Author: Jonatan Kallus
Date: 2017-01-17

# A Brief Introduction to ROPE

#### Jonatan Kallus

#### 2017-01-17

This is a brief introduction to the use of ROPE (resampling of penalized estimates). For a mathematical description of the methods, and a comprehensive simulation study, I refer to the soon to be available article. ROPE performs edge selection with controlled false detection rate in graphical modeling of high-dimensional data.

ROPE models variable selection counts from some consistent method for variable selection, that has been applied to random subsamples of a data matrix. Methods for variable selection in high-dimensional problems have a regularization parameter that tunes the size of the penalty for letting additional variables enter the model. By varying the penalty for each subsample we get a matrix `W` of selection counts. Each column corresponds to a variable and each row corresponds to a level of penalization. Rows should be ordered from lowest to highest penalization, and the sequence of penalizations should be linearly spaced. All values in `W` be non-negative and at most equal to the number of resamples that were performed. The method produces a q-value for each variable, so that selecting all variable with q-value less than 0.05 will yield a selection where approximately 0.05 of the selected variables are false positives.

## An example

Let `X` be a matrix of `n` observations (rows) and `p` variables (columns). Suppose we want to select a graphical model for `X` that has an edge between to variables if they are significantly correlated given all other variables. In this setting we must separate data variables from model variables. I.e. the graphical model that we are estimating have \(d=p(p-1)/2\) variables. Let us call the model variables edges hereafter. The following code uses `glasso` as variable selection method to construct `W`.

    lambda <- seq(0.05, 0.5, 0.025)
    B <- 500
    n <- dim(x)[1]
    p <- dim(x)[2]

    W <- matrix(0, length(lambda), p*(p-1)/2)
    for (i in 1:B) {
      bootstrap <- sample(n, n, replace=TRUE)
      for (j in 1:length(lambda)) {
        selection <- glasso::glasso(cov(x[bootstrap, ]), lambda[j])
        selection <- sign(abs(selection$wi) + t(abs(selection$wi)))
        selection <- selection[upper.tri(selection)]
        W[j, ] <- W[j, ] + selection
      }
    }

Now, `W` contains in each column the number of times an edge was selected for each of the penalty steps. The above is just one way to construct `W`, ROPE is applicable also for other selection methods and other kinds of models. Before using `W` to make an FDR controlled selection, we need to find a penalization interval where the distribution of counts for variables that should not be selected is separated from the distribution of counts for variables that should be selected. ROPE supplies the `explore` function to find such a range.

    install.packages('rope')
    result <- rope::explore(W, B)

This will construct a histogram for each level of penalization, check which histograms that are U-shaped and estimate how separated the distributions are for each level. `explore` returns estimates of separation for each level of penalization until it reaches a penalty level where the histogram is not U-shaped. Now, the user needs to find a range of penalization that ends at the highest level for which histograms are U-shaped, and starts at a location such that the separation has one approximate maximum.

    plot(result$pop.sep)

Let us say that we found such a range to be level with indices 5 to 15. Then we apply `rope` to these counts.

    selected.indices <- 5:15
    lambda <- lambda[selected.indices]
    W <- W[selected.indices, ]
    result <- rope::rope(W, B)

Now, `result` contains q-values for each edge. If we are interested in which edges that should be selected at an FDR of approximately 0.1, we check for q-values below 0.1.

    selected.edges <- result$q < 0.1

This concludes a basic example of the use of `rope` for FDR controlled variable selection. It is recommended to use `rope::plotrope` to examine the results of `rope::explore` and `rope::rope` to make sure that the statistical model of selection counts fits the supplied data.

## Differently structured data

ROPE is well suited to select graphical models. For such models, it is natural to store variables (edges) as a matrix rather than as vector, to keep track of the pair of nodes that each edge connects. For this reason, `rope` contains convenience wrappers `rope::exploregraph` and `rope::ropegraph`. They work just like `explore` and `rope`, but instead of our `length(lambda)` times `p*(p-1)/2` matrix `W`, they take a list of the same length as `lambda` of symmetric `p` times `p` matrices. Furthermore, this package contains the functions `symmetric.matrix2vector` and `vector2symmetric.matrix` to convert between these two ways of storing variable selection counts.