R: Empirical Bayes thresholding on a sequence

ebayesthresh {EbayesThresh} | R Documentation

## Empirical Bayes thresholding on a sequence

### Description

Given a sequence of data, performs Empirical Bayes thresholding, as discussed in Johnstone and Silverman (2004).

### Usage

    ebayesthresh(x, prior = "laplace", a = 0.5, bayesfac = FALSE,
    sdev = NA, verbose = FALSE, threshrule = "median", universalthresh = TRUE,
    stabadjustment)

### Arguments

`x` | Vector of data values.
`prior` | Specification of prior to be used conditional on the mean being nonzero; can be `"cauchy"` or `"laplace"`.
`a` | Scale factor if Laplace prior is used. Ignored if Cauchy prior is used. If, on entry, `a = NA` and `prior = "laplace"`, then the scale parameter will also be estimated by marginal maximum likelihood. If `a` is not specified then the default value 0.5 will be used.
`bayesfac` | If `bayesfac = TRUE`, then whenever a threshold is explicitly calculated, the Bayes factor threshold will be used.
`sdev` | The sampling standard deviation of the data `x`. If, on entry, `sdev = NA`, then the standard deviation will be estimated using the median absolute deviation from zero, as `mad(x, center = 0)`. If a single value is passed to `sdev`, sampling standard deviation will be the same for all observations. A vector of the same length as data sequence can be passed to allow heterogeneous standard deviation, currently only for Laplace prior.
`verbose` | Controls the level of output. See below.
`threshrule` | Specifies the thresholding rule to be applied to the data. Possible values are `"median"` (use the posterior median); `"mean"` (use the posterior mean); `"hard"` (carry out hard thresholding); `"soft"` (carry out soft thresholding); `"none"` (find various parameters, but do not carry out any thresholding).
`universalthresh` | If `universalthresh = TRUE`, the thresholds will be upper bounded by universal threshold; otherwise, the thresholds can take any non-negative values.
`stabadjustment` | If `stabadjustment = TRUE`, the vectors of standard deviations and data values will be divided by the mean of standard deviations in case of inefficiency caused by large value of standard deviation. For heterogeneous sampling standard deviation only; ignored if standard deviation is homogeneous.

### Details

It is assumed that the data vector (x_1, …, x_n) is such that each x_i is drawn independently from a normal distribution with mean θ_i and variance σ_i^2 (σ_i is the same in the homogeneous case). The prior distribution of each θ_i is a mixture with probability 1-w of zero and probability w of a given symmetric heavy-tailed distribution. The mixing weight, and possibly a scale factor in the symmetric distribution, are estimated by marginal maximum likelihood. The resulting values are used as the hyperparameters in the prior.

The true effects θ_i can be estimated as the posterior median or the posterior mean given the data, or by hard or soft thresholding using the posterior median threshold. If hard or soft thresholding is chosen, then there is the additional choice of using the Bayes factor threshold, which is the value such thatthe posterior probability of zero is exactly half if the data value is equal to the threshold.

### Value

If `verbose = FALSE`, a vector giving the values of the estimates of the underlying mean vector.

If `verbose = TRUE`, a list with the following elements:

`muhat` | the estimated mean vector (omitted if `threshrule = "none"`)
`x` | the data vector as supplied
`threshold.sdevscale` | the threshold as a multiple of the standard deviation `sdev`
`threshold.origscale` | the threshold measured on the original scale of the data
`prior` | the prior that was used
`w` | the mixing weight as estimated by marginal maximum likelihood
`a` | (only present if Laplace prior used) the scale factor as supplied or estimated
`bayesfac` | the value of the parameter `bayesfac`, determining whether Bayes factor or posterior median thresholds are used
`sdev` | the standard deviations of the data as supplied or estimated
`threshrule` | the thresholding rule used, as specified above

### Author(s)

Bernard Silverman

### References

Johnstone, I. M. and Silverman, B. W. (2004) Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. Annals of Statistics, 32, 1594–1649.

Johnstone, I. M. and Silverman, B. W. (2004) EbayesThresh: R software for Empirical Bayes thresholding. Journal of Statistical Software, 12.

Johnstone, I. M. (2004) 'Function Estimation and Classical Normal Theory' ‘The Threshold Selection Problem’. The Wald Lectures I and II, 2004. Available from http://www-stat.stanford.edu/~imj.

Johnstone, I. M. and Silverman, B. W. (2005) Empirical Bayes selection of wavelet thresholds. Annals of Statistics, 33, 1700–1752.

The papers by Johnstone and Silverman are available from http://www.bernardsilverman.com.

See also http://www-stat.stanford.edu/~imj for further references, including the draft of a monograph by I. M. Johnstone.

### See Also

`tfromx`, `threshld`

### Examples

    # Data with homogeneous sampling standard deviation using
    # Cauchy prior.
    eb1 <- ebayesthresh(x = rnorm(100, c(rep(0,90),rep(5,10))),
                         prior = "cauchy", sdev = NA)

    # Data with homogeneous sampling standard deviation using
    # Laplace prior.
    eb2 <- ebayesthresh(x = rnorm(100, c(rep(0,90), rep(5,10))),
                         prior = "laplace", sdev = 1)

    # Data with heterogeneous sampling standard deviation using
    # Laplace prior.
    set.seed(123)
    mu <- c(rep(0,90), rep(5,10))
    sd <- c(rep(1, 40), rep(3, 60))
    x  <- mu + rnorm(100, sd = sd)

    # With constraints on thresholds.
    eb3 <- ebayesthresh(x = x, prior = "laplace", a = NA, sdev = sd)

    # Without constraints on thresholds. Observe that the estimates with
    # constraints on thresholds have fewer zeroes than the estimates without
    # constraints.
    eb4 <- ebayesthresh(x = x, prior = "laplace", a = NA, sdev = sd,
                         universalthresh = FALSE)
    print(sum(eb3 == 0))
    print(sum(eb4 == 0))

    # Data with heterogeneous sampling standard deviation using Laplace
    # prior.
    set.seed(123)
    mu <- c(rep(0,90), rep(5,10))
    sd <- c(rep(1, 40), rep(5,40), rep(15, 20))
    x  <- mu + rnorm(100, sd = sd)

    # In this example, infinity is returned as estimate when some of the
    # sampling standard deviations are extremely large. However, this can
    # be solved by stabilizing the data sequence before the analysis.
    eb5 <- ebayesthresh(x = x, prior = "laplace", a = NA, sdev = sd)

    # With stabilization.
    eb6 <- ebayesthresh(x = x, prior = "laplace", a = NA, sdev = sd,
                        stabadjustment = TRUE)

[Package EbayesThresh version 1.4-12 Index]