Causality Preserving Chaotic Transformation and Classification using Neurochaos Learning
Authors: Harikrishnan N B
Aditi Kathpalia
Nithin Nagaraj
Venue: NeurIPS 2022 Accept (2022)
Keywords: Neurochaos Learning
Granger Causality
Compression-Complexity Causality
Coupled Auto Regressive Processes
Coupled Chaotic Maps
Causal Machine Learning
Transfer Learning

Abstract: Discovering cause and effect variables from observational data is an important but challenging problem in science and engineering. In this work, a recently proposed brain inspired learning algorithm namely-\emph{Neurochaos Learning} (NL) is used for the classification of cause and effect time series generated using coupled autoregressive processes, coupled 1D chaotic skew tent maps, coupled 1D chaotic logistic maps and a real-world prey-predator system. In the case of coupled skew tent maps, the proposed method consistently outperforms a five layer Deep Neural Network (DNN) and Long Short Term Memory (LSTM) architecture for unidirectional coupling coefficient values ranging from $0.1$ to $0.7$. Further, we investigate the preservation of causality in the feature extracted space of NL using Granger Causality for coupled autoregressive processes and Compression-Complexity Causality for coupled chaotic systems and real-world prey-predator dataset. Unlike DNN, LSTM and 1D Convolutional Neural Network, it is found that NL preserves the inherent causal structures present in the input timeseries data. These findings are promising for the theory and applications of causal machine learning and open up the possibility to explore the potential of NL for more sophisticated causal learning tasks.

Causality Preserving Chaotic Transformation and Classification using Neurochaos Learning
Harikrishnan N. B.
Department of Computer Science & Information Systems and APPCAIR BITS Pilani K. K. Birla Goa Campus, India
srrrikrishnammbbgo bits-pilani.ace ac.
Aditi Kathpalia
Department of Complex Systems
Institute of Computer Science of the Czech Academy of Sciences
Prague, Czech Republic
kathpalia@cs cas.cz
Nithin Nagaraj
Consciousness Studies Programme National Institute of Advanced Studies Bengaluru, 560012, Karnataka, India nithinGnias res.in in
Abstract
Discovering cause and effect variables from observational data is an important but challenging problem in science and engineering. In this work, a recently proposed brain inspired learning algorithm numely-Neuroche classification cause and effect time series generated using coupled autoregressive processes, coupled ID chaotic skew tent maps, coupled ID chaotic logistic maps and real-world prey- preedtto system. In the case of coupled skew tent maps, the proposed method consistently outperforms five layer Deep Neural Network (DNN) and Long Short Term Memory (LSTM) architecture for unidirectional coupling coefficient values ranging from 0.1 to 0.7. Further, we investigate the preservation of causality in the feature extracted space of NL using Granger Causal- Causality
Learning (NL) is used for the
ity for coupled autoregressive processes and Compression-Complleii for coupled chaotic systems and real-world prey-predator dataset. Unlike DNÑ, LSTM and ID Convolutional Neural Network, is found that NL preserves the inherent causal structures present in the input timeseries data. These findings are promising for the theory and applications of causal machine learning and open up the possibility to explore the potential of NL for more sophisticated causal learning tasks.
Introduction
Despite the success of Machine Learning (ML) and Deep Learning (DL) algorithms in the field of natural language processing [I. computer vision (2), speech recognition [3]. these algorithms face difficulty in interpretability and trustworthiness. One of the main reasons for this is the fact that they are merely discovering associations between 'input' and 'output' in the name of 'learning'. However. discovering associations alone is insufficient as an explanation to aid in the decision making process. Decision-making in everyday human existence relies heavily on reasoning and causal inference.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Hence, incorporation of causal ability, which goes over and beyond associations (or correlations). has become an important objective in current machine learning research
For time-series data, causality has been mathematically defined by Wiener S According to Wiener, "a time series X causes a time series Y, if the past values of X contain information that help prediet Y above and beyond the information contained in the past values of Y alone". This definition was later formulations. Granger Causality (GC) 6
realized through number of saathemmaiicllllgriihii was the pioneering causality estimation technique developed for time-series data. GC has been followed up with a number of extensions and inspirations, some of which are, iiformatioo-thhoreeftt causality as defined by Transfer Entropy 7 and data-compressiin
based causality, as defined by Compression Compression-Completty Causality (CCC) [8] While GC has primarily been formulated for linear (autoregressive) seress,ries the latter are valid for non linear applications. These methods have been
successfully employed in econometries 10. climatology
12 neuroscience
13 etc.
Conventional ML algorithms have not had much success in causal learning or causality based classification from time-series data [15]. They are either employed in combination with an existing time-series based causality detection technique []6 17 or specialized methods assuming underlying causal models have been developed in ML.
20 in order incorporate causality learning abilities
In this work, we focus causality detection from time-series data (without assumption of any causal model) and use a recently proposed brain inspired learning algorithm namely Neurochaos Learning [22] (NL.) to learn generalized causal patterns from time series data. NI. draws its inspiration from the chaotic firing of neurons in the brain 23 NL is a rival architecture to ANNs and has shown promise in classification tasks (especially in the low training sample regime), many-a-times outperforming state-of-the-art methods. NL maps the input data into a high dimensional space which enables efficient classification. In this sense, NL shows similarity to Reservoir Computing [24] which also employs nonlinear mappings of input data. However, NL is fundamentally different from Reservoir Computing in terms of motivation, methodology. and working. NL fundamentally uses the Topological Transitivity property of chaos (25) and Stochastic Resonance [22) for learning classification tasks. Given that NL is still nascent stage of development, is not well known in the Al community and most of its properties are largely unexplored. Hence, the objectives of the present study are to investigate the following:
01: The efficacy of NI in cause-effect classification and compare the same with Deep Neural Network (DNN), ID Convolutional Neural Network (ID CNN), and Long Short Term Memory (LSTM).
02: Does success in cause-effect classification imply preservation of causality?
O3: Can NL. use a transfer learning framework for cause-effect classification?
We do not build a novel causal ML algorithm, but rather explore if the existing NL architecture which does classification, has the capability to classify based on 'causal' information in the data. As a first step, we check its ability distinguish between cause and effect time-series data. To elaborate, the problem of bivariate causality detection is considered, where the algorithm is trained to classify the 'causal' or 'driver' time- seme-series variable as the cause and the time-series variable that is 'affected' or 'driven' by the former as the effect. We do not use any existing causality estimation method as an aid to the NL algorithm for this purpose. Further, for 02, we check whether features extracted from the learning architecture (in the above cause-effect classification task), preserve causality as measured by an existing time-series causality estimation method. This is important because it determines whether NI. is doing causality informed classification or not. The motivation behind many specialized causal learning algorithms that have been recently proposed is generalized learning, as failure in the case of distribution shifts continues to be one of the most important limitations of traditional ML algorithms 126 27]. Hence, 03 becomes an important objective to be looked at for an algorithm attempting to learn causal representations.
We find that a general NL architecture outperforms a five layer DNN, and LSTM architecture in cause-effect classification. Further, the features extracted using NL are found to preserve the cause- effect relationship present in input data. This, however, was not the case for DNN, ID CNN and LSTM, probably because the classification results were not causally informed. The performance comparison of NI. with ID CNN and L.TT is extensively studied in the supplementary material. The findings demonstrate that even a general NL architecture is capable of some basic causal learning

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and hence promising for developing more sophisticated causal ML algorithms required for different tasks.
The sections in the paper are arranged as follows: Section 2 describes the method used to do the cause-effect classification. Section 3 provides details of the simulated data used to earry out the experiments. Section 4 deals with experiments, results and discussions on simulated and real world prey-predator dataset. Section 5 addresses the limitations and scope for future work. The concluding remarks are provided in Section 6.
2 Neurochaos Learning
Neurochaos Learning (NL) is a novel brain inspired neuronal learning algorithm that has been recently
proposed. The authors in [25] 28 state that NL is inspired from the chaotic firings of neurons in the
brain and has mainly two architectures: (a) ChaosNet (28), (b) ChaosFEX+MLL In [29], the authors
employ ChaosNet for continual learning. In another work 30 the authors propose deep ChaosNet for action recognition in videos.
Inspired by these recent developments, in this work, we employ ChaosNet architecture for the classification of cause-effect from observational data. The architecture consists of input layer of Generalized Lüroth Series (GLS) neurons which are one-dimensional (1D) chaotic skew tent maps described as follows:
sso<b, b, b
T(x)
(I)
where the skewness of the map is controlled by the parameter b (0 < b < 1). Upon arrival of the input data/ stimulus (Zy, k-th data point in a time series), the k-th chaotic GLS neuron in the input layer starts firing (from the initial value q) until the chaotic neural trace of the neuron reaches the neighbourhood of the corresponding stimulus (Ix)). The number of chaotic GLS neurons in the input layer is equal to the number of input stimuli (number of data points in time series). From the neural
trace thus generated from each chaotic GLS neuron, the following features are extracted:
1. Firing time (N): The amount of time the chaotic neural trace takes recognise the input stimulus.
2. Firing rate (R): Fraction of time the chaotic neural trace is above the discrimination threshold b as recognize the stimulus.
3. Energy (E); For the chaotic neural trace y(t) with firing time N, energy is defined as:
F
10
4. Entropy (H): For the chaotic neural trace y(t), we first compute the binary symbolic sequence Sym(t) as follows:
0, y(b) < b Sym(t:) = -11 1. b b v(t)) < 1,
where i = 1 to N (firing time). We then compute Shannon Entropy of Sym(t) follows:
(SSm) = Σ pi log)(p.) bits,
(4)
where P1 and P2 refers to the probabilities of the symbols 0 and occurring in Sym(t) respectively.
For each input value Is (stimulus) of data instance of class e is mapped to 4D vector [Nay: Rey En,, H.,). The collection of these 4D vectors forms the ChaosFEX feature space. If the input data consists of only one stimulus (one time point from a given time series) with Z classes, the mean representation vector of the c-th class (with m data instances) is given by
3

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m ME RY H5]. Since there is generally more than one stimulus or time point in time series, this procedure is repeated for each time point.
In the case of ChaosNet, the classifier computes the cosine similarity of the ChaosFEX features extracted from the test sample with the pre-computed mean representation vectors (consisting of mean values of ChaosFEX features for each stimuli) from the training set of each class. The predicted class assigned the label corresponding to the maximum cosine similarity. detailed explanation of the Chaoslet and its working is provided in [25]. In this work, the ChaosNet architecture of NL.
Datasets
To evaluate the efficacy of ChaosNet and deep learning for the classification of cause-effect, we used simulated datasets from (a) Coupled autoregressive (AR) processes, (b) Coupled 1D chaotic maps in master-slave configuration (1D skew tent maps and ID logistic maps) and real-world dataset from (c) prey- prey-predator system.
3.1
Coupled AR processes
The governing equations for the coupled AR processes are the following:
M(I) M(I - 1)
(5)
S(t) = az 1) ()
(6)
where M and S(t) are the independent and the dependent (or the cause and effect) time series respectively at time aj 0.8, 02 = 0.9, the noise intensity > 0.03 and r(t) the i.dd additive
gaussian noise drawn from standard normal distribution. The coupling coefficient 17 is varied from 0 in steps of 0.1. We generate 1000 independent random trials for each value of 17. Each of the data
instances are of length 2000, after removing the initial 500 samples (transients) from the time series.
3.2
Coupled ID Chaotic maps in Master-Slave configuration
3.2.1
Coupled Skew-tent maps
The governing equations used to generate the master and slave time series for the coupled ID skew-tent maps are the following:
M(I) = TT 1)), S(t) = (1-1) (1- M
where Mt) is the master (cause) and S(t) is the slave (effect) system. (t) influences the dynamics
of the slave system (equation (). The coupling coefficient given by v) is varied from 0.9 with a step size of 0.1. (t). and (2) are skew tent maps with skewness b = 0.65, and by = 0.47 respectively. The initial values are chosen randomly for the master-slave system in the interval (0, 1). We generate 1000 independent random trials for each value of 17. Each of the data instances are of length 2000, after removing the initial 500 samples (transients) from the time series.
3.2.2 Coupled Logistic maps
The ID Logistic map is a widely used model to study population dynamics [1. The governing dynamics for coupled logistic maps in master-slave configuration is given by:
M(1) = I, (M(( - 1)), S(t) = (1 (1- Ls - 1)) Mt -1),
(10)
The coupling coefficient 7) varied from 0 to 0.9. (t) A Li 1) L, 1)), and
(2(t) = A₂ Lt(t - 1)(1- Lz(t - 1)), where A = 4 and A₂ = 3.82. The attractor for this coupled
dynamical system provided Figure 5b
For both systems, 1000 data instances (M), S()) are generated and grouped as class-0 (t): Cause) and class-1 (S((): Effect) respectively. Each of the data instances are of length 2000, after removing the initial 500 samples (transients) from the time series.
Table I gives details of the train-test split for the classification tasks for all the simulated datasets.

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Table I: Train- Test distribution for the simulated datasets.
Class
Traindata
Testdata
Class-0
801
199
Class-I
799
201
Total 1600
400
4 Experiments, Results and Discussions
In this section, we begin with a description of hyperparameter tuning for NL and DL followed by a demonstration of causality preservation by ChaosFEX for coupled AR processes, skew tent maps and logistic maps (and the failure of DL). Also, macro Fi-scores for ChaosNet and DL for the cause-effect classification for each are plotted. For all results in this paper, software implementation is performed using Python 3, scikit-learn [2], keras [33], ChaosFEX toolbox [22], Multivariate
Granger Causality (MVGC) toolbox [34], CCC toolbox [8] and MATLABB Comparison of NL. with ID CNN and LSTM extensively studied in the supplementary material.
4.1 Hyperparameter tuning for NL
Every MI. algorithm has of hyperparameters that needs be tuned for efficient performance. In the case of ChaosNet. there are three hyperparameters- initial neural activity (g), discrimination threshold (b), and noise intensity (E) [22]. The hyperparameter tuning done only once with the traindata corresponding to 0.4 (Table separately for the coupled AR processes and coupled skew tent maps.
For a fixed value of b = 0.499, and c = 0.171, was varied from 0.01 to 0.98 with a stepsize of 0.01 for both coupled AR processes and coupled chaotic skew tent maps. In the case of coupled AR processes, a maximum average macro Fl-score = coupled skew tent maps, maximum average macro F1-score 1.0 was obtained for the following values of [0. 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 five-fold cross validation using traindata. We choose 0.56 for further experiments.
0.605 is obtained for q = 0.78. In the case of
4.2 Deep Learning Parameters
A five layer Deep Learning architecture was used to evaluate the efficacy of cause-effect classification. The number of nodes the input layer = 2000, followed by first hidden layer with 5000 neurons and sigmoid activation function. The output from this layer is passed to second hidden layer with 500 neurons and ReLU activation function. This is followed by 100 neurons with ReLU activation function in the third hidden layer. The fourth hidden layer contains 30 neurons with ReLU activation function. The output layer contains 2 neurons with softmax activation function. Training was done for 30 epochs.
4.3 Preservation of Granger Causality for coupled AR processes under chaotic transformation
Accurate estimation of causality for coupled AR processes is ideally done by Granger Causality (GC) since GC models time series as AR processes. This the reason GC is very popular in causal analysis of financial time series, climatology and neuroscience. We extract ChaosFEX features after chaotic transformation of the input time series as described in Section 2. It is important to verify whether GC preserved under such nonlinear transformation. To test this, we perform the following experiment. For 0.78, b = 0.499, and 0.171, the firing time has been extracted from ChaosFEX for time series from coupled AR processes. The GC vs. coupling coefficient plot for firing time depicted in Figure La reveals that indeed GC is nicely preserved. The GC values shown here are obtained from 50 random trials] This indicates the reliability of the chaotic transformation of NL in preserving granger
'The maximum model order setting in the MVGC toolbox was set to 30 for ChaosFEX features and to 20 for DL features.
5

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causality and hence very desirable in applications which employ GC. Note that such property is not available for DL (Figure making NL very attractive candidate for causal ML applications. The experimental results pertaining to the comparative performance evaluation of NI. and DL. for cause-effect classification of coupled AR processes is provided Section 5 under limitations.
0.12
M causes iiiiing Time) (firing Time)
0.0007
causes S (DL) S tauses M (DL)
0.10
0.0006
GC (F-statistic)
(F-statistic)
0.0005
0.08
0.0004
0.06
0.0003
0.04
0.0002
0.02
0.0001
0.00
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Coupling Coefficient
Coupling Coefficient
(a)
(b)
Figure 1: (a) GC coupling coefficient for the firing time feature extracted from the coupled AR processes. The ChaosFEX settings are q 0.78, b 0.499, and e = 0.171. The GC F-statistic is computed from 50 trials. (b) GC coupling coefficient for DL features extracted from the fourth hidden layer of a five layer neural network. The GC F-statistic is computed from 50 trials.
4.4 Classification of Cause-Effect for Coupled Skew-Tent maps in Master-Slave Configuration
In this section, we compare the efficacy of NI. ChaosNet architecture with five layer DNN archi- tecture in cause-effect classification (objective O1). A binary classification problem is formulated, to classify whether a given time-series cause or an effect. The performance of ChaosNet and five layer DNN (DL) for varying coupling coefficient (17 is depicted in Figure
ChaosNet and DL give identical performance macro Fi-score 1.0) for 7) values up 0.5. However, for 1 = [0.6, 0.7]. ChaosNet outperforms DL. Beyond > 0.6, the synchronization error < 0.013 indicating that the two time series are practically identical. Hence, classification fails as there is essentially nothing to distinguish between the two time series owing to synchronization.
4.5 Preservation of causality in ChaosFEX feature space
To check if causality is preserved in the ChaosFEX feature space of unidirectionally coupled skew- tent maps, we use the measure application non linear time series, where often GC can face issues. Figure 2b shows the CCC estimates for original (raw) time series. The estimates plotted are averaged over 50 trials with CCC parameters set L 100, w 15, 50, values from the master to the slave increases with increasing coupling and begins to decrease as the time-series become synchronized and effectively no transfer of information can be detected. As discussed in [8] CCC can take negative values and its magnitude denotes the strength of causation. CCC values the direction of causation from slave to master are much lower magnitude and remain close to zero.
Causality (CCC) { CCC is ideal for
As expected, the magnitude of CCC
CCC for the corresponding firing time feature of ChaosFEX for these coupled maps is depicted in Figure 2c These values are also averaged over 50 trials and computed with CCC parameters set 15, 5 60, B
Here, master slave CCC does not perfectly preserve the increasing trend with increasing values of coupling, however decreases just the estimates for raw data, when the processes proceed to synchronization. The slave to master estimates for the coupling range 0.1 - 0.9 are quite low in magnitude and remain close to zero as expected. Even though the estimates for zero coupling are not very close zero or take exactly the same value and the increasing trend for increasing coupling is not perfectly preserved, CCC estimates in the direction for which coupling exists and opposite are well differentiated and hence can be said that ChaosFEX features do a reasonably good job in preserving causality even for skew chaotic tent maps. Surrogate
to L 120,
"hheee were chosen using the selection criteria described in g

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F1 F1-score, Synchronization Error
1.0
0.005
0.000
0.6
-0.005
0.4
00.010
F1-Scre
0.2
Synchrorization Error
00015
0.0
0.020
causes causes M
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
Coupling Coefficient
Coupling Coefficient
(b)
0.012
0.20
M causes S (D1)
0.010
0.15
0.008
Coc
0.006
M causes 5 driiig Time!
0.10
0.004
(Fining
0.002
0.05
0.000
0.00
-0.002
0.2
0.4
0.6
0.8
0.2 Coupling Coefficient
0.6
0.8
Coupling Coefficient
(c)
(d)
Figure 2: (ab Performance comparison of ChaosNet and five layer DNN for the classification of cause-effect for ID coupled skew tent map master-slave configuration. (b) CCC Coupling Coefficient for the raw data corresponding to ID chaotic skew tent map in master-slave configuration. c CCC vs Coupling Coefficient for firing time (ChaosFEX feature) corresponding to ID chaotic coupled skew tent maps in master-slave configuration. d CCC vs Coupling Coefficient for features extracted from the second last layer of five layer deep neural network corresponding 1D chaotic coupled skew tent maps in master-slave configuration.
based causality analysis might help to reveal more adequate picture of the differentiation and of the existence of causality, but out of the scope of this work.
4.6 Transfer Learning for Cause-Effect Classification
We have demonstrated the possibility of cause-effect classification for coupled chaotic maps master- slave configuration. However, is interesting to explore whether can transfer this 'learning' to scenarios where the master-slave systems are different from the ones for which the method was trained. Specifically,
shall change the skewness of both the master and slave systems from the original parameter values used in the training phase. A more drastic case of transfer learning would be 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). These would help us determine what extent the learning is generalizable for both NL and DL.
We consider the following cases for transfer causal learning:
Case I: Train with master-slave coupled skew tent map system (b₁ 0.65 b2 = 0.47) and test with master-slave coupled skew tent map system with b 0.6 and 0.4 (classification results are in Figure 3a). The attractor for skew tent map master slave testdata with 0.6 and by = 0.4 is provided in Figure 3b)
Case Train with master-slave coupled skew tent map system (b, 0.65, 0.47) and test with master-slave coupled skew tent map system with b, 0.1 and by = 0.3 (classification results are in Figure 4a). The attractor for skew tent map master slave testdata with 6 0.1 and b2 0.3 is provided in Figure

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1.0
1.0
Slave
0.8
0.8
F2-score
0.6
0.6
0.4
0.4
0.2
0.2
es F1 Score
0.0
DL F1-Score
0.0
0.2
0.4 Coupling Coefficient
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
Mn. Sn
Transfer learning for case I: comparative performance of ChaosNet and five layer DNN evaluated using macro Fi-score for in the range 0.9 with stepsize of 0.1. (b) Case Attractor for the coupled 1D chaotic skew tent maps in master slave configuration with by = 0.6 and b2 0.4.
Figure 3:
1.0
Master
Slave
0.8
0.8
11-corre
0.6
0.4
0.4
0.2
0.2
ChaosNat F1-Scare
F1-Score
0.0
0.2
0.4
0.6
0.8
0.0
0.0
0.2
0.4
0.6
1.0
Coupling Coefficient
Mn. S.
(a)
(b)
Figure Transfer Learning for Case comparative performance of ChaosNet and five layer DNN evaluated using macro Fl-score for r in the range to 0.9 with stepsize of 0.1. (b) Case II: Attractor for the coupled ID chaotic skew tent maps in master slave configuration with b = 0.1 and
b2 0.3.
Case III: Train with skew map master-slave coupled skew tent map system (b) 0.65,
b = 0.47) and test with logistic map master-slave system with A = 4.0 and A₂ = 3.82
(Figure (a). The attractor for logistic map master slave testdata with A₁ 4.0 and
A2 3.82 is provided in Figure 5b
1.0
1.0
Slave
0.8
0.8
F1-ccore
0.6
0.6
0.4
0.4
0.2
0.2
Chaoshat F1-Scare F1-Score
0.0
0.0
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0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
Coupling Coefficient
Mn. Sn
(a)
Figure 5: (a) Transfer I Learning for Case III: comparative performance of ChaosNet and five layer DNN evaluated using macro Fl-score for 7 the range 0 to 0.9 with a stepsize of 0.1. (b) Case III: Attractor for the coupled ID logistic maps in master-slave configuration with A 4.0 and
A = 3.82.

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In the case of testing with data generated from different models, ChaosNet completely outperforms DL for Case (Figure Baj and Case II (Figure 4a) for the entire range of 7). For Case III (Figure ChaosNet and DI. shows similar trends (with DI. outperfomiig ChaosNet for some values of 1)). A high performance of ChaosNet classification shows the separability of the mean representation vectors of cause and effect.
4.7 Real Data
The efficacy of ChaosFEX features in cause-effect preservation was evaluated on real world dataset from prey-predator system as well. The data consists of 71 data points of predator (Didinium nasutum) and prey (Paramecium aurelia) populations [35] 36]. This is a system of bidirectional causation as the predator population directly influences the prey population and then itself gets influenced by a change the prey population. is expected that the direct causal influence from the predator to the prey should be higher than in the opposite direction.
For our analysis, initial 9 transients were removed. With the remaining 62 data points, CCC values are computed for the raw data and ChaosFEX feature (firing time). The parameters of CCC used 15, 5 - 4, 8. In the case of ChaosFEX, firing time feature
for the data are L 40, was extracted for the following NL hyperparameters:
parameters chosen for ChaosFEX are L 40, 15, 0.56, 5 = 0.499, and = 0.1. The CCC B The results for the cause-effect preservation for the raw data and ChaosFEX firing time feature is provided in Table 2 CCC rightly captures the higher causal influence from predator to prey population and finds lower influence in the opposite direction, for both raw data and ChaosFEX feature firing time. As only single data instance of predator-por prey time-series available, is not possible train a DL network and use its features for causality estimation in this case.
Table 2: Cause-effect preservation of the prey-predator real world data using CCC.
Class
CCC (rawdata)
CCC (firing time)
DL
Predator + Prey
0.1
0.0484
Unable to compute
Prey Predator
-0.0210
0.0050
Unable compute
5 Limitations
In the case of coupled ID chaotic maps, NI. consistently performed well up 1 = 0.5 for classi- fication. However, the same is not true for the classification of data generated from coupled AR processes. For q = 0.78, In the same figure, it can be seen that DL performance is worse than NI We have used the exact same architecture for DL as we have used for the cause-effect classification of data from coupled chaotic skew-tent maps in master-slave configure (section 4.2).
0.499, and
0.171, the classification results are depicted in Figure
maximum macro Fl-score 0.656 was obtained for 1) 1.0 implying that ChaosNet was not able to find mean representation vectors that could separate the two classes. Choosing more sophisticated similariy metric) could solve this problem and
classifier for NI. (instead of the simplistic improve classification results. We shall explore these possibilities in future study.
In this research, have shown the classification and preservation of causality for unidirectional causation of two variables. A detailed study needs to be undertaken for the classification and causal discovery of coupled high dimensional systems and real world datasets in the future.
"These were chosen using the selection criteria described (8] "We have performed some amount of hyperparmater tuning for DL architecture, however more extensive tuning needs to be performed.

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1.0
ChacsNct-F1 Score
DL-F1-Scare
fc-score, MSE
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Coupling Coefficient
Figure 6: Fl-score coupling coefficient for the classification of the coupled AR processes using
ChaosNet.
Conclusion
In this work, Neurochaos Learning architecture ChaosNet has been used the classification of cause and effect variables from time series generated from coupled chaotic maps and stochastic autoregressive processes. ChaosNet outperforms a five layer deep learning architecture and LSTM in the case of both chaotic tent maps and AR processes (objective 01). In the case of AR processes, ID CNN performs better than ChaosNet for several values of coupling. Whereas, in the case of coupled chaotic skew-tent map, ChaosNet outperforms all the other methods including 1D CNN. Causality testing using Granger Causality (for coupled AR processes) and Compression- Complexxty Causality (for coupled chaotic systems and for real-world prey-predator system) the firing times extracted from the chaotic neural traces reveals the preservation of cause-effect in the NL feature extracted space (objective 02). Features extracted from DNN, ID CNN and LSTM failed to preserve the cause-effect relationship as measured by GC and CCC for coupled AR processes and skew tent map master slave system. This implies that the classification was not causally informed despite 1D CNN showing superior classification performance in some cases. Further, the efficacy of the proposed method was observed in transfer learning of the classification of cause-elfect 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 03). This motivates future research direction of NL in lifelong learning framework, classification of cause-effect using ChaosNet on real world datasets and building more sophisticated causal learning algorithms using NL for specific tasks.
The preservation of causality can be attributed to the rich properties of the nonlinear chaotic transfor- mation of GLS neurons in NI (ChaosNet).. Unlike traditional ANNs, NI. is intrinsically nonlinear deterministic algorithm that performs a point-by connt-y--ooit chaotic transformation, in fact, nonlinear embedding of the input raw features in high dimensional space. Deterministic Chaos combines and determinism. The ergodic, 'random-
the best of both the worlds 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 in the chaotic feature space for efficient classification. The codes used in this study are available here: cew.aannnnneuuuueeeeeeettee effecee--ffct... prrfffcct-peeseevtiono---
aneeddo-randomeet
https://ghh tht.bbb..tt....... com/ Has//Harikkk www.ikiiissnnnnnn/aaeaaa http://asseeeeeee
Acknowledgements
The authors gratefully acknowledge the National Institute of Advanced Studies Consciousness Studies Programme Tata Trusts Project on Kashmir Saivism, Causality Information Theory TETMMUM/NNNIAA/20177000 80051-a 2018 March to 2022 March. AK is thankful to the funding provided by the Czech Science Foundation, Project No. GA19-160663 and by the Czech Academy of Sciences, Praemium Academiae awarded to M. Paluš.
10

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12

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as URL)? [No]
(b) Did you specify all the training details (e.g., data splits. hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [N/A]
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [N/A]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
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13


Reviews:
[{"review_id": "yGvvr6A5u6d", "reviewer": "Reviewer_Uusg", "summary": "In this paper, the authors utilized a recently proposed Neurochaos Learning method for the classification of cause-effect. To verify the brain-inspired learning method, multiple experiments have been conducted on synthetic and real-world datasets. Compared to a five-layer Neural Network, the utilized NL method achieves superior performance.   ", "questions": "It would be great if the author could address the issues raised in the weakness section. ", "limitations": "Yes, the authors fairly discussed the limitations of the method. The potential negative impact may not be applicable to this study. ", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:  \n1. The idea of applying a brain-inspired learning method is interesting and has not been extensively explored yet, this could potentially bring novel insights complementary to the mainstream data-driven model design.\n\n2. The experimental results seem plausible and reflect the feasibility of the NL method compared to a baseline neural network.\n\n\nWeakness:\n\n1 Causality: I think the main drawback of this manuscript is the discussion of causality. In line 25, the authors claim that causality has been mathematically defined by Wiener et.al.. it would be nice to explicitly give the definition here, as reviewers may not familiar with this definition. Importantly, the nuance of causality definition varies from literature [1]  to literature [2]. Without presenting the exact definition of causality quoted in this paper and discussing related definitions, it makes the readers hard to understand the main idea. In terms of 'classification of cause-effect', I am not sure if this terminology makes sense or not. What does it mean by classifying cause-effect (later causality detection is brought in line 38)? I believe the authors should discuss its connection to causal variable identification. This also relates to the fact the study is conducted on observational data.  \n\n[1] Peters J, Janzing D, Schölkopf B. Elements of causal inference: foundations and learning algorithms[M]. The MIT Press, 2017.\n[2] Hernán M A, Robins J M. Causal inference. 2010.\n\n2 Unclear model design: The model architecture and learning details are fragmented or missing. The authors could either provide a plot of model illustration, pseudo-code table, or code repository. Considering that Neurochaos Learning is not a well-known method, it is important to demonstrate integrated details to facilitate reproductivity.\n\n3 Experimental design: The experiments regarding Coupled autoregressive (AR) processes and Coupled 1D chaotic maps etc. don't seem to be well-motivated. Could the authors reason why particularly using such setting to investigate cause-effect of time series. Lastly, the comparison to a five-layer neural network seems to be less convincing, given the rapid developments of deep learning architectures.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "eaJDasM7t_R", "reviewer": "Reviewer_JiUB", "summary": "This paper use Neurochaos Learning architecture for the classification task of cause-effect for data generated from coupled chaotic maps. Results show that NL preserves causality under a chaotic transformation and can successfully classify cause and effect time-series. One more exciting finding is that even a general NL architecture is capable of some essential causal learning. Hence, promising development of more sophisticated causal ML algorithms is required for different tasks.\n", "questions": "1. Lack of novelty: Neurochaos Learning is not first proposed in this work, which may hurt the contribution of this paper.\n2. For O3 in #53, transfer learning is not very important in this setting, and a more detailed analysis of the proposed method itself may be helpful. Or the reviewer may want to see why transfer learning needs to be addressed.", "limitations": "The authors claimed well in the limitations parts.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. This paper introduces the Neurochaos Learning architecture (NL) into the causality domain, which can be seen with more applications.\n2. The proposed method consistently outperforms a five-layer Deep Neural Network architecture, which is convincing.\n\nWeakness:\n1. Lack of novelty: Neurochaos Learning is not first proposed in this work, which may hurt the contribution of this paper.\n2. For O3 in #53, transfer learning is not very important in this setting, and a more detailed analysis of the proposed method itself may be helpful.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "HcnejdkUR9S", "reviewer": "Reviewer_eVRu", "summary": "The authors have shown that the brain inspired learning algorithm- Neurochaos learning is capable of performing the classification of cause-effect in four different coupled systems namely, coupled autoregressive processes, coupled 1D chaotic skew tent maps, and coupled 1D chaotic logistic maps and predator-prey.  They generated simulated datasets and performed experiments using the Master-slave configuration on three simulated and one real dataset. In addition, the authors showed that Granger causality is preserved under chaotic transformation for the coupled systems. The classification performance of the coupled configuration is compared with 5-layered DL architecture. ", "questions": "1. Authors have only used F1-score for the performance comparison. I would like to understand from the authors why are the other metrics were not computed like class-wise accuracy or precision. \n2. As the training data derived from the different coupled systems are balanced (2000), the authors have experimented on a balanced data set. However it will be important to see how the model performs on unbalanced dataset. \n3. Adaptive learning rate – how will the model perform is  a setting where the continuity of tke loss landscape is leveraged instead of treating the Learning rate as a tunable parameter.\n4. Authors have experimented on data points of predator (Didinium  nasutum) and prey (Paramecium aurelia) populations,  I suggest experimenting more on datasets like Predator prey:  Lynx and snowshoe hare, Wolves vs rabbits.\n5.\tThe predator prey systems are natural test beds for neuro chaos learning because they are governed by coupled dynamical systems. In this context can the authors comment on the stability of the learning process?\n", "limitations": "NL is not a causal ML algorithm. If the input data has causal structure, NL preserves that structure.  If the problem is not framed as classification of cause effect , then NL requires causality testing methods like GC, CCC etc to identify which is cause and which is effect.\nThe following are the suggestions:\n1. Cause-Effect preservation is an important property for the interpretability of the algorithm. This can be used to understand why the algorithm works. Interpretability of Neurochaos features + ML algorithms can be further investigated using this property. \n2. Application of NL in the classification of breast cancer curated dataset with heavy class imbalance may be investigated. Particularly I would like to see a discussion on how NL can be used to detect anomalies where the minority class instances are possibly anomalous, like in the breast cancer data, for specific survival year cut offs.", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. Authors have indicated a classification method of cause and effect for any time series data using the NL algorithm. The method shows a fair performance on the simulated and real datasets when their F1-scores were compared. Clearly, F1-scores for ChaosNet were better than DL.\n2. The manuscript also demonstrates the preservation of Granger Causality in the ChaosFEX feature space.\n3. Usage of “Transfer learning” like approach by testing the algorithm on a dataset completely different from the one it was trained on. \n\nThe aim in the paper is not to build a causal ML algorithm, but rather to explore if the existing NL architecture which does classification can also distinguish between cause and effect data. The study answers this in the affirmative and slightly outperforms DNN in this regard. When we ask an important question regarding features such as whether features extracted from the learning architecture (in the cause-effect classification task) in both DNN and NL preserve causality as measured by Granger Causality (GC) and Compression-Complexity Causality (CCC). This is important because it determines whether NL is doing a causality informed classification or not. For this the authors used GC and CCC to check for causal relationship between coupled AR processes and master-slave chaotic skew tent-map system respectively (Note: In the cause-effect classification task they are not using GC and CCC, but the plain NL architecture which was used for classification). In the experiments, they show unambiguously the failure of GC for DNN features (features extracted from the layer just before the output layer), owing to the failure of cause-effect preservation. Whereas in the case of ChaosNet, using GC and CCC, the cause-effect relationship in the input data was faithfully preserved. This is a verification of causality informed classification - i.e., the output from the input layer of NL preserves cause-effect relationship. This is a very important and novel finding for developing causal NL\nalgorithms in the future (and not been shown in any of the previously published work on NL).\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "HQ_a4mzga6L", "reviewer": "Reviewer_UvGj", "summary": "This paper discusses Neurochaos learning and evaluates how it performs on a classification task where it is asked to identify the cause and the effect.\n\nIn brief, Neurochaos learning is this: a 1D skew tent map is a chaotic process. Given an input $x$, we run the process until it comes within $\\epsilon$ of $x$. Various statistics for the process are then computed and these are used as the feature representation of $x$. Consider a multi-class classification task. The training data is used to create average-representations, one per class. Classification on the test data is done by computing the cosine similarity with the per-class representations.\n\nThe authors consider the task of classifying whether a time series is a cause or effect of another time series. In this task, the authors show that NL representations are found to preserve causality -- meaning, for example, that in the NL feature space, Granger causality increases with more coupled systems, as it should. \n\nThroughout the paper, NL and Deep Learning (DL) are compared. In general, the authors find that for this task, NL and DL achieve similar accuracy, but NL is argued to be more desirable, because of the causality-preserving properties discussed above.", "questions": "- Table 1: Why aren't class-0 and class-1 the same number? Aren't the causes and effects generated in pairs?\n- line 89: Is the discrimination threshold $b$ the same as the skewness parameter $b$ in equation (1)?\n- Section 3.1: What is the train/test split for the AR processes? Or is Table 1 applicable for sections 3.1 and 3.2?\n- Section 3: Can you explain briefly what the classification task is? I have two guesses: (1) the classifier is given 2000 steps of a time series and must decide if the time series is the cause or effect. (2) the classifier is given 1000 steps of one time-series and 1000 time-steps of another time-series. The steps are concatenated into a 2000 dimensional vector. The classifier must decide if the first 1000 steps are the cause or the effect. Are either of these correct?\n- Section 4.1: How is the data fed as input to ChaosNet? The input is a 2000 dimensional vector. Is each element of the vector fed as stimulus to a single GLS neuron? Or is there a k-dimensional generalization of the 1D neuron presented in section 2?\n- line 173: Why were these hyper-parameters chosen? \n- Figure 1b: What is the Granger causality computed between? I have two guesses: (1) It is computed between and input time series and the vector of neuron values in the fourth layer (2) It is computed between the average neuron value between the cause and effect time series, after being pushed through the DNN\n- Figure 1: What is the difference between \"M causes S\" and \"S causes M\"? Is the training data swapped? It's not mentioned in line 173, and unless I'm missing something, it's the first time the terms have been used. My best guess is that \"M causes S\" is the GC computed between M and S in NL feature-space, where M is the cause and S is the effect. Is this correct?\n\n## minor things\n- Line 97: Delete \"For\"", "limitations": "Limitations are adequately discussed.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n- Reading about Neurochaos Learning was interesting\n- The authors are honest about the limitations of Neurochaos learning and do not omit plots or experiments where DL is shown to perform better. I think this is commendable.\n\n## Weaknessess\n- In the introduction, Deep Learning is criticized for having poor interpretability and trustworthiness. But the paper does not present a convincing case that Neurochaos Learning is much better on those fronts. In particular, section 2 and section 4.3 describe the basics of NL prediction and parameter-tuning, but not much interpretation of these are given. It's hard to see how NL can be said to /learn/. From the description given in section 2, the whole process seems to have more in common with non-parametric clustering methods.\n- For comparison with Deep Learning, a 5 layer dense feed forward net is used. But given that the task is time-series classification, I feel a more fair comparison would be an RNN, Transformer, or even 1D CNN\n- I don't think the results show conclusively that NL captures causality better than DL. Figure 2a shows that Deep learning outperforms NL in as many cases as it underperforms\n- The paper claims that the NL representations exhibit the same causal relationships that are present in the input data, but the results presented (figure 2c, figure 5a) don't seem to conclusively show that. Additionally, no baselines are given for a transformation that does /not/ preserve causality, so the quality of the results are hard to judge in figure 2c. \n-  My main concern is with significance. The task domain seems extremely tailored to the type of thing NL would be good at. I would be more convinced of the usefulness of NL for causal tasks if it were clear how these methods could be applied to more complex problems. As it is, it's hard to see how this approach would be able to handle, e.g., language modeling.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept