The `fmeffects`

package computes, aggregates, and
visualizes forward marginal effects (FMEs) for supervised machine
learning models. Put simply, an FME is the change in a model’s predicted
value for a given observation if the feature is changed by a certain
value. Read the article
on how FMEs are computed or the methods paper
for more details. Our website is the best way
to find all resources.

There are three main functions:

`fme()`

computes FMEs for a given model, data, feature(s) of interest, and step size(s).`came()`

can be applied subsequently to find subspaces of the feature space where FMEs are more homogeneous.`ame()`

provides an overview of the prediction function w.r.t. each feature by using average marginal effects (AMEs).

Let’s look at data from a bike sharing usage system in Washington,
D.C. (Fanaee-T and Gama, 2014). We are interested in predicting
`count`

(the total number of bikes lent out to users).

```
## 'data.frame': 731 obs. of 10 variables:
## $ season : Factor w/ 4 levels "winter","spring",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ year : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
## $ holiday : Factor w/ 2 levels "yes","no": 2 2 2 2 2 2 2 2 2 2 ...
## $ weekday : Factor w/ 7 levels "Monday","Tuesday",..: 7 1 2 3 4 5 6 7 1 2 ...
## $ workingday: Factor w/ 2 levels "yes","no": 2 2 1 1 1 1 1 2 2 1 ...
## $ weather : Factor w/ 3 levels "clear","misty",..: 2 2 1 1 1 1 2 2 1 1 ...
## $ temp : num 8.18 9.08 1.23 1.4 2.67 ...
## $ humidity : num 0.806 0.696 0.437 0.59 0.437 ...
## $ windspeed : num 10.7 16.7 16.6 10.7 12.5 ...
## $ count : int 985 801 1349 1562 1600 1606 1510 959 822 1321 ...
```

FMEs are a model-agnostic interpretation method, i.e., they can be
applied to any regression or (binary) classification model. Before we
can compute FMEs, we need a trained model. In addition to generic
`lm`

-type models, the `fme`

package supports 100+
models from the `mlr3`

, `tidymodels`

and
`caret`

libraries. Let’s try a random forest using the
`ranger`

algorithm:

```
set.seed(123)
library(mlr3verse)
library(ranger)
task = as_task_regr(x = bikes, target = "count")
forest = lrn("regr.ranger")$train(task)
```

FMEs can be used to compute feature effects for both numerical and
categorical features. This can be done with the `fme()`

function. The most common application is to compute the FME for a single
numerical feature, i.e., a univariate feature effect. The variable of
interest must be specified with the `features`

argument. This
is a named list with the feature names and step lengths. The step length
is chosen to be the number deemed most useful for the purpose of
interpretation. Often, this could be a unit change, e.g.,
`features = list(feature_name = 1)`

. As the concept of
numerical FMEs extends to multivariate feature changes as well,
`fme()`

can be asked to compute a multivariate feature
effect.

Assume we are interested in the effect of temperature on bike sharing
usage. Specifically, we set the step size to 1 to investigate the FME of
an increase in temperature by 1 degree Celsius (°C). Thus, we compute
FMEs for `features = list("temp" = 1)`

.

Note that we have specified `ep.method = "envelope"`

. This
means we exclude observations for which adding 1°C to the temperature
results in the temperature value falling outside the range of
`temp`

in the data. Thereby, we reduce the risk of model
extrapolation.

The black arrow indicates direction and magnitude of the step size.
The horizontal line is the average marginal effect (AME). The AME is
computed as a simple mean over all observation-wise FMEs. We can extract
relevant aggregate information from the `effects`

object:

`## [1] 56.07705`

Therefore, on average, a temperature increase of 1°C is associated
with an increase in predicted bike sharing usage by roughly 56. As can
be seen in the plot, the observation-wise effects seem to vary along the
range of `temp`

. The FME tends to be positive for temperature
values between 0 and 17°C and negative for higher temperature values
(>17°C).

For a more in-depth analysis, we can inspect individual FMEs for each observation in the data (excluding extrapolation points):

```
## Key: <obs.id>
## obs.id fme
## <int> <num>
## 1: 1 175.09480
## 2: 2 236.55453
## 3: 3 80.43792
## 4: 4 91.82632
## 5: 5 189.47642
## 6: 6 130.20006
```

Multivariate feature effects can be considered when one is interested
in the combined effect of two or more numeric features. Let’s assume we
want to estimate the effect of a decrease in temperature by 3°C,
combined with a decrease in humidity by 10 percentage points, i.e., the
FME for `features = list(temp = -3, humidity = -0.1)`

:

```
effects2 = fme(model = forest,
data = bikes,
features = list(temp = -3, humidity = -0.1),
ep.method = "envelope")
```

For bivariate effects, we can plot the effects in a way similar to univariate effects (for more than two features, we can plot only the histogram of effects):

The plot for bivariate FMEs uses a color scale to indicate direction and magnitude of the estimated effect. We see that a drop in both temperature and humidity is associated with lower predicted bike sharing usage especially on days with medium temperatures and medium-to-low humidity. Let’s check the AME:

`## [1] -115.917`

It seems that a combined decrease in temperature by 3°C and humidity by 10 percentage points seems to result in slightly lower bike sharing usage (on average). However, a quick check of the standard deviation of the FMEs implies that effects are highly heterogeneous:

`## [1] 488.7228`

Therefore, it could be interesting to move the interpretation of
feature effects from a global to a regional perspective via the
`came()`

function.

The non-linearity measure (NLM) is a complimentary tool to an FME. Any numerical, observation-wise FME is prone to be misinterpreted as a linear effect. To counteract this, the NLM quantifies the linearity of the prediction function for a single observation and step size. A value of 1 indicates linearity, a value of 0 or lower indicates non-linearity (similar to R-squared, the NLM can take negative values). A detailed explanation can be found in the FME methods paper.

We can compute and plot NLMs alongside FMEs for univariate and
multivariate feature changes. Computing NLMs can be computationally
demanding, so we use `furrr`

for parallelization. To
illustrate NLMs, let’s recompute the first example of an increase in
temperature by 1 degree Celsius (°C) on a subset of the bikes data:

```
effects3 = fme(model = forest,
data = bikes[1:200,],
feature = list(temp = 1),
ep.method = "envelope",
compute.nlm = TRUE)
```

Similarly to the AME, we can extract an Average NLM (ANLM):

`## [1] 0.2142`

A value of 0.2 indicates that a linear effect is ill-suited to describe the change of the prediction function along the multivariate feature step. This means we should be weary of interpreting the FME as a linear effect.

If NLMs have been computed, they can be visualized alongside FMEs
using `with.nlm = TRUE`

:

Equivalently, let’s compute an example with bivariate FMEs with NLMs:

For a categorical feature, the FME of an observation is simply the
difference in predictions when changing the observed category of the
feature to the category specified in `features`

. For
instance, one could be interested in the effect of rainy weather on the
bike sharing demand, i.e., the FME of changing the feature value of
`weather`

to `rain`

for observations where weather
is either `clear`

or `misty`

:

```
##
## Forward Marginal Effects Object
##
## Step type:
## categorical
##
## Feature & reference category:
## weather, rain
##
## Extrapolation point detection:
## none, EPs: 0 of 710 obs. (0 %)
##
## Average Marginal Effect (AME):
## -732.3323
```

An AME of -732 implies that holding all other features constant, a
change to rainy weather can be expected to reduce bike sharing usage by
732.

For categorical feature effects, we can plot the empirical distribution
of the FMEs:

In a similar way, we can consider interactions of categories from
different features. For example, consider the average combined effect of
a clear sky on the weekend, i.e., `weather = "clear"`

and
`workingday = "no"`

:

`## [1] 340.274`

For an informative overview of all feature effects in a model, we can
use the `ame()`

function:

```
## Feature step.size AME SD 0.25 0.75 n
## 1 season winter -940.2072 476.0509 -1305.8796 -619.0434 550
## 2 season spring 148.6925 576.2312 -235.1632 658.2236 547
## 3 season summer 307.5089 555.6694 -41.2263 756.2584 543
## 4 season fall 528.2633 584.9771 48.5054 1123.6323 553
## 5 year 0 -1908.809 641.5265 -2376.3932 -1532.705 366
## 6 year 1 1796.2415 516.7806 1426.7262 2178.6711 365
## 7 holiday no 178.5752 230.7029 90.2962 200.3034 21
## 8 holiday yes -124.7559 162.7055 -179.9942 -16.4992 710
## 9 weekday Sunday 158.9936 193.102 17.1484 253.5031 626
## 10 weekday Monday -163.6149 224.2337 -277.9808 -8.2284 626
## 11 weekday Tuesday -119.3857 201.5994 -204.9901 12.8987 626
## 12 weekday Wednesday -43.3697 177.8772 -114.6719 63.841 627
## 13 weekday Thursday 17.6793 165.7388 -67.1236 93.1163 627
## 14 weekday Friday 54.8807 164.595 -35.2388 121.9946 627
## 15 weekday Saturday 104.975 170.8217 3.2623 184.0102 627
## 16 workingday no -42.1138 140.1271 -144.6964 65.2166 500
## 17 workingday yes 44.8021 156.6879 -63.3893 151.9648 231
## 18 weather misty -238.8897 323.0394 -430.6845 -83.7385 484
## 19 weather clear 374.068 327.8023 146.8156 486.0376 268
## 20 weather rain -732.3323 352.015 -995.5234 -481.4628 710
## 21 temp 1 55.8542 169.3016 -24.8871 102.1538 731
## 22 humidity 0.01 -19.8753 61.7175 -36.4396 9.8367 731
## 23 windspeed 1 -25.8064 78.4991 -56.3924 13.3787 731
```

This computes the AME for each feature included in the model, with a
default step size of 1 for numeric features (or, 0.01 if their range is
smaller than 1). For categorical features, AMEs are computed for all
available categories. Alternatively, we can specify a subset of features
and step sizes using the `features`

argument:

```
overview = ame(model = forest,
data = bikes,
features = list(weather = c("rain", "clear"), humidity = 0.1),
ep.method = "envelope")
overview$results
```

```
## Feature step.size AME SD 0.25 0.75 n
## 1 weather rain -732.33231 352.015 -995.52343 -481.46275 710
## 2 weather clear 374.06801 327.80233 146.81561 486.03759 268
## 3 humidity 0.1 -200.10366 309.63612 -298.44643 2.79348 697
```

Again, note that we advise to set `ep.method = "envelope"`

so we avoid model extrapolation.

We can use `came()`

on a specific FME object to compute
subspaces of the feature space where FMEs are more homogeneous. Let’s
take the effect of a decrease in temperature by 3°C combined with a
decrease in humidity by 10 percentage points, and see if we can find
three appropriate subspaces.

```
##
## PartitioningCtree of an FME object
##
## Method: partitions = 3
##
## n cAME SD(fME)
## 726 -115.91698 488.7228 *
## 315 -384.61383 384.7205
## 223 -15.70148 488.2430
## 188 215.42053 387.9813
## ---
## * root node (non-partitioned)
##
## AME (Global): -115.917
```

As can be seen, the CTREE algorithm was used to partition the feature space into three subspaces. The standard deviation (SD) of FMEs is used as a criterion to measure homogeneity in each subspace. We can see that the SD is substantially smaller in two of the three subspaces when compared to the root node, i.e., the global feature space. The conditional AME (cAME) can be used to interpret how the expected FME varies across the subspaces. Let’s visualize our results:

In this case, we get a decision tree that assigns observations to a
feature subspace according to the season (`season`

) and the
humidity (`humidity`

). The information contained in the boxes
below the terminal nodes are equivalent to the summary output and can be
extracted from `subspaces$results`

. The difference in the
cAMEs across the groups means the expected ME varies substantially in
direction and magnitude across the subspaces. For example, the cAME is
highest on summer days. It is lowest on days in spring, fall or winter
when the humidity is below 66%.

Fanaee-T, H. and Gama, J. (2014). Event labeling combining ensemble detectors and background knowledge. Progress in Artificial Intelligence 2(2): 113–127

Vanschoren, J., van Rijn, J. N., Bischl, B. and Torgo, L. (2013). Openml: networked science in machine learning. SIGKDD Explorations 15(2): 49–60