Simulated data, real problem

Simulated data

Let’s consider a following problem, the model is defined as

\[ y = x_1 * x_2 + x_2 \]

But \(x_1\) and \(x_2\) are correlated. How XAI methods work for such model?

# predict function for the model
the_model_predict <- function(m, x) {
 x$x1 * x$x2 + x$x2
}

# correlated variables 
N <- 50
set.seed(1)
x1 <- runif(N, -5, 5)
x2 <- x1 + runif(N)/100
df <- data.frame(x1, x2)

Explainer for the models

In fact this model is defined by the predict function the_model_predict. So it does not matter what is in the first argument of the explain function.

library("DALEX")
explain_the_model <- explain(1,
                      data = df,
                      predict_function = the_model_predict)

#> Preparation of a new explainer is initiated
#>   -> model label       :  numeric  (  default  )
#>   -> data              :  50  rows  2  cols 
#>   -> target variable   :  not specified! (  WARNING  )
#>   -> predict function  :  the_model_predict 
#>   -> predicted values  :  No value for predict function target column. (  default  )
#>   -> model_info        :  package Model of class: numeric package unrecognized , ver. Unknown , task regression (  default  ) 
#>   -> model_info        :  Model info detected regression task but 'y' is a NULL .  (  WARNING  )
#>   -> model_info        :  By deafult regressions tasks supports only numercical 'y' parameter. 
#>   -> model_info        :  Consider changing to numerical vector.
#>   -> model_info        :  Otherwise I will not be able to calculate residuals or loss function.
#>   -> predicted values  :  numerical, min =  -0.1726853 , mean =  7.70239 , max =  29.16158  
#>   -> residual function :  difference between y and yhat (  default  )
#>   A new explainer has been created!

Ceteris paribus

Use the ceteris_paribus() function to see Ceteris Paribus profiles. Clearly it’s not an additive model, as the effect of \(x_1\) depends on \(x_2\).

library("ingredients")
library("ggplot2")

sample_rows <- data.frame(x1 = -5:5,
                          x2 = -5:5)

cp_model <- ceteris_paribus(explain_the_model, sample_rows)
plot(cp_model) +
  show_observations(cp_model) +
  ggtitle("Ceteris Paribus profiles")

Dependence profiles

Lets try Partial Dependence profiles, Conditional Dependence profiles and Accumulated Local profiles. For the last two we can try different smoothing factors

pd_model <- partial_dependence(explain_the_model, variables = c("x1", "x2"))
pd_model$`_label_` = "PDP"

cd_model <- conditional_dependence(explain_the_model, variables = c("x1", "x2"))
cd_model$`_label_` = "CDP 0.25"

ad_model <- accumulated_dependence(explain_the_model, variables = c("x1", "x2"))
ad_model$`_label_` = "ALE 0.25"

plot(ad_model, cd_model, pd_model) +
  ggtitle("Feature effects - PDP, CDP, ALE")

cd_model_1 <- conditional_dependence(explain_the_model, variables = c("x1", "x2"), span = 0.1)
cd_model_1$`_label_` = "CDP 0.1"

cd_model_5 <- conditional_dependence(explain_the_model, variables = c("x1", "x2"), span = 0.5)
cd_model_5$`_label_` = "CDP 0.5"

ad_model_1 <- accumulated_dependence(explain_the_model, variables = c("x1", "x2"), span = 0.5)
ad_model_1$`_label_` = "ALE 0.1"

ad_model_5 <- accumulated_dependence(explain_the_model, variables = c("x1", "x2"), span = 0.5)
ad_model_5$`_label_` = "ALE 0.5"

plot(ad_model, cd_model, pd_model, cd_model_1, cd_model_5, ad_model_1, ad_model_5) +
  ggtitle("Feature effects - PDP, CDP, ALE")

Dependence profiles in groups

And now, let’s see how the grouping factor works

# add grouping variable
df$x3 <- factor(sign(df$x2))
# update the data argument
explain_the_model$data = df

# PDP in groups
pd_model_groups <- partial_dependence(explain_the_model, 
                                      variables = c("x1", "x2"), 
                                      groups = "x3")
plot(pd_model_groups) +
  ggtitle("Partial Dependence")

# ALE in groups
ad_model_groups <- accumulated_dependence(explain_the_model, 
                                      variables = c("x1", "x2"), 
                                      groups = "x3")
plot(ad_model_groups) +
  ggtitle("Accumulated Local")

# CDP in groups
cd_model_groups <- conditional_dependence(explain_the_model, 
                                      variables = c("x1", "x2"), 
                                      groups = "x3")
plot(cd_model_groups) +
  ggtitle("Conditional Dependence")

Session info

sessionInfo()

#> R version 4.2.2 (2022-10-31)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Monterey 12.5.1
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
#> 
#> locale:
#> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] ggplot2_3.4.0     ranger_0.14.1     ingredients_2.3.0 DALEX_2.4.2      
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.9       highr_0.10       bslib_0.4.2      compiler_4.2.2  
#>  [5] pillar_1.8.1     jquerylib_0.1.4  tools_4.2.2      digest_0.6.31   
#>  [9] lattice_0.20-45  jsonlite_1.8.4   evaluate_0.19    lifecycle_1.0.3 
#> [13] tibble_3.1.8     gtable_0.3.1     pkgconfig_2.0.3  rlang_1.0.6     
#> [17] Matrix_1.5-1     cli_3.6.0        rstudioapi_0.14  yaml_2.3.6      
#> [21] xfun_0.36        fastmap_1.1.0    withr_2.5.0      stringr_1.5.0   
#> [25] knitr_1.41       vctrs_0.5.1      sass_0.4.4       grid_4.2.2      
#> [29] glue_1.6.2       R6_2.5.1         fansi_1.0.3      rmarkdown_2.19  
#> [33] farver_2.1.1     magrittr_2.0.3   scales_1.2.1     htmltools_0.5.4 
#> [37] colorspace_2.0-3 labeling_0.4.2   utf8_1.2.2       stringi_1.7.12  
#> [41] munsell_0.5.0    cachem_1.0.6