Keywords

1 Introduction

In this decade, it is not necessary to have technical knowledge for the investment since the automatic algorithms to sell/buy investment destination have been developed with artificial intelligence (AI). However, these kinds of mechanical trading systems may not support variations realized in future because the systems were developed with use of time series data of the investment outlets in the past, or time sequences generated by stochastic processes to verify the systems. Therefore, we considered applying the generative adversarial network (GAN), which has attracted attention in the field of image generation, to time series of the exchange rate. Learning the properties of variations in the exchange rate whose factors are not elucidated in detail by GAN, the pseudo exchange rates were generated to use as the data for verification of the mechanical trading system.

In previous study, measuring similarity (stationarity, fractality, and degree of determinism) of variations in the exchange rates to the pseudo-exchange rates generated by GANs, we compared Winner processes with the GANs [1]. From the viewpoint of stationarity, the similarity of sequences in the pseudo exchange rates were higher than those generated by the Winner processes, and high scores in the similarity were resulted from both sequences in terms of degree of determinism.

February 2, 2018 marked the day of the largest Dow Jones Industrial Average decline [2]. The exact reasons for this decline remain unknown, but one is thought to be the use of artificial intelligence (AI), or automated trading algorithms, to continuously sell shares (mechanical trading) [3]. In recent years, mechanical trading has not been limited to major hedge funds. For example, many financial institutions (such as securities firms) provide mechanical investment services to general consumers. Mechanical investment offers the advantage of preventing the consumer from having to manage the question of what and when to buy. However, there is the risk that mechanical investment may not be able to predict future fluctuations. In general, automated trading algorithms are tested against past fluctuations to evaluate their effectiveness, but there is the possibility of overfitting if the system solely relies on past fluctuations. Overfitting is a common problem with machine learning. Thus, to prevent this problem, a test may be performed on a large number of time series generated by using a stochastic process. Many researchers have performed studies in which stock prices are considered stochastic processes [4, 5]. However, a time series generated by a stochastic process does not reflect actual stock prices and exchange rate fluctuations, making it unreliable. Therefore, in this study, we used stochastic process-generated time series data.

Current research also includes the use of neural networks to classify images [6]. In 2014, the concept of generative adversarial networks (GAN) was proposed by Goodfellow et al. [7]. Additionally, the amount of research that entails using neural networks to generate images has been increasing. In this study, G is defined as a network that generates simulative sequences (SSs, i.e., fake data) from input noise, and Network D distinguishes whether the data generated by the generator G is the desired real data (i.e., true data). The generator G learns as the discriminator D mistakes the SS with the true data, and D is trained to correctly distinguish between the true and SS. Repeated training of Networks G and D (Fig. 1) can result in the generation of a large amount of SSs if the output of G can generate data that are very similar to the true data.

Fig. 1.
figure 1

GAN model.

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Therefore, we thought that we could improve the reliability of automated trading algorithms by examining various fluctuation patterns using a GAN-generated pseudo time series.

This study was purposed with designing an artificially intelligent GAN by generating one-dimensional time series data that are similar to the true exchange rate data. We evaluated the pseudo exchange rates, as generated by a model based on stationarity [8], fractality, and the degree of determinism [9], in comparison with the actual exchange rate and time series data generated by a stochastic process.

In this study, we apply this AI system to numerical simulations of bio-signals such as stabilograms.

2 Model Design

GAN learning is reported to be difficult to stabilise. In fact, we designed and trained several models in an effort to generate pseudo exchange rates, but there were few models that exhibited stable learning.

Alec Radford et al. provided some suggestions regarding how to stabilise GAN learning [10]. In this study, we used the hyperbolic tangent function (i.e., tanh) as the activation function in the output layer of the generator. With the exception of the output layer/fully connected layer, LeakyReLU [11] was used to design the discriminators, each of which act as an activation function. To design a model, various parameters need to be set. Moreover, because the accuracy of the generated pseudo exchange rate is dependent on the parameters, it is necessary to optimise various parameter settings. Neural network model optimisation often entails using the accuracy rate as an objective variable for classification and prediction [12]. However, the purpose of this study was to identify the characteristics of exchange rate fluctuation, and generate a pseudo time series that simulates the fluctuations. Therefore, GAN learning cannot be evaluated based on the degree of similarity between the actual exchange rate and generated pseudo exchange rate. In addition, it is difficult to describe the characteristics of exchange rates because the factors and systems of exchange rates are not clearly understood. Therefore, an objective variable must be defined. In this study, we discovered two metrics that would facilitate network training. These metrics are as follows: 1) the total output errors for the generator and discriminator training data must be small, and 2) lower output error values for the generator and discriminator training data indicates stable learning. Thus, we developed and optimised the optimisation function (1), which takes into account these two metrics:

$$ OptimisationFunction\left( {G_{Loss} ,D_{Loss} } \right) = ln\frac{{G_{Loss} + D_{Loss} }}{{D_{Loss} /G_{Loss} }} $$
(1)

\( G_{Loss} \) is the training error for the generator network, and \( D_{Loss} \) is the training error for the discriminator network.

In this study, the number of convolutional layers (i.e., four), number of filters per layer (16, 32, 64, 128), and filter sizes (1–10) were set to minimise the value of the optimisation function for the generator model (Fig. 2). The value was evaluated after the parameters were optimised. It should be noted that, because of the high computational expense, the discriminator model was not optimised (Fig. 3).

Fig. 2.
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Generator

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Fig. 3.
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Discriminator

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3 Empirical Study in Stabilometry

We studied the effects of stereoscopic video clips on the elderly. In addition to radial motion, body sway was simultaneously measured while the young and the elderly viewed stereoscopic video clips. The results showed that, in the elderly, the equilibrium function is affected by tracking the visual target in stereoscopic video clips.

3.1 Experiment 1

As a basic study, the stabilometry was conducted for 238 elderly people that stood with Romberg posture on a gravicorder GS3000 (Anima Corp. Ltd., Tokyo). Stabilograms were recorded at 20 Hz sampling with their eyes open/closed for 60 s, respectively. This experiment was approved by the Ethics Committee of Graduate School of Information Science, Nagoya University.

3.2 Experiment 2

In this experiment, the experiment was conducted for twelve healthy volunteers (6 young and 6 elderly) that were 22.5 ± 1.0 yrs. (mean ± standard deviation) and 75.0 ± 8.2 yrs. of age, respectively. Beforehand, the experiment was fully explained to the subjects that could view stereoscopically, and written consent was obtained. The experiment was also approved by the Ethics Committee of the Department of Human and Artificial Intelligent Systems in the Graduate School of the Engineering University of Fukui (No. H2019003). Stabilometry and radial movements were simultaneously measured and recorded at 100 Hz and 60 Hz sampling rates, respectively.

Stereoscopic images used for this experiment was recreated based on Sky Crystal (Olympus Memory Works Corp, Tokyo) with permission (Figs. 4). The 3D video clips:

Fig. 4.
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Visual stimulus; a normal image (a), an image with static regulation of backgrounds (b)

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  • VC1) A normal 3D video clip with full backgrounds (Fig. 4a)

  • VC2) A 3D video clip with the static regulation of backgrounds (Fig. 4b)

On a liquid crystal display (LCD), 55UF8500-JB (LG Electronics, Seoul), were played in visual pursuit for 60 s or in the peripheral vision for 60 s in a dark room. In the VC1, the peripheral visual field was compulsory narrowed. An order effect was herein excluded in the protocol of this experiment in which a test with the subjects’ eyes closed was conducted after each simultaneous measurement.

In this experiment, the stabilometry was conducted by using a Wii balance board (Nintendo, Kyoto). Typical example of stabilograms were shown in Fig. 5a.

Fig. 5.
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Typical Stabilograms; data for 1 min with eyes open (a), simulation patterns after 10,000 step (b), simulation patterns after 20,000 step (c), simulation patterns after 30,000 step (d).

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Also, we used an eye mark recorder, EMR-9 (Nac Image Technology, Tokyo) to measure the radial movement. The position of the viewpoint for each sampling time is composed of x-y coordinate [pix]. Total locus length, area of radial sway, and total locus length per unit area were evaluated as well as the analysis of the body sway. Also, we performed statistical tests for each analytical index. The significance level was set to be 0.05.

4 Results and Consideration

The elderly voluntary participated in this study. Their stabilograms were recorded while standing with Romberg posture (Fig. 5a). In previous studies, the mathematical models of the body sway have been described by stochastic processes on the basis of the following properties for each component;

  1. (i)

    Markov property.

  2. (ii)

    non-anomalous diffusion.

In stabilograms, variables x (right designated as positive) and y (anterior designated as positive) are regarded to be independent [13]. A linear stochastic differential equation (Brownian motion process) has been proposed as a mathematical model to describe body sway [14,15,16]. To describe the individual body sway, we show that it is necessary to extend the following nonlinear stochastic differential equations:

$$ \frac{\partial x}{dt} = - \frac{\partial }{\partial x}U_{x} \left( x \right) + \mu_{x} w_{x} \left( t \right), $$
(2)
$$ \frac{\partial y}{dt} = - \frac{\partial }{\partial y}U_{y} \left( y \right) + \mu_{y} w_{y} \left( t \right), $$
(3)

where \( w_{x} \left( t \right) \) and \( w_{y} \left( t \right) \) express the white noise [17]. The following formulas describes the relationship between the distribution in each direction, \( G_{x} \left( x \right) \) and \( G_{y} \left( y \right), \) and the temporal averaged potential constituting the stochastic differential equations (SDEs):

$$ U_{x} \left( x \right) = - \frac{{\mu_{x}^{2} }}{2}{ \ln }G_{x} \left( x \right) + const., $$
(4)
$$ U_{y} \left( y \right) = - \frac{{\mu_{y}^{2} }}{2}{ \ln }G_{y} \left( y \right) + const. $$
(5)

The variance of stabilograms depends on the temporal averaged potential function (TAPF) with several minimum values when it follows the Markov process (i) without abnormal dispersion (ii). SDEs can represent movements within local stability with a high-frequency component near the minimal potential surface, where a high density at the measurement point is expected. In the numerical analysis of Eqs. (2)–(3), the SDE was rewritten to the difference equation in which the term \( w_{x} \left( t \right) \) or \( w_{y} \left( t \right) \) was substituted into pseudorandom numbers produced by the white Gaussian noise [17] or the 1/f noise [18].

In the experiment 1, we have succeeded in findings of the mathematical models of the body sway in the elderly with use of the GANs. It was confirmed that training was not stable in the created GAN-model because of the small number of real data. Therefore, two-dimensional noise was firstly generated by the independent Winer processes; 1,000,000 kinds of time sequences were provided for each component by the

Wiener processes. Substituting the noise into the generator G, fine-training has been secondly conducted for the optimal parameter of the neural network as a generator/discriminator of the GANs (Table 1 and 2). After the fine-training, training was performed using 204 stabilograms measured in the experiment 1–2.

Table 1. Optimal parameter of the neural network as a generator of the GANs
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Table 2. Optimal parameter of the neural network as a discriminator of the GANs
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Using the stabilograms stated in the last section, machine learning was thirdly conducted to generate simulative stabilograms by the generator G and to distinguish these fake data with measured stabilograms. Lastly, all stabilograms were evaluated by the translation error estimated in accordance with the Wayland algorithm [19].

According to the Wayland algorithm, values of the translation error were distributed around 0.7, which was resulted from both components of measured stabilograms (Fig. 5a). As shown in Figs. 6, these values were greater than those estimated from x-component sequences in the simulative stabilograms after iterations (<20,000 steps) on the deep-learning of the GANs (Fig. 5b–c). The values of simulative translation error were distributed around 0.7 after 60,000–70,000 iterations, which did not depend on the component. However, we could find decrease in the values of simulative translation error with increase of iterations (>70,000 steps).

Fig. 6.
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Translation error estimated from the following x-component in each embedding space; measured stabilograms for 1 min (a), simulation patterns after 20,000 step (b), simulation patterns after 70,000 step (c), simulation patterns after 100,000 step (d).

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In the experiment 2, we recorded the radial motion while viewing stereoscopic video clip. The radial motion of the elderly was quantitatively different from that of the young. We herein applied this AI system to numerical simulations of stabilograms. In the next step, this AI system can be applied to numerical simulations of figure patterns of the radial motion in order to evaluate the anomalous radial motion due to the deterioration of visual function with aging. Also, the AI system can be applied to numerical simulations of bio-signals such as time sequences measured by 3D motion capture, electrocardiograms (ECGs), and electrogastrograms (EGGs). This study shows an example of the application.