Multifractality of time series

Posted on October 27, 2011.

One of the conclusions of fractal geometry is a fact that fractals unlike traditional Euclidean shapes lack characteristic scale. Those “fractured” objects are self-similar – defining geometry is clearly visible on multitude of scales. It is known that self-similarity is observed not only in formally defined geometric objects, such as Sierpinsky triangle or Koch snowflake, but also in the surrounding nature. One of my most favorite examples is a comparison of tree, its branches and a leaf (for more inspiring examples see introduction of Fractals section) – they all have branching structure and something green filling the extra space in between.

The interesting thing, in context of the topic in focus, is that one can extend fractal formalism beyond formal or natural geometric shapes. It is also noticed that some of the natural processes exhibit fractal features in their time series! It is known that geoelectrical processes 1, heartbeat 2 and even human gait 3 time series posses this feature. While financial market, frequently analyzed on Physics of Risk website, time series are also no exception 4, 5. Though the aforementioned time series are much more complex – they exhibit not monofractality (single manner self-similar behavior as the aforementioned formal geometric fractals do), but multifractality!

Fractality of time series

In Physics, but not only in Physics, scale invariance is mostly associated with power law dependencies. One can put this idea down mathematically as:

$657d4e052c6f401ac0b6e251a225880e T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

where $27d503c89be9c66a34cf9be0ff813fa5 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is some time series, $0cc175b9c0f1b6a831c399e269772661 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ some scale shift, $c1d9f50f86825a1a2302ec2449c17196 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ characteristic scaling exponent. Equality of the left hand side and right hand side of equation doesn’t need to be strict – in case of stochastic processes it might represent purely statistical similarity 6.

One of the most well known statistically self-similar processes is Brownian motion, which also known as Wiener process. It is known that Wiener process obeys Gaussian distribution with time dependent standard deviation, which increases overtime as square root of elapsed time – $09bfefd3fb1bee5a276258a558853532 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . Thus we above relation holds for variance of Wiener process with $0903c020b53450a5cfcfdde1c9b81fc9 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . It is worthy note that there is extension of this process known as fractional Brownian motion, which may be represented by different values of characteristic scaling exponent 6.

The problem of monofractality in the time series still remain as Wiener process, or its generalizations, is still described by single scaling exponent. While multifractal time series would be described by a set of scaling exponents 6, 7. Nevertheless our understanding of fractality of time series should have improved as now we should be able to understand fractals, whose dimensions are not limited to spatial ones. Using this example we have also understood what statistical self-similarity stands for – it doesn’t stand for strictly repeating shapes (as for example in aforementioned Sierpinsky triangle case), it stands for repetition of statistical properties, which describe observed random shapes.

As we have mentioned before we will be interested in processes with broad $c1d9f50f86825a1a2302ec2449c17196 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ spectrum. One can obtain it using varying methods 6, 8, while we will limit ourselves to presentation of only one, yet very common and popular, method – MF-DFA method 7.

Multifractal detrended fluctuation analysis (MF-DFA)

To begin with (step 1) the analysis of time series using MF-DFA method one should obtain the profile of time series:

$523beb3e07d5869cec721054c2d90f31 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

here $415290769594460e2e485922904f345d T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is profile, $9dd4e461268c8034f5c8564e155c67a6 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is the analyzed time series, $1f0e9c7544b970e78bbc9c5c00d3dce2 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is its mean value, while $8d9c307cb7f3c4a32822a51922d1ceaa T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ stands for the length of the series. Subtraction of mean is not necessary, yet it might facilitate (depends on the method used for detrending) calculations in further steps.

$signal profile Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 1. Step 1: stochastic time series (a) and its profile (b). Stochastic time series generated using standard Wiener process.

Afterwards (step 2) one should split profile series into separate non-overlaping segments with size $03c7c0ace395d80182db07ae2c30f034 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . After this operation one should have $f7cb8c1c2ad3b2fc3875d69d9495eb64 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ segments. For the most $03c7c0ace395d80182db07ae2c30f034 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ there will be some leftover points, whose number is to small to form another segment. If we are not willing to lose information contained in them we should repeat the same process, namely splitting, from the end of the profile series. In such case we effectively double the number of segments – one should now have $0bc6c1c4012dec15abff2b551f7de140 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ segments.

During the third step trends in all segments are estimated. Trends should be approximated by polynomial of the selected order. As there is no restriction for possible selection, one can choose any positive integer. MF-DFA title is appended based on the selected order. Thus if one uses parabolic (or square) fits, then one can say that MF-DFA2 is used. As it is problematic, in terms of computer resource and evaluation time, to fit data with higher order polynomials, in practice polynomials of first to third order are considered. For the sake of simplicity and ease of presentation we use MF-DFA1 method, namely we use linear least-square fits to detect and remove trends in the profile series.

$fitted profile Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 2. Steps 2 and 3: we have split the profile (red curve) into s=100 sized segments (splits are marked with dashed line) and estimated their trends (blue curve). The profile has remained the same as in Fig. 1., while the profile fits in each given segment are linear.

When the trends are known (step 4) we can remove them by subtracting them from the profile series. Mathematically for the segments, which were formed the start of the series, $b8e722e54d5655aab2ca771690aba892 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , this can be expressed as:

$264c8fd468fad4357947015f51890161 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

while for the remaining segments, $8016c7a8e7c5f061869545b49f947385 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , same thing can be done like this:

$4ce3be9a4a9f5ab053edcd0d149a3f80 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

$f2ns Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 3. Step 4: F_ν²(100) for the selected segments. Profile and its splits remain the same as in Fig. 2.

All what remains (step 5) is to average obtained fluctuations functions over all segments:

$c2994789be545d497caa410dd3a01339 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

In the above $7694f4a66316e53c8cdd9d9954bd611d T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ stands for generalized coefficient, which is the one enabling us to recover multifractal features – it is also the only difference from the original detrended fluctuation analysis (DFA) method 7. It is worthy to note that $7694f4a66316e53c8cdd9d9954bd611d T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is a real number and if its value approaches zero the above relation diverges. Thus in such case one must substitute ordinary averaging procedure with exponential averaging:

$56621b4207b6ad4d11dc24d47cbaa585 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Finally (step 6) by changing $7694f4a66316e53c8cdd9d9954bd611d T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ and $03c7c0ace395d80182db07ae2c30f034 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , while once again iterating through steps 2, 3, 4 and 5, one obtains $77cd033d17fd2e5261e5e15348ce4cfa T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ curves. From their plots on log-log scale one should be able to recover power law relations – $432fe7c3b16ff02f96395d12ae248469 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ (here $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is a generalized exponent, which is related to the generalized Hurst exponents as $26b9977eb6cd4b0ca79b15f826617bb7 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ (for non-stationary $300ce7f443f70c4d48f77a6f62cf9a96 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ time series) or $e480c1c429697b1631ba0f7a7b1d4447 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ (for stationary series, $ec28cb300d245509573327d1fd20122c T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ )). If generalized exponents are constant or almost constant, then time series can be considered monofractal. While if $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ dependence (or in other words spectrum) is rich, then time series can be considered multifractal.

$gen hurst spectra Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 4. Steps 5 and 6: deviations from trends estimated for different segment sizes (a) and generalized exponent spectrum (b). Relations obtained from time series based on one presented in Fig. 1. (original series extended to 32768 points).

In the above figures we have illustrated different MF-DFA steps by analyzing standard Wiener process. Thus the final result, Fig. 4, namely the observed monofractality of time series was not unexpected. Though at first glance it might occur that signal is multifractal as width of $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ spectrum is non-zero as it should be according to the theory. This discrepancy is caused by the finite size of analyzed time series. Actually we have attempted to obtain $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ spectrum in case of original, 1000 point wide, time series, but the obtained $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ spectrum was unexpectedly broad – 0.2 (approximately 15% of the mean $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ ). While the multifractal analysis of the same yet extended, 32768 points wide, series (as you can see in Fig. 4) revealed narrower spectrum – 0.04 (approximately 2.5% of the mean). Literature 6, 7 suggests that in the infinite limit spectrum would converge to single dot.

Furthermore as we see in Fig. 5 $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ spectrum of mutifractal time series is far more broader – 1.1 (approximately 90% of the mean). Curves presented in Fig. 5 were obtained by analyzing time series generated by the Agent based herding model of financial markets. Multifractal properties of this model were already studied with two different methods – GHHCF 9 and the very same MF-DFA 10. Also note the difference in scaling of $77cd033d17fd2e5261e5e15348ce4cfa T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ (Fig. 5 (a) and Fig. 4 (a)).

$gen hurst spectra multi Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 5. Multifractal features of Agent based herding model of financial markets: deviations from trends estimated for different segment sizes (a) and generalized exponent spectrum (b). Model parameters: ε₁=ε₂=1, Δt=0.001, τ(y)=1/y.

Additional information visible in generalized exponent spectrum

Actually MF-DFA is a generalization of an older DFA method. In case of $ed7e87e89723823cc5feb4fe84e41254 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ MF-DFA produces exactly the same results as ordinary DFA method would 7 – $92a3f3a7176e3dc13bd35f03cae6ab8f T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ equals exponent $2510c39011c5be704182423e3a695e91 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ as it is understood in the original DFA framework. Therefore interpretations which have originated from DFA 11 might be also applied to the results obtained with MF-DFA.

Judging from $92a3f3a7176e3dc13bd35f03cae6ab8f T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ time series might be 11:

anti-correlated, if $93a5602d720205db8d11795c0493ff54 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ .
uncorrelated (be related to white noise), if $eade42bdb73d00fedddb4396e35e8a5e T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ .
positively correlated, if $ae2390d6bbc86d8ce44412636e8225bc T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ .
strongly positively correlated, if $dfb4ca7f3779f714365668498b72d6e7 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . Or in other words exhibit so-called pink, or $84d2f3bf60636d49e0540dd1a342d883 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , noise. Such time series should exhibit other long range memory related properties. Note that we have obtained such results with time series from Agent based herding model of financial markets (see Fig. 5).
non-stationary or similar to random walk, if $e2e72a02337a14a14c356791c4eac7e0 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ .
related to Brown noise or Wiener process, if $2989744480900f4a12138b80157ac2ea T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . See Fig. 4.

The above discussion can might be briefly mathematically expressed as 11:

$43fffd1613923fc83b29a7a78cbbc646 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

here $ae539dfcc999c28e25a0f3ae65c1de79 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is negative correlation function exponent, while $b0603860fcffe94e5b8eec59ed813421 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is negative power spectral density exponent. Note that the above relations are strict in no way – they might hold for some short regions of dependence, of for example frequences (in case of power spectral density).

It is important to note that actual $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ might be negative, while MF-DFA is able to detect only its positive values. Though one can extend the detectable range of $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ of repeating step 1 few times. After each repetition detected generalized exponents increases by 1, $7e72a6ee85bc1944045846315faca68f T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ (here $d12f5fca0206a10e535c7668f6f70c42 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is detected generalized exponent, $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ actual generalized exponent and $7b8b965ad4bca0e41ab51de7b31363a1 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is a number of step 1 repetition times), allowing one to analyze strongly anti-correlated time series. This property of generalized exponent also suggests what might be observed for $1022ae48b1a3ac7ddd80fef5607f599c T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ .

This discussion also leads to another interesting conclusion – if step 1 is skipped results obtained from MF-DFA method should also coincide with results obtained from GHHCF method. For example Brownian-like time series would have $92a3f3a7176e3dc13bd35f03cae6ab8f T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ equal to 0.5. Though in such case it would a problem to differentiate between Pink and White noises. The corresponding time series would be almost indistinguishable – both very different dynamics would produce the same value of $92a3f3a7176e3dc13bd35f03cae6ab8f T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , namely zero. Thus obtaining time series profile is very important step needed to be able to distinguish Pink and White noises.

Singularity, Hölder exponent, spectrum

Singularity spectrum is an alternative way to characterize multifractal series. It was derived to accompany standard textbook box counting formalism 8. Box counting formalism is a relatively simple and for a long time very popular method to determine fractal features of analyzed objects (including spatial ones). Thus singularity spectrum has become a default way to characterize multifractal series.

Singularity spectrum, $7b30095cebdfbd41e84eadb432f616b2 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ , can be obtained by transforming generalized scaling function defined in terms of box counting formalism, $22f515484a8d32745e29335b3f8089f0 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ . Note that $6003ba675c5ffaa1b8512967aeb41e73 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ may also be referred to as a scaling function (or scaling exponent), but it is defined in the terms of MF-DFA or DFA, not in the terms of box counting formalism. Though it is known that both scaling exponents are related 7:

$3ec23d3ea6a57098fe7d493ac59c120d T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

By transforming, using Legendre transform, $22f515484a8d32745e29335b3f8089f0 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ one obtains 7:

$dad9d9f3d93d17e9e590d70bf33aa4b7 T 000000 0 ordinary Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

here $7b7f9dbfea05c83784f8b85149852f08 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ is Hölder exponent, which describes the strength of singularity, while $7b30095cebdfbd41e84eadb432f616b2 T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ denotes the dimension of the subset of the series described by certain Hölder exponent 7. In Fig. 6 we clearly see large difference between the detected multifractality of the analyzed models/processes – singularity spectrum of Agent based herding model of financial markets time series has broad spectrum of Hölder exponents, while Wiener process has very thin spectrum.

$holder spectra Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$

Fig. 6. Singularity spectrum: standard Wiener process (red curve) and Agent based herding model of financial markets (blue curve). Parameters of Agent based herding model of financial markets were set as follows: ε₁=ε₂=1, Δt=0.001, τ(y)=1/y.

In scientific literature singularity spectrum is also frequently called Hölder exponent spectrum 6.

Applet

GUI of the applet bellow is reminiscent of the previous applet, which was published together with the text on Agent based herding model of financial markets. Thus there is no point in discussing its input parameters. Furthermore most of the input parameters are directly related to the model discussed in the same text.

The only important differences are that this applet outputs multifractality spectra and that it allows to deform model time series. It is possible to remove correlations by shuffling time series and distort the underlying distribution (values are remapped to the $ccfcd347d0bf65dc77afe01a3306a96b T 000000 0 inline Multifractality of time series stochastic interactive models fractals econophysics multifractality Kirman model$ value region). Deformation of time series is very important and useful function as multifractal features might be observed due to correlations (model dynamics) and broad underlying distribution 7. It is also known that multifractality can be influenced by the number of agents in the agent based model 12, though we don’t analyze this as it is already firmly confirmed that in this model dependence of multifractality on the number of agents is negligible 10.

Above you should see Java applet. If you do not see it, then please make sure that you have JRE installed and that your browser has Java enabled. Also make sure that you are running newest available JRE version. Newest JRE version can be downloaded from http://java.com/getjava.

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Major topics covered Econophysics, Fractals, Interactive models, Stochastic models. Keywords Kirman model, multifractality.

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