Reservoir computing models based on spiking neural P systems for time series classification

Reservoir computing models based on spiking neural P systems for time series classification✩ Hong Peng a,∗, Xin Xiong a, Min Wu a, Jun Wang b, Qian Yang a, David Orellana-Martín c, Mario J. Pérez-Jiménez c aSchool of Computer and Software Engineering, Xihua University, Chengdu, 610039, China bSchool of Electrical Engineering and Electronic Information, Xihua University, Chengdu, 610039, China cResearch Group of Natural Computing, Department of Computer Science and Artificial Intelligence, University of Seville, Sevilla, 41012, Spain A R T I C L E I N F O Keywords: Recurrent neural networks Reservoir computing Nonlinear spiking neural P systems Time series classification A B S T R A C T Nonlinear spiking neural P (NSNP) systems are neural-like membrane computing models with nonlinear spiking mechanisms. Because of this nonlinear spiking mechanism, NSNP systems can show rich nonlinear dynamics. Reservoir computing (RC) is a novel recurrent neural network (RNN) and can overcome some shortcomings of traditional RNNs. Based on NSNP systems, we developed two RC variants for time series classification, RC-SNP and RC-RMS-SNP, which are without and integrated with reservoir model space (RMS), respectively. The two RC variants use NSNP systems as the reservoirs and can be easily implemented in the RC framework. The proposed two RC variants were evaluated on 17 benchmark time series classification datasets and compared with 16 state-of-the-art or baseline classification models. The comparison results demonstrate the effectiveness of the proposed two RC variants for time series classification tasks. 1. Introduction Spiking neural P (SNP) systems (Ionescu, Păun, & Yokomori,2006) are a class of neural-like membrane computing models (Pˇ aun, Rozenberg, & Salomaa,2010), inspired by the spiking phenomenon of biological neurons. An SNP system is composed of a directed graph of neurons, where each neuron has a data unit and set of rules (including spiking rules and forgetting rules). The data unit is used to express the neuron’s internal state and is evolved by spiking/forgetting rules. Therefore, spiking/forgetting rules characterize the dynamic behavior of SNP systems. In recent decades, a variety of biological mechanisms of neurons have been introduced to propose many variants of SNP systems. These biological mechanisms include astrocytes (Pˇ aun,2007), anti-spike (Pan & Pˇ aun,2009), multiple channels (Lv et al.,2021;Peng et al.,2017), polarization (Wu, Pˇ aun, Zhang, & Pan,2017), dynamic threshold (Peng, Wang, Pérez-Jiménez, & Riscos-Núñez,2019), nonlinear coupled mechanism (Peng & Wang,2019), delay on synapses (Song, Luis, Peng, Wang, & Pérez-Jiménez,2021), inhibitory rules (Peng, Li, et al.,2020), dendrite (Peng, Bao, et al.,2020), autapses (Song, Valencia-Cabrera, Peng, & Wang,2021), plasmids (Cabarle et al.,2021), structural plasticity (de la Cruz, Cabarle, Macababayao, Adorna, & Zeng,2021), and ✩This work was partially supported by the National Natural Science Foundation of China (No. 62076206, 62176216), and the Research Fund of Sichuan Science and Technology Project (No. 2022ZYD0115), China. ∗Corresponding author. more. Moreover, several variants have been applied in real-life applications, including fault diagnosis (Peng et al.,2018), image fusion (Li, Peng, Luo, et al.,2021;Li, Peng, & Wang,2021;Li, Peng, Wang, & Huang,2020;Peng, Li, Yang, & Wang,2021), image segmentation (Cai et al.,2022), edge detection (Xian et al.,2023;Yan, Zhang, Luo, Peng, & Wang,2022), function optimization (Zhang, Rong, Neri, & Pérez- Jiménez,2014;Zhu et al.,2021), time series prediction (Liu, Long, Peng, et al.,2022;Liu, Long, Yang, et al.,2022;Liu, Peng, et al.,2023; Long, Liu, Peng, Wang, & Yang,2022;Long, Liu, Peng, Yang, et al., 2022;Long, Lugu, et al.,2022), financial forecasting (Zhang, Yang, Liu, Peng, & Wang,2023), sentiment analysis (Huang, Liu, et al.,2023; Huang, Peng, et al.,2023;Liu, Huang, Yang, Peng, & Wang,2023), and supervisory control (Chen, Peng, Wang, & Hao,2022). Nonlinear SNP (NSNP) systems (Peng, Lv, et al.,2020) are nonlinear variants of SNP systems, where nonlinear spiking rules are introduced. The nonlinear spiking rules contain nonlinear spike consumption and nonlinear spike generation, known as a nonlinear spiking mechanism. Therefore, the nonlinear spiking mechanism is an identifiable feature of NSNP systems that distinguishes them from existing SNP systems and variants. Due to the identifiable feature, NSNP systems can show nonlinear dynamics, so they can be used to characterize nonlinear E-mail address: [email protected] (H. Peng). https://doi.org/10.1016/j.neunet.2023.10.041 Received 31 January 2023; Received in revised form 12 September 2023; Accepted 25 October 2023 Available online 28 October 2023 0893-6080/ 275 H. Peng et al. dynamic systems. Moreover, each neuron in NSNP systems has an internal state unit, which can remember the state of the previous time, and its state can be evolved by nonlinear spiking rules. Therefore, NSNP systems are particularly suitable for processing sequence data. This is the second feature of NSNP systems. This study focused on how to use the two features of NSNP systems to develop a novel model for time series classification tasks. 1.1. Related work Time series data widely exist in all aspects of real life, such as medical processes (Rahhal et al.,2016), economic prediction (Gharehbaghi & Lindén,2017), industrial processes (Cortés-Ibáñez et al., 2020), intelligent transportation (Gupta, Gupta, Biswas, & Dutta,2020), and behavior recognition (body posture Li, Li, & Fu,2018 and behavior state Gupta, Gupta, Biswas, & Dutta,2021). Time series data is a group of observation series sorted in chronological order. Time series can be divided into univariate time series (UTS) and multivariate time series (MTS) according to the number of variables. Because MTS contains more information than UTS, the analysis of time series data, especially MTS analysis, is still a hotspot in the field of data mining and also plays an important role in the fields of science, technology, and finance. Time series classification (TSC) has been a relatively active research topic in recent decades and has important applications in financial classification (Mori, Mendiburu, Miranda, & Lozano,2019;Wan & Si, 2017), traffic classification (Xiao et al.,2019), medical diagnosis (Chaovalitwongse, Pottenger, Wang, Fan, & Iasemidis,2011), and other fields. MTS classification has applications in speech analysis (Trentin, Scherer, & Schwenker,2015) and motion recognition (Hunt & Parry, 2015). However, while processing MTS classification, it is difficult to represent the complex dynamic features in MTS samples. In this study, we focused on using dynamic models to establish classifiers, processing MTS through model order, and using state sequence generated over time to achieve classification tasks. In recent years, many TSC-based methods have been proposed, which can be summarized into three categories: (1) distance-based, (2) feature-based methods, and (3) RC model-based methods. C. and Vercellis (2010) proposed a time series classification method based on discrete support vector machines with time expansion and fixed cardinal bending distance, called TDVM. Mei, Liu, Wang, and Gao (2016) proposed a DTW method based on Markov distance, which established an accurate relationship between each variable and its corresponding category, and calculated the local distance between vectors in MTS, referred to as MDDTW. Górecki and Łuczak (2015) proposed a method using DTS parameter derivative DTW distance and nearest neighbor rule for classification, called DDDTW. For feature-based methods, Baydogan and Runger (2015) proposed a symbol representation-based classifier, called SMTS. Karim, Majumdar, Darabi, and Harford (2019) proposed a method to combine long short-term memory (LSTM) and CNN with extrusion and excitation blocks, called LSTM-FCN. Moreover, Bagnall, Lines, Hills, and Bostrom (2015) proposed a feature detection method by an integrated method (COTE), and Zhao, Lu, Chen, Liu, and Wu (2017) used a convolutional neural network (CNN) for MTS classification. For RC model-based methods, Wang, Wang, and Liu (2016) proposed a method based on echo state network and adaptive differential evolution algorithm (CADE). Ma et al. (2016) proposed a classifier that introduced a spatio-temporal aggregation algorithm into the output layer of echo state networks, called (FESN). Bianchi, Scardapane, Løkse, and Jenssen (2021) proposed a reservoir classification method based on MTS representation. 1.2. Our contribution Reservoir computing (RC) is a family of recurrent neural networks (RNNs), and a significant feature that distinguishes existing RNNs is that the recurrent part (reservoir) is generated randomly and then remains fixed. Moreover, the reservoir provides a rich pool of dynamic features for solving a large variety of tasks. In principle, any dynamic system can be used as the reservoir of RC models. Due to their nonlinear spiking mechanism, NSNP systems can show rich nonlinear dynamics. Therefore, NSNP systems were introduced to develop two RC variants for time series classification tasks: the RC-SNP and RC-RMS-SNP models. The contributions of this work can be summarized as follows: (1) NSNP systems are introduced to propose an RC model for time series classification tasks, called the RC-SNP model. The use of an NSNP reservoir is an essential difference between the RC-SNP and existing RC models. (2) To more effectively extract the representation of time series, reservoir model space (RMS) is introduced to propose a variant, called the RC-RMS-SNP model. The RMS representations can be extracted using ridge regression to predict the next reservoir state. (3) The two proposed RC models are easily implemented in the framework of RC models, and ridge regression algorithms are used to train the parameters of the readout layer, and softmax is used to implement time series classification. (4) Experiments conducted on extensive time series datasets demonstrate the effectiveness of NSNP systems as the reservoir in RC models for time series classification tasks. The remainder of this paper is organized as follows. Section 2 briefly reviews time series classification problems, reservoir computing, and model space representation. Section 3describes the proposed RC models based on SNP systems. The experimental results and analysis are presented in Section 4. Finally, conclusions are drawn in Section 5. 2. Preliminaries 2.1. Time series classification problems Time series classification (TSC) is a task that classifies unlabeled samples given some labeled samples. Suppose that = {(𝐗1,𝐲1), …,(𝐗𝑖,𝐲𝑖),…,(𝐗𝑁,𝐲𝑁)} is a time series dataset with 𝑁samples, where 𝐗𝑖= (𝐱𝑖(1),…,𝐱𝑖(𝑡),…,𝐱𝑖(𝑡max))𝑇is the 𝑖th sample, and 𝐱𝑖(𝑡) ∈ 𝑅𝐷, 𝑡= 1,2,…, 𝑡max;𝐲𝑖∈𝑅𝑁𝑐denotes the label of the 𝑖th sample; 𝐷is the time series dimension at each time step; 𝑡max denotes time series length; and 𝑁𝑐indicates number of categories. Usually, a classifier for TSC can be viewed as a combination of an encoder and decoder. The encoder is used to generate a representation of the input. Moreover, the decoder is a discriminative model that computes the posterior probability of the output given the encoder representation. Because RNNs are particularly suitable to process sequential data, they can be used as the encoder for TSC problems. Given a𝐷-dimensional input time series 𝐗= (𝐱(1),…,𝐱(𝑡),…,𝐱(𝑡max))𝑇, the state updating equation can be described as follows: 𝐮(𝑡) = 𝑓(𝐱(𝑡),𝐮(𝑡− 1); 𝜃𝑒𝑛)(1) where 𝐮(𝑡)denotes the state of the RNN model at time 𝑡, which is associated with the current input 𝐱(𝑡)and previous state 𝐮(𝑡− 1), and 𝜃𝑒𝑛 is a parameter of the encoder. Let 𝐔= (𝐮(1),𝐮(2),…,𝐮(𝑡max))𝑇be a state sequence of the RNN at time 1,2,…, 𝑡max.𝐔can be used to extract the representation 𝐫𝐗=𝑟(𝐔) of the input 𝐗. To reconstruct the original input, an RNN often uses the last state, that is, 𝐫𝐗=𝐮(𝑡max). The decoder is used to map the representation 𝐫𝐗to the output space, which is a set of class labels 𝐲for TSC task, 𝐲=𝑔(𝐫𝐗;𝜃𝑑𝑒)(2) where 𝑔(⋅)can be a feedforward neural network or linear model, and 𝜃𝑑𝑒 denotes the parameters of the decoder. 276 H. Peng et al. For a fully trainable RNN, given a set of MTS, 𝐗1,𝐗2,…,𝐗𝑁, and the corresponding labels 𝐲1,𝐲2,…,𝐲𝑁, the parameters 𝜃𝑒𝑛 and 𝜃𝑑𝑒 for the encoder and decoder, respectively, can be optimized by minimizing the following empirical cost: 𝜃∗ 𝑒𝑛, 𝜃∗ 𝑑𝑒 = arg min 𝜃𝑒𝑛,𝜃𝑑𝑒 1 𝑁 𝑁 ∑ 𝑛=1 𝑙(𝐲𝑛, 𝑔(𝑟(𝑓(𝐗𝑛,𝐔)))) (3) where 𝑙(⋅,⋅)is a generic loss function, and it can be cross entropy over the labels for TSC problems. Thus, the backpropagation through time (BPTT) algorithm can be used to determine the optimal solution, 𝜃∗ 𝑒𝑛, 𝜃∗ 𝑑𝑒. For TSC problems, due to the larger 𝑇(meaning a deeper recurrent architecture), the trainable RNN often suffers from some disadvantages, e.g., gradient dispersion. Thus, some variants of RNNs have been developed to overcome the disadvantages, such as LSTM (Hochreiter & Schmidhuber,1997) and gated recurrent unit (GRU) (Cho et al.,2014). 2.2. Reservoir computing and model space representation RC is a novel variant of RNN (Lukovševičious & Jaeger,2009). However, RC differs from the existing RNN models in terms of designs and learning techniques: (i) the reservoir represents nonlinear dynamics, which realizes the encoding of the input; the decoder is a readout layer, which can be a feedforward neural network or linear model; (ii) RC performs the computational separation of the reservoir and readout; weights of the reservoir are initialized randomly and remain unchanged during the training, and only the weights in the readout can be trained. Echo state networks (ESNs) are a family of RC models for regression problems, including leaky integrator ESN (Jaeger, Lukovševičious, Popovici, & Siewert,2007). The state updating equation of the encoder (or reservoir) can be given by 𝐮(𝑡) = (1 − 𝜆)𝐮(𝑡− 1) + 𝜆tanh(𝐖𝑖𝑛𝐱(𝑡) + 𝐖𝐮(𝑡− 1) + 𝐛)(4) where parameters of the encoder are 𝜃𝑒𝑛 = {𝐖𝑖𝑛,𝐖,𝐛},𝐖𝑖𝑛 is the input- to-reservoir weight matrix, 𝐖is the recurrent reservoir weight matrix, 𝐛is the bias-to-reservoir weight vector, and 𝜆∈ [0,1] is the leaker parameter. The parameters of the encoder are initialized randomly and remain unchanged during the computation. In the ESN, the decoder is expressed by 𝐲=𝑔(𝐫𝐗) = 𝐖𝑜𝑢𝑡𝐫𝐗+𝜃𝑜𝑢𝑡 (5) The parameters of the decoder 𝜃𝑑𝑒 = {𝐖𝑜𝑢𝑡, 𝜃𝑜𝑢𝑡}can be learnt by minimizing a ridge regression loss function 𝐖∗𝑜𝑢𝑡, 𝜃∗ 𝑜𝑢𝑡 = arg min 𝐖𝑜𝑢𝑡,𝜃𝑜𝑢𝑡 1 2‖(𝐖𝑜𝑢𝑡𝐫𝐗+𝜃𝑜𝑢𝑡) − 𝐲‖2+𝜆‖𝐖𝑜𝑢𝑡‖2(6) In the ESN, the representation 𝐫𝐗is the output model space, i.e., 𝐫𝐗= 𝐮(𝑡). Thus, the decoder can be described by 𝐲(𝑡) = 𝐖𝑜𝑢𝑡𝐫𝐗+𝜃𝑜𝑢𝑡 =𝐖𝑜𝑢𝑡𝐮(𝑡) + 𝜃𝑜𝑢𝑡 (7) 3. Proposed RC models based on NSNP systems 3.1. NSNP systems SNP systems are neural-like membrane computing models, and NSNP systems (Peng, Lv, et al.,2020) are nonlinear variants of SNP systems. The NSNP systems in Peng, Lv, et al. (2020) are used as a theoretical computing model, and is used to study the computational completeness. To handle time series classification problems, the theoretical computing model is usually needs to be extended and some constraints relaxed. For convenience, we still refer to this modification as NSNP systems. The nonlinear spiking mechanism is a remarkable feature that distinguish NSNP from SNP systems. The nonlinear spiking mechanism is composed of two parts: nonlinear spike consumption and nonlinear spike generation. Under the control of nonlinear spiking mechanism, NSNP systems can show a rich pool of nonlinear dynamics. Fig. 1. (a) NSNP system 𝛱. (b) Neuron 𝜎𝑖and its adjacent neurons. Fig. 2. Proposed RC-SNP model. Definition 1. An NSNP system is defined by 𝛱= (𝑂, 𝜎1, 𝜎2,…, 𝜎𝑚, 𝑠𝑦𝑛, 𝐱) where (1) 𝑂= {𝑎}is an alphabet (𝑎is called the spike). (2) 𝜎𝑖= (𝑢𝑖, 𝑟𝑖)the is 𝑖th spiking neuron, 𝑖= 1,2,…, 𝑚, where (a) 𝑢𝑖∈𝑅is the initial state of neuron 𝜎𝑖. (b) 𝑟𝑖is a nonlinear spiking rule of the form 𝑎𝑔(𝑢)→𝑎𝑓(𝑢), where 𝑔(𝑢)and 𝑓(𝑢)are linear or nonlinear functions; 𝑢=𝑢𝑖+𝑥𝑖, where 𝑥𝑖is the 𝑖th external input. (3) 𝑠𝑦𝑛 ⊆ {1,2,…, 𝑚} × {1,2,…, 𝑚}(synapses). (4) 𝐱= (𝑥1, 𝑥2,…, 𝑥𝐷)is the external input of the system. Fig. 1 shows an NSNP system 𝛱, which is composed of 𝑚neurons, 𝜎1, 𝜎2,…, 𝜎𝑚. From a topological perspective, this NSNP system is a directed graph with a global topological structure. Moreover, the NSNP system is a multi-input multi-output (MIMO) system, and each neuron is both an input and output neuron. In NSNP systems, neurons are homogeneous, i.e., each neuron has a state unit 𝑢𝑖and nonlinear spiking rule 𝑎𝑔(𝑢)→𝑎𝑓(𝑢). Suppose that 𝑢𝑖(𝑡)is the state of neuron 𝜎𝑖at time 𝑡. The semantics of the spiking rule can be described as follows. When neuron 𝜎𝑖fires, the spikes with value 𝑔(𝑢)are consumed (and spikes with value 𝑢𝑖−𝑔(𝑢)remain), and then the spikes with value 𝑓(𝑢)are generated and sent to the successor neurons and regarded as its external 277 H. Peng et al. Fig. 3. Proposed RC-RMS-SNP model. output ℎ𝑖, where 𝑢=𝑢𝑖(𝑡− 1) + 𝑥(𝑡),𝑢𝑖(𝑡− 1) is its state at last time 𝑡− 1, and 𝑥(𝑡)is the current external input. Thus, the state updating equation and output equation for neuron 𝜎𝑖are given by 𝑢𝑖(𝑡) = 𝑢𝑖(𝑡− 1) − 𝑔(𝑢𝑖(𝑡− 1) + 𝑥(𝑡))+ 𝑚 ∑ 𝑗=1,𝑖≠𝑗 𝑓(𝑢𝑗(𝑡− 1) + 𝑥(𝑡))(8) ℎ𝑖(𝑡) = 𝑓(𝑢𝑖(𝑡− 1) + 𝑥(𝑡))(9) where 𝑖= 1,2,…, 𝑚. To simplify the complexity of model construction, we consider 𝑔(𝑢)=−𝑓(𝑢). Thus, we have 𝑢𝑖(𝑡) = 𝑢𝑖(𝑡− 1) + 𝑚 ∑ 𝑗=1 𝑓(𝑢𝑗(𝑡− 1) + 𝑥(𝑡))(10) Consider 𝑥 =∑𝐷 𝑙=1 𝑥𝑙to be external input; then, the state updating equation for neuron 𝜎𝑖is rewritten as follows: 𝑢𝑖(𝑡) = 𝑢𝑖(𝑡− 1) + 𝑚 ∑ 𝑗=1 𝑓(𝑢𝑗(𝑡− 1) + 𝐷 ∑ 𝑙=1 𝑥𝑙(𝑡))(11) The output equation for neuron 𝜎𝑖is given by ℎ𝑖(𝑡) = 𝑓(𝑢𝑖(𝑡− 1) + 𝐷 ∑ 𝑙=1 𝑥𝑙(𝑡))(12) The NSNP system is further parameterized. Thus, the state updating equation for neuron 𝜎𝑖can be rewritten by 𝑢𝑖(𝑡) = 𝛼⋅𝑢𝑖(𝑡− 1) + 𝑚 ∑ 𝑗=1 𝑤1 𝑖𝑗 ⋅𝑓(𝛽⋅𝑢𝑗(𝑡− 1) + 𝐷 ∑ 𝑙=1 𝑤2 𝑖𝑙 ⋅𝑥𝑙(𝑡))(13) where 𝛼,𝛽,𝑤1 𝑖𝑗 , and 𝑤2 𝑖𝑙 are the parameters, and 𝑖= 1,2,…, 𝑚. The output equation for neuron 𝜎𝑖can be rewritten by ℎ𝑖(𝑡) = 𝑓(𝛾⋅𝑢𝑖(𝑡− 1) + 𝐷 ∑ 𝑙=1 𝑤3 𝑖𝑙 ⋅𝑥𝑙(𝑡))(14) where 𝛾and 𝑤3 𝑖𝑙 are the parameters, and 𝑖= 1,2,…, 𝑚. Let 𝐮(𝑡) = [𝑢1(𝑡), 𝑢2(𝑡),…, 𝑢𝑚(𝑡)]𝑇,𝐱(𝑡) = [(𝑥1(𝑡), 𝑥2(𝑡),…, 𝑥𝐷(𝑡)]𝑇, and 𝐡(𝑡)=[ℎ1(𝑡), ℎ2(𝑡),…, ℎ𝑚(𝑡)]𝑇. Therefore, the state updating equation for the system can be expressed by 𝐮(𝑡) = 𝛼⋅𝐮(𝑡− 1) + 𝑊 𝑓(𝛽⋅𝐮(𝑡− 1) + 𝑊𝑖𝑛𝐱(𝑡))(15) where 𝛼∈ (0,1] and 𝛽∈ (0,1] are two scalar parameters, and 𝑊= (𝑤1 𝑖𝑗 )𝑚×𝑚and 𝑊𝑖𝑛 = (𝑤2 𝑖𝑗 )𝑛×𝑚are two parameter matrices. The output of the system can be expressed by 𝐡(𝑡) = 𝑓(𝛾⋅𝐮(𝑡− 1) + 𝑊′ 𝑜𝑢𝑡𝐱(𝑡))(16) where 𝛾∈ (0,1] is a scalar parameter, and 𝑊′ 𝑜𝑢𝑡 = (𝑤3 𝑖𝑗 )𝑛×𝑚is a parameter matrix; in this study, we set 𝑓(⋅) = tanh(⋅). In this study, The NSNP systems after specialization and parameterization are called the extended NSNP systems. It is important to note that compared to the original NSNP systems (Peng, Lv, et al.,2020), the extended NSNP systems are a simplified and expanded version. In the extended version, neurons are homogeneous: each neuron has a state unit and the same state updating equation. Eqs. (15) and (16) are the mathematical models of the extended NSNP systems. 3.2. RC-SNP model Based on the extended NSNP systems, we developed an RC model for TSC problems, called RC-SNP. Fig. 2 shows the framework of the RCSNP model. The RC-SNP model is composed of an input layer, reservoir, output model space, and readout layer. The input layer, reservoir, and output model space constitute the encoder, and the decoder is the readout layer. Different from existing RC models, the reservoir of the RC-SNP model is an NSNP system. Due to the nonlinear spiking mechanism, NSNP systems can show rich dynamics. Therefore, NSNP systems can be used as reservoirs. For the reservoir based on NSNP systems, Eq. (15) provides its state equating equation, where 𝜃𝑒𝑛 = {𝐖𝑖𝑛,𝐖}are the parameters of the encoder. As usual, the parameters of the encoder are initialized randomly and remain unchanged during the computation. The state sequence of the reservoir (or NSNP system), 𝐔= (𝐮(1),𝐮(2),…, 𝐮(𝑡max))𝑇, exhibits a rich pool of dynamics. In the RC-SNP model, the last state is used as the representation of the input, that is, 𝐫𝐗=𝐮(𝑡max). For the RC-SNP model, the decoder is a linear regression model, and softmax is used to implement the classification of MTS. The parameters of the decoder 𝜃𝑑𝑒 = {𝐖𝑜𝑢𝑡, 𝜃𝑜𝑢𝑡}can be trained by a ridge regression algorithm. 3.3. RC-RMS-SNP model In this section, the RC-SNP model is further improved to present a variant, called RC-RMS-SNP, where the RMS is introduced to replace the output model space. Fig. 3 shows the framework of the RC-RMS- SNP model. The RC-RMS-SNP model also consists of four components: input layer, reservoir, RMS, and readout layer. The encoder includes the input layer, reservoir, and RMS, while the readout layer is the decoder. The main difference between the RC-RMS-SNP and RC-SNP models is the use of the RMS. In the following, we mainly describe the RMS. In RC models, the generation ability of the reservoir is based on a large number of various dynamics, which are generated by the inputs. To predict the next input value, diverse dynamics are chosen according to the prediction horizon of interest. Therefore, the RMS has been introduced to generate the presentation of the input 𝐫𝐗for MTS (Bianchi et al.,2021). In the RMS, each MTS is expressed by the parameters of a linear model, which predicts the next reservoir state by considering all the reservoir dynamics. 278 H. Peng et al. Table 1 Details of 17 MTS data sets. Data set Train Test Length #Classes #Variables Source KICK 16 10 [274, 841] 2 62 UEA WALK 28 16 [128, 1918] 2 62 UEA CMU 29 29 [127, 580] 2 62 UEA RobotFail 100 64 [15, 15] 4 6 UCI ECG 100 100 [39, 152] 2 2 UCR Libras 180 180 [45, 45] 15 2 UCI uWave 200 427 [315, 315] 8 3 UCR PEMS 267 173 [144, 144] 7 963 UCI Jp. Vow. 270 370 [7, 29] 9 12 UCI Wafer 298 896 [104, 198] 2 6 UCR Ch.Traj. 300 2558 [109, 205] 20 3 UCI Dist Phal 400 139 [80, 80] 3 1 UCR Chlo Conc 467 3840 [166, 166] 3 1 UCR Swedish Leaf 500 625 [128, 128] 15 1 UCR NetFlow 803 534 [50, 994] 13 4 UEA Auslan 1140 1425 [45, 136] 95 22 UCI Arab. Dig. 6600 2200 [4, 93] 10 13 UCI To obtain the representation 𝐫𝐗, the reservoir (or NSNP system) is used to process each MTS. The one step-ahead prediction model that predicts the next reservoir state is considered as follows: 𝐮(𝑡+ 1) = 𝑈𝑢𝐮(𝑡) + 𝑢0(17) The parameters 𝑈𝑢and 𝑢0can be solved by minimizing a ridge regression loss function 𝜃∗ 0= arg min 𝐔𝑢,𝑢0 1 2‖(𝐔𝑢𝐮(𝑡) + 𝑢0−𝐮(𝑡+ 1))‖2+𝜆‖𝐔𝑢‖2(18) Thus, the representation of the MTS is 𝐫𝐗=𝜃𝑢= [𝑣𝑒𝑐(𝑈𝑢); 𝑢0], which is the RMS representation. The RMS representation characterizes a generative model of the reservoir state sequence, (𝑝(𝐮(𝑡max),𝐮(𝑡max − 1)),…,𝐮(1); 𝐫𝐗). Similar to the RC-SNP model, the RC-RMS-SNP model uses a linear regression model, and softmax is used to implement the classification of MTS. Therefore, the usual ridge regression algorithm is used to learn the parameters of the decoder. The two proposed RC models were implemented in machine learning platform ‘‘Skilearn’’ by using Python language, and a ridge regression algorithm was used to train the parameters. In the proposed two RC models, NSNP systems are used as the reservoir. Therefore, as usual in the existing RC models, we ensure the echo state property of the reservoir by setting appropriate parameter values. These parameters include spectral radius, connection rate, input scaling, etc. The typical settings of these reservoir parameters can be found in Section 4.2. 4. Experimental results and analysis In this study, we implemented two RC models: RC-SNP and RC-RMS- SNP. The two RC models were evaluated on 17 benchmark datasets for time series classification tasks and compared with 16 state-of-the-art or baseline classifiers. 4.1. Data sets We used 17 open benchmark datasets for time series classification. The datasets were retrieved from UCI,1UCR, and UEA,2which have different numbers of variables and classes and different value ranges. There are three UTS: Distal Phalanx Outline (Dist Phal), Chlorine Concentration (Chlo Conc), and Swedish Leaf. The other 14 datasets are MTS: Kick versus Punch (KICK), Walk versus Run (WALK), CMUsubject16 (CMU), Robotic Arm Failure (RobotFail), Electrocardiography 1https://archive.ics.uci.edu/datasets 2https://www.timeseriesclassification.com/dataset.php Table 2 Hyper-parameters used in the experiments. Data set 𝛼 𝛽 Connection rate Input scaling KICK 0.3 0.1 0.3 0.1 WALK 0.1 0.1 0.25 0.15 CMU 1 0.9 0.8 0.1 RobotFail 1 1 0.2 0.1 ECG 0.3 0.5 0.25 0.15 Libras 0.1 0.9 0.25 0.15 uWave 0.9 0.1 0.25 0.15 PEMS 0.9 0.1 0.5 0.1 Jp. Vow. 0.2 1.0 0.25 0.15 Wafer 0.8 0.2 0.25 0.15 Ch.Traj. 0.6 0.5 0.25 0.15 Dist Phal 1 0.1 0.5 0.2 Chlo Conc 0.1 1.0 0.1 0.4 Swedish Leaf 0.5 0.6 0.8 0.2 NetFlow 0.9 0.6 0.25 0.15 Auslan 0.7 0.3 0.25 0.15 Arab. Dig. 0.4 0.9 0.25 0.15 (ECG), Libras, UWave Gesture Library (uWave), Performance Measurements (PEMS), Japanese Vowels (Jp. Vow.), Wafer, Character Trajectory (Ch. Traj.), Network Flow (NetFlow), Australian Sign Language (Auslan), and Arabic Digits (Arab. Dig.). The details of the datasets are provided in Table 1. In the experiments, each time series was first processed as a three-dimensional tensor and was input to the model for training and prediction. 4.2. Parameter configuration In experiments, the parameters for the RC-SNP and RC-RMS-SNP models were set as follows: (i) the size of NSNP reservoir is 800, spectral radius is 0.99, connection rate is in the range (0,1], and input scaling is in (0,1]; (ii) 𝛼∈ [0,1],𝛽∈ [0,1], noise level is 0.0001, and parameter of dropout neurons is 5.0; (iii) the regularization parameter for ridge regression to perform classification task is 𝜆= 5.0, and the regularization parameter for ridge regression to perform the prediction task is 𝜆= 5.0. Moreover, a simple grid search method is used for the parameter values of 𝛼,𝛽, connection rate, and input scaling for each MTS, as shown in Table 2. For each dataset, we conducted 10 independent random initializations for the NSNP reservoir and then performed the classification tasks. We calculated the average and standard deviation values of the 10 classification results. 4.3. Performance indice To evaluate the proposed RC models and compare them with existing MTS classifiers, the accuracy rate was used as a performance index. 279 H. Peng et al. Table 3 Comparison results of the proposed models with other classifiers in terms of accuracy rate. Data set LSTM GRU DWT-1NN rmESN bi-rmESN KICK 45.67 ±5.46 35.83 ±5.97 70 75.02 ±4.83 100.0 ±0.00 WALK 75.57 ±0.13 75.57 ±0.13 100 100.0 ±0.00 100.0 ±0.00 CMU 55.17 ±13.79 54.79 ±9.52 100 98.27 ±1.72 94.83 ±1.72 RobotFail 64.69 ±3.22 63.75 ±2.30 68.75 64.38 ±1.17 56.88 ±1.25 ECG 76.20 ±4.26 81.20 ±3.49 84 85.20 ±0.75 85.20 ±0.40 Libras 68.22 ±2.62 71.56 ±4.60 88.33 88.11 ±1.43 86.00 ±0.65 uWave 72.52 ±1.71 79.49 ±2.65 89.46 88.88 ±0.52 90.51 ±1.16 PEMS 85.57 ±1.57 89.67 ±1.51 70.52 70.40 ±3.79 72.37 ±2.02 Jp. Vow. 92.70 ±1.36 94.00 ±2.21 93.51 97.83 ±0.50 97.62 ±0.46 Wafer 96.32 ±3.70 98.41 ±0.86 95.09 97.78 ±0.29 97.01 ±0.40 Ch.Traj. 37.10 ±14.62 70.79 ±17.71 95.78 97.36 ±0.24 97.00 ±0.11 Dist Phal 70.94 ±2.93 72.66 ±1.29 74.82 75.57 ±1.32 75.23 ±0.73 Chlo Conc 60.42 ±1.10 60.85 ±1.13 62.6 85.60 ±0.41 83.72 ±0.62 Swedish Leaf 89.58 ±00.71 88.24 ±01.62 81.72 94.51 ±00.80 96.25 ±0.66 NetFlow 91.84 ±1.39 93.13 ±2.25 92.08 96.96 ±0.54 96.75 ±0.50 Auslan 1.05 ±0.00 1.05 ±0.00 85.61 97.25 ±0.25 97.64 ±0.35 Arab. Dig. 96.61 ±0.69 95.98 ±2.91 – 98.12 ±0.21 96.46 ±0.44 Data set dr-rmESM dr-rmESM dr-rmESM RC-SNP RC-RMS-SNP KICK 50.23 ±1.98 60.94 ±0.01 60.50 ±0.01 67.00 ±4.58 70.00 ±0.00 WALK 100.0 ±0.00 100.0 ±0.00 100.0 ±0.00 100.00 ±0.00 100.0 ±0.00 CMU 100.0 ±0.00 98.27 ±1.72 96.55 ±0.01 98.97 ±1.58 92.76 ±3.60 RobotFail 66.25 ±1.88 62.81 ±1.17 63.75 ±1.17 68.91 ±1.77 61.56 ±3.29 ECG 83.80 ±1.60 83.40 ±1.62 84.00 ±1.10 80.50 ±3.61 90.40 ±1.11 Libras 86.67 ±0.79 87.22 ±1.76 84.78 ±1.03 71.39 ±1.37 89.33 ±1.55 uWave 89.39 ±1.45 84.86 ±0.83 86.54 ±1.48 88.11 ±1.02 94.51 ±0.35 PEMS 69.02 ±3.48 69.48 ±3.86 68.67 ±2.77 76.24 ±2.94 72.31 ±2.56 Jp. Vow. 97.99 ±0.65 98.14 ±0.44 96.97 ±0.63 97.30 ±0.40 98.14 ±0.35 Wafer 96.85 ±0.65 95.60 ±0.82 95.69 ±1.00 98.46 ±0.24 99.20 ±0.14 Ch.Traj. 97.52 ±0.54 96.58 ±0.67 97.59 ±0.23 93.03 ±0.34 98.48 ±0.21 Dist Phal 76.52 ±0.70 75.54 ±1.02 75.52 ±2.00 74.60 ±0.56 76.04 ±0.65 Chlo Conc 85.07 ±1.36 79.68 ±0.69 81.21 ±2.63 60.71 ±0.84 80.56 ±0.57 Swedish Leaf 95.56 ±0.66 96.64 ±00.77 95.47 ±00.86 83.54 ±01.17 93.79 ±00.47 NetFlow 96.11 ±0.66 97.27 ±0.47 97.12 ±0.69 78.09 ±0.00 93.01 ±1.16 Auslan 88.70 ±1.38 73.47 ±4.77 85.75 ±0.87 71.99 ±1.54 97.76 ±0.32 Arab. Dig. 96.10 ±0.35 95.39 ±0.52 96.02 ±0.76 93.83 ±0.45 99.13 ±0.16 The accuracy rate is defined by 𝐴𝑐𝑐𝑢𝑟𝑎𝑡𝑒 𝑟𝑎𝑡𝑒 =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 𝑇 𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑠𝑡 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 (19) 4.4. Comparison results The proposed two RC models were first compared with 8 RNN- like time series classifiers on 17 benchmark time series classification datasets. The 8 RNN-like classifiers were LSTM (Hochreiter & Schmidhuber,1997), GRU (Cho et al.,2014), DWT-1NN (Mei et al.,2016), rmESN (Bianchi et al.,2021), bi-rmESN (Bianchi et al.,2021), drrmESN(Max) (Bianchi et al.,2021), dr-rmESN(ReLU) (Bianchi et al., 2021), and dr-rmESN(KAF) (Bianchi et al.,2021), where ‘‘bi’’ indicates bi-directional reservoir, ‘‘dr’’ represents deep MLP used as readout, and ‘‘ReLU’’, ‘‘Maxout’’, and ‘‘KAF’’ represent the respective activation function being used in deep MLPs. Table 3 provides the comparison results of the two proposed RC models with 8 RNN-like time series classifiers in terms of the accuracy rate. The results of the 8 time series classifiers were retrieved from their original references, and their hyper-parameters can be found in the original references. From Table 3, we can observe that the proposed RC-SNP model achieved the best classification accuracy on the RobotFail and WALK datasets, and for most other datasets, it also achieved results close to the optimal classification performance. The proposed RC-RMS-SNP model is an improvement of the RC-SNP model with the embedding of the RMS. From Table 3, it can be seen that (i) the proposed RCRMS-SNP model achieves the best classification accuracy on 9 datasets, including WALE, ECG, Libras, uWave, Jp. Vow., Wafer, Ch. Traj., Auslan, and Arab. Dig; (ii) the difference between the proposed RCRMS-SNP model and dr-rmESN(Max) model with the best performance on the CMU dataset is 7.24%; (iii) the difference between the proposed RC-RMS-SNP model and GRU model with the best performance on the PEMS dataset is 17.36%; (iv) the classification accuracies obtained by the proposed RC-RMS-SNP model on the other five datasets are close to the results of the corresponding optimal models, and the differences are less than 6%. In the experiments, we also compared the proposed RC models with 8 baseline models. Among the compared models, TDVM (C. & Vercellis, 2010) and DD𝐷𝑇 𝑊 (Górecki & Łuczak,2015) are two distance-based models; SMTS (Baydogan & Runger,2015), LSTM-FCN (Karim et al., 2019), and CNN (Zhao et al.,2017) are three feature-based models; and DESN, C𝐴𝐷𝐸 (Wang et al.,2016), and FESN (Ma et al.,2016) are three RC-like models. Table 4 presents the comparison results of the two proposed RC models with 8 baseline models on nine datasets in terms of accuracy rate. The results of the 8 baseline models were retrieved from their original references, and their hyper-parameters can be found in the original references. It can be observed from Table 4 that (i) the proposed RC-RMS-SNP model achieved the best classification accuracy on the ECG, Wafer, and Auslan datasets; (ii) the proposed RC-RMS-SNP model obtained results close to the optimal classification performance for each of the Ch.Traj., Arab. Dig. and Jp. Vow datasets. However, for the CMU, Libras, and uWave datasets, the proposed RC-RMS-SNP model still underperforms the best classification accuracy. Fig. 4(a) shows the average classification performances of the proposed RC models and 8 compared models. Overall, the proposed RCRMS-SNP, rmESN, and bi-rmESM models show high average classification performance; however, the performance of the proposed RC-RMS- SNP model is slightly lower than that of the bi-rmESM model. From Table 3, we can see that this is related to the fact that the proposed models do not show excellent performance on three UTS datasets (Dist Phal, Chlo Conc, and Swedish Leaf). This indicates that the proposed RC models need to be further improved. Fig. 4(b) shows a comparison of the average computing time on 17 benchmark time series data-sets for the proposed RC models and 280 H. Peng et al. Table 4 Comparison results of the proposed models with other models in terms of accuracy rate. Data set TDVM DD𝐷𝑊 𝑇 SMTS LSTM-CNN CNN ECG – 85.5 86.6 86 86.04 Wafer – 98.1 99 – 98.07 Ch.Traj. – 99.1 – 100 98.25 Arab. Dig. – 99.8 – – 98.56 Jp. Vow. 96.6 98 96.5 – 97.66 Auslan – 81.5 97.3 97 72.83 CMU – 96.3 99.3 100 93.81 Libras – 95 – 97 93.61 Data set DESN C𝐴𝐷𝐸 FESN RC-SNP RC-RMS-SNP ECG 86.04 90 86.54 80.50 90.4 Wafer 98.16 98.3 98.07 98.46 99.2 Ch.Traj. 98.11 98 98.18 93.03 98.48 Arab. Dig. 99.16 99.2 99.23 93.83 99.13 Jp. Vow. 98.13 99.4 98.18 97.30 98.14 Auslan 78.48 75.3 81.44 71.99 97.76 CMU 95.00 100 96.90 98.97 92.76 Libras 94.72 95.8 95.56 71.39 89.33 Fig. 4. Comparison of the proposed RC models with other models on all datasets in terms of (a) average performance and (b) average computing time. five compared RC-like models. It can be observed in Fig. 4(b) that the proposed RC models have relatively short average computing time. This illustrates that the proposed RC models are computationally efficient. 5. Conclusions This paper proposed two novel RC variants: the RC-SNP and RCRMS-SNP models. The distinguishing feature of the two RC variants from existing RC models is the use of an NSNP system as the reservoir. Moreover, different from the RC-SNP model, the RC-RMS-SNP model uses RMS to more effectively extract the representation of time series. The proposed RC variants can easily be implemented in the RC framework, and the usual ridge regression algorithm can be used to extract the model space representation and train the parameters of the decoder (readout layer). To evaluate the proposed RC variants, 17 time series classification datasets were used in the experiments. The proposed RC variants were compared with 16 state-of-the-art or baseline classification models. The experimental results demonstrate that the proposed RC variants are competent for time series classification tasks and have certain advantages. From these experiments, we found that on a few datasets, the proposed RC models did not perform the best. This indicates that these two models require further improvement. In the membrane computing community, different variants of SNP systems have been proposed, abstracted from different biological mechanisms, such as SNP systems with autapses, multiple channels, inhibition rules, etc. Due to their different mechanisms, these variables exhibit different dynamic characteristics. Therefore, in future work, we plan to introduce these variants to improve the performance of the proposed RC models. In addition, there are some related theoretical issues, such as the echo state property. These theoretical issues are worth studying in future work. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability The data that has been used is confidential. References Bagnall, A., Lines, J., Hills, J., & Bostrom, A. (2015). 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