Detrending Moving Average Algorithm: from finance to genom. materials

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1 Detrending Moving Average Algorithm: from finance to genomics and disordered materials Anna Carbone Physics Department Politecnico di Torino September 27, 2009

2 Publications 1. Second-order moving average and scaling of long-range correlated series E. Alessio, A. Carbone, G. Castelli, and V. Frappietro, Eur. Phys. Jour. B 27, 197 (2002). 2. Scaling of long-range correlated noisy signals: application to financial markets A. Carbone and G. Castelli, Proc. of SPIE, 407, 5114 (2003). 3. Analysis of the clusters formed by the moving average of a long-range correlated stochastic series A. Carbone, G. Castelli and H. E. Stanley, Phys. Rev. E 69, (2004). 4. Time-Dependent Hurst Exponent in Financial Time Series A. Carbone, G. Castelli and H. E. Stanley, Physica A 344, 267 (2004). 5. Directed self-organized critical patterns emerging from fractional Brownian paths A. Carbone and H. E. Stanley, Physica A 340, 544 (2004). 6. Spatio-temporal complexity in the clusters generated by fractional Brownian paths A. Carbone and H.E. Stanley, Proc. of SPIE 5471, 1 (2004). 7. Quantifying signals with power-law correlations L. Xu, P. Ch. Ivanov, C. Zhi, K. Hu, A. Carbone, H. E. Stanley, Phys. Rev. E 71, (2005). 8. Scaling properties and entropy of long range correlated series A. Carbone and H.E. Stanley, Physica A 384, 21 (2007). 9. Tails and Ties A. Carbone, G. Kaniadakis, and A.M. Scarfone, Eur. Phys. J. B 57, (2007). 10. Where do we stand on econophysics? A. Carbone, G. Kaniadakis and A.M. Scarfone, Physica A 382, xi-xiv (2007). 11. Detrending Moving Average (DMA) Algorithm: a closed form approximation of the scaling law S. Arianos and A. Carbone, Physica A 382, 9 (2007). 12. Algorithm to estimate the Hurst exponent of high-dimensional fractals A. Carbone, Phys. Rev. E 76, (2007). 13. Cross-correlation of long-range correlated series S. Arianos and A. Carbone, J. Stat. Mech: Theory and Experiment P03037, (2009).

Analysis of the clusters formed by the moving average of a long-range correlated stochastic series A. Carbone, G. Castelli and H. E. Stanley, Phys. Rev. E 69, 026105 (2004). 4.

3 The Efficient Market Hypothesis...and Its Critics A Random Walk Down Wall Street First edition published in the 70 s by Burton Malkiel on the wave of the academic success of the influential survey on the Efficient Market Hypothesis (EMH) by Eugene Fama (1970). A Non-Random Walk Down Wall Street First edition published in 1999 by Andrew Lo and Craig MacKinley

Burton Malkiel on the wave of the academic success of the influential survey on the

4 A Moving Average Walk... Down Wall Street

5 Surveys on the use of Technical Analysis by Finance Professionals Horizons min % max% intraday 50% 100% 1 week 50% 80% 1 month 40% 70% 3 months 30% 50% 6 months 20% 30% 1 year 2% 25% 1 Charts, Noise and Fundamentals in the London Foreign Exchange Market. H.L. Allen and M.P. Taylor, The Economic Journal, 100, (1990) 2 Technical trading rule profitability and foreign exchange intervention, B. LeBaron, Journal of International Economics 49, 125 (1999) 3 The Obstinate Passion of foreign Exchange Professional: Technical Analysis. Lukas Menkhoff and Mark P. Taylor, Journal of Economic Literature 45, 4 (2007)

Taylor, The Economic Journal, 100, 49-59 (1990) 2 Technical trading rule profitability and foreign exchange intervention, B.

6 Fractional Brownian or Levy Walk H > 0.5 positive correlation (persistence) H = 0.5 Random walk H < 0.5 negative correlation (anti-persistence)

7 Detrending Moving Average Algorithm (DMA) 20 ~ y n (i) σ 2 DMA = 1 N N i=n [ ] 2 y(i) ỹ n (i) \L y(i) FOXVWHU ỹ n (i) = 1 n n y(i k) k=0 σ 2 DMA n2h i c (j) L i c (j+1) τ j =i c (j+1)-i c (j) 2680

FOXVWHU ỹ n (i) = 1 n n y(i k) k=0 σ 2 DMA n2h 10

8 Detrending Moving Average Algorithm (DMA) σ 2 DMA n2h The log-log plot is a straight-line whose slope can be used to calculate the Hurst exponent H of the time series.

9 Detrending Moving Average Algorithm (DMA) Self-similarity of a time series x(t) σdma 2 is generalized variance for nonstationary signals. It can be derived from the auto-correlation function C xx (t, τ), which measures the self-similarity of a signal: σ xx (t, τ) [x(t) x(t)][x (t + τ) x (t + τ)] By taking τ = 0 and x = x(t), x n (i) = 1 n n k=0 x(i k) the autocovariance C xx (t, τ) reduces to the function σdma S. Arianos and A. Carbone Physica A 382, 9 (2007)

It can be derived from the auto-correlation function C xx (t, τ), which measures the self-similarity of a signal: σ

10 DAX (Deutscher AktienindeX) Prices (a) P(t) Log Returns (b) r(t) = lnp(t + t ) lnp(t) Volatility σ T (t) 2 = 1 T T 1 t=1 [r(t) r(t) T ] 2 (c) with T = 300min (one half trading day) (d) with T = 660min (one trading day)

11 FIB30 futures: PRICES The FIB30 is a future contract on the MIB30 index, which considers the thirty firms with higher capitalization and trading (the top 30 blue-chip index) of the MIBTEL ( (Since 2004 it is named S&P MIB index).

capitalization and trading (the top 30 blue-chip index) of the

12 FIB30 futures: VOLATILITIES T=660min (1 trading day) T=6600min (10 trading days) σ DMA Fract. Brown. Motion H= n

trading days) σ DMA 10-5 10-6 10-7 Fract.

13 Cross-Similarity of two stochastic series x(t) and y(t) In many cases, the coupling between two systems (time series) might be of interest. A measure of cross-correlation can be implemented 5 by defining: σ xy (t, τ) [x(t) x(t)][y (t + τ) ỹ (t + τ)] Now it must be τ 0. Moreover x = x(t) and y = y(t) x n (i) = 1 n n k=0 x(i k) and ỹ n(i) = 1 n n k=0 y(i k). 5 S. Arianos and A.Carbone, J. Stat. Mech.: Theory and Experiment P03037, (2009).

A measure of cross-correlation can be implemented 5 by defining: σ xy (t, τ) [x(t) x(t)][y (t + τ) ỹ (t + τ)]

14 Cross-Similarity of return and volatility σ 2 DMA nh 1+H 2 The log-log plot is a straight-line. The slope is given by the sum of the Hurst exponents H 1 and H 2 of the two time series. 6 6 S. Arianos and A.Carbone, J. Stat. Mech: Theory and Experiment P03037, (2009).

The slope is given by the sum of the Hurst exponents H 1 and H 2

15 Leverage effect: DAX DMA Cross-correlation as a function of the lag τ for the return and volatility of the DAX series. 7 7 S. Arianos and A.Carbone, J. Stat. Mech: Theory and Experiment P03037, (2009).

16 Scaling law of the moving average clusters 8 9 n=200 n=600 y(t) y(t) n=1000 area n=2000 t duration t Anna Carbone 8 A.Carbone, Physics Department G.Castelli Politecnico anddih.e. TorinoStanley, Detrending Phys. Moving Rev. Average E, 69, Algorithm: from (2004) finance to genom

Castelli Politecnico anddih.e. TorinoStanley, www.polito.

17 Scaling law of the cluster lengths l 10 5 (a) (a) l 10 2 ψ l H τ H l τ ψ l ψ l = 1

6 10 1 H 0.4 0.2 10 0 10 0 10 1 10 2 10 3 τ 0.0 0.0 0.1 0.

18 Scaling law of the cluster areas s (b) 1.9 (b) s ψ s H H τ H s τ ψs ψ s = 1 + H

3 10-1 10-2 H 1.2 1.1 1+H 10-3 10 0 10 1 10 2 10 3 τ 1.0 0.0 0.1 0.

19 Scaling law of the pdf of the cluster length and duration H= β=2-h ) P(τ) 10-3 β x x10-5 (a) τ (a) H P(τ, n) τ β F (τ, n) with β = 2 H the fractal dimension of the time series and F (τ, n) defined as: F(τ, n) = e τ/τ.

3 1x10-5 (a) 10-6 10 0 10 1 10 2 10 3 τ 1.2 1.1 (a) 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.

20 Scaling law of the pdf of the cluster area γ=2/(1+h) P(s) (b) H=0.80 H=0.20 H= s γγγ (b) H with γ = 2/(1 + H) and: P(s, n) s γ F (s, n) F(s, n) = e s/s.

50 10 0 10 1 10 2 10 3 s γγγ 1.5 1.4 1.3 1.2 1.1 (b) 1.0 0.0 0.1 0.2 0.

21 Self-organized criticality of the moving average clusters t addition of particles/energy y j t y addition of particles/energy Brownian path fractional Brownian path j+1 Brownian path cluster lifetime j+2 smoothed fractional Brownian path cluster diameter l dissipation of particles/energies dissipation of particles/energy Moving Average Clusters SOC Clusters l τ ψ l ψ l = 1 ψ l = 1 s τ ψ s ψ s = 1 + H ψ s = 3/2 P(l, n) l α α = 2 H α = 3/2 P(τ, n) τ β β = 2 H β = 3/2 P(s, n) s γ γ = 2/(1 + H) γ = 4/3

22 Entropy of long-range correlated time series 10 1 l n n=500 A n=1000 n=2000 n=3000 S(l,n) A' A'' D log l n= l S(l, n) µ(l,n) P(l, n) log P(l, n). S(l, n) = S 0 + D log l + l n. 10 A. Carbone and H.E. Stanley Physica A 385, 21 (2007)

23 Genome Heterogeneity G A T C C T T G A A G C G C C C C C A A G G G C A T C T T C T 1. The sequence of the nucleotide bases ATGC is mapped to a numeric sequence. 2. If the base is a purine (A,G) is mapped to +1, otherwise if the base is a pyrimidine (C,T) is mapped to The sequence of +1 and 1 steps is summed and a random walk y(x) is obtained.

24 Genome Heterogeneity % A, C, G, T % A, C, G, T % A, C, G, T 50 CHR1 40 n=2 A 30 T 20 C 10 G CHR1 n=4 40 A 30 T 20 C 10 G CHR1 40 n=10 A 30 T 20 C 10 G length (bp)

25 Genome Heterogeneity % A, C, G, T % A, C, G, T % A, C, G, T 50 CHR14 A 40 n=2 T C 0 G CHR14 n=4 A T C 0 G CHR14 40 A n=10 T C G length (bp)

26 DMA for high-dimensional fractals (e.g. selfsimilar networks, rough surfaces) 11 Fractal surfaces 0.5 f(i 1,i 2 ) H= f(i 1,i 2 ) H= (b) i i (a) i i A. Carbone Phys. Rev. E 76, (2007)

27 DMA for high-dimensional fractals (e.g. selfsimilar networks, rough surfaces) 12 Moving Average Surface for the fractal surface with H = ~ f(i 1,i 2 ) n 1 x n 2 =15 x ~ f(i 1,i 2 ) n 1 x n 2 = 25 x (a) i i (b) i i A. Carbone Phys. Rev. E 76, (2007)

28 DMA for high-dimensional fractals (e.g. selfsimilar networks, rough surfaces) 13 DMA (c) N 1 x N 2 =4096 x s The plot of σ 2 DMA vs. S = (n n2 2 ) is a straight line. It corresponds to the scaling relation: i.e.: σ 2 DMA [ (n n2 2 ) ] 2H σ 2 DMA S2H 13 A. Carbone Phys. Rev. E 76, (2007)

29 THANK YOU!

How To Calculate A Non Random Walk Down Wall Street

How To Calculate A Non Random Walk Down Wall Street Detrending Moving Average Algorithm: a Non-Random Walk down financial series, genomes, disordered materials... Anna Carbone Politecnico di Torino www.polito.it/noiselab September 13, 2013 The Efficient