Accumulated Local Effects (ALE) were initially developed as a model-agnostic
approach for global explanations of the results of black-box machine
learning algorithms. 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 rewrites the original code from the `{ALEPlot}`

package for calculating ALE data and it completely 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. https://doi.org/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)
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
```

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
)
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 ...
```

Interpretable machine learning (IML) techniques like ALE should be applied not on training subsets nor on test subsets but on a final deployment model after training and evaluation. This final deployment should be trained on the full dataset to give the best possible model for production deployment. (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.

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 general 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 machine learning algorithm.

We train a GAM model to predict diamond prices:

```
# Create a GAM model with flexible curves to predict diamond prices.
# (In testing, mgcv::gam actually performed better than nnet.)
# 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
```

Before starting, we recommend that you enable progress bars to see how long procedures will take. Simply run the following code at the beginning of your R session:

```
# Run this in an R console; it will not work directly within an R Markdown or Quarto block
progressr::handlers(global = TRUE)
progressr::handlers('cli')
```

If you forget to do that, the `{ale}`

package will do it
automatically for you with a notification message.

`ale()`

function for generating ALE data and plotsThe core function in the `{ale}`

package is the
`ale()`

function. Consistent with tidyverse conventions, its
first argument is a dataset. Its second argument is a model object–any R
model object that can generate numeric predictions is acceptable. By
default, it generates ALE data and plots on all the input variables used
for the model. To change these options (e.g., to calculate ALE for only
a subset of variables; to output the data only or the plots only rather
than both; or to use a custom, non-standard predict function for the
model), see details in the help file for the function:
`help(ale)`

.

The `ale()`

function returns a list with various elements.
The two main ones are `data`

, containing the ALE x intervals
and the y values for each interval, and `plots`

, containing
the ALE plots as individual `ggplot`

objects. Each of these
elements is a list with one element per input variable. The function
also returns several details about the outcome (y) variable and
important parameters that were used for the ALE calculation. Another
important element is `stats`

, containing ALE-based
statistics, which we describe in a separate vignette.

```
# Simple ALE without bootstrapping
ale_gam_diamonds <- ale(
diamonds, gam_diamonds,
parallel = 2 # CRAN limit (delete this line on your own computer)
)
```

By default, most core functions in the `{ale}`

package use
parallel processing. However, this requires explicit specification of
the packages used to build the model, specified with the
`model_packages`

argument. (If parallelization is disabled
with `parallel = 0`

, then `model_packages`

is not
required.) See `help(ale)`

for more details.

To access the plot for a specific variable, we can call it by its
variable name as an element of the `plots`

element. These are
`ggplot`

objects, so they are easy to manipulate. For
example, to access and print the `carat`

ALE plot, we simply
call `ale_gam_diamonds$plots$carat`

:

To iterate the list and plot all the ALE plots, we provide here some
demonstration code using the `patchwork`

package for
arranging multiple plots in a common plot grid using
`patchwork::wrap_plots()`

. We need to pass the list of plots
to the `grobs`

argument and we can specify that we want two
plots per row with the `ncol`

argument.

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 built. As
mentioned above, this assumes that IML analysis is carried out on a
final deployment model selected after training and evaluating the model
hyperparameters on distinct subsets. When samples are too small for
this, we provide a different bootstrapping method,
`model_bootstrap()`

, 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 of the test set.

```
# 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(
diamonds_small_test, gam_diamonds,
# Normally boot_it should be set to 100, but just 10 here for a faster demonstration
boot_it = 10,
parallel = 2 # CRAN limit (delete this line on your own computer)
)
# Bootstrapping produces confidence intervals
patchwork::wrap_plots(ale_gam_diamonds_boot$plots, ncol = 2)
```

In this case, the bootstrapped results are mostly similar to single
(non-bootstrapped) ALE result. 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.

Another advantage of ALE is that it provides data for two-way
interactions between variables. This is implemented with the
`ale_ixn()`

function. Like the `ale()`

function,
`ale_ixn()`

similarly requires an input dataset and a model
object. By default, it generates ALE data and plots on all possible
pairs of input variables used for the model. However, an ALE interaction
requires at least one of the variables to be numeric. So,
`ale_ixn()`

has a notion of x1 and x2 variables; the x1
variable must be numeric whereas the x2 can be of any input datatype. To
change the default options (e.g., to calculate interactions for only
certain pairs of variables), see details in the help file for the
function: `help(ale_ixn)`

.

```
# ALE two-way interactions
ale_ixn_gam_diamonds <- ale_ixn(
diamonds, gam_diamonds,
parallel = 2 # CRAN limit (delete this line on your own computer)
)
```

Like the `ale()`

function, the `ale_ixn()`

returns a list with one element per input x1 variable, as well as a
`.common_data`

element with details about the outcome (y)
variable. However, in this case, each variable’s element consists of a
list of all the x2 variables for which the x1 interaction is calculated.
Each x2 element then has two elements: the ALE data for that variable
and a `ggplot`

plot object that plots that ALE data. In the
interaction plots, the x1 variable is always shown on the x axis and the
x2 variable on the y axis.

Again, we provide here some demonstration code to plot all the ALE
plots. It is a little more complex this time because of the two levels
of interacting variables in the output data, so we use the
`purrr`

package to iterate the list structure.
`purrr::walk()`

takes a list as its first argument and then
we specify an anonymous function for what we want to do with each
element of the list. We specify the anonymous function as
`\(.x1) {...}`

where `.x1`

in our case represents
each individual element of `ale_ixn_gam_diamonds$plots`

in
turn, that is, a sublist of plots with which the x1 variable interacts.
We print the plots of all the x1 interactions as a combined grid of
plots with `patchwork::wrap_plots()`

, as before.

```
# Print all interaction plots
ale_ixn_gam_diamonds$plots |>
# extract list of x1 ALE outputs
purrr::walk(\(.x1) {
# plot all x2 plots in each .x1 element
patchwork::wrap_plots(.x1, ncol = 2) |>
print()
})
```

Because we are printing all plots together with the same
`patchwork::wrap_plots()`

statement, 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. For example, we can print the interaction plot between
carat and depth by referring to it thus:
`ale_ixn_gam_diamonds$plots$carat$depth`

.

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 falling in the middle grey band (the median band), which indicates that any interactions would not shift the price outside the middle 5% of its values. In other words, there is no meaningful interaction effect.

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 they have 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.