Introduction to the ale package

Chitu Okoli

April 7, 2025

Accumulated Local Effects (ALE) were initially developed as a model-agnostic approach for global explanations of the results of black-box machine learning algorithms (Apley, Daniel W., and Jingyu Zhu. ‘Visualizing the effects of predictor variables in black box supervised learning models.’ Journal of the Royal Statistical Society Series B: Statistical Methodology 82.4 (2020): 1059-1086 doi:10.1111/rssb.12377). ALE has at least two primary advantages over other approaches like partial dependency plots (PDP) and SHapley Additive exPlanations (SHAP): its values are not affected by the presence of interactions among variables in a model and its computation is relatively rapid. This package reimplements the original algorithm from the {ALEPlot} package and reimplements the plotting of ALE values. It also extends the original ALE concept to add bootstrap-based confidence intervals and ALE-based statistics that can be used for statistical inference.

For more details, see Okoli, Chitu. 2023. “Statistical Inference Using Machine Learning and Classical Techniques Based on Accumulated Local Effects (ALE).” arXiv. doi:10.48550/arXiv.2310.09877.

This vignette demonstrates the basic functionality of the {ale} package on standard large datasets used for machine learning. A separate vignette is devoted to its use on small datasets, as is often the case with statistical inference. (How small is small? That’s a tough question, but as that vignette explains, most datasets of less than 2000 rows are probably “small” and even many datasets that are more than 2000 rows are nonetheless “small”.) Other vignettes introduce ALE-based statistics for statistical inference, show how the {ale} package handles various datatypes of input variables, and compares the {ale} package with the reference {ALEPlot} package.

We begin by loading the necessary libraries.

library(ale)
#> 
#> Attaching package: 'ale'
#> The following object is masked from 'package:base':
#> 
#>     get
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union

diamonds dataset

For this introduction, we use the diamonds dataset, included with the {ggplot2} graphics system. We cleaned the original version by removing duplicates and invalid entries where the length (x), width (y), or depth (z) is 0.

# Clean up some invalid entries
diamonds <- ggplot2::diamonds |> 
  filter(!(x == 0 | y == 0 | z == 0)) |> 
  # https://lorentzen.ch/index.php/2021/04/16/a-curious-fact-on-the-diamonds-dataset/
  distinct(
    price, carat, cut, color, clarity,
    .keep_all = TRUE
  ) |> 
  rename(
    x_length = x,
    y_width = y,
    z_depth = z,
    depth_pct = depth
  )

# Optional: sample 1000 rows so that the code executes faster.
set.seed(0)
diamonds_sample <- ggplot2::diamonds[sample(nrow(ggplot2::diamonds), 1000), ]

summary(diamonds)
#>      carat               cut        color       clarity       depth_pct    
#>  Min.   :0.2000   Fair     : 1492   D:4658   SI1    :9857   Min.   :43.00  
#>  1st Qu.:0.5200   Good     : 4173   E:6684   VS2    :8227   1st Qu.:61.00  
#>  Median :0.8500   Very Good: 9714   F:6998   SI2    :7916   Median :61.80  
#>  Mean   :0.9033   Premium  : 9657   G:7815   VS1    :6007   Mean   :61.74  
#>  3rd Qu.:1.1500   Ideal    :14703   H:6443   VVS2   :3463   3rd Qu.:62.60  
#>  Max.   :5.0100                     I:4556   VVS1   :2413   Max.   :79.00  
#>                                     J:2585   (Other):1856                  
#>      table           price          x_length         y_width      
#>  Min.   :43.00   Min.   :  326   Min.   : 3.730   Min.   : 3.680  
#>  1st Qu.:56.00   1st Qu.: 1410   1st Qu.: 5.160   1st Qu.: 5.170  
#>  Median :57.00   Median : 3365   Median : 6.040   Median : 6.040  
#>  Mean   :57.58   Mean   : 4686   Mean   : 6.009   Mean   : 6.012  
#>  3rd Qu.:59.00   3rd Qu.: 6406   3rd Qu.: 6.730   3rd Qu.: 6.720  
#>  Max.   :95.00   Max.   :18823   Max.   :10.740   Max.   :58.900  
#>                                                                   
#>     z_depth      
#>  Min.   : 1.070  
#>  1st Qu.: 3.190  
#>  Median : 3.740  
#>  Mean   : 3.711  
#>  3rd Qu.: 4.150  
#>  Max.   :31.800  
#>

Here is the description of the modified dataset.

Variable Description
price price in US dollars ($326–$18,823)
carat weight of the diamond (0.2–5.01)
cut quality of the cut (Fair, Good, Very Good, Premium, Ideal)
color diamond color, from D (best) to J (worst)
clarity a measurement of how clear the diamond is (I1 (worst), SI2, SI1, VS2, VS1, VVS2, VVS1, IF (best))
x_length length in mm (0–10.74)
y_width width in mm (0–58.9)
z_depth depth in mm (0–31.8)
depth_pct total depth percentage = z / mean(x, y) = 2 * z / (x + y) (43–79)
table width of top of diamond relative to widest point (43–95) 
str(diamonds)
#> tibble [39,739 × 10] (S3: tbl_df/tbl/data.frame)
#>  $ carat    : num [1:39739] 0.23 0.21 0.23 0.29 0.31 0.24 0.24 0.26 0.22 0.23 ...
#>  $ cut      : Ord.factor w/ 5 levels "Fair"<"Good"<..: 5 4 2 4 2 3 3 3 1 3 ...
#>  $ color    : Ord.factor w/ 7 levels "D"<"E"<"F"<"G"<..: 2 2 2 6 7 7 6 5 2 5 ...
#>  $ clarity  : Ord.factor w/ 8 levels "I1"<"SI2"<"SI1"<..: 2 3 5 4 2 6 7 3 4 5 ...
#>  $ depth_pct: num [1:39739] 61.5 59.8 56.9 62.4 63.3 62.8 62.3 61.9 65.1 59.4 ...
#>  $ table    : num [1:39739] 55 61 65 58 58 57 57 55 61 61 ...
#>  $ price    : int [1:39739] 326 326 327 334 335 336 336 337 337 338 ...
#>  $ x_length : num [1:39739] 3.95 3.89 4.05 4.2 4.34 3.94 3.95 4.07 3.87 4 ...
#>  $ y_width  : num [1:39739] 3.98 3.84 4.07 4.23 4.35 3.96 3.98 4.11 3.78 4.05 ...
#>  $ z_depth  : num [1:39739] 2.43 2.31 2.31 2.63 2.75 2.48 2.47 2.53 2.49 2.39 ...
summary(diamonds$price)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>     326    1410    3365    4686    6406   18823

Interpretable machine learning (IML) techniques like ALE should be applied on the same dataset that was used to train the model. An explanation is an explanation of a trained model and a trained model is intrinsically linked to the dataset on which it is trained. (When a dataset is too small to feasibly split into training and test sets, then the ale package has tools to appropriately handle such small datasets.

Modelling with generalized additive models (GAM)

ALE is a model-agnostic IML approach, that is, it works with any kind of machine learning model. As such, {ale} works with any R model with the only condition that it can predict numeric outcomes (such as raw estimates for regression and probabilities or odds ratios for classification). For this demonstration, we will use generalized additive models (GAM), a relatively fast algorithm that models data more flexibly than ordinary least squares regression. It is beyond our scope here to explain how GAM works (you can learn more with Noam Ross’s excellent tutorial), but the examples here will work with any statistical or machine learning algorithm.

We train a GAM model to predict diamond prices:

# Create a GAM model with flexible curves to predict diamond prices.
# Smooth all numeric variables and include all other variables.
gam_diamonds <- mgcv::gam(
  price ~ s(carat) + s(depth_pct) + s(table) + s(x_length) + s(y_width) + s(z_depth) +
    cut + color + clarity,
  data = diamonds
  )
summary(gam_diamonds)
#> 
#> Family: gaussian 
#> Link function: identity 
#> 
#> Formula:
#> price ~ s(carat) + s(depth_pct) + s(table) + s(x_length) + s(y_width) + 
#>     s(z_depth) + cut + color + clarity
#> 
#> Parametric coefficients:
#>              Estimate Std. Error  t value Pr(>|t|)    
#> (Intercept)  4436.199     13.315  333.165  < 2e-16 ***
#> cut.L         263.124     39.117    6.727 1.76e-11 ***
#> cut.Q           1.792     27.558    0.065 0.948151    
#> cut.C          74.074     20.169    3.673 0.000240 ***
#> cut^4          27.694     14.373    1.927 0.054004 .  
#> color.L     -2152.488     18.996 -113.313  < 2e-16 ***
#> color.Q      -704.604     17.385  -40.528  < 2e-16 ***
#> color.C       -66.839     16.366   -4.084 4.43e-05 ***
#> color^4        80.376     15.289    5.257 1.47e-07 ***
#> color^5      -110.164     14.484   -7.606 2.89e-14 ***
#> color^6       -49.565     13.464   -3.681 0.000232 ***
#> clarity.L    4111.691     33.499  122.742  < 2e-16 ***
#> clarity.Q   -1539.959     31.211  -49.341  < 2e-16 ***
#> clarity.C     762.680     27.013   28.234  < 2e-16 ***
#> clarity^4    -232.214     21.977  -10.566  < 2e-16 ***
#> clarity^5     193.854     18.324   10.579  < 2e-16 ***
#> clarity^6      46.812     16.172    2.895 0.003799 ** 
#> clarity^7     132.621     14.274    9.291  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Approximate significance of smooth terms:
#>                edf Ref.df       F  p-value    
#> s(carat)     8.695  8.949  37.027  < 2e-16 ***
#> s(depth_pct) 7.606  8.429   6.758  < 2e-16 ***
#> s(table)     5.759  6.856   3.682 0.000736 ***
#> s(x_length)  8.078  8.527  60.936  < 2e-16 ***
#> s(y_width)   7.477  8.144 211.202  < 2e-16 ***
#> s(z_depth)   9.000  9.000  16.266  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> R-sq.(adj) =  0.929   Deviance explained = 92.9%
#> GCV = 1.2602e+06  Scale est. = 1.2581e+06  n = 39739

ALE object for ALE data

The core object in the {ale} package is the {S7} ALE object. effect stores the ALE data and, optionally, ALE statistics and bootstrap data for one or more categories. The first argument to the ALE() constructor is a model object–any R model object that can generate numeric predictions is acceptable. By default, it generates 1D (or “first order”) ALE on all the variables in the dataset that was used to train the model, if the dataset is included in the model object. If not, the dataset can be specified with the data argument. It can optionally create ALE only for specified variables and interactions using the x_cols argument. To change these options (e.g., to calculate ALE for only a subset of variables; to output the data only or to use a custom, non-standard predict function for the model), see details in the help file for the object: help(ALE). In this introduction, we only demonstrate The basics of retrieving and plotting ALE data. For details on ALE statistics see the dedicated vignette on that topic.

# Simple ALE without bootstrapping
ale_gam_diamonds <- ALE(gam_diamonds)

By default, most core functions in the {ale} package use parallel processing. If your computer has problems with this, you can disable parallelization with the argument parallel = 0.

To access the plot for a specific variable, we must first create an ALEPlots object by calling the plot() method on the ALE object which internally generates ggplot objects with the full flexibility of {ggplot2}:

# Print a plot by entering its reference
diamonds_plots <- plot(ale_gam_diamonds)

To retrieve a specific variable plot, you can use the get() method of the ALEPlots object. For example, to access and print the carat ALE plot, we can simply refer to get(diamonds_plots, 'carat'):

# Print a plot by entering its reference
get(diamonds_plots, 'carat')

To display all the ALE plots in an ALEPlots object, we can simply call its print() or plot() methods. Behind the scenes, they use the {patchwork} package to arrange multiple plots in a common plot grid using patchwork::wrap_plots(), so we can pass arguments from that function. For example, we can specify that we want two plots per row with the ncol argument:

# Print all plots
plot(diamonds_plots, ncol = 2)

Bootstrapped ALE

One of the key features of the ALE package is bootstrapping of the ALE results to ensure that the results are reliable, that is, generalizable to data beyond the sample on which the model was trained. As mentioned above, this assumes that IML analysis is carried out on a model whose hyperparameters were determined by cross-validation. When samples are too small for cross-validation, we provide a different approach by bootstrapping the entire model with a ModelBoot object, explained in the vignette for small datasets.

Although ALE is faster than most other IML techniques for global explanation such as partial dependence plots (PDP) and SHAP, it still requires some time to run. Bootstrapping multiplies that time by the number of bootstrap iterations. Since this vignette is just a demonstration of package functionality rather than a real analysis, we will demonstrate bootstrapping on a small subset of the test data. This will run much faster as the speed of the ALE algorithm depends on the size of the dataset. So, let us take a random sample of 200 rows.

# Bootstraping is rather slow, so create a smaller subset of new data for demonstration
set.seed(0)
new_rows <- sample(nrow(diamonds), 200, replace = FALSE)
diamonds_small_test <- diamonds[new_rows, ]

Now we create bootstrapped ALE data and plots using the boot_it argument. ALE is a relatively stable IML algorithm (compared to others like PDP), so 100 bootstrap samples should be sufficient for relatively stable results, especially for model development. Final results could be confirmed with 1000 bootstrap samples or more, but there should not be much difference in the results beyond 100 iterations. However, so that this introduction runs faster, we demonstrate it here with only 10 iterations.


ale_gam_diamonds_boot <- ALE(
  model = gam_diamonds, 
  data = diamonds_small_test, 
  # Normally boot_it should be set to at least 100, but just 10 here for a faster demonstration
  boot_it = 10
)
#> Loading required package: intervals

# Bootstrapping produces confidence intervals
plot(ale_gam_diamonds_boot) |> 
  print(ncol = 2)

In this case, the bootstrapped results are mostly similar to single (non-bootstrapped) ALE results. In principle, we should always bootstrap the results and trust only in bootstrapped results. The most unusual result is that values of x_length (the length of the diamond) from 6.2 mm or so and higher are associated with lower diamond prices. When we compare this with the y_width value (width of the diamond), we suspect that when both the length and width (that is, the size) of a diamond become increasingly large, the price increases so much more rapidly with the width than with the length that the width has an inordinately high effect that is tempered by a decreased effect of the length at those high values. This would be worth further exploration for real analysis, but here we are just introducing the key features of the package.

ALE interactions

Another advantage of ALE is that it provides data for 2D interactions between variables. This is also implemented with the ALE() constructor. When the d2 element of the x_cols list argument is set to TRUE, ALE() generates ALE data on all possible 2D interactions from the input dataset. To change the default options (e.g., to calculate interactions for only certain pairs of variables), see details in the help file for the object: help(ALE).

# ALE two-way interactions
ale_2D_gam_diamonds <- ALE(
  gam_diamonds,
  x_cols = list(d2 = TRUE)
)

The plot() method similarly creates 2D ALE plots from the ALE object. The subset() method of ALEPlots extracts a new ALEPlots object with only the selected variables or interaction terms:

diamonds_2D_plots <- plot(ale_2D_gam_diamonds)

diamonds_2D_plots |>
  # Select all 2D interactions that involve 'carat'
  subset(list(d2_all = 'carat')) |> 
  print(ncol = 2)

Because we are printing all plots together, some of them might appear vertically distorted because each plot is forced to be of the same height. For more fine-tuned presentation, we would need to refer to a specific plot. The ale package supports the standard R formula notation for specifying variables. For example, we can print the interaction plot between carat and depth by referring to it thus: get(diamonds_2D_plots, ~ carat:clarity).

get(diamonds_2D_plots, ~ carat:clarity)

This is not the best dataset to use to illustrate ALE interactions because there are none here. This is expressed in the graphs by the ALE y values all being grey, the middle range of data. In the plots, the darker squares indicate the relative percentage of actual data in each interaction intersection. So, there is very little actual data for 0.2 carats; there is much more higher carat ranges.

Note that ALE interactions are very particular: an ALE interaction means that two variables have a composite effect over and above their separate independent effects. So, of course x_length and y_width both have effects on the price, as the one-way ALE plots show, but their interaction has no additional composite effect. To see what ALE interaction plots look like in the presence of interactions, see the ALEPlot comparison vignette, which explains the interaction plots in more detail.