# Simulated data, real problem

#### Przemyslaw Biecek

#### 2023-01-15

# 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