---
title: "Positive Manifold (Sign Restriction)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Positive Manifold (Sign Restriction)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# 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
```{r setup, warning=FALSE, message=FALSE}
library(GGMncv)
library(corrplot)
```

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

```{r}
corrplot::corrplot(cor(ptsd), method = "shade")
```

Notice that all of the correlations are positive.

## Partial Correlations
Here are the partial correlations:

```{r}
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`:

```{r, message=FALSE, warning=FALSE}
# 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.

```{r}
# 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}
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))
```
