the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
19 Mar 2021
19 Mar 2021
Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part I. Method development and toy model demonstration
- 1International Max Planck Research School on Earth System Modelling, Hamburg, Germany
- 2Max Planck Institute for Meteorology, Hamburg, Germany
- 3Ludwig-Maxmillians-Universität München, Munich, Germany
- 1International Max Planck Research School on Earth System Modelling, Hamburg, Germany
- 2Max Planck Institute for Meteorology, Hamburg, Germany
- 3Ludwig-Maxmillians-Universität München, Munich, Germany
Abstract. Existent methods to identify linear response functions from data require tailored perturbation experiments, e.g. impulse or step experiments. And if the system is noisy, these experiments need to be repeated several times to obtain a good statistics. In contrast, for the method developed here, data from only a single perturbation experiment at arbitrary perturbation is sufficient if in addition data from an unperturbed (control) experiment is available. To identify the linear response function for this ill-posed problem we invoke regularization theory. The main novelty of our method lies in the determination of the level of background noise needed for a proper estimation of the regularization parameter: This is achieved by comparing the frequency spectrum of the perturbation experiment with that of the additional control experiment. The resulting noise level estimate can be further improved for linear response functions known to be monotonic. The robustness of our method and its advantages are investigated by means of a toy model. We discuss in detail the dependence of the identified response function on the quality of the data (signal-to-noise ratio) and on possible nonlinear contributions to the response. The method development presented here prepares in particular for the identification of carbon-cycle response functions in Part II of this study. But the core of our method, namely our new approach to obtain the noise level for a proper estimation of the regularization parameter, may find applications in solving also other types of linear ill-posed problems.
Guilherme L. Torres Mendonça et al.
Status: closed
-
RC1: 'Comment on npg-2021-9', Anonymous Referee #1, 24 Mar 2021
- AC1: 'Reply on RC1', Guilherme Torres Mendonça, 31 Mar 2021
-
RC2: 'Comment on npg-2021-9', Anonymous Referee #2, 28 Mar 2021
Report on "Identification of linear response functions from arbitrary perturbation experiments in the presence of noise: Part I. Method development and toy model demonstration” by Torres Mendonça et al
The authors suggest a numerical method that estimates the linear response using only a single trajectory of the perturbed system together with a time series of the unperturbed system. Abandoning ensemble averages the usual response formula acquires an additional noise term. Their method consists of several steps. Firstly they introduce standard Tikhonov regularization. The regularization parameter is chosen to match the (unknown) noise level. In a second step the noise level is determined from the control trajectory of the unperturbed system. The authors discuss several additional ways how to optimise the regularization parameter by carefully determining what they call high and low frequency contributions and using a possible monotonic behaviour of the response function.
The authors perform careful and detailed numerical analysis of a simple toy model. They compare their results with existing methods based on ensemble averages. I particularly liked that they carefully discussed and illustrated the range of validity of their method and showing under what conditions it can be broken, for example by studying the addition of nonlinear effects.
I believe that this is a valuable piece of research that will be beneficial for the community, and I recommend publication after some minor issues have been addressed. I list the following comments the authors may want to incorporate:
** in the introduction the authors state that typically methods estimating the response rely on some “prior information”. They state that their method does not need prior information. That seems not quite true (they assume monotonicity, Eqn (8), Picard condition etc).
** following up on my previous point, their method requires a few assumptions on the underlying dynamical system under consideration such as (8) and what they call the Picard condition. It would be nice to have these assumptions listed somewhere.
** it is well known that one can formulate regularization such as ridge regression in a Bayesian framework where the regularization corresponds to a prior. Could the authors comment on what the meaning of this prior is in their context?
** introducing the noise $\eta$ in (5) can be justified by the central limit theorem, I assume, which could be mentioned.
** although I appreciated that the authors went through some trouble in explaining the details necessary to understand their approach, the manuscript could gain by being more succinct. For example, the sentence right after Eqn (24) just reiterates what has been described before.
** in the introduction the authors give a nice account of the use of linear response theory in the climate sciences. Their exposition, however, might give the false impression that linear response should be expected. Whereas it is now proven that systems driven by noise satisfy linear response theory (Hairer and Majda 2010), the situation for deterministic systems as initiated by Ruelle is far more complicated. Viviane Baladi and co-workers in fact showed that very simple dynamical systems such as the logistic map do not obey linear response. Moreover, examples of dynamical systems in the climate sciences are known that exhibit a rough parameter dependency (Chekroun et al 2014). The question of how to reconcile the fact that generic high-dimensional dynamical systems satisfy linear response theory even when their individual microscopic constituents do not, was addressed by Wormell and Gottwald (2018, 2019). The following references are relevant for this discussion:
Chekroun et al, PNAS 111 (2014), 1684-1690
Hairer and Majda, Nonlinearity 23 (2010), 909
Baladi and Smania, Nonlinearity 21 (2008), 677–711
Baladi and Smania, Ergodic Theory Dyn. Syst. 30 (2010), 1–20
Baladi, “Linear response, or else,” in ICM Seoul 2014, Proceedings, Volume III (ICM, 2014), 525–545, (e-print arXiv:1408.2937)
Baladi, Benedick, and Schnellmann, Invent. Math. 201 (2015), 773–844
Wormell and Gottwald, J Stat Physics 172 (2018), 1479–1498
Wormell and Gottwald, Chaos 29 (2019), 113127** there are other recent methods dealing with response theory from a numerical point of view, either detecting it or calculating the response, which the authors may want to include:
Gottwald, Wormell and Wouters, Physica D 331 (2016), 89-101Ni and Wang, Journal of Computational Physics 347 (2017), 56-77
Ni, "Approximating linear response by non-intrusive shadowing algorithms” (2020), arXiv:2003.09801 [math.DS]
** page 15, after Eqn (31): “Since almost every linear system can be diagonalised, we assume” —> “We assume ..”** page 17, l439, delete “extremely”- AC2: 'Reply on RC2', Guilherme Torres Mendonça, 08 Apr 2021
-
CC1: 'Comment on npg-2021-9', Valerio Lucarini, 10 Apr 2021
Dear authors,
I have greatly appreciated your paper and the idea behind it. The application developed in Part II is also very interesting. I have a comment I would recommend you to consider. When you introduce Eqs. 7 and 8 you are making the (very important) assumption that all the eigenvalues of the unperturbed transfer operator are real, whereas they are in general complex (with each complex number accompanied by its conjugate). Complex conjugate pairs allow for the presence of oscillating terms in the Green function. You might see a longer explanation of this in Tantet et al. 2020 and Lucarini 2018.. In other terms, you are allowing for purely relaxation behvaiour in your system. This might well work out fine for some applications, but not on other ones (where more complex feedbacks are present).
Best Regards,
Valerio Lucarini
Tantet, A., Chekroun, M.D., Neelin, J.D. et al. Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation. J Stat Phys 179, 1449–1474 (2020). https://doi.org/10.1007/s10955-019-02444-8
Lucarini, V. Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands. J Stat Phys 173, 1698–1721 (2018). https://doi.org/10.1007/s10955-018-2151-5
-
AC3: 'Reply on CC1', Guilherme Torres Mendonça, 14 Apr 2021
Dear Prof. Lucarini,We appreciate your comments. We fully agree with your remark that our Eq. (8) assumes real relaxation times \tau and thereby rules out oscillatory behaviour. We point out this limitation in the discussions section, L822-824. Also referee #2 asked us to make the assumptions underlying our approach more transparent. Following that suggestion we will mention in the introduction and also when introducing Eq. (8) that the respective eigenvalues are assumed to be real. This indeed introduces an important limitation, namely to the class of overdamped systems to which presumably the global carbon cycle belongs, which is our main application in Part II of this study. We emphasize however that, as discussed in L804-824, Eq. (8) is not essential for our RFI method, so that by dropping this assumption (thus recovering the response function pointwise) one could in principle still apply the method to more general systems.Nevertheless, as also discussed in L804-824, assuming Eq. (8) has some advantages over recovering the response function pointwise. Therefore it would be nice to be able apply our RFI method by extending this assumption to include complex eigenvalues. But at the moment it is not clear to us how this could be done.Best regards,Guilherme L. Torres Mendonça, Julia Pongratz and Christian H. Reick
-
AC3: 'Reply on CC1', Guilherme Torres Mendonça, 14 Apr 2021
Status: closed
-
RC1: 'Comment on npg-2021-9', Anonymous Referee #1, 24 Mar 2021
- AC1: 'Reply on RC1', Guilherme Torres Mendonça, 31 Mar 2021
-
RC2: 'Comment on npg-2021-9', Anonymous Referee #2, 28 Mar 2021
Report on "Identification of linear response functions from arbitrary perturbation experiments in the presence of noise: Part I. Method development and toy model demonstration” by Torres Mendonça et al
The authors suggest a numerical method that estimates the linear response using only a single trajectory of the perturbed system together with a time series of the unperturbed system. Abandoning ensemble averages the usual response formula acquires an additional noise term. Their method consists of several steps. Firstly they introduce standard Tikhonov regularization. The regularization parameter is chosen to match the (unknown) noise level. In a second step the noise level is determined from the control trajectory of the unperturbed system. The authors discuss several additional ways how to optimise the regularization parameter by carefully determining what they call high and low frequency contributions and using a possible monotonic behaviour of the response function.
The authors perform careful and detailed numerical analysis of a simple toy model. They compare their results with existing methods based on ensemble averages. I particularly liked that they carefully discussed and illustrated the range of validity of their method and showing under what conditions it can be broken, for example by studying the addition of nonlinear effects.
I believe that this is a valuable piece of research that will be beneficial for the community, and I recommend publication after some minor issues have been addressed. I list the following comments the authors may want to incorporate:
** in the introduction the authors state that typically methods estimating the response rely on some “prior information”. They state that their method does not need prior information. That seems not quite true (they assume monotonicity, Eqn (8), Picard condition etc).
** following up on my previous point, their method requires a few assumptions on the underlying dynamical system under consideration such as (8) and what they call the Picard condition. It would be nice to have these assumptions listed somewhere.
** it is well known that one can formulate regularization such as ridge regression in a Bayesian framework where the regularization corresponds to a prior. Could the authors comment on what the meaning of this prior is in their context?
** introducing the noise $\eta$ in (5) can be justified by the central limit theorem, I assume, which could be mentioned.
** although I appreciated that the authors went through some trouble in explaining the details necessary to understand their approach, the manuscript could gain by being more succinct. For example, the sentence right after Eqn (24) just reiterates what has been described before.
** in the introduction the authors give a nice account of the use of linear response theory in the climate sciences. Their exposition, however, might give the false impression that linear response should be expected. Whereas it is now proven that systems driven by noise satisfy linear response theory (Hairer and Majda 2010), the situation for deterministic systems as initiated by Ruelle is far more complicated. Viviane Baladi and co-workers in fact showed that very simple dynamical systems such as the logistic map do not obey linear response. Moreover, examples of dynamical systems in the climate sciences are known that exhibit a rough parameter dependency (Chekroun et al 2014). The question of how to reconcile the fact that generic high-dimensional dynamical systems satisfy linear response theory even when their individual microscopic constituents do not, was addressed by Wormell and Gottwald (2018, 2019). The following references are relevant for this discussion:
Chekroun et al, PNAS 111 (2014), 1684-1690
Hairer and Majda, Nonlinearity 23 (2010), 909
Baladi and Smania, Nonlinearity 21 (2008), 677–711
Baladi and Smania, Ergodic Theory Dyn. Syst. 30 (2010), 1–20
Baladi, “Linear response, or else,” in ICM Seoul 2014, Proceedings, Volume III (ICM, 2014), 525–545, (e-print arXiv:1408.2937)
Baladi, Benedick, and Schnellmann, Invent. Math. 201 (2015), 773–844
Wormell and Gottwald, J Stat Physics 172 (2018), 1479–1498
Wormell and Gottwald, Chaos 29 (2019), 113127** there are other recent methods dealing with response theory from a numerical point of view, either detecting it or calculating the response, which the authors may want to include:
Gottwald, Wormell and Wouters, Physica D 331 (2016), 89-101Ni and Wang, Journal of Computational Physics 347 (2017), 56-77
Ni, "Approximating linear response by non-intrusive shadowing algorithms” (2020), arXiv:2003.09801 [math.DS]
** page 15, after Eqn (31): “Since almost every linear system can be diagonalised, we assume” —> “We assume ..”** page 17, l439, delete “extremely”- AC2: 'Reply on RC2', Guilherme Torres Mendonça, 08 Apr 2021
-
CC1: 'Comment on npg-2021-9', Valerio Lucarini, 10 Apr 2021
Dear authors,
I have greatly appreciated your paper and the idea behind it. The application developed in Part II is also very interesting. I have a comment I would recommend you to consider. When you introduce Eqs. 7 and 8 you are making the (very important) assumption that all the eigenvalues of the unperturbed transfer operator are real, whereas they are in general complex (with each complex number accompanied by its conjugate). Complex conjugate pairs allow for the presence of oscillating terms in the Green function. You might see a longer explanation of this in Tantet et al. 2020 and Lucarini 2018.. In other terms, you are allowing for purely relaxation behvaiour in your system. This might well work out fine for some applications, but not on other ones (where more complex feedbacks are present).
Best Regards,
Valerio Lucarini
Tantet, A., Chekroun, M.D., Neelin, J.D. et al. Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation. J Stat Phys 179, 1449–1474 (2020). https://doi.org/10.1007/s10955-019-02444-8
Lucarini, V. Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands. J Stat Phys 173, 1698–1721 (2018). https://doi.org/10.1007/s10955-018-2151-5
-
AC3: 'Reply on CC1', Guilherme Torres Mendonça, 14 Apr 2021
Dear Prof. Lucarini,We appreciate your comments. We fully agree with your remark that our Eq. (8) assumes real relaxation times \tau and thereby rules out oscillatory behaviour. We point out this limitation in the discussions section, L822-824. Also referee #2 asked us to make the assumptions underlying our approach more transparent. Following that suggestion we will mention in the introduction and also when introducing Eq. (8) that the respective eigenvalues are assumed to be real. This indeed introduces an important limitation, namely to the class of overdamped systems to which presumably the global carbon cycle belongs, which is our main application in Part II of this study. We emphasize however that, as discussed in L804-824, Eq. (8) is not essential for our RFI method, so that by dropping this assumption (thus recovering the response function pointwise) one could in principle still apply the method to more general systems.Nevertheless, as also discussed in L804-824, assuming Eq. (8) has some advantages over recovering the response function pointwise. Therefore it would be nice to be able apply our RFI method by extending this assumption to include complex eigenvalues. But at the moment it is not clear to us how this could be done.Best regards,Guilherme L. Torres Mendonça, Julia Pongratz and Christian H. Reick
-
AC3: 'Reply on CC1', Guilherme Torres Mendonça, 14 Apr 2021
Guilherme L. Torres Mendonça et al.
Model code and software
Supplementary material for “Identification of linear response functions from arbitrary perturbation experiments in the presence of noise - Part I. Method development and toy model demonstration” Torres Mendonca, G., Pongratz, J., and Reick, C. H. https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_3288741
Guilherme L. Torres Mendonça et al.
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