# Positive Manifold (Sign Restriction)

# Introduction

For some research questions, there might be expectations in regards to the directions of the edges. For example, in symptom networks, all relations are often hypothesized to be positive (i.e., positive manifold). In turn, any negative relations are thought to be spurious.

In GGMncv, it is possible to estimate the conditional dependence structure, given that all edges in the graph are positive (sign restriction).

## Packages

    library(GGMncv)
    library(corrplot)

## Correlations

The `ptsd` dataset includes 20 post-traumatic stress symptoms. The following visualizes the correlation matrix:

    corrplot::corrplot(cor(ptsd), method = "shade")

Notice that all of the correlations are positive.

## Partial Correlations

Here are the partial correlations:

    pcors <- -cov2cor(solve(cor(ptsd))) + diag(ncol(ptsd))

    corrplot::corrplot(pcors,
                       method = "shade")

Notice that some relations went to essentially zero (white), whereas other changed direction altogether.

## GGM

Here the conditional dependence structure is selected via `ggmncv`:

    # fit model
    fit <- GGMncv::ggmncv(cor(ptsd),
                          n = nrow(ptsd),
                          progress = FALSE,
                          penalty = "atan")

    # plot graph
    plot(GGMncv::get_graph(fit),
         edge_magnify = 10,
         node_names = colnames(ptsd))

Notice a few negatives are included in the graph.

## Sign Restriction

Here the graph is re-estimated, with the constraint that all of negative edges in the above plot are actually zero.

    # set negatives to zero (sign restriction)
    adj_new <- ifelse(fit$P <= 0, 0, 1)

    check_zeros <- TRUE

    # track trys
    iter <- 0

    # iterate until all positive
    while(check_zeros){
      iter <- iter + 1
      fit_new <- GGMncv::constrained(cor(ptsd), adj = adj_new)
      check_zeros <- any(fit_new$wadj < 0)
      adj_new <- ifelse(fit_new$wadj <= 0, 0, 1)
    }

    # make graph object
    new_graph <- list(P = fit_new$wadj,
                      adj = adj_new)
    class(new_graph) <- "graph"

    # plot graph
    plot(new_graph,
         edge_magnify = 10,
         node_names = colnames(ptsd))

The graph now only includes positive edges. Note this is not the same as simply removing the negative relations, as, in this case, this is the maximum likelihood estimate for the inverse covariance matrix.

Note also `new_graph` is making the graph class so that it can be plotted with `plot`.

## Alternative Approach

The above essentially takes the selected graph, and then re-estimates it with the constraint that the negative edges are zero. Perhaps a more sophisticated approach is to select the graph with those constraints.

This can be implemented with:

    R <- cor(ptsd)
    n <- nrow(ptsd)
    p <- ncol(ptsd)

    # store fitted models
    fit <- ggmncv(R = R,
                  n = n,
                  progress = FALSE,
                  store = TRUE,
                  n_lambda = 50)

    # all fitted models
    # sol: solution
    sol_path <- fit$fitted_models

    # storage
    bics <- NA
    Thetas <- list()

    for(i in seq_along(sol_path)){

      # positive in wi is a negative partial
      adj_new <- ifelse(sol_path[[i]]$wi >= 0, 0, 1)

      check_zeros <- TRUE

      # track trys
      iter <- 0

      # iterate until all positive
      while(check_zeros){
        iter <- iter + 1
        fit_new <- GGMncv::constrained(R, adj = adj_new)
        check_zeros <- any(fit_new$wadj < 0)
        adj_new <- ifelse(fit_new$wadj <= 0, 0, 1)
    }

      bics[i] <- GGMncv:::gic_helper(
        Theta = fit_new$Theta,
        R = R,
        n = n,
        p = p,
        type = "bic",
        edges = sum(fit_new$Theta[upper.tri(fit_new$Theta)] != 0)
      )

      Thetas[[i]] <- fit_new$Theta
    }

    # select via minimizing bic
    # (then convert to partial correlatons)
    pcors <- -(cov2cor(Thetas[[which.min(bics)]]) - diag(p))

    # make graph class
    new_graph <- list(P = pcors,
                      adj = ifelse(pcors == 0, 0, 1))
    class(new_graph) <- "graph"

    # plot graph
    plot(new_graph,
         edge_magnify = 10,
         node_names = colnames(ptsd))