Santa Fe Time Series Research Articles

Performance-enhanced time-delayed photonic reservoir computing system using a reflective semiconductor optical amplifier.

We propose a time-delayed photonic reservoir computing (RC) architecture utilizing a reflective semiconductor optical amplifier (RSOA) as an active mirror. The performance of the proposed RC structure is investigated by two benchmark tasks, namely the Santa Fe time-series prediction task and the nonlinear channel equalization task. The simulation results show that both the prediction and equalization performance of the proposed system are significantly improved with the contribution of RSOA, with respect to the traditional RC system using a mirror. By increasing the drive current of the RSOA, the greater nonlinearity of the RSOA gain saturation is achieved, as such the prediction and equalization performance are enhanced. It is also shown that the proposed RC architecture shows a wider consistency interval and superior robustness than the traditional RC structure for most of the measured parameters such as coupling strength, injection strength, and frequency detuning. This work provides a performance-enhanced time-delayed RC structure by making use of the nonlinear transformation of the RSOA feedback.

Enhanced optoelectronic reservoir computation using semiconductor laser with double delay feedbacks.

We numerically explored the enhanced performance and physical mechanism of semiconductor laser (SL) based reservoir computation (RC) with double optoelectronic feedback (DOEF). One-step and multistep Santa Fe time series predictions were used as standard test benchmarks in this work. We found that in the optimized parameter region the normalized mean square error (NMSE) of an SL-based RC under DOEF is smaller than an SL-based RC with single optoelectronic feedback (SOEF). In addition, the performance improvement is more obvious for multistep prediction, which is particularly suitable for more complex tasks that requires a higher memory capability (MC). The enriched node states (optical intensity of the virtual nodes for each sample) and the enhanced MC of the proposed DOEF were verified by a comparison to SOEF under the optimized feedback strength. The influence of the feedback strength and the delay difference on the NMSE and the MC was also investigated. Our study should be helpful in the design of a high-performance optoelectronic RC based on an SL.

All-optical reservoir computing system based on polarization dynamics

Reservoir computing (RC) is a simplified recurrent neural network and can be implemented by using a nonlinear system with delay feedback, thus it is called delay-based RC. Various nonlinear nodes and feedback loop structures have been proposed. Most of existing researches are based on the dynamical responses in intensity of the nonlinear systems. There are also a photoelectric RC system based on wavelength dynamics and an all-optical RC based on the phase dynamics of a semiconductor laser with optical feedback, as well as so-called polarization dynamics of a vertical cavity surface emitting laser (VCSEL). However, these VCSEL-RCs actually are based on the intensity dynamics of two mutually orthogonal polarization modes, or polarization-resolved intensity dynamics. The RC based on rich dynamical responses in polarization has not yet been found. A semiconductor optical amplifier (SOA) fiber ring laser can produce rich dynamical states in polarization, and is used in optical chaotic secure communication and distributed optical fiber sensing. To further expand the application of polarization dynamics of the SOA fiber ring laser and open up a new direction for the research of optical RC neural network, an all-optical RC system based on polarization dynamics of the ring laser is proposed. The ring laser is used as the reservoir, and the SOA as the nonlinear node. After the input signal is masked according to a synchronization scheme, it is injected into the reservoir by intensity modulation for a continuous wave generated by a superluminescent light emitting diode (SLED). The dynamical response in polarization of the ring laser is detected by a polarizer and a photodetector. The influences of the SOA operation current, output power of the SLED and attenuation of a variable optical attenuator (VOA) in the fiber loop on the polarization dynamic characteristic (mainly referring to the output degree of polarization) of the laser are analyzed experimentally. The fading memory and nonlinear response of the RC system based on the polarization dynamic response and intensity dynamic response are compared experimentally. The influences of output power of the SLED and attenuation of the VOA on fading memory, consistency and separation of the RC system based on the two kinds of dynamic responses are investigated experimentally. Thus the range of the VOA attenuation is determined. The network performance of the polarization dynamics RC system is evaluated by processing a Santa Fe time series prediction task and a multi-waveform recognition task. The normalized mean square error can be as low as 0.0058 for the time series prediction task, and the identification rate can be as high as 100% for the recognition task under the appropriate system parameters and only 30 virtual nodes. The experimental results show that the polarization dynamics RC system has good prediction performance and classification capability, which are comparable to the existing RC system based on intensity dynamics of the ring laser. The system can be expected to process two tasks in parallel when the polarization dynamics and intensity dynamics are used at the same time.

Characteristic Study on Parallel Reservoir Computation Based on VCSEL Dynamics With Optoelectronic Feedback

We proposed a new reservoir computation (RC) structure using polarization dynamics of vertical cavity surface emitting lasers (VCSELs) with optoelectronic feedback. Santa-Fe time-series prediction and nonlinear channel equalization were used to evaluate its parallel processing capability. We numerically discussed the effects of feedback strength, injection strength, frequency detuning, injection current and scaling factors on the RC performance of the system. The results show that, the operation range of negative feedback is larger than that of positive feedback; under appropriate feedback and lager injection strengths, good RC performance can be achieved for a wider injection current; with continuous variation of feedback strength from negative to positive values, the optimized location for frequency detuning moves from positive to negative values. We also evaluated the processing rate of proposed RC structure. The related study is helpful for the design and optimization of VCSEL-based RC under optoelectronic feedback.

Enhanced Prediction Performance of Reservoir Computing Based on Mutually Delay-Coupled Semiconductor Lasers via Parameter Mismatch

As an efficient information processing method, reservoir computing (RC) is essential to artificial neural networks (ANNs). Via the Santa Fe time series prediction task, we numerically investigated the effect of the mismatch of some critical parameters on the prediction performance of the RC based on two mutually delay-coupled semiconductor lasers (SLs) with optical injection. The results show that better prediction performance can be realized by setting appropriate parameter mismatch scenarios. Especially for the situation with large prediction errors encountered in the RC with identical laser parameters, a suitable parameter mismatch setting can achieve computing performance improvement of an order of magnitude. Our research is instructive for the hardware implementation of laser-based RC, where the parameter mismatch is unavoidable.

Adaptive time-delayed photonic reservoir computing based on Kalman-filter training.

We propose an adaptive time-delayed photonic reservoir computing (RC) structure by utilizing the Kalman filter (KF) algorithm as training approach. Two benchmark tasks, namely the Santa Fe time-series prediction and the nonlinear channel equalization, are adopted to evaluate the performance of the proposed RC structure. The simulation results indicate that with the contribution of adaptive KF training, the prediction and equalization performance for the benchmark tasks can be significantly enhanced, with respect to the conventional RC using a training approach based on the least-squares (LS). Moreover, by introducing a complex mask derived from a bandwidth and complexity enhanced chaotic signal into the proposed RC, the performance of prediction and equalization can be further improved. In addition, it is demonstrated that the proposed RC system can provide a better equalization performance for the parameter-variant wireless channel equalization task, compared with the conventional RC based on LS training. The work presents a potential way to realize adaptive photonic computing.

Microring resonators with external optical feedback for time delay reservoir computing.

Microring resonators (MRRs) are a key photonic component in integrated devices, due to their small size, low insertion losses, and passive operation. While the MRRs have been established for optical filtering in wavelength-multiplexed systems, the nonlinear properties that they can exhibit give rise to new perspectives on their use. For instance, they have been recently considered for introducing optical nonlinearity in photonic reservoir computing systems. In this work, we present a detailed numerical investigation of a silicon MRR operation, in the presence of external optical feedback, in a time delay reservoir computing scheme. We demonstrate the versatility of this compact, passive device, by exploiting different operating regimes and solving computing tasks with diverse memory requirements. We show that when large memory is required, as it occurs in the Narma 10 task, the MRR nonlinearity does not play a significant role when the photodetection nonlinearity is involved, while the contribution of the external feedback is significant. On the contrary, for computing tasks such as the Mackey-Glass and the Santa Fe chaotic timeseries prediction, the MRR and the photodetection nonlinearities contribute both to efficient computation. The presence of optical feedback improves the prediction of the Mackey-Glass timeseries while it plays a minor role in the Santa Fe timeseries case.

Experimental Investigation of an Optical Reservoir Computing System Based on Two Parallel Time-Delay Reservoirs

We experimentally investigate the performances of an optical reservoir computing (RC) system based on two parallel time-delay reservoirs composed of two semiconductor lasers (SLs) subject to optical feedback. In such a system, the information being processed is split into two parts to send into two reservoirs through directly modulating the pump currents of two SLs, and the temporal output of the two SLs are sampled and taken as the virtual node states for training and testing. Via Santa Fe time series prediction task and multi-waveform recognition task, the performances of the proposed RC system are investigated and compared with those of the system based on one reservoir. The results show that the system based on two parallel reservoirs behaves better performance and stronger parameter robustness than that based on one reservoir. Moreover, through analyzing the dependence of the system performances on the number of virtual node states actually used for readout, the potential data processing rate (DPR) of the system is evaluated. For processing a prediction task under guaranteeing the normalized mean square error below 0.1 and a recognition task under guaranteeing the signal error rate below 0.005, the potential DPR of the proposed RC system can achieve 200 MSa/s, which is twice the DPR of the system with only one reservoir.

Design of parallel reservoir computing by mutually-coupled semiconductor lasers with optoelectronic feedback

Via implementing parallel tasks including Santa-Fe time series prediction and nonlinear channel equalization, we studied the characteristics of reservoir computing (RC) using two mutually-coupled semiconductor lasers (MC-SLs) with optoelectronic feedback. The influence of system parameters (injection current, coupling strength, scaling factor and coupling time-delays) on RC performance have been investigated in detail. Results show that, the optimized performance is always achieved around the edge of bifurcation points; positive feedback is more preferred when the scaling factor exceeds a certain level; under optimized parameters, both of lower normalized mean squared error (NMSE) and symbol error rate (SER) can be achieved to perform parallel tasks, due to the richer dynamics and states of virtual nodes. The effect of system parameters on RC performance was also discussed. Our study would be helpful for the design and optimization of proposed parallel RC with optoelectronic feedback.

Extreme learning machine ensemble model for time series forecasting boosted by PSO: Application to an electric consumption problem

Ensemble Model is a tool that has attracted attention due to its capability to improve the outcome performance of emerging techniques to solve the modelling and classifying problem. However, the feasibility of applying intelligent algorithms to build the Ensemble Model presents a challenge of its own. In this work, an Extreme Learning Machine ensemble is applied to the Time Series modelling problem. We develop a thorough study of the models that will be candidates to compose the ensemble, obtaining statistical results of optimal topologies to solve each Time Series problem. The proposed method for the ensemble is the weighted averaging method, where the parameters (weights) are tuned with the Particle Swarm Optimization algorithm. Lastly, the ensemble is tested first in the well known Santa Fe Time Series Competition benchmark. Given the obtained satisfactory results, the ensemble is secondly tested in a real problem of Spain’s electric consumption forecasting. It is demonstrated that the PSO is a suitable algorithm to optimize Extreme Learning Machine ensemble and that the proposed strategy can obtain good results in both Time Series problems.

Performance optimization of a reservoir computing system based on a solitary semiconductor laser under electrical-message injection.

A simple reservoir computing (RC) system based on a solitary semiconductor laser under an electrical message injection is proposed, and the performances of the RC are numerically investigated. Considering the lack of memory capacity (MC) in such a system, some auxiliary methods are introduced to enhance the MC and optimize the performances for processing complex tasks. In the pre-existing method, the input information is the current input data combined with some past input data in a weighted sum in the input layer (named as M-input). Another auxiliary method (named as M-output) is proposed to introduce the output layer for optimizing the performances of the RC system. The simulated results demonstrate that the MC of the system can be improved after adopting the auxiliary methods, and the effectiveness under adopting the M-input integrated with the M-output (named as M-both) is the most significant. Furthermore, we analyze the system performances for processing the Santa Fe time series prediction task and the nonlinear channel equalization (NCE) task after adopting the above three auxiliary methods. Results show that the M-input is the most suitable for the prediction task while the M-both is the most appropriate for the NCE task.

Parallel information processing using a reservoir computing system based on mutually coupled semiconductor lasers

Via the nonlinear channel equalization and the Santa-Fe time series prediction, the parallel processing capability of a reservoir computing (RC) system based on two mutually coupled semiconductor lasers is demonstrated numerically. The results show that, for parallel processing the prediction tasks of two Santa-Fe time series with rates of 0.25 GSa/s, the minimum prediction errors are 3.8 × 10−5 and 4.4 × 10−5, respectively. For parallel processing two nonlinear channel equalization tasks, the minimum symbol error rates (SERs) are 3.3 × 10−4 for both tasks. For parallel processing a nonlinear channel equalization and a Santa-Fe time series prediction, the minimum SER is 6.7 × 10−4 for nonlinear channel equalization, and the minimum prediction error is 4.6 × 10−5 for Santa-Fe time series prediction.

Performance Enhancement of a Delay-Based Reservoir Computing System by Using Gradient Boosting Technology

Gradient boosting technology has been proved to be an effective scheme for enhancing the performances of spatially distributed reservoir computing (RC) systems. In this work, a gradient boosting scheme by combining two reservoirs is proposed and numerically investigated in a delayed-based RC system. The original reservoir in the delayed-based RC system is a vertical-cavity surface-emitting laser (VCSEL) under polarization-rotated optical feedback (PR-OF), and it is trained on the desired output. The other VCSEL under PR-OF or polarization-preserved optical feedback (PP-OF) is supplemented to be an extra reservoir, which is trained on the remaining error of the original reservoir. Via Santa-Fe time series prediction task and 10th-order nonlinear autoregressive moving average (NARMA10) task, the performances of the delay-based RC system are evaluated before and after supplementing the extra reservoir, and then the effectiveness of the gradient boosting technology in the delayed RC system can be analyzed. The simulated results demonstrate that adopting gradient boosting technology is effective in a delay-based RC system. Comparatively speaking, the enhanced effect is more obvious under taking a VCSEL with PR-OF as the extra reservoir.

Parallel information processing by a reservoir computing system based on a VCSEL subject to double optical feedback and optical injection.

In this work, we propose a scheme of reservoir computing (RC) for processing a Santa-Fe time series prediction task and a signal classification task in parallel, and the performances of the RC have been numerically investigated. For this scheme, a vertical-cavity surface-emitting laser (VCSEL) simultaneously subject to double optical feedback and optical injection is utilized as a nonlinear node, and the parallel information processing of the RC system is implemented based on the dynamical responses of X polarization component (X-PC) and Y polarization component (Y-PC) in the VCSEL. Considering that two different feedback frames (polarization-preserved optical feedback (PP-OF) or polarization-rotated optical feedback (PR-OF)) may be adopted in two feedback loops, four feedback combination cases are numerically analyzed. The simulated results show that the parallel processing ability of the proposed RC system depends on the feedback frames adopted in two loops. After comprehensively evaluating the parallel processing performances of the two tasks under different feedback combinations, the best parallel processing performance can be achieved by adopting PP-OFs in both two feedback loops. Under some optimized operation parameters, this proposed RC system can realize the lowest prediction error of 0.0289 and the lowest signal classification error of 2.78 × 10-5.

Performance optimization research of reservoir computing system based on an optical feedback semiconductor laser under electrical information injection.

Via Santa Fe time series prediction and nonlinear channel equalization tasks, the performances of a reservoir computing (RC) system based on an optical feedback semiconductor laser (SL) under electrical information injection are numerically investigated. The simulated results show that the feedback delay time and strength seriously affect the performances of this RC system. By adopting a current-related optimized feedback delay time and strength, the RC can achieve a good performance for an SL biased within a wide region of 1.1-3.5 times its threshold. The prediction errors are smaller than 0.01 when implementing the Santa Fe tests, and the symbol error rates (SERs) are very low on the order of 10-5 for accomplishing nonlinear channel equalization tests under a signal-to-noise ratio (SNR) of 32 dB. Moreover, under a given RC performance level, the information processing rate of the RC can be improved by increasing the SL current.

Effects of Some Operation Parameters on the Performance of a Reservoir Computing System Based on a Delay Feedback Semiconductor Laser With Information Injection by Current Modulation

Via a Santa Fe time series prediction task, the performance of a reservoir computing (RC) system based on a delay feedback semiconductor laser (SL) under current modulation is experimentally investigated and evaluated in detail. Experimental results show that some key operation parameters including the modulation index, the feedback ratio and the SL bias current, seriously affect the prediction performance of this RC system. By optimizing parameter settings, the degradation of the RC performance with the increase of the SL bias current can be efficiently alleviated, and the RC system can still realize a good performance with a NMSE smaller than 0.1 even for the SL biased at a high level of three times threshold current Ith. This research is an improvement to a previously experimental report in which such RC system can achieve a good performance only under a SL bias level near I th .

Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers

Abstract A reservoir computing (RC) system based on two mutually delay coupled semiconductor lasers (MDC-SLs) subject to optical injection is proposed, and the chaotic time series prediction and signal classification performance for such a system is numerically investigated via the Santa-Fe time-series prediction task and the signal classification task, respectively. The simulation results indicate that, when adopting this system to process ultra-fast data with rate up to 1 Gbps, the best prediction error 5 . 1 × 1 0 − 5 ± 5 . 2 × 1 0 − 6 and signal recognition error 5 . 5 × 1 0 − 4 ± 8 . 8 × 1 0 − 5 can be achieved under optimized operation parameters.

Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection.

A reservoir computing (RC) system based on a semiconductor laser (SL) with double optical feedback and optical injection is proposed, and the prediction performance of such a system is numerically investigated via Santa Fe Time-Series Prediction task. The simulation results indicate that the RC system can yield a good prediction performance. Through optimizing some relevant operating parameters, ultra-fast information processing rates up to Gb/s level can be realized for the prediction error is below 3%.

Intrinsic dimension estimation of manifolds by incising balls

Dimensionality reduction is a very important tool in data mining. Intrinsic dimension of data sets is a key parameter for dimensionality reduction. However, finding the correct intrinsic dimension is a challenging task. In this paper, a new intrinsic dimension estimation method is presented. The estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball. The method is compared with the previous dimension estimation methods. Experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the Santa Fe time series competition, and the results show that the new method is accurate and robust.

Complex-valued forecasting of wind profile

This paper presents a novel approach for the simultaneous modelling and forecasting of wind signal components. This is achieved in the complex domain by using novel neural network algorithms and architectures. We first perform a signal nonlinearity and component-dependent analyses, which suggest the use of modular complex-valued recurrent neural networks (RNNs). This RNN-based modelling rests upon a combination of nonlinearity, complexity and internal memory and allows for the multiple step ahead forecasting of the wind signal in its complex form (speed and direction). The approach is first verified on benchmark Data Set A ( NH 3 laser data) of the Santa Fe Time Series Prediction Competition together with artificial data generated by chaotic Mackey–Glass equations, and then applied to the real-world wind measurements. Simulations support the proposed architecture and algorithms.