A RANDOMNESS TEST FOR FINANCIAL TIME SERIES

Transcription

1 A RANDOMNESS TEST FOR FINANCIAL TIME SERIES W Adrián Risso * Abstract A randomness test is generated using tools from symbolic dynamics, and the theory of communication The new thing is that neither normal distribution nor symmetric probability distribution, nor variance process is necessary to be assumed Even more, traditional independent identically normal white noise is nested Asymmetric probability distributions and more complex process like GARCH can also be tested The test is used in different financial asset returns taken from NYSE S&P, and Dow Jones indexes, and the years treasure notes returns are also tested Therefore we reject the null hypothesis of randomness for daily data In addition, some series reject the null hypothesis when using weekly and monthly data When assuming an asymmetric probability distribution as model for the returns hypothesis of randomness is rejected in the case of daily returns, and also for some cases using weekly and monthly data This could suggest that asset returns follow a complex nonlinear dynamics, or there exist some kind of anomalies in the financial market We also believe in the importance of the test, for instance, in checking if the residual of an ARIMA follow a white noise process no matter this process is normal distributed or not Introduction Bachelier () was the first proposing stock prices follow a Brownian motion These stock prices would reflect all available information according to the efficient market hypothesis suggested by Fama () Therefore the better prediction of future prices are the present prices Since then, difference in stock prices has been modelled as white noise process, implying that stock prices are random walk processes This assumption was criticized due to the existence of some well known stylized facts, Mandelbrot () considers that financial returns have a long memory, then stock prices should be modelled using a fractal Brownian motion Peters () () gives evidence of fractability in financial markets using Hurst exponent as a measure of persistence Moreover, it seems that constant variance hypothesis in financial returns is not supported by empirical evidence In fact Mandelbrot () proposes an stable paretian distribution in order to model the asset returns which implies an infinite variance Since then, different models have been proposed permitting the variance to change, an example is the ARCH model (see Engel ()), and the Markov switching models (see Hamilton ()) Lo and McKinlay (), using the variance ratio test, found that financial returns behaviour would not be random This fact would suggest the existence of different anomalies, Singal () surveys all the anomalies found until now In this context, the present work has two principal objectives At first we tried to demonstrate that stock market asset returns do not behave as any type of white noise processes, and then, Brownian motion theory is not applied in this cases We try to do this, taking the daily decreases and increases and seeing the behaviour of the combination for,,, and day decreases and increases Since theory says that returns are completely random, combinations of * Università degli Studi di Siena address: risso@unisiit I would like to acknowledge Doyle Farmer, Roberto Renò, and Lionello Punzo for their very helpful comments, and suggestions

2 different day decreases and increases should have the same probability among them It means it would not be expected to find combinations of decreases and increases more probable than others For instance imagine that means decrease in one day returns and means increase Imagine also that we have the following daily time series of codified returns: We have days of decreases and increases, if the process in completely random in day the probability of decreases should be / =/ and the same for increasing Moreover if the process is random, in days we have possibilities and the probability should be / for each possible case Reasoning in this manner, the probability of an event composed by combination of n days should be -n, in case of having a random process In order to do this we develop a test of randomness based in the symbolic dynamic and the concept of entropy The developing of a test of randomness not only for asset returns but also a general powerful test of randomness will be the second objective In fact the test of randomness does not need the assumption of normality, and it permits the variance to follow different processes, like a GARCH process, or even an infinite variance like in the case of the paretian distributions suggested by Mandelbrot () Section explains what symbolic time series analysis and symbolic dynamic are, in section a definition of Shannon entropy as a measure of the uncertainty is given People familiar with these two concepts could skip the latter sections Section proceeds to describe our randomness test, and in section real data from NYSE is taken and contrasted with the new statistic In section we assume a more general probability distribution for the returns were possible asymmetry is considered Finally section concludes ) Symbolic Dynamics and Symbolic Time Series Analysis (STSA) The earliest developments in symbolic dynamics began with the study of the complex behaviour of dynamical system In, Hadamard developed a symbolic description of sequence in geodesic flows on surfaces of negative curvature Morse and Hedlund () were the first using the term symbolic dynamic Their idea was that a symbolic trajectory T was formed by symbols taken from a finite set of generating symbols subject to rules of admissibility which are dependent on the underlying Poincaré fundamental group The trajectory taken with one of its symbols is called a symbolic element Symbolic elements are analogues of line elements on an ordinary trajectory A metric is assigned to the space of symbolic elements and this space turns out to be perfect, compact, and totally disconnected As Piccardi () remarks, Symbolic dynamic is a term which denotes theoretical investigation and is well distinct from its experiment-oriented approach known as STSA During the last years the STSA methodology has been applied in the qualitative analysis of time series in different fields of the sciences Nonetheless, few application in economics has been done using this type of series Symbolization involves transformation of raw time-series measurements into a series of discrete symbols that are processed to extract information about the generation process As Keller and Wittfeld () note the idea behind this method is simple: instead of considering the exact state of a system at some time, one is interested in a coarse-grained description The state space is decomposed into a small number of pieces, and states being contained in the same piece are identified The STSA is still developing in different ways, however we will focus on the approach given by Finney, and Daw () ) Symbolization We will proceed to describe the STSA methodology based on Finney and Daw (), Finney, Daw, and Halow (), and Daw, Finney and Tracy () Given a continuous or discrete time series (for instance, let us consider a series of daily asset returns) it is necessary to define the number of symbols n Transformation consists in the partition of the original time series in one or many selected thresholds, values of the original series which are in one part of the partition will be codified with a symbol as is shown in figure

3 Fig: Process of symbolizing of a Time Series Taken from Daw, Finney, Tracy () Typically, the discretization partition is defined in such a way that the individual occurrence of each symbol is equiprobable with all others In the dynamic transformation, arithmetic differences of adjacent data points define the symbolic values We symbolize a positive difference as and a negative difference as Such a differenced symbolization scheme is relatively insensitive to extreme noise spikes in the data After symbolization, the next step in identification of temporal patterns is the construction of symbol sequences (words in the symbolic-dynamics terminology) If each possible sequence is represented in terms of a unique identifier, the end result will be a new time series often referred to as a symbolic-sequence series (or code series) Symbolic-sequence construction has at least outward similarities to time-delay embedding Again, if the process is random all the possible words of the same length will have the same probability Symbol-sequence construction has also been described in terms of symbolic trees For a fixed sequence length of L successive symbols, the total number of branches is n L, and thus the number of possible sequences increases exponentially with tree depth There is no methodology to choose the correct number of L Daw, Finney, and Tracy () assert that a recent improvement to the fixed-sequence-length approach is the implementation of context trees by Kennel and Mees (, ) This approach allows some of the possible sequences (ie branches) to be shortened to reflect reduced predictability over long times Modification of the length of individual branches depends on information theoretic measures that indicate how efficiently the observed dynamics are predicted (ie, how well the symbolization scheme compresses the available information) For a given dynamical system, all sequences are not realizable Such nonocurring sequences are called forbidden words Various statistics can be used to characterize the symbol-sequence histograms For example, we can define a modified Shannon entropy as: H S = pi, L log pi, L log N i where, N is the total number of sequences in the time series, i is a symbol-sequence index of sequence vector length L, and p i,l is the relative frequency of symbol sequence i For random data H S should be equal to, whereas for nonrandom data it should be between and

4 There exist many works in sciences using this special form of symbolization, see Daw, Finney and Tracy () for a review in Astrophysics, Biology, Medicine, Fluid Flow, Chemistry, Mechanical Systems, Artificial intelligence, Control, and Communication However, this approach has been scarcely applied in economics, Brida () considers an application of STSA to economic time series, Brida, and Punzo () apply the symbolic time series using dynamic regimes Nevertheless, the approach has not been applied in finances Other types of symbolic time series analysis have been done Keller and Lauffer (), Keller and Wittfeld (), and Keller and Sinn () use symbolic time series from the ECG (electroencephalogram) In the latter paper the approach ordinal analysis of time series is developed but is a very incipient methodology They assert that this method allows to recognize structure in a system and to discriminate and classify different states In order to get reliable statements, it is necessary to develop models and ordinal statistical characteristics beyond the permutation entropy Tiňo, et al in Abu-Mostafa, et al () use a different approach in order to predict the volatility in the Dow Jones Industrial Average They find that there appears to indeed lie some potential in using a symbolic dynamics approach to volatility forecasting, in which continuous values are replaced by symbols through quantization They mark that there have not been many applications of symbolic methods to modelling financial time series However, it was found that discretization of financial time series can potentially filter the data and reduce the noise Even more, the symbolic nature of the pre-processed data enables one to interpret the predictive models as list of clear rules But, there are shortcomings: ) The determination of the number of quantization intervals (symbols) and their cut values ad hoc; ) In such situation due to nonstationarity issues many symbols make the statistics poor, in fact (Giles et al ()) indicates that the predictive model achieved the best performance with binary inputs streams In fact Giles et al () use an hybrid neural network for different foreign exchange rates, the direction of change for the next day is predicted with a % error rate ) Shannon Entropy as a measure of uncertainty Clausius () introduced the concept of entropy as a measure of the amount of energy in a thermodynamic system But we shall focus in the meaning given by Information Theory In fact, Shannon () considers entropy as an useful measure of uncertainty in the context of communication theory where the maximum values is taken by a completely random event For instance, if we consider an English Book is an obvious fact that certain combinations of letters are more probable than other The latter comes from the fact that English has a probability distribution If we take a page of an English book we will see that events of three letters like THE will appear more frequently than combinations like XCV In fact the combination THE has a larger probability in English language than XCV, however the two combinations will have the same probability in a random process of letters Knowing this we can increase efficiency codifying more probably events with simple codes and no so probable others with more complex codes Therefore Shannon measure will give a value for the English less than the value for a completely random process This idea will be fundamental in our analysis, because if a time series of increases and decreases in stock market is completely random as theory asserts their entropy should be larger than a non-random process Then the Normalized Shannon entropy will be our measure of randomness that we will use in the stock market asset returns The entropy of a random variable is a measure of the uncertainty of the random variable; it is a measure of the amount of information required on the average to describe the random variable We will first introduce the concept of entropy showing its required properties: ) It should be a function of P=(p,p,,p n ) in this manner we can write H=H(p,p,,p n )=H(P),where P is probability distribution of the events ) It should be a continuous function of p,p,,p n Small changes in p,p,,p n should cause a small change in H n ) It should not change when the outcomes are rearranged among themselves Kurths, Schwarz, Witt, Krampe, Abel () use this approach to measure signal analysis Their application in cardiology, detecting High-Risk-Patients for Sudden Cardiac Death is very interesting

5 ) It should not change if an impossible outcome is added to the probability scheme ) It should be minimum and possibly zero when there is no uncertainty ) It should be maximum when there is maximum uncertainty which arises when the outcomes are equally likely so that H n should be maximum when p =p = =p n =/n ) The maximum value of H n should increase as n increases Shannon () suggested the following measure: H n (p,p,,p n )=- p i log p i Logarithms to base are used then entropy is measured in bits This measure satisfies all properties mentioned above and takes the maximum when all events are equally likely The latter is easily seen if we solve the following Lagrange equation: - p i log p i - λ( p i =) Since the function is concave its local maximum is also a global maximum Note also that when p i = we have log= which is proved by continuity since xlogx as x Thus adding terms of zero probability does not change the entropy In order to clarify the concept of Shannon, let's take an event with two possibilities and their respective probabilities p and q=-p The Shannon Entropy will be defined by H=-(plog(p)+qlog(q)) Fig shows graphically the shape of the function, note that the maximum is given when the probability is for each event This cases corresponds to a random event (like flipping a coin) Instead, a certain event (when probability of one event is ) will produce an entropy equal to Fig: Shape of the Shannon entropy function Note that maximum happens when the process is random (p=) This idea of maximum randomness when entropy is maximum, and certainty when the entropy is zero will be basic in the future of this work In general, Khinchin () showed that any measure satisfying all the properties must take the following form: -k p i log p i Where k is an arbitrary constant In particular we will be interested in the normalized Shannon Entropy, therefore we will take k=/log (n) This will be used when comparing events of different lengths We said that Shannon entropy is maximum when all outcomes are equally probable This is consistent with Laplace's principle of insufficient reason that unless there is information to the contrary, all outcomes should be considered equally probable

6 ) Test of Randomness R based on the normalized Shannon Entropy As it was mentioned before we are interested in a powerful test in order to see if the stock market asset returns are random events as the theory asserts Therefore we will use the Normalized Shannon Entropy which gives a continuous measure from (a completely certain event) to (a completely random event) as fundamental input in our test We can obtain different data series from NYSE for more than days We are also interested in daily increases and decreases in financial returns, in the symbolic dynamic conception it means that we define the partition in Thus we will have basically possible events in one day, the return of an asset increases or decreases Being this a random event as theory asserts probability of each event would be obtaining a maximum Entropy equals to Even more if we take consecutives days (a word of letters according to STSA) in the stock market we have equally likely events ( days increase in returns, days of decreases, first day increases and second decreases, and the converse) the entropy should not be different from Even more if the event is random, taking different consecutive days, these words should have the same probability among them In general if we take n consecutive days of these random events the possibilities increase at the rate of ⁿ, for a random event the probability of finding a word of n days should be in this case -n The conception of the test is very simply We simulate daily random increases and decreases in the stock market as flipping a coin, assuming as theory says that process is random In order to do this we used MatLab random number generator of s and s, where will mean decrease, and increase Note that no probability distribution is assumed, symmetry is not required, and no assumption about variance is held Thus the generality of the test for random events is obtained, and the test could be used not only for the asset returns but also each time we are interested in checking if a time series is a white noise, for instance though in the residual of an ARIMA model The only requisite here is that and be equally probable Thus, we simulated times returns changes, and we computed the correspondent entropy (H), giving always a number near to as it was expected In this manner we obtain H for day, days, days,, and consecutive days No difference should be found in H for choosing,,,, or consecutive days, but since the sample is limited is expected that as the days increase, H reduces However, theoretically a random event of consecutive days has no variation from the maximum entropy value Using the Entropy simulated values of a random process we are able to construct our variable, it will be R=-H, It means that is a random process and is completely certain event Now we construct the corresponding simulated distribution for a data sample Figure,,,, and show these simulated distributions Rday Percent Rday

7 Rdays Percent Rdays Rdays Percent Rdays Rdays Percent Rdays

8 Rdays Percent Rdays The following is to take real data from stock market, codifying the stock market asset returns in increases and decreases according to the STSA and compute the Entropy, and then R Therefore comparing the R with the table critical value using a p-value of % we are able to reject the null hypothesis about randomness of the process when the statistic is at the right of the critical value Then H₀)R= means that our null hypothesis is that the process is random Table : Critical Values at % for statistic R=-H (where H is Entropy) Sample= Rday Rdays Rdays Rdays Rdays Source: Based on the obtained results ) Standard & Poor, Dow Jones and years treasure notes interest rate ) Daily data We obtained data series from the S&P, Dow Jones and the year treasure notes interest rate Taking daily data for asset returns, the years treasure note interest rate difference, the Dow Jones, and the S&P index differences, and codifying the series according to increases and decreases we computed the corresponding entropies until consecutive days event We describe the procedure because is an useful manner to show how we have to proceed to compute the estimator r(t)=ln(p(t))-ln(p(t-)) where r(t) is the financial return at the moment t and P(t) is its respective price After that we proceed to codify the data according to the partition, therefore for positive values we assign, and for the negative ones > < = ) ( ) ( ) ( t r if t r if t s Once we have the daily sequence of s and s (or decreases and increases) we will be interested in the combination of these events in days,,, and days Of courses the combinations of s and s increases according to L, where n is the quantity of days Thus L= ( days) means that we have ³= possible cases, if the r(t) is white noise then all these combinations should be equally probable producing an entropy equal to

9 Having the codified data, we compute the indicator R for,,,, and days For instance in days we have possibilities: (,,),(,,),(,,),(,,),(,,),(,,),(,,), and (,,) So we proceed to compute the cases and we calculate the R in the following way: R( ) H () = + log ( L n n n log + + log ) n n n n n Where n i is the quantity of the i-th event in the time series, and n is the total of events in the time series Table shows the results Table : Randomness test based on Normalized Shannon Entropy (R=-H) ( days) Financial Return R-day R-days R-days R-days R-days Alcoa Inc Boeing Co Caterpillar Inc Coca Cola Co Du Pont EI Eastman Kodak Co General Electric Co General Motors Co Hewlett Packard Co IBM Walt Disney Co S&P Dow Jones years treasure notes Source: Based on the obtained results Note that all the R values are greater than the critical value at %, so we can reject the null hypothesis that financial returns are completely random, for instance they do not behave as white noise process Therefore there are combination of events that are more probable than others This fact could be suggesting that the process would have a complex nonlinear dynamic that seems to be completely random or the existence of some kind of anomaly It is important to consider that we selected a particular partition, it is possible the existence of another partition or partitions remarking more important facts about the asset returns dynamic Since a methodology for the selection of the correct partition is not available, more work is necessary in order to discover the possible existence of a simpler dynamic in the returns ) Weekly data In order to check whether the random process is rejected taking other period of time, we also consider weekly data for the same assets obtaining more than observations In the same way as we did with the daily data we proceed to do the simulations for weeks obtaining the critical values in Table Table : Critical Values at % for statistic R=-H (where H is Entropy) Sample= R-week R-weeks R-weeks R-weeks R-weeks Source: Elaborated in base to obtained results Computing the R- statistic for the assets as we did in the section we obtain as result that null hypothesis of randomness is rejected in the case of indexes (S&P, Dow Jones), and for the year treasure notes More For instance, n is the quantity of time that event (,,) is seen in the codified time series

10 difficult is the case of particular assets, where only in the case of Coca Cola Co, General Electric Co, and IBM the null hypothesis is rejected The results are shown in Table Table : Randomness test based on Normalized Shannon Entropy (R=-H) ( weeks) Financial Return R-week R-weeks R-weeks R-weeks R-weeks Alcoa Inc Boeing Co Caterpillar Inc Coca Cola Co * * * Du Pont EI Eastman Kodak Co General Electric Co * General Motors Co Hewlett Packard Co IBM * * * Walt Disney Co S&P * * * * * Dow Jones * * * * * years treasure notes * * * * * indicates rejection of the null hypothesis Source: Based on the obtained results ) Monthly data We also tested the behaviour of the financial returns using monthly data Then, we proceeded to simulate times series of points The critical values at % are showed in table Table : Critical Values at % for statistic R=-H (where H is Entropy) Sample= R-month R-months R-months R-months R-months Source: Based on the obtained results Table shows the results for the R-statistics As we can see randomness is rejected in case of S&P, and Dow Jones indexes Note that also is rejected the null hypothesis for Coca Cola Co, and Walt Disney Co While it is not possible to reject randomness for the other financial returns Table : Randomness test based on Normalized Shannon Entropy (R=-H) ( months) Financial return R-month R-months R-months R-months R-months Alcoa Inc Boeing Co Caterpillar Inc Coca Cola Co * * * * * Du Pont EI Eastman Kodak Co General Electric Co General Motors Co Hewlett Packard Co IBM Walt Disney Co * * * * *

11 S&P * * * * * Dow Jones * * * * * years treasure notes * indicates rejection of the null hypothesis Source: Based on the obtained results ) Changing a traditional assumption about returns Let us assume now that returns have an asymmetric probability distribution For instance think that positive returns are more likely than negative ones In that case we should change the symbolization in order to use our test It is well known that the median is a good measure of the probability distribution centre when we have asymmetric distribution Then we will test if returns follow a random process with an asymmetric distribution If the latter is true, doing a symbolization that takes % percent of probability by a side and % in the other side would produce combinations equally probable This is possible using the median (we call µ) as a threshold We know that we are going far from the traditional assumption but we want to show that even in this case the test is useful if we do the correct partition This mean we have to take a threshold (in this case µ) defining two regions with % percent of probability as is shown in Fig Fig: Asymmetric probability distribution for r We have defined the partition in µ, since the process is random different combinations of s and s should be random Therefore we can do the symbolization using the function defined bellow if s ( t) = if r( t) < µ r( t) > µ Being a random process the combination of these symbols for different moments of time should be equally probable as it was explained before, and the test could be used in the same way as before Table is a summary of the results that changes when we do this symbolization Table : Randomness test assuming an asymmetric distribution Daily R R R R R S&P * * * * Dow Jones * * * * Weekly R R R R R Coca Cola Co * * * General Electric Co * Hewlett Packard Co IBM * * * Walt Disney Co S&P Dow Jones year treasure notes * * * *

12 Monthly R R R R R Alcoa Inc Boeing Co * Caterpillar Inc Coca Cola Co Du Pont EI Eastman Kodak Co General Electric Co General Motors Co Hewlett Packard Co Walt Disney Co S&P Dow Jones years treasure notes * indicates rejection of the null hypothesis ( Remember null hypothesis is: Ho) R= ) Source: Based on the obtained results As we can see using daily data we can reject the null hypothesis of randomness (R=) in all the cases so equal as in the symmetric case Using weekly data we can reject the randomness in the cases of Coca Cola Co, General Electric Co, IBM, and the year treasure notes, as we did in the symmetric case But now we cannot reject the randomness in case of the S&P and Dow Jones When we use monthly data Boeing Co seems to be the only one who does not present a random process In all the other case we cannot reject the randomness hypothesis ) Conclusion Modelling asset returns still assumes a white noise process This idea comes from Bachelier () who modelled asset prices as a Brownian motion Fama () connects this random process with the efficient market hypothesis, it means if the market is perfect so there is no place for forecasting However empirical evidence exists showing that the asset returns would not be random, see for instance Peter () who uses Hurst coefficient in order to show this, Mandelbrot () asserts that normality assumption is not correct at the moment of modelling financial time series, instead of this he proposes the stable paretian distribution Lo and McKinlay () shows that asset returns would not be random process according to their variance ratio test, and Singal () shows all the known anomalies in the stock market We tried to show that financial series are not well modelled assuming a white noise process In order to do this we used a new approach that comes from symbolic dynamics and information theory The method developed is powerful since it does not require neither the assumption of normality, nor constant variance Even more the latter process is nested in our test, since it recognizes this and others, for instance think of a GARCH process, or and asymmetric distribution The only requirement for this random test is a mean probability distribution In this sense the test recognizes a general white noise It was shown that in all the cases the null hypothesis of randomness was rejected for daily data Both cases were tested, assuming a symmetric mean distribution or assuming an asymmetric distribution This suggests that some events are more probable, and therefore the efficient market hypothesis would not be working Even more when we try to test the randomness using different periods (weekly and monthly data), randomness is also rejected for S&P, Dow Jones, and Coca Cola Co in the case of assuming a symmetric mean distribution The latter suggests that random walk would not be a good model in order to represent this kind of financial returns A possible next step would be to take for instance to make the test using symbols (big decrease, small decrease, small increase, and big increase returns)

13 We believe also that the test could be applied as a general randomness test, for instance imagine testing the residuals of an ARIMA process This test could be important detecting whether the residuals are a white noise no matter if they are normal distributed or not References Bachelier, L (): Theory of Speculation, in PCootner, ed, The Random Character of Stock Market Prices Cambridge, MA: MIT Press, Brida, JG, (): Symbolic Dynamics in Multiple Regime Economic Models, Tesi di Dottorato di Ricerca in Economia Politica, Dipartimento di Economia Politica Università degli Studi di Siena Brida, JG, Punzo, LF (): Symbolic time series analysis and dynamic regimes, Structural Change and Economic Dynamics,, pp - Clausius, R (): The nature of the motion we call heat, translated in Stephen G Brush (ed), Kinetic Theory, Daw, CS,Finney, CEA, Tracy, ER (): A review of symbolic analysis of experimental data, Review of Scientific Instruments February -- Volume, Issue, pp - Engle, Robert F (): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation ; Econometrica, Vol, No (Jul, ), pp - Fama (): The Behavior of Stock Market Prices, Journal of Business, Vol, pp - Finney, Daw, Green (): Symbolic Time-Series Analysis of Engine Combustion measurements, SAE paper No Finney, Daw, Nguyen, Hallow (): Symbolic Sequences Statistics for Monitoring Fluidization, International Mechanical Engineering Congress & Exposition Hamilton (): A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle, Econometrica,, pp - Keller, Lauffar, Heinz (): Symbolic Analysis of High-Dimensional Time Series International Journal of Bifurcation and Chaos, pp - Keller, Wittfeld, Hatharina (): Distances of Time Series Components by Means of Symbolic Dynamics International Journal of Bifurcation and Chaos, pp - Keller, Sinn, (): Ordinal Analysis of Time Series Physica A, pp - Khinchin (): Mathematical Foundations of Information Theory Courier Dover Publications Kurths, Schwartz, Krampe, Abel (): Measures of Complexity in Signal Analysis, in Chaotic, Fractal, and Nonlinear Signal Processing Woodbury, New York pp - Lo; Andrew, W, MacKinlay, A, Craig (): Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test,The Review of Financial Studies, Vol, No (Spring, ), pp - Mandelbrot (): The Variation of Certain Speculative Prices The Journal of Business, Vol, No (Oct ), pp - Morse, M, Hedlund, GA (): Symbolic Dynamics, American Journal of Mathematics, Vol, No (Oct, ), pp - Peters, Edgard, E (): Fractal Market Analysis, first edition, Wiley Finances Edition Peters, Edgard, E (): Chaos and order in the capital markets, second edition, Wiley Finances Edition Piccardi, Carlo (): On the Control of Chaotic System via Symbolic time series analysis, Chaos, Vol, No, pp - Shannon, Claude (): A Mathematical theory of communication, Bell Sys Tech Journal, : -, -,

14 Singal, V (): Beyond the Random Walk: A Guide to Stock Market Anomalies and Low-Risk Investing Oxford University Press Tiňo, Peter, Schittenkopf, C, Dorffner, G, Dockner, EJ (): A Symbolic Dynamics Approach to Volatility Prediction, in Abu-Mostafa, et al () Computational Finance () MIT