Original: D YNAMIC
Revised: D YNAMIC

Original: T IME
Revised: T IME

Original: L AG
Revised: L AG

Original: R EGRESSION :
Revised: R EGRESSION :

Original: P REDICTING
Revised: P REDICTING

Original: W HAT & W HEN
Revised: W HAT & W HEN

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: This paper tackles a new regression problem, called Dynamic Time-Lag Regression (DTLR), where a cause signal drives an effect signal with an unknown time delay.
Revised: This paper tackles a new regression problem, called Dynamic Time-Lag Regression (DTLR), where a cause signal drives an effect signal with an unknown time delay.

Original: The motivating application, pertaining to space weather modelling, aims to predict the near-Earth solar wind speed based on estimates of the Sun’s coronal magnetic ﬁeld.
Revised: The motivating application, pertaining to space weather modelling, aims to predict the near-Earth solar wind speed based on estimates of the Sun’s coronal magnetic ﬁeld.

Original: DTLR differs from mainstream regression and from sequence-to-sequence learning in two respects: ﬁrstly, no ground truth (e.g., pairs of associated subsequences) is available; secondly, the cause signal contains much information irrelevant to the effect signal (the solar magnetic ﬁeld governs the solar wind propagation in the heliosphere, of which the Earth’s magnetosphere is but a minuscule region).
Revised: DTLR differs from mainstream regression and from sequence-to-sequence learning in two respects: ﬁrstly, no ground truth (e.g., pairs of associated subsequences) is available; secondly, the cause signal contains much information irrelevant to the effect signal (the solar magnetic ﬁeld governs the solar wind propagation in the heliosphere, of which the Earth’s magnetosphere is but a minuscule region).

Original: A Bayesian approach is presented to tackle the speciﬁcs of the DTLR problem, with theoretical justiﬁcations based on linear stability analysis.
Revised: A Bayesian approach is presented to tackle the speciﬁcs of the DTLR problem, with theoretical justiﬁcations based on linear stability analysis.

Original: A proof of concept on synthetic problems is presented.
Revised: A proof of concept on synthetic problems is presented.

Original: Finally, the empirical results on the solar wind modelling task improve on the state of the art in solar wind forecasting.
Revised: Finally, the empirical results on the solar wind modelling task improve on the state of the art in solar wind forecasting.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: A signiﬁcant body of work in machine learning concerns the modeling of spatio-temporal phenomena (Shi and Yeung, 2018; Rangapuram et al., 2018), including the causal analysis of time series Peters et al.
Revised: A signiﬁcant body of work in machine learning concerns the modeling of spatio-temporal phenomena (Shi and Yeung, 2018; Rangapuram et al., 2018), including the causal analysis of time series Peters et al.

Original: (2017), with applications ranging from markets (Pennacchioli et al., 2014) to bioinformatics (Brouard et al., 2016) to climate (Nooteboom et al., 2018).
Revised: (2017), with applications ranging from markets (Pennacchioli et al., 2014) to bioinformatics (Brouard et al., 2016) to climate (Nooteboom et al., 2018).

Original: This paper focuses on the problem of modeling the temporal dependency between two spatio-temporal phenomena, where the latter one is caused by the former one (Granger, 1969; Runge, 2018) with a non-stationary time delay.
Revised: This paper focuses on the problem of modeling the temporal dependency between two spatio-temporal phenomena, where the latter one is caused by the former one (Granger, 1969; Runge, 2018) with a non-stationary time delay.

Original: The motivating application domain is that of space weather.
Revised: The motivating application domain is that of space weather.

Original: The sun, a perennial source of charged energetic particles, is at the origin of geomagnetic phenomena within the sun-earth system.
Revised: The sun, a perennial source of charged energetic particles, is at the origin of geomagnetic phenomena within the sun-earth system.

Original: Speciﬁcally, the sun ejects charged particles into the surrounding space in all directions and some of these particle clouds, a.k.a.
Revised: Speciﬁcally, the sun ejects charged particles into the surrounding space in all directions and some of these particle clouds, a.k.a.

Original: solar wind , reach the Earth’s vicinity.
Revised: solar wind , reach the Earth’s vicinity.

Original: High speed solar wind is a major threat for the modern world, causing severe damages to e.g., satellites, telecommunication infrastructures, under sea pipelines, among others.
Revised: High speed solar wind is a major threat for the modern world, causing severe damages to e.g., satellites, telecommunication infrastructures, under sea pipelines, among others.

Original: A key prediction task thus is to forecast the speed of the solar wind in the vicinity of the Earth (Munteanu et al., 2013; Haaland et al., 2010; Reiss et al., 2019), sufﬁciently early to emit an alarm and be able to prevent the damage to the best possible extent.
Revised: A key prediction task thus is to forecast the speed of the solar wind in the vicinity of the Earth (Munteanu et al., 2013; Haaland et al., 2010; Reiss et al., 2019), sufﬁciently early to emit an alarm and be able to prevent the damage to the best possible extent.

Original: Formally the goal is to model the dependency between heliospheric observations (available at light speed), referred to as cause series , and the solar wind speed series recorded at the Lagrangian point L 1 (a point on the Sun-Earth line 1.5 million kilometers away from the Earth), referred to as effect series .
Revised: Formally the goal is to model the dependency between heliospheric observations (available at light speed), referred to as cause series , and the solar wind speed series recorded at the Lagrangian point L 1 (a point on the Sun-Earth line 1.5 million kilometers away from the Earth), referred to as effect series .

Original: The key difﬁculty is that the time lag between an input and its effect, the solar wind recorded at L 1 , varies from circa 2 to 5 days depending on, among many factors, the initial direction of emitted particles and their energy.
Revised: The key difﬁculty is that the time lag between an input and its effect, the solar wind recorded at L 1 , varies from circa 2 to 5 days depending on, among many factors, the initial direction of emitted particles and their energy.

Original: Would the lag be constant, the solar wind prediction problem would boil down to a mainstream regression problem.
Revised: Would the lag be constant, the solar wind prediction problem would boil down to a mainstream regression problem.

Original: The challenge here is to predict, from the solar image x ( t ) at time t the value y ( t + ∆ t ) of the solar wind speed reaching the earth at time t + ∆ t where both the value y ( t + ∆ t ) and the time lag ∆ t depend on x ( t ) .
Revised: The challenge here is to predict, from the solar image x ( t ) at time t the value y ( t + ∆ t ) of the solar wind speed reaching the earth at time t + ∆ t where both the value y ( t + ∆ t ) and the time lag ∆ t depend on x ( t ) .

Original: Related work.
Revised: Related work.

Original: Indeed, the modeling of dependencies among time series has been intensively tackled (see e.g., Zhou and Sornette (2006); Runge (2018)).
Revised: Indeed, the modeling of dependencies among time series has been intensively tackled (see e.g., Zhou and Sornette (2006); Runge (2018)).

Original: When considering varying time lag, many approaches rely on dynamic time warping (DTW) (Sakoe and Chiba, 1978).
Revised: When considering varying time lag, many approaches rely on dynamic time warping (DTW) (Sakoe and Chiba, 1978).

Original: For instance,
Revised: For instance,

Original: DTW is used in Gaskell et al.
Revised: DTW is used in Gaskell et al.

Original: (2015), taking a Bayesian approach to achieve the temporal alignment of both series under some restricting assumptions (considering slowly varying time lags and linear relationships between the cause and effect time series).
Revised: (2015), taking a Bayesian approach to achieve the temporal alignment of both series under some restricting assumptions (considering slowly varying time lags and linear relationships between the cause and effect time series).

Original: More generally, the use of DTW in time series analysis relies on simplifying assumptions on the cause and effect series (same dimensionality and structure) and builds upon available cost matrices for the temporal alignment.
Revised: More generally, the use of DTW in time series analysis relies on simplifying assumptions on the cause and effect series (same dimensionality and structure) and builds upon available cost matrices for the temporal alignment.

Original: Also related is sequence-to-sequence learning (Sutskever et al., 2014), primarily aimed to machine translation.
Revised: Also related is sequence-to-sequence learning (Sutskever et al., 2014), primarily aimed to machine translation.

Original: While Seq2Seq modelling relaxes some former assumptions (such as the ﬁxed or comparable sizes of the source and target series), it still relies on the known segmentation of the source series into disjoint units (the sentences), each one being mapped into a large ﬁxed-size vector using an LSTM; and this vector is exploited by another LSTM to extract the output sequence.
Revised: While Seq2Seq modelling relaxes some former assumptions (such as the ﬁxed or comparable sizes of the source and target series), it still relies on the known segmentation of the source series into disjoint units (the sentences), each one being mapped into a large ﬁxed-size vector using an LSTM; and this vector is exploited by another LSTM to extract the output sequence.

Original: Attention-based mechanisms Graves (2013); Bahdanau et al.
Revised: Attention-based mechanisms Graves (2013); Bahdanau et al.

Original: (2015) alleviate the need to encode the full source sentence into a ﬁxed-size vector, by learning the alignment and allowing the model to search for the parts of the source sentence relevant to predict a target part.
Revised: (2015) alleviate the need to encode the full source sentence into a ﬁxed-size vector, by learning the alignment and allowing the model to search for the parts of the source sentence relevant to predict a target part.

Original: More advanced attention mechanisms (Kim et al., 2017; Vaswani et al., 2017) reﬁne the way the source information is leveraged to produce a target part.
Revised: More advanced attention mechanisms (Kim et al., 2017; Vaswani et al., 2017) reﬁne the way the source information is leveraged to produce a target part.

Original: But to our best knowledge, the end-to-end learning of the sequence-to-sequence modelling relies on the segmentation of the source and target series, and the deﬁnition of associated pairs of segments (e.g. the sentences).
Revised: But to our best knowledge, the end-to-end learning of the sequence-to-sequence modelling relies on the segmentation of the source and target series, and the deﬁnition of associated pairs of segments (e.g. the sentences).

Original: Our claim is that the regression problem of predicting both what the effect is and when the effect is observed, called Dynamic TimeLag Regression (DTLR), constitutes a new ML problem:
Revised: Our claim is that the regression problem of predicting both what the effect is and when the effect is observed, called Dynamic TimeLag Regression (DTLR), constitutes a new ML problem:

Original: With respect to the modeling of dependencies among time series, it involves stochastic dependencies of arbitrary complexity; the relationship between the cause and the effect series can be non-linear (the what model).
Revised: With respect to the modeling of dependencies among time series, it involves stochastic dependencies of arbitrary complexity; the relationship between the cause and the effect series can be non-linear (the what model).

Original: Furthermore, the time lag phenomenon (the when model) can be non smooth (as opposed to e.g. Zhou and Sornette (2006)).
Revised: Furthermore, the time lag phenomenon (the when model) can be non smooth (as opposed to e.g. Zhou and Sornette (2006)).

Original: With respect to sequence-to-sequence translation, a main difference is that the end-to-end training of the model cannot rely on pairs of associated units (the sentences), adversely affecting the alignment learning.
Revised: With respect to sequence-to-sequence translation, a main difference is that the end-to-end training of the model cannot rely on pairs of associated units (the sentences), adversely affecting the alignment learning.

Original: Lastly, and most importantly, in the considered DTLR problem, even if the cause series has high information content, only a small portion of it is relevant to the prediction of the effect series.
Revised: Lastly, and most importantly, in the considered DTLR problem, even if the cause series has high information content, only a small portion of it is relevant to the prediction of the effect series.

Original: On one hand, the cause series might be high dimensional (images) whereas the effect series is scalar; on the other hand, the cause series governs the solar wind speed in the whole heliosphere and not just in near-Earth space.
Revised: On one hand, the cause series might be high dimensional (images) whereas the effect series is scalar; on the other hand, the cause series governs the solar wind speed in the whole heliosphere and not just in near-Earth space.

Original: In addition to avoiding typically one or two orders of magnitude expansion of an already large input signal dimension, inserting the time-lag inference explicitly in the model can also potentially improve its interpretability.
Revised: In addition to avoiding typically one or two orders of magnitude expansion of an already large input signal dimension, inserting the time-lag inference explicitly in the model can also potentially improve its interpretability.

Original: Organization of the paper.
Revised: Organization of the paper.

Original: The Bayesian approach proposed to tackle the speciﬁcs of the DTLR regression problem is described in section 2; the associated learning equations are discussed, followed by a stability analysis and a proof of consistency (section 3).
Revised: The Bayesian approach proposed to tackle the speciﬁcs of the DTLR regression problem is described in section 2; the associated learning equations are discussed, followed by a stability analysis and a proof of consistency (section 3).

Original: The algorithm is detailed in section 4.
Revised: The algorithm is detailed in section 4.

Original: The experimental setting used to validate the approach is presented in section 5; the empirical validation on toy problems and on the real-world problem are discussed in section
Revised: The experimental setting used to validate the approach is presented in section 5; the empirical validation on toy problems and on the real-world problem are discussed in section

Original:  P ROBABILISTIC D YNAMICALLY D ELAYED R EGRESSION 2. P OSITION OF THE PROBLEM
Revised:  P ROBABILISTIC D YNAMICALLY D ELAYED R EGRESSION 2. P OSITION OF THE PROBLEM

Original: Given two time series, the cause series x ( t ) ( x ( t ) ∈ X ⊂ R D ) and the observed effect series y ( t ) , the sought model consists of a mapping f ( . ) which maps each input pattern x ( t ) to an output y ( φ ( t )) , and a mapping g ( . ) which determines the time delay φ ( t ) − t between the input and output patterns: y ( φ ( t ))
Revised: Given two time series, the cause series x ( t ) ( x ( t ) ∈ X ⊂ R D ) and the observed effect series y ( t ) , the sought model consists of a mapping f ( . ) which maps each input pattern x ( t ) to an output y ( φ ( t )) , and a mapping g ( . ) which determines the time delay φ ( t ) − t between the input and output patterns: y ( φ ( t ))

Original: =
Revised: =

Original: f [ x ( t )] (1) φ ( t ) =
Revised: f [ x ( t )] (1) φ ( t ) =

Original: t + g [ x ( t )] (2) with f : X → R , and g : X → R + , where t ∈ R + represents the continuous temporal domain.
Revised: t + g [ x ( t )] (2) with f : X → R , and g : X → R + , where t ∈ R + represents the continuous temporal domain.

Original: The input signal x ( t ) is possibly high dimensional and contains the hidden cause to the effect y ( t )
Revised: The input signal x ( t ) is possibly high dimensional and contains the hidden cause to the effect y ( t )

Original: ∈ R ; y ( t ) is assumed to be scalar in the remainder of the paper.
Revised: ∈ R ; y ( t ) is assumed to be scalar in the remainder of the paper.

Original: Function g : X → R + represents the time delay between inputs and outputs.
Revised: Function g : X → R + represents the time delay between inputs and outputs.

Original: Vectors are written using bold fonts.
Revised: Vectors are written using bold fonts.

Original: As said, Eqs 1-2 deﬁne a regression problem that differs from standard regression in two ways: Firstly, the time lag g [ x ( t )] is non-stationary as it depends on x ( t ) .
Revised: As said, Eqs 1-2 deﬁne a regression problem that differs from standard regression in two ways: Firstly, the time lag g [ x ( t )] is non-stationary as it depends on x ( t ) .

Original: Secondly, g [ x ( t )] is unknown, i.e. it is not recorded explicitly in the training data.
Revised: Secondly, g [ x ( t )] is unknown, i.e. it is not recorded explicitly in the training data.

Original: Assumption.
Revised: Assumption.

Original: For the sake of the model identiﬁability and computational stability, the time warping function φ ( t ) =
Revised: For the sake of the model identiﬁability and computational stability, the time warping function φ ( t ) =

Original: t + g [ x ( t )] is assumed to be sufﬁciently regular w.r.t.
Revised: t + g [ x ( t )] is assumed to be sufﬁciently regular w.r.t.

Original: t .
Revised: t .

Original: Formally, φ ( . ) is assumed to be continuous 2 .
Revised: Formally, φ ( . ) is assumed to be continuous 2 .

Original: 2. P ROBABILISTIC D YNAMIC T IME -L AG REGRESSION
Revised: 2. P ROBABILISTIC D YNAMIC T IME -L AG REGRESSION

Original: For practical reasons, cause and effect series are sampled at constant rate, and thereafter noted x t and y t with t in N .
Revised: For practical reasons, cause and effect series are sampled at constant rate, and thereafter noted x t and y t with t in N .

Original: Accordingly, mapping g maps x t onto a ﬁnite set T of possible time lags, with
Revised: Accordingly, mapping g maps x t onto a ﬁnite set T of possible time lags, with

Original: T = { ∆ t min . . . , ∆ t max } an integer interval deﬁned from domain knowledge.
Revised: T = { ∆ t min . . . , ∆ t max } an integer interval deﬁned from domain knowledge.

Original: The unavoidable error due to the discretization of the continuous time lag is mitigated along a probabilistic model, associating to each cause x , a set of predictors ˆ y ( x ) =
Revised: The unavoidable error due to the discretization of the continuous time lag is mitigated along a probabilistic model, associating to each cause x , a set of predictors ˆ y ( x ) =

Original: { ˆ y i ( x ) , i ∈ T } and a probability distribution ˆ p ( x ) on T estimating the probability of delay of the effects of x .
Revised: { ˆ y i ( x ) , i ∈ T } and a probability distribution ˆ p ( x ) on T estimating the probability of delay of the effects of x .

Original: Overall, the DTLR solution is sought as a probability distribution conditioned on cause x , mixture of Gaussians 3 centered on the predictors ˆ y i ( x ) , where the mixture weights are deﬁned from ˆ p ( x ) .
Revised: Overall, the DTLR solution is sought as a probability distribution conditioned on cause x , mixture of Gaussians 3 centered on the predictors ˆ y i ( x ) , where the mixture weights are deﬁned from ˆ p ( x ) .

Original: More formally, letting y t denote the vector of random variables { y t
Revised: More formally, letting y t denote the vector of random variables { y t

Original: + i , i ∈ T } :
Revised: + i , i ∈ T } :

Original: [Equation]
Revised: [Equation]

Original: with Σ =
Revised: with Σ =

Original: Diag ( σ i ( τ ) 2 ) the diagonal matrix of variance parameters attached to each time-lag i ∈ T .
Revised: Diag ( σ i ( τ ) 2 ) the diagonal matrix of variance parameters attached to each time-lag i ∈ T .

Original: Two simpliﬁcations are made for the sake of the analysis.
Revised: Two simpliﬁcations are made for the sake of the analysis.

Original: Firstly, the stochastic time lag is modelled as the vector ( τ i ) , i ∈ T of binary latent variables, where τ i indicates whether x t drives y t
Revised: Firstly, the stochastic time lag is modelled as the vector ( τ i ) , i ∈ T of binary latent variables, where τ i indicates whether x t drives y t

Original: + i ( τ i = 1 ) or not ( τ i = 0 ).
Revised: + i ( τ i = 1 ) or not ( τ i = 0 ).

Original: The assumption that every cause has a single effect is modelled by imposing:
Revised: The assumption that every cause has a single effect is modelled by imposing:

Original: [Equation]
Revised: [Equation]

Original: From constraint (4), probability distribution ˆ p ( x ) thus is sought as the vector (ˆ p i ( x )) for i in T , summing to 1,
Revised: From constraint (4), probability distribution ˆ p ( x ) thus is sought as the vector (ˆ p i ( x )) for i in T , summing to 1,

Original: such that ˆ p
Revised: such that ˆ p

Original: i ( x ) stands for the probability of the effect of x t = x to occur with delay i .
Revised: i ( x ) stands for the probability of the effect of x t = x to occur with delay i .

Original: The second simplifying assumption is that the variance σ 2 i ( τ ) of predictor ˆ y i does not depend on x , by setting:
Revised: The second simplifying assumption is that the variance σ 2 i ( τ ) of predictor ˆ y i does not depend on x , by setting:

Original: [Equation]
Revised: [Equation]

Original: with σ 2 a default variance and α ij ≥ 0 a matrix of non-negative real parameters.
Revised: with σ 2 a default variance and α ij ≥ 0 a matrix of non-negative real parameters.

Original: This particular formulation supports the tractable analysis of the posterior probability of τ i (in supplementary material).
Revised: This particular formulation supports the tractable analysis of the posterior probability of τ i (in supplementary material).

Original: The fact that x can inﬂuence y i through predictor ˆ y i ( x ) even when τ i = 0 reﬂects an indirect inﬂuence due to the auto-correlation of the y series.
Revised: The fact that x can inﬂuence y i through predictor ˆ y i ( x ) even when τ i = 0 reﬂects an indirect inﬂuence due to the auto-correlation of the y series.

Original: This inﬂuence comes with a higher variance, enforced by making α ij a decreasing function of | i − j | .
Revised: This inﬂuence comes with a higher variance, enforced by making α ij a decreasing function of | i − j | .

Original: More generally, a large value of α ii compared to α ij for i (cid:54) =
Revised: More generally, a large value of α ii compared to α ij for i (cid:54) =

Original: j corresponds to a small auto-correlation time of the effect series
Revised: j corresponds to a small auto-correlation time of the effect series

Original: [Equation]
Revised: [Equation]

Original: 2. L EARNING CRITERION
Revised: 2. L EARNING CRITERION

Original: The joint distribution is classically learned by maximizing the log likelihood of the data, which can here be expressed in closed form.
Revised: The joint distribution is classically learned by maximizing the log likelihood of the data, which can here be expressed in closed form.

Original: Let us denote respectively the dataset and parameters as { ( x , y ) } data and θ =
Revised: Let us denote respectively the dataset and parameters as { ( x , y ) } data and θ =

Original: (ˆ y , ˆ p , σ, α ) .
Revised: (ˆ y , ˆ p , σ, α ) .

Original: From Eq.
Revised: From Eq.

Original: (3) the conditional probability q i ( x , y ) def =
Revised: (3) the conditional probability q i ( x , y ) def =

Original: P ( τ i = 1 | x , y ) reads: q i ( x , y ) =
Revised: P ( τ i = 1 | x , y ) reads: q i ( x , y ) =

Original: [Equation]
Revised: [Equation]

Original: with normalization constant
Revised: with normalization constant

Original: [Equation]
Revised: [Equation]

Original: The log-likelihood then reads (intermediate calculations in supplementary material, appendix A):
Revised: The log-likelihood then reads (intermediate calculations in supplementary material, appendix A):

Original: [Equation]
Revised: [Equation]

Original: where E data denotes averaging over the dataset.
Revised: where E data denotes averaging over the dataset.

Original: For notational simplicity, the time index t is omitted in the following and the empirical averaging on the data is noted E data .
Revised: For notational simplicity, the time index t is omitted in the following and the empirical averaging on the data is noted E data .

Original: The hyper-parameters σ and matrix α of the model are obtained by optimizing L :
Revised: The hyper-parameters σ and matrix α of the model are obtained by optimizing L :

Original: [Equation]
Revised: [Equation]

Original: In addition the optimal ˆ y and ˆ p reads:
Revised: In addition the optimal ˆ y and ˆ p reads:

Original: [Equation]
Revised: [Equation]

Original: where the above conditional empirical averaging operates as an averaging over samples close to x .
Revised: where the above conditional empirical averaging operates as an averaging over samples close to x .

Original: These are self-consistent equations, since q i ( x , y ) depends on the parameters σ 2 and α ij , ˆ y and ˆ p .
Revised: These are self-consistent equations, since q i ( x , y ) depends on the parameters σ 2 and α ij , ˆ y and ˆ p .

Original: The proposed algorithm detailed in section 4 implements the saddle point method deﬁned from Eqs (7,5,8,9): alternatively, hyper-parameters σ and α ij are updated from Eq.
Revised: The proposed algorithm detailed in section 4 implements the saddle point method deﬁned from Eqs (7,5,8,9): alternatively, hyper-parameters σ and α ij are updated from Eq.

Original: (7) based on the current ˆ y i and ˆ p i ; and predictors ˆ y i and mixture weights ˆ p i are updated according to Eqs (8) and (9) respectively.
Revised: (7) based on the current ˆ y i and ˆ p i ; and predictors ˆ y i and mixture weights ˆ p i are updated according to Eqs (8) and (9) respectively.

Original:  T HEORETICAL ANALYSIS
Revised:  T HEORETICAL ANALYSIS

Original: The proposed DTLR approach is shown to be consistent and analyzed in the simple case where α is a diagonal matrix ( α ij = αδ ij ).
Revised: The proposed DTLR approach is shown to be consistent and analyzed in the simple case where α is a diagonal matrix ( α ij = αδ ij ).

Original: 3. L OSS FUNCTION AND RELATED OPTIMAL PREDICTOR
Revised: 3. L OSS FUNCTION AND RELATED OPTIMAL PREDICTOR

Original: Let us assume that the hyper-parameters of the model have been identiﬁed together with predictors ˆ y i ( x ) and weights
Revised: Let us assume that the hyper-parameters of the model have been identiﬁed together with predictors ˆ y i ( x ) and weights

Original: ˆ p
Revised: ˆ p

Original: i ( x ) .
Revised: i ( x ) .

Original: These are leveraged to achieve the prediction of the effect series.
Revised: These are leveraged to achieve the prediction of the effect series.

Original: For any given input x , the sought eventual predictor is expressed as (ˆ y ( x ) , ˆ I ( x )) where ˆ I ( x ) is the predicted time lag and ˆ y ( x ) the predicted value.
Revised: For any given input x , the sought eventual predictor is expressed as (ˆ y ( x ) , ˆ I ( x )) where ˆ I ( x ) is the predicted time lag and ˆ y ( x ) the predicted value.

Original: The associated L 2 loss is:
Revised: The associated L 2 loss is:

Original: [Equation]
Revised: [Equation]

Original: Then it comes:
Revised: Then it comes:

Original: Proposition 3.1.
Revised: Proposition 3.1.

Original: With same notations as in Eq.
Revised: With same notations as in Eq.

Original: (3), with α ij = αδ ij , α > 0 , the optimal composite predictor ( y (cid:63) , I (cid:63) ) is given by
Revised: (3), with α ij = αδ ij , α > 0 , the optimal composite predictor ( y (cid:63) , I (cid:63) ) is given by

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: In supplementary material, Appendix C.
Revised: In supplementary material, Appendix C.

Original: 3. L INEAR STABILITY ANALYSIS
Revised: 3. L INEAR STABILITY ANALYSIS

Original: The saddle point (Eqs 7, 5, 8, 9) admits among others a degenerate solution, corresponding to ˆ p
Revised: The saddle point (Eqs 7, 5, 8, 9) admits among others a degenerate solution, corresponding to ˆ p

Original: i ( x ) = 1 / |T | , α ij = 0 for all pairs ( i, j ) , with σ 2 = σ 20 .
Revised: i ( x ) = 1 / |T | , α ij = 0 for all pairs ( i, j ) , with σ 2 = σ 20 .

Original: Informally the model converges toward this degenerate trivial solution when there is not enough information to build specialized predictors ˆ y i .
Revised: Informally the model converges toward this degenerate trivial solution when there is not enough information to build specialized predictors ˆ y i .

Original: Let us denote ∆ y 2 i (
Revised: Let us denote ∆ y 2 i (

Original: [Equation]
Revised: [Equation]

Original: ) =
Revised: ) =

Original: [Equation]
Revised: [Equation]

Original: y i − ˆ y i ( x ) (cid:1) 

Revised: y i − ˆ y i ( x ) (cid:1)

Original: the square error made by predictor ˆ y i for x , and
Revised: the square error made by predictor ˆ y i for x , and

Original: [Equation]
Revised: [Equation]

Original: the average of MSE over the set of the predictors ˆ y i , i ∈ T .
Revised: the average of MSE over the set of the predictors ˆ y i , i ∈ T .

Original: Let us investigate the conditions under which the degenerate solution may appear, by computing the
Revised: Let us investigate the conditions under which the degenerate solution may appear, by computing the

Original: Hessian of the log-likelihood and its eigenvalues.
Revised: Hessian of the log-likelihood and its eigenvalues.

Original: Under the simplifying assumption
Revised: Under the simplifying assumption

Original: [Equation]
Revised: [Equation]

Original: the model involves 2 |T | functional parameters ˆ y and ˆ p and two hyper-parameters α and r =
Revised: the model involves 2 |T | functional parameters ˆ y and ˆ p and two hyper-parameters α and r =

Original: σ 2 /σ 20 .
Revised: σ 2 /σ 20 .

Original: After the computation of the Hessian (in supplementary material, Appendix B) the system involves three key statistical quantities, two global ones:
Revised: After the computation of the Hessian (in supplementary material, Appendix B) the system involves three key statistical quantities, two global ones:

Original: [Equation]
Revised: [Equation]

Original: and a local |T | -vector of components
Revised: and a local |T | -vector of components

Original: [Equation]
Revised: [Equation]

Original: Up to a constant, C 1 represents the covariance between the latent variables { τ i } and the normalized predictor errors.
Revised: Up to a constant, C 1 represents the covariance between the latent variables { τ i } and the normalized predictor errors.

Original: C 1 smaller than one indicates a positive correlation between the latent variables and small errors; the smaller the better.
Revised: C 1 smaller than one indicates a positive correlation between the latent variables and small errors; the smaller the better.

Original: For the degenerate solution, i.e. q = q 0 uniform, C 1 [ q 0 ] = 1 and
Revised: For the degenerate solution, i.e. q = q 0 uniform, C 1 [ q 0 ] = 1 and

Original: C 2 [ q 0 ] represents the default variability among the prediction errors.
Revised: C 2 [ q 0 ] represents the default variability among the prediction errors.

Original: u i [ x , q ] informally measures the quality of predictor ˆ y
Revised: u i [ x , q ] informally measures the quality of predictor ˆ y

Original: i relatively to the other ones at x .
Revised: i relatively to the other ones at x .

Original: More precisely, a negative value of u i [ x , q ] indicates that ˆ y
Revised: More precisely, a negative value of u i [ x , q ] indicates that ˆ y

Original: i is doing better than average in the neighborhood of x .
Revised: i is doing better than average in the neighborhood of x .

Original: At a saddle point the parameters are given by:
Revised: At a saddle point the parameters are given by:

Original: [Equation]
Revised: [Equation]

Original: The predictors ˆ y are decoupled from the rest whenever they are centered, which we assume.
Revised: The predictors ˆ y are decoupled from the rest whenever they are centered, which we assume.

Original: So the analysis can focus on the other parameters.
Revised: So the analysis can focus on the other parameters.

Original: If ˆ p is ﬁxed a saddle point is stable iff
Revised: If ˆ p is ﬁxed a saddle point is stable iff

Original: [Equation]
Revised: [Equation]

Original: In particular, the degenerate solution is unstable if
Revised: In particular, the degenerate solution is unstable if

Original: [Equation]
Revised: [Equation]

Original: Note that for ∆
Revised: Note that for ∆

Original: y i ( x ) iid centered with variance σ 20 and relative kurtosis κ (conditionally to x ) one has C 2 =
Revised: y i ( x ) iid centered with variance σ 20 and relative kurtosis κ (conditionally to x ) one has C 2 =

Original: (2 + κ )(1 − 1 / | T | ) .
Revised: (2 + κ )(1 − 1 / | T | ) .

Original: Therefore, whenever ∆ y 2 i ( x ) ﬂuctuates and the relative curtosis is non-negative, the degenerate solution is unstable and will thus be avoided.
Revised: Therefore, whenever ∆ y 2 i ( x ) ﬂuctuates and the relative kurtosis is non-negative, the degenerate solution is unstable and will thus be avoided.

Original: If ˆ p is allowed to evolve (after Eq.
Revised: If ˆ p is allowed to evolve (after Eq.

Original: the degenerate trivial solution becomes unstable as soon as
Revised: the degenerate trivial solution becomes unstable as soon as

Original: C 2 [ q 0 ] is non-zero, due to the fact that the gradient points in the opposite direction to u ( x ) (with d ˆ p ( x ) ∝ − C 2 [ q 0 ] u ( x ) ), thus rewarding the predictors with lowest errors by increasing their weights.
Revised: C 2 [ q 0 ] is non-zero, due to the fact that the gradient points in the opposite direction to u ( x ) (with d ˆ p ( x ) ∝ − C 2 [ q 0 ] u ( x ) ), thus rewarding the predictors with lowest errors by increasing their weights.

Original: The system is then driven toward other solutions, among which the localized solutions of the form: ˆ p
Revised: The system is then driven toward other solutions, among which the localized solutions of the form: ˆ p

Original: i ( x ) =
Revised: i ( x ) =

Original: δ i,I ( x ) , with an input dependent index
Revised: δ i,I ( x ) , with an input dependent index

Original: I ( x ) ∈ T .
Revised: I ( x ) ∈ T .

Original: As shown (in supplementary material, Appendix C) the maximum likelihood localized solution also minimizes the loss function (Eq. 10).
Revised: As shown (in supplementary material, Appendix C) the maximum likelihood localized solution also minimizes the loss function (Eq. 10).

Original: The stability of such localized solutions and the existence of other (non-localized) solutions is left for further work.
Revised: The stability of such localized solutions and the existence of other (non-localized) solutions is left for further work.

Original:  T HE DTLR ALGORITHM
Revised:  T HE DTLR ALGORITHM

Original: The DTLR algorithm learns both regression models ˆ y ( x ) and ˆ p ( x ) from series x t and y t , using alternate optimization of the model parameters and the model hyper-parameters α and σ 2 , after Eqs (7,5,8,9).
Revised: The DTLR algorithm learns both regression models ˆ y ( x ) and ˆ p ( x ) from series x t and y t , using alternate optimization of the model parameters and the model hyper-parameters α and σ 2 , after Eqs (7,5,8,9).

Original: The model search space is that of neural nets, parameterized by their weight vector θ .
Revised: The model search space is that of neural nets, parameterized by their weight vector θ .

Original: The inner optimization loop updates θ using mini-batch based stochastic gradient descent.
Revised: The inner optimization loop updates θ using mini-batch based stochastic gradient descent.

Original: At the end of each epoch, after all minibatches have been considered, the outer optimization loop computes hyper-parameters α and σ 2 on the whole data.
Revised: At the end of each epoch, after all minibatches have been considered, the outer optimization loop computes hyper-parameters α and σ 2 on the whole data.

Original: Initialization of α and σ it ←− 0 ; while it < max do while epoch do θ ←− Optimize ( L ( θ, α, σ 2 )) ; end σ 2 ←− σ 20 | T |− C 1 [ q ] | T |−; α ←− | T | | T |− 1 1 − C 1 [ q ] C 1 [ q ] ; end
Revised: Initialization of α and σ it ←− 0 ; while it < max do while epoch do θ ←− Optimize ( L ( θ, α, σ 2 )) ; end σ 2 ←− σ 20 | T |− C 1 [ q ] | T |−; α ←− | T | | T |− 1 1 − C 1 [ q ] C 1 [ q ] ; end

Original: Result: Model parameters θ = { ˆ y , ˆ p } , hyper-parameters α, σ
Revised: Result: Model parameters θ = { ˆ y , ˆ p } , hyper-parameters α, σ

Original: The algorithm code is available in supplementary material and will be made public after the reviewing period.
Revised: The algorithm code is available in supplementary material and will be made public after the reviewing period.

Original: The initialization of hyper-parameters α and σ is settled using preliminary experiments (same setting for all considered problems: α ∼ U (0 . 75 , 2) ; σ 2 ∼ U (10 − 5 , 5) ).
Revised: The initialization of hyper-parameters α and σ is settled using preliminary experiments (same setting for all considered problems: α ∼ U (0 . 75 , 2) ; σ 2 ∼ U (10 − 5 , 5) ).

Original: The neural architecture implements predictors ˆ y ( x ) and weights ˆ p ( x ) on the top of a same feature extractor from input x .
Revised: The neural architecture implements predictors ˆ y ( x ) and weights ˆ p ( x ) on the top of a same feature extractor from input x .

Original: In the experiments, the architecture of the feature extractor is a 2-hidden layer fully connected network.
Revised: In the experiments, the architecture of the feature extractor is a 2-hidden layer fully connected network.

Original: On the top of the feature extractor are the single layer ˆ y and ˆ p models, each with | T | output neurons, with | T | the size of the chosen domain for the time lag.
Revised: On the top of the feature extractor are the single layer ˆ y and ˆ p models, each with | T | output neurons, with | T | the size of the chosen domain for the time lag.

Original:  E XPERIMENTAL S ETTING
Revised:  E XPERIMENTAL S ETTING

Original: The goal of experimental validation is threefold.
Revised: The goal of experimental validation is threefold.

Original: A ﬁrst issue regards the accuracy of DTLR, measured from the mean absolute error (MAE), root mean square error (RMSE) and Pearson correlation of the learned DTLR model ( y (cid:63) ( x t ) , I (cid:63) ( x )) .
Revised: A ﬁrst issue regards the accuracy of DTLR, measured from the mean absolute error (MAE), root mean square error (RMSE) and Pearson correlation of the learned DTLR model ( y (cid:63) ( x t ) , I (cid:63) ( x )) .

Original: DTLR is compared to the natural baseline deﬁned as the regressor with constant time lag, ˆ y ¯∆ ( x t ) , with ¯∆ being the average of all possible time lags in T .
Revised: DTLR is compared to the natural baseline deﬁned as the regressor with constant time lag, ˆ y ¯∆ ( x t ) , with ¯∆ being the average of all possible time lags in T .

Original: The predictions of DTLR and the baseline are compared with the ground truth value of the effect series.
Revised: The predictions of DTLR and the baseline are compared with the ground truth value of the effect series.

Original: However the predicted time lag I (cid:63) ( x t ) can only be assessed if the ground truth time-lag relationship is known.
Revised: However the predicted time lag I (cid:63) ( x t ) can only be assessed if the ground truth time-lag relationship is known.

Original: In order to do so, three synthetic problems of increasing difﬁculty are deﬁned below.
Revised: In order to do so, three synthetic problems of increasing difﬁculty are deﬁned below.

Original: Secondly, the stability and non-degeneracy of the learned model are assessed from the statistical quantities σ 0 and C 1 (section 3.2), compared to the degenerate solution ˆ p
Revised: Secondly, the stability and non-degeneracy of the learned model are assessed from the statistical quantities σ 0 and C 1 (section 3.2), compared to the degenerate solution ˆ p

Original: i ( x ) = 1 / |T | .
Revised: i ( x ) = 1 / |T | .

Original: For C 1 < 1 , the model accurately specializes the found predictors ˆ p i .
Revised: For C 1 < 1 , the model accurately specializes the found predictors ˆ p i .

Original: Lastly, and most importantly, DTLR is assessed on the solar wind prediction problem, and compared to the best state of the art in space weather.
Revised: Lastly, and most importantly, DTLR is assessed on the solar wind prediction problem, and compared to the best state of the art in space weather.

Original: Synthetic Problems.
Revised: Synthetic Problems.

Original: Four synthetic problems of increasing difﬁculty are generated using Stochastic Langevin Dynamics .
Revised: Four synthetic problems of increasing difﬁculty are generated using Stochastic Langevin Dynamics .

Original: In all problems, the cause signal x t ∈ R 10 and the effect signal y t are generated as follows (with η = 0 .
Revised: In all problems, the cause signal x t ∈ R 10 and the effect signal y t are generated as follows (with η = 0 .

Original: 02 , s 2 = 0 .
Revised: 02 , s 2 = 0 .

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: with time-lag mapping g ( x t ) ranges in a time interval with width 20 (except for problem I where |T | = 15 ).
Revised: with time-lag mapping g ( x t ) ranges in a time interval with width 20 (except for problem I where |T | = 15 ).

Original: The complexity of the synthetic problems is governed by the amplitude and time-lag functions f and g (more in appendix, Table 2):
Revised: The complexity of the synthetic problems is governed by the amplitude and time-lag functions f and g (more in appendix, Table 2):

Original: Problem
Revised: Problem

Original: [Equation]
Revised: [Equation]

Original: Other
Revised: Other

Original: I v
Revised: I v

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: II
Revised: II

Original: [Equation]
Revised: [Equation]

Original: III √
Revised: III √

Original: [Equation]
Revised: [Equation]

Original: IV
Revised: IV

Original: v
Revised: v

Original: [Equation]
Revised: [Equation]

Original: Solar Wind Speed Prediction.
Revised: Solar Wind Speed Prediction.

Original: The challenge of predicting solar wind speed from heliospheric data is due to the non-stationary propagation time of the solar plasma through the interplanetary medium.
Revised: The challenge of predicting solar wind speed from heliospheric data is due to the non-stationary propagation time of the solar plasma through the interplanetary medium.

Original: For the sake of a fair comparison with the best state of the art Reiss et al.
Revised: For the sake of a fair comparison with the best state of the art Reiss et al.

Original: (2019), the same experimental setting is used.
Revised: (2019), the same experimental setting is used.

Original: The cause series x t includes the solar magnetic ( ﬂux tube expansion ,
Revised: The cause series x t includes the solar magnetic ( ﬂux tube expansion ,

Original: FTE) and the coronal magnetic ﬁeld strength estimates produced by the current sheet source surface (Zhao and Hoeksema, 1995) model, exploiting the hourly magnetogram data recorded by the Global
Revised: FTE) and the coronal magnetic ﬁeld strength estimates produced by the current sheet source surface (Zhao and Hoeksema, 1995) model, exploiting the hourly magnetogram data recorded by the Global

Original: Oscillation Network Group from 2008 to 2016.
Revised: Oscillation Network Group from 2008 to 2016.

Original: The effect series, the hourly solar wind data is available from the OMNI data base from the Space Physics Data Facility 5 .
Revised: The effect series, the hourly solar wind data is available from the OMNI data base from the Space Physics Data Facility 5 .

Original: After domain knowledge, the time-lag ranges from 2 to 5 days, segmented in six-hour segments (thus |T | = 12 ).
Revised: After domain knowledge, the time-lag ranges from 2 to 5 days, segmented in six-hour segments (thus |T | = 12 ).

Original: For the i -th segment, the "ground truth" solar wind y i is set to its median value over the 6 hours.
Revised: For the i -th segment, the "ground truth" solar wind y i is set to its median value over the 6 hours.

Original: DTLR is validated using a nine fold cross-validation (Table 3 in appendix), where each fold is a continuous period corresponding to a solar rotation.
Revised: DTLR is validated using a nine fold cross-validation (Table 3 in appendix), where each fold is a continuous period corresponding to a solar rotation.

Original:  E MPIRICAL VALIDATION
Revised:  E MPIRICAL VALIDATION

Original: Table 1 summarizes the DTLR performance on the synthetic and solar wind problems (detailed results are provided in the appendix).
Revised: Table 1 summarizes the DTLR performance on the synthetic and solar wind problems (detailed results are provided in the appendix).

Original: [Table]
Revised: [Table]

Original: 6. S YNTHETIC PROBLEMS
Revised: 6. S YNTHETIC PROBLEMS

Original: On the easy Problem
Revised: On the easy Problem

Original: I
Revised: I

Original: [Table]
Revised: [Table]

Original: DTLR predicts the correct time lag for 97 .
Revised: DTLR predicts the correct time lag for 97 .

Original: 93% of the samples.
Revised: 93% of the samples.

Original: The higher value of σ 0 in problems I and II compared to the other problems is explained from the higher variance in the effect series y ( t ) .
Revised: The higher value of σ 0 in problems I and II compared to the other problems is explained from the higher variance in the effect series y ( t ) .

Original: On Problem II, DTLR accurately learns the inverse relationship between x t , g ( x t ) and y t on average.
Revised: On Problem II, DTLR accurately learns the inverse relationship between x t , g ( x t ) and y t on average.

Original: The time lag is overestimated in the regions with low time lag (with high velocity), which is blamed on the low sample density in this region, due to the data generation process.
Revised: The time lag is overestimated in the regions with low time lag (with high velocity), which is blamed on the low sample density in this region, due to the data generation process.

Original: Interestingly, Problems
Revised: Interestingly, Problems

Original: III and IV are better handled by DTLR, despite a more complex dynamic time lag relationship.
Revised: III and IV are better handled by DTLR, despite a more complex dynamic time lag relationship.

Original: In both latter cases however, the model tends to under-estimate the time lag in the high time lag region and conversely to over-estimate it in the low time lag region.
Revised: In both latter cases however, the model tends to under-estimate the time lag in the high time lag region and conversely to over-estimate it in the low time lag region.

Original: 6. T HE SOLAR WIND PROBLEM
Revised: 6. T HE SOLAR WIND PROBLEM

Original: DTLR ﬁnds an operational solar wind model (Table 1), though the signiﬁcantly higher difﬁculty of the solar wind problem is witnessed by the C 1 value close to the degenerate value 1.
Revised: DTLR ﬁnds an operational solar wind model (Table 1), though the signiﬁcantly higher difﬁculty of the solar wind problem is witnessed by the C 1 value close to the degenerate value 1.

Original: The detailed comparison with the state of the art Reiss et al.
Revised: The detailed comparison with the state of the art Reiss et al.

Original: (Fig. 1, Left) shows that DTLR improves on the current best state of the art (on all variants including ensemble approaches, and noting that median models are notoriously hard to beat).
Revised: (Fig. 1, Left) shows that DTLR improves on the current best state of the art (on all variants including ensemble approaches, and noting that median models are notoriously hard to beat).

Original: (Fig. 1, Right) shows the good correlation between the predicted solar wind 7 and the measured solar wind.
Revised: (Fig. 1, Right) shows the good correlation between the predicted solar wind 7 and the measured solar wind.

Original: [Table]
Revised: [Table]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original:  D ISCUSSION AND PERSPECTIVES
Revised:  D ISCUSSION AND PERSPECTIVES

Original: The contribution of the paper is twofold.
Revised: The contribution of the paper is twofold.

Original: A new ML setting, Dynamic Time Lag Regression has been deﬁned, aimed at the modelling of varying time-lag dependency between time series.
Revised: A new ML setting, Dynamic Time Lag Regression has been deﬁned, aimed at the modelling of varying time-lag dependency between time series.

Original: The introduction of this new setting is motivated by an important scientiﬁc and practical problem from the domain of space weather, an open problem for over two decades.
Revised: The introduction of this new setting is motivated by an important scientiﬁc and practical problem from the domain of space weather, an open problem for over two decades.

Original: Secondly, a Bayesian formalization has been proposed to tackle the DTLR problem, relying on a saddle point optimization process.
Revised: Secondly, a Bayesian formalization has been proposed to tackle the DTLR problem, relying on a saddle point optimization process.

Original: A closed form analysis of the training procedure stability under simplifying assumptions has been conducted, yielding a practical alternate optimization formulation, implemented in the DTLR algorithm.
Revised: A closed form analysis of the training procedure stability under simplifying assumptions has been conducted, yielding a practical alternate optimization formulation, implemented in the DTLR algorithm.

Original: This algorithm has been successfully validated on synthetic and real-world problems, although some bias toward the mean has been detected in some cases.
Revised: This algorithm has been successfully validated on synthetic and real-world problems, although some bias toward the mean has been detected in some cases.

Original: On the methodological side, this work opens a short term perspective (handling the bias) and a longer term perspective, extending the proposed nested inference procedure and integrating the model selection step within the inference architecture.
Revised: On the methodological side, this work opens a short term perspective (handling the bias) and a longer term perspective, extending the proposed nested inference procedure and integrating the model selection step within the inference architecture.

Original: The challenge is to provide the algorithm with the means of assessing online the stability and/or the degeneracy of the learning trajectory.
Revised: The challenge is to provide the algorithm with the means of assessing online the stability and/or the degeneracy of the learning trajectory.

Original: Regarding the motivating solar wind prediction application, a next step consists of enriching the data sources and the description of the cause series x t , typically by directly using the solar images.
Revised: Regarding the motivating solar wind prediction application, a next step consists of enriching the data sources and the description of the cause series x t , typically by directly using the solar images.

Original: Another perspective is to consider other applications of the general DTLR setting, e.g. considering ﬁne-grained modelling of diffusion phenomena.
Revised: Another perspective is to consider other applications of the general DTLR setting, e.g. considering ﬁne-grained modelling of diffusion phenomena.

Original: A PPENDIX A L OG LIKELIHOOD OF THE LATENT MODEL (3) A. D IRECT COMPUTATION
Revised: A PPENDIX A L OG LIKELIHOOD OF THE LATENT MODEL (3) A. D IRECT COMPUTATION

Original: Due to the single effect constraint (4) the mixture model (3) can be expressed simply as
Revised: Due to the single effect constraint (4) the mixture model (3) can be expressed simply as

Original: [Equation]
Revised: [Equation]

Original: Let θ def =
Revised: Let θ def =

Original: (ˆ y , ˆ p , σ, α ) denote the parameters of the model and consider the probability that predictor
Revised: (ˆ y , ˆ p , σ, α ) denote the parameters of the model and consider the probability that predictor

Original: ˆ y i is the good one conditionally to a pair of observation ( x
Revised: ˆ y i is the good one conditionally to a pair of observation ( x

Original: [Equation]
Revised: [Equation]

Original: y ) :
Revised: y ) :

Original: [Equation]
Revised: [Equation]

Original: | x, y )
Revised: | x, y )

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: This gives immediately
Revised: This gives immediately

Original: [Equation]
Revised: [Equation]

Original: A. L ARGE DEVIATION ARGUMENT
Revised: A. L ARGE DEVIATION ARGUMENT

Original: Even though the log likelihood can be obtained by direct summation, for sake of generality we show how this can result from a large deviation principle.
Revised: Even though the log likelihood can be obtained by direct summation, for sake of generality we show how this can result from a large deviation principle.

Original: Assume that the number of learning samples tends to inﬁnity
Revised: Assume that the number of learning samples tends to inﬁnity

Original: [Equation]
Revised: [Equation]

Original: and so that in a small volume dv = dxd y around a given joint conﬁguration ( x, y ) , the number of data N x, y becomes large.
Revised: and so that in a small volume dv = dxd y around a given joint conﬁguration ( x, y ) , the number of data N x, y becomes large.

Original: Restricting the likelihood to this subset of the data yields the following:
Revised: Restricting the likelihood to this subset of the data yields the following:

Original: [Equation]
Revised: [Equation]

Original: Upon introducing the relative frequencies:
Revised: Upon introducing the relative frequencies:

Original: [Equation]
Revised: [Equation]

Original: satisfying
Revised: satisfying

Original: [Equation]
Revised: [Equation]

Original: the sum over the τ ( m ) i is replaced by a sum over these new variables, with the summand obeying a large deviation principle
Revised: the sum over the τ ( m ) i is replaced by a sum over these new variables, with the summand obeying a large deviation principle

Original: [Equation]
Revised: [Equation]

Original: where the rate function reads
Revised: where the rate function reads

Original: [Equation]
Revised: [Equation]

Original: Taking the saddle point for q i yield as a function of ( x, y ) expression (7).
Revised: Taking the saddle point for q i yield as a function of ( x, y ) expression (7).

Original: Inserting this into F and taking the average over the data set yields the log likelihood (5) with opposite sign:
Revised: Inserting this into F and taking the average over the data set yields the log likelihood (5) with opposite sign:

Original: [Equation]
Revised: [Equation]

Original: A. S ADDLE POINT EQUATIONS
Revised: A. S ADDLE POINT EQUATIONS

Original: Now we turn to the self-consistent equations relating the parameters θ of the model at a saddle point of the log likelihood function.
Revised: Now we turn to the self-consistent equations relating the parameters θ of the model at a saddle point of the log likelihood function.

Original: First, the optimization of the predictors ˆ y yields:
Revised: First, the optimization of the predictors ˆ y yields:

Original: [Equation]
Revised: [Equation]

Original: Then the optimization of ˆ p gives:
Revised: Then the optimization of ˆ p gives:

Original: [Equation]
Revised: [Equation]

Original: with λ ( x ) a Lagrange multiplier to insure
Revised: with λ ( x ) a Lagrange multiplier to insure

Original: that (cid:80) i ˆ p
Revised: that (cid:80) i ˆ p

Original: i ( x ) = 

Revised: i ( x ) =

Original: This gives
Revised: This gives

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: Hence (cid:88)
Revised: Hence (cid:88)

Original: [Equation]
Revised: [Equation]

Original: in order to fulﬁll the normalization constraint
Revised: in order to fulﬁll the normalization constraint

Original: [Equation]
Revised: [Equation]

Original: yielding ﬁnally expression (9).
Revised: yielding ﬁnally expression (9).

Original: Finally the optimization of α reads:
Revised: Finally the optimization of α reads:

Original: [Equation]
Revised: [Equation]

Original: A PPENDIX B P ROOF OF P ROPOSITION 3. 
Revised: A PPENDIX B P ROOF OF P ROPOSITION 3.

Original: Given I ( x ) a candidate index function we associate the point-like measure p
Revised: Given I ( x ) a candidate index function we associate the point-like measure p

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: Written in terms of p the loss function reads
Revised: Written in terms of p the loss function reads

Original: [Equation]
Revised: [Equation]

Original: Under (3) (with α ij = αδ ij ) the loss is equal to
Revised: Under (3) (with α ij = αδ ij ) the loss is equal to

Original: [Equation]
Revised: [Equation]

Original: The minimization w.r.t.
Revised: The minimization w.r.t.

Original: ˆ y yields
Revised: ˆ y yields

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: In turn, as a function of p i the loss being a convex combination, its minimization yields
Revised: In turn, as a function of p i the loss being a convex combination, its minimization yields

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Combining these equations (16,17,18) we get
Revised: Combining these equations (16,17,18) we get

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof.
Revised: which concludes the proof.

Original: A PPENDIX C S TABILITY ANALYSIS
Revised: A PPENDIX C S TABILITY ANALYSIS

Original: The analysis is restricted for simplicity to the case α ij = αδ ij .
Revised: The analysis is restricted for simplicity to the case α ij = αδ ij .

Original: The log likelihood as a function of r = σ 2 /σ 20 and β = α/r after inserting the optimal q = q ( x, y ) reads in that case
Revised: The log likelihood as a function of r = σ 2 /σ 20 and β = α/r after inserting the optimal q = q ( x, y ) reads in that case

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: and where λ ( x ) is a Lagrange multiplier which has been added to impose the normalization of ˆ p
Revised: and where λ ( x ) is a Lagrange multiplier which has been added to impose the normalization of ˆ p

Original: [Equation]
Revised: [Equation]

Original: The gradient reads
Revised: The gradient reads

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: This leads to the following relation at the saddle point:
Revised: This leads to the following relation at the saddle point:

Original: [Equation]
Revised: [Equation]

Original: Let us now compute the Hessian.
Revised: Let us now compute the Hessian.

Original: It is easy to see that the block corresponding to the predictors ˆ y decouples from the rest as soon as these predictors are centered.
Revised: It is easy to see that the block corresponding to the predictors ˆ y decouples from the rest as soon as these predictors are centered.

Original: Denoting
Revised: Denoting

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: i [
Revised: i [

Original: [Equation]
Revised: [Equation]

Original: There are two blocks in this Hessian, the one corresponding to r and β and the one corresponding to derivatives with respect to ˆ p
Revised: There are two blocks in this Hessian, the one corresponding to r and β and the one corresponding to derivatives with respect to ˆ p

Original: i .
Revised: i .

Original: The stability of the ﬁrst one depends on the sign of C 2 [ q ] − 2 C 21 [ q ] for |T | large while the second block is always stable as being an average of the exterior product of the vector ( q 1 ( x, y ) /
Revised: The stability of the ﬁrst one depends on the sign of C 2 [ q ] − 2 C 21 [ q ] for |T | large while the second block is always stable as being an average of the exterior product of the vector ( q 1 ( x, y ) /

Original: ˆ p 1 ( x ) , . . . , q |T | ( x, y ) /
Revised: ˆ p 1 ( x ) , . . . , q |T | ( x, y ) /

Original: ˆ p |T | ( x )) by itself.
Revised: ˆ p |T | ( x )) by itself.

Original: At the degenerate point α = 0 , r = 1 , ˆ p
Revised: At the degenerate point α = 0 , r = 1 , ˆ p

Original: i
Revised: i

Original: = 1 / |T | the Hessian simpliﬁes as follows.
Revised: = 1 / |T | the Hessian simpliﬁes as follows.

Original: Denote
Revised: Denote

Original: [Equation]
Revised: [Equation]

Original: ∈T d
Revised: ∈T d

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: e
Revised: e

Original: [Equation]
Revised: [Equation]

Original: 2 ( x ) a given vector of perturbations, decomposed onto a set of unit tangent vectors, { e 1 and e 2 } being respectively associated to r and β , while e i ( x ) associated to ˆ p
Revised: 2 ( x ) a given vector of perturbations, decomposed onto a set of unit tangent vectors, { e 1 and e 2 } being respectively associated to r and β , while e i ( x ) associated to ˆ p

Original: i ( x ) for all i ∈ T and x ∈ X .
Revised: i ( x ) for all i ∈ T and x ∈ X .

Original: Denote u = (cid:88)
Revised: Denote u = (cid:88)

Original: i ∈T
Revised: i ∈T

Original: [Equation]
Revised: [Equation]

Original: v
Revised: v

Original: [Equation]
Revised: [Equation]

Original: = (cid:88)
Revised: = (cid:88)

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: e i
Revised: e i

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Original: With these notations the Hessian reads:
Revised: With these notations the Hessian reads:

Original: [Equation]
Revised: [Equation]

Original: In fact we are interested in the eigenvalues of H in the subspace of deformations which conserve the norm of ˆ p , i.e. orthogonal to v ( x ) , thereby given by
Revised: In fact we are interested in the eigenvalues of H in the subspace of deformations which conserve the norm of ˆ p , i.e. orthogonal to v ( x ) , thereby given by

Original: [Equation]
Revised: [Equation]

Original: In this subspace the Hessian reads
Revised: In this subspace the Hessian reads

Original: [Equation]
Revised: [Equation]

Original: where M is the number of data points, resulting from the fact that (cid:88)
Revised: where M is the number of data points, resulting from the fact that (cid:88)

Original: [Equation]
Revised: [Equation]

Original: because E data ( ·| x ) as a function of x is actually a point-wise function on the data.
Revised: because E data ( ·| x ) as a function of x is actually a point-wise function on the data.

Original: If | u | 2 > 0 or if | u | = 0 and 1 + |T | ( C 2 / 2 − 1) > 0 there is at least one positive eigenvalue.
Revised: If | u | 2 > 0 or if | u | = 0 and 1 + |T | ( C 2 / 2 − 1) > 0 there is at least one positive eigenvalue.

Original: Let Λ be such an eigenvalue.
Revised: Let Λ be such an eigenvalue.

Original: After eliminating dr and dβ from the eigenvalue equations in dη , the deformation along this mode veriﬁes
Revised: After eliminating dr and dβ from the eigenvalue equations in dη , the deformation along this mode veriﬁes

Original: [Equation]
Revised: [Equation]

Original: which corresponds to increasing r and α while decreasing for each x the ˆ p i having the highest mean relative error
Revised: which corresponds to increasing r and α while decreasing for each x the ˆ p i having the highest mean relative error

Original: u i [ x ] .
Revised: u i [ x ] .

Original: Concerning solutions for which
Revised: Concerning solutions for which

Original: [Equation]
Revised: [Equation]

Original: is concentrated on some index ˆ I ( x ) , the analysis is more complex.
Revised: is concentrated on some index ˆ I ( x ) , the analysis is more complex.

Original: In that case C 2 [ p ] = 0 and
Revised: In that case C 2 [ p ] = 0 and

Original: C 1 [ p ] > 0 .
Revised: C 1 [ p ] > 0 .

Original: The ( r, β ) sector has 2 negative eigenvalues, while the ˆ p block is ( − ) a covariance matrix, so it has as well negative eigenvalues.
Revised: The ( r, β ) sector has 2 negative eigenvalues, while the ˆ p block is ( − ) a covariance matrix, so it has as well negative eigenvalues.

Original: The coupling between these two blocks could however in principle generate in some cases some instabilities.
Revised: The coupling between these two blocks could however in principle generate in some cases some instabilities.

Original: Still, the log likelihood of such solutions reads
Revised: Still, the log likelihood of such solutions reads

Original: [Equation]
Revised: [Equation]

Original: so we get the following optimal solution
Revised: so we get the following optimal solution

Original: [Equation]
Revised: [Equation]

Original: A PPENDIX D E XPERIMENTS : A DDITIONAL D ETAILS
Revised: A PPENDIX D E XPERIMENTS : A DDITIONAL D ETAILS

Original: Here we provide some additional details and context to the experimental validation of the DTLR methodology described in section 5.
Revised: Here we provide some additional details and context to the experimental validation of the DTLR methodology described in section 5.

Original: Table 2 provides some information about the datasets used in the synthetic and solar wind prediction problems 8 .
Revised: Table 2 provides some information about the datasets used in the synthetic and solar wind prediction problems 8 .

Original: Sections D.1.1 and D.1.2 give additional plots for evaluating the experimental results.
Revised: Sections D.1.1 and D.1.2 give additional plots for evaluating the experimental results.

Original: For the solar wind prediction task, the solar wind data was mapped into standardized Gaussian space using a quantile-quantile and inverse probit mapping.
Revised: For the solar wind prediction task, the solar wind data was mapped into standardized Gaussian space using a quantile-quantile and inverse probit mapping.

Original: Nine fold cross-validation was performed using splits as speciﬁed in table 3.
Revised: Nine fold cross-validation was performed using splits as speciﬁed in table 3.

Original: To compare the DTLR results with the state of the art solar wind forecasting, we used results from Reiss et al.
Revised: To compare the DTLR results with the state of the art solar wind forecasting, we used results from Reiss et al.

Original: (2019, Table 1).
Revised: (2019, Table 1).

Original: Since Reiss et al.
Revised: Since Reiss et al.

Original: compared the various forecasting methods on only one solar rotation (ﬁrst row of table 3), comparing these results with DTLR can be considered as a preliminary examination.
Revised: compared the various forecasting methods on only one solar rotation (ﬁrst row of table 3), comparing these results with DTLR can be considered as a preliminary examination.

Original: Nevertheless, the results presented in table 1a show encouraging signs for the competitiveness and usefulness of the DTLR method.
Revised: Nevertheless, the results presented in table 1a show encouraging signs for the competitiveness and usefulness of the DTLR method.

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: D. S UPPLEMENTARY P LOTS D.1. S YNTHETIC P ROBLEMS
Revised: D. S UPPLEMENTARY P LOTS D.1. S YNTHETIC P ROBLEMS

Original: [Figure]
Revised: [Figure]

Original: i
Revised: i

Original: [Figure]
Revised: [Figure]

Original: O u t
Revised: O u t

Original: [Figure]
Revised: [Figure]

Original: t
Revised: t

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: i t y
Revised: i t y

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: D.1. S OLAR W IND P REDICTION
Revised: D.1. S OLAR W IND P REDICTION

Original: lllllllllllllllllllllllllllllllllllll lll llllll llllll lll llllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l l l ll l lllll l l llll lll l ll l llllllll l lll ll l l llll l ll lllllllll lll ll l l llll l llllll llllllllllllllllll llllll l l l lllllllllllllll ll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l lll l lllll l lll llllll l llllllllll l l llll lll lllllll l ll l l lllll llllllllllllllll ll llllllllllllllllllllllllll lllll lllllllllllll lllllllllllllllll lllllllllllllllllllllllllllllllll llllllllllllll l lllllllllllll l l llllll l l l l lllllllllll
Revised: lllllllllllllllllllllllllllllllllllll lll llllll llllll lll llllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l l l ll l lllll l l llll lll l ll l llllllll l lll ll l l llll l ll lllllllll lll ll l l llll l llllll llllllllllllllllll llllll l l l lllllllllllllll ll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l lll l lllll l lll llllll l llllllllll l l llll lll lllllll l ll l l lllll llllllllllllllll ll llllllllllllllllllllllllll lllll lllllllllllll lllllllllllllllll lllllllllllllllllllllllllllllllll llllllllllllll l lllllllllllll l l llllll l l l l lllllllllll

Original: l l l l l l l l l lllllll llllllllll l l
Revised: l l l l l l l l l lllllll llllllllll l l

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: l l l ll llllllll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Revised: l l l ll llllllll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

Original: ll lllllllllllllllllllllllllll llllllllllll l l l l l llll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll l l l lllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Revised: ll lllllllllllllllllllllllllll llllllllllll l l l l l llll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll l l l lllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

Original: [Figure]
Revised: [Figure]

Original: lllllllll lllllll llllllll lllllllllllllll l lllllllll lll l l llll l
Revised: lllllllll lllllll llllllll lllllllllllllll l lllllllll lll l l llll l

Original: [Figure]
Revised: [Figure]

Original: ll lllll l l l l llll l l l l lll lll
Revised: ll lllll l l l l llll l l l l lll lll

Original: lll l l ll lllllllllllll llll l l llll l l l l lllll llll lllllll llllllllllllllllllllllllllllllllllllllllllllllllll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll l lllll ll l l l ll lll l l l llllllll l l l l l l l l llllll l llllllllllllllllllllllllllllllllllllllllllllllll llllllll llllllllllll ll ll lll l lllllllllllllllllllllllllllllllllll llllllllll llllll llllll llllllll llllllllllllllllllllllllllllllllll
Revised: lll l l ll lllllllllllll llll l l llll l l l l lllll llll lllllll llllllllllllllllllllllllllllllllllllllllllllllllll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll l lllll ll l l l ll lll l l l llllllll l l l l l l l l llllll l llllllllllllllllllllllllllllllllllllllllllllllll llllllll llllllllllll ll ll lll l lllllllllllllllllllllllllllllllllll llllllllll llllll llllll llllllll llllllllllllllllllllllllllllllllll

Original: [Figure]
Revised: [Figure]

Original: A PPENDIX E N EURAL N ETWORK A RCHITECTURE D ETAILS
Revised: A PPENDIX E N EURAL N ETWORK A RCHITECTURE D ETAILS

Original: Problem
Revised: Problem

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Revised: (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)

Original: [Figure]
Revised: [Figure]