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KAN4TSF: Are KAN and KAN-based models Effective for Time Series Forecasting

创建日期：2024-09-11 16:54:19

所属分类：计算机computer science

标签： KAN4TSF

Are KAN and KAN-based models Effective for Time Series Forecasting

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KAN4TSF: Are KAN and KAN-based models

Effective for Time Series Forecasting

Abstract

Time series forecasting is a crucial task that predicts the future values of variables

based on historical data. Time series forecasting techniques have been developing

in parallel with the machine learning community, from early statistical learning

methods to current deep learning methods. Although existing methods have made

significant progress, they still suffer from two challenges. The mathematical

theory of mainstream deep learning-based methods does not establish a clear

relation between network sizes and fitting capabilities, and these methods often lack

interpretability. To this end, we introduce the Kolmogorov-Arnold Network (KAN)

into time series forecasting research, which has better mathematical properties

and interpretability. First, we propose the Reversible Mixture of KAN experts

(RMoK) model, which is a KAN-based model for time series forecasting. RMoK

uses a mixture-of-experts structure to assign variables to KAN experts. Then, we

compare performance, integration, and speed between RMoK and various baselines

on real-world datasets, and the experimental results show that RMoK achieves the

best performance in most cases. And we find the relationship between temporal

feature weights and data periodicity through visualization, which roughly explains

RMoK’s mechanism. Thus, we conclude that KAN and KAN-based models

(RMoK) are effective in time series forecasting. Code is available at KAN4TSF:

https://github.com/2448845600/KAN4TSF.

1 Introduction

Time series forecasting (TSF) is the task of using historical data to predict future states of variables.

This research area includes a broad scope of applications, such as financial investment, weather

forecasting, traffic estimation, and health management Bi et al. [2023], Gao et al. [2023], Savcisens

et al. [2024], Han et al. [2024a]. The machine learning community’s progress has long inspired time

series forecasting technology: the popularity of early statistical learning methods gave rise to SVR

and ARMIA, while the development of deep learning introduced MLP and Transformer into time

series forecasting. At present, various time series forecasting methods cover almost all deep learning

network architectures, such as RNN, CNN, Transformer, and MLP Nie et al. [2023], Wu et al. [2023],

Han et al. [2024b]. The forecasting models derived from different network architectures have their

own advantages in forecasting performance, running speed, and resource usage.

Although deep learning-based models have made notable progress in time series forecasting, there

are still several challenges. The universal approximation theorem (UAT), which is the mathematical

foundation of most mainstream forecasting models, cannot provide a guarantee on the necessary

network sizes (depths and widths) to approximate a predetermined continuous function with specific

Preprint. Under review.

arXiv:2408.11306v1 [cs.LG] 21 Aug 2024accuracy. And this theory can only achieve an approximation rather than a representation. The

limitations of UAT have become the sword of Damocles hanging over time series forecasting.

Furthermore, the prediction mechanism of existing models is black-box, resulting in a lack of

interpretability. These nontransparent methods are suspected of being suitable for tasks that require a

low tolerance for errors, such as medicine, law and finance.

Kolmogorov-Arnold Network (KAN) Liu et al. [2024a], which is based on the Kolmogorov-Arnold

representation theorem (KART), has become a novel approach to solving the above challenges. On the

one hand, KART proves that a multivariate continuous function can be represented as a combination

of finite univariate continuous functions. This theorem establishes the relationship between network

size and input shape under the premise of representation. On the other hand, KAN offers a pruning

strategy that simplifies the trained KAN into a set of symbolic functions, enabling the analysis

of specific modules’ mechanisms, thereby significantly enhancing the network’s interpretability.

In addition, KAN’s function fitting idea is consistent with the properties of time series, such as

periodicity and trend, which is conducive to embedding prior knowledge into the network structure

and improving the performance of the network.

Despite being a relatively recent proposal, KAN, which employs a trainable 1D B-spline functions to

convert incoming signals, has already sparked numerous efforts to improve or broaden its capabilities.

Some studies propose KAN’s variants which replace the B-splines with Chebyshev polynomials SS

[2024], wavelet functions Bozorgasl and Chen [2024], Jacobi polynomials Aghaei [2024], ReLU

functions Qiu et al. [2024], etc., to accelerate training speed and improve network performance.

Other studies introduce KAN with existing popular network structures for various applications. For

example, ConvKAN Bodner et al. [2024] and GraphKAN Zhang and Zhang [2024], Xu et al. [2024]

are proposed for image processing and graph processing. In summary, KANs have been extensively

empirically studied in vision and language Azam and Akhtar [2024], Yu et al. [2024]. However,

existing studies lack a KAN-based model that considers time series domain knowledge, making it

impossible to verify whether KAN is effective in time series forecasting.

To this end, we aim to propose a KAN-based model for the time series forecasting task and evaluate its

effectiveness from four perspectives: performance, integration, speed, and interpretability. First, we

propose the Reversible Mixture of KAN Experts model (RMoK), a KAN-based time series forecasting

model that uses multiple KAN variants as experts and a gating network to adaptively assign variables

to specific experts for prediction. RMoK is implemented as a single-layer network because we

hope that it can have similar performance and better interpretability than existing methods. Then,

we use a unified training and evaluation setting to compare the performance of RMoK and current

popular baselines on seven real-world datasets. The experimental results show that RMoK achieves

state-of-the-art (SOTA) performance in most cases. Subsequently, we conduct a comprehensive

empirical study on KAN-based models, including the comparison between KAN and Linear, the

effect of integrating KANs with the Transformer, and the speed of the KAN-based models. Finally,

we discuss the interpretability of RMoK using the example of weather prediction. We visualize the

weights of temporal features at different time steps in KAN and find the correlation between the

weight distribution and the periodicity of the data.

To sum up, the contributions of this work include:

• To the best of our knowledge, this is the first work that comprehensively discusses the

effectiveness of the booming KANs for time series forecasting.

• To validate our claims, we propose the Reversible Mixture of KAN Experts model, which

uses a single layer of the mixture of KAN experts to keep a balance between performance

and interpretability.

• We fairly compare the performance between RMoK and baselines on seven real-world

datasets, and the experimental results show that RMoK achieves the best performance in

most cases. And we also conduct a comprehensive empirical study on KAN-based models

about integration and speed.

• We mine the relationship between time feature weights and data periodicity through visual

ization, which roughly explains the mechanism of RMoK.

In summary, compared with the baselines in terms of performance, integration, speed, and inter

pretability, we conclude that KAN is effective in time series forecasting.

2Linear

KAN

Figure 1: The computational process of Linear and KAN layers under a certain input and output

dimension.

2 Problem Defintion

In multivariate time series forecasting, given historical data X = [X1, · · · , XT ] ∈ R T ×C , where T

is the time steps of historical data and C is the number of variates. The time series forecasting task is

to predict Y = [XT +1, · · · , XT +P ] ∈ R P ×C during future P time steps.

3 Related Work

3.1 Time-Series Forecasting Models

Although Transformer-based methods have almost become the standard in CV and NLP, various

network architectures (such as Transformer, CNN, and MLP) are competing in time series forecasting

recently.

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