Continuous Evolution Pool: Taming Recurring Concept Drift in Online Time Series Forecasting

## Abstract

Abstract Recurring concept drift, a type of concept drift in which previously observed data patterns reappear after some time, is one of the most prevalent types of concept drift in time series. As time progresses, concept drift occurs and previously encountered concepts are forgotten, thereby leading to a decline in the accuracy of online predictions. Existing solutions mainly employ parameter updating techniques to delay forgetting; however, this may result in the loss of some previously learned knowledge while neglecting the exploration of knowledge retention mechanisms. To retain all conceptual knowledge and fully utilize it when the concepts recur, we propose the C ontinuous E volution P ool ( CEP ), a pooling mechanism that stores different instances of forecasters for different concepts. Our method first selects the forecaster nearest to the test sample and then learns the features from its neighboring samples—a process we refer to as the retrieval . If there are insufficient neighboring samples, it indicates that a new concept has emerged, and a new model will evolve from the current nearest sample to the pool to store the knowledge of the concept. Simultaneously, the elimination mechanism will enable outdated knowledge to be cleared to ensure the prediction effect of the forecasters. Experiments on different architectural models and eight real datasets demonstrate that CEP effectively retains the knowledge of different concepts. In the scenario of online forecasting with recurring concepts, CEP significantly enhances the prediction results.

## 1 Introduction

Accurate time series forecasting—a fundamental task in fields such as finance, energy management, traffic prediction, and environmental monitoring—empowers better decision-making, optimized resource allocation, and effective risk management (Jin et al., 2024). However, in practical applications, online time series forecasting frequently encounters a substantial challenge known as *concept drift*. This refers to the change in the relationship between input variables and their true values over time. In the context of time series, as time elapses, the underlying patterns and behaviors governing the data can shift, making previously learned models less effective. *Recurring concept drift* is a particularly prevalent and intricate form of concept drift in time series data shown in Figure [1](#S1.F1)(a). It is characterized by the periodic reappearance of certain concepts after a period of absence. For instance, seasonal patterns may cause similar consumption patterns to recur annually or monthly in electricity consumption time series. Nevertheless, as the time series evolves, the model may forget these previously encountered concepts during the non-recurrence periods, leading to a decline in the accuracy of online predictions examplified in Figure [1](#S1.F1)(b).

Figure: Figure 1: Concept recurrence is a notable phenomenon in time series data. For instance, in (a), the second concept recurs following the emergence of the fourth concept. During online updates and concept drifts, forecasters often suffer from forgetting previously learned concepts, as in (b). To address this challenge and effectively track recurring concepts, the Continuous Evolution Pool employs an evolving and retrieval strategy, assigning distinct forecasters to manage different concepts.
Refer to caption: /html/2506.14790/assets/x1.png

Various methods have been devised to tackle concept drift in the realm of online time series forecasting, but each has its own set of limitations. Methods based on Experience Replay (ER) (Chaudhry et al., 2019) such as DER++ (Buzzega et al., 2020), store historical information in a buffer for learning. However, over time and with concept drift, the samples stored in the memory can become obsolete, and the restricted memory size may lead to the forgetting of once emerged concepts, thus degrading performance when adapting to the evolving data distribution. FSNet (Pham et al., 2023), drawing inspiration from the Complementary Learning Systems theory, endeavors to balance rapid adaptation to novel changes and the retrieval of past knowledge. However, in scenarios with sample delays, it encounters forecasting biases, hampering its capacity to accurately capture time series patterns. It also struggles to effectively learn relevant features during intricate concept drifts. OneNet (Zhang et al., 2023) attempts to address concept drift by dynamically updating and integrating models through a reinforcement learning based approach. Nevertheless, it grapples with handling long-term dependencies in time series data, which causes inaccuracies in predicting future trends. In highly volatile data scenarios, its weight adjustment strategy fails to promptly adapt to rapid data changes.

Existing solutions for addressing concept drift in online time series forecasting mainly center around parameter updating techniques. These methods strive to make the model adapt to the changing data distribution by constantly adjusting the model’s parameters. Although, to some extent, these methods can delay the forgetting of past concepts, they have multiple shortcomings. First, parameter updating might lead to the loss of some valuable previously learned knowledge. As the model tries to fit the new data, it could overwrite or distort information related to past concepts. Second, these methods generally overlook the exploration of effective knowledge retention mechanisms. Specifically, they do not explicitly store and utilize the knowledge of different concepts in a way that allows for easy retrieval and reuse when those concepts reappear. Third, experiments on real-world datasets have demonstrated that neural networks can effectively learn under gradual distribution shifts without requiring elaborate techniques (Read, 2018). In the context of concept abruption (Read, 2018), particularly in scenarios involving recurring concepts, there remains considerable room to improve the performance of existing methods. This is primarily because these methods have difficulty effectively capturing and adapting to sudden changes in data patterns, resulting in suboptimal forecasting accuracy. In light of these limitations, two crucial questions arise:

To address these limitations and effectively manage recurring concept drift in online time series forecasting, we propose the Continuous Evolution Pool (CEP). This is a novel pooling mechanism designed to store multiple forecasters corresponding to different concepts. Its core idea involves partitioning samples from distinct distributions and continuously updating separate models for each evolving concept. When a new test sample arrives, CEP selects the forecaster in the pool that is nearest to this sample for prediction, concurrently learning from the features of its neighboring samples. If the neighboring samples are insufficient, this indicates the emergence of a new concept, prompting the evolution of a new model from the closest available sample. This new model is then added to the pool to retain knowledge of the emerging concept. Additionally, CEP employs an elimination mechanism that removes outdated knowledge and filters noisy data, thereby maintaining high forecasting accuracy. The main contributions of this paper are as follows:

- 1.
We identify recurring concept drift as a significant challenge in online time series forecasting. Existing methods that rely on parameter updating frequently forget previously learned concepts during periods of non-recurrence and lack efficient mechanisms for knowledge retention. This results in reduced prediction accuracy when past concepts reappear.
- 2.
We propose the Continuous Evolution Pool (CEP), a novel mechanism specifically designed to address recurring concept drift. By storing and retrieving forecasters associated with distinct concepts, CEP effectively mitigates knowledge loss inherent in traditional methods, enabling the model to leverage previously acquired knowledge more effectively.
- 3.
Through extensive experiments involving multiple real-world datasets and various neural network architectures, we demonstrate that CEP substantially enhances online prediction accuracy in scenarios characterized by recurring concept drift. CEP consistently outperforms existing methods, particularly in managing complex time series data and delay scenarios.

## 2 Related Work

Concept Drift. Unlike conventional time series analysis, where concept drift might be less critical, it is a central concern in online time series forecasting. Suppose the input variables $x$ and their ground truth $y$ follow a distribution $p_{t}$ over time as Equation ([1](#S2.E1)). For two different time points $t_{1}$ and $t_{2}$, the distribution shift can be formally defined by Equation ([2](#S2.E2)) (Read, 2018).

$$ $(x,y)\sim p_{t}(x,y)$ (1) $$

$$ $p_{t_{1}}(y|x)\neq p_{t_{2}}(y|x)$ (2) $$

Concept drift was formally defined (Tsymbal, 2004), with later work emphasizing the need for continuous adaptation in data streams (Read, 2018). Early models exhibited catastrophic forgetting and lacked rapid learning, hindering real-time adaptation and rarely utilizing deep neural networks. Subsequent work includes the AdaRNN (Du et al., 2021), which struggled with complex relationships, RevIN (Kim et al., 2022) designed for distribution shift, and the exploration into online drift detection (Wan et al., 2024). Despite these advancements, significant challenges remain in online time series forecasting, necessitating continuous research.

Online Time Series Forecasting. Unlike traditional methods, online time series forecasting requires continuous learning as new data arrives. Early models (Huszár, 2018; Kirkpatrick et al., 2018) suffered from forgetting past knowledge and lacked rapid adaptation, rarely utilizing deep neural networks. As the field evolved, FSNet (Pham et al., 2023) improved adaptation speed to recent data. To better capture complex relationships and address concept drift, OneNet (Zhang et al., 2023) was introduced, combining cross-variable/cross-time modeling and online ensembling. Despite these advances, challenges persist, requiring ongoing research for improved performance and adaptability.

## 3 Methodology

In this section, we introduce the proposed CEP. It helps partition samples from different distributions, enabling the updating of different models for evolution. We describe how to utilize the dynamic evolution mechanism for the recurring concept scenario. For Q1, the evolution mechanism is introduced that assign the different concepts for different forecasters. For Q2, some adaptation techniques such as Elimination are introduced to manage the previously learned concepts. The detailed notations are summarized in Appendix Table [4](#A1.T4).

Data Format. A univariate time series is represented as $x=(x_{1},x_{2},\ldots,x_{N})$. Consider a look-back window of length $\mathbf{L}$, which is denoted as $\mathbf{x}_{i}=(x_{i},x_{i+1},\ldots,x_{i+\mathbf{L}-1})$. The corresponding ground truth $\mathbf{y}_{i}$ with a forecasting horizon of $\mathbf{H}$ is defined as $\mathbf{y}_{i}=(x_{i+\mathbf{L}},x_{i+\mathbf{L}+1},\ldots,x_{i+\mathbf{L}+\mathbf{H}-1})$. Each pair of $\mathbf{x}_{i}$ and $\mathbf{y}_{i}$ forms an instance pair expressed as $\mathcal{D}_{i}=(\mathbf{x}_{i},\mathbf{y}_{i})$.

Problem Formulation. In contrast to traditional time series forecasting, online forecasting is partitioned into two distinct stages: (1) the warm-up stage and (2) the online stage. The dataset was divided into the two stages at a specific ratio. Once the dataset $x$ is split into these two stages, sample pairs are constructed for each of them. The warm-up stage setting is consistent with traditional time series forecasting. In the online stage, the step size of the instance pair is set as forecasting horizon in Figure [2](#S3.F2). This setting effectively prevents the forecasting of the ground truth in the overlapping parts, which is in line with the setup of delayed feedback adopted in previous research (Pham et al., 2023; Zhang et al., 2023). The batch size and epoch for the online stage are both set to $1$. For *univariate time series without auxiliary information from other channels*, it is necessary to focus on the conceptual changes of the single channel itself. Previous research (Zhang et al., 2023) has shown that time-channel independent online forecasting can reduce interference from too many channels.

Figure: Figure 2: Delayed feedback setting. The blue region represents the input, while the yellow region denotes the corresponding ground truth. In the case of no delay (depicted in the third row), the model is tasked with predicting a portion of the ground truth. This situation is considered *unfair* as it does not accurately reflect the real-world scenario where a delay is often present, potentially leading to misleading evaluation results.
Refer to caption: /html/2506.14790/assets/x2.png

### 3.1 Warm-Up Stage

Pool Initialization. We first initialize a pool set $Pl=\{(f(\theta_{1}),\mathbf{g}_{1})\}$ with a single element. Here, $f(\theta_{1})$ represents the initialized ancestor forecaster with parameter $\theta_{1}$, and $\mathbf{g}_{1}$ is the corresponding gene vector that stores the features of the instances. The gene vector is composed of two parts with a gene ratio $\tau_{g}$ as defined in Equation ([3](#S3.E3)): the global gene vector $\tilde{\mathbf{g}}$ and the local gene vector $\check{\mathbf{g}}$. The global gene $\tilde{\mathbf{g}}$ is designed to capture and record long-term macroscopic features, which are crucial for forecasting. In contrast, the local gene $\check{\mathbf{g}}$ is responsible for documenting short-term features that are strongly correlated with the current time. During the warm-up stage, which has the same task as typical time series forecasting, CEP employs the ancestor forecaster to make forecasts and accumulate knowledge for the online stage with recurring concepts.

$$ $\mathbf{g}=\tau_{g}*\check{\mathbf{g}}+(1-\tau_{g})*\tilde{\mathbf{g}}$ (3) $$

Adaptive Updating. Upon the arrival stream of the instance $\mathbf{x}_{i}$, the gene $\mathbf{g}_{1}$ undergoes updates in in Figure [3](#S3.F3)(a). The genes of the forecaster will be adaptively updated as the instance stream arrives and will remain stable after a period of time. Instances are endowed with genes. Forecaster genes are updated by leveraging instance genes capable of recognizing recurring concepts. The instance gene $\mathbf{x}$ is precisely defined by the mean $\mu$ and the standard deviation $\sigma$ as presented in Equation ([4](#S3.E4)). When the look-back window is too long, the most recent scope time step $S$ for the instance is limited. To rapidly adapt to short-term fluctuations in the data, the model employs a weighted approach with local ratio $\tau_{l}$ for updating the local genes, as described in Equation ([5](#S3.E5)). Meanwhile, the global genes are updated using the statistical formulas in Equation ([6](#S3.E6)), which accurately calculate the mean and variance based on all previous samples, thereby allowing for the estimation of the overall trend.

$$ $\mathbf{g}_{x}=\text{Gene}(x)=(\mu(\mathbf{x}[-S:]),\sigma(\mathbf{x}[-S:]))$ (4) $$

$$ $\check{\mathbf{g}}\leftarrow\tau_{l}*\mathbf{g}_{x}+(1-\tau_{l})*\check{\mathbf{g}}$ (5) $$

$$ $\tilde{\mathbf{g}}\leftarrow(\frac{n*\check{\mathbf{g}}_{\mu}+\mathbf{g}_{x,\mu}}{n+1},\sqrt{\frac{n*\check{\mathbf{g}}_{\sigma}^{2}+\mathbf{g}_{x,\sigma}^{2}+(\check{\mathbf{g}}_{\sigma}-g_{x,\sigma})^{2}}{n+1}})$ (6) $$

Additionally, *adaptive updating is not only limited to the warm-up stage but also applies to the online stage*. In the case of continuous concept drift shown in Section [4.2](#S4.SS2), the forecaster can adapt naively. This should not lead to the evolution of more forecasters, causing the samples to be dispersed, and making it difficult for all forecasters to be well-trained. The updating behaviour can enable the forecaster to follow the slight and continuous changes in the data.

Figure: Figure 3: Illustration of the proposed CEP mechanism: (a) Adaptive Updating: Continuously update the position of the evolved forecaster in the gene space as new instances arrive. (b) Nearest Evolution: If the instance surpasses the forecaster’s evolution threshold, evolve and replicate the closest forecaster at the instance. (c) Optimizer Adjustment: Adjust the learning rate for the shifted concept to ensure accurate adaptation. (d) Forecaster Elimination: The forecaster $f_{4}$ associated with the rarely-occurring noise concept may be removed due to prolonged inactivity. (e) Nearest Retrieval: Identify the nearest forecaster in the gene space when an input instance is encountered.
Refer to caption: /html/2506.14790/assets/x3.png

### 3.2 Online Stage

Nearest Retrieval. Upon the arrival of a new instance $\mathbf{x}$, the gene $\mathbf{g}_{x}$ of this instance is computed. Subsequently, to match recurring concepts, the forecaster closest to the instance’s gene is retrieved, as depicted in Figure [3](#S3.F3)(e). The distance between genes is simply measured by Euclidean distance, as presented in Equation ([7](#S3.E7)). The nearest forecaster $f(\theta_{N})$ defined in Equation ([8](#S3.E8)) will be retrieved for either splitting or forecasting.

$$ $d(\mathbf{g}_{x},\mathbf{g})=\left\lVert\mathbf{g}_{x}-\mathbf{g}\right\rVert_{2}$ (7) $$

$$ $(f(\theta_{N}),\mathbf{g}_{N})=\underset{(f(\theta_{i}),\mathbf{g}_{i})\in Pl}{\text{argmin}}\ d(\mathbf{g}_{x},\mathbf{g}_{i})$ (8) $$

Nearest Evolution. Given that the instances in online time series forecasting arrive incrementally, the features of different concepts may exhibit instability during the initial stages. To ensure the stability of the gene, each forecaster in the pool $Pl$ is assigned a safety period $\tau_{safe}$. During the online phase, if the number of forecasting instances completed by $f(\theta_{N})$ is fewer than the safety period, no splitting operation is executed. Conversely, when the number of completed forecasting instances by $f(\theta_{N})$ reaches or exceeds the safety period, the splitting threshold $\tau$ is determined. Once this threshold is exceeded, it implies that a distribution shift has occurred in the current instance. The threshold is established to detect the recurring concepts. The threshold criteria $\tau_{\mu}$ and $\tau_{\sigma}$ for the mean and standard deviation are defined in Equation ([9](#S3.E9)) and Equation ([10](#S3.E10)), respectively. If any element in the gene surpasses the threshold, the evolution process, as described in Equation ([11](#S3.E11)), is initiated. Upon evolution, a split forecaster is added to the evolution pool $Pl$. The newly created forecaster inherits the parameters of the nearest forecaster $f(\theta_{N})$, and simultaneously, its corresponding gene inherits the gene $g_{x}$ of the input instance, as shown in Figure [3](#S3.F3)(b).

Subsequently, the split forecaster will be tasked with forecasting the instance $\mathbf{x}$ and updating its parameters. *It is important to note that in online forecasting, the normalization layer typically maintains the current concept. For CEP, the evolution mechanism serves a similar purpose.* On one hand, the evolution mechanism ensures that the samples learned by the forecasters in the pool belong to individual concepts. On the other hand, CEP does not focus on the internal design of the forecaster, aiming to enhance generalization and accommodate a wider range of forecasters.

$$ $|\mathbf{g}_{x,\mu}-\mathbf{g}_{N,\mu}|\leq\tau_{\mu}*\mathbf{g}_{N,\sigma}$ (9) $$

$$ $\mathbf{g}_{x,\sigma}\leq\tau_{\sigma}*\mathbf{g}_{N,\sigma}$ (10) $$

$$ $Pl\leftarrow Pl\cup\{(f(\theta_{N}),\mathbf{g}_{x})\}$ (11) $$

Forecaster Elimination. In time series analysis, it is not uncommon to encounter situations where specific concepts do not recur over an extended duration. When there is a significant temporal gap between concepts, the previously employed forecaster fails to capture the local information pertinent to the current period. As a result, if a forecaster remains idle for an extended period, it will be removed from the pool, as illustrated in Figure [3](#S3.F3)(d). Eliminating these idle forecasters serves two important purposes. Firstly, it mitigates the potential errors stemming from time series fluctuations. Such fluctuations could otherwise trigger the erroneous evolution of forecasters, leading to inaccurate predictions. Secondly, it conserves storage space within the pool, ensuring efficient resource utilization. To implement this elimination process, a threshold $\tau_{e}$ is established. Let $n_{wait}$ represent the number of instances that have occurred since the last usage of the forecaster, and $n_{pred}$ denote the total number of instances forecasted by the forecaster. If $n_{wait}$ exceeds the forecaster threshold, as defined in Equation ([12](#S3.E12)), the forecaster will be expunged from the pool, as shown in Figure [3](#S3.F3)(d). Similar to the concept drift phenomenon observed during forecaster updates, highly volatile noise data can cause the emergence of inaccurate forecasters. *The elimination mechanism plays a crucial role in promptly identifying and removing these erroneously evolved forecasters.* Moreover, it acts as a safeguard, preventing noise data from corrupting normal forecasters and maintaining the overall integrity of the forecasting system.

$$ $n_{wait}>\tau_{e}*n_{pred}$ (12) $$

Optimizer Adjustment. The continuous evolution pool is a dynamically adjusting forecaster set. Owing to the purification and elimination processes of the forecasters, the parameters within the optimizer will change dynamically. To optimize the new parameters, the optimizer can employ inspiration strategies: Reinitialize a parameter group using the parameters of the nearest forecaster. When the concept shift in the time series is not gradual but abrupt, certain techniques should be adopted to enable the evolving forecaster to adapt more accurately to new concept, as depicted in Figure [3](#S3.F3)(c). During the initial parameter update, adjust learning rate $lr$ based on $lr=\tau_{lr}*lr_{raw}$, where $\tau_{lr}$ denotes the adjustment ratio of the learning rate. Subsequently, gradually and smoothly restore the learning rate, as defined in Equation ([13](#S3.E13)), where $t_{lr}$ represents the inspiration time.

$$ $lr\leftarrow max(lr_{raw},\tau_{lr}^{-\frac{1}{t_{lr}}}*lr)$ (13) $$

Gradient Abandonment. Consider an instance pair $\mathcal{D}=(\mathbf{x},\mathbf{y})$. Once the forecaster within the evolution pool makes a forecast using the input $\mathbf{x}$, the corresponding ground truth $\mathbf{y}$ is obtained. The crucial step is to ascertain whether the gene $\mathbf{g}_{y}$ of the ground truth will trigger the subsequent split, as defined by Equation ([9](#S3.E9)) and Equation ([10](#S3.E10)). If it is determined that $\mathbf{g}_{y}$ will lead to the next split, the forecaster will refrain from parameter updating. The underlying reasoning is that a distribution shift is anticipated to occur in the subsequent time step. By abstaining from parameter updates under such circumstances, it can effectively preclude the instances with the shifted concept from corrupting the original forecaster. This approach ensures the integrity and accuracy of the forecaster’s performance in relation to the original data distribution.

### 3.3 Hyperparameter Configuration

The thresholds in our user-driven algorithm are configurable, allowing performance to be tuned; for example, the mean threshold $\tau_{\mu}$ in Equation ([9](#S3.E9)) influences splitting frequency, with lower values promoting more aggressive divisions similar to temperature. This study employs empirically determined parameters. Specifically, $\tau_{\mu}$ is set to $3$, aligning our splitting criterion with the statistical convention of a $3\sigma$ rule, commonly used in Z-test to detect significant deviations from an established mean based on its standard deviation. This facilitates a statistically grounded approach to identifying shifts warranting new forecaster evolution. At the same time, we conducted sensitivity tests for different thresholds to provide a reference for the performance of the CEP in Appendix Section [D.7](#A4.SS7).

## 4 Experiments

### 4.1 Experiment Setting

Datasets. During the experiments, we employed a diverse array of datasets to comprehensively evaluate the time series forecasting model. These datasets encompass: the ECL dataset (Trindade, 2015), the ETT dataset (comprising 4 distinct subsets) Zhou et al. (2021), the Exchange dataset (Wu et al., 2021), the Traffic dataset (Wu et al., 2021), and the WTH dataset (Wu et al., 2021).

Baselines. In our experiments, we evaluated multiple baselines across the domains of continual learning, time series forecasting, and online learning. The Experience Replay (ER) method (Chaudhry et al., 2019) stores historical data in a buffer and interleaves it with new samples during the learning process. DER++ (Buzzega et al., 2020) incorporates a knowledge distillation strategy to enhance performance. FSNet (Pham et al., 2023) utilizes a fast-slow learning mechanism to capture both short-term changes and long-term patterns in online time series forecasting. OneNet (Zhang et al., 2023) improves time series forecasting under concept drift through online ensembling, which combines multiple models to adapt to evolving data distributions and boost performance. For time series forecasting, we screened models based on various backbones, including: Crossformer (Zhang and Yan, 2023), DLinear (Zeng et al., 2023), TimesNet (Wu et al., 2023), PatchTST (Nie et al., 2023), SegRnn (Lin et al., 2023), iTransformer (Liu et al., 2024), and TimeMixer (Wang et al., 2024). In all benchmarks, the length of the look-back window was set to $\mathbf{L}=60$, and the forecast horizon was varied as $\mathbf{H}\in\{1,30,60\}$. The metric used for evaluation were Mean Squared Error (MSE).

### 4.2 Main Results

#### 4.2.1 Gradual Concept Shift

An experiment was carried out to predict $\mathbf{H}=1$, *aiming to verify that the baseline models can inherently adapt to gradual distribution shifts.* The radar plot depicting the experimental results is presented in Figure [5](#S4.F5). Analysis of the plot reveals that the existing online forecasting methods do not uniformly outperform others across the given dataset. The experience replay-based method (Buzzega et al., 2020) achieved state-of-the-art (SOTA) results on four datasets, while the SOTA results for the remaining four datasets were obtained by naive models. FSNet (Pham et al., 2023) and OneNet (Zhang et al., 2023) leverage gradient information. However, the continuous nature of the distribution shift causes the learning of relevant features to be less effective. Moreover, the forecasting errors of all models are within an acceptable range and are relatively low. Consequently, *in the scenario of continuous distribution shift with a small forecasting horizon, elaborate techniques to mitigate the impact of concept shift are unnecessary.*

Figure: Figure 4: Result of gradual concept shift. Baselines are naive models. The full results are presented in Appendix Table [7](#A4.T7).
Refer to caption: /html/2506.14790/assets/x4.png

#### 4.2.2 Comparison with Naive Forecasting Models

Experiment results are presented in Table [1](#S4.T1). The proposed CEP generally reduces the MSE in forecasting across diverse datasets and model architectures. This reduction in MSE indicates that CEP effectively captures and utilizes the features of different concepts, thereby enhancing the performance of online time series forecasting in scenarios with recurring concept drift. Importantly, CEP’s effectiveness is not limited to a single architecture or dataset, demonstrating its broad applicability. Notably, the Crossformer model (Zhang and Yan, 2023) experiences a particularly significant improvement in univariate online forecasting scenarios, with the MSE reduction exceeding 25%. This substantial improvement can be attributed to the unique characteristics of Crossformer, which leverages both time and variable dimensions in its design. In contrast, models like PatchTST (Nie et al., 2023) rely solely on the time dimension, potentially limiting their ability to adapt to complex concept drifts. However, in rare cases, CEP may not yield the expected improvements. One possible reason is that, to ensure a fair comparison, a single set of hyperparameters was used across all experiments. This approach, while maintaining fairness, may not be optimal for every dataset, as different datasets may require tailored hyperparameter settings to achieve the best performance.

**Table 1: The averaged MSE errors of different forecasters when using CEP are presented. Enhanced and reduced outcomes are marked with and respectively; enhancements exceeding 15% are highlighted in . The best and second-best performances are highlighted. The full results are presented in Appendix Table [8](#A4.T8).**
| Data | TimeMixer | +CEP | iTransformer | +CEP | PatchTST | +CEP | DLinear | +CEP | SegRNN | +CEP | TimesNet | +CEP | Crossformer | +CEP | TCN | +CEP |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ECL | 0.294 | 0.271 | 0.275 | 0.256 | 0.330 | 0.291 | 0.307 | 0.299 | 0.278 | 0.274 | 0.313 | 0.280 | 0.530 | 0.442 | 0.355 | 0.316 |
| - | -7.84% | - | -7.09% | - | -11.97% | - | -2.61% | - | -1.44% | - | -10.70% | - | -16.70% | - | -11.00% |  |
| ETTh1 | 0.337 | 0.324 | 0.328 | 0.326 | 0.331 | 0.323 | 0.337 | 0.308 | 0.304 | 0.300 | 0.355 | 0.333 | 0.636 | 0.465 | 0.454 | 0.445 |
| - | -3.86% | - | -0.61% | - | -2.27% | - | -8.75% | - | -1.48% | - | -6.20% | - | -26.83% | - | -2.09% |  |
| ETTh2 | 2.713 | 2.526 | 2.547 | 2.518 | 2.667 | 2.467 | 2.717 | 2.630 | 2.629 | 2.584 | 2.642 | 2.564 | 7.743 | 7.440 | 3.201 | 2.811 |
| - | -6.91% | - | -1.14% | - | -7.52% | - | -3.22% | - | -1.73% | - | -2.95% | - | -3.91% | - | -12.18% |  |
| ETTm1 | 0.778 | 0.747 | 0.777 | 0.738 | 0.762 | 0.720 | 0.849 | 0.751 | 0.718 | 0.715 | 0.818 | 0.820 | 2.439 | 2.267 | 0.973 | 0.798 |
| - | -3.92% | - | -5.02% | - | -5.52% | - | -11.54% | - | -0.49% | - | 0.31% | - | -7.05% | - | -17.99% |  |
| ETTm2 | 0.231 | 0.232 | 0.241 | 0.236 | 0.255 | 0.245 | 0.242 | 0.234 | 0.233 | 0.227 | 0.248 | 0.241 | 0.416 | 0.347 | 0.266 | 0.257 |
| - | 0.43% | - | -2.07% | - | -4.12% | - | -3.31% | - | -2.58% | - | -2.82% | - | -16.71% | - | -3.20% |  |
| Exchange | 0.358 | 0.344 | 0.376 | 0.361 | 0.354 | 0.346 | 0.412 | 0.381 | 0.296 | 0.294 | 0.470 | 0.393 | 2.168 | 1.533 | 0.516 | 0.401 |
| - | -3.91% | - | -3.86% | - | -2.26% | - | -7.53% | - | -0.68% | - | -16.40% | - | -29.29% | - | -22.31% |  |
| Traffic | 0.512 | 0.500 | 0.521 | 0.492 | 0.677 | 0.587 | 0.628 | 0.628 | 0.795 | 0.783 | 0.559 | 0.563 | 0.668 | 0.645 | 0.721 | 0.711 |
| - | -2.44% | - | -5.48% | - | -13.30% | - | 0.00% | - | -1.51% | - | 0.72% | - | -3.45% | - | -1.32% |  |
| WTH | 0.359 | 0.356 | 0.355 | 0.354 | 0.367 | 0.353 | 0.342 | 0.341 | 0.342 | 0.342 | 0.370 | 0.376 | 0.399 | 0.394 | 0.381 | 0.381 |
| - | -0.84% | - | -0.42% | - | -3.81% | - | -0.29% | - | -0.15% | - | 1.49% | - | -1.13% | - | 0.00% |  |

#### 4.2.3 Comparison with Online Forecasting Models

Experiment results are presented in Table [2](#S4.T2). ER, DER++, FSNet, and OneNet did not yield satisfactory performance in delayed scenarios. Experience replay methods (Chaudhry et al., 2019; Buzzega et al., 2020) reuse instances stored in memory to update the model. However, as time progresses, due to the limited memory capacity, the samples in memory may fail to retain the concepts that emerged earlier because of concept drift. FSNet (Pham et al., 2023) is prone to forecasting biases, an inevitable consequence of the delayed arrival of samples. OneNet (Zhang et al., 2023) takes short-term information into account to adjust model parameters, thus optimizing forecasting results to some degree in delayed scenarios. To preserve the knowledge of different concepts comprehensively, the proposed CEP employs a pooling mechanism to assign different concepts to distinct models. In delay-feedback scenarios, nearly all the improvements achieved by CEP significantly outperform those of previous time series methods. Specifically, the MSE is reduced by over $20\%$. This suggests that in delay scenarios, especially those with recurring concepts, previous online methods do not perform as effectively as anticipated and may even be inferior to the naive model shown in Table [2](#S4.T2). In contrast, CEP can effectively manage recurring concepts.

**Table 2: The average MSE error of previous online forecasting methods. The previous methods all integrated TCN according to the original settings, and CEP’s forecaster is also TCN. Enhanced and reduced outcomes are marked with and respectively; enhancements exceeding 15% are highlighted in . The best and second-best performances are highlighted. Full results are presented in Appendix Table [9](#A4.T9).**
| Data | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| TCN | 0.355 | 0.454 | 3.201 | 0.973 | 0.266 | 0.516 | 0.721 | 0.381 |
| ER | 0.636 | 0.498 | 3.346 | 1.358 | 0.329 | 3.531 | 1.062 | 0.572 |
| 79.41% | 9.58% | 4.53% | 39.57% | 23.73% | 584.87% | 47.40% | 50.33% |  |
| DER++ | 0.588 | 0.461 | 3.150 | 1.199 | 0.308 | 3.051 | 1.009 | 0.549 |
| 65.73% | 1.54% | -1.61% | 23.23% | 15.82% | 491.85% | 40.04% | 44.15% |  |
| FSNet | 0.863 | 0.452 | 3.800 | 1.562 | 0.349 | 4.770 | 1.559 | 0.754 |
| 143.30% | -0.44% | 18.70% | 60.53% | 31.26% | 825.32% | 116.38% | 98.16% |  |
| OneNet | 0.401 | 0.439 | 3.505 | 1.114 | 0.294 | 1.162 | 0.679 | 0.421 |
| 13.12% | -3.30% | 9.50% | 14.49% | 10.73% | 125.41% | -5.76% | 10.64% |  |
| CEP | 0.316 | 0.445 | 2.811 | 0.798 | 0.257 | 0.401 | 0.711 | 0.381 |
| -11.00% | -2.09% | -12.18% | -17.99% | -3.20% | -22.31% | -1.32% | 0.00% |  |

### 4.3 Visualization

Figure [8](#S4.F8) depicts the forecasting results of the naive model and the CEP. After the concept reappeared (instance 2-14), the MSE of the CEP did not increase notably in Figure [8](#S4.F8), while the MSE of the naive model rose significantly after the concept shift. Moreover, we visualized the operational details of the CEP. In Figure [8](#S4.F8), the forecasts of different concepts in the CEP are marked with distinct colors, and the CEP promptly and precisely identified different concepts.

Figure: Figure 6: Visualization of forecasting results
Refer to caption: /html/2506.14790/assets/img/vis_value.png

### 4.4 Computation Complexity

We have collected the performance metrics of online methods and CEP, including model size and running time, and visualized them in Figure [5](#S4.F5). It can be observed that FSNet does not yield positive results, and OneNet integrates two FSNet models, resulting in a larger number of model parameters. Moreover, due to the large time delay, other methods have a negative impact on the naive TCN model. Only CEP improves the prediction accuracy of TCN and maintains a similar number of parameters to the naive TCN at the end of the online stage. This is attributed to CEP’s elimination mechanism, which removes outdated models. Although CEP may have many forecasters in a short period of time, leading to a temporary large memory occupation, in the long run, the elimination mechanism ensures the space efficiency of CEP. Detailed analysis of the performance of CEP is presented in Appendix [C](#A3).

### 4.5 Model Analysis

Simultaneously, we carried out elaborate ablation experiments on CEP. The experimental results are presented in Table [3](#S4.T3). Taking TCN as an instance, if the forecaster does not undergo evolution, the prediction error will increase to a certain degree. Other mechanisms are devised to render the evolution more congruent with actual data. Thus, the impacts of different mechanisms on diverse datasets may differ. It can be observed that the fast gene and the slow gene exhibit distinct performances on various datasets. Consequently, CEP integrates the effects of both and updates accordingly. The same principle holds for other mechanisms. Comprehensive ablation results and other experimental details, encompassing the outcomes of parameter sensitivity, are provided in Appendix [D](#A4).

**Table 3: The MSE error of the ablation experiments with forecaster TCN. The results are the mean values of different backbone forecasters. The best and second-best performances are highlighted. Full results are presented in Appendix Table [10](#A4.T10).**
| Dataset | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $\mathbf{H}$ | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 |
| CEP | 0.266 | 0.365 | 0.384 | 0.505 | 2.173 | 3.449 | 0.631 | 0.965 | 0.206 | 0.308 | 0.262 | 0.539 | 0.698 | 0.724 | 0.313 | 0.448 |
| w/o Evolution (Naive TCN) | 0.300 | 0.409 | 0.378 | 0.530 | 2.536 | 3.866 | 0.748 | 1.198 | 0.205 | 0.326 | 0.381 | 0.650 | 0.710 | 0.731 | 0.317 | 0.444 |
| w/o Fast Gene | 0.271 | 0.365 | 0.400 | 0.516 | 2.209 | 3.626 | 0.631 | 0.965 | 0.209 | 0.308 | 0.363 | 0.676 | 0.698 | 0.724 | 0.313 | 0.441 |
| w/o Slow Gene | 0.265 | 0.379 | 0.384 | 0.505 | 2.305 | 3.461 | 0.631 | 1.087 | 0.206 | 0.308 | 0.270 | 0.528 | 0.698 | 0.724 | 0.310 | 0.427 |
| w/o Elimination | 0.283 | 0.392 | 0.384 | 0.505 | 2.186 | 3.421 | 0.631 | 0.965 | 0.202 | 0.308 | 0.266 | 0.539 | 0.698 | 0.724 | 0.313 | 0.448 |
| w/o Inspiration | 0.273 | 0.379 | 0.333 | 0.456 | 2.316 | 3.577 | 0.656 | 0.930 | 0.204 | 0.311 | 0.283 | 0.512 | 0.698 | 0.724 | 0.313 | 0.438 |
| w/o Gradient Abandonment | 0.289 | 0.379 | 0.382 | 0.500 | 2.187 | 3.547 | 0.598 | 0.931 | 0.205 | 0.313 | 0.270 | 0.536 | 0.698 | 0.724 | 0.313 | 0.443 |

## 5 Conclusion

Recurring concept drift is common in time series data, with naive models in online forecasting often losing previously learned concept info during long-term predictions and parameter updates, making it hard for them to predict effectively when concepts reoccur. The CEP mechanism addresses this by assigning different concept instances to separate forecasters, reducing knowledge forgetting, evolving a new forecaster from the nearest existing one when a new concept appears using it as prior knowledge, and incorporating an elimination mechanism to avoid reusing noisy and outdated concepts. Experiments first looked at gradual concept shift with a small forecasting horizon, where the naive model performed similarly to previous online methods with low errors, suggesting it can adapt to continuous shifts without complex techniques. For long-horizon forecasting, CEP significantly boosts prediction performance across common model architectures, with over 25% improvement for Crossformer cross-variable models, while FSNet, OneNet, and experience replay methods can’t effectively deal with the forgetting problem due to time delays. In multivariate time series, different variables’ concept shifts may not be synchronized, and future research could explore efficiently using multivariate information and maintaining prediction performance during partial concept shifts.

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## Appendix A Notations

In this section, we provide a summary of the notation used in the main paper, particularly in the method section, for quick reference. Table [4](#A1.T4) lists the key symbols and their corresponding descriptions.

**Table 4: Summary of important notations.**
| Notation | Description | Hyperparameter |
| --- | --- | --- |
| $p$ | Probability distribution of time series | ✗ |
| $x$ | Original time series |  |
| $\mathbf{x}_{i}$ | Input window |  |
| $\mathbf{y}_{i}$ | Ground truth window |  |
| L | Input window length |  |
| H | Forecasting horizon |  |
| $\mathcal{D}_{i}$ | Input and ground truth window pair: Data pair |  |
| $\mathcal{D}$ | Data pairs |  |
| $\widetilde{\mathcal{D}}_{i}$ | Data pairs of concept $i$ |  |
| $Pl$ | Forecaster Pool |  |
| $f(\cdot)$ | Forecaster |  |
| $\theta_{i}$ | Forecaster parameter of concept $i$ |  |
| $\mathbf{g}_{i}$ | Gene of concept $i$ |  |
| $\check{\mathbf{g}}$ | Global gene |  |
| $\tilde{\mathbf{g}}$ | Local gene |  |
| $\mathbf{g}_{x}$ | Sample gene |  |
| $\mu$ | Mean |  |
| $\sigma$ | Standard deviation |  |
| $S$ | Scope of $\mu$ and $\sigma$ |  |
| $n$ | Number of historical samples |  |
| $d(\cdot,\cdot)$ | Distance of gene |  |
| $N$ | Nearest neighbor |  |
| $n_{wait}$ | Number of wait time of forecaster |  |
| $n_{pred}$ | Number of prediction of forecaster |  |
| $lr$ | Learning rate |  |
| $\tau_{g}$ | Global gene ratio | ✓ |
| $\tau_{l}$ | Fast gene ratio |  |
| $\tau_{\mu}$ | Mean threshold for splitting |  |
| $\tau_{\sigma}$ | Standard deviation threshold for splitting |  |
| $\tau_{safe}$ | Intialization time of forecaster |  |
| $\tau_{e}$ | Elimination threshold of forecaster |  |
| $\tau_{lr}$ | Adjustment ratio of learning rate |  |

## Appendix B Technical Details

The motivation behind continuous learning is to continuously learn the characteristics of emerging concepts. Specifically, Experience Replay [Chaudhry et al., 2019] replays samples with different labels. This ensures that the trained model does not forget the categories it has previously learned. In contrast, the continuous evolution pool adopts a distinct strategy. It allows different models to learn from different data distributions.

Static Scenario. Consider a dataset $x$. Suppose the set of generated instance pairs is denoted as $\mathcal{D}=\{\mathcal{D}_{1},\mathcal{D}_{2},\ldots,\mathcal{D}_{N}\}$. The continuous evolution pool operates by essentially partitioning the set $\mathcal{D}$. Specifically, instance pairs sharing the same distribution are grouped together into a single set $\widetilde{\mathcal{D}}_{i}=\{\mathcal{D}_{j}|\mathcal{D}_{j}\sim p_{i}\}$. The subsets $\widetilde{\mathcal{D}}_{i}$ constructed in this way exhibit no distribution drift within the set. By learning an initial model for these different subsets, incremental learning can be efficiently performed, as shown in Algorithm [1](#algorithm1) and Algorithm [2](#algorithm2).

Online Scenario. However, in an online scenario, the data poses three main challenges. First, the data is imbalanced. Second, due to privacy requirements, instance pairs cannot be stored during the online learning process. Third, instance pairs from different distributions do not arrive in a sequential order. When new instances of a particular distribution arrive, the output needs to be promptly generated, and a model that has not been initialized with the new data cannot be used. Therefore, in Algorithm [3](#algorithm3) and Algorithm [4](#algorithm4), a model corresponding to the most similar distribution that has already occurred should be selected as the base model. In online time series, since the data is continuous, some mechanisms specific to time series need to be incorporated. As mentioned in Section [3](#S3), CEP is presented in Algorithm [5](#algorithm5).

Figure: Algorithm 1 Discontinuous Evolution Pool: Training Stage

Figure: Algorithm 2 Discontinuous Evolution Pool: Test Stage

Figure: Algorithm 3 Lite Continuous Evolution Pool: Warm-up Stage

Figure: Algorithm 4 Lite Continuous Evolution Pool: Online Stage

Figure: Algorithm 5 Continuous Evolution Pool

## Appendix C Complexity Analysis

An in-depth efficiency analysis of online evolutionary pools has been carried out. Considering the final output structure is identical to the original model, the time complexity per prediction step is the same as that of the original model, denoted as $O(k)$, where $k$ represents the computational operations required by the original model for a single input instance.

Regarding spatial complexity, this can be seen as a limitation of the online evolutionary pool. Assuming the evolutionary pool does not shrink, the spatial complexity is $O(|\widetilde{\mathcal{D}}|\cdot k)$. Here, $|\widetilde{\mathcal{D}}|$ represents the number of distinct subsets into which the dataset is theoretically divided, and $k$ is the space required to store the parameters of a single model with the same structure as the original model. This is because multiple models with the same structure are trained on these subsets. As the number of distinct subsets $|\widetilde{\mathcal{D}}|$ grows, the spatial complexity increases linearly, which can lead to significant memory consumption and potential resource bottlenecks.

Nevertheless, there is no need to overly worry about the expansion of spatial complexity. This is because the elimination mechanism can effectively address this issue. The elimination mechanism will be elaborated in the context of online time series forecasting. By removing models that are no longer needed or that have low utility, the elimination mechanism can keep the size of the evolutionary pool under control, thereby reducing the overall spatial complexity and making the online evolutionary pool more efficient and practical. Figure [9](#A3.F9) is the visualization of the number of forecasters stored in the CEP during the online stage, which can specifically reflect the spatial complexity in the experiment.

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/traj/ECL_30.png

Figure: (k) Exchange, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/traj/Exchange_30.png

## Appendix D Experiment Details

### D.1 Dataset

During the experiments, we used a diverse range of datasets to comprehensively assess the time series forecasting model. ETT, which covers two years, monitors the oil temperature and six power load characteristics. ETTh2 logs data hourly, while ETTm1 records at 15-minute intervals, presenting multiple granularities for the analysis of electricity-related time series. Electricity (ECL) dataset, which monitors the electricity consumption of 321 clients from 2012 to 2014, facilitates the exploration of intricate consumption patterns. Weather (WTH) monitored 11 climatic aspects at around 1,600 spots throughout the United States every hour from 2010 to 2013. Traffic, sourced from the California Department of Transportation, supplies hourly freeway occupancy rates in the San Francisco Bay Area, which is beneficial for traffic flow forecasting. Exchange, spanning from 1990 to 2016 and containing the daily exchange rates of eight countries, is instrumental in predicting the fluctuations in the foreign exchange market [Wu et al., 2021]. All the datasets were used with a single channel with the recurring concept shift in Table [5](#A4.T5). In the experiment, the ratio of the data set in the warm-up stage and online stage is $25:75$.

**Table 5: Dataset statistics.**
| Data | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Feature | MT_104 | LULL | LUFL | LULL | MUFL | OT | 16 | DewPointFarenheit |
| Timestep | 26,304 | 14,400 | 14,400 | 14,400 | 14,400 | 7,588 | 17,544 | 35,064 |

### D.2 Experimental Setting

The experiments were conducted on a computer equipped with an AMD Ryzen 9 7950X CPU, which boasts 16 cores and a clock speed of 5.1 GHz. This computer is furnished with 64 GB of memory, and all experiments were executed using PyTorch. The default hyperparameters of CEP are presented in Table [6](#A4.T6). The details regarding the hyperparameter sensitivity are provided in Section [D.7](#A4.SS7).

**Table 6: Hyparameters of CEP in the experiments.**
| Hyperparameter | $\tau_{\mu}$ | $\tau_{\sigma}$ | $\tau_{g}$ | $\tau_{l}$ | $\tau_{safe}$ | $\tau_{e}$ |
| --- | --- | --- | --- | --- | --- | --- |
| Value | 3.0 | / | 0.8 | 0.2 | 15 | 1.5 |

### D.3 Full Experiment Result of Gradual Concept Shift

When the forecasting horizon $\mathbf{H}=1$, each input instance updates only one time step. This leads to only minor alterations in the overall input data, resulting in a gradual concept shift. We conducted experiments with $\mathbf{H}=1$ using previous online methods and naive forecasters with different common backbones. Additionally, we applied the CEP to the naive forecasters. The specific experimental results are presented in Table [7](#A4.T7). From the experimental results, it is evident that the online time series forecasting methods did not demonstrate an absolute advantage over the naive forecasters. Both achieved state-of-the-art (SOTA) performance in four datasets. Moreover, CEP effectively reduced the prediction errors of the naive forecasters across the majority of datasets.

**Table 7: Full MSE errors of gradual concept shift experiments. Enhanced and reduced outcomes are marked with and respectively; enhancements exceeding 15% are highlighted in . The best and second-best performances are highlighted.**
| Data | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| DER++ | 0.111 | 0.082 | 0.666 | 0.086 | 0.048 | 0.145 | 0.400 | 0.065 |
| ER | 0.128 | 0.084 | 0.680 | 0.088 | 0.049 | 0.150 | 0.446 | 0.084 |
| FSNet | 0.297 | 0.093 | 0.769 | 0.123 | 0.063 | 0.814 | 0.567 | 0.093 |
| OneNet | 0.113 | 0.084 | 0.726 | 0.114 | 0.060 | 0.249 | 0.309 | 0.067 |
| Crossformer | 0.122 | 0.126 | 1.193 | 0.253 | 0.088 | 0.082 | 0.255 | 0.093 |
| Crossformer+CEP | 0.121 | 0.124 | 1.109 | 0.241 | 0.146 | 0.081 | 0.209 | 0.073 |
| Enhancement | -0.82% | -1.59% | -7.04% | -4.74% | 65.91% | -1.22% | -18.04% | -21.51% |
| TimesNet | 0.078 | 0.112 | 0.845 | 0.170 | 0.122 | 0.063 | 0.297 | 0.074 |
| TimesNet+CEP | 0.116 | 0.105 | 0.880 | 0.163 | 0.072 | 0.091 | 0.312 | 0.077 |
| Enhancement | 48.72% | -6.25% | 4.14% | -4.12% | -40.98% | 44.44% | 5.05% | 4.05% |
| DLinear | 0.160 | 0.109 | 1.026 | 0.218 | 0.069 | 0.077 | 0.251 | 0.061 |
| DLinear+CEP | 0.160 | 0.109 | 1.026 | 0.223 | 0.069 | 0.077 | 0.251 | 0.061 |
| Enhancement | 0.00% | 0.00% | 0.00% | 2.29% | 0.00% | 0.00% | 0.00% | 0.00% |
| PatchTST | 0.107 | 0.117 | 1.024 | 0.218 | 0.084 | 0.103 | 0.389 | 0.092 |
| PatchTST+CEP | 0.085 | 0.090 | 0.960 | 0.193 | 0.068 | 0.057 | 0.245 | 0.076 |
| Enhancement | -20.56% | -23.08% | -6.25% | -11.47% | -19.05% | -44.66% | -37.02% | -17.39% |
| iTransformer | 0.116 | 0.117 | 0.988 | 0.258 | 0.080 | 0.081 | 0.309 | 0.126 |
| iTransformer+CEP | 0.148 | 0.115 | 0.955 | 0.202 | 0.084 | 0.090 | 0.299 | 0.099 |
| Enhancement | 27.59% | -1.71% | -3.34% | -21.71% | 5.00% | 11.11% | -3.24% | -21.43% |
| TimeMixer | 0.067 | 0.097 | 0.708 | 0.112 | 0.064 | 0.049 | 0.262 | 0.052 |
| TimeMixer+CEP | 0.067 | 0.091 | 0.723 | 0.118 | 0.061 | 0.039 | 0.258 | 0.052 |
| Enhancement | 0.00% | -6.19% | 2.12% | 5.36% | -4.69% | -20.41% | -1.53% | 0.00% |
| TCN | 0.078 | 0.084 | 0.782 | 0.122 | 0.057 | 0.056 | 0.194 | 0.050 |
| TCN+CEP | 0.069 | 0.078 | 0.701 | 0.094 | 0.052 | 0.031 | 0.181 | 0.044 |
| Enhancement | -11.54% | -7.14% | -10.36% | -22.95% | -8.77% | -44.64% | -6.70% | -12.00% |

### D.4 Full Experiment Result of Naive Forecasters

We conducted experiments with forecasting horizons set to $\mathbf{H}\in\{30,60\}$. In each instance, at least half of the total input length is updated, preventing a gradual shift in the distribution. Consequently, the phenomenon of concept recurrence becomes more pronounced. The detailed experimental results are presented in Table [8](#A4.T8). These results demonstrate that CEP substantially enhances the prediction performance of the naive forecaster. To more intuitively illustrate the forecasting effect of CEP, the forecasting results of CEP and the naive forecasters are depicted in Figure [10](#A4.F10).

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/naive_value/ECL_30.png

Figure: (e) ETTh2, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/naive_value/ETTh2_30.png

**Table 8: Full MSE errors of different forecasters when using CEP are presented. Enhanced and reduced outcomes are marked with and respectively; enhancements exceeding 15% are highlighted in .**
| Data | $\mathbf{H}$ | TimeMixer | +CEP | iTransformer | +CEP | PatchTST | +CEP | DLinear | +CEP | SegRNN | +CEP | TimesNet | +CEP | Crossformer | +CEP | TCN | +CEP |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ECL | 30 | 0.273 | 0.244 | 0.255 | 0.232 | 0.300 | 0.263 | 0.275 | 0.264 | 0.242 | 0.237 | 0.287 | 0.253 | 0.416 | 0.368 | 0.300 | 0.266 |
| - | -10.62% | - | -9.02% | - | -12.33% | - | -4.00% | - | -2.07% | - | -11.85% | - | -11.54% | - | -11.33% |  |  |
| 60 | 0.314 | 0.297 | 0.295 | 0.279 | 0.360 | 0.318 | 0.339 | 0.334 | 0.314 | 0.311 | 0.339 | 0.306 | 0.644 | 0.515 | 0.409 | 0.365 |  |
| - | -5.41% | - | -5.42% | - | -11.67% | - | -1.47% | - | -0.96% | - | -9.73% | - | -20.03% | - | -10.76% |  |  |
| ETTh1 | 30 | 0.286 | 0.273 | 0.279 | 0.274 | 0.283 | 0.272 | 0.289 | 0.259 | 0.252 | 0.247 | 0.302 | 0.283 | 0.569 | 0.375 | 0.378 | 0.384 |
| - | -4.55% | - | -1.79% | - | -3.89% | - | -10.38% | - | -1.98% | - | -6.29% | - | -34.09% | - | 1.59% |  |  |
| 60 | 0.388 | 0.375 | 0.377 | 0.378 | 0.378 | 0.374 | 0.385 | 0.356 | 0.356 | 0.352 | 0.408 | 0.383 | 0.702 | 0.555 | 0.530 | 0.505 |  |
| - | -3.35% | - | 0.27% | - | -1.06% | - | -7.53% | - | -1.12% | - | -6.13% | - | -20.94% | - | -4.72% |  |  |
| ETTh2 | 30 | 2.252 | 2.031 | 1.949 | 1.959 | 2.119 | 1.919 | 2.069 | 2.051 | 1.866 | 1.842 | 2.073 | 2.011 | 6.744 | 6.558 | 2.536 | 2.173 |
| - | -9.81% | - | 0.51% | - | -9.44% | - | -0.87% | - | -1.29% | - | -2.99% | - | -2.76% | - | -14.31% |  |  |
| 60 | 3.174 | 3.020 | 3.145 | 3.077 | 3.215 | 3.014 | 3.365 | 3.208 | 3.392 | 3.325 | 3.211 | 3.117 | 8.742 | 8.322 | 3.866 | 3.449 |  |
| - | -4.85% | - | -2.16% | - | -6.25% | - | -4.67% | - | -1.98% | - | -2.93% | - | -4.80% | - | -10.79% |  |  |
| ETTm1 | 30 | 0.619 | 0.612 | 0.610 | 0.589 | 0.618 | 0.569 | 0.656 | 0.586 | 0.557 | 0.551 | 0.684 | 0.682 | 2.002 | 1.827 | 0.748 | 0.631 |
| - | -1.13% | - | -3.44% | - | -7.93% | - | -10.67% | - | -1.08% | - | -0.29% | - | -8.74% | - | -15.64% |  |  |
| 60 | 0.936 | 0.882 | 0.943 | 0.886 | 0.905 | 0.870 | 1.042 | 0.916 | 0.879 | 0.878 | 0.951 | 0.958 | 2.876 | 2.707 | 1.198 | 0.965 |  |
| - | -5.77% | - | -6.04% | - | -3.87% | - | -12.09% | - | -0.11% | - | 0.74% | - | -5.88% | - | -19.45% |  |  |
| ETTm2 | 30 | 0.225 | 0.224 | 0.210 | 0.205 | 0.221 | 0.216 | 0.202 | 0.198 | 0.201 | 0.196 | 0.224 | 0.212 | 0.324 | 0.302 | 0.205 | 0.206 |
| - | -0.44% | - | -2.38% | - | -2.26% | - | -1.98% | - | -2.49% | - | -5.36% | - | -6.79% | - | 0.49% |  |  |
| 60 | 0.237 | 0.240 | 0.272 | 0.267 | 0.289 | 0.273 | 0.281 | 0.269 | 0.265 | 0.258 | 0.272 | 0.270 | 0.508 | 0.391 | 0.326 | 0.308 |  |
| - | 1.27% | - | -1.84% | - | -5.54% | - | -4.27% | - | -2.64% | - | -0.74% | - | -23.03% | - | -5.52% |  |  |
| Exchange | 30 | 0.257 | 0.249 | 0.272 | 0.247 | 0.259 | 0.235 | 0.305 | 0.272 | 0.193 | 0.192 | 0.308 | 0.257 | 1.645 | 1.129 | 0.381 | 0.262 |
| - | -3.11% | - | -9.19% | - | -9.27% | - | -10.82% | - | -0.52% | - | -16.56% | - | -31.37% | - | -31.23% |  |  |
| 60 | 0.459 | 0.439 | 0.479 | 0.475 | 0.449 | 0.457 | 0.518 | 0.489 | 0.398 | 0.395 | 0.631 | 0.528 | 2.691 | 1.937 | 0.650 | 0.539 |  |
| - | -4.36% | - | -0.84% | - | 1.78% | - | -5.60% | - | -0.75% | - | -16.32% | - | -28.02% | - | -17.08% |  |  |
| Traffic | 30 | 0.486 | 0.471 | 0.487 | 0.455 | 0.673 | 0.584 | 0.614 | 0.614 | 0.680 | 0.669 | 0.525 | 0.520 | 0.634 | 0.612 | 0.710 | 0.698 |
| - | -3.09% | - | -6.57% | - | -13.22% | - | 0.00% | - | -1.62% | - | -0.95% | - | -3.47% | - | -1.69% |  |  |
| 60 | 0.538 | 0.528 | 0.554 | 0.529 | 0.680 | 0.589 | 0.641 | 0.641 | 0.910 | 0.897 | 0.592 | 0.605 | 0.701 | 0.677 | 0.731 | 0.724 |  |
| - | -1.86% | - | -4.51% | - | -13.38% | - | 0.00% | - | -1.43% | - | 2.20% | - | -3.42% | - | -0.96% |  |  |
| WTH | 30 | 0.316 | 0.315 | 0.311 | 0.312 | 0.324 | 0.310 | 0.295 | 0.295 | 0.298 | 0.297 | 0.327 | 0.328 | 0.347 | 0.339 | 0.317 | 0.313 |
| - | -0.32% | - | 0.32% | - | -4.32% | - | 0.00% | - | -0.34% | - | 0.31% | - | -2.31% | - | -1.26% |  |  |
| 60 | 0.401 | 0.396 | 0.399 | 0.395 | 0.410 | 0.396 | 0.388 | 0.386 | 0.386 | 0.386 | 0.413 | 0.423 | 0.450 | 0.449 | 0.444 | 0.448 |  |
| - | -1.25% | - | -1.00% | - | -3.41% | - | -0.52% | - | 0.00% | - | 2.42% | - | -0.22% | - | 0.90% |  |  |

### D.5 Full Experiment Result of Online Methods

All online methods incorporate the same TCN component. Therefore, the performance of online methods, the naive TCN, and the TCN utilizing the CEP is compared that the full experimental results are presented in Table [9](#A4.T9). An intriguing phenomenon is that, in comparison to the naive TCN, online methods nearly fail in most scenarios. A plausible explanation is that, owing to delayed feedback, the information acquired by online methods is delayed and fails to adapt promptly to concept changes. The CEP significantly enhances the performance of TCN, effectively reducing prediction errors across a wide range of datasets.

**Table 9: Full MSE error of previous online forecasting methods. Enhanced and reduced outcomes compared to TCN are marked with and respectively; enhancements exceeding 15% are highlighted in . The best and second-best performances are highlighted.**
| Data | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $\mathbf{H}$ | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 |
| TCN | 0.300 | 0.409 | 0.378 | 0.530 | 2.536 | 3.866 | 0.748 | 1.198 | 0.205 | 0.326 | 0.381 | 0.650 | 0.710 | 0.731 | 0.317 | 0.444 |
| ER | 0.558 | 0.714 | 0.348 | 0.647 | 2.551 | 4.141 | 0.902 | 1.814 | 0.252 | 0.405 | 3.105 | 3.956 | 1.241 | 0.883 | 0.515 | 0.629 |
| 86.00% | 74.57% | -7.94% | 22.08% | 0.59% | 7.11% | 20.59% | 51.42% | 22.93% | 24.23% | 714.96% | 508.62% | 74.79% | 20.79% | 62.46% | 41.67% |  |
| DER++ | 0.554 | 0.621 | 0.325 | 0.597 | 2.290 | 4.009 | 0.819 | 1.579 | 0.241 | 0.374 | 1.771 | 4.331 | 1.194 | 0.824 | 0.500 | 0.597 |
| 84.67% | 51.83% | -14.02% | 12.64% | -9.70% | 3.70% | 9.49% | 31.80% | 17.56% | 14.72% | 364.83% | 566.31% | 68.17% | 12.72% | 57.73% | 34.46% |  |
| FSNet | 0.867 | 0.858 | 0.379 | 0.525 | 3.015 | 4.584 | 1.112 | 2.012 | 0.292 | 0.405 | 3.840 | 5.700 | 1.693 | 1.425 | 0.752 | 0.756 |
| 189.00% | 109.78% | 0.26% | -0.94% | 18.89% | 18.57% | 48.66% | 67.95% | 42.44% | 24.23% | 907.87% | 776.92% | 138.45% | 94.94% | 137.22% | 70.27% |  |
| OneNet | 0.382 | 0.420 | 0.364 | 0.514 | 2.624 | 4.386 | 0.976 | 1.252 | 0.245 | 0.343 | 0.833 | 1.491 | 0.720 | 0.638 | 0.363 | 0.479 |
| 27.33% | 2.69% | -3.70% | -3.02% | 3.47% | 13.45% | 30.48% | 4.51% | 19.51% | 5.21% | 118.64% | 129.38% | 1.41% | -12.72% | 14.51% | 7.88% |  |
| CEP | 0.266 | 0.365 | 0.384 | 0.505 | 2.173 | 3.449 | 0.631 | 0.965 | 0.206 | 0.308 | 0.262 | 0.539 | 0.698 | 0.724 | 0.313 | 0.448 |
| -11.33% | -10.76% | 1.59% | -4.72% | -14.31% | -10.79% | -15.64% | -19.45% | 0.49% | -5.52% | -31.23% | -17.08% | -1.69% | -0.96% | -1.26% | 0.90% |  |

### D.6 Ablation Study

In the ablation experiment, we tested the mechanisms in the CEP. These mechanisms include evolution, using only fast genes, using only slow genes, elimination, inspiration, and gradient abandonment. We tested all the forecasters and took the average of the results. The specific experimental results are in Table [10](#A4.T10). The most basic evolution mechanism in the CEP played a major role when compared to the performance without CEP. Other mechanisms perform partially well for two reasons. First, the performance of different forecasters can vary under the influence of CEP, and second, the use of uniform default parameters limits the effectiveness of the datasets with different characteristics.

**Table 10: Full ablation study results. All results are the averaged MSE of different backbone forecasters. The best and second-best performances are highlighted.**
| Evolution | Fast<br>Gene | Slow<br>Gene | Elimination | Inspiration | Gradient<br>Abandonment | ECL | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Exchange | Traffic | WTH |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 | 30 | 60 |  |  |  |  |  |  |
| ✗ | 0.294 | 0.377 | 0.330 | 0.441 | 2.701 | 4.014 | 0.812 | 1.216 | 0.227 | 0.306 | 0.453 | 0.784 | 0.601 | 0.668 | 0.317 | 0.411 |  |  |  |  |  |
| ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | 0.287 | 0.368 | 0.289 | 0.404 | 2.494 | 3.786 | 0.688 | 1.056 | 0.232 | 0.287 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |
| ✓ | 0.288 | 0.373 | 0.296 | 0.407 | 2.499 | 3.743 | 0.706 | 1.066 | 0.228 | 0.286 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |  |
| ✓ | ✗ | 0.283 | 0.364 | 0.297 | 0.411 | 2.714 | 4.060 | 0.731 | 1.114 | 0.225 | 0.287 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |
| ✓ | 0.284 | 0.370 | 0.306 | 0.416 | 2.711 | 3.983 | 0.756 | 1.133 | 0.222 | 0.285 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |  |
| ✓ | ✗ | ✗ | 0.285 | 0.372 | 0.289 | 0.409 | 2.494 | 3.786 | 0.688 | 1.056 | 0.224 | 0.287 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |
| ✓ | 0.285 | 0.373 | 0.296 | 0.411 | 2.499 | 3.743 | 0.706 | 1.066 | 0.222 | 0.284 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |  |
| ✓ | ✗ | 0.269 | 0.342 | 0.297 | 0.414 | 2.714 | 4.060 | 0.731 | 1.114 | 0.224 | 0.287 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |
| ✓ | 0.275 | 0.342 | 0.306 | 0.418 | 2.711 | 3.983 | 0.756 | 1.133 | 0.221 | 0.285 | 0.448 | 0.775 | 0.578 | 0.649 | 0.314 | 0.408 |  |  |  |  |  |
| ✓ | ✗ | ✗ | ✗ | ✗ | 0.285 | 0.355 | 0.287 | 0.400 | 2.436 | 3.693 | 0.688 | 1.055 | 0.223 | 0.287 | 0.321 | 0.629 | 0.578 | 0.649 | 0.311 | 0.402 |  |
| ✓ | 0.286 | 0.357 | 0.288 | 0.400 | 2.427 | 3.670 | 0.706 | 1.069 | 0.222 | 0.284 | 0.324 | 0.614 | 0.578 | 0.649 | 0.312 | 0.405 |  |  |  |  |  |
| ✓ | ✗ | 0.281 | 0.342 | 0.295 | 0.409 | 2.608 | 3.822 | 0.731 | 1.131 | 0.223 | 0.287 | 0.329 | 0.660 | 0.578 | 0.649 | 0.308 | 0.401 |  |  |  |  |
| ✓ | 0.281 | 0.347 | 0.296 | 0.410 | 2.599 | 3.813 | 0.756 | 1.145 | 0.223 | 0.285 | 0.332 | 0.634 | 0.578 | 0.649 | 0.309 | 0.404 |  |  |  |  |  |
| ✓ | ✗ | ✗ | 0.276 | 0.362 | 0.287 | 0.400 | 2.442 | 3.730 | 0.688 | 1.055 | 0.221 | 0.287 | 0.329 | 0.640 | 0.578 | 0.649 | 0.315 | 0.403 |  |  |  |
| ✓ | 0.281 | 0.369 | 0.288 | 0.400 | 2.431 | 3.680 | 0.706 | 1.069 | 0.221 | 0.284 | 0.335 | 0.629 | 0.578 | 0.649 | 0.315 | 0.407 |  |  |  |  |  |
| ✓ | ✗ | 0.273 | 0.345 | 0.295 | 0.409 | 2.599 | 3.850 | 0.731 | 1.131 | 0.221 | 0.287 | 0.334 | 0.674 | 0.578 | 0.649 | 0.312 | 0.404 |  |  |  |  |
| ✓ | 0.277 | 0.348 | 0.296 | 0.410 | 2.584 | 3.816 | 0.756 | 1.145 | 0.220 | 0.285 | 0.341 | 0.650 | 0.578 | 0.649 | 0.313 | 0.408 |  |  |  |  |  |
| ✓ | ✗ | ✗ | ✗ | 0.287 | 0.358 | 0.287 | 0.400 | 2.415 | 3.683 | 0.688 | 1.056 | 0.224 | 0.287 | 0.368 | 0.624 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |
| ✓ | 0.287 | 0.361 | 0.288 | 0.400 | 2.405 | 3.647 | 0.706 | 1.066 | 0.223 | 0.284 | 0.366 | 0.621 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |  |  |  |
| ✓ | ✗ | 0.281 | 0.349 | 0.295 | 0.409 | 2.592 | 3.833 | 0.731 | 1.114 | 0.224 | 0.287 | 0.366 | 0.656 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |  |  |
| ✓ | 0.281 | 0.352 | 0.296 | 0.410 | 2.576 | 3.796 | 0.756 | 1.133 | 0.224 | 0.285 | 0.363 | 0.657 | 0.578 | 0.649 | 0.314 | 0.410 |  |  |  |  |  |
| ✓ | ✗ | ✗ | 0.276 | 0.371 | 0.287 | 0.400 | 2.401 | 3.744 | 0.688 | 1.056 | 0.221 | 0.287 | 0.365 | 0.624 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |  |
| ✓ | 0.277 | 0.374 | 0.288 | 0.400 | 2.396 | 3.671 | 0.706 | 1.066 | 0.221 | 0.284 | 0.363 | 0.621 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |  |  |  |
| ✓ | ✗ | 0.268 | 0.342 | 0.295 | 0.409 | 2.578 | 3.889 | 0.731 | 1.114 | 0.221 | 0.287 | 0.358 | 0.656 | 0.578 | 0.649 | 0.314 | 0.409 |  |  |  |  |
| ✓ | 0.266 | 0.341 | 0.296 | 0.410 | 2.568 | 3.817 | 0.756 | 1.133 | 0.220 | 0.285 | 0.355 | 0.657 | 0.578 | 0.649 | 0.314 | 0.410 |  |  |  |  |  |

### D.7 Hyperparameter Sensitivity

CEP incorporates multiple mechanisms, each governed by distinct hyperparameters that regulate the system response. It is important to note that the efficacy of these mechanisms, and therefore the optimal settings for their associated hyperparameters, can be highly dataset-dependent. The utility of a specific mechanism is contingent upon the particular characteristics of the dataset, meaning not all components will necessarily be beneficial in every data scenario. To comprehensively evaluate the impact of these hyperparameters and illustrate this dependency, we carried out experiments across all datasets and for two forecasting horizons $\textbf{H}\in\{30,60\}$. The detailed experimental results are presented in Figures [11](#A4.F11) through [17](#A4.F17).

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x6.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x22.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x38.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x54.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x70.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x86.png

Figure: (a) ECL, $\textbf{H}=30$
Refer to caption: /html/2506.14790/assets/x102.png

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 4
presentation: 2
questions: 1- It is not clear why "recurring" concept drift is any of particularity as compared to normal concept drift. It is just some concepts that appear and reappear. I don't see any special traitment in the proposed mechanism that deals with "recurring".

2- I don't understand why there is a warm-up stage. Yes, I understand that you start with one forecaster and there is a process tu tune it to fit to some initial samples. This is not fundamentally different from creation of any new forecaster in the "Online stage".
soundness: 2
strengths: The proposed mechanism is relatively simple and is general which allows it to be easily coupled with any forecaster, 

The experimentation on forecasting performance improvement is extensive both in terms of the forecasting models compared and the dataset used.
summary: This paper deals with concept drift problem in time series forecasting. It proposes a pooling mechanism to associate each forecaster to a concept. This mechanism includes creation of new forecaster for new concept, elimination of forecasters corresponding to outdated concepts. The proposed mechanism can be associated with any forecaster models. Experimental results suggest that it allows to improve the forecasting  performance in the presence of concept drifts.
weaknesses: The simplicity of the concepts managed by the proposed mechanism is also a major limitation of this work, In fact, a concept in this work correspond basically to the means and standard deviation of the time series in a look-back window. Concepts with very different patterns may have the same statistical properties of means and standard deviation.

Anohter limitation is that the proposed mechanism is for single time series. Multiple time series exhibitete even more complex pattens for concepts.

Some key parameters such as the splliting parameter $tau$ is difficult to fix. There is a lack of discussion about how to chose the values for these parameters.

Tables and figures are not always well explaned. For instance, it it not clear what the percentage means in Table 2. The three settings in Figure 2 are not well explained neither. The quality of writing of this paper is between fair and poor.

---

confidence: 1
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. The paper defines a "gene" using the mean and standard deviation of a data window for proactive concept identification. While computationally efficient, this might not capture complex temporal patterns or shapes. The lightweight alternative or additional features (e.g., trend, periodicity, or simple shape descriptors) could be explored in future work to enhance the robustness.

2. The paper employs a hard assignment strategy, matching a data window to a single forecaster. A concern is that this might perform poorly during transitional phases between concepts. Please explain it in detail.

3. The mechanism for removing idle forecasters uses a fixed period, which may forget concepts with long recurrence intervals. Have you considered implementing a more adaptive elimination strategy to maintain the performance?

4. The method is explicitly designed for recurring concept drift, which may inherently limit its efficiency on tasks with non-recurring patterns. It is needed to consider other types of concept drift.
soundness: 3
strengths: 1.The paper has a clear structure and logical flow. It first identifies the challenge of recurring concept drift and systematically analyzes the limitations of existing methods. Subsequently, the paper proposes the core solution, the CEP framework, and progressively elaborates on its key mechanisms.

2.The method is highly innovative. Instead of relying on the traditional passive approach of adapting to drift based on prediction error, the paper innovatively proposes a proactive detection mechanism based on "genes".

3. The paper not only details the algorithmic aspects of CEP but also provides an in-depth discussion of its key advantages for practical deployment, such as avoiding "catastrophic forgetting" by retaining specialized forecasters and managing computational resources through its elimination mechanism.

4.The experimental design is thorough and provides comprehensive validation. The paper not only tests its method on multiple real-world datasets with diverse characteristics but also applies it as a universal framework to several mainstream forecasting models, proving its general applicability and effectiveness.

5.The rich visualization analysis and detailed ablation studies clearly reveal the internal working mechanisms of CEP and the necessity of its components, thereby strengthening the persuasiveness of the conclusions.
summary: The paper proposes an innovative framework named the "Continuous Evolution Pool" (CEP) to address recurring concept drift in online time series forecasting. At its core, the framework introduces "genes" to proactively identify data concepts and dynamically maintains a specialized pool of forecasters. The strengths of the work lie in its clear structure, an innovative proactive drift detection paradigm, and comprehensive experimental validation across multiple datasets and foundational models; rich visualizations and ablation studies also provide strong support for its conclusions. However, defining a "gene" solely by its mean and standard deviation may fail to capture more complex data distributions, while the hard assignment strategy for forecasters may perform poorly in transitional concept regions. The fixed elimination period for forecasters could also struggle to mitigate "catastrophic forgetting" for concepts with long recurrence periods. Furthermore, the method is primarily designed for periodic tasks, limiting its effectiveness in non-recurring drift scenarios, and the theoretical derivation on the effectiveness of Euclidean distance lacks direct experimental verification. In summary, while the paper is well-structured and the method is novel, the robustness and theoretical validation of its core mechanisms require further investigation and experimental support.
weaknesses: 1.The paper defines a "gene" as the mean and standard deviation of a data window. Although this method is simple and computationally efficient, it may not be sufficient to capture the distribution within the window, making it unable to detect all types of concept drift.

2.When a data window is in a transitional stage between two concepts, it may be difficult to select an appropriate forecaster for prediction.

3.The proposed method performs exceptionally well on data with recurring patterns, but its efficiency on non-recurring tasks is limited.

4.The paper derives that if the concepts are normally distributed, maximizing the log-likelihood is approximately equivalent to minimizing the Euclidean distance. However, this derivation is not supported by experimental evidence, thus the effectiveness of using Euclidean distance is not verified.

5.The paper removes forecasters that have been idle for a certain period. However, this fixed elimination period means that it cannot mitigate catastrophic forgetting for concepts whose recurrence period exceeds the set threshold.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. The elimination criterion (Eq. 12) uses a fixed ratio . Could this accidentally remove a forecaster for a rare-but-important concept (e.g., annual seasonality in a short evaluation window)? Have you considered a more sophisticated retention policy (e.g., based on historical recurrence intervals)?

2. You justify the gene $\mathbf{g}_x = (\mu, \sigma)$ well from a statistical moment perspective. Did you experiment with incorporating other simple, efficient-to-compute features into the gene (e.g., a crude measure of trend or a single autocorrelation coefficient)? If so, what were the results? If not, could such an extension be a straightforward way to handle a wider class of concept drifts?

3. The performance on the Traffic dataset suggests a limitation with your fixed $\tau_{\mu}=3$ threshold for datasets with subtle, non-recurring drifts. Your practical guide suggests tuning $\tau_{\mu}$ lower for such cases. Did you run experiments on Traffic with a lower $\tau_{\mu}$, and if so, did performance recover? This would be a powerful point to demonstrate the method's adaptability.

4. While you state the inference time is identical to the base forecaster, the evolution, retrieval, and elimination operations do introduce overhead. Could you provide quantitative data (e.g., average percentage increase in total runtime compared to a naive model) to give readers a concrete sense of the practical computational cost?
soundness: 3
strengths: **Originality:** The formulation of the "Continuous Evolution Pool" is a fresh and compelling idea. The gene-based, proactive concept identification is a significant shift from reactive, error-based adaptation strategies.

**Quality & Significance:** The empirical validation is thorough and convincing. The method's ability to significantly improve performance across diverse model architectures and under challenging delay scenarios is a major strength. 

**Clarity:** The core algorithm is presented clearly, and the extensive appendix provides valuable theoretical justifications (regret analysis, statistical grounding) and practical guides.

**Practicality:** The method is model-agnostic and comes with a practical hyperparameter tuning guide, enhancing its potential for real-world adoption.
summary: This paper addresses the challenging problem of recurring concept drift in online time series forecasting. The authors identify a key limitation of existing methods—catastrophic forgetting of past concepts due to continuous parameter updates—and propose a novel framework called the Continuous Evolution Pool (CEP).

The core idea is to maintain a dynamic pool of forecasters, each specialized for a distinct data concept. CEP operates through three main mechanisms:

Retrieval: For a new input instance $\mathbf{x}_t$, its "gene" $\mathbf{g}_x = (\mu(\mathbf{x}_t), \sigma(\mathbf{x}_t))$ is computed, and the closest forecaster in the gene space is selected for prediction.

Evolution: If the instance's gene deviates significantly from the nearest forecaster's gene (based on a statistically motivated threshold, e.g., $|\mathbf{g}{x,\mu} - \mathbf{g}{N,\mu}| > \tau_{\mu} \cdot \mathbf{g}_{N,\sigma}$), a new forecaster is "evolved" by splitting from the nearest one, effectively capturing a new or recurring concept.

Elimination: Inactive forecasters are pruned to manage pool size and remove outdated knowledge.

The authors demonstrate through extensive experiments that CEP significantly boosts the performance of various base forecasters (e.g., TCN, PatchTST) and substantially outperforms strong online learning baselines, especially under delayed feedback scenarios.
weaknesses: **Theoretical vs. Empirical Grounding:** While the appendix provides a regret analysis, it remains somewhat high-level. A more formal bound on the identification regret, perhaps under specific assumptions about the separation between concepts in the gene space, would strengthen the theoretical analysis.

**Limitations of the Gene Representation:** The gene, while effective, is a relatively simple statistical summary. The paper acknowledges its limitation on the Traffic dataset (with frequent, low-magnitude fluctuations) but could discuss this more deeply. Are there types of concept drift (e.g., changes in auto-correlation structure, frequency content) that this gene would be inherently blind to? A brief discussion or experiment hinting at this would be valuable.

**Multivariate extension:** The method is evaluated only on univariate settings. Many real-world time series are multivariate, and concept drift may affect channels asynchronously. How CEP scales to this setting is unclear.

**Experiment Analysis:** The paper could provide more insight into why certain strong baselines like FSNet and OneNet fail so dramatically in the delayed setting (Table 2). A brief qualitative analysis or hypothesis in the main text would add depth beyond just reporting the performance gap.

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confidence: 4
contribution: 1
initial_rating: 2
presentation: 2
questions: See previous section
soundness: 2
strengths: - The paper addresses the important challenge of recurring concept drift in non-stationary time series
- The framework introduces mechanisms for addition, deletion, activation of recurrent
summary: The aim of the paper is to address the possible time evolution (i.e., concept drift) in time series forecasting. This is a crucial issue since (fairly) all the time series are directly or indirectly time-dependent. In more detail, the paper introduces a framework, called Continuous Evolution Pool (CEP), for the time series forecasting under recurrent concept drift. The framework introduces a feature vector called gene (being the representative of the time series) and mechanisms to add, reactive and remove concept drift. When a new concept arrives, it is inserted in the pool of concepts and it reactivated when the same gene is identified, so as to reduce the catastrophic forgetting effects.
weaknesses: - Limited feature representation: The gene vector relies only on mean and variance, which may be insufficient for distinguishing complex temporal concepts. What about considering moments or frequency-domain statistics to provide richer representations?

- Choice of distance metric: The model compares genes via Euclidean distance. However, for time series similarity, techniques such as DTW (Dynamic Time Warping) or Mahalanobis distance might better capture temporal and statistical dependencies. The authors should discuss this limitation.

- What about the splitting threshold? I guess it is a crucial parameter to set

- What exactly is nwait? Is it a time count or number of missed updates? Its calibration seems critical to stability.

- The value of L is fixed across all datasets. Is this empirically optimal or arbitrary? It might influence both performance and drift detection frequency.

- What about considering not just threshold for detecting changes? I would suggest to consider change detection mechanisms such as CUSUM, etc..

- What if the detection of a new concept is missed? This is a False Negative detection that could severely induce issues in the characterization of the error. I guess thresholds are not robust enough for detection. 

- I would have expected to see zero-shot learning foundation models for time series forecasting (e.g., Chronos, Tirex) in the experimental section.


Decision: Reject