IEEE Workshop on Nonlinear Circuit Networks
December 3-4, 2021

Chaotic Data Classification by Using Attractors
Including Time Information
Aika Ohno, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering of Tokushima University
2-1 Minami-Josanjima, Tokushima-shi, Tokushima-ken 770-8506, Japan
Email: {a-ohno, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Chaotic data can be seen in various kinds of datasuch as biological signal. It is important to analyze chaotic dataand know slight difference of chaotic parameters with better
accuracy. As a method of chaos theory, time series data canbe embedded into attractors which can be shown as images.
In addition, recent years, neural network is used for image
classification of various kinds of data.
In this study, we classify chaotic time series data by using
embedded attractors images and residual networks which is one
kind of neural networks, and we propose embedded attractorsthat include time information for dataset. Then, we discuss thesimulation results by using confusion matrixes.

Classification of chaotic time series data by using reconstructed attractors with neural networks has been studied [5].
However, reconstructed attractors would be lost the time
information. From that perspective, we investigate the effectof using attractor that include time information for dataset.
By proposing this method, our study object is finding effective
preprocessing for chaotic time series data for neural networks.
II. C HAOS T HEORY AND E MBEDDED ATTRACTORS
By using Takens’ embedding theorem, time series data x(t)can be embedded into n-dimensional vector v(t) with timedelay τ as shown in Eq. (1).

I. I NTRODUCTIONRecently, neural network has been got a lot of attention, andit achieves significant results especially on image recognition.
It is used for various kinds of tasks such as image recognition, prediction, segmentation and natural language processing.
Also, it is applied for a lot of kinds of data and fields not
only academic fields but also industrial and medical fields.
Image recognition by neural network is surpass human error
rate [1]. Thereby, neural networks can recognize images evenhuman cannot classify. Residual Networks (ResNet) is oneof the most popular networks for image classifications [2].
ResNet is the winner network of ILSVRC 2015, and it is thenetworks that enable to have deep layered networks without
vanishing gradient problem.
Chaotic characteristics can be observed in various kinds oftime series data such as biological signals and stock pricefluctuations, and chaotic theory is applied for these dataanalysis. Chaos means the signals that seems to be disorder,
but actually it is deterministic. It has features sensitivity of
initial conditions and non-periodicity. Analytics of pulse wave
is one of the application example of chaos theory. Some studies
show different feelings have different statement of chaos [3].
Then, proper recognition of the differences of chaos parameters enables to estimate their bodies and feeling conditions
with better accuracies. Therefore, it is important to recognize
chaotic features and classify depends on the differences ofchaotic statements. However, as chaos seems to be disorder, it
is difficult to recognize parameters differences by only seeing
time series data. Therefore, Takens’ embedding theorem isused, and some chaos characteristics that are difficult to beseen in time series data can be shown [4].

v(t) = (x(t), x(t + τ ), x(t + 2τ ), ..., x(t + (n − 1)τ )) (1)1-dimension (1D) time series data is converted into 3dimensions (3D) time delay coordinate system, and it is shownin Fig. 1. This 3D object is called attractor.

Fig. 1.

Reconstructed 3D attractor from time series data.

III. R ESIDUAL N ETWORKS
ResNet is commonly used for the classification of image
data. In this study, ResNet50 that has 50 layers is used.
Figure 2 shows the structure of ResNet50, and it is pretrainedby ImageNet dataset. To make the figure simple, the repetitionsof batch normalization and activation between convolution- 69 -layers are omitted. Dotted lines indicate the process to increase
dimensions with a stride of 2.

Fig. 2.

understanding, axes and labels are shown in Fig. 3 and 4, but
actual images of dataset does not include axes and labels andcolor bars. Dataset images have only attractors.

The structure of ResNet50.

Fig. 3.

The reconstructed 3D plot attractor.

IV. DATASETAs a chaotic data, Lorenz data which is famous chaotic
equations is used. These equations are shown the following
Eq. (2).
 dx
 dt = σ(y − x)dydt = x(ρ − z) − y dz
dt = xy − βz(2)

In this study, as the parameters, σ = 10 and β = 8/3 areused, and the dataset has three classes ρ = 27, 28, 29. The
data have different initial values. X-axis data is used, and thelength of each time series data is 500 time steps from time
step 1000 to 1500. These are embedded with time delay τ =10.
Train data and test data are divided randomly. Train datahas 1800 data and test data has 1200 data in total of 3 classes,and each train and test data have the same ratio of 3 classes.
Each image size is 224 × 224.
V. P ROPOSED M ETHOD

Fig. 4.

The reconstructed 3D plot attractor includes time information.

Figure 5 and Fig. 6 shows the another 3D line attractors.
It is drawn in 3D phase space which is different from aforementioned 3D plot attractors that express 3rd dimension bycoloring of plots. Figure 5 shows normal 3D line attractor,and Fig. 6 shows proposed 3D line attractor includes timeinformation.

By using Takens’ embedding theorem, time information islost. In our proposed method, time information will be addedon reconstructed attractors.
3rd dimension’s information x(t + 2τ ) is replaced by time
information t. Equation (3) shows the vector when time series
data is embedded into 3-dimension vector.
v(t) = (x(t), x(t + τ ), x(t + 2τ ))(3)Equation (4) shows the vector when includes time information that we propose.
v(t) = (x(t), x(t + τ ), t)(4)

Both of the 3rd dimensions of the vectors are expressed bycoloring of plot as shown in Fig 4. Figure 3 shows normal 3D
plot attractor with color plotting, and Fig. 4 shows proposed
3D plot attractor that includes time information. For easier- 70 -Fig. 5.

The reconstructed 3D line attractor.

Figure 8 shows the confusion matrix of proposed method
3D plot attractors dataset include time information.

Fig. 6.

The reconstructed 3D line attractor includes time information.

VI. S IMULATION R ESULTS
Batch size is 16, and reduce learning rate method is used.
Initial learning rate is 0.0001.
Table 1 shows the simulation results of 3D plot attractor
dataset, these are the average score of 3 times simulations.
TABLE IS IMULATION RESULT OF 3D PLOT ATTRACTOR DATASET
3D plot attractors
Proposed method

Train accuracy
0.92500.9570

Test accuracy
0.85470.8845The proposed method shows better accuracy than 3D plot
attractors dataset in both of train and test accuracies.
Figure 7 shows the confusion matrix when 3D plot attractors
dataset was used. Confusion matrix shows how many images
are predicted which labels, and it shows also which actuallabels.

Fig. 8. The confusion matrix of 3D plot attractors include time information.

There is more mis-labeled images on related ρ = 28 whichis the same tendency with 3D plot attractors dataset. The mostfrequent mis-labeled category is true ρ = 28 predicted ρ = 29.
Figure 9 shows differential matrix between 3D plot attractors dataset and proposed method. Differential matrix Cm
is calculated by Eq. (5). Cm1 is the confusion matrix of
3D attractors dataset, and Cm2 is the confusion matrix ofproposed 3D attractors include time information dataset.
Cm = Cm1 − Cm2

Fig. 9.

Fig. 7.

The confusion matrix of 3D plot attractors.

From Fig. 7, there is more mis-labeled images on related ρ= 28. Categories of true ρ = 28 predicted ρ = 29 and true ρ= 29 predicted ρ = 28 has more mis-labeled than others.

(5)

The differential matrix of two kinds of 3D plot attractors dataset.

From Fig 9, the mis-labeled categories true label ρ = 27
predicted ρ = 28 and true label ρ = 29 predicted ρ = 28 have
improved well compared with other mislabeled categories. On
the other hand, true label ρ = 28 predicted ρ = 29 has lessimprovement than others, even though it has more mis-labeledimages in both datasets.

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Table 2 shows the simulation results of 3D line attractors
dataset, these are the average score of 3 times simulations.

Figure 12 shows differential matrix between 3D line attractors dataset and proposed method. As same as the aforementioned differential matrix, it is calculated by Eq. (5).

TABLE IIS IMULATION RESULT OF 3D LINE ATTRACTORS
3D line attractors
Proposed method

Train accuracy
0.97330.9818

Test accuracy
0.9292
0.9739Train accuracy of the proposed method is slightly better,and test accuracy of the proposed method shows better result
than original 3D line attractors dataset.
Figure 10 shows the confusion matrix when 3D line attractors dataset was used.

Fig. 12.

The differential matrix of two kinds of 3D plot attractors dataset.

From Fig 12, the mis-labeled categories true label ρ = 27
predicted ρ = 28 and true label ρ = 29 predicted ρ = 28 have
improved well compared with other mislabeled categories.
This is the same tendency with 3D plot attractors dataset.
VII. C ONCLUSIONS

Fig. 10.

The confusion matrix of 3D line attractors.

There is more mis-labeled images on ρ = 28 as well as 3D
plot attractors dataset. 4 categories related on ρ = 28 havealmost same amount of mis-labeled images.
Figure 11 shows the confusion matrix when 3D attractors
line that include time information dataset was used.

In this study, we classified chaotic time series data Lorenz
data by ResNet, and we proposed the method that use 3Dattractors includes time information. Our proposed method
showed better accuracy than normal 3D embedded attractors
on both way of 3D plot attractors and 3D line attractors.
Then, we discussed the simulation result by using confusion
matrixes, and it is found that the mis-labeled categories truelabel ρ = 27 predicted ρ = 28 and true label ρ = 29 predictedρ = 28 have improved well by our proposed method. For
our future works, we would like to study other preprocessing
method to improve accuracy.
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Proceedings of the IEEE conference on computer vision and patternrecognition. 2016.
[3] Oyama-Higa, Mayumi, et al. ”Development of a Self-Check System for
Mental Health Using a Pulse Wave Mouse.” ICSOFT (ISDM/EHST/DC).
2007.
[4] K. Aihara, T. Ikeguchi, Y. Yamada and M. Komuro, “Fundamentals andapplications of chaotic time series analysis”, Industrial book, p.23, 2000.
[5] Chen, Yen-Lin, et al. ”High-Dimensional Phase Space Reconstruction
with a Convolutional Neural Network for Structural Health Monitoring.”
Sensors 21.10 (2021): 3514.

Fig. 11. The confusion matrix of 3D line attractors include time information.

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