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The practical fruits of multifractality are not precisely known yet but in some fields including finance interesting features of this phenomenon were shown (see, e.g., |
) that rise hope for interesting future applications connected with risk analysis. Therefore the questions regarding accuracy, applicability and reliability of multifractal measurements are crucial for proper analysis and interpretation of obtained results. |
Since the seminal paper by Kantelhardt, et.al., we know that multifractal properties one observes may appear not only as result of existing long-range nonlinear autocorrelations but also from the presence of fat tails in probability distributions of data or from linear autocorrelations present in shorter (finite) time ... |
). In fact the mutual interaction and interplay between these three sources of multifractal effects leads to observable multifractal spectrum. It is a nontrivial task to determine generally how these three ingredients relatively influence the measured multifractal features. We need some general method and analytical or... |
. They may significantly change its observed multifractal properties for all data lengths. Only the multifractality The expected level of multifractal artifacts existing due to finite-size effects and linear autocorrelations in time series was described generally from the quantitative point of view in series of papers |
In this paper we will make the similar quantitative analysis of spurious multifractality caused by different types of broad probability distribution of data including also asymmetric fat-tailed distributions. The latter ones are expected to occur in some real systems including financial ones |
It is worth to notice that in fact nonlinear effects also produce broad distribution of data what in turn influences multifractal phenomena in a way of specific feedback. Hence the statement that ”true” multifractality is generated only by nonlinear effects is somehow misleading. Nevertheless, we try to identify in thi... |
We will use the multifractal detrended fluctuation analysis (MFDFA) within this paper. MFDFA is now commonly accepted technique in searching for multifractal properties of data in time series. MFDFA has been applied so far in diversified scientific problems like, e.g., seismology |
, cosmology , biology , meteorology , medicine , music , geophysics , and mainly finances This technique is reported to have an advantage over the other known approach based on wavelets |
. Since it is described elsewhere (see e.g., ) we will only briefly recall it here. The main steps of MFDFA go as follows. Let [MATH] be a signal profile, where [MATH] is an analyzed time series and [MATH] denotes averaging over all [MATH] ’s. One divides [MATH] into [MATH] non-overlapping segments of length [MATH] (ti... |
[MATH] in each segment [MATH] and [MATH] -th order fluctuation function is calculated according to: [EQUATION] where [MATH] . In this paper, to make the results more readable, we use [MATH] , and apply the scaling range: [MATH] [MATH] (for synthetic data), [MATH] (for real data). For a signal with fractal characteristi... |
[EQUATION] where [MATH] is a generalized Hurst exponent. The bi- or multifractal stationary signals have [MATH] profile as a decreasing function of [MATH] ; if [MATH] the signal is called monofractal. |
Often the multifractal properties are presented in the Hölder language as the multifractal singularity spectrum [MATH] The singularity spectrum [MATH] can be calculated according to the following relations |
[EQUATION] where [MATH] is called the singularity (Hölder) exponent. The wealth of multifractality present in time series can be defined as a spread of the generalized Hurst exponent [MATH] . It is considered as dependent on the [MATH] parameter range |
[EQUATION] where [MATH] and [MATH] are respectively the minimal and the maximal value of the real deformation parameter [MATH] taken into account (usually the symmetric range [MATH] is proposed). |
The degree of multifractality can be also estimated by measuring the width of [MATH] spectrum [EQUATION] In the limit [MATH] both multifractal characteristics in Eqs.(4) and (5) coincide. |
Note, that in order to distinguish the deformation multifractal parameter [MATH] from the parameter used in ” [MATH] -deformed” fat tailed Tsallis distribution, which will also be used in this paper, the latter one will be denoted further on by [MATH] |
The paper is organized as follows. In section 2 we investigate the quantitative effect of broad distribution on multifractal characteristics of data within MFDFA. Two cases of fat tailed distributions are considered: stable ones with infinite variance (the Levy type of PDF or CDF) with the attractor made by Levy distri... |
Influence of symmetric and asymmetric broad distributions of synthetic data on registered multifractal outcomes To analyze various features of multifractality we will use time series of uncorrelated data drawn from [MATH] Gaussian distribution |
as well as time series of empirical, usually nonlinearly correlated data. For all data, we shall explore quantitatively the impact of heavy tailed asymmetric and symmetric probability distributions on multifractality and compare it with the effect of linear and nonlinear correlations present in a signal of finite lengt... |
The fat tails discovered in the real probability distributions in many complex systems including stock and money market and the complex character of the underlying temporal correlations indicate that the conventional concept of ergodicity may break down in the real dynamics. Under such conditions the generalized formal... |
may offer an appropriate framework to generate the corresponding time series. In non-extensive approach one is capable to pass in a compact and very economic way through all intermediate cases of fat tailed distributions just by altering the value of one parameter [MATH] as described below. For this reason we use time ... |
[EQUATION] The cumulative form of [MATH] Gaussian distribution is defined as follows [EQUATION] where, the [MATH] and [MATH] signs correspond to the right and left wings of the distribution, while |
[EQUATION] [EQUATION] [EQUATION] [MATH] [MATH] [MATH] [MATH] and [MATH] is the Gauss hypergeometric function. This type of family distributions develops asymptotically (for large [MATH] ) a power law behavior, contrary to exponential behavior characteristic for normal (Gaussian) distribution. Thus, for cumulative [MATH... |
[EQUATION] holds. Importantly, distributions of the uncorrelated [MATH] Gaussian signals, depending on [MATH] , are either in the Gaussian attractor regime (for [MATH] ) or Levy attractor regime (for [MATH] ). It means that sum of independent [MATH] random variables satisfying such distributions undergoes respectively ... |
In order to investigate the possible impact of symmetric (asymmetric) broad distributions on multifractal effects, we generated time series with symmetric and asymmetric distribution according to Eq.( ). It was assumed that the asymmetric distribution is one for which the right tail (positive fluctuations) has a normal... |
the spurious multifractal spread [MATH] due to presence of FSE is limited then to [MATH] for [MATH] and [MATH] for [MATH] Examples of the fluctuation function ( [MATH] ), the multifractal spectrum ( [MATH] ) and the generalized Hurst exponent ( [MATH] ) (both for [MATH] from the Gaussian and Levy regimes) for symmetric... |
For all [MATH] a good power-law dependence of [MATH] was observed for the scales [MATH] . A small variability of the slope coefficients, both for symmetric and asymmetric case in Gaussian regime indicate a monofractal character of the analyzed data in this region. Examples for [MATH] and [MATH] are given for this case ... |
It is seen at the level of [MATH] for [MATH] where the value of [MATH] , i.e., the righthanded edge of spurious multifractal spectrum is clearly shifted to the left in the case of asymmetrical distribution (compare middle-bottom panels in Figs. and ). |
While in the Gaussian regime the effect of (a)symmetry of probability distribution does not significantly affect multifractal properties ( [MATH] ), the remarkable differences are visible in the Levy area. For a symmetric case, we observe the dependence [MATH] in excellent agreement with theoretical prediction, i.e., [... |
. In contrast, when one of the tails of the distribution has a shape of normal PDF (the asymmetric case) [MATH] , where [MATH] (see Fig. ). The smaller value of this exponent in the latter case indicates that asymmetrical distribution significantly depletes the multifractal nature of the analyzed data. When [MATH] decr... |
To look more closely at the impact of the phenomenon of symmetry, asymmetry and fat tails of probability distributions on measured multifractal features, we study the generalized Hurst exponent dependent on [MATH] parameter ( [MATH] ) and the properties of its spread [MATH] |
Fig. shows the main results obtained for variety of fat tailed PDF. From the perspective of many [MATH] , it can also be seen that the Hurst exponent profile [MATH] strictly depends on [MATH] (or [MATH] ) and the (a)symmetry of distributions. The stronger effect is obviously seen in the Levy attractor regime. The value... |
[MATH] for the asymmetric case and [MATH] for the symmetric case respectively. To make this analysis more exhaustive from quantitative point of view, we have shown the spread [MATH] vs the moment [MATH] in logarithmic scale in Fig. . This scale more clearly distinguishes several ranges of [MATH] parameter for which the... |
Looking first at distributions from Gaussian attractor (top panels of Fig. ) we see that for small range of deformation parameter [MATH] dependence [MATH] on the maximal moment [MATH] used to calculate such spurious multifractal spread may be well approximated by a power law |
[EQUATION] where [MATH] for all distributions in this attractor. The corresponding values of [MATH] coefficients and [MATH] exponents are collected in Table 1. Note that plots for [MATH] and [MATH] correspond to PDF close to normal distribution so that shown result recreates in fact the spurious multifractality [MATH] ... |
The case of broad PDF from Levy attractor regime is different (see bottom panels in Fig. ). The saturation of multifractal spread [MATH] occurs at the same level of [MATH] independent on the shape of PDF for asymmetric case. Contrary, for the symmetric case the level of saturation depends on [MATH] exponent and is give... |
Application for detecting multifractal components of real financial signals In this section we will apply the previous general findings to study multifractal ingredients of real empirical financial data. In the beginning we shall present an application of findings from the previous section to multifractal analysis of p... |
We created unweighted index [MATH] based on high frequency price data from the companies listed in Dow Jones Industrial Average (DJIA). The time interval between consecutive records for all companies has been chosen as [MATH] . We considered price returns of 30 companies (AA, AIG, AXP, BA, BAC, CAT, CSCO, CVX, DD, DIS,... |
[MATH] As another standard procedure, we calculated normalized and centered returns [MATH] defined as [MATH] where [MATH] is the standard deviation of returns over the period [MATH] and [MATH] denotes a time average. In addition, all overnight returns have been removed, because they cover a much longer time interval in... |
For comparison we also present a systematic study of such characteristics for the Polish stock market index WIG20 (Warszawski Index Gieldowy - Warsaw Stock Market Index) over the period Nov. 17, 2001 – Feb. 13, 2018 for the time lags [MATH] . This market is commonly classified as still emerging but at least with no dou... |
The cumulative distribution function (CDF) of [MATH] of moduli of DJIA and WIG20 price returns collected from the whole period specified above is shown in Fig. and Fig. . We present the distributions of moduli of the returns because the distributions of positive and negative fluctuations turned out to be almost symmetr... |
It can be seen that the tails of all distributions are relatively thick and vanish according to the power law [MATH] . The fat tails of distributions clearly indicate that the nature of the moduli of logarithmic price returns importantly differs from the Gaussian one. Moreover, from the [MATH] Gaussians point of view, ... |
In order to check the possible impact of phenomena like: linear and non-linear correlations, FSE and the effect of broad probability distributions on multifractal character of the considered real data for different time-lags, we first calculated the generalized Hurst exponent and its spread [MATH] in a range of moments... |
the profile [MATH] and the spread [MATH] for the synthetic series of the same length (denoted further on as [MATH] ) but drawn from Gaussian distribution and with the same level of linear autocorrelations as empirical series [MATH] . This can be done with the help of Fourier filtering method (see, ref. |
for details). The series [MATH] contributes to multifractal spectrum only with spurious multifractality related to short length of data (FSE) and to the involved linear autocorrelations. Its spread [MATH] is given as |
[EQUATION] where [MATH] and [MATH] is the main Hurst exponent. The values of all parameters in calculations are taken from ref. Finally, we calculated the spread [MATH] of spurious multifractality related to effects of broad data distribution only. This was based on synthetic series (labeled as [MATH] ) generated in se... |
The latter series has a probability distribution consistent with real data but neither linear nor non-linear autocorrelations are present in it. We also assumed that the series [MATH] does not involve the spurious multifractality caused by FSE because it is much longer ( [MATH] data points) than the real data length an... |
Having this in mind one can evaluate the upper threshold of multifractal effects associated only with all spurious ingredients and finally the ”true” multifractality related only with non-linear correlations which is the most interesting. These results are provided in Fig. 13 (for DJIA) and Fig. 14 (for WIG20). |
Apart from the initial multifractal spread [MATH] made for empirical data within MFDFA, we show in this figure also the ”true” multifractal spread for these data [MATH] . The latter one corresponds to multifractal content of examined series after all spurious effects induced by FSE or by broad distribution of data are ... |
It is worth mentioning that the tails of PDF of absolute returns for WIG20 are found thinner and more deformed than in the case of DJIA – in particular for the minute time scale. This, in turn, influences [MATH] values which are smaller than corresponding ones for DJIA index. Nevertheless the ’true’ multifractal conten... |
As a final example we performed the similar analysis for other kind of data taking the exchange ratios of three currencies from Forex with respect to USD: EUR/USD, GBP/USD and RUB/USD in the period 01.01.2014–31.12.2016 (with the trading hours 8:00 - 22:00) with time-lags of exchange returns [MATH] min ( [MATH] data po... |
In all considered cases the ’true’ multifractal content of data connected with nonlinear properties is seriously reduced with respect to naive description before the spurious effects are subtracted (compare the first and the last column in Table 3). The highest relative influence of spurious effect of heavy tail in PDF... |
[EQUATION] [EQUATION] Concluding remarks The goal of this article was the detailed quantitative analysis of spurious multifractal effects induced by the presence of broad distribution of data in time series. To make this analysis more close to practical application we analyzed PDF with heavy tails [MATH] Gaussian distr... |
This paper has been divided into two major and in some way related parts. The first part concerned the analysis of synthetic data [MATH] Gaussians distributions. In this part we have explored quantitatively the impact of heavy tailed symmetric and asymmetric probability distributions on multifractality. We assumed that... |
Regarding analysis of the whole multifractal spectrum of data from broad distributions we showed that the spurious multifractal effects induced by this kind of PDF can be well described quantitatively by a power law linking the spurious multifractal spread [MATH] expressed in Hurst language with the multifractal deform... |
Going in this direction we finally provided examples from stock market (DJIA and WIG20) and money market (Forex) indicating the real multifractal content of empirical signal in time series against its spurious constituents. The main conclusion to be drawn from the real data analysis presented in the second part of the ... |
Therefore, one should be very careful drawing conclusions from the multifractal analysis and interpretation of the observed multifractal spread in any complex system. In particular, from a practical point of view, the effects that we quantitatively described can have applications in modeling and forecasting the widely ... |
[EQUATION] Acknowledgement This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge (University of Rzeszów). |
# Source: arxiv 1805.11954 # Title: Long Short-Term Memory Networks for CSI300 Volatility Prediction with Baidu Search Volume # Sections: all # Downloaded: 2026-03-03T04:47:24.852547+00:00 |
Long Short-Term Memory Networks for CSI300 Volatility Prediction with Baidu Search Volume Abstract. Intense volatility in financial markets affect humans worldwide. Therefore, relatively accurate prediction of volatility is critical. We suggest that massive data sources resulting from human interaction with the Interne... |
Corresponding author: Ren-Jie Han, 512910603@qq.com 1. Introduction Volatility prediction in financial markets is of great practical and theoretical interest. Volatility plays crucial roles in financial markets, such as in derivative pricing, portfolio risk management, and hedging strategies. Therefore, it is demanding... |
According to Herbert Simon, actors begin their decision-making process by attempting to gather information . Nowadays, information gathering often consists of searching online sources. Hence search volumes of key words related to finance may reveal market sense and and focus of investors. Similar to the Google Trends, ... |
In this study, we investigate the intriguing possibility of analyzing search query data from Baidu Index, modeled by Long Short-Term Memory neural network, to show the feasibility of predicting the stock market volatility through the search volumes. To our best knowledge, there has been no previous attempt to deploy LS... |
The increasing volumes of ’big data’ reflecting various aspects of our everyday activities represent a vital new opportunity for scientists to address fundamental questions about the complex world we inhabit |
. Baidu index and Google search volumes, do not only reflect aspects of the current state of the economy, but may also provide some insight into future trends in the behavior of economic participants. Yu, Zhang (2012), taking Baidu search terms as an agent variable of personal investor concern, found that Baidu search ... |
. Baidu search volume is also used as a proxy variable for individual investors’ attention in Zhao, Lu and Wang(2013). The authors analyzed the relationship between Baidu search volumes and the stock returns of 1301 stocks in Growth Enterprises Market Board, and found that there is a positive correlation between search... |
. Da et.al(2011), proposed a measure of investor attention using search frequency in Google (Search Volume Index(SVI)), provided evidence that SVI captures the attention of retail investors, and found the relation between investor attention and asset prices |
. Using historic data from the period between Jan 2004 and Feb 2011, T Preis et.al(2013), found that detectable increases in Google search volumes for keywords relating to financial markets before stock market falls |
Prediction tasks on financial time series are notoriously difficult, primarily driven by the high degree of noise and the generally accepted, semi-strong form of market efficiency |
. Meanwhile, there are plenty of well-known capital market anomalies that are in stark contrast with the notion of market efficiency. In the past years, initial evidence has been established that machine learning techniques are capable of identifying (non-linear) structures in financial market data |
. Petersen, A et.al(2017) applied LSTM networks to all S&P 500 constituents from 1992 until 2015 2. Data description and preprocessing |
In this work, we study the CSI 300 index based on publicly available daily data comprising high, low, open, close, and close prices. Daily returns [MATH] are evaluated as the log difference of the close price, while daily volatility [MATH] is estimated using the high, low, open and close prices in the following equatio... |
[EQUATION] [EQUATION] We remark that this definition is the best among all quadratic combination under some criteria Starting from June 1st 2006, Baidu has been collecting the daily volume of searches from personal computer related to various aspects of macroeconomics. This database is available to the public as the Ba... |
and have shown correlations between Baidu index and the equity market. In this work, we use this trend data as a representation of the public interest in various macroeconomic factors. |
For this study, we include 28 domestic trends which are listed in Table 1 with their abbreviations. We use [MATH] to denote the aggregated data, |
[EQUATION] We split the whole data set into a training set (80%) and a test set (20%). The training set ranges from 1-June-2006 to 17-July-2015 while the test set ranges from 20-July-2015 to 27-Oct-2017. Additionally, it is worth noting here that all these 30 time series are stationary in the sense that their unit-root... |
Preprocessing the time series with different observation interval and normalization window may cause corresponding difference of causality between the input and output. Let [MATH] be the observation interval |
[EQUATION] [EQUATION] [EQUATION] In this study, we aim to predict volatility, so we denote the next period volatility by [MATH] [EQUATION] |
We use moving average values to normalize the above observed data. With a look-back window [MATH] , a time series [MATH] is normalized to [MATH] defined by |
[EQUATION] Each combination of [MATH] and [MATH] should determine an observation and normalization scheme with its unique predictive power. We denote these schemes as [MATH] and the resulting data as [MATH] In principle, one may apply learning models on each scheme and evaluate the accuracy of prediction on a validatio... |
Let us make a brief introduction to mutual information. For any discrete random variable pair [MATH] , let [MATH] be the joint probability function of [MATH] . Let [MATH] and [MATH] be the marginal probability function of [MATH] and [MATH] respectively. The mutual information between [MATH] and [MATH] is defined as |
[EQUATION] Assuming conditional independence between the input variables in [MATH] , the mutual information can be broken down into a sum of the individual components of [MATH] with [MATH] Therefore we choose [MATH] to maximize |
[EQUATION] One may try to use the time series [MATH] to empirically compute [MATH] according to ( 2.6 ). However, note that the values in [MATH] should be unique due the accuracy of real data. More precisely, one has |
[EQUATION] Then a direct calculation gives [EQUATION] which only depends on the sample size. Thus applying mutual information in this way can not reveal any relation between [MATH] and [MATH] . Considering this, we divide the data into small groups and regard values in each group as one point. Precisely, if [MATH] is a... |
[EQUATION] Similarly, divide the interval [MATH] evenly into [MATH] subintervals [MATH] . Then we define the marginal law function as |
[EQUATION] and the joint law function as [EQUATION] Then we define the empirical mutual information between series [MATH] and [MATH] as |
[EQUATION] Note that [EQUATION] which indicates we should not take too large [MATH] . In our study, we take [MATH] Figure shows the mutual information for different combination of [MATH] . Clearly, when the normalization window [MATH] is around 5, the mutual information is maximized. Another obvious phenomenon is that,... |
[EQUATION] Note that [MATH] corresponds to weekly return and volatility, which is also a consideration of our choice. 3. Methods |
LSTM networks, introduced by Hochreiter and Schmidhuber (1997) and were furthered in the following years by Gers et al. (2000) and Graves and Schmidhuber (2005), belong to the class of recurrent neural networks (RNNs). LSTM networks are designed to learn long-term dependencies and are capable of vanishing and exploding... |
LSTM networks contain an input layer, one or more hidden layers, and an output layer. The number of explanatory variables (feature space) equal to the number of neurons in the input layer. The output space is determined by the quantity of neurons in the output layer. The hidden layer(s) contains the memory cell, which ... |
In figure [MATH] stands for the input vector at time step t, the information flow [MATH] and the volatility estimation [MATH] are computed from the former step. [MATH] and [MATH] are weight matrices, [MATH] and [MATH] are bias vectors; [MATH] represent the values of each gate; [MATH] and [MATH] are the cell gate and ca... |
At step 1, the previous cell state [MATH] is determined by the LSTM layer how much it should be forgotten. Given [MATH] , the bias term of the forget gate, [MATH] can be computed as |
[EQUATION] where the function [MATH] is defined by [EQUATION] At step 2, the LSTM layer determines which information should be added to the network’s cell states: |
[EQUATION] [EQUATION] Here [MATH] In the last step, the output [MATH] is computed through the following two equations: [EQUATION] |
[EQUATION] Through ( 3.12 ) and ( 3.13 ) the volatility [MATH] will be predicted. The fundamental of this time series forecasting is |
[EQUATION] We apply the deep learning library Keras to estimate the coefficients by training in python 3. Specifically, the lag of the LSTM is set at 50, and the bach contains 5 examples, time step is 5. The objective loss function we choose in the model is mean absolute percent error (MAPE). When we set the epochs at ... |
[EQUATION] 4. Results and discussion In figure , the observed volatility together with the predicted values is plotted. As we can see in the figure, the predicted values fit the actual volatility in decent accuracy, especially when the actual volatility is small. |
As we have indicated in the last chapter, MAPE as the loss function, is shown in table . In terms of mean square error (MSE), the LSTM also performs better than the benchmark model. |
Our LSTM model avoids significant over-fitting as the MAPE evaluated in the test set is 17%, which is close to the MAPE (%15.6) in the training set. The MSE of the LSTM model is 2% which is far smaller than the benchmark model, as listed in table |
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