1 Introduction
Many machine learning (ML) interpretation methods (see [18, 10] for an overview) are based on making predictions on perturbed input features, e.g., by permuting feature values. The partial dependence plot (PDP) [9] and permutation feature importance (PFI) [8]
perturb individual features without conditioning on the remaining features, i.e., feature values are changed while ignoring the joint distribution. If features are dependent, such perturbations will cause predictions that extrapolate to areas of the feature space with low density. Extrapolation can result in misleading interpretations
[12] (see Fig. 1).An obvious approach to avoid extrapolation would be to perturb a feature conditional on all other features and thereby preserve the joint distribution. The interpretation of conditional feature effect and importance differ from the unconditional variants (see Fig. 2). The conditional effect of a feature is a mixture of its unconditional effect and the unconditional effects of all dependent features. The conditional PFI must be interpreted as the importance of a feature given the other features. If two features are highly dependent, their conditional importance is lower than their unconditional importance, because their shared information can be substituted by the other feature. For global interpretation methods, there is a tradeoff between avoiding extrapolation and unconditional interpretation of feature effects and importance.
1.0.1 Contributions:
We propose novel, modelagnostic variants of the conditional PDP and conditional PFI based on interpretable subgroups. Our approach is based on constructing subgroups in which the feature of interest is independent from other features and values within the groups are permuted. Subgroup permutation greatly reduces extrapolation while maintaining unconditional interpretation within the subgroups. Furthermore, we introduce a data fidelity measure that quantifies the ability of an interpretation method to preserve the data distribution. Using simulated and real data, we show that conditional subgroup permutation achieves stateoftheart data fidelity. We compare our conditional subgroup PFI with the true cPFI in a simulation and demonstrate stateoftheart performance. In an application, we illustrate how our conditional PDP and PFI can reveal new insights into the ML model and the data.
2 Notation and Background
We consider ML prediction functions , where is a model prediction and is a
dimensional feature vector. We define
as an observed single feature (vector) and to refer to theth feature as a random variable. With
we refer to complementary feature space . We refer to the value of the th feature value from the th instance as and to the tuples as data.Permutation Feature Importance (PFI) for a feature
is estimated as the average increase in prediction loss when the feature is permuted in training or test data:
(1) 
where is a permutation of and the number of repeated permutations. Numerous variations of this formulation exist. Breiman [3]
proposed the PFI for random forests, which is computed from the outofbag samples of individual trees. Subsequently, Fisher et. al
[Fisher2018] introduced a modelagnostic PFI version.The Partial Dependence Plot (PDP) [9] describes the average effect of the jth feature on the prediction. The PDP evaluated at feature value is:
(2) 
3 Related Work
Conditional PDP.
The marginal plot (MPlot) [2] averages the predictions locally on the feature grid and mixes effects of dependent features (see Fig. 2).
Hooker (2007) [11] proposed a functional ANOVA decomposition with hierarchically orthogonal components.
The decomposition requires access to the joint distribution of the data.
The approach has the undesirable property that in e.g. a linear model the coefficients are not recovered when features are correlated.
Accumulated Local Effect (ALE) plots by Apley and Zhu [2] reduce extrapolation by accumulating the finite differences computed within intervals of the feature of interest.
Interpretations of ALE plots are, by definition, only locally valid.
Furthermore, there is no satisfactory approach to derive ALE plots for categorical features, since ALE requires ordered feature values.
Our proposed approach can handle categorical features.
Another PDP variant based on stratification was proposed by [19].
However, this stratified PDP describes only the data and is independent of the model.
Conditional PFI.
Strobl et. al [24] proposed a conditional PFI for the random forest.
While [24] relies on the splits of the underlying random forest trees and permutes the features within these subgroups, we construct the subgroups explicitly from the conditional distribution of the features in a modelagnostic way.
Hooker and Mentch [12] suggested four methods for conditional feature importance: Conditional Variable Importance, Dropped Variable Importance, PermuteAndRelearn Importance and the ConditionandRelearn Importance.
Our proposed method is a variant of the Conditional Variable Importance measure based on interpretable subgroups.
Knockoffs are random variables which are ”copies” of the original features that preserve the joint distribution but are otherwise independent of the prediction target.
Knockoffs can be used to replace feature values for conditional feature importance computation.
Candes et. al [4] proposed knockoffs based on the correlation structure of the features.
Others have proposed to use generative adversarial networks for generating knockoffs [22].
Knockoffs are not transparent with respect to how they condition on the features, while we report interpretable subgroups.
4 Conditional Subgroups
PFI and PDPs are based on sampling from marginal feature distributions which causes extrapolation when features are dependent [12].
Conditional variants of PFI and PDPs (see Section 3) avoid extrapolation by sampling from distributions conditional on the remaining features.
However, with conditional PFI and conditional PDP the data dependencies between and can influence the sample, leading to an intepretation that mixes properties of the model with properties of the dataset [15].
We suggest approaching the dependent feature problem by constructing an interpretable grouping such that the feature of interest is independent of the remaining features within each subgroup, i.e. .
Sampling from the groupwise marginal distribution reduces extrapolation (Fig. 3).
Within a group, samples from the marginal and the conditional distribution coincide.
The grouping consequently enables (1) the application of standard PFI and PDPs within each group without extrapolation and (2) sampling from the global conditional distribution and using groupwise permutation.
With our approach we exploit these properties to derive both (1) groupwise unconditional and (2) global conditional interpretations.
The groupwise unconditional PFIs and PDPs can be seen as a decomposition of the global conditional interpretation.
To get a good approximation of the marginal distribution in a group, the group should contain sufficient observations.
Moreover, the groupings should be humanintelligible.
Existing approaches that model the conditional distribution for interpretation [4, 24, 1] do not provide such a coarse, explicit interpretable grouping.
Transformation trees:
We use transformation trees [14] to model the conditional distribution of the feature of interest given features .
This approach partitions the feature space so that the distribution of within the resulting subgroups is homogeneous, i.e. the groupwise parameterization of the modeled distribution is independent of .
By specifying a maximum tree depth or the minimum number of observations within a node, the granularity of the partitioning can be traded off with the homogeneity of distributions within a partition.
Partitions can be described with the conditions that determine its boundaries, e.g. in form of the decision path.
We leverage this partitioning to construct an interpretable grouping .
The new variable can be calculated by assigning every observation the indicator of the partition that it lies in (meaning for with the group variable’s value is defined as ).
In theory, the approach faces two challenges.
First, not every distribution can be perfectly partitioned into homogeneous and interpretable parts (e.g. in the case of linear Gaussian dependencies).
However, the granularity of the grouping can be adjusted using the model’s hyper parameters.
As empirical results show, the method’s performance is equal to or better than existing approaches in groundtruth evaluations (Section 6.3).
Second, the distribution we specify for the model needs to be able to capture the dependencies between and , for to hold.
However, the approach is in principle agnostic to the specified distribution and the default transformation family of distributions is very general, as empirical results suggest [14]. In most settings, it is therefore reasonable to assume
For more detailed explanations of transformation trees please refer to [14].
For the remainder of this paper, we have set the minimum number of observations in a node to 30, used Bernstein polynomials of degree five for the transformation function and the Normal distribution as target distribution. We denote the subgroups by
, where is the kth subgroup for feature j, with groups in total for the jth feature. The subgroups are disjoint: and . Let be a subset of that refers to the data subset belonging to the subgroup .4.1 Conditional Permutation Feature Importance
We estimate the PFI of feature within a subgroup as , where refers to the permutation of within the subgroup . Algorithm 1 describes the cPFI estimation for one feature in detail on unseen data.
The algorithm has two outcomes: We get importance values for feature for each subgroup () and a global conditional feature importance (). The latter is equivalent to the weighted average of subgroup importances regarding the number of observations within each subgroup (Appendix 0.A).
The cPFI needs the same amount of model evaluations as the PFI ().
4.2 Conditional Partial Dependence Plot (cPDP)
The conditional PDP has a different interpretation than the unconditional PDP, as the motivating example shows (Fig. 2). The proposed PDP variant solves the problem of extrapolation while allowing an unconditional interpretation within each subgroup. Since within groups the marginal and conditional distribution coincide, we compute the for each group using the (unconditional) standard PDP formula in Equation 2. This results in multiple PDPs per feature, which can be displayed together in the same plot as in Fig. 8.
Again, we do not only get the groupwise result. We can aggregate subgroup PDPs to yield the conditional PDP (cPDP). A proof is given in Appendix 0.B.
We restrict each subgroup to the interval
. For our visualization, we suggest to plot the PDPs similar to boxplots, where the dense center quartiles are indicated with a bold line (see Fig.
4).The subgroups PDPs do not break if features are independent.
Theorem 4.1
When feature is independent of features , each subgroup PDP has the same expectation as the unconditional PDP, and an
times larger variance, where
and are the number of observations in the data and the subgroup .The proof is shown in Appendix 0.C.
Equivalence in expectation and higher variance under independence of and holds true even if the partitions would be randomly chosen.
Assuming we perform permutations in both settings, both the PDP and the set of subgroup need evaluations, since (and worst case if evaluated at each value).
5 Data and Model Fidelity
5.1 Data Fidelity
PDP and PFI work by data perturbation, prediction and subsequent aggregation [23]. We define a measure of data fidelity to quantify the ability to preserve the joint distribution under perturbation.
Definition 1 (Data Fidelity)
Data fidelity is the degree to which a perturbation of feature preserves the joint distribution of , i.e. the degree to which
This definition is similar to a property required for knockoffs, see e.g. [4]. Based on data , perturbations create a new dataset which is to be compared to the original data distribution. In this twosample testscenario, the maximum mean discrepancy (MMD) can be used to compare whether two samples come from the same distribution. We propose to measure the data fidelity with the empirical MMD:
(3) 
where is the original dataset and a dataset with perturbed . As kernel
we used the radial basis function kernel for all experiments. We require the features to be scaled to a mean of zero and a standard deviation of one. Categorical features are onehot encoded. For parameter
of the radial basis function kernel, we chose the median L2distance between data points.5.2 Model Fidelity
Model fidelity has been defined as how close the predictions of an explanation method are to the ML model [21].
Similarly, we define model fidelity for feature effects as the mean squared error between model prediction and the prediction by the partial function (which depends only on feature ) defined by the feature effect method.
For a given data instance, the predicted outcome from an, e.g., PDP is the yaxis value at the observed value.
(4) 
where is a feature effect function such as ALE or PDP. In order to evaluate ALE plots, they have to be adjusted such that they are on a comparable scale to a PDP [2], i.e., .
6 Evaluation
6.1 Data Fidelity Evaluation
We evaluated how different types of perturbations affect the data fidelity measure (based on MMD) for numerous datasets (see Table 1).
wine  satellite  wind  space  pollen  quake  

No. of rows  6497  6435  6574  3107  3848  2178 
No. of features  12  37  15  7  6  4 
We used 40% of the data to fit the transformation trees to find the subgroups. The remaining 60% were split in half. One half remained unchanged, while we perturbed one of the features in the other half. Then, we computed the MMD comparing the two datasets. We sampled a subset of observations from the PDP / PFI perturbation dataset by permuting once for each observation , so that we get a perturbed dataset of the same size as the original data . This differs from the PDP definitions where for each observation is replaced by a set of grid values. According to [6], PFI and PDP can be formulated with the same underlying feature replacement strategy, either by replacing the feature values using predefined grid points (as usually done in PDP) or by permuting the feature values (as usually done in PFI). For the perturbation in subgroups, we permuted once within each subgroup. For the intervalbased perturbation of ALE plots, we used a grid based on 30 quantiles that determine the intervals. We averaged two MMD computations: Once when moving each observation to the left border of the containing ALE interval, and once to the right border. For ModelX knockoffs [4] we replaced the feature with its knockoff. We repeated the experiment 30 times with different random seeds.
Fig. 5 shows that the PDP/PFI type of perturbation has a low data fidelity (high MMD) compared to all other approaches. ModelX knockoffs and conditional subgroup permutation (with many groups) have the best data fidelity. Even splitting with a maximum depth of only 1 (two subgroups) strongly improves data fidelity. The deeper the trees are, the more subgroups are found and the better the data fidelity. Ranked across all features and datasets, the average rankings show that deep conditional subgroups even outperform ALE, see Table 2.
BL  xKO  CS10  CS5  ALE  CS2  CS1  Perm  
Mean ranks  2.06  2.97  2.99  3.74  4.23  5.61  6.68  7.72 
6.2 Model Fidelity Evaluation
In this section we evaluate the model fidelity of PDP, ALE and subgroup PDPs. We trained random forests (500 trees), linear models and knearest neighbours models (k = 7) on various datasets (Table 1). 70% of the data were used to train the models and the transformation trees. 30% of the data were used to evaluate model fidelity. For each model and each dataset, we measured model fidelity between effect prediction and model prediction (Equation 4), averaged across observations and features. Table 3 shows that the model fidelity of ALE and PDP is similar, while the subgroup PDPs have the best model fidelity. This is interesting since the grouping is neither based on the model nor the real target, but solely on the conditional dependence structure of the features.
pollen  quake  satellite  space  wind  wine  

PDP  9.56  0.03  4.77  0.12  43.86  0.73 
ALE  9.56  0.03  4.77  0.12  43.86  0.73 
cPDP(1)  9.56  0.03  4.43  0.07  28.81  0.70 
cPDP(2)  8.08  0.03  3.18  0.05  24.15  0.67 
cPDP(5)  7.13  0.03  2.73  0.03  18.94  0.60 
cPDP(10)  7.06  0.03  2.34  0.02  17.93  0.59 
Median model fidelity averaged over features in a random forest for various datasets. The cPDPs always had a lower loss (i.e. higher model fidelity) than PDP and ALE. The loss monotonically decreases with increasing maximum tree depth for subgroup construction. Using different models (knn or linear model) produced similar results, see Appendix
0.E.6.3 Conditional Feature Importance Evaluation
We computed the true conditional feature importance for following simulated linear model: , where with .
In the simulations, we varied the correlation between and , while remained independent.
We repeated the experiments 30 times and sampled 1000 data points in each repetition.
We examined two experimental settings. In setting (I) we assumed that our machine learning model recovered the true linear regression model
. We measured the absolute distance for the true cPFI of feature (derivation in Appendix 0.D) and the cPFI based on subgroups with different tree depths for subgroup generation. In setting (II) we trained a random forest (with 100 conditional inference trees [13], mtry = 2 and maxdepth = 10). For this random forest we computed the cPFI with various methods (computed on the random forest) and compared it to the true cPFI (based on the data generating process). We compared our subgroup cPFI approach, the random forest based cPFI by Strobl et. al [24], and ModelX knockoffs [4].Fig. 6 shows that (I) the deeper the transformation trees (and the more subgroups), the better the true cPFI is approximated and (II) our subgroupbased cPFI approach is equal or superior to the stateofthe art.
7 Application
On the following practical application we demonstrate that subgroup based conditional variants are a valuable tool to understand model and data beyond insights given by PFI, PDPs or ALE plots.
We trained a random forest to predict daily bike rentals [7] with given weather and seasonal information.
The data (, ) was divided into 70% training and 30% test data.
The features in the bike data are dependent.
For example, the correlation between temperature and humidity is 0.13
.
The data contains both categorical and numerical features and we are interested in the multivariate, nonlinear dependencies, thus correlation is an inadequate measure of dependence.
We therefore indicate the degree of dependence by showing the extent to which we can predict each feature from all other features in Table 4.
Per feature, we trained a random forest to predict the feature from all other features.
Random forests can capture nonlinear dependencies and interactions and work reasonably well without tuning.
We measured the proportion of loss explained to quantify the dependence of the respective feature on all other features.
For numerical features we used the Rsquared measure.
For categorical features we computed , where is the mean misclassification error, the true class, the classification function of the random forest and the most common class in the training data.
We divided the training data into two folds and trained the random forest on one half.
Then we computed the proportion of explained loss on the other half and vice versa.
Finally we averaged the results.
season  yr  holiday  weekday  workingday  weathersit  temp  hum  windspeed 
45%  5%  38%  14%  100%  42%  66%  42%  10% 
To construct the subgroups, we set the maximum tree depth to 2, i.e. we limited the number of possible subgroups to 4. We compared the unconditional and conditional PFI for the bike rental predictions, see Fig. 7.
The most important feature, according to PFI, was the temperature. Temperature is less important when we condition on season and humidity. To get a deeper understanding of the temperature effect, we examined the effect plots (see Fig. 8). Both ALE and PDP show a monotonous increase of predicted bike rentals up until 25 C and a decrease beyond that. The PDP shows a weaker negative effect of very high temperatures which might be caused by extrapolation: High temperatures days are combined with e.g. winter. A limitation of the ALE plot is that we can only interpret it locally. In contrast, our subgroups are explicit about the subgroup conditions in which the interpretation of the PDP is valid and shows the distributions in which the feature effect may be interpreted. The PDPs in subgroups reveal a more nuanced picture: For dry summer days, increasing temperature mostly has a negative effect on the predicted number of bike rentals.
The change in intercepts of the subgroup PDP can be interpreted as the effect of the grouping features (season and humidity).
The slope can be interpreted as the temperature effect within a subgroup.
We also demonstrate the subgroup PDPs for the season, a categorical feature.
Fig. 9 shows both the PDP and our subgroup PDPs.
The normal PDP shows that on average there is no difference between spring, summer and fall and only slightly less bike rentals in winter.
The PDP with four subgroups conditional on temperature shows that the unconditional PDP is misleading.
8 Discussion
We proposed the conditional PDP and the conditional PFI, both based on subgroups. This research addresses the inherent conflict between extrapolation and an unconditional interpretation. Our subgroupbased approach unites the best of both worlds: It reduces extrapolation and makes the conditioning explicit through interpretable subgroups, while allowing unconditional interpretation within the subgroups. We have shown that permuting data within subgroups greatly improves data fidelity compared to unconditional permutation. For this purpose we introduced a data fidelity measure based on the maximum mean discrepancy. As a surprising finding, the model fidelity of the subgroup PDPs is better than that of ALE or PDPs. In a simulation we showed that our conditional PFI exceeds or is equal to the stateoftheart for conditional PFI. The measure of data fidelity can be used for other interpretation methods as well. For local explanation methods such as LIME [21] or Shapley Values [17] it could be adapted to measure local data fidelity.
Computational Details. All experiments were conducted using mlr [16] and R [20]. The code for all experiments is available at https://github.com/compstatlmu/paper_2019_dependent_features/.
Acknowledgements. This work is funded by the Bavarian State Ministry of Science and the Arts in the framework of the Centre Digitisation.Bavaria (ZD.B) and supported by the German Federal Ministry of Education and Research (BMBF) under Grant No. 01IS18036A. The authors of this work take full responsibilities for its content.
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Appendix 0.A Decompose cPFI into subgroup PFIs
Assuming a perfect construction of , it holds that and also that (as is a compression of ). Therefore
(5) 
When we sample the replacement for an from the marginal within a group (, e.g. via permutation) we also sample from the conditional . Every datapoint from the global sample can therefore equivalently be seen as a sample from the marginal within the group, or as a sample from the global conditional.
As follows, the weighted sum of marginal subgroup PFIs coincides with the cPFI.
(6)  
(7)  
(9) 
Appendix 0.B Decompose cPDPs into subgroup PDPs
As we know that we see that the conditional PDP as defined below can be seen as a point wise local sum of marginal PDPs. We denote that we consider a conditional distribution with by denoting at the end of a line.
(10)  
(11)  
(12)  
(13)  
(14)  
(15)  
(16) 
We need
in order to construct the cPDP from the groupwise PDPs. These probabilities can be approximated, but cannot be trivially derived analytically, as they depend on the unknown distributions of
within the groups.If we would construct the groups by partitioning of (and modelling of the conditional of ), the term would evaluate to or and the aggregation would be straightforward to perform.
Appendix 0.C Expectation and Variance of the PDP in a Subgroup
We show that under feature independence the PDP and a PDP in an arbitrary subgroup have the same expected value and the subgroup PDP has a higher variance.
Proof
Appendix 0.D Groundtruth cPFI
The data has the following distribution:
And with . We assume an ML model For any data point, the squared loss is:
The expectation for this is:
For the permutation feature importance, we permute one of the features. The following formula shows permutation of :
(17)  
(18)  
(19)  
(20) 
So for a single data point, the feature importance is:
The expected value of this is:
(21)  
(22)  
(23) 
Finally the expected conditional permutation feature importance becomes:
(24)  
(25) 
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