effect of coarse graining on detrended fluctuation analysis several studies have investigated the scaling behavior in naturally occurring biological and physical processes using techniques such as detrended fluctuation analysis dfa data acquisition is an inherent part of these studies and maps the continuous process into digital data the resulting digital data is discretized in amplitude and time and shall be referred to as coarse grained realization in the present study since coarse graining precedes scaling exponent analysis it is important to understand its effects on scaling exponent estimators such as dfa in this brief communication k means clustering is used to generate coarse grained realizations of data sets with different correlation properties namely anti correlated noise long range correlated noise and uncorrelated noise it is shown that the coarse graining can significantly affect the scaling exponent estimates it is also shown that scaling exponent can be reliably estimated even at low levels of coarse graining and the number of the clusters required varies across the data sets with different correlation properties

co ordinate transformations for second order systems part i general transformations abstract when the dynamics of any general second order system are cast in a state space format the initial choice of the state vector usually comprises one partition representing system displacements and another representing system velocities co ordinate transformations can be defined which result in more general definitions of the state vector this paper discusses the general case of co ordinate transformations of state space representations for second order systems it identifies one extremely important subset of such coordinate transformations namely the set of structure preserving transformations for second order systems and it highlights the importance of these it shows that one particular structure preserving transformation results in a new system characterized by real diagonal matrices and presents a forceful case that this structure preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems in preference to the eigenvalue eigenvector solutions conventionally accepted

dynamic scaling approach to study time series fluctuations we propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface growth problem in this framework the fluctuation sampling time interval plays the role of time variable whereas the physical time is treated as the analog of spatial variable in this way we found that the fluctuations of many real world time series satisfy the analog of the family viscek dynamic scaling ansatz this finding permits us to use the powerful tools of kinetic roughening theory to classify model and forecast the fluctuations of real world time series

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

long term memory in extreme returns of financial time series abstract it is well known that while daily price returns of financial markets are uncorrelated their absolute values volatility are long term correlated here we provide evidence that certain subsequences of the returns themselves also exhibit long term memory these subsequences consist of maxima or minima of returns in consecutive time windows of r days our analysis shows that for both stocks and currency exchange rates long term correlations are significant for r 4 we argue that this long term memory which is similar to that observed in volatility clustering sheds further insight on price dynamics that might be used for risk estimation

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

is non gaussianity sufficient to produce long range volatile correlations abstract scaling analysis of the magnitude series volatile series has been proposed recently to identify possible non linear multifractal signatures in the given data y ashkenazy et al phys rev lett 86 2001 1900 y ashkenazy et al physica a 323 2003 19 t kalisky y ashkenazy s havlin phys rev e 72 2005 011913 in this article correlations of volatile series generated from stationary first order linear feedback process with gaussian and non gaussian innovations are investigated while volatile correlations corresponding to gaussian innovations exhibited uncorrelated behavior across all time scales those of non gaussian innovations showed significant deviation from uncorrelated behavior even at large time scales the results presented raise the intriguing question whether non gaussian innovations can be sufficient to realize long range volatile correlations

universal scaling behaviors of meteorological variables volatility and relations with original records abstract volatility series defined as the magnitude of the increments between successive elements of five different meteorological variables over china are analyzed by means of detrended fluctuation analysis dfa for short universal scaling behaviors are found in all volatility records whose scaling exponents take similar distributions with similar mean values and standard deviations to reconfirm the relation between long range correlations in volatility and nonlinearity in original series dfa is also applied to the magnitude records defined as the absolute values of the original records the results clearly indicate that the nonlinearity of the original series is more pronounced in the magnitude series

modeling heart rate variability in healthy humans a turbulence analogy many complex systems share similar statistical characteristics in this letter a turbulence analogy is proposed for the long term heart rate variability of healthy humans based on such an analogy the equivalence of an inertial range is found and a cascade model which captures the statistical properties of the heart rate data is given

volatility of linear and nonlinear time series previous studies indicated that nonlinear properties of gaussian distributed time series with long range correlations u i can be detected and quantified by studying the correlations in the magnitude series ensuremath mid u i ensuremath mid the volatility however the origin for this empirical observation still remains unclear and the exact relation between the correlations in u i and the correlations in ensuremath mid u i ensuremath mid is still unknown here we develop analytical relations between the scaling exponent of linear series u i and its magnitude series ensuremath mid u i ensuremath mid moreover we find that nonlinear time series exhibit stronger or the same correlations in the magnitude time series compared with linear time series with the same two point correlations based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series the nonlinear multifractal time series is generated by multiplying a long range correlated time series that represents the magnitude series with uncorrelated time series that represents the sign series mathrm sgn u i we apply our techniques on daily deep ocean temperature records from the equatorial pacific the region of the el nin o phenomenon and find i long range correlations from several days to several years with 1 f power spectrum ii significant nonlinear behavior as expressed by long range correlations of the volatility series and iii broad multifractal spectrum

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

effects of non stationarity on the magnitude and sign scaling in the multi scale vertical velocity increment abstract we study the scaling properties of multi scale increment series from the stationary and non stationary vertical velocity records by the detrended fluctuation analysis dfa and series decomposition methods and show the impact of non stationarity on the magnitude and sign scaling properties firstly we show it is difficult to define a significant scaling range for stationary records themselves secondly for increment series with lag one the nonlinear correlation found in magnitude fluctuation i e volatility fluctuation is obvious for both cases and over the smaller scale range the linear correlation found usually in the sign series is the same for both cases however over the larger scale range the difference is sharp and marked nonlinear correlation is also incorporated for the non stationary sign series thirdly with the lags in increment increasing there exist two scaling ranges in both the magnitude and sign fluctuations the nonlinear correlation in the magnitude series over both small and large scale ranges decreases with the lags increasing for the sign series the linear correlation is dominant over both small and large scale ranges for stationary case however the nonlinear correlation can be found over the large scale ranges for non stationary case and will decrease with lags increasing the nonlinear correlation found in the sign series over large scale range is resulted from the modulation of the large scale structures found in the non stationary data which will incorporate parts of the magnitude information into the sign series

magnitude and sign of long range correlated time series decomposition and surrogate signal generation we systematically study the scaling properties of the magnitude and sign of the fluctuations in correlated time series which is a simple and useful approach to distinguish between systems with different dynamical properties but the same linear correlations first we decompose artificial long range power law linearly correlated time series into magnitude and sign series derived from the consecutive increments in the original series and we study their correlation properties we find analytical expressions for the correlation exponent of the sign series as a function of the exponent of the original series such expressions are necessary for modeling surrogate time series with desired scaling properties next we study linear and nonlinear correlation properties of series composed as products of independent magnitude and sign series these surrogate series can be considered as a zero order approximation to the analysis of the coupling of magnitude and sign in real data a problem still open in many fields we find analytical results for the scaling behavior of the composed series as a function of the correlation exponents of the magnitude and sign series used in the composition and we determine the ranges of magnitude and sign correlation exponents leading to either single scaling or to crossover behaviors finally we obtain how the linear and nonlinear properties of the composed series depend on the correlation exponents of their magnitude and sign series based on this information we propose a method to generate surrogate series with controlled correlation exponent and multifractal spectrum

a shift and deflate technique for quadratic matrix polynomials abstract a technique for deflating two eigenvalues 1 2 of an n n quadratic matrix polynomial a z is proposed the method requires the knowledge of a right eigenvector corresponding to 1 and of a left eigenvector corresponding to 2 this technique consists of two stages the shift of 1 and 2 to 0 and respectively and the deflation of 0 and the final result is the construction of an n 1 n 1 quadratic matrix polynomial which shares with a z all the eigenvalues except for 1 and 2 the particular case of palindromic quadratic matrix polynomials is treated

appropriateness of correlated first order auto regressive processes for modeling daily temperature records abstract the present study investigates linear and volatile nonlinear correlations of first order auto regressive process with uncorrelated ar 1 and long range correlated car 1 gaussian innovations as a function of the process parameter in the light of recent findings a kiraly i m janosi phys rev e 65 2002 0511021 we discuss the choice of car 1 in modeling daily temperature records we demonstrate that while car 1 is able to capture linear correlations it is unable to capture nonlinear volatile correlations in daily temperature records

surrogate analysis of volatility series from long range correlated noise detrended fluctuation analysis dfa c k peng s v buldyrev a l goldberger s havlin f sciortino m simons h e stanley nature 356 1992 168 of volatility series has been proposed to identify possible nonlinear multifractal signatures in the given empirical sample y ashkenazy p ch ivanov s havlin c k peng a l goldberger h e stanley phys rev lett 86 2001 1900 y ashkenazy s havlin p ch ivanov c k peng v schulte frohlinde h e stanley physica a 323 2003 19 t kalisky y ashkenazy s havlin phys rev e 72 2005 011913 long range volatility correlation can be an outcome of static as well as dynamical nonlinearity in order to argue in favor of dynamical nonlinearity surrogate testing is used in conjunction with volatility analysis y ashkenazy p ch ivanov s havlin c k peng a l goldberger h e stanley phys rev lett 86 2001 1900 y ashkenazy s havlin p ch ivanov c k peng v schulte frohlinde h e stanley physica a 323 2003 19 t kalisky y ashkenazy s havlin phys rev e 72 2005 011913 in this brief communication surrogate testing of volatility series from long range correlated monofractal noise and their static invertible nonlinear transforms is investigated long range correlated noise is generated from fractional auto regressive integrated moving average farima 0 d 0 with gaussian and non gaussian innovations we show significant deviation in the scaling behavior between the empirical sample and the surrogate counterpart at large time scales in the case of farima 0 d 0 with non gaussian innovations whereas no such discrepancy was observed in the case of gaussian innovations the results encourage cautious interpretation of surrogate analysis of volatility series in the presence of non gaussian innovations

deflating quadratic matrix polynomials with structure preserving transformations given a pair of distinct e s l 1 l 2 of an nbyn quadratic matrix polynomial q l with nonsingular leading coefficient and their corresponding ev s we show how to transform q l into a quadratic of the form twobytwoa qd l 0 0 q l having the same e s as q l with qd l an n 1 times n 1 quadratic matrix py and q l a scalar quadratic py with roots l 1 and l 2 this block diagonalization cannot be achieved by a similarity transformation applied directly to q l unless the ev s corresponding to l 1 and l 2 are parallel we identify conditions under which we can construct a family of 2n times 2n elementary similarity transformations that a are rank two modifications of the identity matrix b act on linearizations of q l c preserve the block structure of a large class of block symmetric linearizations of q l thereby defining new quadratic matrix polynomials q 1 l that have the same e s as q l d yield quadratics q 1 l with the property that their ev s associated with l 1 and l 2 are parallel and hence can subsequently be deflated by a similarity applied directly to q 1 l this is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action

nonlinear strength quantifier based on phase correlation abstract distinguishing linear series from nonlinear ones is of great importance for not only understanding the processes generating these series but also their modeling and prediction there are many methods and their nonlinear quantifiers to reach this goal but no one is powerful enough to deal with any given series in this study the artificially generating series with given tailored phase correlation strength is applied to test the performance of some widely used methods and their nonlinear quantifiers given in the previous studies both advantages and shortcomings are shown for these methods and their nonlinear metrics especially spurious results from these methods and their nonlinear metrics can be found for some cases much weak or strong phase correlations in order to avoid these spurious results modified phase correlation quantifiers are proposed at the same time multiple stripes found in the phase map can be explained from the view point of enhanced nonlinear strength

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

on the multifractal effects generated by monofractal signals we study quantitatively the level of false multifractal signal one may encounter while ana lyzing multifractal phenomena in time series within multifractal detrended fluctuation analysis mf dfa the investigated effect appears as a result of finite length of used data series and is additionally amplified by the long term memory the data eventually may contain we pro vide the detailed quantitative description of such apparent multifractal background signal as a threshold in spread of generalized hurst exponent valuesh or a threshold in the width of multifractal spectrum below which multifractal properties of the system are only apparent i e do not exist despite 6 0 orh 6 0 we find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent its strength decays according to power law with respect to the length of time series the influ ence of basic linear and nonlinear transformations applied to initial data in finite time series with various levels of long memory is also investigated this provides additional set of semi analytical results the obtained formulas are significant in any interdisciplinary application of multifractality including physics financial data analysis or physiology because they allow to separate the true multifractal phenomena from the apparent artificial multifractal effects they should be a helpful tool of the first choice to decide whether we do in particular case with the signal with real multiscaling properties or not

magnitude and sign scaling in power law correlated time series a time series can be decomposed into two sub series a magnitude series and a sign series here we analyze separately the scaling properties of the magnitude series and the sign series using the increment time series of cardiac interbeat intervals as an example we 6nd that time series having identical distributions and long range correlation properties can exhibit quite di8erent temporal organizations of the magnitude and sign sub series from the cases we study it follows that the long range correlations in the magnitude series indicate nonlinear behavior speci6cally our results suggest that the correlation exponent of the magnitude series is a monotonically increasing function of the multifractal spectrum width of the original series on the other hand the sign series mainly relates to linear properties of the original series we also show that the magnitude and sign series of the heart interbeat interval series can be used for diagnosis purposes c

spectral decomposition of real symmetric quadratic λ matrices and its applications spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions the canonical form often facilitates discussions which otherwise would be complicated and involved spectral decomposition is of fundamental importance in many applications the well known glr theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees but its development is based on complex numbers this paper modifies the glr theory for the special application to real symmetric quadratic matrix polynomials q a mx2 cx k m nonsingular subject to the specific restriction that all matrices in the representation be realvalued it is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which in turn can be constructed from the spectral data applications to a variety of challenging inverse problems are discussed

scaling sensitivity and stability in the numerical solution of quadratic eigenvalue problems the most common way of solving the quadratic eigenvalue problem qep l 2 m l d k x 0 is to convert it into a linear problem l x y z 0 of twice the dimension and solve the linear problem by the qz algorithm or a krylov method in doing so it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution we discuss these issues for three particular linearizations the standard companion linearization and two linearizations that preserve symmetry in the problem for illustration we employ a model qep describing the motion of a beam simply supported at both ends and damped at the midpoint we show that the above linearizations lead to poor numerical results for the beam problem but that a two parameter scaling proposed by fan lin and van dooren cures the instabilities we also show that half of the eigenvalues of the beam qep are pure imaginary and are eigenvalues of the undamped problem our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations the main conclusions of which are summarized as well as arguing that scaling should routinely be used we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors

random cascades on wavelet dyadic trees we introduce a new class of random fractal functions using the orthogonal wavelet transform these functions are built recursively in the space scale half plane of the orthogonal wavelet transform cascading from an arbitrary given large scale towards small scales to each random fractal function corresponds a random cascading process referred to as a w cascade on the dyadic tree of its orthogonal wavelet coefficients we discuss the convergence of these cascades and the regularity of the so obtained random functions by studying the support of their singularity spectra then we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process we illustrate all our results on various numerical examples

co ordinate transformations for second order systems part ii elementary structure preserving transformations abstract it has been shown in a previous paper that there is a real valued transformation from the generaln degree of freedom second order system to a second order system characterized by diagonal matrices an immediate extension of this fact is that for any second order system there is a set of real valued transformations thestructure preserving transformations which transform this system to a different second order system having identical characteristic behaviour there are several possible reasons why it may be very useful to achieve a particular structure in the transformed system it is obvious that a diagonal structure is extremely useful and a method has been devised for determining the diagonalizing transformation from the solution of the usual complex eigenvalue eigenvector problem this paper begins by outlining the usefulness of some other structures then it defines a class of elementary structure preserving co ordinate transformations that transform from one n degree of freedom second order system to another the termelementary is applied because any one of these transformations is the minimum rank modification of the identity transformation the changes occurring in the system matrices as a result of the application of one such elementary transformation transpire to be very simple in form they are low rank and they can be computed very efficiently this paper provides the fundamental tools to enable the design of structure preserving co ordinate transformations which transform a second order system originally characterized by three general matrices in stages into a mathematically similar second order system characterized by three diagonal matrices the procedure by which the individual elementary transformations are obtained is still under development and it is not discussed in this paper however an illustration is given of a five degree of freedom self adjoint system being transformed into tridiagonal form

simultaneous tridiagonalization of two symmetric matrices we show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations no assumptions are made on the non singularity or definiteness of the two matrices the reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via householder reflectors two algorithms are proposed one using non orthogonal rank one modifications of the identity matrix and the other more costly but more stable using a combination of householder reflectors and non orthogonal rank one modifications of the identity matrix with minimal condition numbers each of these tridiagonalization processes requires o n 3 arithmetic operations and respects the symmetry of the problem we illustrate and compare the two algorithms with some numerical experiments

attempt to distinguish long range temporal correlations from the statistics of the increments by natural time analysis most probable value of 1 of the shuffled data is found to be approximately equal to that of the original data the difference most likely arising from temporal correlation finally it is found that the differential entropy associated with the probability p1 maximizes for around 1 6 1 7 which is comparable to the value determined experimentally in diverse phenomena e g solar flares icequakes dislocation glide in stressed single crystals of ice etc it also agrees with the b value in the gutenberg richter law of earthquakes in addition the case of multiplicative cascades is studied in the natural time domain

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

preservation of long range temporal correlations under extreme random dilution abstract many natural time series exhibit long range temporal correlations that may be characterized by power law scaling exponents however in many cases the time series have uneven time intervals due to for example missing data points noisy data and outliers here we study the effect of randomly missing data points on the power law scaling exponents of time series that are long range temporally correlated the fourier transform and detrended fluctuation analysis dfa techniques are used for scaling exponent estimation we find that even under extreme dilution of more than 50 the value of the scaling exponent remains almost unaffected random dilution is also applied on heart interbeat interval time series it is found that dilution of 70 80 of the data points leads to a reduction of only 8 in the scaling exponent it is also found that it is possible to discriminate between healthy and heart failure subjects even under extreme dilution of more than 90

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

nonextensivity and natural time the case of seismicity nonextensive statistical mechanics pioneered by tsallis has recently achieved a generalization of the gutenberg richter law for earthquakes this remarkable generalization is combined here with natural time analysis which enables the distinction of two origins of self similarity i e the process memory and the process increments infinite variance by using also detrended fluctuation analysis for the detection of long range temporal correlations we demonstrate the existence of both temporal and magnitude correlations in real seismic data of california and japan natural time analysis reveals that the nonextensivity parameter q in contrast to some published claims cannot be considered as a measure of temporal organization but the tsallis formulation does achieve a satisfactory description of real seismic data for japan for q 1 66 when supplemented by long range temporal correlations