=Paper=
{{Paper
|id=Vol-3793/paper12
|storemode=property
|title=Online Explainable Ensemble of Tree Models Pruning
for Time Series Forecasting
|pdfUrl=https://ceur-ws.org/Vol-3793/paper_12.pdf
|volume=Vol-3793
|authors=Amal Saadallah
|dblpUrl=https://dblp.org/rec/conf/xai/Saadallah24
}}
==Online Explainable Ensemble of Tree Models Pruning
for Time Series Forecasting==
https://ceur-ws.org/Vol-3793/paper_12.pdf
Online Explainable Ensemble of Tree Models Pruning
for Time Series Forecasting⋆
Amal Saadallah
Lamarr Institute for Machine Learning and AI, Dortmund, Germany
Abstract
Tree-based models are commonly used in time series forecasting due to their inherent interpretability,
which makes them preferable to more complex black-box models. However, simple tree-based models are
prone to overfitting, limiting their applicability in real-world scenarios. Ensembles of tree-based models
are employed to mitigate this, but ensemble pruning is challenging, especially in the presence of dynamic
time series data and concept drift. In this paper, we use TreeSHAP, a tree-specific explainability tool,
to perform online tree-based ensemble pruning that adapts dynamically to changes in the time series,
addressing the concept drift issue. Empirical evaluations on real-world time series datasets demonstrate
that our method performs on par with or better than state-of-the-art techniques. In future research,
we plan to automate the determination of the optimal number of clusters for ensemble pruning by
leveraging ensemble properties like diversity, accuracy, and stability. This automation aims to enhance
both the flexibility and explainability of the model selection process. Given that this work is in its early
stages, we seek feedback and collaboration with experts to create a robust and explainable framework
for ensemble-based time series forecasting.
Keywords
Tree Models, Online Ensemble Pruning, TreeSHAP, Time Series Forecasting, Concept-drift, Explainability
1. Introduction
Time series forecasting is crucial for real-time planning and decision-making across various
fields like traffic management, weather prediction, and financial markets. However, it is also
one of the most challenging tasks due to the complex and dynamic nature of time series data,
which often involves non-stationary variations and is susceptible to concept drift [1]. This
makes accurate forecasting inherently difficult, necessitating models that can adapt to changing
data patterns [2, 3, 4, 5, 6, 7]. Given these challenges, explainability in forecasting models has
become increasingly important, especially for safety-critical applications. Tree-based models
are often favored for their intrinsic explainability, but identifying appropriate models for specific
time series requires adaptability due to time-varying characteristics. Decision Trees and their
ensembles, like Random Forests and Gradient-boosted Trees, are commonly used for time series
forecasting. However, these models can struggle with dynamic data since they typically operate
Late-breaking work, Demos and Doctoral Consortium, colocated with The 2nd World Conference on eXplainable Artificial
Intelligence: July 17–19, 2024, Valletta, Malta
⋆
This research has partly been funded by the Federal Ministry of Education and Research of Germany and the state
of North-Rhine-Westphalia as part of the Lamarr-Institute for Machine Learning and Artificial Intelligence.
*
Corresponding author.
$ amal.saadallah@tu-dortmund.de (A. Saadallah)
0000-0003-2976-7574 (A. Saadallah)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
in a static manner, not inherently considering variations in the underlying time series. In
addition, combining multiple models into ensembles can improve forecasting accuracy, but
at the cost of explainability. To address these issues, we propose an online ensemble pruning
approach for time series forecasting, where the ensemble members are selected based on an
adaptive clustering procedure that uses TreeSHAP values to group models with similar modeling
paradigms. This methodology not only ensures diversity within the ensemble but also allows for
an explainable selection process by indicating which aspects of the time series data contribute
most to the predictions.
In our future research, one key goal is to automatically determine the optimal number of
clusters, which corresponds to the ideal number of trees or ensemble members in the ensemble.
This would involve using ensemble properties such as diversity, accuracy, and stability to guide
the selection of the most suitable cluster count. By automating this process, we aim to improve
the ensemble’s flexibility and effectiveness in adapting to dynamic time series data. Moreover,
we intend to deepen the explainability aspect of our approach by explicitly demonstrating that
selecting models based on different TreeSHAP values aligns with distinct modeling paradigms
and hypotheses. This could be achieved by visualizing or analyzing how these varying TreeSHAP
values translate into different interpretations of the underlying data, providing insights into
the rationale behind model selection. Given that this is early-stage work, we plan to engage
with experts in the field to exchange ideas and gather feedback. Collaboration with specialists
will be instrumental in refining our methodology for selecting the optimal number of trees
and enhancing explainability. By incorporating diverse perspectives, we hope to develop a
robust and transparent approach that addresses the complexities of time series forecasting while
maintaining clarity in model selection and ensemble pruning. This collaborative effort will
contribute to building a reliable framework for ensemble-based forecasting, with a particular
emphasis on explainability and adaptability.
2. Methodology
Our proposed method uses TreeSHAP for online ensemble pruning using model clustering. First,
we define the used notation. Second, we describe Shapley values with a focus on TreeSHAP
values [8]. Third, we show how we generate the candidate tree-based models. Finally, we
demonstrate how TreeSHAP values are used for model clustering to allow for efficient ensemble
pruning and how the whole process is made adaptive to the changes in the time series.
2.1. Preliminaries
A time series 𝑋 is a temporal sequence of values, where 𝑋1:𝑡 = {𝑥1 , 𝑥2 , · · · , 𝑥𝑡 } is a sequence
of 𝑋 until time 𝑡 and 𝑥𝑖 is the value of 𝑋 at time 𝑖. Denote with T = {𝑇 1 , 𝑇 2 , · · · , 𝑇 𝑀 } the
pool of 𝑀 tree-based models trained to approximate a true unknown function 𝑓 that generated
1 2 𝑀
𝑋. Let 𝑥 ^ 𝑡+ℎ = (𝑥^ 𝑇𝑡+ℎ , 𝑥 ^ 𝑇𝑡+ℎ ) be the vector of forecast values of 𝑋 at a future time
^ 𝑇𝑡+ℎ , · · · , 𝑥
instant 𝑡 + ℎ, ℎ ≥ 1 (i.e. 𝑥𝑡+ℎ ) by each of the models in T. An ensemble model 𝑇¯ T of T at time
instant 𝑡 + ℎ can be formally expressed as a convex combination of the forecasts of the models in
𝑗 𝑗 𝑗
^ 𝑡+ℎ ) = 𝑀
T: 𝑇¯ T (𝑥 ^ 𝑇𝑡+ℎ where 𝑤𝑡+ℎ ∈ [1, 𝑀 ] are the ensemble weights. The weights
∑︀
𝑗=1 𝑤𝑡+ℎ 𝑥
are constrained to be positive and sum to one. In addition, it can be seen from the notation that
the weights are time-dependent. This is one of the requirements in online ensemble learning,
where the weights are required to be set in a timely manner to cope with the dynamic nature
of the time series and the time-changing performance of the ensemble members [5, 6]. The
goal of dynamic online ensemble pruning is to identify the subset of models S ⊂ T that should
compose the ensemble at each time step 𝑡 + ℎ such that the expected prediction error of the
pruned ensemble is reduced compared the full ensemble 𝑇¯ T for each forecast.
)︀2 )︀2
𝑎𝑟𝑔𝑚𝑎𝑥S⊂T E 𝑥𝑡+ℎ − 𝑇¯ T (𝑥 ^ 𝑡+ℎ ) |𝑋1:𝑡+ℎ−1 − E 𝑥𝑡+ℎ − 𝑇¯ S (𝑥 ^ 𝑡+ℎ ) |𝑋1:𝑡+ℎ−1 (1)
[︀(︀ ]︀ [︀(︀ ]︀
2.2. TreeSHAP Ensemble Learning
2.2.1. Ensemble Pruning
We divide the time series 𝑋1:𝑡 into 𝑋𝜔𝑡𝑟𝑎𝑖𝑛 = {𝑥1 , 𝑥2 , · · · , 𝑥𝑡−𝜔 } and 𝑋𝜔𝑣𝑎𝑙 =
{𝑥𝑡−𝜔+1 , 𝑥𝑡−𝜔+2 , · · · , 𝑥𝑡 }, with 𝜔 a provided window size. 𝑋𝜔𝑡𝑟𝑎𝑖𝑛 is used for training the
models in T and 𝑋𝜔𝑣𝑎𝑙 is used to compute the TreeSHAP values. For each tree-based model
𝑇 𝑗 ∈ 𝒯 , for each observation 𝑥𝑡−𝜔+𝑘 ∈ 𝑋𝜔𝑣𝑎𝑙 with 𝑘 ∈ [1, 𝜔], we compute a TreeSHAP value
𝜑𝑗𝑖 (𝑥𝑡−𝜔+𝑘 ) for each lagged value, i.e., 𝑖 ∈ [1, 𝑙𝑗 ], where 𝑙𝑗 is the number of lags on which the
model 𝑇 𝑗 is trained. Then, we aggregate absolute SHAP values over all the observations in
𝑋𝜔𝑣𝑎𝑙 to acquire SHAP-based lag importance 𝐼𝑖𝑗 for each lag 𝑖 ∈ [1, 𝑙𝑗 ] using the model 𝑇 𝑗 :
𝜔
1 ∑︁ 𝑗
𝐼𝑖𝑗 = |𝜑𝑖 (𝑥𝑡−𝜔+𝑘 )|, ∀ 𝑖 ∈ [1, 𝑙𝑗 ], ∀ 𝑇 𝑗 ∈ 𝒯 (2)
𝜔
𝑘=1
Each model 𝑇 𝑗 ∈ 𝒯 can then be characterized by a vector I𝑗 = {𝐼1𝑗 , 𝐼2𝑗 , · · · 𝐼𝑙𝑗𝑗 }. The models
can thus be clustered using their SHAP-based lag importance vectors I𝑗 . However, different
models in 𝒯 might be trained using different lag values. As a result, the length of the vectors
I𝑗 can vary between 𝑙𝑚𝑖𝑛 and 𝑙𝑚𝑎𝑥 . It exists clustering distance measure that can handle
vectors of different lengths [9]. However, we are mainly interested in grouping models based
on the way they represent the relationship between the input lagged values and the output.
Therefore, we assume that the models that are trained using a lag value 𝑙𝑗 lower than 𝑙𝑚𝑎𝑥
ignore the importance and the contribution of lagged features that are greater than 𝑙𝑗 . In
other words, if the mode 𝑇 𝑗 is trained on 𝑙𝑗 ≤ 𝑙𝑚𝑎𝑥 , for each 𝑖, such that 𝑙𝑗 ≤ 𝑖 ≤ 𝑙𝑚𝑎𝑥 , the
value of its corresponding SHAP-based lag importance 𝐼𝑖𝑗 on 𝑖 is set to zero. In this manner,
we bring all the vectors I𝑗 for all the models 𝑇 𝑗 ∈ 𝒯 to the same length 𝑙𝑚𝑎𝑥 , and we use
K-means with Euclidean distance for model clustering. Models belonging to different clusters
are expected to have different modeling paradigms of the contributions of different lagged
values to the predictions, which contributes to boosting the ensemble diversity. We select only
cluster representatives to take part in the ensemble. We simply select the closest model to each
cluster center.
2.2.2. Ensemble Adaptation
Streaming time series data is prone to significant changes, leading to concept drifts . To account
for these shifts, the selection of ensemble members must be updated, allowing for the inclusion of
models that can better address newly emerging patterns. Concept drift is detected by monitoring
deviations in the mean of the time series over time, using the Hoeffding Bound to evaluate if
these deviations are significant. If a drift is detected, an alarm is triggered, the TreeSHAP-based
model clustering is updated, and the ensemble is adjusted to reflect the new patterns in the data.
3. Experiments
Our method is denoted in the following as OEP-TT: Online explainable Ensemble Pruning of
Tree models for Time series forecasting.
3.1. Experimental Setup
We use 100 univariate time series datasets from various application domains, including financial,
weather, and synthetic data. These datasets are provided by the Monash Forecasting Repository
[10]. We process each time series 𝑋 by using the first 50% for training (𝑋𝜔𝑡𝑟𝑎𝑖𝑛 ), the following
25% for validation (𝑋𝜔𝑣𝑎𝑙 ) and the remaining 25% for testing. Due to this way of splitting the
time series, we discard series that are shorter than 250 to allow enough training and validation
data. All experiments have been performed on consumer hardware, namely on a 2022 MacBook
Pro in R.
3.2. OEP-TT Setup
Tree-based models set-up: We construct a pool 𝒯 of tree-based models using different
parameter settings that are summarized in Table 1. The list of parameters and their value ranges
Tree-based Model Configurations
Decision Tree (DT) Maximum Depth 𝑑𝑚𝑎𝑥 ∈ {4, 8, 16}
Random Forest (RF) Number of trees 𝑛𝑡𝑟𝑒𝑒𝑠 ∈ {50, 100, 150, 200}
Num. of variables sampled 𝑚𝑡𝑟𝑦 ∈ {3, 5, 7}
at each split
Minimum size of terminal nodes 𝑛𝑜𝑑𝑒𝑠𝑖𝑧𝑒 ∈ {5, 10, 15}
Gradient Boosted DT Number of trees 𝑛𝑡𝑟𝑒𝑒𝑠 ∈ {50, 100, 150, 200}
(GBDT) Maximum depth of each tree 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛.𝑑𝑒𝑝𝑡ℎ ∈ {5, 7, 15}
Shrinkage parameter 𝑠ℎ𝑟𝑖𝑛𝑘𝑎𝑔𝑒 ∈ {0.001, 0.01, 0.1}
eXtreme Gradient Max number of iterations 𝑛𝑟𝑜𝑢𝑛𝑑𝑠 ∈ {50, 100, 150, 200}
Boosting (Xgboost) Step size of each boosting step 𝑒𝑡𝑎 ∈ {0.001, 0.01, 0.1}
Maximum Depth 𝑚𝑎𝑥.𝑑𝑒𝑝𝑡ℎ ∈ {5, 7, 15}
Light GBM (LGBM) Metric 𝑚𝑒𝑡𝑟𝑖𝑐 ∈ {𝐿1 , 𝐿2 }-Regularization
Max number of iterations 𝑛𝑢𝑚𝑖 𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 ∈ {50, 100}
Maximum depth of each tree 𝑚𝑎𝑥𝑑 𝑒𝑝𝑡ℎ ∈ {5, 7, 15}
Table 1
Hyper-parameters values of the tree-based models. Different configurations are generated by taking
some combinations of these hyper-parameters as described in the last column.
in Table 1 is not exhaustive, and further parameters and values can be considered to generate
more base learners. We also vary the lag parameter 𝑙 on which the tree-based models are trained,
i.e., 𝑙 ∈ {3, 5, 7, 10, 15, 20}. Considering different combinations of all the parameters, we train
a total of 294 tree-based models.
OEP-TT set-up: OEP-TT has also a number hyper-parameters: 𝑀 is the Size of the Pool
of the tree-based models T: 294, 𝜔 is the Size of the validation time window :25% of the data
length, |𝑆| is the Number of final selected models: 6.
3.3. State-of-the-Art Methods Setup
We compare OEP-TT against State-of-the-Art (SoA) methods for online ensemble pruning, tree-
based ensembles, and time series forecasting in general. These models include: Auto-Regressive
Integrated Moving Average (ARIMA) [11], Exponential Smoothing (ETS) [11], Long Short-Term
Memory (LSTM) [12], Multi-Layer Perceptron (MLP) [12], Convolutional Neural Network with
LSTM (CNN-LSTM, Bi-LSTM) [13], Random Forest (RF) [14], Gradient-Boosted Decisions
Trees (GBDT) [15], eXtrem Gradient Boosting (XGBoost) [16], and Light Gradient-Boosting
Machine (LGBM) [17].
To enable a fair comparison with OEP-TT, we feed to these ensemble pruning methods
the same pool of tree-based models T that was used for OEP-TT: Ens: Ensemble of all the
base modes in T; OCL [5]: Online drift-aware clustering of the tree-based models in T using
covariance-based clustering; OTOP [5]: Online drift-aware Top best-performing tree-based
models ranking using temporal correlation analysis; DEMSC [5]: Dynamic Ensemble Members
Selection using Clustering: Online drift-aware Top best-performing models ranking using
temporal correlation analysis combined with covariance-based clustering; ADE [18, 19] was
recently developed for an online dynamic ensemble of forecasters construction. A meta-learning
strategy that specializes the tree-based models across the input time series. A sequential
weighting schema is developed to automatically select ensemble members by setting their
weights to zero.
We also compare OEP-TT to its variants: OEP-TT-ST: Static variant of OEP-TT. Pruning is
decided at the initial forecasting instant and kept fixed along testing; OEP-TT-Per: Pruning is
updated periodically in a blind manner (i.e. without considering the occurrence of the drift).
3.4. Results
3.4.1. Predictive Performance
Table 2 presents the average ranks and their deviation for OEP-TT and its variants and SoA
methods for time series forecasting and online ensemble pruning. For the paired comparison, we
compare our method OEP-TT against each of the other methods. We counted wins and losses
for each dataset using the RMSE scores. We use the non-parametric Wilcoxon Signed Rank test
to compute significant wins and losses, which are presented in parenthesis (significance level
0.05). In the results in Table 2, OEP-TT outperforms almost all the baseline methods in terms
of ranks and wins/loses in pairwise comparison.
In this part, we show how initially OEP-TT supports explainability for the reason behind
specific tree-based model selection to construct the ensemble at a specific time instant or interval,
for model performance, and for the importance of input lagged time series observations.
Method Avg. Rank Std Deviation Wins Losses
ETS 16.60 4.53 89 (74) 11 (9)
ARIMA 15.39 5.90 88 (79) 12 (10)
MLP 13.48 5.04 80 (71) 20 (20)
LSTM 11.03 3.35 78 (70) 22 (21)
CNN-LSTM 7.29 2.68 70 (68) 30 (18)
Bi-LSTM 10.32 4.35 75 (74) 25 (20)
RF 10.76 5.04 75 (72) 25 (20)
GBDT 11.84 3.35 78 (78) 22 (18)
LGBM 12.61 2.68 80 (78) 20 (18)
Xgboost 15.54 2.68 87 (84) 13 (12)
ADE 12.68 3.61 79 (75) 21 (21)
OTOP 14.15 3.78 84 (81) 16 (15)
Ens 7.03 3.90 70 (601) 30 (28)
DEMSC 3.13 3.11 58 (57) 42 (30)
OCL 3.02 3.67 63 (61) 37 (27)
OEP-TT-St 2.74 2.85 54 (52) 46 (41)
OEP-TT-Per 1.95 2.78 - (-) - (-)
OEP-TT 2.18 2.19 - (-) - (-)
Table 2
Comparison (in terms of average rank achieved over 100 datasets) between our method and the baselines
The rank column presents the average rank and its standard deviation across different time series. An
average rank of 1 means the model was the best performing on all time series.
3.4.2. Explainability Aspects
Figure 1 shows the TreeSHAP values clusters of the Saugeen River Flow data set. The dots
on each lag value stand for the TreeSHAP values taken by the models belonging to the same
cluster, while the line connects the mean values to show the TreeSHAP values for each lag
of the representative selected model on each cluster (only for visualization purposes). Note
that we show the name of the model and the value of the first hyper-parameter plus the lag
value on which it is trained to distinguish between selected models belonging to the same
family of tree-based models, e.g., RF200(Lag10) and RF50(Lag7). It can be seen that on different
clusters different patterns of lagged values contributions to the target time series observations
are observed. This confirms that our clustering procedure promotes ensemble diversity by
enforcing the selection of models that have different modeling paradigms and distinct views
on the importance of specific lag values. For example, while models in cluster 6 favor higher
lag values and emphasize the contribution of their corresponding value to the output forecast
value, models in cluster 5 are built on the assumption of restricting the memory of the models
to lower lag values (𝑙 = 3). We can notice that in 3 clusters out of 6, models rely on restricted
lagged values (clusters 2, 3, and 5). Even with this limited width of memory, i.e., historical data,
they can excel in terms of predictive performance.
Clustered TreeShap values
1 2
1.00 1.00
0.75 0.75
0.50 0.50
0.25 0.25
0.00 0.00
3 4
Selected Models
1.00 1.00
TreeShap Value
RF200(Lag10)
0.75 0.75
GBM(Lag5)
0.50 0.50 XGboost(Lag3)
RF50(Lag7)
0.25 0.25
LGBM(Lag3)
0.00 0.00 RF150(Lag15)
5 6
1.00 1.00
0.75 0.75
0.50 0.50
0.25 0.25
0.00 0.00
lag15 lag14 lag13 lag12 lag11 lag10 lag9 lag8 lag7 lag6 lag5 lag4 lag3 lag2 lag1 lag15 lag14 lag13 lag12 lag11 lag10 lag9 lag8 lag7 lag6 lag5 lag4 lag3 lag2 lag1
Lag value
Figure 1: Comparison of TreeSHAP-based models clusters on the Saugeen River Flow data set.
4. Concluding Remarks and Future Work
This paper introduces OEP-TT a novel method for online adaptive ensemble of tree-based
models pruning. Through the use of TreeSHAP values, we are able to gain insight into its
decision-making process, both for model selection, as well as for the input time series points
relevance. We showed the advantages of OEP-TT on 100 real-world datasets, both in terms of
predictive performance as well as its explainability aspects. In future work, we plan to extend
our method to hybrid model pools by using the most efficient Shapley value estimation methods
for each model family, such as TreeSHAP for tree-based models, DeepSHAP [8] for Neural
Networks, as well as KernelSHAP [8] for remaining models, tune the size of the ensemble and
dive further into the explainability aspects. Given that this is early-stage work, we plan to
engage with experts in the field to exchange ideas and gather feedback.
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