Original: Causality Preserving Chaotic Transformation and Classification using Neurochaos Learning
Revised: Causality Preserving Chaotic Transformation and Classification using Neurochaos Learning

Original: Abstract
Revised: Abstract

Original: Discovering cause-effect from observational data is an important but challengingproblem in science and engineering.
Revised: Discovering cause-effect from observational data is an important but challengingproblem in science and engineering.

Original: In this work, a recently proposed brain inspiredlearning algorithm namely-
Revised: In this work, a recently proposed braininspired learning algorithm namely-

Original: Neurochaos Learning (NL) is used for the classificationof cause-effect from coupled autoregressive processes, coupled 1D chaotic skewtent maps, coupled 1D chaotic logistic maps and a real-world prey-predator system.
Revised: Neurochaos Learning (NL) is used for theclassification of cause-effect from coupled autoregressive processes, coupled 1Dchaotic skew tent maps, coupled 1D chaotic logistic maps and a real-world preypredator system.

Original: In the case of coupled skew tent maps, the proposed method consistently outperforms a five layer Deep Neural Network  architecture for coupling coefficient valuesranging from 0 .
Revised: In the case of coupled skew tent maps, the proposed methodconsistently outperforms a five layer Deep Neural Network (DNN) and Long Short Term Memory (LSTM) architecture for coupling coefficient values ranging from 0 .

Original: 1 to 0 .
Revised: 1to 0 .

Original: Further, we investigate the preservation of causality in thefeature extracted space of NL using Granger Causality for coupled autoregressiveprocesses and Compression-Complexity Causality for coupled chaotic systems andreal-world prey-predator dataset.
Revised: Further, we investigate the preservation of causality in the feature extractedspace of NL using Granger Causality for coupled autoregressive processes and Compression-Complexity Causality for coupled chaotic systems and real-worldprey-predator dataset.

Original: It is found that NL preserves causality under achaotic transformation and can successfully classify cause and effect time-series.
Revised: Unlike DNN, LSTM and 1D Convolutional Neural Network,it is found that NL preserves the inherent causal structures present in the inputtimeseries data.

Original: This finding is promising for the theory and applications of causal machine learningand opens up the possibility to explore the potential of NL for more sophisticatedcausal learning tasks.
Revised: This finding is promising for the theory and applications of causalmachine learning and opens up the possibility to explore the potential of NL formore sophisticated causal learning tasks.

Original:  Introduction
Revised:  Introduction

Original: Despite the success of Machine Learning (ML) and Deep Learning (DL) algorithms in the field ofnatural language processing [1], computer vision [2], speech recognition [3], these algorithms facedifficulty in interpretability and trustworthiness.
Revised: Despite the success of Machine Learning (ML) and Deep Learning (DL) algorithms in the field ofnatural language processing [1], computer vision [2], speech recognition [3], these algorithms facedifficulty in interpretability and trustworthiness.

Original: One of the main reasons for this is the fact that theyare merely discovering associations between ‘input’ and ‘output’ in the name of ‘learning’.
Revised: One of the main reasons for this is the fact that theyare merely discovering associations between ‘input’ and ‘output’ in the name of ‘learning’.

Original: However,discovering associations alone is insufficient as an explanation to aid in the decision making process.
Revised: However,discovering associations alone is insufficient as an explanation to aid in the decision making process.

Original: Decision-making in everyday human existence relies heavily on reasoning and causal inference.
Revised: Decision-making in everyday human existence relies heavily on reasoning and causal inference.

Original: Hence, incorporation of causal ability, which goes over and beyond associations (or correlations),has become an important objective in current machine learning research [4].
Revised: Hence, incorporation of causal ability, which goes over and beyond associations (or correlations),has become an important objective in current machine learning research [4].

Original: For time-series data, causality has been mathematically defined by Wiener [5].
Revised: For time-series data, causality has been mathematically defined by Wiener [5].

Revised: According to Wiener,“a time series X causes a time series Y , if the past values of X contain information that help predict Yabove and beyond the information contained in the past values of Y alone".

Original: This mathematicaldefinition was later realized through a number of mathematical/algorithmic formulations.
Revised: This definition was laterrealized through a number of mathematical/algorithmic formulations.

Original: Granger Causality (GC) [6] was the pioneering causality estimation technique developed for time-seriesdata.
Revised: Granger Causality (GC) [6]was the pioneering causality estimation technique developed for time-series data.

Original: GC has been followed up with a number of extensions and inspirations, some of which are,information-theoretic causality as defined by Transfer Entropy [7] and data-compression basedcausality, as defined by Compression-Complexity Causality [8].
Revised: GC has beenfollowed up with a number of extensions and inspirations, some of which are, information-theoreticcausality as defined by Transfer Entropy [7] and data-compression based causality, as defined by Compression-Complexity Causality [8].

Original: While GC has primarily beenformulated for linear time-series, the latter are valid for non-linear applications.
Revised: While GC has primarily been formulated for linear timeseries, the latter are valid for non-linear applications.

Original: These methods havebeen successfully employed in econometrics [9, 10], climatology [11, 12], neuroscience [13, 14] etc.
Revised: These methods have been successfully employedin econometrics [9, 10], climatology [11, 12], neuroscience [13, 14] etc.

Original: Conventional ML algorithms have not had much success in causal learning or causality basedclassification from time-series data [15].
Revised: Conventional ML algorithms have not had much success in causal learning or causality basedclassification from time-series data [15].

Original: They are either employed in combination with an existingtime-series based causality detection technique [16, 17] or specialized methods assuming underlying causal models have been developed [18, 19, 20, 21], in order to incorporate causality learning abilitiesin ML.
Revised: They are either employed in combination with an existingtime-series based causality detection technique [16, 17] or specialized methods assuming underlyingcausal models have been developed [18, 19, 20, 21], in order to incorporate causality learning abilitiesin ML.

Original: In this work, we focus on causality detection from time-series data  and use a recently proposedbrain inspired learning algorithm namely Neurochaos Learning [22] (NL) to learn generalized causalpatterns from time series data.
Revised: In this work, we focus on causality detection from time-series data (without assumption of anycausal model) and use a recently proposed brain inspired learning algorithm namely Neurochaos Learning [22] (NL) to learn generalized causal patterns from time series data.

Original: NL draws its inspiration from the chaotic firing of neurons in thebrain [23].
Revised: NL draws its inspirationfrom the chaotic firing of neurons in the brain [23].

Original: NL is a rival architecture to ANNs and has shown promise in classification tasks (especiallyin the low training sample regime), many-a-times outperforming state-of-the-art methods.
Revised: NL is a rival architecture to ANNs and hasshown promise in classification tasks (especially in the low training sample regime), many-a-timesoutperforming state-of-the-art methods.

Original: NL mapsthe input data into a high dimensional space which enables efficient classification.
Revised: NL maps the input data into a high dimensional space whichenables efficient classification.

Original: In this sense, NL shows similarity to Reservoir Computing [24] which also employs nonlinear mappings of inputdata.
Revised: In this sense, NL shows similarity to Reservoir Computing [24]which also employs nonlinear mappings of input data.

Original: However, NL is fundamentally different from Reservoir Computing in terms of motivation,methodology, and working.
Revised: However, NL is fundamentally differentfrom Reservoir Computing in terms of motivation, methodology, and working.

Original: NL fundamentally uses the Topological Transitivity property of chaos [25]and Stochastic Resonance [22] for learning classification tasks.
Revised: NL fundamentallyuses the Topological Transitivity property of chaos [25] and Stochastic Resonance [22] for learningclassification tasks.

Original: Given that NL is still at a nascentstage of development, it is not well known in the AI community and most of its properties are largelyunexplored.
Revised: Given that NL is still at a nascent stage of development, it is not well known inthe AI community and most of its properties are largely unexplored.

Original: Hence, the objectives of this present study are to investigate the following:
Revised: Hence, the objectives of thispresent study are to investigate the following:

Original: O1 :
Revised: O1 :

Original: The efficacy of NL in cause-effect classification and compare the same with Deep  Neural Network (DNN).
Revised: The efficacy of NL in cause-effect classification and compare the same with Deep NeuralNetwork (DNN), 1D Convolutional Neural Network (1D CNN), and Long Short Term

Original: O2 : Does success in cause-effect classification imply preservation of causality? O3 : Can NL use a transfer learning framework for cause-effect classification?
Revised: Memory (LSTM) .O2 : Does success in cause-effect classification imply preservation of causality?O3 : Can NL use a transfer learning framework for cause-effect classification?

Original: We do not build a novel causal ML algorithm, but rather explore if the existing NL architecturewhich does classification, has the capability to classify based on ‘causal’ information in the data.
Revised: We do not build a novel causal ML algorithm, but rather explore if the existing NL architecture whichdoes classification, has the capability to classify based on ‘causal’ information in the data.

Original: As a first step, we check its ability to distinguish between cause and effect time-series data.
Revised: As a firststep, we check its ability to distinguish between cause and effect time-series data.

Revised: To elaborate, theproblem of bivariate causality detection is considered, where the algorithm is trained to classify the‘causal’ or ‘driver’ time-series variable as the cause and the time-series variable that is ‘affected’ or‘driven’ by the former as the effect .

Original: Wedo not use any existing causality estimation method as an aid to the NL algorithm for this purpose.
Revised: We do not use any existing causality estimation method as an aidto the NL algorithm for this purpose.

Original: Further, for O2 , we check whether features extracted from the learning architecture (in the abovecause-effect classification task), preserve causality as measured by an existing time-series causalityestimation method.
Revised: Further, for O2 , we check whether features extracted from thelearning architecture (in the above cause-effect classification task), preserve causality as measured byan existing time-series causality estimation method.

Original: This is important because it determines whether NL is doing a causality informedclassification or not.
Revised: This is important because it determines whether NL is doing a causality informed classification or not.

Revised: The motivation behind many specializedcausal learning algorithms that have been recently proposed is generalized learning as failure in thecase of distribution shifts continues to be one of the most important limitations of traditional MLalgorithms [26, 27].

Revised: Hence, O3 becomes an important objective to be looked at for an algorithmattempting to learn causal representations.

Original: We find that a general NL architectureslightly outperforms a general DL architecture (five layer DNN) in cause-effect classification.
Revised: We find that a general NL architecture outperforms a five layer DNN, and LSTM architecture incause-effect classification.

Original: Further, the features extracted using NL are found to preservethe cause-effect relationship present in input data.
Revised: Further, the features extracted using NL are found to preserve the causeeffect relationship present in input data.

Original: This, however, was not the case for DNN, probablybecause the classification results were not causally informed.
Revised: This, however, was not the case for DNN, 1D CNN and LSTM, probably because the classification results were not causally informed.

Revised: The performancecomparison of NL with 1D CNN and LSTM is extensively studied in the supplementary material.

Original: The findings demonstrate that even ageneral NL architecture is capable of some basic causal learning and hence promising for developingmore sophisticated causal ML algorithms required for different tasks.
Revised: The findings demonstrate that even a general NL architecture is capable of some basic causal learningand hence promising for developing more sophisticated causal ML algorithms required for differenttasks.

Original: The sections in the paper are arranged as follows:
Revised: The sections in the paper are arranged as follows:

Original: Section 2 describes the method used to do thecause-effect classification.
Revised: Section 2 describes the method used to do thecause-effect classification.

Original: Section 3 provides details of the simulated data used to carry out theexperiments.
Revised: Section 3 provides details of the simulated data used to carry out theexperiments.

Original: Section 4 deals with experiments, results and discussions on simulated and real worldprey-predator dataset.
Revised: Section 4 deals with experiments, results and discussions on simulated and real worldprey-predator dataset.

Original: Section 5 addresses the limitations and scope for future work.
Revised: Section 5 addresses the limitations and scope for future work.

Original: The concludingremarks are provided in Section 6.
Revised: The concludingremarks are provided in Section 6.

Original:  Neurochaos Learning
Revised:  Neurochaos Learning

Original: Neurochaos Learning (NL) is a novel brain inspired neuronal learning algorithm that has been recentlyproposed.
Revised: Neurochaos Learning (NL) is a novel brain inspired neuronal learning algorithm that has been recentlyproposed.

Original: The authors in [25, 26] claim that NL is inspired from the chaotic firings of neurons in thebrain and has mainly two architectures: (a) ChaosNet [26], (b) ChaosFEX+ML.
Revised: The authors in [25, 28] claim that NL is inspired from the chaotic firings of neurons in the brain and has mainly two architectures: (a) ChaosNet [28], (b) ChaosFEX+ML.

Original: In [27], the authorsemploy ChaosNet for continual learning.
Revised: In [29], the authorsemploy ChaosNet for continual learning.

Original: In another work [28], the authors propose deep ChaosNetfor action recognition in videos.
Revised: In another work [30], the authors propose deep ChaosNetfor action recognition in videos.

Original: Inspired by these recent developments, in this work, we employ ChaosNet architecture for theclassification of cause-effect from observational data.
Revised: Inspired by these recent developments, in this work, we employ ChaosNet architecture for theclassification of cause-effect from observational data.

Original: The architecture consists of an input layer of
Revised: The architecture consists of an input layer of

Original: Generalized Lüroth Series (GLS) neurons which are one-dimensional (1D) chaotic skew tent mapsdescribed as follows:
Revised: Generalized Lüroth Series (GLS) neurons which are one-dimensional (1D) chaotic skew tent mapsdescribed as follows:

Original: [Equation]
Revised: [Equation]

Original: where the skewness of the map is controlled by the parameter b ( 0 < b < 1 ).
Revised: where the skewness of the map is controlled by the parameter b ( 0 < b < 1 ).

Original: Upon arrival of the inputdata/ stimulus ( x k ) , the chaotic GLS neurons in the input layer starts firing (from the initial value q ) until the chaotic neural trace of the neurons reaches the ϵ neighbourhood of the correspondingstimulus.
Revised: Upon arrival of the inputdata/ stimulus ( x k ) , the chaotic GLS neurons in the input layer starts firing (from the initial valueq ) until the chaotic neural trace of the neurons reaches the ϵ neighbourhood of the correspondingstimulus.

Original: From the neural trace thus generated, the following features are extracted:
Revised: From the neural trace thus generated, the following features are extracted:

Original: Firing time ( N ) :
Revised: Firing time ( N ) :

Original: The amount of time the chaotic neural trace takes to recognise the input
Revised: The amount of time the chaotic neural trace takes to recognise the input

Original: stimulus.
Revised: stimulus.2.

Original: Firing rate ( R ) :
Revised: Firing rate ( R ) :

Original: Fraction of time the chaotic neural trace is above the discriminationthreshold b so as to recognize the stimulus.
Revised: Fraction of time the chaotic neural trace is above the discriminationthreshold b so as to recognize the stimulus.3.

Original: Energy ( E ) :
Revised: Energy ( E ) :

Original: For the chaotic neural trace y ( t ) with firing time N , energy is defined as:
Revised: For the chaotic neural trace y ( t ) with firing time N , energy is defined as:

Original: [Equation]
Revised: [Equation]

Original: Entropy ( H ) :
Revised: Entropy ( H ) :

Original: For the chaotic neural trace y ( t ) , we first compute the binary symbolic
Revised: For the chaotic neural trace y ( t ) , we first compute the binary symbolic

Original: sequence Sym ( t ) as follows:
Revised: sequence Sym ( t ) as follows

Original: [Equation]
Revised: [Equation]

Original: where i = 1 to N (firing time)
Revised: where i = 1 to N (firing time)

Original: We then compute Shannon Entropy of Sym ( t ) as follows:
Revised: We then compute Shannon Entropy of Sym ( t ) as follows:

Original: [Equation]
Revised: [Equation]

Original: where p 1 and p 2 refers to the probabilities of the symbols 0 and 1 occurring in Sym ( t )respectively.
Revised: where p 1 and p 2 refers to the probabilities of the symbols 0 and 1 occurring in Sym ( t )respectively.

Original: For each input value s (stimulus) of a data instance of class k is mapped to a 4D vector[ N s , R s , E s , H s ] .
Revised: For each input value s (stimulus) of a data instance of class k is mapped to a 4D vector[ N s , R s , E s , H s ] .

Original: The collection of these 4D vectors forms the ChaosFEX feature space.
Revised: The collection of these 4D vectors forms the ChaosFEX feature space.

Original: If theinput data is 1D with Z classes, the mean representation vector of the k -th class is given by
Revised: If theinput data is 1D with Z classes, the mean representation vector of the k -th class is given by1 m [ (cid:80) mi

Original: [Equation] i , (cid:80) [Equation] (cid:80) 
Revised: =1 N i , (cid:80) mi =1 R i , (cid:80) mi

Original: [Equation]

Original: =1 E i , (cid:80) [Equation] =1 H i ] .
Revised: =1 E i , (cid:80) mi =1 H i ] .

Original: In the case of ChaosNet , the classifier computes the cosine similarity of the ChaosFEX featuresextracted from the test sample with the pre-computed mean representation vectors (consisting ofmean values of ChaosFEX features for each stimuli) of each class.
Revised: In the case of ChaosNet , the classifier computes the cosine similarity of the ChaosFEX featuresextracted from the test sample with the pre-computed mean representation vectors (consisting ofmean values of ChaosFEX features for each stimuli) of each class.

Original: The predicted class is assignedthe label corresponding to the maximum cosine similarity.
Revised: The predicted class is assignedthe label corresponding to the maximum cosine similarity.

Original: A detailed explanation of the ChaosNetarchitecture and its working is provided in [25].
Revised: A detailed explanation of the ChaosNetarchitecture and its working is provided in [25].

Original:  Datasets
Revised:  Datasets

Original: To evaluate the efficacy of ChaosNet and deep learning for the classification of cause-effect, we usedsimulated datasets from (a) Coupled autoregressive (AR) processes, (b) Coupled 1D chaotic maps inmaster-slave configuration (1D skew tent maps and the 1D logistic maps) and real-world dataset froma (c) prey-predator system.
Revised: To evaluate the efficacy of ChaosNet and deep learning for the classification of cause-effect, we usedsimulated datasets from (a) Coupled autoregressive (AR) processes, (b) Coupled 1D chaotic maps inmaster-slave configuration (1D skew tent maps and the 1D logistic maps) and real-world dataset froma (c) prey-predator system.

Original: 3.1 Coupled AR processes
Revised: 3.1 Coupled AR processes

Original: The governing equations for the coupled AR processes are the following:
Revised: The governing equations for the coupled AR processes are the following:

Original: [Equation]
Revised: [Equation]

Original: where M ( t ) and S ( t ) are the independent and the dependent (or the cause and effect) time seriesrespectively;
Revised: where M ( t ) and S ( t ) are the independent and the dependent (or the cause and effect) time seriesrespectively;

Original: a 1 = 0 . 8 , a 2 = 0 .
Revised: a 1 = 0 . 8 , a 2 = 0 .

Original: 9 , the noise intensity γ = 0 .
Revised: 9 , the noise intensity γ = 0 .

Original: 03 and r ( t ) is the i.i.d additive gaussiannoise drawn from a standard normal distribution.
Revised: 03 and r ( t ) is the i.i.d additive gaussiannoise drawn from a standard normal distribution.

Original: The coupling-coefficient η is varied from 0 to 1in steps of 0 .
Revised: The coupling-coefficient η is varied from 0 to 1in steps of 0 .

Original: We generated 1000 independent random trials for each value of η .
Revised: We generated 1000 independent random trials for each value of η .

Original: Each of the datainstances are of length 2000 , after removing the initial 500 samples (transients) from the time series.
Revised: Each of the datainstances are of length 2000 , after removing the initial 500 samples (transients) from the time series.

Original: 3.2 Coupled 1D Chaotic maps in Master-Slave configuration
Revised: 3.2 Coupled 1D Chaotic maps in Master-Slave configuration

Original: 3.2.1 Coupled Skew-tent maps
Revised: 3.2.1 Coupled Skew-tent maps

Original: The governing equations used to generate the master and slave time series for the coupled 1Dskew-tent maps are the following:
Revised: The governing equations used to generate the master and slave time series for the coupled 1Dskew-tent maps are the following:

Original: [Equation]
Revised: [Equation]

Original: where M ( n ) is the master (cause) and S ( n ) is the slave (effect) system.
Revised: where M ( n ) is the master (cause) and S ( n ) is the slave (effect) system.

Original: M ( n ) influences thedynamics of the slave system (equation 8).
Revised: M ( n ) influences thedynamics of the slave system (equation 8).

Original: The coupling coefficient is given by η is varied from 0to 0 .
Revised: The coupling coefficient is given by η is varied from 0to 0 .

Original: 9 with a step size of 0 . 1 .
Revised: 9 with a step size of 0 . 1 .

Original: T 1 ( n ) , and T 2 ( n ) are skew tent maps with skewness b 1 = 0 .
Revised: T 1 ( n ) , and T 2 ( n ) are skew tent maps with skewness b 1 = 0 .

Original: 65 , andb 2 = 0 .
Revised: 65 , andb 2 = 0 .

Original: 47 respectively.
Revised: 47 respectively.

Original: The initial values are chosen randomly for the master-slave system in theinterval (0 , 1) .
Revised: The initial values are chosen randomly for the master-slave system in theinterval (0 , 1) .

Original: We generated 1000 independent random trials for each value of η .
Revised: We generated 1000 independent random trials for each value of η .

Original: Each of the datainstances are of length 2000 , after removing the initial 500 samples (transients) from the time series.
Revised: Each of the datainstances are of length 2000 , after removing the initial 500 samples (transients) from the time series.

Original: 3.2.2 Coupled Logistic maps
Revised: 3.2.2 Coupled Logistic maps

Original: The 1D Logistic map is a widely used model to study population dynamics [29].
Revised: The 1D Logistic map is a widely used model to study population dynamics [31].

Original: The governingdynamics for coupled logistic maps in master-slave configuration is given by:
Revised: The governingdynamics for coupled logistic maps in master-slave configuration is given by:

Original: [Equation]
Revised: [Equation]

Original: The coupling coefficient η is varied from 0 to 0 .
Revised: The coupling coefficient η is varied from 0 to 0 .

Original: L 1 ( n ) = 4 · L 1 ( n − 1)(1 − L 1 ( n − 1)) , and
Revised: L 1 ( n ) = 4 · L 1 ( n − 1)(1 − L 1 ( n − 1)) , and

Original: L 2 ( n ) = 3 .
Revised: L 2 ( n ) = 3 .

Original: · L 2 ( n − 1)(1 − L 2 ( n − 1)) .
Revised: · L 2 ( n − 1)(1 − L 2 ( n − 1)) .

Original: The attractor for this coupled dynamical system isprovided in Figure 5b.
Revised: The attractor for this coupled dynamical system isprovided in Figure 5b.

Original: For both systems, 1000 data instances ( M ( n ) , S ( n ) ) are generated and grouped as class-0 ( M ( n ) :
Revised: For both systems, 1000 data instances ( M ( n ) , S ( n ) ) are generated and grouped as class-0 ( M ( n ) :

Original: Cause) and class-1 ( S ( n ) : Effect) respectively.
Revised: Cause) and class-1 ( S ( n ) : Effect) respectively.

Original: Table 1 gives details of the train-test split for theclassification task.
Revised: Table 1 gives details of the train-test split for theclassification task.

Original: [Table]
Revised: [Table]

Original:  Experiments, Results and Discussions
Revised:  Experiments, Results and Discussions

Original: In this section, we begin with a description of hyperparameter tuning for NL and DL followed bya demonstration of causality preservation by ChaosFEX for coupled AR processes (and the failureof DL).
Revised: In this section, we begin with a description of hyperparameter tuning for NL and DL followed by ademonstration of causality preservation by ChaosFEX for coupled AR processes, skew tent mapsand logistic maps (and the failure of DL).

Original: Subsequently, macro F1-scores for ChaosNet and DL for the cause-effect classification foreach η are plotted.
Revised: Also, macro F1-scores for ChaosNet and DL for thecause-effect classification for each η are plotted.

Original: For all results in this paper, software implementation is performed using Python3, scikit-learn [30], keras [31], ChaosFEX toolbox [22], Multivariate Granger Causality (MVGC)toolbox [32], CCC toolbox [8] and MATLAB.
Revised: For all results in this paper, software implementationis performed using Python 3, scikit-learn [32], keras [33], ChaosFEX toolbox [22], Multivariate Granger Causality (MVGC) toolbox [34], CCC toolbox [8] and MATLAB.

Revised: Comparison of NL with1D CNN and LSTM is extensively studied in the supplementary material.

Original: 4.1 Hyperparameter tuning for NL
Revised: 4.1 Hyperparameter tuning for NL

Original: Every ML algorithm has a set of hyperparameters that needs to be tuned for efficient performance.
Revised: Every ML algorithm has a set of hyperparameters that needs to be tuned for efficient performance.

Original: In the case of ChaosNet , there are three hyperparameters - initial neural activity ( q ), discriminationthreshold ( b ), and noise intensity ( ϵ ) [22].
Revised: In the case of ChaosNet , there are three hyperparameters - initial neural activity ( q ), discriminationthreshold ( b ), and noise intensity ( ϵ ) [22].

Original: The hyperparameter tuning is done only once with thetraindata corresponding to η = 0 .
Revised: The hyperparameter tuning is done only once with thetraindata corresponding to η = 0 .

Original: 4 (Table 1) separately for the coupled AR processes and coupledskew tent maps.
Revised: 4 (Table 1) separately for the coupled AR processes and coupledskew tent maps.

Original: For a fixed value of b = 0 .
Revised: For a fixed value of b = 0 .

Original: 499 , and ϵ = 0 .
Revised: 499 , and ϵ = 0 .

Original: 171 , q was varied from 0 .
Revised: 171 , q was varied from 0 .

Original: 01 to 0 .
Revised: 01 to 0 .

Original: 98 with a stepsize
Revised: 98 with a stepsize

Original: of0 .
Revised: of0 .

Original: 01 for both coupled AR processes and coupled chaotic skew tent maps.
Revised: 01 for both coupled AR processes and coupled chaotic skew tent maps.

Original: In the case of coupled ARprocesses, a maximum average macro F1-score = 0 .
Revised: In the case of coupled ARprocesses, a maximum average macro F1-score = 0 .

Original: 605 is obtained for q = 0 .
Revised: 605 is obtained for q = 0 .

Original: In the case of coupled skew tent maps, a maximum average macro F1-score = 1 . 0 was obtained for the followingvalues of q = [0.16, 0.26, 0.27, 0.28, 0.29, 0.30, 0.31, 0.32, 0.34, 0.36, 0.37, 0.38, 0.48, 0.51, 0.52,0.56, 0.57, 0.72, 0.76, 0.77, 0.78, 0.79, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.91, 0.92, 0.93,0.94, 0.95, 0.96, 0.98] in a five-fold cross validation using traindata.
Revised: In the case ofcoupled skew tent maps, a maximum average macro F1-score = 1 . 0 was obtained for the followingvalues of q = [0.16, 0.26, 0.27, 0.28, 0.29, 0.30, 0.31, 0.32, 0.34, 0.36, 0.37, 0.38, 0.48, 0.51, 0.52,0.56, 0.57, 0.72, 0.76, 0.77, 0.78, 0.79, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.91, 0.92, 0.93,0.94, 0.95, 0.96, 0.98] in a five-fold cross validation using traindata.

Original: We choose q = 0 .
Revised: We choose q = 0 .

Original: 56 for furtherexperiments.
Revised: 56 for furtherexperiments.

Original: 4.2 Deep Learning Parameters
Revised: 4.2 Deep Learning Parameters

Original: A five layer Deep Learning architecture was used to evaluate the efficacy of cause effect classification.
Revised: A five layer Deep Learning architecture was used to evaluate the efficacy of cause effect classification.

Original: The number of nodes in the input layer = 2000 , followed by first hidden layer with 5000 neuronsand sigmoid activation function.
Revised: The number of nodes in the input layer = 2000 , followed by first hidden layer with 5000 neuronsand sigmoid activation function.

Original: The output from this layer is passed to second hidden layer with500 neurons and ReLU activation function.
Revised: The output from this layer is passed to second hidden layer with500 neurons and ReLU activation function.

Original: This is followed by 100 neurons with Relu activationfunction in the third hidden layer.
Revised: This is followed by 100 neurons with Relu activationfunction in the third hidden layer.

Original: The fourth hidden layer contains 30 neurons with ReLu activationfunction.
Revised: The fourth hidden layer contains 30 neurons with ReLu activationfunction.

Original: The output layer contains 2 neurons with softmax activation function.
Revised: The output layer contains 2 neurons with softmax activation function.

Original: Training was donefor 30 epochs.
Revised: Training was donefor 30 epochs.

Original: 4.3 Preservation of Granger Causality for coupled AR processes under a chaotictransformation
Revised: 4.3 Preservation of Granger Causality for coupled AR processes under a chaotic

Revised: transformation

Original: Accurate estimation of causality for coupled AR processes is ideally suited for Granger Causality(GC)
Revised: Accurate estimation of causality for coupled AR processes is ideally suited for Granger Causality(GC)

Original: since GC models time series as AR processes.
Revised: since GC models time series as AR processes.

Original: This is the reason GC is very popular in causalanalysis of financial time series, climatology and neuroscience.
Revised: This is the reason GC is very popular in causalanalysis of financial time series, climatology and neuroscience.

Original: ChaosFEX features are extracted aftera chaotic transformation of the input time series.
Revised: ChaosFEX features are extracted aftera chaotic transformation of the input time series.

Original: It is important to verify whether Granger Causalityis preserved under such a nonlinear transformation.
Revised: It is important to verify whether Granger Causalityis preserved under such a nonlinear transformation.

Original: To test this, we perform the following experiment.
Revised: To test this, we perform the following experiment.

Original: For q = 0 .
Revised: For q = 0 .

Original: 78 , b = 0 .
Revised: 78 , b = 0 .

Original: 499 , and ϵ = 0 .
Revised: 499 , and ϵ = 0 .

Original: 171 , the firing time has been extracted from ChaosFEX for timeseries from coupled AR processes.
Revised: 171 , the firing time has been extracted from ChaosFEX for timeseries from coupled AR processes.

Original: The GC vs. coupling coefficient plot for firing time depicted in
Revised: The GC vs. coupling coefficient plot for firing time depicted in

Original: Figure 1a reveals that indeed Granger Causality is nicely preserved.
Revised: Figure 1a reveals that indeed Granger Causality is nicely preserved.

Original: The GC values shown here areobtained from 50 random trials 1 .
Revised: The GC values shown here areobtained from 50 random trials 1 .

Original: This indicates the reliability of the chaotic transformation of NL inpreserving granger causality and hence very desirable in applications which employ GC.
Revised: This indicates the reliability of the chaotic transformation of NL inpreserving granger causality and hence very desirable in applications which employ GC.

Original: Note thatsuch a property is not available for DL (Figure 1b) making NL a very attractive candidate for causal
Revised: Note thatsuch a property is not available for DL (Figure 1b) making NL a very attractive candidate for causal

Original: ML applications.
Revised: ML applications.

Original: [Figure]
Revised: The experimental results pertaining to the comparative performance evaluation forcause-effect classification for coupled AR processes is provided in Section 5 under limitations.

Original: 4.4 Classification of Cause Effect for Coupled Skew-Tent maps in Master-Slave Configuration
Revised: 4.4 Classification of Cause Effect for Coupled Skew-Tent maps in Master-Slave Configuration

Original: In this section, we compare the efficacy of NL - ChaosNet architecture with a five layer DNN architecture in cause-effect classification (objective O1 ).
Revised: In this section, we compare the efficacy of NL - ChaosNet architecture with a five layer DNN architecture in cause-effect classification (objective O1 ).

Original: A binary classification problem is formulated, to classify whether a given time-series is a cause or an effect.
Revised: A binary classification problem is formulated, toclassify whether a given time-series is a cause or an effect.

Original: The performance of ChaosNet and fivelayer DNN (DL) for varying coupling coefficient ( η ) is depicted in Figure 2a.
Revised: The performance of ChaosNet and fivelayer DNN (DL) for varying coupling coefficient ( η ) is depicted in Figure 2a.

Original: [Figure]

Original: ChaosNet and DL give identical performance (a macro F1-score = 1 . 0 ) for η values up to 0 .
Revised: ChaosNet and DL give identical performance (a macro F1-score = 1 . 0 ) for η values up to 0 .

Original: However, for η = [0 . 6 , 0 . 7] , ChaosNet outperforms DL.
Revised: However, for η = [0 . 6 , 0 . 7] , ChaosNet outperforms DL.

Original: Beyond η > 0 . 6 , the synchronization error< 0 .
Revised: Beyond η > 0 . 6 , the synchronization error< 0 .

Original: 013 indicating that the two time series are practically identical.
Revised: 013 indicating that the two time series are practically identical.

Original: Hence, classification fails asthere is essentially nothing to distinguish between the two time series owing to synchronization. 
Revised: Hence, classification fails asthere is essentially nothing to distinguish between the two time series owing to synchronization. [Figure] [Figure]

Original: 4.5 Preservation of causality in ChaosFEX feature space
Revised: 4.5 Preservation of causality in ChaosFEX feature space

Original: To check if causality is preserved in the ChaosFEX feature space of unidirectionally coupled skewtent maps, we use the measure Compression-Complexity Causality (CCC) [8].
Revised: To check if causality is preserved in the ChaosFEX feature space of unidirectionally coupled skewtent maps, we use the measure Compression-Complexity Causality (CCC) [8].

Original: CCC is ideal forapplication to non-linear time series, where often GC can face issues.
Revised: CCC is ideal forapplication to non-linear time series, where often GC can face issues.

Original: Figure 2b shows the CCCestimates for original (raw) time series.
Revised: Figure 2b shows the CCCestimates for original (raw) time series.

Original: The estimates plotted are averaged over 50 trials with
Revised: The estimates plotted are averaged over 50 trials with

Original: CCC parameters 2 set to L = 100 , w = 15 , δ = 50 , B = 4 .
Revised: CCC parameters 2 set to L = 100 , w = 15 , δ = 50 , B = 4 .

Original: As expected, the magnitude of CCCvalues from the master to the slave increases with increasing coupling and begins to decrease asthe time-series become synchronized and effectively no transfer of information can be detected.
Revised: As expected, the magnitude of CCCvalues from the master to the slave increases with increasing coupling and begins to decrease asthe time-series become synchronized and effectively no transfer of information can be detected.

Original: Asdiscussed in [8], CCC can take negative values and its magnitude denotes the strength of causation.
Revised: Asdiscussed in [8], CCC can take negative values and its magnitude denotes the strength of causation.

Original: CCC values in the direction of causation from slave to master are much lower in magnitude andremain close to zero.
Revised: CCC values in the direction of causation from slave to master are much lower in magnitude andremain close to zero.

Original: CCC for the corresponding firing time feature of ChaosFEX for these coupled maps is depicted in
Revised: CCC for the corresponding firing time feature of ChaosFEX for these coupled maps is depicted in

Original: Figure 2c.
Revised: Figure 2c.

Original: These values are also averaged over 50 trials and computed with CCC parameters setto L = 120 , w = 15 , δ = 60 , B = 2 .
Revised: These values are also averaged over 50 trials and computed with CCC parameters setto L = 120 , w = 15 , δ = 60 , B = 2 .

Original: Here, master to slave CCC does not perfectly preserve theincreasing trend with increasing values of coupling, however decreases just as the estimates for rawdata, when the processes proceed to synchronization.
Revised: Here, master to slave CCC does not perfectly preserve theincreasing trend with increasing values of coupling, however decreases just as the estimates for rawdata, when the processes proceed to synchronization.

Original: The slave to master estimates for the couplingrange 0 .
Revised: The slave to master estimates for the couplingrange 0 .

Original: 1 − 0 .
Revised: 1 − 0 .

Original: 9 are quite low in magnitude and remain close to zero as expected.
Revised: 9 are quite low in magnitude and remain close to zero as expected.

Original: Even though theestimates for zero coupling are not very close to zero or take exactly the same value and the increasingtrend for increasing coupling is not perfectly preserved, CCC estimates in the direction for whichcoupling exists and the its opposite are well differentiated and hence it can be said that ChaosFEXfeatures do a reasonably good job in preserving causality even for skew chaotic tent maps.
Revised: Even though theestimates for zero coupling are not very close to zero or take exactly the same value and the increasingtrend for increasing coupling is not perfectly preserved, CCC estimates in the direction for whichcoupling exists and the its opposite are well differentiated and hence it can be said that ChaosFEXfeatures do a reasonably good job in preserving causality even for skew chaotic tent maps.

Original: Surrogatebased causality analysis might help to reveal a more adequate picture of the differentiation and of theexistence of causality, but is out of the scope of this work.
Revised: Surrogatebased causality analysis might help to reveal a more adequate picture of the differentiation and of theexistence of causality, but is out of the scope of this work.

Original: 4.6 Transfer Learning for Cause-Effect Classification
Revised: 4.6 Transfer Learning for Cause-Effect Classification

Original: We have demonstrated the possibility of cause-effect classification for coupled chaotic maps in masterslave configuration.
Revised: We have demonstrated the possibility of cause-effect classification for coupled chaotic maps in masterslave configuration.

Original: However, it is interesting to explore whether we can transfer this ‘learning’ toscenarios where the master-slave systems are different from the ones for which the method wastrained.
Revised: However, it is interesting to explore whether we can transfer this ‘learning’ toscenarios where the master-slave systems are different from the ones for which the method wastrained.

Original: Specifically, we shall change the skewness of both the master and slave systems from theoriginal parameter values used in the training phase.
Revised: Specifically, we shall change the skewness of both the master and slave systems from theoriginal parameter values used in the training phase.

Original: A more drastic case of transfer learning wouldbe to test on an entirely different nonlinear map, for example, coupled logistic maps without training afresh (using the same learned parameters as the coupled skew tent maps).
Revised: A more drastic case of transfer learning wouldbe to test on an entirely different nonlinear map, for example, coupled logistic maps without trainingafresh (using the same learned parameters as the coupled skew tent maps).

Original: These would help usdetermine to what extent the learning is generalizable for both NL and DL.
Revised: These would help usdetermine to what extent the learning is generalizable for both NL and DL.

Original: We consider the following cases for transfer causal learning:
Revised: We consider the following cases for transfer causal learning:

Original: • Case
Revised: • Case

Original: I: Train with master-slave coupled skew tent map system ( b 1 = 0 .
Revised: I: Train with master-slave coupled skew tent map system ( b 1 = 0 .

Original: 65 , b 2 = 0 .
Revised: 65 , b 2 = 0 .

Original: and test with master-slave coupled skew tent map system with b1 = 0 .
Revised: and test with master -slave coupled skew tent map system with b

Original: 6 and b 2 = 0 .
Revised: = 0 . and b 2 = 0 .

Original: 4(classification results are in Figure 3a).
Revised: 4(classification results are in Figure 3a).

Original: The attractor for skew tent map master slave testdata with b 1 = 0 .
Revised: The attractor for skew tent map master slave testdatawith b 1 = 0 .

Original: 6 and b 2 = 0 . 4 is provided in Figure 3b.
Revised: 6 and b 2 = 0 . 4 is provided in Figure 3b.

Original: [Figure]
Revised: [Figure]

Original: • Case II: Train with master-slave coupled skew tent map system ( b 1 = 0 .
Revised: • Case II: Train with master-slave coupled skew tent map system ( b 1 = 0 .

Original: 65 , b 2 = 0 .
Revised: 65 , b 2 = 0 .

Original: and test with master-slave coupled skew tent map system with b 1 = 0 . 1 and b 2 = 0 .
Revised: and test with master-slave coupled skew tent map system with b 1 = 0 . 1 and b 2 = 0 .

Original: (classification results are in Figure 4a).
Revised: (classification results are in Figure 4a).

Original: The attractor for skew tent map master slave testdata with b 1 = 0 . 1 and b 2 = 0 .
Revised: The attractor for skew tent map master slave testdatawith b 1 = 0 . 1 and b 2 = 0 .

Original: 3 is provided in Figure 4b.
Revised: 3 is provided in Figure 4b.

Revised: • Case III:

Revised: Train with skew tent map master-slave coupled skew tent map system ( b 1 = 0 .

Revised: b 2 = 0 . 47 ) and test with logistic map master-slave system with A 1 = 4 . 0 and A 2 = 3 .

Original: [Figure]{+82(Figure 5a). The attractor for logistic map master slave testdata with A 1 = 4 . 
 andA 2 = 3 . 82 is provided in Figure 5b.+}
Revised: {+82(Figure 5a). The attractor for logistic map master slave testdata with A 1 = 4 . 
 andA 2 = 3 . 82 is provided in Figure 5b.+}

Original: In the case of testing with data generated from different models, ChaosNet completely outperforms
Revised: In the case of testing with data generated from different models, ChaosNet completely outperforms

Original: DL for Case I
Revised: DL for Case I

Original: (Figure 3a) and Case II (Figure 4a) for the entire range of η .
Revised: (Figure 3a) and Case II (Figure 4a) for the entire range of η .

Original: For Case III (Figure 5a),
Revised: For Case III (Figure 5a),

Original: [Figure] ChaosNet and DL shows similar trends (with DL outperforming ChaosNet for some values of η ). 
Revised: ChaosNet and DL shows similar trends (with DL outperforming ChaosNet for some values of η ). A [Figure]

Original: Ahigh performance of ChaosNet in classification shows the separability of the mean representationvectors of cause and effect.
Revised: [Figure] high performance of ChaosNet in classification shows the separability of the mean representationvectors of cause and effect [Figure]

Original: 4.7 Real Data
Revised: 4.7 Real Data

Original: The efficacy of ChaosFEX features in cause-effect preservation was evaluated on a real world datasetfrom a prey-predator system as well.
Revised: The efficacy of ChaosFEX features in cause-effect preservation was evaluated on a real world datasetfrom a prey-predator system as well.

Original: The data consists of 71 data points of predator (Didiniumnasutum) and prey (Paramecium aurelia) populations [33, 34].
Revised: The data consists of 71 data points of predator (Didiniumnasutum) and prey (Paramecium aurelia) populations [35, 36].

Original: This is a system of bidirectionalcausation as the predator population directly influences the prey population and then itself getsinfluenced by a change in the prey population.
Revised: This is a system of bidirectionalcausation as the predator population directly influences the prey population and then itself getsinfluenced by a change in the prey population.

Original: It is expected that the direct causal influence from thepredator to the prey should be higher than in the opposite direction.
Revised: It is expected that the direct causal influence from thepredator to the prey should be higher than in the opposite direction.

Original: For our analysis, initial 9 transients were removed.
Revised: For our analysis, initial 9 transients were removed.

Original: With the remaining 62 data points, CCC valuesare computed for the raw data and ChaosFEX feature (firing time).
Revised: With the remaining 62 data points, CCC valuesare computed for the raw data and ChaosFEX feature (firing time).

Original: The parameters of CCC 3 usedfor the raw data are L = 40 , w = 15 , δ = 4 , B = 8 .
Revised: The parameters of CCC 3 usedfor the raw data are L = 40 , w = 15 , δ = 4 , B = 8 .

Original: In the case of ChaosFEX, firing time featurewas extracted for the following NL hyperparameters: q = 0 .
Revised: In the case of ChaosFEX, firing time featurewas extracted for the following NL hyperparameters: q = 0 .

Original: 56 , b = 0 .
Revised: 56 , b = 0 .

Original: 499 , and ϵ = 0 .
Revised: 499 , and ϵ = 0 .

Original: The CCCparameters chosen for ChaosFEX are L = 40 , w = 15 , δ = 4 , B = 4 .
Revised: The CCCparameters chosen for ChaosFEX are L = 40 , w = 15 , δ = 4 , B = 4 .

Original: The results for the cause-effectpreservation for the raw data and ChaosFEX firing time feature is provided in Table 2.
Revised: The results for the cause-effectpreservation for the raw data and ChaosFEX firing time feature is provided in Table 2.

Original: CCC rightlycaptures the higher causal influence from predator to prey population and finds a lower influence inthe opposite direction, for both raw data and ChaosFEX feature - firing time.
Revised: CCC rightlycaptures the higher causal influence from predator to prey population and finds a lower influence inthe opposite direction, for both raw data and ChaosFEX feature - firing time.

Revised: As only a single datainstance of predator-prey time-series is available, it is not possible to train a DL network and use itsfeatures for causality estimation in this case.

Original: [Table]
Revised: [Table]

Revised: [Table]

Original:  Limitations
Revised:  Limitations

Original: In the case of coupled 1D chaotic maps, NL consistently performed well up to η = 0 .
Revised: In the case of coupled 1D chaotic maps, NL consistently performed well up to η = 0 .

Original: 5 for classification.
Revised: 5 for classification.

Original: However, the same is not true for the classification of data generated from coupled ARprocesses.
Revised: However, the same is not true for the classification of data generated from coupled ARprocesses.

Original: For q = 0 .
Revised: For q = 0 .

Original: 78 , b = 0 .
Revised: 78 , b = 0 .

Original: 499 , and ϵ = 0 .
Revised: 499 , and ϵ = 0 .

Original: 171 , the classification results are depicted in Figure 6.
Revised: 171 , the classification results are depicted in Figure 6.

Original: In the same figure, it can be seen that DL performance is worse than NL 4 .
Revised: In the same figure, it can be seen that DL performance is worse than NL 4 .

Original: We have used the exactsame architecture for DL as we have used for the cause-effect classification of data from coupledchaotic skew-tent maps in master-slave configure (section 4.2).1.0 Coupling Coefficient 0. F 1 - s c o r e , M
Revised: We have used the exactsame architecture for DL as we have used for the cause-effect classification of data from coupledchaotic skew-tent maps in master-slave configure (section 4.2).

Original: [Figure]

Original: ChaosNet-F1-Score DL-F1-Score

Original: [Figure]
Revised: [Figure]

Original: A maximum macro F1-score = 0.656 was obtained for η = 1 . 

Revised: A maximum macro F1-score = 0.656 was obtained for η = 1 .

Original: implying that ChaosNet was not ableto find mean representation vectors that could separate the two classes.
Revised: implying that ChaosNet was not ableto find mean representation vectors that could separate the two classes.

Original: Choosing a more sophisticatedclassifier for NL (instead of the simplistic cosine-similarity metric) could solve this problem andimprove classification results.
Revised: Choosing a more sophisticatedclassifier for NL (instead of the simplistic cosine-similarity metric) could solve this problem andimprove classification results.

Original: We shall explore these possibilities in a future study.
Revised: We shall explore these possibilities in a future study.

Original: In this research, we have shown the classification and preservation of causality for unidirectionalcausation of two variables.
Revised: In this research, we have shown the classification and preservation of causality for unidirectionalcausation of two variables.

Original: A detailed study needs to be undertaken for the classification and causaldiscovery for coupled high dimensional systems and real world datasets in the future.
Revised: A detailed study needs to be undertaken for the classification and causaldiscovery for coupled high dimensional systems and real world datasets in the future.

Original:  Conclusion
Revised:  Conclusion

Original: In this work, Neurochaos Learning architecture - ChaosNet has been used in the classification ofcause-effect for data generated from coupled chaotic maps.
Revised: In this work, Neurochaos Learning architecture - ChaosNet has been used in the classification ofcause-effect for data generated from coupled chaotic maps.

Original: ChaosNet outperforms a five layer deeplearning architecture  in several cases (objective O1 ).
Revised: ChaosNet outperforms a five layer deeplearning architecture and LSTM in several cases (objective O1 ).

Original: In the experiments, ChaosNet consistentlyoutperformed a five layer DNN upto a coupling coefficient of 0.7.
Revised: In the experiments, ChaosNetconsistently outperformed a five layer DNN upto a coupling coefficient of 0.7.

Original: Causality testing using Granger Causality (for coupled AR processes) and Compression-Complexity Causality (for coupled chaoticsystems and for a real-world prey-predator system) on the firing times extracted from the chaoticneural traces reveals the preservation of cause-effect in the NL feature extracted space (objective O2 ).
Revised: Causality testingusing Granger Causality (for coupled AR processes) and Compression-Complexity Causality (forcoupled chaotic systems and for a real-world prey-predator system) on the firing times extractedfrom the chaotic neural traces reveals the preservation of cause-effect in the NL feature extractedspace (objective O2 ).

Revised: Features extracted from DNN, 1D CNN and LSTM failed to preserve thecause-effect relationship as measured by GC and CCC for coupled AR processes and skew tent mapmaster slave system.

Original: Further, the efficacy of the proposed method was observed in transfer learning of the classificationof cause-effect from the master-slave time series generated from different chaotic unimodal maps(skew-tent maps with different skews and logistic map with different parameters) (objective O3 ).
Revised: Further, the efficacy of the proposed method was observed in transfer learningof the classification of cause-effect from the master-slave time series generated from different chaoticunimodal maps (skew-tent maps with different skews and logistic map with different parameters)(objective O3 ).

Original: Thismotivates future research direction of NL in lifelong learning framework, classification of cause-effectusing ChaosNet on real world datasets and building more sophisticated causal learning algorithmsusing NL for specific tasks.
Revised: This motivates future research direction of NL in lifelong learning framework,classification of cause-effect using ChaosNet on real world datasets and building more sophisticatedcausal learning algorithms using NL for specific tasks.

Original: The preservation of causality can be attributed to the rich properties of the nonlinear chaotic transformation of GLS neurons in NL ( ChaosNet ).
Revised: The preservation of causality can be attributed to the rich properties of the nonlinear chaotic transformation of GLS neurons in NL ( ChaosNet ).

Original: Unlike traditional ANNs, NL is intrinsically a nonlineardeterministic algorithm that performs a point-by-point chaotic transformation, in fact, a non-linearembedding of the input raw features in to a high dimensional space.
Revised: Unlike traditional ANNs, NL is intrinsically a nonlineardeterministic algorithm that performs a point-by-point chaotic transformation, in fact, a non-linearembedding of the input raw features in to a high dimensional space.

Original: Deterministic Chaos combinesthe best of both the worlds - pseudo-randomness and determinism .
Revised: Deterministic Chaos combinesthe best of both the worlds - pseudo-randomness and determinism .

Original: The ergodic, ‘random-like’structure of the chaotic neural traces enables an effective transformation of the input data (stimuli)preserving causality that is inherent in the input space and at the same time ensuring separability inthe chaotic feature space for efficient classification.Checklist For all authors..
Revised: The ergodic, ‘random-like’structure of the chaotic neural traces enables an effective transformation of the input data (stimuli)preserving causality that is inherent in the input space and at the same time ensuring separability inthe chaotic feature space for efficient classification.

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