Time Series Analysis with Noise-Mixing Effects
Using Neural Networks
Takuya Nakamura, Ryosuke Shimizu, Yoko Uwate and Yoshufumi Nishio
Dept. of Electrical and Electric Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-5806, Japane-mail:{takuya, shimizu, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— One-dimensional convolutional neural networks (1DCNNs) are used for time series analysis. However, noise mixed inthe data can interfere with time series analysis. We propose amethod of injecting noise into the original data and theautocorrelation function data for feature extraction of training
data. The test accuracy of neural network is compared with theoriginal data and the autocorrelation function data.

Step 2. The data to be trained is replaced withautocorrelation functions from the original data.
Step 3. The autocorrelation function is trained on the 1dResNet.
A.

Autocorrelation Function
The autocorrelation function (ACF) is defined as theproduct of time t and the data shifted by k from t in time series
data. The correlation between the current data and the data ofthe past time shifted by k is examined.
The autocorrelation function is expressed by the following Eq.
(1).

Keywords; neural networks; time series analysis;

I. INTRODUCTION
There are many phenomena in the world that have time-series
characteristics, such as temperature and seismic waves. Manystudies have been conducted to analyze the characteristics of
those time series. As a method to observe those characteristics,
many time series have been converted into data and time series
analysis using neural networks (NN) such as recurrent neuralnetworks (RNN) and one-dimensional convolutional neuralnetworks (1D-CNN) have been studied in recent years.
However, the data often contain information that is not needed
for analysis, called noise. If the amount of noise mixed in thedata is too large, time series analysis becomes complicated andvery troublesome. In this study, the classification accuracy of theneural network is investigated when noise is injected into thetraining data.

∑∑

̅

̅

̅

1

n is the sampling number and k is the time lag.
Figure 1 shows the example of the original data. Figure 2 shows
autocorrelation of the original data.

II. ONE-DIMENSIONAL CONVOLUTIONAL NEURAL NETWORKS
Convolutional neural network (CNN) is a structure that adds
a process called convolution and pooling to conventional NN.
CNN is mainly used for image recognition, natural language
processing, and time series analysis. However, since images are
two-dimensional data in image recognition, the number of input
data increases and the computational cost increases. Therefore,
the use of a one-dimensional convolutional neural network (1DCNN), which allows the number of data to be input as it is,makes it possible to reduce the computational cost. In addition,a 1d-residual network (1d-ResNet) is used to prevent gradientloss problems.

Figure 1. Original data.

III. PROPOSED METHODSIn this study, the following approach is taken to checkclassification accuracy in the presence of mixed noise.
Step 1. White noise is added to the original data and theautocorrelation function.

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Figure 2. Autocorrelation function (ACF).

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B.
White NoiseWhite noise is used in this case. One of the characteristics ofwhite noise is that the energy is uniformlypresent in all
frequency bands. White noise is often mixed in with data suchas audio. We investigate the learning of 1d-ResNet when white
noise is added to the training data.

be seen that as the noise rate increases, the classification
accuracy decreases from approximately 64[%] to 62[%] in bothcases. Although there are some areas where the original data
has higher classification accuracy, overall, the classification
accuracy is higher using the autocorrelation function.
TABLE II.

IV. VERIFICATION SIMULATION MODEL
In this study, two types of classification accuracy are tested
using original data for training data and using an autocorrelation
function, and original data is used for test data. A dropout layer
is also used to prevent over-training.
The 1d-ResNet is used as the classification model 1d-resnet
is used to solve the gradient loss problem, which is a problem
of multilayer networks, by learning residuals. Figure 3 shows
the structure of residual learning. Convolution(1 × 1, 128)
means convolution layer with 1×1 filter and 128 channel.

ACCURACY OF PROPOSED METHOD AND
AUTOCORRELATION FUNCTION

Rate of noiseClassification
accuracy(ACF)[%]Classification
accuracy(Original)[%]

0.0

64.81

64.52

0.1

64.31

64.30

0.2

64.37

63.00

0.364.73

64.98

0.4

64.30

64.45

0.5

63.83

62.88

0.6

64.65

61.27

0.7

63.13

62.56

0.8

63.66

61.63

0.9

62.09

62.71

1.0

62.65

62.11

VII. CONCLUSION
In this study, 1d-ResNet is trained on training datacontaining white noise using an autocorrelation function. Theresults showed that the classification accuracy is higher when
the autocorrelation function is used. As a future work, we would
like to conduct similar validation with algorithms other than 1dResNet.

Figure 3. Structure of residual learning.

V.

DATASETAs a chaotic data, Lorenz data, which is famous chaotic
equation is used. These equations are shown the following Eq.
(2).
⎧⎪

REFERENCES
[1]2⎨
⎪
⎩

[2]

[3]

In this study, as the parameters, σ = 10 and β = 8/3 are used,
and the dataset has threee classes that have the different ρ, andrecognize the parameters differences by using 1d-resnet.Table
I shows the number of the train data and test data.

[4]

T. Furugaki, Y. Uwate and Y. Nishio, “Time Sries Classification Using
Autocorrelation Function as Training Data in 1D-CNN”, 2020.
K. He, X. Zhang, S. Ren, J. Sun, “Deep resiual learning for imagerecognition”, In Proceeedings of the IEEE. Conference on computer
vision and pattern recognition, pp.770-778, 2016.
T. Furugaki, “Influence of Surrogate Data and Autocorrelation Function
on Classification Accuracy of Neural Networks”, 2021.
M. Matsugu, K. Mori, Y. Mitari and Y. Kaneda, “Subject independent
facial expression recognition with robust face detection usnig a
convolutional neural network”, Neural networks 16, vol.5, pp.555-559,
2003.

TABLE I. THE NUMBER OF THE TRAIN DATA AND TEST DATA
DataACF

Original

Train data

1800

1800

Test data

1200

1200

VI. SIMULATION RESULTS
Table II shows the autocorrelation function and theclassification accuracy of the original data, verified with noise
rates from 0.0 to 1.0 in increments of 0.1. From Table II, it can

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