Interaction importance

[RoelVerbelen]

I think it would be useful to have a function that computes/visualises the relative importance of interaction effects.

Here's an example for an xgboost model where SHAP interaction values are available:

library(shapviz)
library(tidyverse)
library(xgboost)

set.seed(3653)
x <- c("carat", "cut", "color", "clarity")
dtrain <- xgb.DMatrix(data.matrix(diamonds[x]), label = diamonds$price)
fit <- xgb.train(params = list(learning_rate = 0.1), data = dtrain, nrounds = 65)

# Explanation data
dia_small <- diamonds[sample(nrow(diamonds), 2000), ]

# shapviz object with SHAP interaction values
shp_i <- shapviz(
  fit, X_pred = data.matrix(dia_small[x]), X = dia_small, interactions = TRUE
)

# Interaction importance
shap_interactions <- apply(2 * abs(shp_i$S_inter), c(2, 3), mean)
shap_interactions[lower.tri(shap_interactions, diag = TRUE)] <- NA
as.data.frame.table(shap_interactions, responseName = "interaction_strength") %>% 
  filter(!is.na(interaction_strength)) %>% 
  arrange(desc(interaction_strength))
#>    Var1    Var2 interaction_strength
#> 1 carat clarity            600.07087
#> 2 carat   color            412.44253
#> 3 color clarity            188.35864
#> 4 carat     cut             98.98317
#> 5   cut clarity             23.92846
#> 6   cut   color             17.94669

^{Created on 2023-10-24 with reprex v2.0.2}

Ideally, this function would also work, based on some heuristics, for models that don't have SHAP interaction values available. I don't think using the heuristics in potential_interactions() (weighted squared correlations) willl work here as it doesn't take the amount of variation of the SHAP values in each bin into account, so the current interaction importance values are not comparable across features.

Maybe switching to the modelled part of the variation would work (and note that this also addresses #119): in each bin, fit a linear regression model and compute the mean of the absolute values of the fitted values minus the overall mean. I believe this boils down to the SHAP importance metric for a linear regression model with one feature. Doing so brings it on a scale that's comparable across bins and across features (differente vs in potential_interactions()).

Here's a code example to illustrate what I mean more clearly:

# shapviz object without interactions
shp <- shapviz(
  fit, X_pred = data.matrix(dia_small[x]), X = dia_small, interactions = FALSE
)

# Replace correlation measure with modelled variation measure
# Swapping out the function `r_sq` with `mod_var` 
potential_interactions_modelled <- function(obj, v) {
  stopifnot(is.shapviz(obj))
  S <- get_shap_values(obj)
  S_inter <- get_shap_interactions(obj)
  X <- get_feature_values(obj)
  nms <- colnames(obj)
  v_other <- setdiff(nms, v)
  stopifnot(v %in% nms)
  
  if (ncol(obj) <= 1L) {
    return(NULL)
  }
  
  # Simple case: we have SHAP interaction values
  if (!is.null(S_inter)) {
    return(sort(2 * colMeans(abs(S_inter[, v, ]))[v_other], decreasing = TRUE))
  }
  
  # Complicated case: we need to rely on modelled variation based heuristic
  mod_var <- function(s, x) {
    sapply(x, 
           function(x) {
             tryCatch({
               mean(abs(stats::lm(s ~ x)$fitted - mean(s)))
             }, error = function(e) {
               return(NA)
             })
           })
  }
  n_bins <- ceiling(min(sqrt(nrow(X)), nrow(X) / 20))
  v_bin <- shapviz:::.fast_bin(X[[v]], n_bins = n_bins)
  s_bin <- split(S[, v], v_bin)
  X_bin <- split(X[v_other], v_bin)
  w <- do.call(rbind, lapply(X_bin, function(z) colSums(!is.na(z))))
  modelled_variation <- do.call(rbind, mapply(mod_var, s_bin, X_bin, SIMPLIFY = FALSE))
  sort(colSums(w * modelled_variation, na.rm = TRUE) / colSums(w), decreasing = TRUE)
}

# Current implementation with SHAP interaction values
potential_interactions(shp_i, v = "cut")
#>    carat  clarity    color 
#> 98.98315 23.92846 17.94669

# Current implementation based on heuristics 
potential_interactions(shp, v = "cut")
#>      carat    clarity      color 
#> 0.49739669 0.07223855 0.04243011

# Suggested implementation based on heuristics 
potential_interactions_modelled(shp, v = "cut")
#>    carat  clarity    color 
#> 35.23818 14.73922 10.66934

# Current implementation with SHAP interaction values
potential_interactions(shp_i, v = "carat")
#>   clarity     color       cut 
#> 600.07087 412.44253  98.98317

# Current implementation based on heuristics 
potential_interactions(shp, v = "carat")
#>   clarity     color       cut 
#> 0.5301601 0.1545190 0.1121987

# Suggested implementation based on heuristics 
potential_interactions_modelled(shp, v = "carat")
#>  clarity    color      cut 
#> 248.3854 177.3795 132.6098

# Function to create table with ranked interaction variables
table_potential_interactions <- function(predictor) {
  pi <- potential_interactions_modelled(shp, predictor)
  tibble(var1 = predictor, var2 = names(pi), interaction_strength = pi)
}

# Interaction importance
map(x, table_potential_interactions) %>% 
  bind_rows() %>% 
  arrange(desc(interaction_strength))
#> # A tibble: 12 × 3
#>    var1    var2    interaction_strength
#>    <chr>   <chr>                  <dbl>
#>  1 carat   clarity                248. 
#>  2 clarity carat                  186. 
#>  3 carat   color                  177. 
#>  4 carat   cut                    133. 
#>  5 color   carat                  128. 
#>  6 clarity color                   89.3
#>  7 color   clarity                 55.6
#>  8 clarity cut                     48.7
#>  9 cut     carat                   35.2
#> 10 color   cut                     25.0
#> 11 cut     clarity                 14.7
#> 12 cut     color                   10.7

Note that this analysis is not symmetric, but I don't think that's an issue as the table above is informative: it suggests you to split out var1 effects by var2 and hence look at PD plots or SHAP dependence plots for var1 by different segments of var2.

[mayer79]

Great stuff, thanks a lot. Regarding the first part, we already have:

sv_interaction(shp_i, kind = "no")

#              carat   clarity     color       cut
# carat   3034.55635 600.07087 412.44253  98.98317
# clarity  600.07089 631.56112 188.35863  23.92845
# color    412.44249 188.35864 420.76788  17.94669
# cut       98.98315  23.92846  17.94669 110.39928

sv_interaction(..., kind = "bar") does not exist yet and could be used to make some sort of barplot, e.g., in the form "a:b", "a:c" etc., similar to the pairwise interaction plot in https://github.com/mayer79/hstats

[RoelVerbelen]

Thanks @mayer79!

sv_interaction(..., kind = "bar") would be a great and intuitive implementation for this exhibit. Ideally, I'd like it to work for models that don't have SHAP interaction values as well, by relying on the heuristic.

[mayer79]

SHAP interactions are additive and fair, just like normal SHAP values. I currently don't want to do as if our heuristics would satisfy any of these properties. We might pick up the idea later, though.

[RoelVerbelen]

That's fair, thanks for considering!