mean) gTheta <- pchisq(tau, df = 10) - pchisq(c(0, tau[-length(tau)]), df = 10) gTheta <- gTheta / sum(gTheta) simData <- data.frame(theta = tau, gTheta = gTheta, Mean = mean, StdDev = sd, Bias = bias, CoefVar = sd / mean) table1 <- transform(simData, gTheta = 100 * gTheta, Mean = 100 * Mean, StdDev = 100 * StdDev, Bias = 100 * Bias)
The table below summarizes the results for some chosen values of $\theta .$
knitr::kable(table1[c(5, 10, 15, 20, 25), ], row.names=FALSE)
Although, the coefficient of variation of $\hat{g}(\theta)$ is still large, the $g(\theta)$ estimates are reasonable.
We can compare the empirical standard deviations and biases of $g(\hat{\alpha})$ with the approximation given by the formulas in the paper.
library(ggplot2) library(cowplot) theme_set(theme_get() + theme(panel.grid.major = element_line(colour = "gray90", size = 0.2), panel.grid.minor = element_line(colour = "gray98", size = 0.5))) p1 <- ggplot(data = as.data.frame(results[[1]]$stats)) + geom_line(mapping = aes(x = theta, y = SE.g), color = "black", linetype = "solid") + geom_line(mapping = aes(x = simData$theta, y = simData$StdDev), color = "red", linetype = "dashed") + labs(x = expression(theta), y = "Std. Dev") p2 <- ggplot(data = as.data.frame(results[[1]]$stats)) + geom_line(mapping = aes(x = theta, y = Bias.g), color = "black", linetype = "solid") + geom_line(mapping = aes(x = simData$theta, y = simData$Bias), color = "red", linetype = "dashed") + labs(x = expression(theta), y = "Bias") plot_grid(plotlist = list(p1, p2), ncol = 2)
The approximation is quite good for the standard deviations, but a little too small for the biases.
Here we are given the word counts for the entire Shakespeare canon in
the data set bardWordCount
. We assume the $i$th distinct word
appeared $X_i \sim Poisson(\Theta_i)$ times in the canon.
data(bardWordCount) str(bardWordCount)
We take the support set $\mathcal{T}$ for $\Theta$ to be equally spaced on the log-scale and the sample space for $\mathcal{X}$ to be $(1,2,\ldots,100).$
lambda <- seq(-4, 4.5, .025) tau <- exp(lambda)
Using a regularization parameter of c0=2
we can deconvolve the data
to get $\hat{g}.$
result <- deconv(tau = tau, y = bardWordCount, n = 100, c0=2) stats <- result$stats
The plot below shows the Empirical Bayes deconvoluation estimates for the Shakespeare word counts.
ggplot() + geom_line(mapping = aes(x = lambda, y = stats[, "g"])) + labs(x = expression(log(theta)), y = expression(g(theta)))
The quantity $R(\alpha)$ in the paper (Efron, Biometrika 2015) can be
extracted from the stats
list; in this case for a regularization
parameter of c0=2
we can print its value:
print(result$S)
The stats
list contains other estimates quantities as well.
As noted in the paper citing this package, about
r 100 * round(stats[161, "G"], 2)
percent of the total mass of
$\hat{g}$ lies below $\Theta = 1$, which is an underestimate. This can
be corrected for by defining
$$
\tilde{g} = c_1\hat{g} / (1 - e^{-\theta_j}),
$$
where $c_1$ is the constant that normalizes $\tilde{g}$.
When there is truncation at zero, as is the case here, the
deconvolveR
package now returns an additional column in stats[,
"tg"]
which contains this correction for thinning. (The default
invocation of deconv
assumes zero truncation for the Poisson family,
argument ignoreZero = FALSE
).
d <- data.frame(lambda = lambda, g = stats[, "g"], tg = stats[, "tg"], SE.g = stats[, "SE.g"]) indices <- seq(1, length(lambda), 5) ggplot(data = d) + geom_line(mapping = aes(x = lambda, y = g)) + geom_errorbar(data = d[indices, ], mapping = aes(x = lambda, ymin = g - SE.g, ymax = g + SE.g), width = .01, color = "blue") + labs(x = expression(log(theta)), y = expression(g(theta))) + ylim(0, 0.006) + geom_line(mapping = aes(x = lambda, y = tg), linetype = "dashed", color = "red")
We can now plot the posterior estimates.
gPost <- sapply(seq_len(100), function(i) local({tg <- d$tg * result$P[i, ]; tg / sum(tg)})) plots <- lapply(c(1, 2, 4, 8), function(i) { ggplot() + geom_line(mapping = aes(x = tau, y = gPost[, i])) + labs(x = expression(theta), y = expression(g(theta)), title = sprintf("x = %d", i)) }) plots <- Map(f = function(p, xlim) p + xlim(0, xlim), plots, list(6, 8, 14, 20)) plot_grid(plotlist = plots, ncol = 2)
As a check, one can perform a parametric bootstrap using $$ y^{*} \sim Mult_n(N, {\mathbf f}) $$ with the MLE $\hat{\mathbf f} = {\mathbf f}(\hat{\alpha})$. We use 200 bootstrap replcates to compute the standard errors for $\hat{g}$ and compare them to the theoretical values in the plot below. The agreement is pretty good.
set.seed(1783) B <- 200 fHat <- as.numeric(result$P %*% d$g) fHat <- fHat / sum(fHat) yStar <- rmultinom(n = B, size = sum(bardWordCount), prob = fHat) gBoot <- apply(yStar, 2, function(y) deconv(tau = tau, y = y, n = 100, c0 = 2)$stats[, "g"]) seG <- apply(gBoot, 1, sd) ggplot(data = d) + geom_line(mapping = aes(x = lambda, y = SE.g, color = "Theoretical", linetype = "Theoretical")) + geom_line(mapping = aes(x = lambda, y = seG, color = "Bootstrap", linetype = "Bootstrap")) + scale_color_manual(name = "Legend", values = c("Bootstrap" = "black", "Theoretical" = "red")) + scale_linetype_manual(name = "Legend", values = c("Bootstrap" = "solid", "Theoretical" = "dashed")) + theme(legend.title = element_blank()) + labs(x = expression(log(theta)), y = expression(sigma(hat(g))))
Suppose then that a previously unknown Shakespearean corpus of length $t\times C$ were found, $C\times 900,000$ the length of the known canon. Assuming a Poisson process model with intensity $\Theta_i$ for word $i$, the probability that word $i$ did not appear in the canon but does appear in the new corpus is $$ e^{-\Theta_i}\left(1-e^{-\Theta_it}\right); $$ yielding after some work, an estimate for $R(t)$, the expected number of distinct new words found, divided by $N$, the observed number of distinct words in the canon: $$ R(t)=\sum_{j=1}^m\hat{g}jr_j(t), $$ with $$ r_j=\frac{e^{-\theta{(j)}}}{1-e^{-\theta_{(j)}}}\left(1-e^{-\theta_{(j)}t}\right). $$
We can compute the $R(t)$ and plot it as follows.
gHat <- stats[, "g"] Rfn <- function(t) { sum( gHat * (1 - exp(-tau * t)) / (exp(tau) - 1) ) } r <- sapply(0:10, Rfn) ggplot() + geom_line(mapping = aes(x = 0:10, y = r)) + labs(x = "time multiple t", y = expression(R(t)))
And the (speculative) doubling time for Shakespeare's vocabulary is easy to compute too.
print(uniroot(f = function(x) Rfn(x) - 1, interval = c(2, 4))$root)
Consider a data set that is generated via the following mechanism.
$$ z_i \sim N(\mu_i, 1), \mbox{ $i = 1,2,\ldots, N = 10,000$} $$
with
$$ \mu_i= \begin{cases} 0, & \mbox{with probability .9}\ N(-3, 1), & \mbox{with probability .1}\ \end{cases} $$
set.seed(129023) N <- 10000 pi0 <- .90 data <- local({ nullCase <- (runif(N) <= pi0) muAndZ <- t(sapply(nullCase, function(isNull) { if (isNull) { mu <- 0 c(mu, rnorm(1)) } else { mu <- rnorm(1, mean = -3) c(mu, rnorm(1, mean = mu)) } })) data.frame(nullCase = nullCase, mu = muAndZ[, 1], z = muAndZ[, 2]) })
Below is a histogram of the data ($z$ values) and that of the $\Theta$s.
p1 <- ggplot(mapping = aes(x = data$z)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..) ), color = "brown", bins = 60, alpha = 0.5) + labs(x = "z", y = "Density") p2 <- ggplot(mapping = aes(x = data$mu)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..) ), color = "brown", bins = 60, alpha = 0.5) + labs(x = expression(theta), y = "Density") plot_grid(plotlist = list(p1, p2), ncol = 2)
Now we deconvolve this using $\mathcal{T} = (-6, -5.75,\ldots, 3)$ and a spike at zero and a fifth-degree polynomial.
tau <- seq(from = -6, to = 3, by = 0.25) atomIndex <- which(tau == 0) result <- deconv(tau = tau, X = data$z, deltaAt = 0, family = "Normal", pDegree = 5)
The estimates and the standard errors of the penalized MLE $\hat{g}$
with c0 = 1
are shown below.
knitr::kable(result$stats)
Per the above table, the estimated probability of $\mu = 0$ is r round(result$stats[atomIndex, "g"], 3)
$\pm$ r round(result$stats[atomIndex, "SE.g"] , 3)
with a bias of
about r round(result$stats[atomIndex, "Bias.g"] , 3)
.
We can now plot the $g$-estimate removing the atom at 0.
gData <- as.data.frame(result$stats[-atomIndex, c("theta", "g")]) gData$g <- gData$g / sum(gData$g) ggplot(data = gData) + geom_line(mapping = aes(x = theta, y = g)) + geom_line(mapping = aes(x = theta, y = dnorm(theta, mean = -3)), color = "red") + labs(x = expression(theta), y = expression(g(theta)))
The density approximation is not accurate at all, however, the posterior estimates for the g's are similar to what one obtains by the Benjamini-Yekutieli procedure as shown below.
p <- pnorm(data$z) orderP <- order(p) p <- p[orderP]
## FCR q <- 0.05 R <- max(which(p <= seq_len(N) * q / N)) discIdx <- orderP[1:R] disc <- data[discIdx, ] cat("BY_q procedure discoveries", R, "cases,", sum(disc$nullCase), "actual nulls among them.\n")
alphaR <- 1 - R * q / N zAlpha <- qnorm(alphaR, lower.tail = FALSE) zMarker <- max(disc$z) xlim <- c(-7.6, 0.0) ylim <- c(-10, 0.0) BY.lo <- c(xlim[1] - zAlpha, xlim[2] - zAlpha) BY.up <- c(xlim[1] + zAlpha, xlim[2] + zAlpha) Bayes.lo <- c(0.5 * (xlim[1] - 3) - 1.96 / sqrt(2), 0.5 * (xlim[2] - 3) - 1.96 / sqrt(2)) Bayes.up <- c(0.5 * (xlim[1] - 3) + 1.96 / sqrt(2), 0.5 * (xlim[2] - 3) + 1.96 / sqrt(2))
d <- data[order(data$mu), ] muVals <- unique(d$mu) s <- as.data.frame(result$stats) indices <- findInterval(muVals, s$theta) + 1 gMu <- s$g[indices] st <- seq(min(data$z), -2.0, length.out = 40) gMuPhi <- sapply(st, function(z) gMu * dnorm(z - muVals)) g2 <- apply(gMuPhi, 2, function(x) cumsum(x)/sum(x)) pct <- apply(g2, 2, function(dist) approx(y = muVals, x = dist, xout = c(0.025, 0.975))) qVals <- sapply(pct, function(item) item$y)
ggplot() + geom_line(mapping = aes(x = xlim, y = BY.lo), color = "blue") + geom_line(mapping = aes(x = xlim, y = BY.up), color = "blue") + geom_line(mapping = aes(x = xlim, y = Bayes.lo), color = "magenta", linetype = "dashed") + geom_line(mapping = aes(x = xlim, y = Bayes.up), color = "magenta", linetype = "dashed") + geom_point(mapping = aes(x = disc$z, y = disc$mu), color = "red") + geom_point(mapping = aes(x = disc$z[disc$nullCase], y = disc$mu[disc$nullCase]), color = "orange") + geom_line(mapping = aes(x = rep(zMarker, 2), y = c(-10, 1))) + geom_line(mapping = aes(x = st, y = qVals[1, ]), color = "brown") + geom_line(mapping = aes(x = st, y = qVals[2, ]), color = "brown") + labs(x = "Observed z", y = expression(mu)) + annotate("text", x = -1, y = -4.25, label = "BY.lo") + annotate("text", x = -1, y = 1.25, label = "BY.up") + annotate("text", x = -7.5, y = -6.1, label = "Bayes.lo") + annotate("text", x = -7.5, y = -3.4, label = "Bayes.up") + annotate("text", x = -2.0, y = -1.75, label = "EB.lo") + annotate("text", x = -2.0, y = -3.9, label = "EB.up") + annotate("text", x = zMarker, y = 1.25, label = as.character(round(zMarker, 2)))
In the first normal example, the $\theta$ distribution had a significant atom at 0 and the rest of the density was smeared around -3. We now investigate what happens when the $\theta$ distribution is clearly bimodal. Below is a histogram of the the $\theta$ and alongside a histogram of the data, generated using $X_i \sim N(\theta_i, 1)$.
p1 <- ggplot(mapping = aes(x = disjointTheta)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..) ), color = "brown", bins = 60, alpha = 0.5) + labs(x = expression(theta), y = "Density") set.seed (2332) z <- rnorm(n = length(disjointTheta), mean = disjointTheta) p2 <- ggplot(mapping = aes(x = z)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..) ), color = "brown", bins = 60, alpha = 0.5) + labs(x = "z", y = "Density") plot_grid(plotlist = list(p1, p2), ncol = 2)
We deconvolve the data, using various values for the spline degrees of freedom.
tau <- seq(from = -4, to = 6, by = 0.2) plots1 <- lapply(2:8, function(p) { result <- deconv(tau = tau, X = z, family = "Normal", pDegree = p) g <- result$stats[, "g"] ggplot(mapping = aes(x = disjointTheta)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..)), color = "brown", bins = 60, alpha = 0.2) + geom_line(mapping = aes(x = tau, y = g), color = "blue") + labs(x = expression(theta), y = "Density", title = sprintf("DF = %d", p)) })
Choosing the degrees of freedom to be 7, we now examine the effect of regularization parameter $c0$.
plots2 <- lapply(c(0.5, 1, 2, 4, 8, 16, 32), function(c0) { result <- deconv(tau = tau, X = z, family = "Normal", pDegree = 6, c0 = c0) g <- result$stats[, "g"] ggplot(mapping = aes(x = disjointTheta)) + geom_histogram(mapping = aes(y = ..count.. / sum(..count..)), color = "brown", bins = 60, alpha = 0.2) + geom_line(mapping = aes(x = tau, y = g), color = "blue") + labs(x = expression(theta), y = "Density", title = sprintf("C0 = %.1f", c0)) }) plots <- mapply(function(x, y) list(x, y), plots1, plots2) plot_grid(plotlist = plots, ncol = 2)
The quantity $S(\alpha)$ is a measure of the strength of the penalty
term $c_0$ and is returned by deconv
in the result list as a named
item S
. Below we vary $c_0$ and degrees of freedom from 5 to 7 to
gauge the effect of the regularization.
c0_values <- c(.5, 1, 2, 4, 8, 16, 32) stable <- data.frame( c0 = c0_values, `DF 5` = sapply(c0_values, function(c0) deconv(tau = tau, X = z, family = "Normal", pDegree = 5, c0 = c0)$S), `DF 6` = sapply(c0_values, function(c0) deconv(tau = tau, X = z, family = "Normal", pDegree = 6, c0 = c0)$S), `DF 7` = sapply(c0_values, function(c0) deconv(tau = tau, X = z, family = "Normal", pDegree = 7, c0 = c0)$S) ) knitr::kable(stable)
A DF of 6 or 7 captures the bimodality and a choice of $c_0 < 4$ would avoid excessive penalization.
The dataset surg
contains data on intestinal surgery on 844 cancer
patients. In the study, surgeons removed satellite nodes for later
testing. The data consists of pairs $(n_i, X_i)$ where $n_i$ is the
number of satellites removed and $X_i$ is the number found to be
malignant among them.
We assume a binomial model with $X_i \sim Binomial(n_i, \theta_i)$ with $\theta_i$ being the probability of any one satellite site being malignant for the $i$th patient.
We take $\mathcal{T} = (0.01, 0.02,\ldots, 0.09)$, so $m = 99.$ We take $Q$ to be the default 5-degree natural spline with columns standardized to mean 0 and sum of squares equal to 1. The penalization parameter is set to 1. The figure below shows the estimated prior density of $g(\theta)$.
tau <- seq(from = 0.01, to = 0.99, by = 0.01) result <- deconv(tau = tau, X = surg, family = "Binomial", c0 = 1) d <- data.frame(result$stats) indices <- seq(5, 99, 5) errorX <- tau[indices] ggplot() + geom_line(data = d, mapping = aes(x = tau, y = g)) + geom_errorbar(data = d[indices, ], mapping = aes(x = theta, ymin = g - SE.g, ymax = g + SE.g), width = .01, color = "blue") + labs(x = expression(theta), y = expression(paste(g(theta), " +/- SE")))
The complete table of estimates and standard errors is also available.
knitr::kable(d[indices, ], row.names = FALSE)
The empirical Bayes estimate of the prior distribution puts most of its mass on the small values as can be seem below.
cat(sprintf("Mass below .20 = %0.2f\n", sum(d[1:20, "g"]))) cat(sprintf("Mass above .80 = %0.2f\n", sum(d[80:99, "g"])))
The posterior distribution of $\theta_i$ given $(n_i, X_i)$ is computed using Bayes rule as
$$ \hat{g} (\theta | X_i = x_i, n_i) = \frac{g_{\hat{\alpha}} (\theta) {n_i \choose x_i} \theta^{x_i} (1 - \theta)^{n_i-x_i}} {f_{\hat{\alpha}}(n_i, x_i)} $$
where the denominator is given by
$$ f_\alpha(n_i, x_i) = \int_0^1{n_i \choose x_i} \theta^{x_i}(1-\theta)^{n_i-x_i}g_\alpha(\theta)\,d\theta. $$
with the mle $\hat{\alpha}$ in place of $\alpha$.
Since $g(\theta)$ is discrete, the integrals are mere sums as shown below.
theta <- result$stats[, 'theta'] gTheta <- result$stats[, 'g'] f_alpha <- function(n_k, x_k) { ## .01 is the delta_theta in the Riemann sum sum(dbinom(x = x_k, size = n_k, prob = theta) * gTheta) * .01 } g_theta_hat <- function(n_k, x_k) { gTheta * dbinom(x = x_k, size = n_k, prob = theta) / f_alpha(n_k, x_k) }
We plot a few posterior distributions.
g1 <- g_theta_hat(x_k = 7, n_k = 32) g2 <- g_theta_hat(x_k = 3, n_k = 6) g3 <- g_theta_hat(x_k = 17, n_k = 18) ggplot() + geom_line(mapping = aes(x = theta, y = g1), col = "magenta") + ylim(0, 10) + geom_line(mapping = aes(x = theta, y = g2), col = "red") + geom_line(mapping = aes(x = theta, y = g3), col = "blue") + labs(x = expression(theta), y = expression(g(paste(theta, "|(x, n)")))) + annotate("text", x = 0.15, y = 4.25, label = "x=7, n=32") + annotate("text", x = 0.425, y = 4.25, label = "x=3, n=6") + annotate("text", x = 0.85, y = 7.5, label = "x=17, n=18")
The empirical Bayes posterior estimate $\theta^{EB}$ for patients 34, 40, and 679 in the dataset for example is
cat(sprintf("Empirical Bayes Estimate: %f\n", 0.01 * sum(theta * g2)))
As a check on the estimates of standard error and bias provided by
deconv
, we compare the results with what we obtain using a
parametric boostrap.
The boostrap is run as follows. For each of 1000 runs, 844 simulated
realizations $\hat{\Theta}^$ are sampled from density $\hat{g}.$ Each
gave an $X_i^ \sim Binomial(n_i, \Theta_i^*)$ with $n_i$ the $i$th
sample in the original data set. Finally, $\hat{\alpha}$ was computed
using deconv
.
set.seed(32776) B <- 1000 gHat <- d$g N <- nrow(surg) genBootSample <- function() { thetaStar <- sample(tau, size = N, replace = TRUE, prob = gHat) sStar <- sapply(seq_len(N), function(i) rbinom(n = 1 , size = surg$n[i], prob = thetaStar[i])) data.frame(n = surg$n, s = sStar) } bootResults <- lapply(seq_len(B), function(k) { surgBoot <- genBootSample() mat <- deconv(tau = tau, X = surgBoot, family = "Binomial", c0 = 1)$stats mat[, c("g", "Bias.g")] }) gBoot <- sapply(bootResults, function(x) x[, 1]) BiasBoot <- sapply(bootResults, function(x) x[, 2]) indices <- c(seq(1, 99, 11), 99) table2 <- data.frame(theta = tau, gTheta = round(gHat * 100, 3), sdFormula = round(d$SE.g * 100, 3), sdSimul = round(apply(gBoot, 1, sd) * 100, 3), BiasFormula = round(d$Bias.g * 100, 3), BiasSimul = round(apply(BiasBoot, 1, mean) * 100, 3))[ indices, ]
We print out some estimated quantities for comparison.
knitr::kable(table2, row.names = FALSE)
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