Calculate the optimal parameters for a mixture of Weibull Extreme Value Distributions for supervised PCA

Calculate the parameters which minimise the negative log- likelihood of a mixture of two Weibull Extreme Value distributions.

OptimGumbelMixParams(
  max_tControl_vec,
  pathwaySize_vec,
  initialVals = c(p = 0.5, mu1 = 1, s1 = 0.5, mu2 = 1, s2 = 0.5),
  optimMethod = "L-BFGS-B",
  lowerBD = c(0, -Inf, 0, -Inf, 0),
  upperBD = c(1, Inf, Inf, Inf, Inf)
)

Arguments

max_tControl_vec	A vector of the maximum absolute \(t\)-scores for each pathway (returned by the pathway_tControl function) when under the null model. Under the null model, the response vector will have been randomly generated or parametrically bootstrapped.
pathwaySize_vec	A vector of the number of genes in each pathway.
initialVals	A named vector of initial values for the Weibull parameters. The values are \(p\) : The mixing proportion between the Gumbel minimum and Gumbel maximum distributions. This parameter is bounded by \([0, 1]\) and defaults to 0.5. \(\mu_1\) : The mean of the first distribution. This parameter is unbounded and defaults to 1. \(s_1\) : The precision of the first distribution. This parameter is bounded below by 0 and defaults to 0.5. \(\mu_2\) : The mean of the second distribution. This parameter is unbounded and defaults to 1. \(s_2\) : The precision of the second distribution. This parameter is bounded below by 0 and defaults to 0.5.
optimMethod	Which numerical optimization routine to pass to the optim function. Defaults to "L-BFGS-B", which allows for lower and upper bound constraints. When this option is specified, lower and upper bounds for ALL parameters must be supplied.
lowerBD	A vector of the lower bounds on the initialVals. Defaults to c(0, -Inf, 0, -Inf, 0).
upperBD	A vector of the upper bounds on the initialVals. Defaults to c(1, Inf, Inf, Inf, Inf).

Value

A named vector of the estimated values for the parameters which minimize the negative log-likelihood of the mixture Weibull Extreme Value distributions.

Details

The likelihood function is equation (4) in Chen et al (2008): a mixture of two Gumbel Extreme Value probability density functions, with mixing proportion \(p\). Within the code of this function, the values mu1, mu2 and s1, s2 are placeholders for the mean and precision, respectively.

A computational note: the "L-BFGS-B" option within the optim function requires a bounded function or likelihood. We therefore replaced Inf with 10 ^ 200 in the check for boundedness. As we are attempting to minimise the negative log- likelihood, this maximum machine value is effectively +Inf.

See https://doi.org/10.1093/bioinformatics/btn458 for more information.

See also

optim; GumbelMixpValues; pathway_tControl; SuperPCA_pVals

Examples

  # DO NOT CALL THIS FUNCTION DIRECTLY.
  # Use SuperPCA_pVals() instead.

if (FALSE) {
  ###  Load the Example Data  ###
  data("colon_pathwayCollection")


  ###  Simulate Maximum Absolute Control t-Values  ###
  # The SuperPCA algorithm defaults to 20 threshold values; the example
  #   pathway collection has 15 pathways.
  t_mat <- matrix(rt(15 * 20, df = 5), nrow = 15)

  absMax <- function(vec){
    vec[which.max(abs(vec))]
  }
  tAbsMax_num <- apply(t_mat, 1, absMax)


  ###  Calculate Optimal Parameters for the Gumbel Distribution  ###
  OptimGumbelMixParams(
    max_tControl_vec = tAbsMax_num,
    pathwaySize_vec = lengths(colon_pathwayCollection$pathways)
  )
}