Maximum entropy spectral analysis forex

). Configurational entropy has also been applied for spectral analysis and shown to have a better resolution than BESA for autoregressive. Request PDF | A maximum entropy method to assess the predictability of financial and commodity prices | A novel signal processing method for the analysis of. Foreign-Exchange Rate Dynamics: An Empirical Study Using Maximum Entropy Spectral Analysis Academic Article uri icon. View record in Web of Science ®.
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The problem is exacerbated when a decision needs to be made in seconds. It should be noticed that the smaller the time window, the higher the share of noise in the total information observed. Among the methods dealing with the noise reduction in financial data are various technical indicators, combinations of market indicators, and symbolic representation of the data. The last approach is especially useful in the case of decision support systems used to aid the traders in the decision process.

However, one of the main drawbacks of such an approach is a visible limitation of the information delivered to the decision-maker. For instance, Japanese candlesticks visualizing high-low-close-open prices are one of the most popular charts used in the technical analysis.

Renko or Heiken Ashi charts are also in the category of the simplified price representation. The opposite side includes concepts like Ichimoku charts [ 7 ], which extend the information derived for the decision-maker and can be used as a confirmation signal. Other technical indicators fall in the same category, where some additional information is added for the decision-maker, while the simplification of the initial price chart is rarely seen.

Each indicator has a different informative value for the investor, and their selection is more a matter of individual preferences than an objective assessment of effectiveness. In the article, we suggest using entropy as an objective measure of the amount of information that various indicators and different ways of describing the time series carry.

Because there are dozens of different ways to describe time series and dozens of different indicators, in this article, we discuss the narrower problem of symbolic representation of the data based on the relative and discretized price values. It is commonly assumed that the symbolic representation derived for the decision-maker in such a way leads to the limitation of the observed information.

Intuitively, extending symbolic representation and adding new information to a previously generated symbolic chart should increase the amount of information obtained. We use the entropy concept to verify experimentally if the above statement is true. The novelty of the approach consists of treating entropy directly as a measure of the amount of information provided to the recipient in our case, on the example of an investor in the forex market.

We discuss the concept of the symbolic data and briefly recall our proposed representation. Further, we examine whether the size and construction of symbolic data used to describe the market situation affect entropy values and thus indicate a different amount of information obtained for investors. All the above goals are verified within the numerical experiments section including the statistical tests. These experiments are preceded by the methodological description including the data transformation, as well as the entropy concept details.

At the same time, it is worth noting that the numerical experiments, as well as the statistical verification show that the extension of the symbolic representation does not visibly affect the entropy values. The article is organized as follows: In the next section, we present the theoretical background of the research. The literature review is discussed as well. The third section is focused on the various symbolic data representations. The fourth section includes the numerical experiments and the statistical verification of the generated results.

The discussion is included after the numerical experiments. Two last sections include the short summary and details of future research. Entropy can be treated as a measure of the complexity of time series in a variety of fields. Today, we observe growing interest in using entropy in various areas and a growing number of publications related to measures of complexity and entropy in particular.

Among many propositions, permutation entropy PE introduced by Bandt and Pompe [ 8 ] was designated to investigate time series. The advantages of PE have made it widely used, and the modifications proposed in the literature increased its usefulness in new areas of research.

In our research, we adopt an enhanced time-dependent pattern entropy method introduced in [ 9 ] see also [ 10 , 11 ] that reduces variations to binary symbolic dynamics and considers the pattern of symbols in a sliding temporal window. Permutation entropy as a time series complexity measure belongs to the wider family of ordinal and symbolic methods. Because permutation patterns can have different lengths, the parameter n is set.

Next, values in each vector are permuted in increasing order:. The relative frequency of each permutation pattern is calculated as follows:. In general, higher PE indicates that the process described by the time series is more complex and unpredictable. During subsequent research, additional parameters were added to the measure.

The original assumption about comparing the nearest neighbors in permutation patterns was replaced with the second parameter: time delay [ 12 , 13 , 14 ] between neighboring equidistant time points that are to be compared. Later, in [ 12 ], C. He recommended using n for which n! PE has numerous advantages. PE is conceptually simple in the sense that it does not presuppose any model, and as a consequence, it has a minimal set of parameters.

PE is invariant to nonlinear monotonous transformations, and in comparison to other measures, permutation entropy does not require a time series with a large number of elements [ 11 , 12 ]. From a technical point of view, PE is very easy to compute.

In particular, it does not require any numerical optimization; it is computationally extremely fast and does not need preprocessing, which makes it suitable for big datasets [ 13 ]. Thanks to these advantages, PE has been applied in various domains. A comprehensive summary of permutation entropy itself and its applications can be found in recent surveys [ 15 , 16 , 17 ].

Permutation patterns offer also a new research tool, i. If the pattern does not appear in the time series, such a pattern is referred to as forbidden. The number of possible n -length patterns is known and is equal to n! The number of forbidden patterns in relation to a total number of patterns F p n! The advantage of forbidden patterns analysis is that it can be used even for small datasets as if particular patterns appear frequently in the smaller dataset, the meaning of this fact arises.

Due to this, it is used in financial time series analysis [ 18 , 19 , 20 ]. This dependency was also used in [ 21 ] for comparison of emerging and mature stock markets. PE has also its limitations. The values themselves are not taken into consideration, which suggests that PE is not focused on the degree to which the neighboring elements differ from each other.

In Figure 1 , a few patterns are presented. Symbolic representation generated with the use of discretization described in detail further in the next section can be used to minimize the impact of the noise on the data. Thus, after initial preprocessing, the discretized values can be treated as the symbols in the time series. To overcome this drawback, a few modifications have been proposed. Liu and Yue [ 22 ] proposed fine-grained permutation entropy FGPE , which not only retains all the advantages and merits of PE, but also improves the performance for detecting the dynamical change of time series by introducing an additional factor to quantify the difference between the neighboring values.

In such a case, the patterns identical from the point of view of PE could be further discriminated. Other modifications of PE were proposed in [ 23 , 24 ]. Although PE is considered one of the best measures of the complexity of time series, it is worth mentioning that other types of entropy are also used.

A comprehensive review of different entropy definitions and their application can be found in [ 25 ]. Initially, entropy was used to study the dynamics of physical systems, but interest in using entropy in financial time series investigations grew particularly after , when the financial crisis occurred, while the number of indicators did not signal any danger incoming.

Since entropy is an indicator of complexity and unpredictability, in relation to financial variables, low entropy means it can be predicted, while high entropy indicates process randomness and high uncertainty. For this reason, financial time series are the subject of many entropy studies, and searching for financial risk indicators is still an urgent problem [ 26 ].

Many research works that used entropy for financial time series can be found [ 27 , 28 , 29 , 30 ] for instance. Bentes and Menezes [ 31 ] used the concept of Tsallis entropy, which constitutes a possible generalization of the Boltzmann—Gibbs or Shannon entropy to investigate the volatility of seven indexes.

It was also used in a comparative analysis of stock markets before the financial crisis in and [ 10 ]. A review of the application of entropy in finance can be found in [ 32 , 33 , 34 ]. Financial time series are characterized by small, but frequent and rapid changes of values that make them volatile, chaotic, multifractal, and temporally asymmetric [ 28 ]. Values in these data are unbounded, and from a long-time perspective, they create trends and cycles [ 35 ].

Complexity, disorder, chaos, volatility, etc. These are some of the most important factors influencing the behavior of investors on the market; hence the great interest in measures and methods that describe them. Original financial data are for most cases trend-based.

Thus, the main problem is to estimate the potential price direction. This is true for the long-term investments; however, in the case of short-term, an approach excluding the noise present in the market and at the same time deriving the most important and at the same time non-redundant information is crucial.

It is commonly known that the instrument price is not as important as relative differences between two neighboring values. Additional preprocessing transformation before moving into the symbolic representation is the discretization process. We took into account the frequency and interval discretizations. Both processes are introduced in Figure 2. We investigated the relative changes of instruments; thus, the small changes near the zero value could be common in this case.

This could lead to undesirable situations, where many small intervals for the frequency discretization near the zero value are generated. In that case, we are rather interested in equally-sized intervals derived by the interval discretization.

For most of the experiments, we used the relative data, which basically means that we used the information about the percentage price change between two successive readings. In such a case, most of the observed values were near the zero value representing no change at all between two successive readings. Thus, the interval discretization dividing the analyzed interval was used.

This allowed us to estimate the maximal and minimal relative price change and then divide this range into equal parts. The same situation could not be possible for the frequency discretization, where the vast majority of observations would be located near the zero value small relative changes for the most cases.

This could lead to the situation where the single discretization interval could be very small for example between 0. This situation is shown in Figure 2. Differences between the interval and frequency discretization are very small in the case of the original data, where the trend is included. This is due to the fact that the number of readings is rather uniformly distributed along the whole analyzed range please see the Y axis in Figure 3 a,c.

To sum up, in this case, discretized data do not visibly differ greatly from the original data. However, in the case of the relative data, using the frequency discretization could lead to the situation where the data before the discretization and after the preprocessing are completely different. Our goal was to reduce the noise, where at the same time, the data representation was as close to the original data as possible. Different stages of the data preprocessing. As was emphasized, our biggest concern was the large noise observed on the market followed by the possibility of a trend occurring.

This noise is observed in Figure 3 a. Eventual discretization of the original data presented in Figure 3 b does not solve the problem of excluding the trend. Thus, the discretization process presented in Figure 3 d is followed by deriving the relative data from the original data Figure 3 c. There are several definitions of an asset return [ 36 ].

The asset return defined by 5 is called a simple return, while the continuously compounded return is defined as:. Eventually, the symbolic time series d is built on the basis of past k readings:. Such a symbolic time series with the length of k is further examined on the basis of information derived for the decision-maker this is obtained with the use of the permutation entropy.

In our approach, we also used the second concept of deriving relative changes related not to the difference between two neighboring price values, but rather between the first and k th element in the time series. This leads to the following formula:. We use the following formula to derive the symbolic representation for the data:. In general, the whole range, within which every analyzed value original or relative could be found was divided into equally-sized intervals.

Every interval had some predefined value or symbol , which was used instead of this value. This concept was originally introduced in [ 37 ]. The main difference between the approach introduced in this article and the concept derived in the above work was the relative value calculation called here as the exponential symbolic time series. The present approach denoted further as the symbolic time series took into account the difference between the two neighboring values; while the exponential symbolic time series calculated the difference between the actual price value and the first price value observed in the analyzed time series.

The summary of both approaches can be found in Figure 4. Comparison of the symbolic time series and symbolic exponential time series representation. Transforming data into a symbolic form is one of the tools used in the study of dynamic systems [ 38 ]. The aim of this operation is to provide a simplified picture of complicated dynamics that ensures the preservation of the most important features of the tested object while enabling the use of new methods and accelerating or simplifying calculations.

It is especially useful for nonlinear and chaotic time series [ 39 ]. Symbolization is based on dividing the state space of the examined system into a finite number of elements and describing the trajectories of individual points in accordance with this division [ 14 ]. Symbolization means describing an original time series with a set of symbols from the established alphabet.

Introducing a symbolic description of time series raises the problem of equal values [ 3 ]. Originally, for time series with continuous values, Brandt and Pompe suggested adding a small random perturbation in this case, but while we worked with a symbolic description, this did not apply. In the case of PE, introducing symbolic description aims at providing a more precise description of the time series. The goal of the conducted experiments was the entropy analysis for the currency pairs.

We investigated the impact of the data representation on the entropy value. Having many time series, the most frequent challenge is their clustering and classification. In this case, the entropy of the entire time series was calculated and used for further purposes. In the case of financial data, the more important task was finding patterns and predictions.

In this case, the typical technique is moving-window analysis. Entropy was calculated for each window separately and investigating how entropy changed over time in relation to the original financial time series. At the same time, due to a large number of symbolic representation variants, discretization levels, parameters related to entropy, and other parameters, we were forced to limit the data presented in the further part of this section.

Each single currency pair included values, where every single value reading on the chart was generated at the end of the daily session. Thus, the overall analyzed period covered approximately 10 years: from June to July The above data and time period were selected due to the good availability of high-quality data free from missing values or outliers. Moreover, a selected time period not only covered the different kinds of trends on the market and financial crisis — , but also allowed us to investigate if the proposed entropy-based approach was capable of delivering good-quality information in the case where the situation on the market was not stable.

In all experiments for every currency pair, the permutation entropy was calculated. There is a natural 5 day a week period in the analysis of the financial time series [ 42 ]. Such parameters allowed calculating the entropy value for the 4 elements on the basis of the 27 values in every time window. For the forex financial data, this allowed analyzing exactly a 6 week period, which would be considered rather as a long-term investment, allowing minimizing the possible random noise on the market as much as possible.

Thus, on the one hand, these parameters met the conditions for calculating PE, and on the other hand, they corresponded to the periods of analysis of financial time series used by investors. All presented results applied for all time windows almost windows for every currency pair.

Thus, the first time window included readings starting at 1 reading up to 30; the second window started at 2 and ended at 31, and so on. At the same time, the permutation entropy was calculated for 4 successive elements, thus 1. In all our experiments, we used the D n to mark that the data were discretized with the number of discrete values equal to n. The notation of our data is as follows:.

Experiments related to the entropy analysis for the successive time windows were used to evaluate the impact of the financial data representation on the entropy values. To achieve these goals, we performed the detailed analysis separately for every single currency pair. Moreover, the charts were divided into two parts:. We investigated what was the impact of the proposed symbolic time series representation on the entropy values in this particular case, we limited the number of elements included in the symbolic description to 4.

At the same time, we compared the different discretization levels 5 and 7 strictly for the symbolic series representation. In the first part of the experiments, we analyzed the differences in the entropy obtained with the use of the original data, relative data, symbolic exponential data, and the symbolic series. This was repeated for all four currency pairs and can be seen in Figure 5. The expected entropy values should be as low as possible, which meant that some additional information was obtained.

It seemed that there was little or none at all difference between the original financial data in purple and the symbolic exponential series the green line ; while for the calculation of the relative, discretized values made the entropy reduction especially visible in the middle of the charts. A small fragment of the data was selected to better capture this observation see Figure 6.

Here, we can see that actually, the entropy was decreased for the discretized values. This, in general, was compatible with the intuition that moving towards the discretized values, some information was lost. Permutation entropy values for the selected time window fragment. However, by extending the symbolic representation, some additional information was added; thus, the differences in the entropy values between the symbolic representation of different lengths should differ.

We investigated this in Figure 7 , where we can see the comparison of the symbolic representation with different numbers of symbols included and different discretization levels. These results were counter-intuitive; thus, we would rather expect that the larger number of symbols included would lead to higher entropy values.

We observed different entropy levels between different discretization levels D 5 and D 7 ; however, there was no visible difference between the symbolic representation length. For the analysis of the entropy distribution for the whole analyzed time window, entropy histograms were calculated. In the case of the developed experiments, entropy had the following range: 0. Thus, the entropy calculated for the whole time window was grouped in the ranges with a difference of 0.

On the charts in Figure 8 and Figure 9 , we can see the entropy histograms, where the vertical axis represents the entropy value, which fit in the given interval, while the horizontal axis represents the deterministically calculated intervals. The figures were generated for every currency pair separately and included two different analyses:. Permutation entropy histograms for the selected time window fragment.

We can conclude that the lower entropy was observed for the cases where the more elements were observed in the intervals. Very often, a right-shifted normal distribution was observed. Thus, it is worth mentioning that using the symbolic representation did not affect the differences in the entropy in comparison to the original data. This was especially interesting, because of the fact that the symbolic representation showed the information not only about the present reading, but also involved the information given in the previous readings.

The information extension affected its higher uniqueness, and the histograms presented on the charts in this section clearly showed that the growth of the information in the case of the symbolic representation did not visibly change the entropy. Tests were conducted for the entropy values grouped in the exact same manner, as was observed in the case of the histograms presented in the previous subsection. The goal of these tests was to evaluate if there existed a statistical difference between the symbolic data representation and other price representations.

Statistical tests were performed for all currency pairs jointly ; however, we divided this procedure into two separate analyses:. Friedman test results and differences of the mean ranks rel. Friedman test results and differences of the mean ranks org.

The hypotheses for the comparison across repeated observations were as follows:. Thus, Hypothesis H0 was rejected, while hypothesis H1 was confirmed. The statistical tests indicated that it was possible to reject the hypothesis H0 and to confirm the hypothesis H1. From the comparison of all representations: original data, symbolic exponential series, and Symbolic Series 3, there were statistical differences; thus, the histograms for all these approaches were different.

We also conducted statistical tests, the main goal of which was to estimate which method statistically would have the lowest entropy value for all analyses. To achieve this goal, we used all collected entropy values for every method, which had an overall analyzed readings for every method. Such low values of the critical difference were related mostly to the number of analyzed readings reaching almost 10, In this case, the tests were used mostly to calculate the mean ranks for the sample.

On this basis, it was observed that the best rank, equal to 1. It was better than the rank for the original data by 0. The results of the numerical experiments from the previous section could be interpreted in two ways: in terms of assessing market measures and in terms of further possibilities of using entropy in market analysis.

Due to the large number of free parameters existing in the proposed symbolic representations, it was very difficult to point out their best values, which visibly affected the quality of the results. Thus, our motivation was to find a way to estimate what was the impact of these parameters on the information acquired from the market.

Similarly, the posterior cepstrum of autocorrelation e p n transform from the posterior spectral density can be expressed as. Doing the integration of both sides of Eq. Equation 13 can be expanded as a set of N linear equations:. Equation 15 enables to solve for the Lagrange multipliers in a straight-forward manner. Thus, the Lagrange multipliers can be estimated from the summation of the prior and posterior autocepstrums.

The prior autocepstrum can be obtained from the observed periodicity of streamflow. The posterior autocepstrum can be estimated from the following recursive function introduced by Nadeu as. It is seen from Eq. Thus, for given N lag autocorrelations, the cepstrum of autocorrelation can be computed up to lag N. It can be noted that for solving parameters of the configurational entropy without prior, the cepstrum e q in Eq.

On the other hand, for the configurational entropy and relative entropy, the autocorrelation is extended with the inverse relationship of Eq. When no prior is given, Eq. Streamflow is forecasted in the manner that autocorrelation function is extended. Thus using BESA, streamflow is forecasted by a linear combination of past series weighted by the coefficients in Eq.

Thus, streamflow is forecasted by. Then Eq. When no prior is given, e p is 0, thus, streamflow forecasted by CESA becomes. Iowa River is a tributary of Mississippi River, which is about km long and has a drainage area of 33, km 2. Two stations are chosen for comparison, one from the upstream and the other from the downstream.

The upstream station of Iowa River has a mean monthly streamflow of 9. As seen from the figure, the spectral density of the upstream station is more likely to be multi-peaked than that of the downstream station. The performance of the three entropy spectral analyses was evaluated by the Itakura-Saito distortion, which is a measure of the perceptual difference between an original spectrum and its estimate. The distortion is defined as. The smaller value represents a better fit. Spectral density estimated by three entropy spectral analysis methods for a an upstream station on Iowa River and b a downstream station on Iowa River.

As seen in Fig. However, for monthly streamflow of Iowa River, the 12 month periodicity is the most important periodicity, and should always be most significant. Streamflow was forecasted by the three entropy spectral analysis methods with a 3 year lead time for an upstream station and a 1 year lead time for a downstream station on Iowa River, as shown in Fig.

Streamflow forecasted using entropy spectral analyses for a an upstream station on Iowa River and b a downstream station on Iowa River. As shown in Fig. But it did not monotonically increase or decrease. After a small drop in the month of May, streamflow increases again in June then drops.

During the low flow season from September to February, another small peak occurs in October or November. The entropy spectral analyses discussed in this paper, though not exactly, forecasted streamflow of the above characteristics and fitted the observations with r 2 of over 0.

The forecasted streamflow by RESA was closest to the observations both for upstream and downstream stations, which led to r 2 higher than 0. The earlier peaks forecasted by BESA missed more streamflow volume during the high flow season than the other methods.

During the low flow season, the advantage of RESA was significant than the others, where streamflow was forecasted close to the observation, as shown in Fig. However, it is noted that BESA had the poorest forecasting during the low-flow season, compared to the other two methods. The forecasted streamflows in November of the second and third lead years were 4. It seemed that the way of forecasting streamflow using the recursive function of cepstrum analysis had an advantage over linear forecasting used by BESA.

The reason is that cepstrum analysis, especially when incorporating prior cepstrum by RESA, helps realize homomorphic characteristics of time series Oppenheim and Schafer , thus is more applicable than linear forecasting. The relative errors in forecasts in versus lead time are shown in Fig. For the downstream station on Iowa River in Fig.

Relative errors by three methods for a an upstream station on Iowa River and b a downstream station on Iowa River. Three entropy spectral analysis methods, developed from Burg entropy, configurational entropy and relative entropy, are reviewed in the paper. The relative entropy spectral analysis yields the highest resolution in estimating the spectral density of observed streamflow of Iowa River.

The relative entropy spectral analysis also provides the highest reliability in streamflow forecasting. When forecasting lead time increases, RESA is more consistent than the other two methods. Comput Stat Data An 56 1 — Holden-Day series in time series analysis.

Holden-Day, San Francisco. Google Scholar. Burg JP Maximum entropy spectral analysis. Water Resour Res — Article Google Scholar. J Hydrol — Hydrol Process 16 2 — Frieden BR Restoring with maximum likelihood and maximum entropy. J Opt Soc Am — Rev Geophys 40 1 — Nature — Elsevier, Burlington, p 1. Technometrics — Kybernetes — IAHS Publ. Stoch Hydrol Hydraul — Labat D Recent advances in wavelet analyses: Part I.

A review of concepts. J Hydrol 1—4 — Signal Process — Phys Rep — Phys Chem Earth 31 18 — Nadeu C Finite length cepstrum modeling - a simple spectrum estimation technique. International Conference on Digital Signal Processing. Florence, Italy, pp — Papademetriou RC Experimental comparison of two information-theoretic spectral estimators.

In: Signal Processing Proceedings, Schroeder MR Linear prediction, extermal entropy and prior information in speech signal analysis and synthesis. Speech Comm — Shore JE Minimum cross-entropy spectral analysis. Naval Research laboratory, Washington, DC. Shore JE Minimum cross-entropy spectral-analysis.

Maximum entropy spectral analysis forex | Maximum likelihood: extracting unbiased information from complex networks. The Levinson-Burg algorithm is a recursive algorithm for estimating prediction coefficients, and improves the original Levinson algorithm by computing forward and backward prediction error together to update the coefficient of next order Collomb ; Lin and Wong Marchenko, V. An analytic derivation of the efficient portfolio frontier. E 98 In the article, we suggest maximum entropy spectral analysis forex entropy as an objective measure of the amount of information that various indicators and different ways of describing the time series carry. |

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Skip to search form Skip to main content Skip to account menu. DOI: Various data examples support this conclusion. View via Publisher. Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations. Figures from this paper. Citation Type. Has PDF. Publication Type. More Filters. Maximum entropy spectral analysis for streamflow forecasting. Highly Influenced. View 5 excerpts, cites methods and background.

Maximum Entropy Spectral Analysis: a case study. View 4 excerpts, cites methods and background. A new structure for sequential detection and maximum entropy spectral estimator for characterization of volcanic seismic signals. View 3 excerpts, cites methods. For example, in spectral analysis the expected peak shape is often known, but in a noisy spectrum the center of the peak may not be clear.

In such a case, inputting the known information allows the maximum entropy model to derive a better estimate of the center of the peak, thus improving spectral accuracy. In the periodogram approach to calculating the power spectra, the sample autocorrelation function is multiplied by some window function and then Fourier transformed.

The window is applied to provide statistical stability as well as to avoid leakage from other parts of the spectrum. However, the window limits the spectral resolution. Maximum entropy method attempts to improve the spectral resolution by extrapolating the correlation function beyond the maximum lag in such a way that the entropy of the corresponding probability density function is maximized in each step of the extrapolation.

The maximum entropy rate stochastic process that satisfies the given empirical autocorrelation and variance constraints is an autoregressive model with independent and identically distributed zero-mean Gaussian input. Therefore, the maximum entropy method is equivalent to least-squares fitting the available time series data to an autoregressive model.

Once the autoregressive coefficients have been determined, the spectrum of the time series data is estimated by evaluating the power spectral density function of the fitted autoregressive model. From Wikipedia, the free encyclopedia. Spectral density estimation method. Categories : Entropy Information theory Statistical signal processing Spectroscopy.

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