Abstract

Optical implementations of reservoir computing systems are very promising because of their high processing speeds and the possibility to process several tasks in parallel. These systems can be implemented using semiconductor lasers subject to optical delayed feedback and optical injection. While the amount of the feedback/injection can be easily controlled, it is much more difficult to control the optical feedback/injection phase. We present extensive numerical investigations of the influence of the feedback/injection phases on laser-based reservoir computing systems with feedback. We show that a change in the phase can lead to a strong reduction in the reservoir computing system performance. We introduce a new readout layer design that -at least for some tasks- reduces this sensitivity to changes in the phase. It consists in optimizing the readout weights from a coherent combination of the reservoir’s readout signal and its delayed version rather than only from the reservoir’s readout signal as is usually done.

© 2016 Optical Society of America

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References

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2015 (1)

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

2014 (5)

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[Crossref] [PubMed]

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: role of phase dynamics,” Opt. Express 22, 8672–8686 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

2013 (4)

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).
[Crossref] [PubMed]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

2012 (5)

2011 (2)

A. Rodan and Peter Tiňo, ”Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

2009 (1)

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

2008 (1)

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

2005 (1)

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

2004 (1)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

2003 (1)

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

2002 (1)

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]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Appeltant, L.

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[Crossref] [PubMed]

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]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

Argyris, A.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Beri, S.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Bienstman, P.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Bogris, A.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Bolk, J.

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bit generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012).
[Crossref] [PubMed]

Brunner, D.

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).
[Crossref] [PubMed]

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

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]

Chembo, Y. K.

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]

Chlouverakis, K.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Dambre, J.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

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]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, “Memory versus non-linearity in reservoirs,” In Proc. Int Neural Networks (IJCNN) Joint Conf, 1–8 (2010).

Danckaert, J.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[Crossref] [PubMed]

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: role of phase dynamics,” Opt. Express 22, 8672–8686 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bit generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012).
[Crossref] [PubMed]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Duport, F.

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and Serge Massar, “All optical reservoir computing,” Opt. Express 20, 22783–22795 (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]

Dutoit, X.

D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, “Memory versus non-linearity in reservoirs,” In Proc. Int Neural Networks (IJCNN) Joint Conf, 1–8 (2010).

Elsäer, W.

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Escalona-Moran, M. A.

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

Fiers, M.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Fischer, I.

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[Crossref] [PubMed]

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

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).
[Crossref] [PubMed]

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]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

García-Ojalvo, J.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

Gavrielides, A.

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Gelens, L.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Green, K.

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Gutierrez, J. M.

Haas, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

Haelterman, M.

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and Serge Massar, “All optical reservoir computing,” Opt. Express 20, 22783–22795 (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]

Hamacher, M.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Heil, T.

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Hicke, K.

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

Jacquot, M.

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]

Jaeger, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

Khoder, M.

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Krauskopf, B.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Larger, L.

Leijtens, X.

Leijtens, X. J. M.

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

Lenstra, D.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

Maass, W.

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]

Markram, H.

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]

Martinenghi, R.

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]

Massar, S.

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]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

Mechet, P.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Mezosi, G.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Mirasso, C. R.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

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).
[Crossref] [PubMed]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[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. Express 20, 3241–3249 (2012).
[Crossref] [PubMed]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

Morthier, G.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Natschläger, T.

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]

Nguimdo, R. M.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: role of phase dynamics,” Opt. Express 22, 8672–8686 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bit generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012).
[Crossref] [PubMed]

Ortín, S.

Paquot, Y.

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]

Pesquera, L.

Rodan, A.

A. Rodan and Peter Tiňo, ”Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).
[Crossref]

Rybalko, S.

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]

Schneider, B.

Schrauwen, B.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

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]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, “Memory versus non-linearity in reservoirs,” In Proc. Int Neural Networks (IJCNN) Joint Conf, 1–8 (2010).

Simpson, T. B.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

Smerieri, A.

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and Serge Massar, “All optical reservoir computing,” Opt. Express 20, 22783–22795 (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]

Sorel, M.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Soriano, M. C.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

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).
[Crossref] [PubMed]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[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. Express 20, 3241–3249 (2012).
[Crossref] [PubMed]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

Syvridis, D.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Tino, Peter

A. Rodan and Peter Tiňo, ”Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).
[Crossref]

Vaerenbergh, T. V.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Van der Sande, G.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: role of phase dynamics,” Opt. Express 22, 8672–8686 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bit generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012).
[Crossref] [PubMed]

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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Vandoorne, K.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

Verschaffelt, G.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: role of phase dynamics,” Opt. Express 22, 8672–8686 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, M. Khoder, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection,” Opt. Lett. 395945–5948 (2014).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bit generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012).
[Crossref] [PubMed]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Verstraeten, D.

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[Crossref]

D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, “Memory versus non-linearity in reservoirs,” In Proc. Int Neural Networks (IJCNN) Joint Conf, 1–8 (2010).

Wieczorek, S.

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013).
[Crossref]

IEEE Trans. Neural Netw. (1)

A. Rodan and Peter Tiňo, ”Minimum complexity echo state network,” IEEE Trans. Neural Netw. 22, 131–144 (2011).
[Crossref]

IEEE Trans. Neural Netw. Learn. Syst. (1)

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26, 3301–3307 (2015).
[Crossref] [PubMed]

Nat. Commun. (3)

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4, 1364 (2013).
[Crossref] [PubMed]

K. Vandoorne, P. Mechet, T. V. Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 4541 (2014).
[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,” Nat. Commun. 2, 468 (2011).
[Crossref] [PubMed]

Neural Comput. (1)

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]

Opt. Express (5)

Opt. Lett. (1)

Photon. Technol. Lett. (1)

M. Khoder, R. M. Nguimdo, G. Verschaffelt, J. Bolk, X. J. M. Leijtens, and J. Danckaert, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” Photon. Technol. Lett. 26, 520–523 (2014).
[Crossref]

Phys. Rep. (1)

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 4161 (2005).
[Crossref]

Phys. Rev Lett. (1)

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]

Phys. Rev. E (1)

T. Heil, I. Fischer, W. Elsäer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: bifurcation scenarios and mechanisms,” Phys. Rev. E 67, 066214 (2003).
[Crossref]

Phys. Rev. Lett. (2)

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

Sci. Rep. (2)

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4, 3629 (2014).
[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]

Science (1)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80 (2004).
[Crossref] [PubMed]

Other (2)

D. Verstraeten, J. Dambre, X. Dutoit, and B. Schrauwen, “Memory versus non-linearity in reservoirs,” In Proc. Int Neural Networks (IJCNN) Joint Conf, 1–8 (2010).

A. S. Weigend and N. A. Gershenfeld, “Time series prediction: Forecasting the future and understanding the past,” ftp://ftp.santafe.edu/pub/Time-Series/Competition (1993).

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Figures (11)

Fig. 1
Fig. 1 (a) Classical RC scheme, (b) RC scheme based on a nonlinear node (laser) with delayed feedback.
Fig. 2
Fig. 2 Scheme of parallel information processing using an SRL with double self-feedback. CW: clockwise mode, CCW: counterclockwise mode, MZM: Mach-Zehnder modulator. S1(t) and S2(t) are the input data streams of the tasks to be processed in the CW and CCW modes, respectively. We use a single random mask generator (i.e identical masks) for the two tasks and a single semiconductor laser (SL) to optically inject the data into each directional mode of the SRL. The optical path is in red, while the electric path is in blue.
Fig. 3
Fig. 3 Orbit diagrams showing the reservoir output extrema as a function of different static phases: (a) ω0T for Δω = 0 and Φ1 = Φ2 = 0; (b) Δω for ω0T = 0.6π and Φ1 = Φ2 = 0; and (c) Φ1 = Φ2 for Δω = 0 and ω0T = 0.6π. These diagrams are obtained by extracting the local extrema from the recorded power |Ecw|2. Note that the orbit diagrams in |Ecw|2 (not shown) are the same.
Fig. 4
Fig. 4 NMSE (a) and SER (b) as function of the static feedback phase ω0T for Δω = 0 and Φ0 = 0 for single processing task (magenta) and for two simultaneous processing tasks (red for task 1 and blue for task 2).
Fig. 5
Fig. 5 (a) Memory capacity as a function of the feedback phase when j is truncated to 10 in Eq. (9); (b) Memory function m(j) for two values of ω0T.
Fig. 6
Fig. 6 (a) Memory capacity as a function of the feedback phase for Td = 0 and Td = 1.6T; (b) Memory function m(j) for different values of Td considering ω0T = 0.6π.
Fig. 7
Fig. 7 Comparison between the SER values obtained by calculating the readout weights from recorded signals |(E1,2(t) + E1,2(t − Td)|2)/2 with Td = 0 (magenta, red, blue) and Td = 1.6T(black, cyan). The symbol (•) corresponds to task 1 while (▼) corresponds to task 2. The symbol (○) indicates the case for which a single task is computed using the CW directional mode. Other conditions are as stated Fig. 4(b)
Fig. 8
Fig. 8 NMSE for (a) single Santa-Fe time series prediction and (b) simultaneous prediction of two independent Santa-Fe time series using the two directional modes of SRLs with double self-feedback as function of the static feedback phase ω0T for Δω = 0 and Φ0 = 0.
Fig. 9
Fig. 9 NMSE for (a) single Santa-Fe time series prediction and (b) simultaneous prediction of two independent Santa-Fe time series using the two directional modes of SRLs with double self-feedback as function of the frequency detuning Δω for ω0T = 0.6π and Φ0 = 0.
Fig. 10
Fig. 10 SER for (a) single NCE task processing and (b) simultaneous processing of two independent NCE tasks using the two directional modes of SRLs with double self-feedback as function of the injection frequency detuning Δω for ω0T = 0.6π and Φ0 = 0. Td = 0 (magenta, blue, red) and Td = 1.6T(black, cyan). The symbol (•) corresponds to task 1 while (○) corresponds to task 2.
Fig. 11
Fig. 11 (a) Evolution of NMSE for two simultaneous Santa-Fe predictions; (b) SER for simultaneous processing of two NCE tasks as a function of the MZM’s static phase Φ1,2 for Δω = −π/2 rad/ns and ω0 = 0.6π.

Equations (10)

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E ˙ c w = κ ( 1 + i α ) [ G c w N 1 ] E c w ( k d + i k c ) E c c w + η c w E c w ( t T ) e i ω 0 T + D c w ξ c w ( t ) + k 1 1 ( t ) ,
E ˙ c c w = κ ( 1 + i α ) [ G c c w N 1 ] E c c w ( k d + i k c ) E c w + η c c w E c c w ( t T ) e i ω 0 T + D c c w ξ c c w ( t ) + k 2 2 ( t ) ,
N ˙ = γ [ μ N ( 1 + G c w | E c w | 2 + G c c w | E c c w | 2 ) ] ,
1 , 2 ( t ) = | 0 | 2 { 1 + e i [ S 1 , 2 ( t ) + Φ 1 , 2 ] } e i Δ ω 1 , 2 t ,
N M S E ( y , y t a r g e t ) = y ( n ) y t a r g e t ( n ) 2 y t a r g e t ( n ) y t a r g e t ( n ) 2 ,
Q ( n ) = 0.08 d ( n + 2 ) 0.12 d ( n + 1 ) + d ( n ) + 0.18 d ( n 1 ) 0.1 d ( n 2 ) + 0.091 d ( n 3 ) 0.05 d ( n 4 ) + 0.04 d ( n 5 ) + 0.03 d ( n 6 ) + 0.01 d ( n 7 ) .
S ( n ) = Q ( n ) + 0.036 Q ( n ) 2 0.011 Q ( n ) 3 + ξ e ( n ) ,
S E R = N u m b e r o f m i s c l a s s i f i e d v a l u e s T o t a l n u m b e r o f t e s t e d v a l u e s .
M C = j = 0 + m ( j ) ,
m ( j ) = [ V ( i j ) V ( i j ) ] [ y j ( i ) y j ( i ) ] | V ( i j ) V ( i j ) | 2 | y j ( i ) y j ( i ) | 2 ,

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