## Optoelectronic reservoir computing: tackling noise-induced performance degradation |

Optics Express, Vol. 21, Issue 1, pp. 12-20 (2013)

http://dx.doi.org/10.1364/OE.21.000012

Acrobat PDF (1154 KB)

### Abstract

We present improved strategies to perform photonic information processing using an optoelectronic oscillator with delayed feedback. In particular, we study, via numerical simulations and experiments, the influence of a finite signal-to-noise ratio on the computing performance. We illustrate that the performance degradation induced by noise can be compensated for via multi-level pre-processing masks.

© 2013 OSA

## 1. Introduction

1. J. L. O’Brien, “Optical quantum computing,” Science **7**, 1567–1570 (2007). [CrossRef]

2. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics **4**, 261 (2010). [CrossRef]

2. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics **4**, 261 (2010). [CrossRef]

3. W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. **14**, 2531–2560 (2002). [CrossRef] [PubMed]

5. D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks **20**, 391–403 (2007). [CrossRef] [PubMed]

6. M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science **321**, 48–50 (2008). [CrossRef] [PubMed]

3. W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. **14**, 2531–2560 (2002). [CrossRef] [PubMed]

7. W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. **69**, 593–616 (2004). [CrossRef]

8. K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express **16**, 11182—11192 (2008). [CrossRef] [PubMed]

9. K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks **22**, 1469–1481 (2011). [CrossRef]

10. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. **2**, 468 (2011). [CrossRef]

10. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. **2**, 468 (2011). [CrossRef]

11. L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express **20**, 3241–3249 (2012). [CrossRef] [PubMed]

13. R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. **108**, 244101 (2012). [CrossRef] [PubMed]

14. D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. **8**, 257–258 (2012). [CrossRef]

11. L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express **20**, 3241–3249 (2012). [CrossRef] [PubMed]

12. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. **2**, 287 (2012). [CrossRef] [PubMed]

11. L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express **20**, 3241–3249 (2012). [CrossRef] [PubMed]

*τ*(delay time) and multiplied by an input mask (pre-processing) before it is injected into the nonlinear oscillator [10

10. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. **2**, 468 (2011). [CrossRef]

15. J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos **20**, 037101 (2010). [CrossRef] [PubMed]

16. J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. **2**, 514 (2012). [CrossRef] [PubMed]

## 2. Optoelectronic feedback scheme

17. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun. **30**, 257–261 (1979). [CrossRef]

19. L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique **5**, 669–681 (2004). [CrossRef]

**20**, 3241–3249 (2012). [CrossRef] [PubMed]

^{2}nonlinear transformation is provided by a Mach-Zehnder modulator (MZM). The output of the MZM is delayed via a 4 km long optical fiber and, after optical detection, acts as a nonlinear driving force term at the input of a first order low pass filter with a cut-off frequency of

*f*= 1/2

_{c}*πT*∼ 663 KHz (DC preserving feedback). The overall gain of the nonlinear feedback plays the role of the bifurcation parameter

_{R}*β*, which can be easily tuned by the light intensity level, seeding optically the MZM. For RC purposes, the nonlinearity gain is adjusted such that the steady state is stable when the oscillator is autonomous (no external information added). The electronic feedback signal serves as the radio-frequency drive of the MZM, closing the delayed feedback loop (delay time 20.82

*μ*s). This optoelectronic oscillator exhibits the dynamical regimes typically observed for the Ikeda dynamics, including a period doubling route to chaos [19

19. L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique **5**, 669–681 (2004). [CrossRef]

*u*(

_{I}*s*), by the following dynamical equation [11

**20**, 3241–3249 (2012). [CrossRef] [PubMed]

18. L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. **34**, 594–601 (1998). [CrossRef]

*x*is a dimensionless dynamical variable that corresponds to the voltage applied to the MZM by the low pass filter (X in Fig. 2) with the following renormalization

*V*∼ 6.5V is the voltage required to modulate the MZM over one

_{π}*π*-range. In this manner,

*x*is normalized such that it corresponds to the unit free argument of the sin

^{2}nonlinearity. The time in normalized units is

*s*(

*s*=

*t/T*), with

_{R}*T*= 240 ns being the oscillator response time. Parameter

_{R}*β*is the nonlinearity gain (total gain in the oscillation loop), and the delay time in normalized units is denoted as

*τ*. The delay time in the experimental realization is

*τ*′ = 20.82

*μ*s, i.e.

*τ*=

*τ*′/

*T*= 86.75 in normalized time units. Here, we consider a delay reservoir with 400 virtual nodes, i.e. the virtual node spacing is

_{R}*τ*/400 ∼ 0.2 as suggested in [10

**2**, 468 (2011). [CrossRef]

*γ*are the offset phase of the nonlinearity and the input scaling, respectively. The external input signal

*u*(

_{I}*s*) is introduced as a modulation to the nonlinearity, corresponding to the normalized input signal Ui(t) in Fig. 2. In order to keep a consistent time reference,

*u*(

_{I}*s*) is also delayed in Eq. (1). The offset phase Φ can be tuned to adjust the rest state (stable steady state) around which the external input information is generating nonlinear transients, via modulation of the MZM voltage.

*β*is of particular importance for the dynamical characterization of the optoelectronic oscillator in the absence of external input, with

*β*being typically employed as the bifurcation parameter in many experimental studies. In particular, this system starts to oscillate when

*β*> 1 [18

18. L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. **34**, 594–601 (1998). [CrossRef]

20. Y. Kouomou Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. **32**, 2571–2573 (2007). [CrossRef] [PubMed]

*β*< 1, the system is stable (DC voltage) and the steady state of

*x*depends on the offset phase Φ. The nonlinearity gain can be controlled in the experiments by varying the power emitted by the laser diode in Fig. 2. Assuming a linear dependence of the power with the injection current, we can define the nonlinearity gain as follows: where

*I*is the laser injection current,

*I*= 14 mA is the laser threshold current and

_{th}*I*

_{β1}= 88 mA is the current at which the system starts to oscillate (

*β*= 1).

## 3. Multi-valued masking scheme

22. A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks **22**, 131–144 (2011). [CrossRef]

**2**, 468 (2011). [CrossRef]

23. U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH_{3} laser”, Phys. Rev. A **40**, 6354–6365 (1989). [CrossRef]

24. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/andreas/Time-Series/SantaFe.html (1993).

22. A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks **22**, 131–144 (2011). [CrossRef]

25. L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing **51**, 321–339 (2003). [CrossRef]

*u*(

*t*+ 1), based on its previous values up to time

*t*. To this end, we sequentially feed the RC system with the input stream

*u*(

*t*),

*u*(

*t*− 1),

*u*(

*t*− 2)...

*u*(

*t*−

*n*), already pre-processed with the input mask, and we try to predict the value one time step in the future,

*u*(

*t*+ 1). In particular, the time-series that we use for the prediction task consists of 4000 samples of the Santa Fe laser dataset normalized to zero mean and unit variance. Out of the Santa Fe laser time-series, 75% of the samples (3000) are used for training and the remaining 25% (1000) are used for the prediction and the evaluation of the prediction error (testing). The training procedure, which is carried out off-line, consists of a standard multiple linear regression. The independent variables for the regression are the responses of the oscillator to the pre-processed inputs, sampled at the virtual node positions. The corresponding target values are the inputs samples one time step in the future. In the training procedure, an output weight is assigned to each virtual node, such that the weighted sum of all the virtual nodes values approximates the desired target value as closely as possible. The weights obtained in the training procedure are later used to test the prediction error with the remaining (untrained) input samples. Four different random partitions of the original data are evaluated for training and testing in order to obtain a statistically significant prediction error.

*β*= 0.8 and

*γ*= 0.45 as a function of the offset phase of the nonlinearity Φ when two-valued and six-valued input masks are implemented. In the absence of noise, the binary mask (blue dashed line) yields a minimum prediction error of about 1%. In the presence of 10 bits quantization noise, however, the prediction error for a six-valued mask (solid black line) is significantly lower than for the binary mask (solid red line), over the entire parameter range, with a minimum error of about 2%. We have checked that increasing the number of discrete values in the input mask does not improve the performance for the given SNR further. Figure 4(b) presents a summary of the numerical results for the NMSE prediction error around an optimum region of operation (Φ = −0.65

*π*) in the

*β*−

*γ*plane for a six-valued input mask. An extended region of low prediction error (NMSE<3%) can be identified at the upper right part of the

*β*−

*γ*plane. The NMSE rapidly degrades for

*β*> 1.

**20**, 3241–3249 (2012). [CrossRef] [PubMed]

*β*−

*γ*values for 10 quantization bits. It is worth noting that it is possible to isolate the individual effects of each optimizing step via the numerical simulations (see Fig. 5). First of all, the numerical parameter scan allows for the optimization of the phase offset (Φ), the nonlinearity gain (

*β*) and the input scaling (

*γ*), yielding a prediction error ∼6% with a two-valued mask and 8 bits quantization. Second, the use of a six-valued mask further reduces the error from ∼6% to ∼3% with 8 bits quantization. Finally, an additional increase in the SNR from 8 to 10 quantization bits, which can be obtained e.g. from oversampling and subsequent averaging, results in a decrease from ∼3% to ∼2% in the prediction error, as shown in Fig. 5.

## 4. Experimental evaluation

21. M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. **36**, 2212–2214 (2011). [CrossRef] [PubMed]

*β*= 0.8 and

*γ*= 0.

23. U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH_{3} laser”, Phys. Rev. A **40**, 6354–6365 (1989). [CrossRef]

24. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/andreas/Time-Series/SantaFe.html (1993).

22. A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks **22**, 131–144 (2011). [CrossRef]

## 5. Conclusion

**22**, 131–144 (2011). [CrossRef]

24. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/andreas/Time-Series/SantaFe.html (1993).

25. L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing **51**, 321–339 (2003). [CrossRef]

**20**, 3241–3249 (2012). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. L. O’Brien, “Optical quantum computing,” Science |

2. | H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics |

3. | W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput. |

4. | H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science |

5. | D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks |

6. | M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science |

7. | W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci. |

8. | K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express |

9. | K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks |

10. | L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun. |

11. | L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express |

12. | Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. |

13. | R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett. |

14. | D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys. |

15. | J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos |

16. | J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep. |

17. | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun. |

18. | L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron. |

19. | L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique |

20. | Y. Kouomou Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. |

21. | M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. |

22. | A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks |

23. | U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH |

24. | A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/andreas/Time-Series/SantaFe.html (1993). |

25. | L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(200.3050) Optics in computing : Information processing

(250.4745) Optoelectronics : Optical processing devices

**ToC Category:**

Optics in Computing

**History**

Original Manuscript: August 15, 2012

Revised Manuscript: November 1, 2012

Manuscript Accepted: December 16, 2012

Published: January 2, 2013

**Citation**

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, "Optoelectronic reservoir computing: tackling noise-induced performance degradation," Opt. Express **21**, 12-20 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-12

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### References

- J. L. O’Brien, “Optical quantum computing,” Science7, 1567–1570 (2007). [CrossRef]
- H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics4, 261 (2010). [CrossRef]
- W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002). [CrossRef] [PubMed]
- H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science304, 78–80 (2004). [CrossRef] [PubMed]
- D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks20, 391–403 (2007). [CrossRef] [PubMed]
- M. Rabinovich, R. Huerta, and G. Laurent, “Transient dynamics of neural processing,” Science321, 48–50 (2008). [CrossRef] [PubMed]
- W. Maass and H. Markram, “On the computational power of recurrent circuits of spiking neurons,” J. Comput. Syst. Sci.69, 593–616 (2004). [CrossRef]
- K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Campenhout, “Towards optical signal processing using photonic reservoir computing,” Opt. Express16, 11182—11192 (2008). [CrossRef] [PubMed]
- K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Networks22, 1469–1481 (2011). [CrossRef]
- L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nature Commun.2, 468 (2011). [CrossRef]
- L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express20, 3241–3249 (2012). [CrossRef] [PubMed]
- Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep.2, 287 (2012). [CrossRef] [PubMed]
- R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev Lett.108, 244101 (2012). [CrossRef] [PubMed]
- D. Woods and T. J. Naughton, “Photonic neural networks,” Nature Phys.8, 257–258 (2012). [CrossRef]
- J. P. Crutchfield, L. D. William, and S. Sudeshna, “Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems beyond the digital hegemony,” Chaos20, 037101 (2010). [CrossRef] [PubMed]
- J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,” Sci. Rep.2, 514 (2012). [CrossRef] [PubMed]
- K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Commun.30, 257–261 (1979). [CrossRef]
- L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron.34, 594–601 (1998). [CrossRef]
- L. Larger, J.-P. Goedgebuer, and V. S. Udalsov, “Ikeda–based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C.R. de Physique5, 669–681 (2004). [CrossRef]
- Y. Kouomou Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett.32, 2571–2573 (2007). [CrossRef] [PubMed]
- M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett.36, 2212–2214 (2011). [CrossRef] [PubMed]
- A. Rodan and P. Tin̂o, “Minimum complexity echo state network,” IEEE Trans. Neural Networks22, 131–144 (2011). [CrossRef]
- U. Huebner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser”, Phys. Rev. A40, 6354–6365 (1989). [CrossRef]
- A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/andreas/Time-Series/SantaFe.html (1993).
- L. Cao, “Support vector machines experts for time series forecasting”, Neurocomputing51, 321–339 (2003). [CrossRef]

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